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Structure-property relationships of modified polyesters probedby solid-state NMR and FTIR spectroscopyCitation for published version (APA):Ziari, M. (2009). Structure-property relationships of modified polyesters probed by solid-state NMR and FTIRspectroscopy. Eindhoven: Technische Universiteit Eindhoven. https://doi.org/10.6100/IR643420

DOI:10.6100/IR643420

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Structure-property relationships of modified polyesters probed by solid-state NMR and FTIR

spectroscopy

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de

rector magnificus, prof. dr. ir. C. J. van Duijn, voor een commissie aangewezen

door het College voor Promoties in het openbaar te verdedigen op woensdag 24 juni 2009 om 16.00 uur

door

Maya Ziari

geboren te Algiers, Algerije

Dit proefschrift is goedgekeurd door de promotoren: prof.dr. J. Gelan en prof.dr. P. J. Lemstra Copromotor: prof.dr. P. Adriaensens

A catalogue record is available from the Eindhoven University of Technology Library

ISBN: 978-90-386-1896-8

Copyright © 2009 by Maya Ziari

Cover design : Valentine Kreykamp

The Dutch Polymer Institute (DPI) financially supported the research described in this thesis, project

337.

An electronic version of this thesis is downloadable in PDF-format from the website of the Eindhoven

University of Technology (http://www.tue.nl/bib).

In Memory of Laurus, Manola, Farah

“The Science of Today is the Technology of Tomorrow”

To my parents

Table of Contents

v

Table of Contents

Summary ix

Chapter 1. Introduction and scope

1.1 General introduction 1

1.2 Modified polyesters 3

1.3 Characterization of polyesters 5

1.4 Scope of the thesis 7

1.5 Outline of the thesis 8

1.6 References 10

Chapter 2. Basics of NMR relaxation for polymer structure investigation

2.1 General approach 11

2.1.1 Definition and physics basis 11

2.1.2 Pulsed NMR and bulk magnetization 13

2.1.3 Laboratory frame and rotating frame 15

2.2 NMR relaxometry 17

2.2.1 Spin relaxation times 17

2.2.2 NMR relaxometry for polymer characterization 21

2.3 References 30

Chapter 3. Morphology and differences in modified polyesters probed by solid-state NMR

3.1 Introduction 34

3.2 Experimental section 35

3.2.1 Sample preparation 35

3.2.2 NMR experiments 37

3.2.3 NMR pulse sequences 37

3.2.4 NMR fitting procedure 39

3.2.5 Software 39

3.2.6 Modulated DSC 39

Table of Contents

vi

3.3 Results and discussion 40

3.3.1 13C-CP MAS experiments 40

3.3.2 1H T1 and T1 relaxation experiments 43

3.3.3 Quantification of each phase 50

3.3.4 Domains sizes 53

3.4 Conclusion 55

3.5 References 57

Chapter 4. Phase composition of modified polyesters probed by solid-state 1H wideline NMR of spin-spin relaxation

4.1 Introduction 60





4.3.1 T2 spin-spin relaxation (HEPS) 65

4.3.2 Analysis and relaxation measurements for sample (BD96DI04) 72

4.3.3 Analysis and relaxation measurements for sample (BD54DI46) 73

4.4 Conclusion 78

4.5 References 79

Chapter 5. Heterogeneous chain dynamics in polyester network probed by 1H static spin-spin relaxation NMR experiments

5.1 Introduction 82




5.2.3 Relaxation background 85

5.2.4 Fitting procedure 87

5.2.5 MDSC experiments 89

5.2.6 DMTA experiments 90

5.2.7 DES experiments 90


Table of Contents

vii

5.3.1 Network heterogeneity and temperature effect on T2 measurement 90

5.3.2 Quantification of rigid and mobile components 102

5.4 Conclusion 107

5.5 References 108

Chapter 6. A FTIR study on the solid-state copolymerization of bis(2-hydroxyethyl)terephthalate and poly(butylene terephthalate) and the resulting copolymers

6.1 Introduction 110


6.3 Theory 112


6.4.1 Assignment of the absorption bands of the pure components 113

6.4.2 Kinetics of the SSP reaction studied by FTIR spectroscopy 114

6.4.3 Temperature dependent experiments 118

6.5 Conclusion 121

6.6 References 123

Appendix 1. Effect of initial estimates and constrains selection in Multivariate Curve Resolution – Alternating Least Squares. Application to low-resolution NMR data 125

Appendix 2. Combining linear and non-linear regression algorithms as an alternative for multivariate curve resolution problems of low selectivity 149

Appendix 3. Calculation of the (mean) number of monomer units between two cross-linked junctions 161

Acknowledgements 165

Curriculum Vitae 169

Summary

ix

Structure-property relationships of modified polyesters probed by solid-state NMR and FTIR spectroscopy

Summary

In polymer science, a better understanding of the relationship between microscopic

structures and macroscopic properties is essential for an intelligent design of new and/or

improved materials. The objective of this thesis is to contribute to the analysis of

structure-property relationships through the use of non-destructive state-of-the-art NMR

and IR analytical methods in order to characterize polyester co-polymers and chemically

cross-linked unsaturated polyesters.

One of the effects of NMR radio-frequency pulses is to cause the spin populations to

deviate from their equilibrium population. The rate at which the spin system returns to

equilibrium informs about the molecular dynamics and the inter-nuclear distances.

Molecular motions induce nuclear relaxation, since atomic fluctuations modify the

magnetic field experienced by a nuclei and its environment. These fluctuations can

occur in a broad range of frequency and depend directly on the chemical structure and

the morphology. There are different relaxation times that can be measured depending on

the scale of the investigated motions. Fast molecular motions can be investigated by

measuring the so called spin-lattice relaxation time (T1), or spin-lattice relaxation in the

rotating frame (T1) while measuring the spin-spin relaxation time (T2) can be used to

look at slower molecular dynamics.

These different techniques of NMR relaxometry and spectroscopy are exploited in this

thesis to study the molecular dynamics and the morphology of three multi-phase

polymeric systems.

The two first multi-phase polymers have been synthesized by adding two co-monomers,

resp. a diol (2,2-bis[4-(2-hydroxyethoxy)phenyl]propane – Dianol 220®) and a

terephthalate (bis(2-hydroxyethyl)terephthalate – BHET), to poly(butylene

terephthalate) (PBT). Because of its high crystallization rate, PBT is employed in a

broad range of applications, which includes injection-molding grades, films, fibers and

adhesives. In order to extend further its performance, PBT is commonly modified so as

to increase its Tg and improve its dimensional stability, impact strength and

Summary

x

compatibility with other polymers, while retaining its high crystallization rate. Among

using various modification methods, which include copolymerization in the melt,

M.A.G. Janssen [PhD. Thesis, 2005, TU/e] has developed a solid-state

(co)polymerization (SSP) technique to prepare copolymers from polycondensation of

PBT with diols and terephthalates comonomers. In this study, state-of-the-art NMR and

IR analytical methods are exploited, to confirm that diols and terephthalates are

incorporated into the amorphous phase of the PBT by SSP without modification of its

crystalline phase, and more generally to better understand the relationships among

molecular motions, polymer morphology, and macroscopic properties of the multi-phase

PBT systems.

For that purpose, static 1H T1ρ solid-state NMR relaxation experiments have been

performed in combination with cross-polarization magic angle NMR (13C CP-MAS) and

static 1H T2-relaxation measurements on the PBT-Dianol copolymer. Multi-component

1D- and 2D-analysis methods of the proton relaxation behavior showed the presence of

‘’three phases’’ corresponding respectively to the crystalline phase, the rigid-amorphous

phase and mobile-amorphous phase of the polymer systems. Copolymers with different

PBT / Dianol ratios were analyzed: lower Dianol content showed that the Dianol

monomer is exclusively incorporated in the amorphous fractions of the copolymer

system, whereas part of the PBT crystalline phase is also converted to amorphous

copolymer phase at higher Dianol content. The fractions of protons associated to the

different relaxation times, characteristic of each phase (crystalline, rigid-amorphous and

mobile-amorphous), have been determined by using a novel multi-variable analysis

method, based on an orthogonal-projection approach (OPA): correlating line-shape

information to relaxation decay has been shown to be very useful to analyze relaxation

data and better understanding of the molecular dynamics and morphology of multi-

phase polyester systems.

Static 1H T2-relaxation measurements (Solid-Echo and Hahn-Echo) were used to better

estimate the crystallinity in the various samples. As expected, the crystallinity decreased

with increased Dianol concentration. Additional cross-polarization 13C CP-MAS NMR

experiments further confirmed that the Dianol is incorporated into both amorphous

fractions (rigid and mobile). During this study, it was clearly demonstrated that, if a too

high concentration of Dianol is incorporated via SSP, the crystallinity in the sample

decreases drastically as compared to pure PBT. In order to obtain enhanced macroscopic

Summary

xi

properties as compared to pure PBT, the modified polyesters should not exceed a certain

concentration of Dianol. Only then, the glass transition temperature will increase while

the crystallinity and high crystallization rate of pure PBT will remain.

Based on the spin-diffusion phenomenon, 1H T1 and 1H T1 relaxation measurements

have been used to estimate lower and upper limits of the phase domain sizes. The

estimated values between 3 and 80 nm for the rigid phase correlate well with

corresponding SAXS data between 5 and 50 nm.

As for the second PBT-BHET copolymer system, the first objective was to follow the

kinetics of incorporation under isothermal conditions. In that way, the development of

the chemical microstructure during the SSP reaction could be examined and the

miscibility of the BHET and the PBT could also be investigated as a function of

different PBT/BHET ratios. The results show that competition patterns take place

depending on the ratio of PBT/BHET: the structures include annealing of PBT,

incorporation of BHET in amorphous PBT and BHET homo-polymerization. At low

BHET concentrations, BHET is incorporated in PBT forming a non-random copolymer,

while for high BHET concentrations a separate PET phase is found and the PET homo-

polymers have sufficiently large chain lengths to crystallize.

An unsaturated polyester resin (UPR-Palatal intermediate) cross-linked with styrene,

was the third system analyzed. The purpose was to evaluate the use of solid-state NMR

characterization in order to extract micro-structural and molecular dynamics information

of complex chemical cross-linked polymer networks. 1H-NMR T2 relaxation

experiments have been performed to investigate the mobility and heterogeneity in the

network. The results of 1H spin-spin NMR relaxation show a strong heterogeneous

dynamic behavior. Also here three types of 1H mobilities were observed above Tg but

the different molecular mobility parts in the polymer network are now correlated to the

inter cross-link chains, the dangling chains ends and the sol fraction. The average

molecular weight between cross-links was investigated and estimated to be only a few

polystyrene monomer units. The data obtained via other characterization techniques,

such DMTA, DES and M-DSC, confirmed these results.

Chapter 1

1

Chapter 1

Introduction and Scope

1.1. General introduction and historical perspectives

Polyesters represent a class of polymers which contain the ester functional group in their

main chain. Although polyesters do exist in nature (natural polyesters have been known

since around 1830), the term polyester generally refers to the large family of synthetic

polyesters (resins), which includes polycarbonates and polyethylene terephthalates

(PET) [1,2].

Polyesters are widely used materials with diverse applications (fibers, plastics,

coatings). They are strong, colorfast, and resistant to corrosion and chemical attack. In

general they have extremely good mechanical properties and are heat resistant.

The first polyester fibers were synthesized by Carothers, also the inventor of nylon

fibers. His work was resumed by a group of British scientists, J.R. Whinfield, J.T.

Dickson, W.K. Birtwhistle, and C.G. Ritchie, who successfully created the first

polyester fiber called Terylene in 1941 [3]. A few years later, Dupont came up with an

other polyester fiber, the Dacron®. Today, polyester fibers are widely used in the textile

industry. For that purpose, they are often spun together with fibers of cotton, producing

a cloth with some of the better properties of each.

Polyesters are polymers made by a polycondensation reaction in which the linkage

between the molecules occurs through the formation of ester groups.

We can distinguish three main categories of polyesters, depending on their properties

and type of applications:

1. The thermoplastic engineering types, such as poly(ethylene terephthalate) (PET),

and the poly(butylene terephthalate) (PBT);

2. The thermoset composites such as unsaturated polyester resins;

3. The aromatic liquid-crystalline polyesters.

Chapter 1

2

Poly(ethylene terephthalate) PET is the most widely used linear polyester. It is prepared

from terephthalic acid and ethylene glycol (35 million tones/annum). Such polymer can

be prepared in bulk or in solution using an excess of ethylene glycol to increase the

esterification rate. This reaction leads first to a low-molecular-weight hydroxyl-

terminated polyester, which is then transesterified with removal of excess glycol to

attain high molecular weight. One should note that terephthalic acid is quite insoluble in

common solvents and sublimes at 300 oC. For this reason, dimethyl terephthalate

(DMT) has been used more frequently to make PET. Another process based on ethylene

oxide instead of ethylene glycol can also be used. Ethylene oxide reacts rapidly with

terephthalic acid to form an intermediate which will undergoes alcoholysis to finally

form the polyester.

PET is a slowly crystallizing polymer and is therefore usually in the amorphous state

and hence transparent, but can become translucent when it is in its semi-crystalline state.

With excellent barrier (CO2 permeation) and mechanical properties, PET is becoming

the polymer of choice for food and beverage packaging applications, even replacing

traditional materials such as metals or glass. The addition of anti-nucleating agents

permits the injection molding of PET and broadens its application range. However, for

an injection molding type of application, polybutylene terephthalate (PBT) is preferably

used. PBT as well is a semi-crystalline polyester, produced at an annual rate of about

400 thousand tons. It is characterized by a high crystallization rate, and good thermal,

mechanical and electrical properties. Such a polyester is polymerized following a two-

stage process. First, bis(4-hydroxybutyl)-terephthalate (BHBT) is formed by trans-

esterification of DMT with 1,4 butanediol, then BHBT further reacts by

polycondensation and leads to PBT. In comparison with PET, PBT has a slightly lower

strength and rigidity, a slightly better impact resistance, and a slightly lower glass

transition temperature. PBT is used for door and window hardware, automobile luggage

racks and body panels, headlight reflectors, and fiber optic cables. It is being used to

replace PVC for cable sheathing, due to a combination of environmental pressures and

outstanding electrical properties across a wide range of temperatures. Last but not least,

PBT is also replacing metals (in conjunction with polycarbonates, PC, as an alloy/blend)

for automotive body panels.

Chapter 1

3

Generally the use of PET and PBT blends broadens the range of application of these

thermoplastic engineering-grade polyesters.

Unsaturated polyester resins (UPR) are the third-largest class of thermoset molding

resins. They constitute the most common polymers used in conjunction with glass fiber

reinforcing. Typical applications are in fiberglass-reinforced shower stalls, boat hulls,

wind turbine blades, construction panels and autobody parts. UPR are prepared in a step

polymerization process from a glycol, typically 1,2 propylene glycol together with both

a saturated and an unsaturated dicarboxylic acid. The unsaturated acid provides sites for

cross-linking along the backbone, while the saturated acids effectively limit the number

of cross-link points. The effect of limiting theses sites is to reduce the brittleness of the

cured resin. They are numerous possible monomers that can be used in the backbone of

the polyester prepolymers. Typical monomer includes 1,2-propanediol, with maleic acid

(usually added as an anhydride) and phtalic acid (again usually added as an anhydride).

Cross-linking is initiated through an exothermic reaction involving organic peroxides,

such as methyl ethyl ketone peroxide or benzoyl peroxide [4].

A relatively new class of polyesters is the so-called liquid-crystalline polyesters which

possess excellent flow properties, but their use is still limited. Hence this class of

polyesters will not be discussed in the thesis [5].

1.2. Modified polyesters

In general, polyesters in bulk do not fulfill all requirements of industrial applications.

Ways to overcome this is to modify the bulk properties by melt-reactive blending, for

instance with other polymers, by copolymerization in the melt or in the solid-state or by

cross-linking the material [6]. In the first case, miscibility of the blend is essential. This

implies keeping two homoploymers in the molten state in the presence of a catalyst to

enhance the trans-esterification reactions [7]. In that way, a fully random copolymer is

obtained [8,9].

Several studies about modification of polyesters can be found in literature, for example,

about the modification of poly(butadiene terephtalate) (PBT) using reactive blending

[9,10].

Chapter 1

4

Generally, polyester is modified through the so-called melt polymerisation (MP). The

principle of MP is based on mixing a diol monomer with a homopolymer in the

presence of a catalyst [11]. The diol monomer has to fulfill some conditions, such as

having a melting temperature (Tm) below that of the homopolyester and being thermally

stable. During the incorporation of the diol monomer, the original diol of the

homopolyester should evaporate. The end product is a modified polyester with diol

monomer and it is a fully random copolymer. The two previous methods of modification

of polyesters are mainly suited for amorphous polymers. For semi-crystalline polyesters,

the situation gets more complicated. In fact, the resulting copolymers have a shorter and

more irregular crystalline structure then the original homopolyester. This implies a

decrease in the melting temperature, crystallinity, and crystallization rate, so that the

final macroscopic properties will be affected.

During the last decade, studies have been reported on ways to modify semi-crystalline

polyesters in a controlled way, so that the crystallization behavior of the original

polyester is retained, while the mechanical properties are enhanced. One that caught our

attention is the so-called solid-state polymerization (SSP) [12,13,14]. SSP is generally

used to increase the molecular weight of semi-crystalline condensates [15]. The

principle of SSP is based on performing a reaction at a temperature well above the glass

transition temperature (Tg) of the original homopolyester and just below its melting

temperature (Tm). In that way the polymer chains in the amorphous phase are highly

mobile and the transesterification reaction with the other polycondensate can occur.

These reactions are generally of the type of outer-outer transesterification reactions [14].

Another method to modify polyesters is through cross-linking. Cross-linking can be

introduced into an assembly of polymer molecules either as the polymerization takes

place or as a separate step after the initial macromolecule has been formed.

Covalent chemical bonds that occur between macromolecules are known as chemical

cross-links. Their presence and their density have a major influence on both chemical

and mechanical properties of the material in which they occur. The presence of cross-

links between macromolecules influences the way in which these materials respond to

heat. As mentioned earlier, uncross-linked polymers will generally melt and flow at

sufficiently high temperatures; they are usually thermoplastics. By contrast, cross-linked

polymers cannot melt because of the constraints of molecular motions introduced by the

Chapter 1

5

cross-links. Instead, at high enough temperature, they will undergo irreversible

deformation. Such polymers will also not dissolve, as the solvatation of chain segments

cannot overcome the effect of the covalent bonds between the macromolecules.

However, depending on the cross-link density, such material may admit significant

amount of solvent, becoming softer and swollen. Such swelling is reversible.

The mechanical properties of the cross-linked polymers depend on the degree of cross-

linking. Heavily cross-linked polymers tend to be very brittle. Indeed, heavily cross-

linked materials contain a dense three-dimensional network of covalent bonds that

implies very little degree of freedom for molecular motion. Hence there is no

mechanism available to allow the material to take up the stress.

Certain commercially important cross-linking reactions are carried out with unsaturated

polymers. One of the widely used ones involves unsaturated polyesters. As already

mentioned in the first part of this chapter, such polyesters are obtained using

bifunctional acids that contain double bounds. In most cases the cross-linking step

consists of adding a solvent monomer (usually styrene) to the unsaturated polyester,

followed by curing the system. The cross-linking reaction is based on a typical radical

polymerization reaction, initiated by the presence of a peroxide, activated by a redox

reaction with cobalt salts (or thermally). This kind of reaction is called curing. In

general, during the curing reaction, the physical properties of the polyester are changing

such as the shear, the torsion modulus, the hardness, the dielectric constant and the

viscosity. The idea is to generate a three-dimensional network with good mechanical

properties and strength impact [16-20].

1.3. Characterization of polyesters

We can divide the modified polyesters in two classes: on the one hand the thermoplastic

materials such as modified polyesters obtained via reactive blending or via MP, or SSP,

and on the other hand the thermoset cross-linked polyesters, which consist of a rigid

network. Depending on the class of polyesters, different characterization techniques are

required. In general, modified polyesters such as modified PBT or PET, can be

characterized with liquid-state Nuclear Magnetic Resonance Spectroscopy (NMR), Size

Exclusion Chromatography (SEC), High-Performance Liquid-Chromatography (HPLC),

Chapter 1

6

Differential Scanning Calorimetry (DSC), and Small-Angle X-ray Scattering (SAXS). 1H liquid-state NMR and solid-state NMR could be used to study the chemical

microstructure [12,14]. Quantitative 13C NMR can also be used to study the sequence

distribution in order to determine the degree of randomness (R) of the copolymer. DSC

could be used to investigate the thermal properties of the material as well as the

morphology. Diffraction techniques can also be used for that purpose. The average

molecular weight (Mw) of the polymer can be determined by chromatographic

techniques. Characterization of polymer networks is a much more complex task. This

has captivated the attention of many polymer scientists, especially during the last two

decades [15,16,18-26]. To be able to improve and optimize the macroscopic properties

of cross-linked resins, it is crucial to develop a thorough understanding at the

microscopic level, including features such as the cross-link density of a network, the

number of dangling chains, loops, and eventual polymer chains unattached to the

network (commonly called the sol fraction). The chemical structure (topology) of cross-

linked networks is extremely complicated [19, 20]. During the formation of a network,

many different structures may be formed, resulting in a very heterogeneous network.

Different initiation, propagation, termination, and combination reactions may occur, as

well as side reactions with other components present in the curing resin system. Many

parameters affect the curing process, including the temperature, the solvent, the

chemical and physical properties of the (pre)-polymers, the additives and the catalysts

present. The final network structure with all its variations, heterogeneities and defects

determines the properties of the network [21].

Not many analytical techniques can be applied to study the chemical structure of

polyester networks. Fourier-Transform InfraRed (FTIR) and High-Resolution Solid-

State Nuclear-Magnetic Resonance (NMR) spectroscopy will yield information on the

chemical composition of the network, such as the number of residual double bonds.

However, this information pertains to an average structure, rather than to the actual

chemical moieties present. In other words, the heterogeneity of the network is not

reflected. Also, it is very difficult to use such spectroscopic methods to obtain accurate

quantitative data on networks.

The morphology of polymeric networks can be studied with X-ray and neutron-

scattering techniques, or with spectroscopic methods. The physical and mechanical

Chapter 1

7

properties of the networks can be studied using thermal and mechanical techniques,

such as Differential Scanning Calorimetry (DSC), Dielectric Spectroscopy (DES) and

Dynamic Thermal Mechanical Analysis (DMTA). Also, solid-state NMR relaxometry

(mainly 1H T2-relaxation) has proven to be a valuable technique for studying various

types of polymeric networks; these include, for example, cross-linked epoxy resins [22,

23], cross-linked poly(styrenes) [24], and cured EPDM [25], as well as network resins

and unsaturated polyesters [26].

1.4. Scope of the thesis

The objective of this project is to study and investigate structure-property relationship of

modified polyester samples using solid-state NMR spectroscopy and infrared

spectroscopy (FTIR) as the main characterization techniques. Three types of modified

polyesters have been studied: two types of modified poly(butylene terephthalates) (PBT)

with diol monomers and an unsaturated polyester network.

We studied the microstructure of modified PBT. Two types of copolymers have been

studied; one copolymer resulting from the incorporation of 2,2-bis[4-(2-

hydroxyethoxy)phenyl]propane monomers also known as Dianol 220 and the other

copolymer resulting from the incorporation of bis(2-hydroxyethyl)terephthalate

monomer also called BHET. Both monomers have been incorporated via solid-state

polymerization (SSP). For the first copolymer a three phases system morphology was to

be demonstrated. For that purpose, 1H T1 and 1H T1ρ experiments were performed in

combination with cross-polarization NMR and 1H T2-relaxation measurements.

Mixtures with different PBT / Dianol ratios were analyzed.

On the second copolymer system, the incorporation of the BHET monomer in PBT via

SSP was investigated. The idea was to follow the kinetics of incorporation under

isothermal conditions. In that way, the development of the chemical microstructure

during the SSP reaction could be examined, and the miscibility of the BHET and the

PBT could also be investigated. Samples with different PBT/BHET ratios were also

analyzed. The kinetics of the SSP reaction was followed using infrared spectroscopy.

The microstructure and thermal behavior of the different copolymer samples were

Chapter 1

8

analyzed using two-dimensional correlation infrared spectroscopy. The presence of PET

homopolymer (blended material) was to be demonstrated.

Furthermore we studied the molecular dynamics of a cross-linked polyester network. It

is well known that the characterization of such chemically cross-link 3D-network is a

very challenging task. 1H-NMR T2 relaxation experiments have been performed to

investigate the mobility and the heterogeneity of the network. Solid-echo and Hahn-

echo pulse sequences have been applied. The data were fitted using combinations of

Gaussian and exponential decays. A 1 dimensional (1D) and a (2 D) approach have been

used for the analyses.

1.5. Outline of the thesis

In Chapter 2, the basic principles of solid-state NMR are reviewed, based on the

classical physics approach. Also NMR relaxation mechanisms are described and related

to the molecular dynamics and the morphology of the polymeric materials.

In Chapter 3, solid-state NMR spectroscopy and relaxometry have been used to

determine the morphology of the final products of the incorporation of a Dianol

monomer into poly(butylene terephthalate) (PBT). A series of these modified polyesters

has been investigated with different dianol / PBT ratios. Solid-state NMR of 1H spin

diffusion measurements, spin-lattice relaxation times T1 and T1 (in the rotating frame)

were used to investigate the morphology in combination with T1 filtered 1H-13C cross-

polarization experiments. The results have been compared to these obtained through

SAXS and TMDSC techniques.

In Chapter 4, static solid-state 1H spin-spin or transverse relaxation (T2) experiments

were used to investigate the rigid fractions of the modified PBT samples. The results

were fitted with a multi-exponential decay model, or a combination of Gaussian and

exponential decays. Solid-echo as Hahn-echo pulse sequences have been applied.

In Chapter 5, heterogeneous chain dynamics in a cured unsaturated polyester resin

(UPR-Palatal intermediate) are investigated using 1H-NMR transverse relaxation

experiments (T2). Different transverse relaxation times (T2) are reported at different

temperatures. T2 measurements were performed on a cross-linked sample in the non-

Chapter 1

9

swollen and the swollen state, at respectively 4.7 and 11.7 Tesla by means of a novel

multi-variable procedure (Appendix 1& 2). Spectral resolution has been implemented

at high field, correlating line-shape information to relaxation decay. The obtained two-

dimensional data sets were analyzed in terms of two decaying components without any

assumption about the underlying line shape. Furthermore, Dynamic Mechanical

Thermal Analysis (DMTA) and Dielectric Spectroscopy (DES) were used to provide

complementary information on the network mobility.

In Chapter 6, the kinetics of incorporation of the bis (2-hydroxyethyl)terephtalate

(BHET) monomer in poly(butylene terephthalate) (PBT) via solid-state polymerization

(SSP) was investigated using infrared spectroscopy. In that way, the development of the

chemical microstructure during the SSP reaction could be examined, and the miscibility

of the BHET and the PBT could also be investigated. Samples with different

PBT/BHET ratios were characterized. The microstructure and thermal behavior of the

different copolymer samples obtained through SSP were analyzed using two-

dimensional correlation infrared spectroscopy. The presence of PET homopolymer

(blended material) was demonstrated.

1.6. References

[1] Van Berkel, R. W. M.; Van Hartingsveldt, E. A. A.; van der Sluijs, C. L. Handbook of Thermoplastics, Marcel Dekker Ed. 1997.

[2] Young, R. J.; Lovell, P. A. Introduction to Polymers, Chapman & Hall 2nd Ed. 1991.

[3] Flory P. J. Principles of Polymer Chemistry, Cornell Press Ithaca 1953.

[4] Coran, A. Y. Encyclopedia of Polymer Engineering, John Willey & Sons 2nd Ed. 1989.

[5] Mucha, M. Colloid &Polymer Science 1991, 269, 1435.

[6] Devaux, J.; Godard, P.; Mercier, J. P.; Touillaux, R.; Dereppe, J. M. J.Polym. Sci., Polym. Phys. 1982, 20, 1881.

[7] Backson, S. C. E.; Kenwright, A. M.; Richards, R. W. Polymer 1995, 36, 1991.

[8] Kim, J. H; Lyoo, W. S.; Ha, W. S. J. Appl. Polym. Sci. 2001, 82, 159.

[9] Marchese, P.; Celli, A; Fiorini, M. J. Polym. Sci., Polym. Phys. 2004, 42, 2821.

[10] Fernandez-Berrediti, M. J.; Iruin, J. J.; Maiza, I. Polymer 1995, 36, 1357.

[11] Berti C.; Colonna, M.; Fiorini, M.; Lorenzetti, C.; Marchese, P. Macromol. Mat. Eng. 2004, 289, 49.

[12] Hait, S. B.; Sivaram, S. Macromol. Chem. Phys. 1998, 199, 2689.

[13] Kimura, M.; Porter .R. S. J.Polym. Sci., Polym. Phys. 1983, 21, 367.

[14] Jansen, M. A. G.; Goossens, J. G. P.; de Wit, G.; Bailly, C.; Koning, C. E. Macromolecules 2005, 38, 2659.

[15] Jansen, M. A. G.; Goossens, J. P. G.; de Wit, G.; Bailly, C.; Koning C. E. Anal. Chim. Acta 2006, 557, 19.

[16] Litvinov, V. M.; Prajna, P. Spectroscopy of Rubbers and Rubbery Materials 2002, 360.

[17] Vilgis, T. A.; Heinrich, G. Coll. Pol. Sci. 1991, 269, 1003.

[18] Harrison, D. P. J.; Yates, W. R.; Johnson, J. F. J. Macromol. Sci. 1985, 25, 481.

[19] Schimmel, K-H.; Heinrich, G. Coll. Pol. Sci. 1992, 242, 1008.

[20] Vilgis, T. A.; Heinrich, G. Macromol. Theory and Sim. 1994, 3, 271.

[21] Woll, R. P. Macromolecule 1993, 26, 1564.

[22] O’Donnell, J. H.; Whittaker A. K. Polymer 1992, 33, 62.

[23] Orza, R. A.; Magusin, P. C. M. M.; Litvinov, V. M.; van Duin, M.; Michels. M. A. J. Macro. Symp. 2007, 40, 8999.

[24] Mohanraj, S.; Ford, W. T. Macromolecules 1985, 18, 351.

[25] Winters, R.; Lugtenburg, J.; Litvinov, V. M. Polymer 2001, 42, 24.

[26] Spyros, A. Journal of Applied Polymer Science 2003, 88, 1881.

10

Chapter 2

11

Chapter 2

Basics of NMR relaxation for polymer structure investigation

2.1. General approach

Nuclear magnetic resonance spectroscopy (NMR) uses the spin resonance phenomenon

to study physical, chemical, and biological properties of matter. As a consequence,

NMR spectroscopy finds applications in several areas of science.

NMR is a very powerful analytical tool to investigate molecular structure-property

relationships of polymer systems [1-8]. NMR can be used, on the one hand, as a

spectroscopic technique to obtain information about packing of the polymer, chemical

structure and conformation of the molecules. On the other hand, NMR can be used as a

relaxometry technique to probe chain dynamics and morphology of polymers [9-12].

This chapter should provide the reader with an understanding of the principles of NMR

from the microscopic, macroscopic and materials (polymer) perspectives. First, the

basic principles of solid-state NMR are reviewed based on the classical approach of

physics, then NMR relaxation mechanisms are described and related to the molecular

dynamics and the morphology of polymeric materials.

2.1.1. Definition and physics basis

Nuclear magnetic resonance is a physical phenomenon of atomic nuclei in a magnetic

field, which can be used to investigate molecular properties of matter [13-17]. Atomic

nuclei of most isotopes, except those with both even proton and neutron numbers,

possess a property called spin. Spin is a quantum mechanical operator I = (Ix, Iy, Iz)

proportional to the angular momentum J = I with Planck’s constant = h/2. I can

have different orientations, but its size is restricted to )1( II where I is an isotope-

dependent spin quantum number.

Chapter 2

12

Nuclei with non-zero spin also have a magnetic moment:

Iμ (2.1)

with the so-called gyromagnetic ratio, which is an isotope dependent property. For

polymer materials the 1H- and 13C-nuclei are the most important. Both have spin

quantum number I = ½ and therefore have two quantum states |> and |> with different

projections <Iz> of I on the magnetic field B0, respectively, m = ½ and m = -½. As

follows from the Zeeman Hamiltonian for a spin interacting with a magnetic field with

size B0:

0BzIH (2.2)

the spin-up state |> has a lower energy E –½ B0 and the spin-down state |> has a

higher energy E ½ B0 . Transitions between the and state can be induced by

radio-frequency (rf) radiation with the proper angular frequency:

/0 EE B0 (2.3)

This is the basis of nuclear magnetic resonance. Instead of this quantum mechanical

picture of NMR, a convenient semi-classical picture is nowadays used for NMR. In

thermodynamic equilibrium, an ensemble of nuclear spins in a magnetic field B0 has a

net alignment of the overall nuclear magnetic moment along B0. A short rf pulse at their

resonance frequency o will tilt this overall magnetization away from B0. After this

pulse, the magnetization will precess around the magnetic field axis with frequency o =

B0 according to the equation of motion for angular momentum J under the influence of

a torque x B0 which tries to align along B0:

0BμJ

dt

d (2.4)

or, after substitution J = I and = I = J, in terms of the magnetic moment :

0Bμμ

dt

d (2.5)

and in terms of the spin operator

0BII

dt

d (2.6)

Chapter 2

13

Figure 2.1.a. Orientation of the magnetic moment () vectors in a static magnetic field B0 (left). Figure 2.1.b. Energy scheme of the nuclei (I=1/2) (right).

According to Planck’s law, the energy necessary to reverse the orientation of the

magnetic moment or to produce a transition between the energy levels equals the

difference in the allowed energy levels. The frequency of the radiofrequency (rf)

irradiation ( 0 ) needed to cause such a transition is proportional to the Larmor

frequency (0). The difference in energy is given by the following equation:

00

2

1

2

1 hν.ωEEΔΕ

(2.7)

2.1.2. Pulsed NMR and bulk magnetization

It is rather cumbersome to describe NMR on a microscopic scale. A macroscopic picture

is more convenient. In an NMR experiment, a large number of nuclei are observed (in

the order of 1023). The vector sum of the magnetization vectors from all the individual

nuclei results in a net or bulk magnetization or M0 that aligns at equilibrium along the

direction of the field B0 along z-axis (Figure 2.2). In order to describe pulsed NMR it is

necessary to talk in terms of the net or bulk magnetization.

As already mentioned above, when a group of spins are placed into a magnetic field B0,

the spin aligns in one of the two possible orientations.

At room temperature, the number of spins in the lower energy level, N, slightly

outnumbers the number in the upper level, N. Boltzmann statistics tells us that:

Chapter 2

14

TE/k -

α

β beN

N (2.8)

E is the energy difference between the spin states; k is the Boltzman's constant,

1.3805x10-23 J/K, and T is the absolute temperature. One could observe that the ratio N

/ N decreases with decreasing temperature resulting in an increased sensitivity.

The value of the net magnetization at equilibrium is given by the Curie’s law as a

function of the static magnetic field B0, the temperature of the system T, the number of

spins in the sample N, the spin I and the gyro-magnetic ratio (equation 2.9):

0B..T.k

).I(I.N.γM

B3

122

0

(2.9)

In the NMR experiment the frequency of the photon is in the radio frequency (rf) range.

In current 1H NMR spectroscopy, 0 ranges between 60 and 1000 MHz.

In pulse NMR the signal is proportional to the population difference between the states.

NMR is in principal a rather insensitive spectroscopic technique because of the small

population differences at the Boltzmann equilibrium state with the small h quanta.

Thanks to the state- of- the- art electronics, very small population differences can be

detected in a very efficient way.

Figure 2.2. Precession of the magnetic moment around a double cone (left). Distributions of all the magnetic moments (), on upper () and lower energy level () with N N. A macroscopic resultant called net magnetization is observed (or bulk magnetization M0 (right)) .

Chapter 2

15

2.1.3. Laboratory frame and rotating frame

In an NMR experiment, transitions are induced between two spin states by irradiating

the nuclei with a superimposed radiofrequency (rf) pulse (also called B1 field) applied

perpendicularly to the original field B0. The net magnetization vector M0 is tipped away

from the z- axis, describing a complex spiral motion going up and down along the z

axis. The circular motion is in a MHz frequency-range (rf), while the nutation is in the

kHz range.

This is the situation in the laboratory frame (xyz). Using a rotating frame (x’y’z), that

rotates with frequency rf, does fix the B1 vector along the x’ axis. In that case, M0

describes a simple circular motion in the z-y’ plane.

To understand how a small magnetic field B1 is able to rotate the magnetization away

from the z-axis even in the presence of a strong magnetic field B0, it is necessary to look

at the motion in a rotating frame of reference R’(x’y’z).

In the laboratory frame R the atomic magnetization vectors precess about B0 at a

constant angle and at an angular frequency 0 . When a rf pulse is applied along the x-

axis (B1) the magnetization will precess around an effective magnetic field Beff at an

angular frequency eff . The rf field B1 oscillates about B0 at an angular frequency rf .

In the rotating frame R’ the magnetization is seen to precess around the static magnetic

field at an apparent frequency ( 0 - rf ).

According to equation (2.1), the static magnetic field will not be B0 anymore but a

reduced magnetic field: ∆B = ( 0 - rf )/. When o = rf, the apparent Larmor

frequency will be 0 and the static magnetic field will vanish as described in Figure 2.3.

In the rotating frame B0 seems to shrink and B1 becomes dominant. The rotation angle

in the plane z-y’ depends on the time the transmitter field B1 is on (the pulse width) p

and its magnitude:

1B p (2.10)

Chapter 2

16

z z

∆B Ω Beff ωeff

x xB1 ω1

Figure 2.3. In the rotating frame, the effective field Beff is the vectorial sum of the reduced field B and the B1 field (left). We can also express the Beff in term of frequencies (right). is defined as the angle between B and Beff.

As observed in Figure 2.4 a 90 pulse rotates the equilibrium magnetization down to the

y’- axis. Stopping the transmitter B1 after a 90° pulse results in a maximal transverse

magnetization rotating about the z- axis and will induce a current in a coil of wire in the

transversal plane. The receiver will detect a current in the x’y’-plane which gives a sine

wave as a function of time. This wave however will decay with a time constant T2* (see

chapter 2.2.1.2.) due to dephasing of the spins. This signal is the free induction decay

(FID). The FID is converted into the frequency domain spectrum by a Fourier

Transformation (FT) (Figure 2.5).

z

y’

x’

B1

My’

z

y’

x’

B1

My’

Figure 2.4. Direction of the net magnetization vector M0 after a 90 x’. A 180 pulse rotates the equilibrium magnetization down to along the z axis.

Chapter 2

17

Figure 2.5. Decay of the transverse magnetization M, in the time (left) and frequency domain after Fourier-transformation, FT, (right).

2.2. NMR relaxometry

2.2.1. Spin Relaxation times

2.2.1.1. Spin-lattice relaxation time (T1)

At equilibrium, the net magnetization vector lies along the direction of the applied

magnetic field B0 and is called the equilibrium magnetization M0. In this configuration,

the z component of magnetization MZ equals M0. MZ is referred to as the longitudinal

magnetization. When a 90°-pulse is applied, the magnetization along B0 disappears and

the population of the two levels becomes the same (or inverted after a 180 pulse) as

seen in Figures 2.6.

The process to return to the Boltzmann equilibrium state along the z-axis is the so-called

spin-lattice relaxation time or longitudinal relaxation time T1.

Inversion recovery (Figure 2.6.b) is mostly used to determine T1 relaxation. In that case,

the magnetization Mz is described by equation (2.11):

(2.11))21( 1/0

Ttz eMM

Chapter 2

18

y’

z

y’

x’

B1

My’

n = n

z

x’

B1

My’

n > n

M-z

z-

(a) (b)

y’

z

y’

x’

B1

My’

n = n

z

x’

B1

My’

n > n

M-z

z-

(a) (b)z

y’

x’

B1

My’

z

y’

x’

B1

My’

n = n

z

x’

B1

My’

n > n

M-z

z-

(a) (b)

Figure 2.6.(a): direction of the net magnetization vector M0 after a 90x’- pulse, so called saturation recovery (left). Figure 2.6.(b): after a 180x’- pulse, so called inversion recovery (right).

Also a 90°-pulse can be used, the so-called saturation recovery (Figures 2.6.a and 2.6.b);

in that case, the return of the Mz magnetization to equilibrium value M0 is described by

the following equation:

(2.12) )1( 1/0

Ttz eMM

The interactions of the magnetization vectors with the changing electromagnetic dipoles

in the lattice result in a transfer of energy from the spin system to the lattice. This spin

lattice relaxation process is an enthalpic process and a first order process that can be

described by the Bloch equation [16] as below:

1

0 )(

T

MM

dt

dM zz (2.13)

As already mentioned, most of T1 relaxation time measurements are performed by

means of the inversion recovery method. Due to the repetitive character of pulse NMR

spectroscopy, T1 is heavily related to its quantitative aspect. Effectively, before applying

a new pulse sequence, it is necessary for Mo to recover its Boltzmann equilibrium.

Therefore, to obtain quantitative results, a delay time of 5 times T1 should be respected

between two successive pulse sequences.

Chapter 2

19

2.2.1.2. Spin-spin relaxation (T2)

Spin-spin relaxation is defined as the decay to zero of the transverse component of M0

(the magnetization My’). When applying a pulse ( pulse), the magnetization M0

along the z- axis is not only flipped away from the direction of B0, but also the

precession movements of the magnetic moments are synchronized (coherence). As in

the case of a oscillatory movement, the magnetic moments precess at the same

frequency and phase. The de-phasing (loss of coherence) starts when B1 = 0. The

precession movements of each spin are no more synchronized, as each spin will

experience a slightly different magnetic field caused by spin-spin interactions. After a

certain time, the projection onto the x’-y’ plane are uniformly distributed around B0 as

seen in Figure 2.7. After a time T2, the transverse components of the magnetizations Mx’

and My’ are reduced to 0.37 M0 (2.15). This time is the so- called spin-spin relaxation

time. What is measured is the lost of coherence between the spins as a function of time.

'90 x

The decay of the x’-y’ magnetization may be written using the Bloch’s equations [16]:

2

''

T

M

dt

dMxx (2.14.1)

2

''

T

M

dt

dMyy (2.14.2)

During spin-spin relaxation the magnetization is transferred from one spin to another

without changing the net energy of the system (flip flop mechanism) that is an entropic

process; no change in population of the energy levels occurs.

The decay of the transverse magnetization My’ is also described by an exponential

equation:

(2.15) 2/0'

Tty eMM

T2 cannot be longer than T1. When the longitudinal magnetization has recovered its

maximal value aligned along the z-axis, the transverse magnetization is 0. The study of

T2 reveals important information about the structure of the matter since T2 determines

the line width of the signal in the frequency spectrum. This line width probes

information about local mobility of the sample as explained in the following section

(2.2.2).

Chapter 2

20

x’ y’ x’ y’

x’ y’ x’ y’

x’ y’

(a) (b)

(c) (d)

(e)

x’ y’ x’ y’

x’ y’ x’ y’

x’ y’

(a) (b)

(c) (d)

(e)

Figure 2.7. Loss of coherence between the spins after a 90 x’ pulse (a, b, c), after a partial dephasing (d) and after complete loss of coherence (e) (My’= 0).

The line width is given by the following relation:

2

2/1.

1

T (2.16)

As the static magnetic field B0 is not exactly homogeneous, field inhomogeneities cause

nuclei (even the ones that are chemically equivalent) to precess with slightly different

Larmor frequencies. Therefore T2* is the spin-spin relaxation time that incorporates field

inhomogeneities:

1/ = 1/T2mat + 1/T2inhom (2.17) 2T

The experimental pulse sequences to determine and T2 are described in Chapter 4. 2T

2.2.1.3. Spin lattice in the rotating frame (T1)

As introduced earlier, a classical pulse will bring the magnetization in the x’y’

plane. If this is immediately followed by a 90 phase-shift of the transmitter, also the B1

field will lie along y’ and will lock the magnetization on the y’-axis as seen in Figure

2.8. The relaxation of the magnetization vector becomes more difficult because of the

competition between B1 and B0. However B0 is a much stronger field, so the

magnetization will relax much more slowly. The time it will take for the spin lock

'90 x

Chapter 2

21

magnetization to relax is the so-called spin-lattice relaxation in the rotating frame (T1).

As for T1, the lattice also influences T1. This is also an enthalpy process. Molecular

motions stimulate this relaxation most efficiently when their frequencies match the

frequencies dictated by the strength of the spin lock field B1 (1 =γ. B1) that is in the

order of tens of kHz. The experimental pulse sequence to determine T1 is described in

Chapter 3.

Figure 2.8 (a, b) 90 x’ pulse and phase shift. (c, d ) spin-lock of the magnetization along y’-axis.

2.2.2. NMR relaxometry for polymer characterization

Besides classical spectroscopy, NMR has long been used to study the dynamics of

polymers since spin relaxation times and line-shapes are extremely sensitive to chain

motions [17-20]. Solid-state NMR experiments probe information about molecular

structure and molecular dynamics. Solid-state NMR probes chain dynamics of polymers

that are orders of magnitude slower as compared to polymers in solutions.

As mentioned above, one of the effects of radio-frequency pulses is to cause the spin

populations to deviate from their Boltzmann equilibrium population. The rate at which

Chapter 2

22

d

xation times that can be measured depending on the scale of the

relaxation time (T2) can be used to look at slower molecular

tric

the energy levels of the nuclear spin state

: quadrupolar interactions, HD: dipolar interactions, HJ :

usion [26, 27]. Both

influences on relaxation are described in the following sections.

the spin system returns to equilibrium tells us about the molecular dynamics and the

inter-nuclear distances. Molecular motions induce nuclear magnetic relaxation since

atomic fluctuations modify the local magnetic field experienced by nuclei in their

environments. These local fluctuations can occur in a broad range of frequencies an

depend directly on the chemical structure and the morphology of the material [21,22].

There are different rela

investigated motions.

Fast molecular motions can be investigated by measuring the so called spin-lattice

relaxation time (T1), or spin-lattice relaxation in the rotating frame (T1) while

measuring the spin-spin

dynamics [23, 24, 25].

For relaxation studies in more solution-like materials (e.g. amorphous polymeric system

above Tg), the relaxation times are very sensitive to molecular motions in the MHz scale

and relaxation is caused by the combination of rapid motions. In the solid-state,

relaxation times are sensitive to the same time scale of motion but also to dynamics on

the kHz scale. Depending on the type of nuclei and their environment, Dipole-Dipole

interactions (D-D), Chemical Shift Anisotropy (CSA) and QUADrupolar interactions

(for spins ≠ 1/2)(QUAD) will be responsible for the relaxation mechanism. In general,

the observed relaxation rate is the sum of all these contributions. The strength of each

interaction is very different, therefore the relaxation mechanism is mostly caused by the

interactions that are the most efficient. In quantum mechanics we will use the

Hamiltonian operator H that contains several terms describing the magnetic and elec

interactions. They lead to the splitting of s:

H = Hz +Hq +Hrf +HD +HJ (2.18)

Hz: Zeeman interactions, Hq

scalar coupling interactions.

In polymer systems, dipolar interactions (DD) between 1H-1H or 13C - 1H are the most

efficient mechanism with a strength around 100 kHz or less. Due to the weak strength,

chemical shift anisotropy of proton nuclei and the J-coupling (100 Hz) can be neglected.

Through these magnetic dipolar interactions, two majors phenomenon will dominate the

spin relaxation in polymer, molecular motions and spin diff

Chapter 2

23

2.2.2.1. Relaxation times as function of the magnetic field strength and the spectral

density of molecular motions

In NMR, transverse relaxation experiments refer to attenuation of single-quantum

coherences, which are entirely absent when thermal equilibrium is reached. As

mentioned above, the losses of coherences in polymers are mainly due to dipolar

coupling. As a consequence of molecular dynamics, the local field produced by the

dipolar interaction changes, leading to a change in the precession frequencies.

For a two-spin system the local field can be totally or partially averaged out, depending

on the frequency of the molecular motion and on the time scale on which the

information is requested,

Bloc = )1cos3(1

. ,2

,3

SISIr

(2.19)

r I,S is the inter-nuclear distance, θI,S the angle between the inter-nuclear vector and the

main field B0. Molecular motions will cause fluctuations of θI,S and depending on the

rate of the fluctuation the relaxation mechanism will be rather fast or rather slow. In

fact, the relaxation times are strongly dependent on the way the dipole-dipole

interactions are varied with molecular motions by the so called correlation time c . This

c corresponds to the time needed to describe a radius (circle divided by 2) and is

representative for the interval between successive reorientations (by vibrational,

rotational or translation motions). It is important to mention that the temperature,

viscosity and molecular weight have an influence on the correlation time c and as a

consequence also on the relaxation times [28].

Figure 2.9 shows the relationship between the relaxation times T1, T1, T2 and the

correlation time c at different field strength. One can observe that when B0 increases,

the T1 minimum increases and moves to shorter values of correlation times.

All the different orientations of the molecule will be spread over a broad range of

frequencies. The distribution of the different reorientations is given by the spectral

density function defined as:

221

2)(

c

cJ

(2.20)

Chapter 2

24

Figure 2.9. Relaxation times (T1 , T2, T 1 ) in function of correlation time c at a two magnetic field strength ( 400MHz and ---20MHz).

Bloembergen, Purcell and Pound [28] describe the dipolar relaxation rates for two equal

spins at a distance r and with isotropic fluctuations as function of the spectral density

function:

LL JJrT

24

10

316

24

1

(2.21)

LL JJJrT

22503

20

316

24

2

(2.22)

16

24

1 2

32

2

5

10

31

JJJrT LL

(2.23)

From these equations, one could see that molecular motions at the Larmor

frequency LJ (with L=0) or at twice the Larmor frequency, LJ 2 in the MHz

region, have an influence on the three relaxation times. The static term J(0) only

contributes to spin-spin relaxation and describes the interactions that cause the loss of

coherence between the spin. The 1J term is only contained in the T1 and depends

on the B1 field (kHz regime).

Chapter 2

25

The line-width of the frequency distribution in the absence of molecular motion (Δ)rl

and the correlation time c of the fluctuating interactions are used to classify molecular

motion. From equation 2.21 to 2.23 and from Figure 2.10 we may distinguish three

motional limits:

- In the complete motional averaging limit ((Δ)rl . c 1) the transverse relaxation

is influenced by non secular (rapidly varying) spin interactions when

0 . c 1. The J (0) values are low for all frequencies of molecular motions.

Therefore, none of the relaxation processes are efficient. This region is known as the

extreme narrowing limit and is located in Figure 2.9 on the left side of the minimum.

All the relaxation times are similar in that region and decrease as c increases.

The fluctuation rate of the spin interaction is fast compared to the transverse relaxation

rate and the Bloch/Wangness/Redfield (BWR) theory is used to describe the situation

(the relaxation functions are described by simple exponentials) [15]. This is generally

observed in mobile amorphous polymeric region above the glass transition, comparable

to the liquid state.

- In a rigid lattice limit and large spin system ((Δ)rl . c 1 ), the transverse

relaxation is determined by secular (low varying) spin interaction, when 0 . c 1,

which tends to be quasi-stationary on the time scale of the transverse relaxation. In this

limit, a high density of low frequency motions is observed. The spectral density at the

Larmor frequency 0 and the T1 decay, which is influenced by motions close to the

Larmor frequency, are less efficient. In this case, the J (0) and J (20) terms (see BPP

equations) are also negligible in T2 relaxation and the term J (0), corresponding to the

static term, becomes predominant, leading the spin-spin relaxation to be dominated by

the very low frequency motions, so by static dipole-dipole interactions. One of the

theories to describe this T2 behavior applied to polymer systems is the Anderson/Weiss

formalism (the relaxation functions are described by Gaussian functions) [28, 29]. This

is often observed below the glass transition in semi-crystalline polymers and in the rigid

crystalline phase.

- ((Δ)rl . c =1 ) defines partial motional averaging limit. In this limit, the J(0) is

high at the Larmor frequency, so the spin lattice relaxation is the most efficient (as seen

in Figure 2.10). The transverse relaxation is determined by both secular and non-secular

Chapter 2

26

motions. The fluctuations are fast but not fast enough to completely average out the

dipolar interactions. The proper theory to describe this situation is also found in the

Anderson/Weiss formalism, which consists of a combination of Gaussian (secular

interactions) and exponential decay functions (non secular interactions).

This limit is often observed in polymeric networks. In general, “the relaxation picture”

in polymer networks (above Tg) exhibits different molecular mobilities in the material,

producing different relaxation signals of different forms and lengths. Physical or

chemical cross-links will show rather slow molecular dynamics while dangling end or

side-chain groups will depict faster molecular dynamics. This diversity of motions

(combination of isotropic and anisotropic motions) leads to describe the relaxation with

a superposition of exponential decays or stretch exponential decay (Weybull)[30] and

Gaussian decays. However, there is still some discussion in the literature on the

mathematical form of this decay [28-32].

Figure 2.10. Plot of the spectral density J() as function of the frequency , for (a) complete motional averaging limit (b) partial motional averaging limit (c) rigid lattice limit.

A more general model that collects for both exponential and Gaussian character is

widely used to describe the relaxation of the transverse magnetization in polymer

networks:

CBcccA TtCTtBttqMTtAtM 222

22 expexp1expexp)( (2.24)

M(t) represents the magnetization function, the factor A represents the fraction of inter-

cross-link chains, B corresponds to the fraction of more mobile components such

Chapter 2

27

dangling chains, and C corresponds to the liquid-like fraction commonly named sol-

fraction. t is the pulse spacing in the pulse experiment. T2A, T2B and T2C are the related

relaxation times, c the apparent correlation time of inter-cross-link orientations, and q

a measure of the mean residual part of the second moment M2 of the dipolar interaction

in the rigid limit (T< Tg).

2.2.2.2. Spin diffusion

In polymers the 1H-nuclei are dipole coupled and will form a sort of dipole-coupled

network. If the perturbation of the magnetization occurs in one volume element, it will

diffuse to the surrounding. That is the spin diffusion phenomenon [26, 27].

Spin diffusion is defined as the transport of spin energy within the lattice by mutual

energy conservative spin flips. As in the major part of diffusion processes, this follows

Fick’s law that predicts how diffusion causes the magnetization field to change with

time; that implies the presence of a magnetization gradient.

2

2 ),(),(

r

tqMD

t

tqM

(2.25)

),( tqM describes the magnetization at the position q and at time t, D is the spin-

diffusion coefficient which depends on the average proton to proton distance as well as

on the dipolar interaction and is about 1 nm2/s in polymers [33] .

For non-isolated nuclei in heterogeneous solids, T1 and T1 relaxation times of 1H-atoms

are in different surroundings and are generally averaged out through spin diffusion. The

spin diffusion process is only efficient in a rigid lattice. In liquid-like samples, spin

diffusion is vanished because of fast isotropization of the molecular motions. In semi-

crystalline polymers, crystalline phases show a different relaxation compared to

amorphous ones, so spin diffusion will be different in the different phases. For such

reason, proton spin-lattice relaxation measurements in the rotating and laboratory frame

(1H T1 and 1H T1) are often used to estimate the lower and upper limit of domains sizes

on the basis of the following equation, valid for supposed 3D nano-domain structures

[33]:

Chapter 2

28

DTL 6 (2.26)

If the average domain size in a polymer blend is smaller than ca. 1 nm, proton-proton

spin diffusion averages out any T1 or T1 relaxation difference. In that case, all protons

decay with the same effective T1 and T1. In contrast, if the domain size is larger than ca.

50 nm, spin diffusion is too slow to average out such differences and the phases will

decay each with their intrinsic (probably different) T1 and T1 values. In the intermediate

range, 1 nm < domain size < 50 nm, we expect to find different effective T1 values and

a single effective T1. The reason is that T1 tends to be 10-100 times longer than T1, so

that spin diffusion, though unable to homogenize T1, is still able to average out T1

differences.

2.2.2.3. Summary of spin relaxation time and application in polymers

1H T1 relaxation is dominated by molecular motions at the Larmor time scale 0.1 – 10 s.

In the liquid state, spin-lattice relaxation T1 is dominated by molecular dynamic while in

solids, because of strong dipolar coupling between protons, spin diffusion is very

effective and will also determine the T1 decay. In polymers below the glass transition

(Tg), T1 will be dominated by spin diffusion (right side of the curve see Figure 2.9) and

all the 1H-nuclei will relax through spin diffusion and will depict the same T1 value,

even though we may depict different regions or phases within the polymer. Above the

Tg. molecular dynamic will also determine the relaxation (left side of the curve Figure

2.9). Generally in pure liquid, 1H T1 will be different for chemically distinct protons as

spin diffusion is vanished. Exactly the same goes for 1H T1 relaxation. However 1H T1

is affected by motions in ms scale (1 – 100 ms), so T1 is mostly shorter then T1 and

relaxation will be averaged out over smaller regions. In polymers 1H T2 relaxation is

easier to interpret, as the spin-spin relaxation time is only determined by local molecular

dynamics. 1H T2 is short 10 μs – 1 s (tenths of microseconds for polymers); therefore

spin diffusion does not have time to average out the relaxation.

Spin-lattice relaxation is mainly used for morphology study of polymeric systems; it

helps to probe information about homogeneity of mixing in a blend or in a copolymer

while spin-spin relaxation is commonly used to characterize polymeric networks (cross-

Chapter 2

29

link density, network heterogeneity) and to quantify the crystalline phase of a semi-

crystalline polymer.

Chapter 2

30

2.3. References

[1] Packer, K. J; Pope, J. M.; Yeung, R. R.; Cudby, M. E. A. J. Polym. Sci. 1984, 22, 589.

[2] Havens, J. R.; Vanderhart, D. L. Macromolecules 1985, 18, 1663.

[3] Hentschel, D. R.; Sillescu, H.; Spiess, H. W. Macromolecules 1981, 14, 1605.

[4] Schmidt-Rohr, K.; Spiess, H. W. Multidimensional Solid-State NMR and polymers, Academic Press 1994.

[5] Miller, J. B. J. Thermal Analysis 1997, 49, 521.

[6] Yu, H.; Natansohn, A.; Singh, M. A.; Plivelic, T. Macromolecules 1999, 32, 7562.

[7] Hedesiu, C.; Demco, D. E.; Kleppinger, R.; Buda, A. A.; Blumich, B.; Remerie, K.; Litvinov, V. M. Polymer 2007, 48, 763.

[8] Yao, Y.-F.; Graf, R.; Spiess, H. W. Macromolecules 2008, 41, 2514.

[9] Wind, M.; Brombacher, L.; Heuer, A.; Graf, R.; Spiess, H. W. Solid State NMR 2005, 27, 132.

[10] Demco, D. E.; Litvinov V. M.; Rata, G.; Popescu, C.; Phan, K-H.; Schmidt, A.; Blumich, B. Macromol. Chem. Phys. 2007, 208, 2085.

[11] Hedesiu, C.; Demco, D. E.; Kleppinger, R.; Poel, G. V.; Remerie, K.; Litvinov, V. M.; Blumich, B.; Steenbakkers, R. Macromol. Mater. Eng. 2008, 293, 847.

[12] Bertmer, M.; Wang, M.; Kruger, M.; Blumich, B.; Litvinov V. M.; Van Es, M.; Chem. Mater. 2007, 19, 1089.

[13] Bovey, F. A.; Nuclear Magnetic Resonance Spectroscopy, Academic Press 1988.

[14] Friebolin, H.; Basic one- and Two-Dimensional NMR Spectroscopy, VCH Wiley 1991.

[15] Bloch, F.; Hansen, W. W.; Packard, M. Phys. Rev. 1946, 69, 127.

[16] Bloch, F. Phys. Rev. 1946, 70, 460.

[17] Litvinov, V. M.; De Prajna, P. Spectroscopy of Rubbery Materials, RAPRA Technology 2002.

[18] O’Donnell, J. H.; Whittaker, A. K. Polymer 1992, 33, 62.




[22] Adriaensens, P.; Storme, L.; Carleer, R.; D’Haen, J.; Gelan, J. Macromolecules 2002, 35, 135.

[23] Heuert, U.; Knorgen, M.; Menge, H.; Scheler, G.; Schneider, H. Polymer Bulletin 1996, 37, 489.

[24] Kuhn, W.; Barth, P.; Denner P.; Muller, R. Solid State Nuclear Magnetic Resonance 1996, 6, 295.

Chapter 2

31

[25] Fisher, E; Grindberg, F.; Kimmich, R. J. of Chem. Phys. 1998, 109, 846.

[26] Wang, J. J. Chem. Phys. 1996, 104, 4850.

[27] McBierty, V. J.; Douglass, D. C. J. Polym. Sci. Macromol. Rev. 1981, 16, 295.

[28] Bloembergen, E. M.; Purcell, R.; Pound, V. Physical Review 1948, 73, 679.

[29] Anderson, P. W.; Weiss, P. R. Rev. Modern. Phys. 1953, 25, 269.

[30] Litvinov, V. M.; Dias, A. A. Macromolecules 1999, 32, 3624.

[31] Simon, G.; Baumann, K.; Gronski, W. Macromolecules 1992, 25, 3629.

[32] Gotlib, Y.; Lifshitz, M. I.; Shevelen, V. A.; LishanskijI, I. S.; Balanina, V. Polym. Sci. USSR 1976 (English translation).

[33] Magusin, P. C. M. M.; Mezari, B.; Van der Mee, L.; Palmans, A. R. A.; Meijer, E. W. Macromol. Symp. 2005, 230, 126.

Chapter 3

33

Chapter 3

Morphology differences in modified polyesters probed by solid-state NMR

spectroscopy

Summary

In this study, solid-state NMR spectroscopy has been used to investigate the

morphology of poly(butylene terephthalate) (PBT)-copolymers obtained by the

incorporation of 2,2-bis[4-(2-hydroxyethoxy)phenyl]propane (Dianol) via solid-state

copolymerization (SSP). A series of such modified polyesters with varying PBT /

Dianol ratios have been investigated in previous work (M. Jansen et al.). The

morphology of the resulting products was shown to consist of a two-phase model, a

crystalline and an amorphous phase, the latter being subdivided into a rigid amorphous

fraction and a mobile amorphous fraction.

Static solid-state 1H NMR T1 measurements (spin-lattice relaxation times in the rotating

frame) were mainly used to investigate the morphology of the modified PBT samples,

together with T1(1H) -filtered 1H-13C cross-polarization experiments (CP-MAS). To be

able to distinguish between the different components of the sample, multivariate

statistics has been used. Also, from T1 (1H) and T1(

1H) measurements, upper and lower

limits of domain sizes were estimated. The results have been compared to small angle

X-Ray scattering (SAXS).

Chapter 3

34

3.1. Introduction

Poly(butylene terephthalate) (PBT) is a semi-crystalline aromatic polyester with a fast

crystallization rate. It is possible to modify PBT properties by reacting it in the melt

with other polycondensates. This modification results in a random copolymer with a

decreased crystallinity [1-6]. However, it is also possible to modify PBT while keeping

its fast crystallization rate. It has been shown in a previous study that solid-state

polymerization (SSP) is an effective method to incorporate diol co-monomers into PBT

while retaining its high crystallization rate and enhancing other properties such as

mechanical properties [7].

The process of SSP is to operate at a reaction temperature just below the melting

temperature of the homopolymer. At this temperature, the polymer-chain segments in

the amorphous phase are mobile enough allowing trans-esterification to occur between

the homopolymer and the co-monomer [7, 8, 9]. However, the polymer-chain segments

in the crystalline phase are insufficiently mobile to participate to the reaction and will

remain unchanged.

This chapter looks at the morphology of modified PBT samples with a Dianol co-

monomer 2,2-bis[4-(2-hydroxyethoxy)phenyl]propane (commonly named Dianol 220)

obtained via SSP by means of solid-state NMR. A series of these modified polyesters has

been investigated with varying Dianol / PBT ratios. These samples were prepared by M.

Jansen [10] who previously showed that the morphology of the resulting products

follows a (two)-phase model, composed of a crystalline phase and an amorphous phase,

the latter subdivided into a rigid amorphous fraction and a mobile amorphous fraction.

Jansen et al. [10] developed a special calculation method showing that only the mobile

amorphous fraction is accessible for incorporation of the Dianol via SSP and that Dianol

is randomly incorporated into this mobile amorphous fraction. Their conclusions were

based on Differential Scanning Calorimetry (DSC) measurements and computation of

the degree of randomness using liquid-NMR spectroscopy.

The properties of materials are strongly related to their morphology. A variety of

techniques, such as thermal analysis, mechanical relaxation, microscopy, light scattering

and computer simulation are available to probe the heterogeneity of multiphase systems.

Each method has its own advantages and disadvantages.

Chapter 3

35

The use of solid-state NMR for investigating morphology and semi-quantitatively

estimate the microstructure of the different phases present in the system have been

demonstrated by many authors [11-22].

The results discussed below illustrate the use of solid-state NMR relaxometry for

investigating nano-structures (morphologies) and dynamics of the PBT modified

samples. The incorporation of the Dianol in the mobile amorphous fraction was also

investigated. Static solid-state NMR measurements of 1H spin relaxation (spin-lattice

relaxation time in the rotating frame, T1) were used to investigate the morphology of

the modified PBT samples, together with T1(1H) -filtered 1H-13C cross-polarization

experiments (CP-MAS). The latter makes it possible to observe the chemical shifts and

the relaxation parameters for each resolvable carbon nucleus and their protons. The

observed chemical shift provides information concerning the conformation and packing

of the polymer while the relaxation parameters indicate its dynamics. Also, upper and

lower limits of domain sizes were established. The results have been compared to small-

angle X-Ray scattering (SAXS).

T1 relaxation measurements probe molecular motions characterized by frequencies in

the mid-kilohertz region. These measurements are used to study phase separation in

semi-crystalline polymers as they allow more spatial averaging then T2 relaxation times.

The molecular motions in a crystalline phase are much more restricted then in an

amorphous matrix. At a temperature between the glass transition temperature (Tg) and

the melting point (Tm) there is a big contrast between the mobility in the crystalline

phase and in the amorphous phase. These two types of molecular structures will

therefore demonstrate different molecular dynamics, hence different T1 relaxation

times [23-28]. Such measurements probe quantitative information, as in general the area

underneath the solid-state NMR signal is proportional to the number of protons in the

system. In that way, it is possible to quantify the number of proton in each phase.

3.2. Experimental Section

3.2.1. Sample preparation

The modified PBT samples were obtained and synthesized by M.A.G. Jansen [10].

Chapter 3

36

The synthesis of PBT-Dianol copolymers by SSP is described in Jansen’s thesis.

Different ratios of PBT and Dianol were used and are reported in Table 3.1. BDx

corresponds to the fraction in mol% of PBT (expressed in 1,4-butanediol units) whereas

Diy corresponds to the fraction in mol% of Dianol in the initial mixture used for the SSP

(BDxDiy)feed . After SSP, the mol fractions slightly differ. This difference may be

attributed to the evaporation of 1,4-butanediol or ethylene glycol during the SSP

reaction. The general chemical structure of the (BDxDiy)ssp copolymer samples is shown

in Figure 3.1:

Diol

T = 180-200 °C

N2

Crystalline part

Amorphous part

DiolDiol

T = 180-200 °C

N2

T = 180-200 °C

N2

Crystalline part

Amorphous part

HOCH2CH2O C

CH3

CH3

OCH2CH2OHCC

O O

OCH2(CH2)2CH2O

n

C

CH3

OCH2(CH2)2CH2O-C OCH2CH2O

o o

-C-

CH3

-OCH2CH2O-C C

o o

m n

C

CH3


o o

-C-

CH3

-OCH2CH2O-C C

o o

m n

C

CH3


o o

-C-

CH3

-OCH2CH2O-CC

CH3


o o

-C-

CH3

-OCH2CH2O-COCH2(CH2)2CH2O-C OCH2CH2O

o o

-C-

CH3

-OCH2CH2O-C C

o o

m n

C

o o

m n

(a) (b)

(c)

(d) (e)

Diol

T = 180-200 °C

N2

Crystalline part

Amorphous part

DiolDiol

T = 180-200 °C

N2

T = 180-200 °C

N2

Crystalline part

Amorphous part

HOCH2CH2O C

CH3

CH3

OCH2CH2OHCC

O O

OCH2(CH2)2CH2O

n

C

CH3


o o

-C-

CH3

-OCH2CH2O-C C

o o

m n

C

CH3


o o

-C-

CH3

-OCH2CH2O-C C

o o

m n

C

CH3


o o

-C-

CH3

-OCH2CH2O-CC

CH3


o o

-C-

CH3


o o

-C-

CH3

-OCH2CH2O-C C

o o

m n

C

o o

m n

Diol

T = 180-200 °C

N2

Crystalline part

Amorphous part

DiolDiol

T = 180-200 °C

N2

T = 180-200 °C

N2

Crystalline part

Amorphous part

HOCH2CH2O C

CH3

CH3

OCH2CH2OHCC

O O

OCH2(CH2)2CH2O

n

C

CH3


o o

-C-

CH3

-OCH2CH2O-C C

o o

m n

C

CH3


o o

-C-

CH3

-OCH2CH2O-C C

o o

m n

C

CH3


o o

-C-

CH3

-OCH2CH2O-CC

CH3


o o

-C-

CH3


o o

-C-

CH3

-OCH2CH2O-C C

o o

m n

C

o o

m n

(a) (b)

(c)

(d) (e)

Figure 3.1. (a) Dianol monomer (b) poly-(butylene terephthalate) (PBT) (c) poly-(butylene terephthalate) (PBT)-copolymers chemical structures; incorporation of diols in PBT via solid-state polymerization, before SSP(d) a physical mixture of PBT and dianol; after SSP(e), PBT copolymer with crystallizable PBT blocks [7].

Chapter 3

37

The original PBT pellets (Mn=15 kg/mol Mw = 34 kg/mol) were provided by GE

Plastics (The Netherlands) and the 2,2-bis [4-(2-hydroxyethoxy)phenyl]propane (Dianol

220) was provided by Air Liquide (France). PBT and Dianol were mixed by dissolution

in 1,1,1,3,3,3-hexafluoro-2-propanol (HFIP) prior to SSP. For all SSP reactions, a

reaction time of 9 hours was used at a reaction temperature of 180 C. Average

molecular weights Mn were obtained using size exclusion chromatography (SEC) [1].

3.2.2. NMR experiments

1H-decoupled 13C NMR spectra were recorded on a Bruker DMX500 spectrometer

using a CP/MAS probe with the observe channel tuned on 13C and the decoupler

channel to 1H. All 13C NMR spectra were recorded by use of standard 1H-13C cross-

polarization with an mplitude modulated contact pulse of 1ms and 5 s delay time.

T1(1H)- filtered CP-MAS spectra were measured with a /2 proton lock pulse

preceding the 1H-13C contact pulse; magic angle spinning was performed at 12 kHz.

Static T1 decays were measured with a /2-spin-lock-hahn-echo sequence at 353 K and

435 K. Some further experiments on samples BD96Di04 and BD54Di46 have been carried

out at U-Hasselt, at ambient temperature, 353 K and at 435 K on a Varian Inova 400

spectrometer using a dedicated wide-line probe equipped with a 5 mm coil. For T1 all

experimental results could be fitted using a mono-exponential decay. For T1

experiments, two and three exponential decays were used. However, the fit with three

components was found to be significantly better.

3.2.3. NMR Pulse sequence

3.2.3.1. Inversion recovery

To determine the proton spin lattice relaxation time T1, the inversion recovery pulse

sequence has been used. The equilibrium magnetization is first disturbed by a 180z rf-

pulse. As a consequence, the magnetization is inverted along the longitudinal direction

(Figure 2.6.b in Chapter 2). Due to the T1 relaxation, the magnetization becomes

smaller, passes through zero and starts to grow back to its original value. However,

Chapter 3

38

igure 3.2. T1(1H)- filtered CP-MAS-pulse sequence

longitudinal magnetization cannot be detected directly, therefore a 90 pulse is applied

for detection.

3.2.3.2. /2-spin-lock- pulse sequence

To determine the static proton spin lattice relaxation time in the rotating frame T1, a

classical 90x’ pulse has been used followed by a 90 phase shift of the transmitter (see

Chapter 2, Figure 2.8). In such a way, the magnetization is spin-locked along the y’-

axis. The B1 field strength used in our experiments is 50 kHz.

3.2.3.3. T1(1H)- filtered CP-MAS-pulse sequence

The pulse sequence starts also with a protons /2-spin-lock pulse sequence. The

magnetization of the proton is spin-locked during a time t. After the time t, the carbons

are brought into contact with protons; this is the cross-polarization. During that time, the

proton magnetization is transferred to the carbon spin system. Afterwards, the carbon

signal is detected under high-power proton coupling transition (Figure 3.2).

F

Chapter 3

39

3.2.4. NMR Fitting procedure (2D analysis)

The NMR signal is acquired as a function of time t (spin locking time):

(3.1)

where Ginhom(

)exp()2|()(),2( 000inhom0 titRGdtS 0 ) reflects the inhomogeneous spectrum arising from residual

anisotropy, chemical shift differences and magnetic field inhomogeneity; R( 0 |2 + t)

is the spin relaxation decay associated with spectral frequency ω0. In the usual approach,

the maximum of the signal at t = 0 is measured as a function of 2 without spectral

resolution. The resulting monotonous decay, a 1D data set, can be analyzed in terms of a

number of exponential components. It is proposed to use the spectral resolution in the

full 2D data set I (2, ) after Fourier transformation in the t dimension (see detail of

2D analysis in Chapter 5). For a mixture of components with mono-exponential

relaxation and well-separated spectral frequencies and relaxation rates, it should be

relatively easy to extract the intrinsic properties from a series of the spectra. In real

cases, however, components may overlap in both the spectral and Laplace domain. To

be able to distinguish between the different components of the sample, a data analysis

approach based on multivariate statistics has been used. This approach uses a

“Multivariate Analysis” to estimate the line shapes of the overlapping proton resonance.

There are several methods to fit the exponential decay model, the one used here, is

based on the so-called LINOL method describes in Appendix 2 (Linear-Non-Linear

fitting).

3.2.5. Software

Home-built routines, written in MATLAB 6.5 (The Mathworks, Natick, MA, USA),

were used for NMR data treatment. The solid-state NMR relaxation data were fitted

using a combination of exponential decay functions with the use of a novel procedure

based on multi-variable analysis.

3.2.6. Modulated DSC

It was shown that modulated DSC (MDSC) allows for deconvolution of signals from

amorphous and crystalline phases. The method has a greater resolution then non-

Chapter 3

40

modulated DSC. Amorphous phases give rise to a reversing heat flow, which is readily

converted into a specific heat capacity (Cp) and its derivative, which highlights the glass

transition temperatures. The Tg’s and the corresponding heat capacities of the modified

PBT samples were measured via (MDSC) using TA Q1000 DSC, equipped with an

auto-sampler and a refrigerated cooling system (RCS). The temperature calibration was

obtained using Indium. The (BDxDiy)ssp copolymer samples were prepared in aluminum

pans (between 10 and 15 mg). An oscillating heat flow signal with a period of 60

seconds and amplitude of 1 C and an underlying heating rate of 1 C/ min was used.

All samples were measured following a specific ageing procedure. The samples were

first heated in normal DSC mode from 0 C to 240 C then cooled and heated up again

in modulated DSC to 140 C. Each sample was kept at this temperature during

respectively 1 hour, 10 hours and 20 hours. After each ageing time, the samples were

cooled to 0 C and reheated using a cooling and heating rates of 10 C/min. The Tg

values were extracted from the signal of the reversing heat flow.

3.3. Results and discussion

3.3.1. 13C-CP-MAS experiments

Figure 3.3 shows 13C-CP-MAS spectra of the PBT homopolymer, Dianol monomer and

PBT copolymer. By comparison it is easy to assign each peak from the copolymer

sample. Generally, the narrow shapes correspond to the PBT while the broader shapes to

the Dianol.

In Figures 3.4 and 3.5, 13C–CP-MAS spectrum recorded at 353 K temperature at

different spin-locking times are shown. The 13C detection allows signals of PBT and

Dianol to be resolved on the basis of chemical shift whereas the decay of the signal

intensity as function of the spin-locking time reflects T1H of each component. At longer

spin-locking time (filter time) the signals corresponding to the most mobile phase of the

polymeric system are removed while the ones corresponding to the more rigid phase of

the system remain. T1(1H)- filtered CP -MAS experiments were recorded after the

application of a /2 pulse on the 1H channel, followed by a variable spin-locking time

then by 13C detection through 13C 1H CP at fixed contact time at 353 K.

Chapter 3

41

200 150 100 50 0 ppm

(a)

(b)

(c)

200 150 100 50 0 ppm

(a)

(b)

(c)

Figure 3.3. 1H- 13C CPMAS spectra of the PBT homopolymer (a), Dianol monomer (b), PBT copolymer (BD54Di46)ssp (c), measured at room temperature .

From the T1(1H)-filtered 13C-CP-MAS spectra of BD76Di24 in Figure 3.4 one can see

that the first spectrum recorded at a filtered time of 10 s contains the signals of both

PBT and Dianol. The main rigid PBT-component stays equally divided over the

complete spectrum while the resonance peaks of the Dianol are decreasing with

increasing filter time. At longer filtered times one can also notice a line-narrowing of

the overlapping peak at 125 ppm.

In Figure 3.5, the spectrum of BD54Di46 at filter time 10 s clearly shows separate

Dianol peaks at about 30 and 115 ppm. Increasing the filter time to 6 ms shows a

disappearance of the Dianol peaks. Also the increase in signal to noise is a consequence

of the increased molecular mobility of the BD54Di46 copolymer.

Chapter 3

42

200 150 100 50 0 ppm

(a)

(b)

(c)

200 150 100 50 0 ppm200 150 100 50 0 ppm

(a)

(b)

(c)

Figure 3.4. T1 -filtered 1H- 13C CPMAS spectra of the copolymer (BD76Di24)ssp at filter times 10 s (a), 3ms (b), 6 ms (c), measured at 353 K.

T1(1H)-filtered 13C-CP-MAS experiments show two types of chains mobility in the

material. The Dianol signals show a relaxation time T1 of about 2 ms. Cross-

polarization tends to overemphasize the rigid polymer phase in 13C - NMR spectra, so

only 1H with sufficiently long T1 values could be detected given the presence of a finite

contact time within the pulse. Since the filtered spectra are based on 1H -13C cross-

polarization, a quantitative analysis is not possible. For more quantitative analysis, static

proton spin-lattice relaxation in the rotating frame T1 experiments were carried out.

The measurements of 1H spin-lattice relaxation in the rotating frame are very sensitive

to motions in the mid-kilohertz region and typically involve main chain segmental

motions of polymers above the glass-transition temperature. Also, such measurements in

principle probe quantitative information, as the area underneath the solid-state NMR

Chapter 3

43

-50250 200 150 100 50 0 ppm

-50250 200 150 100 50 0 ppm

a

b

-50250 200 150 100 50 0 ppm

-50250 200 150 100 50 0 ppm

-50250 200 150 100 50 0 ppm

-50250 200 150 100 50 0 ppm

a

b

(a)

(b)

-50250 200 150 100 50 0 ppm

-50250 200 150 100 50 0 ppm

a

b

-50250 200 150 100 50 0 ppm

-50250 200 150 100 50 0 ppm

-50250 200 150 100 50 0 ppm

-50250 200 150 100 50 0 ppm

a

b

(a)

(b)

Figure 3.5. T1 -filtered 1H- 13C CPMAS spectra of the copolymer (BD54Di46)ssp at filter times 10 s (a) and 6 ms (b), measured at 353 K.

line is proportional to the amount of protons in the system. In that way, it is possible to

quantify the number of protons in each phase of the system.

3.3.2. 1H T1 and 1H T1 relaxation experiments

NMR spin-lattice relaxation times are not only determined by molecular dynamics but

also by spin diffusion where energy is transferred throughout the spin system by

successive spin flips. The NMR spin-lattice relaxation time measurements T1 and T1

are generally used to identify heterogeneity of materials at different levels, T1 at ~ 5 to

>100 nm size and T1 at 1 to 5 nm size. They provide information about the miscibility

and the domain size of various phases inside the polymer system [29-32].

cs but

also by spin diffusion where energy is transferred throughout the spin system by

successive spin flips. The NMR spin-lattice relaxation time measurements T1 and T1

are generally used to identify heterogeneity of materials at different levels, T1 at ~ 5 to

>100 nm size and T1 at 1 to 5 nm size. They provide information about the miscibility

and the domain size of various phases inside the polymer system [29-32].

T1 experiments were first used in this work to semi-quantitatively determine the micro-

domain structures of the materials, based on mobility differences between the crystalline

phase, the rigid amorphous phase and the mobile amorphous phase. T1 values were

T1 experiments were first used in this work to semi-quantitatively determine the micro-

domain structures of the materials, based on mobility differences between the crystalline

phase, the rigid amorphous phase and the mobile amorphous phase. T1 values were

Chapter 3

44

obtained from the analysis of the decay of the magnetization measured at increasing

spin-lock times after the initial spin-locking excitation pulse. According to the

Bloembergen, Purcell and Pound theory (BPP), rigid phases in polymers should have

protons depicting a longer T1 relaxation time than mobile ones [33]. This has been

verified at room temperature for sample BD96Di04 but was not observed in the sample

containing the highest mole fraction of Dianol (BD54Di46)ssp. This is due to a very high

mobility in the sample.

To be able to distinguish between the two amorphous fractions, we recorded data at 353

K and 435 K, (30 and 110 oC) above the Tg of the samples. For the T1H experiments, the

values of the relaxation times decrease at increasing temperatures (at RT and at 162 °C

from 1.15 s to 0.86 ms). For all the samples, only one component fit was needed to fit

the data. The T1H values are located at the right side of the relax–curve of logT1 as a

function of log c (see also Figure 2.9). The relaxation decay is dominated by spin

diffusion longer then the domain sizes.

Log c

Log T 1

Log c

Log T 1

Figure 3.6. Spin-lattice relaxation time in function of correlation time, and distribution of different (BDxDiy)ssp fractions (rigid-solid phase (), intermediate (), mobile (). Plain markers correspond to high amount and hollow ones to low amount).

Chapter 3

45

For all T1 experiments, the data were fitted best with a three components fit at both

temperatures and for all the samples containing Dianol. The T1 values are found to be

much longer for protons in the mobile phases than for the ones in the rigid phase. This

has been verified by recording the data at low and high temperatures. The T1 values for

the samples increase when temperature increases, suggesting at first that the values are

located on the left side of the curve in Figure 3.6. For the proper interpretations of the

T1 values, however, one should mention that during the relaxation in ms, spin diffusion

takes place over a limited distance (few nm). Effectively, spin diffusion will take place

with protons belonging to the same environment (same unit or monomer) and the same

phase, but will only partly reach the protons in the other phases. Even in the rigid

phases, spin diffusion as well as local molecular mobility influence the relaxation decay.

In the most mobile part, mainly local molecular mobility influences the decay. In each

phase, there are mobile protons (xf) and less mobile protons (xs). The T1 values are

dominated by the fastest relaxing protons according to the equation (3.2):

ρT1

1 =

ρs

s

ρf

f

T

x

T

x

11

(3.2)

The proposed T1 values are all at the left side of the relax minimum. In Figure 3.6 we

propose an interpretation for the T1 positions that will also depend on the amount of

fast and slow protons in each fraction.

At 435 K, the best fit of the decay is found to be multi-exponential. One and two phases

decay models perform significantly worse. A three-component model gives a good fit as

shown in Figure 3.7.

Three different values of T1 could be extracted. These three values correspond to three

types of 1H mobility present in the system. The obtained T1 values differ by one to two

orders of magnitude as shown in Table 3.2, the times 0.4, 0.05 and 34 ms would

correspond to an intermediate rigid phase (rigid amorphous phase), the values 10, 1.2, 3

and 0.9 ms to the most immobile solid phase and the longest T1 decays from 12 to 161

ms corresponding to a most mobile amorphous phase. The overall mobility is in good

agreement with Tg values from MDSC results presented in Table 3.1.

Chapter 3

46

Figure. 3.7. T1 relaxation decay curves recorded at 435K (experimental data fitted with three exponential decay model). () correspond to sample (BD100Di00)ssp; () correspond to sample (BD54Di46)ssp ; () correspond to sample (BD96Di04)ssp

Table 3.1. Glass transition temperature (Tg) for the synthesized (BDxDiy)ssp copolymers obtained from MDSC measurements.

(BDxDiy)feed mixtures

(BDxDiy)ssp copolymers

Heating run 1

x y FBD-T, total FDi-T, total Tg

[mol%] [mol%] [mol%] [mol%] [°C]

100 0 100 0 53.5

95 5 96 4 62

80 20 76 24 67.8

65 35 54 46 51

Chapter 3

47

Table 3.2. T1 relaxation times obtained with 3 exponential decay fit at 435 K using 2D data analysis.

FBD-T, total

[mol%] FDi-T, total

[mol%] Crystalline T1i (ms)

Rigid amorphous T1s (ms)

Mobile amorphousT1l (ms)

100 0 10 0.4 45

96 4 1.2 0.05 12

76 24 3 0.05 52

54 46 0.9 34 161

FBD-T, total

[mol%] FDi-T, total

[mol%] Fraction

(%) Fraction

(%) Fraction

(%)

100 0 23 51 26

96 4 7 26 67

76 24 3 35 62

54 46 19 39 42

For all modified PBT samples, three ranges of mobility are observed with similar

relaxation times, except for the sample with the highest concentration of Dianol

(BD54Di46)ssp. This sample shows much higher relaxation time values for the

intermediate and mobile phase while the solid phase has a normal low T1 value of 0.9

ms, close to the most solid phase T1 values of the other copolymers. These T1 values

can be explained by the higher Dianol incorporation in the original PBT non-crystalline

fraction. Also, one can see from Figure 3.8 of the well-resolved 1H NMR spectrum of

the most mobile phase that the proton peaks show high T1 values. There is still

sufficient peak intensity at the long proton locking time of 240 ms.

Table 3.3 shows two components analysis of the proton spectra of the sample

(BD54Di46)ssp at 435 K, resulting in well-separated short and long proton T1 values for

each of the four proton signals. The aromatic peak around 7 ppm and the aliphatic peak

around 4 ppm are representative for the PBT as well as for the Dianol regions while the

signal at 1.6 ppm mostly shows the Dianol domains, represented by the Me groups (part

of it is also due to the CH2 group). A two components fit of the signal at 1.6 ppm results

in T1 values of 24 and 145 ms, corresponding respectively to an intermediate solid

amorphous and a mobile amorphous phase. These values closely resemble the values of

the three components fit in Table 3.2 of 34 and 161 ms respectively for the intermediate

and the mobile phase. The two component analysis of the other three proton signals,

Chapter 3

48

representative for PBT as well as for the Dianol, show a low T1 value around 1 ms for a

solid phase and a high T1 value between 58 and 107 ms for a mobile phase. From the

signal at 1.6 ppm, a value much higher than 1ms is found for T1. This confirms that

during SSP the Dianol monomer is only incorporated in the amorphous phase of the

PBT.

Figure 3.8. NMR signal of the (BD54Di46)ssp copolymer recorded at 435 K using T1 s

relaxation experiments obtained at t = 240 ms (thin line) t = 10 s (broad line).

Table 3.3. T1 relaxation times obtained via peak peaking analysis with 2 exponential decay fit for the sample (BD54Di46)ssp containing the highest amount of Dianol at 435 K.

ppm T1 s (ms) T1l (ms)

7.8 1 87

6.9 0.65 107

4.3 1 58

1.6 24 145

Chapter 3

49

All the NMR relaxometry data have been analyzed with respectively 1D data fits and

2D data fits. For all samples, the 2D analyses seem to show consistency with reality and

are close to the value found by M. Jansen [10]. For the sample (BD54Di46)ssp, the 1D

analysis was preferred. Effectively, the mobility in the sample is much higher and the

resolution drastically increases (already observed in Figure 3.8). The fact that the 2D

analysis takes the chemical shift into account may not always reflect the phase

relaxation. It may also assimilate relaxations of protons in a phase such as side groups

(Me-) to the overall protons relaxation in that same phase relaxation. The 1D data seem

to be closer to the physical reality and also close to what has been obtained using a

dedicated wideline probe with the 400 MHz NMR spectrometer.

Also, one additional argument, showing the strong influence of the Me-groups

contained in the Dianol, is that between 1D and 2D analysis the data are very close for

the most rigid phase while for the intermediate and mobile one, the relaxation times are

different (Table 3.4).

Sample (BD54Di46)ssp contained much more Dianol, so one expects a higher chain

mobility as clearly observed from the much lower Tg value, resulting from MDSC

measurements (Table 3.1), as compared to the other modified samples. This has already

been reported by M. Jansen et al. and is due to the lower number-average molecular

weight (Mn = 8 kg/mol) of the polymer chains [7].

Table 3.4. Rigid fraction, intermediate fraction (rigid amorphous) and mobile fraction obtained from static T1 experiments with 3 exponential decay fit at 500MHz, 400 MHz(1D analysis and 2D analysis).

FBD-T, total

[mol%]

FDi-T, total

[mol%]

Crystalline

T1s (ms)

Rigid

amorphous

T1i (ms)

Mobile amorphous

T1l (ms)

2D (500MHz) 54 46 0.9 34 161

1D (500MHz) 54 46 1.1 23.1 77

1D (400MHz) 54 46 2 9 28

FBD-T, total

[mol%]

FDi-T, total

[mol%] Crystalline

(%)

Rigid amorphous

(%)

Mobile amorphous

(%) 2D (500MHz) 54 46 19 39 42

1D (500MHz) 54 46 9.7 31 50.4

1D (400MHz) 54 46 6.5 38.5 55

Chapter 3

50

Some experiments have also been carried out at 353 K and the obtained T1 values were

compared to the ones found with T1(1H)- filtered CP-MAS experiments. Fitting the

system with a two-exponential model gives a T1 value for the mobile component of

about 3 ms, very close to the one found for the mobile component in the CP-

experiments (2 ms). For all samples the values were in good agreement with those found

using CP. However, the best fit also here was obtained with a three-exponential model

and the T1 value for the most-mobile component was found to be about 10 ms. A

mobility higher than the one found in CP experiments for Dianol supports the

suggestion that the Dianol is not only present in the most-mobile fraction of the system

but also in the intermediate phase (CP is effective for less mobile protons).

Some further experiments have been obtained on a NMR spectrometer at 400 MHz, for

the sample with the least fraction of Dianol and the most fraction of Dianol.

In both cases the best fit was also obtained using a three- exponential decay fit. For

sample (BD96Di04)ssp the results obtained are in good agreement with what has been

obtained using the 500 MHz system. For sample (BD54Di46)ssp discrepancies are

observed, probably due to the influence of the chemical shift on the analysis method to

fit the data (Table 3.4). The number of experimental points taken for experiments will

probably also have an influence of the results.

3.3.3. Quantification of each phase

As already mentioned above, static proton T1 experiments can yield more quantitative

information, as this technique does not involve cross-polarization or magic-angle

spinning. The amount of 1H in each phase was quantified (assuming that for most

samples, the system is still in the diffusion limit with low spin diffusion for the more

mobile amorphous samples). These fractions are reported in Table 3.2 and Table 3.4,

and are obtained with a multiple-exponential decay fit using (for most of them)

multivariate analysis (Appendix 1 and 2):

lis TtCTtBTtAtM expexpexp)( (3.3)

M (t) represents the magnetization function, A represents the fraction of components

with a short relaxation time, B corresponds to the fraction of components with an

Chapter 3

51

intermediate relaxation time and C corresponds to the fraction of components with long

relaxation time; t is the spin lock time.

So far, no much correlation regarding phase quantification has been found between

these data and these obtained by Jansen et al. via DSC and solution NMR reported in

Table 3.5.

In both cases, models are used and different assumptions are made. Also different

techniques probe different information, so that quantification remains a challenge. The

crystallinity obtained by Jansen et. al. was determined via DSC and is thus based on the

amount of heat needed to break the crystal order, whereas the crystallinity determined

with NMR is based on differences in chain mobility within the rigid crystalline phase

and the amorphous one (besides NMR relaxation probes proton fractions while DSC

reports the data to mole fractions). Also T1 relaxation decay is influenced by both spin

diffusion and local molecular dynamic. This will clearly affect the physical

interpretation of each phase in the system and thereby the relative fraction of proton in

each phase. However, Table 3.2 shows that the crystalline fraction is decreasing with

increasing amount of Dianol. The Dianol acts as a reactant for the trans-esterification

reaction in the amorphous phase but also as a solvent (swelling agent) for that same

phase. Furthermore, during the time scale of NMR experiments, some rearrangement in

the amorphous phase may occur, that could explain some of the observed differences.

Table 3.5. (%) of the three fractions present in the system derived from MDSC and solution NMR (Martijn Jansen et al.).

FBD-T, total

[mol%] FBD-T, total

[mol%] Crystalline

(%) Rigid amorphous

(%) Mobile amorphous

(%)

100 0 54 28 18

96 4 50 22 28

76 24 32 8 60

54 46 8 1 91

The samples were then measured at room temperature then heated to 435 K and cooled

down to room temperature and measured again. No significant differences were

observed in the fraction of components and in the relaxation times. In addition,

Chapter 3

52

modulated MDSC experiments have been carried out and all samples were measured

following a specific ageing procedure. The variations in the heat capacity are shown in

Table 3.6. The results depict no variation in heat capacity indicating that no

rearrangement in the amorphous fraction takes place.

Some complementary analyses have been done to help estimate the amount of rigid and

mobile 1H-atoms contained in each different phases of the system. Deconvolution and

integration of the 1H-NMR signal corresponding to the broad part of the raw signal at a

short spin locking time obtained from T1 experiments were done. The results are

reported in Table 3.7 and have been compared to the total rigid fraction found by Jansen

et al (the total rigid fraction being the sum of the crystalline fraction and the rigid

amorphous fraction). The integration of the deconvoluted signal gives values for the

rigid component, very close to those found by Jansen et al. [7,10].

Table 3.6. TMDSC results obtained before and after ageing procedure.

(BDxDiy)ssp copolymers Heatingrun1 No

ageing

Heating run 2 after 1 hour

ageing

Heating run 2

after 10 hours

ageing

Heating run 2

after 20 hours ageing

Cp Cp Cp Cp FBD-T, total

[mol%]

FDi-T, total

[mol%] J/g C

100 0 0.39 0.4 0.39 0.38

96 4 0.42 0.41 0.37 0.40

76 24 0.47 0.48 0.50 0.47

54 46 0.58 0.53 0.59 0.6

If we look at Table 3.8 showing the static NMR line width analysis by peak

deconvolution and integration of the sample (BD54Di46)ssp, we can observe that the

relative area underneath the peak at 1.6 ppm is the same at a spin locking time of 10 s

and at a spin locking time of 240 ms. This clearly suggests that Dianol is not only

contained in the most mobile part. The relative area underneath the peak at 1.6 ppm,

mostly corresponding to the Me-protons of Dianol, would have a much higher value at a

Chapter 3

53

spin locking time of 240 ms, typical for the most mobile amorphous. The two phases

behavior of Dianol has also been noticed before, showing T1 values of 24 ms

(intermediate phase) and 145 ms (mobile phase) in Table 3.3.

Table 3.7. Rrigid fraction obtained by integration of the deconvoluted peak corresponding to the broad part of the raw signal.

FBD-T, total

[mol%]

FDi-T, total

[mol%] Rigid fraction

(%)

100 0 84

96 4 84

76 24 44

54 46 5

Table 3.8. Peak deconvolution and integration from the static line width analysis of sample (BD54Di46)ssp obtained at 435 K and at two different spin locking time.

ppm Area (%)

at 10 s

Area (%)

at 240 ms

7.8 17.5 18

6.9 18.3 26

4.3 31 27

1.6 28 29

3.3.4. Domain sizes

As already mentioned above, proton spin-lattice relaxations in rotating and laboratory

frame measurements (1H T1 and 1H T1) provide information about the level of

heterogeneity (phase morphology) on the nm scale due to the process of proton spin

diffusion [34]. This latter process is not a physical movement of protons but a transfer of

spin energy by successive spin-flips. This process is often modeled as Fickian diffusion.

It allows to judge the degree of size separation and to estimate the size of molecular

domains. In polymeric systems proton spin-lattice relaxations in rotating and laboratory

Chapter 3

54

frame measurements (1H T1 and 1H T1) are often used to estimate lower and upper limit

of domain sizes on the basis of the following equation, valid for lamella structures [35-

37]:

16 TDL s (3.4)

Ds is the spin-diffusion coefficient which depends on the average proton- to- proton

distance as well as on the dipolar interaction.

If the average domain size in a polymer blend is smaller than ca. 1 nm, proton-proton

spin diffusion averages out any T1 or T1 relaxation differences. In that case, all protons

decay with the same effective T1 and T1. In contrast, if the domain size is larger than ca.

50 nm, spin diffusion is too slow to average out such differences and the different

phases will decay each with their intrinsic (probably different) T1 and T1 values. In the

intermediate range, 1 nm < domain size < 50 nm, we expect to find different effective

T1 values and a single effective T1. The reason is that T1 tends to be 10-100 times

longer than T1, so that spin diffusion is thought to be unable to homogenize T1, but is

still able to average out T1 differences. If we look at the SAXS results obtained by

Jansen et al. (5 nm < domain size < 50 nm) shown in Figure 3.9 and the data obtained

through NMR relaxation experiments, we seem to be in good agreement with the above

concept.

Also, T1 values were obtained from the analysis of the recovery of the magnetization

measured at increasing times after the initial inversion pulse. For all samples and at each

studied temperature, the recovery was found to be mono-exponential indicating that spin

diffusion averages the T1 characteristic of protons belonging to different domains.

Assuming a spin-diffusion coefficient Ds of 1 nm2 /ms [26] and considering that the

minimum T1 observed for all samples being about 1 s an upper limit of about 80 nm

based of equation (3.4) is proposed. A lower boundary could also be estimated using the

values of T1 in the rigid phase. The data are reported in Table 3.9. An upper and lower

limit for the mixing scale (domain sizes) may give some idea about the size of the

crystalline and the (rigid)-amorphous domains. The limits in the rigid fractions are in

reasonable agreement with those obtained using SAXS by Jansen et al. [7].

Chapter 3

55

Figure 3.9. Morpbhological parameters obtained from SAXS: long period Lp, average crystal thickness lx, average amorphous layer thickness l ( M. Jansen et al. ).

Table 3.9. Lower boundary of domain size obtained from T1 of the three fractions obtained at 435 K (Li corresponding to the lower boundary limit of the intermediate fraction, Lr corresponding to the lower boundary limit of the rigid fraction, Lm corresponding to the lower boundary limit of the mobile fraction).

FDi-T, total

[mol%]

Li

(nm)

Lr

(nm)

Lm

(nm)

0 1.54 7.7 16.4

4 0.54 2.68 8.48

24 0.54 4.24 17.66

46 7.34 3.4 12.96

3.4. Conclusions

The Dianol was first identified in CP-MAS experiments. CP-MAS experiments are

sensitive only in a rigid limit (because of cross-polarization efficiency) so it follows that

Dianol is not only incorporated in the most-mobile fraction of the sample but also in the

amorphous interphase.

Chapter 3

56

For all samples, 1H T1 experiments do not exhibit a single or dual-exponential decay

behavior, but a much better fit was found using a three exponential decay fit. Such a

result is in good agreement with the three fractions (two phases) demonstrated by Jansen

et al. In a quantitative matter, 1H T1 NMR experiments are not in good agreement with

those found by Jansen et al. The interpretation of 1H T1 experiments is not always

straightforward, especially in this type of polymers where fast molecular dynamics of

rotational motions of (- CH3) groups and 180C flip rotation of benzene groups may

well contribute to spin relaxation, especially at high temperature where spin diffusion is

low. However static 1H line width analysis (deconvolution and integration) was used for

quantification and the results obtained for the rigid fractions are found to be in good

agreement with these found by Jansen et al. [7,10].

A higher mobility than the one found in CP experiments was depicted in 1H T1 NMR

experiments at 353 K. This seems to further confirm that Dianol is not only located in

the most mobile fraction but also in the amorphous interphase.

For quantification of phase morphology into the modified PBT samples, T1 analysis

seems to hold few ambiguities, especially as spin diffusion as well as local mobility

influence the relaxation decay. For such reason T2 relaxation solid-echo and Hahn-echo

analyses for a more accurate quantification of the different fractions were performed and

are presented in the following Chapter 4.

Chapter 3

57

3.6. References


[2] Kim, J. H.; Lyoo, W. S.; Ha, W. S. J. Appl. Polym. Sci. 2001, 82,159.

[3] Marchese, P.; Celli, A.; Fiorini, M. J. Polym. Sci., Polym. Phys. 2004, 42, 2821.

[4] Fernandez-Berrediti, M. J.; Iruin, J .J.; Maiza, I. Polymer 1995, 36, 1357.

[5] Berti, C.; Colonna, M.; Fiorini, M.; Lorenzetti, C.; Marchese, P. Macromol. Mat. Eng. 2004, 289, 49.

[6] Kim, J. P.; Lyoo, W. S.; Ghim, H.D. Polymer 2003, 44, 895.

[7] Jansen, M. A. G.; Goossens, J. G. P.; de Wit, G.; Bailly, C.; Koning C. E. Macromolecules 2005, 38, 2659.

[8] Hait, S. B.; Sivaram, S. Macromol. Chem. Phys. 1998, 199, 2689.

[9] Kimura, M.; Porter, R. S. J. Polym. Sci., Polym. Phys. 1983, 21, 367.

[10] Jansen, M. A. G.; Goossens, J. G. P.; de Wit, G.; Bailly, C.; Koning C. E. Anal. Chim. Acta 2006, 557, 19.


[12] Havens, J. R.; Vanderhart, D .L. Macromolecules 1985, 18, 1663.


[14] Schmidt-Rohr, K.; Spiess, H. W. Multidimensional Solid-State NMR and polymers; Academic Press 1994.



[17] Hedesiu, C.; Demco, D. E.; Kleppinger, R.; Buda, A. A.; Blumich, B.; Remerie, K.; Litvinov, V. M. Polymer 2007, 48, 763.

[18] Yao, Y. F.; Graf, R.; Spiess, H. W. Macromolecules 2008, 41, 2514.

[19] Wind, M.; Brombacher, L.; Heuer, A.; Graf, R.; Spiess, H. W. Solid State NMR 2005, 27,132.

[20] Demco, D. E.; Litvinov V. M.; Rata, G.; Popescu, C.; Phan, K. H.; Schmidt, A.; Blumich, B. Macromol. Chem. Phys. 2007, 208, 2085.

[21] Hedesiu, C.; Demco, D. E.; Kleppinger, R.; Van der Poel, G.; Remerie, K.; Litvinov, V. M.; Blumich, B.; Steenbakkers, R. Macromol. Mater. Eng. 2008, 293, 847.

[22] Bertmer, M.; Wang, M.; Kruger, M.; Blumich, B.; Litvinov V. M. Chem. Mater. 2007, 19, 1089.

[23] Clauss, J.; Schmidt-Rohr, K.; Spiess, H. W. Acta Polym. 1993, 44, 1.

[24] Kenwright, A. M.; Say, B. J. Solid State NMR 1996, 7, 85.

Chapter 3

58

[25] Wang, J.; Jack, K. S.; Natansohn, A. J. Chem. Phys. 1997, 107, 1016.

[26] Magusin, P. C. M. M.; Mezari, B.; Van der Mee, L.; Palmans, A. R. A.; Meijer, E. W. Macromol. Symp. 2005, 230, 126.

[27] Adriaensens, P.; Storme, L.; Carleer, R.; D’Haen, J.; Gelan, J. Macromolecules 2002, 35, 135.

[28] Lequieu, W.; Van de Velde, P.; Du Prez, F. E.; Adriaensens, P.; Storme, L.; Gelan, J. Polymer 2004, 45, 7943.

[29] Goh, S. H.; Siow, K. S. Polym. Bull. 1987, 17, 453.

[30] Hong, J.; Goh, S. H.; Lee, S. H.; Siow, K. S. Polymer 1995, 36, 143.

[31] Hernandez, R.; Perez, E.; Mijangos, C.; Lopez, D. Polymer 2005, 46, 7066.

[32] Vanhaecht, B.; Willem, R.; Biesemans, M.; Goderis, B.; Basiura, M.; Magusin, P. C. M. M.; Dolbnya, I.; Koning, C. E. Macromolecules 2003.


[34] Fedotov, V. D.; Schneider, H. Structure and dynamics of bulk polymers by NMR-methods, Springer 1989.

[35] Douglas, D. C.; Jones, G. P. J. Chem. Phys. 1966, 45, 956.

[36] Wang, J.; Cheung, M. K.; Mi, Y. L. Polymer 2001, 42, 3087.

[37] Wang, J.; Cheung, M. K.; Mi, Y. L. Polymer 2002, 43, 1357.

Chapter 4

59

Chapter 4

Phase composition of modified polyesters probed by solid-state 1H wideline NMR

of spin-spin relaxation

Summary

Solid-state NMR spectroscopy has been used to quantify morphological fractions of

poly(butylene terephthalate) (PBT)-copolymers obtained by the incorporation of 2,2-

bis[4-(2-hydroxyethoxy)phenyl]propane (Dianol) via solid-state copolymerization

(SSP). Different PBT / Dianol ratios have been investigated.

In this study, static solid- state 1H spin-spin relaxation experiments were used to

investigate the rigid fractions of the modified PBT samples. The results were fitted with

multi-exponential decay functions, or a combination of Gaussian and exponential

decays. Solid-Echo as Hahn-Echo pulse sequences have also been applied.

Chapter 4

60

4.1 Introduction

As already mentioned in Chapter 3, PBT is a semi-crystalline polymer, that presents a

broad range of application because of its ease of processability and high crystallization

rate. Nevertheless, the properties of the bulk polymer do not always meet the demand

and requirements, especially at elevated temperature. In order to broaden the range of

product applications, it might be useful to modify PBT in such a way that its high

crystallization rate is retained while its glass transition temperature (Tg) is increased. In

this chapter, modified PBT samples with a diol monomer obtained via solid-state

polymerization are investigated. The samples were synthesized in the solid-state by M.

Jansen et al., exhibing superior properties resulting from the modified microstructures

[1,2]. Jansen et al. also demonstrated that the trans-esterification reaction only occurs in

the amorphous phase of their samples modified in the solid state. Therefore, PBT blocks

are retained.

The diol was selected according to the following boundary conditions [1]:

- The diol should increase the Tg of PBT after incorporation.

- It should be commercially available.

- It should be thermally stable as the SSP reaction temperature is about 180 C to

210 C.

- The melting temperature (Tm) of the Diol should be lower then the one of the

PBT but higher than the glass transition (Tg) of PBT.

- For a trans-esterification reaction to occur between Diol and PBT, a good

miscibility condition of the Dianol is required.

- The diol should be highly reactive.

The 2,2-bis[4-(2-hydroxyethoxy)phenyl]propane (Dianol) was selected, that fulfilled all

the above requirements.

Jansen et al. demonstrated that Dianol was fully incorporated into the amorphous phase,

and that the microstructure of the modified PBT consists of a two-phase system

composed of a crystalline phase and an amorphous one, the latter subdivided into a rigid

amorphous fraction and a mobile amorphous one [2].

In Chapter 3, static solid-state NMR measurements of spin-lattice relaxation times in

the rotating frame T1 and T1 relaxation times were used to estimate 1H spin diffusion

Chapter 4

61

distances and to investigate the morphology of the modified PBT samples, together with

T1(1H) -filtered 1H-13C cross-polarization experiments (CP-MAS). From T1 (1H) and

T1(1H) measurements, an upper and a lower limit of domain sizes were established.

Furthermore, the presence of a three phase system was demonstrated, assigning a

crystalline phase, a rigid amorphous or interfacial phase and a mobile amorphous phase.

The results were found to be in reasonable agreement with Jansen et al. However, a few

discrepancies were found in phase’s quantification, which might be attributed to the use

of different techniques that probe different types of information.

The spin-lattice relaxation T1 processes are determined as well by molecular motions as

by spin-diffusion and they have been proved to be successful to study phase

morphology in many polymeric systems [3-11]. However, for highly crystalline rigid

polymers, the strong static proton dipolar interaction makes the spin-spin relaxation

process dominant in the relaxation. During the very short T2 relaxation process spin-

diffusion cannot happen. For this reason, T2 spin-spin relaxation measurements are

preferred when investigating crystalline and interfacial rigid fractions [12-15].

In this chapter, the results of static T2 (1H) relaxation experiments on the modified PBT

samples are discussed. The data of the transverse relaxation decay times were fitted with

two and three exponential decay models; the relaxation time values and the fractions of

mobile and rigid protons in each phase were extracted. The results have been analyzed

and compared to those calculated by Jansen et al. using modulated DSC (TMDSC). The

results are also compared to the values found by spin lattice relaxation in the rotating

frame as well as to the simulated fractions calculated starting from the 1H NMR

fractions obtained for the pure PBT sample.

Additional T2 (1H) relaxation experiments with a solid-echo detection sequence were

carried out on samples (BD96Di04)ssp and (BD54Di46)ssp at room temperature (RT), 353 K

and 435 K. As an extra reference, some T2 experiments with a Hahn-echo detection

sequence were carried out on the (BD54Di46)ssp sample. Mostly the best fit was found as

a combination of Gaussian and exponential (Lorentzian) decay. In general the values of

crystalline fraction were found in reasonable agreement with those found by Jansen et

al. and these obtained via T1 experiments (see Chapter 3). For the sample

(BD96Di04)ssp the high rigidity of the sample (highest Tg) however makes the

quantification of the crystalline phase challenging, even at 435 K well above Tg.

Chapter 4

62

4.2. Experimental Section


The modified PBT samples were obtained and synthesized by M. Jansen [1].

The synthesis of PBT-Dianol copolymers by SSP is described in Jansen’s thesis.

Different ratios of PBT and Dianol were used and are reported in Chapter 3. BDx

corresponds to the fraction in mol% of PBT (expressed in 1,4-butanediol units) whereas

Diy corresponds to the fraction in mol% of Dianol in the initial mixture used for the

reaction (BDxDiy)feed. After SSP, the initial mol fractions slightly differ from

(BDxDiy)ssp. This difference may be attributed to the evaporation of 1,4-butanediol or

ethylene glycol during the SSP reaction.


The NMR experiments were performed in Eindhoven (TU/e) on a DMX500 Bruker

spectrometer operating at 500 MHz for protons, and a 7-mm probe head was used. The

relaxation experiments were carried out at 435 K using a variable temperature unit. For

such high temperature, calibration was performed using the melting point of organic

compounds. The samples were equilibrated at each temperature for 20 minutes. The

Hahn-Echo Pulse Sequence (HEPS) (see below) was used to record the decay of the

transverse magnetization Mz at first. Later at the U-Hasselt a Solid-Echo Pulse

Sequence (SEPS) developed by Powles and Strange was also applied [16].

4.2.2.1. Hahn-Echo pulse (HEPS), T2 relaxation experiments

A Hahn-Echo (HEPS) measurement is the production of spin echo by radiofrequency

(rf) pulses. The HEPS, consists of a 900 radiofrequency pulse which flips longitudinal

magnetisation Mz into the xy’-plane, whereby the transverse magnetization Mxy’ starts to

precess with the Larmor frequency [17]. After half the echo time τ, the preparation

phase is followed by a refocusing 1800 pulse which serves to generate an echo signal.

The sequence is based on a double-pulse: 900 - τ -1800

- τ – acquisition (Figure 4.1.a).

The echo serves to re-phase all the coherences, which are lost during the time τ between

900 and 1800 pulses due to dephasing along the y’-axis. This dephasing is due to

Chapter 4

63

extrinsic static magnetic field inhomogeneities (i.e. field inhomogeneities of the main

magnetic field) and intrinsic dephasing coming from the examined material.

The real echo-time (TE) is half of the 900 pulse + 2τ + the 1800 pulse. In the following

experiments, the time τ (between two pulses) is varied from 13 s to 170 ms for samples

(BD54Di46)ssp and (BD76Di24)ssp, from 10 s to 4 ms for sample (BD96Di04)ssp and from

13 s to 4 ms for sample (BD100Di00)ssp. The second pulse in the HEPS inverts nuclear

spins and an echo signal is obtained with a maximum at time t = TE after the first pulse.

By varying the pulse spacing, intensity profiles are obtained as function of time τ. The 1H-NMR T2 HEPS relaxation signal is acquired as a function of echo time t starting

from the top of the echo. The experimental data were fitted with a multiple-exponential

decay fit (or Lorentzian L) using multivariate analysis:

lis TtCTtBTtAtM expexpexp)( (4.1)

Where M(t) represents the magnetization function, A is proportional to the fraction of

components with a short relaxation time, B to the fraction of components with an

intermediate relaxation time and C to the fraction of components with long relaxation

time; t = TE is the real echo-time in the HEPS experiment.

Figure 4.1.a. Hahn-Echo pulse sequence for a weakly coupled system, after a single quantum refocus.Solid echo proton wide-line (static) NMR measurements (SEPS) were carried out at ambient temperature, 353 K and 435 K on a Varian Inova 400 spectrometer using a dedicated wide-line probe equipped with a 5 mm coil.

4.2.2.2. Solid echo experiments (SEPS), T2 relaxation experiments

Solid echo proton wide-line (static) NMR measurements were carried out at ambient

temperature, 353 K and 435 K on a Varian Inova 400 spectrometer in a dedicated

wide-line probe equipped with a 5 mm coil.

Chapter 4

64

On-resonance Free Induction Decays (FIDs) were acquired by applying the solid-echo

pulse technique (900x’-tse-900

y’-tse-acquisition) in an effort to overcome the effect of

dead-time of the receiver [5]. The 900 pulse length t90 was set to 1.5 s and spectra were

recorded with a spectral width of 2 MHz (0.5 s dwell time) allowing an accurate

determination of the echo maximum. The solid echo is formed with a maximum at t =

(3t90/2 + 2tse). A preparation delay of 5 times the T1H relaxation decay time was always

respected between successive accumulations to obtain quantitative results.

For the analysis of the T2H-FIDs combinations of Gaussian (G) and Lorentzian

(exponential) (L) fitting methods have been used (see equations 4.5.1 and 4.5.2). For the

two-component analyses a GL-combination (equation 4.5.1) showed a reasonable result

for solid (G) and mobile (L) phases. For the three-component method the GGL-

combination showed the best acceptable result for solid-crystalline (G), solid-

amorphous (G) and mobile-amorphous (L) phases.

For the (BD54Di46)ssp sample at 435 K the Solid-Echo detection method (SEPS), using

the analysis starting at the maximum FID-intensity, produced extra FID-oscillations

because the chemical shifts of the proton signals of the most mobile phase resolved in 3

peaks at 400 MHz (4 peaks at 500 MHz). Therefore, an adapted Solid-Echo experiment

with variable echo-time in the detection was implemented. For the analysis, the total

peak area after FT was used as a function of the variable echo-time. The solid-echo

pulse sequence is shown in Figure 4.1.b.

Figure 4.1.b. Solid echo pulse sequence for a strongly coupled system, after a double quantum refocus.

As an alternative, Hahn-Echo detection pulse sequence (HEPS) has also been carried

out, using the same dedicated wide-line probe.

Chapter 4

65


4.3.1. T2 spin-spin relaxation experiments of Hahn-Echo pulse sequence (HEPS)

It is well known that in the solid state the proton NMR spectrum of semi-crystalline

polymers at temperatures between the glass transition and the crystal melting

temperature is characterized by superimposed broad and narrow resonances. Protons, in

the rigid phase, give rise to dipolar broadened resonances while protons in the

amorphous phase give rise to motional narrowed resonances. As already mentioned in

the introduction, several studies have been done to determine the crystallinity, phase

composition and mobility of semi-crystalline polymers [18-29].

Figure 4.2.a. T2 Relaxation decays curves recorded at 435 K (experimental data fitted with three exponential decay model ). () correspond to sample (BD100Di00)ssp; () correspond to sample (BD96Di04)ssp; ( ) correspond to sample (BD76Di24)ssp, () correspond to sample (BD54Di46)ssp.

Chapter 4

66

In this study, 1H – NMR spin-spin Hahn-Echo relaxation experiments (HEPS) have

been applied to the modified PBT samples. The transverse magnetization decays of

these samples above the glass transition and below the melting temperature were fitted

with a multi-component exponential decay fit (LLL).

For all samples measured at 500 MHz and 435 K, the FID can be decomposed into three

components. The data were best fitted with a three-exponential decay model (Figure

4.2.a) suggesting three types of 1H mobilities in the samples. This can correspond to

mobility of 1H-atoms contained in the crystalline phase (very fast decay, T2s), to the

rigid amorphous fraction (intermediate decay T2i) and to the mobile amorphous fraction

(slow decay T2l) (Table 4.1.a and 4.1.b).

Like the FID, the spectra could be deconvoluted into three components. Broad,

intermediate and narrow lines originate from these three components, the faster

decaying component showing the broader line.

Figure 4.2.b. Relaxation decays curves recorded at 435K (experimental data fitted with two exponentials decay model). () correspond to sample (BD100Di00)ssp; () correspond to sample (BD96Di04)ssp; () correspond to sample (BD76Di24)ssp, () correspond to sample (BD54Di46)ssp.

Chapter 4

67

A fitting with a two-component decay could only be accepted for the (BD54Di46)ssp

sample at 435 K as observed in Figure 4.2.b and Table 4.1.c.

Table 4.1.a. T2 relaxation times obtained with three exponential decay fit (2D analysis) at 435 K.

FBD-T, total FDi-T, total Crystalline T2s (μs)

Rigid amorphous Mobile amorphous [mol%] [mol%] T2i (μs) T2l (ms)

100 0 10 120 1.4

96 4 11 70 0.27

76 24 10 300 1.2

54 46 7000 10.000 41

Table 4.1.b. T2 relaxation times obtained with three exponential decay fit (1D analysis) at 435 K.

FBD-T, total FDi-T, total Crystalline T2s (μs)

Rigid amorphous Mobile amorphous [mol%] [mol%] T2i (μs) T2l (ms)

100 0 16 156 1.08

96 4 9 72 0. 28

76 24 43 391 1.5

54 46 62 1998 9

Table 4.1.c T2 relaxation times obtained with two exponential decay fit (2D analysis) at 435 K.

FBD-T, total

[mol%] FDi-T, total T2l (ms) T2s (s)

Rigid [mol%] Amorphous

100 0 14 0.60

96 4 12 0.17

76 24 80

0.8

54 46 4340 10

As already shown in Chapter 3, the quantitative analysis for the FIDs and wide-line

NMR spectra (after FT) in terms of crystalline, rigid amorphous and mobile amorphous

can be complicated due to a number of reasons.

Chapter 4

68

First, one should perform the experiments at a temperature well above the Tg to probe

more mobility in the sample in order to be able to distinguish different phases. However,

this requirement implies that some phase rearrangements may occur. So the temperature

should be high enough but not too high to prevent annealing effects. This has been

checked by MDSC (results presented in Chapter 3) but also in NMR experiments.

Values of relaxation times and corresponding fractions were found to remain the same

after temperature treatment (heating to 435 K and subsequent cooling).

The T2 relaxation times at 435 K are shown in Table 4.1. The data were fitted with 1D

and 2D methods. The results show that, when more Dianol is incorporated, the 1D and

2D data diverge. As already observed in Chapter 3, the 2D analyses do not seem to

reflect the phase behavior when more Dianol is incorporated. The fact that the 2D

analysis takes the chemical shift into account may not always reflect the phase

relaxation (see Appendix 1 & 2). It may also assimilate relaxation of protons in a phase

such as side groups (methyl group: Me-) to the overall protons relaxation in that phase

relaxation. The 1D data seem to better describe the physical reality. However, a 2D

analysis seems to help when trying to localize the Dianol (see Chapter 3).

For all samples, except for (BD54Di46)ssp, the shorter 1H relaxation times obtained

through 1D and 2D are in the order of a few s, which is the typical order of magnitude

of T2 relaxation of 1H-atoms present in a crystalline phase where molecular motions are

restricted and relaxation decay is very fast as seen in Figure 4.2.a. These samples,

except for (BD54Di46)ssp, also show 1H mobilities with T2-values in the order of 100 s

which could be assigned to 1H in an intermediate rigid phase, corresponding to the rigid

amorphous fraction as mentioned by Jansen et al. [1,2]. The third type of mobility is

attributed to 1H-atoms in the most mobile phase (amorphous phase) and shows a

relaxation time in the order of the ms.

For the sample (BD54Di46)ssp with the highest mole fraction of Dianol, the T2 values

obtained with a 2D fitting, are much higher, the three relaxation times are in the same

order of magnitude (tenths of ms) and, at first sight, a two component fit (LL) seems to

be more realistic. These data should be treated with care, and as mentioned above, for

this sample, a 1D fitting is more realistic.

In highly crystalline polymers, the strong static dipolar interactions make the spin-spin

relaxation process more efficient. In the case of highly amorphous systems such as

Chapter 4

69

(BD54Di46)ssp, the spin-spin relaxation process at 435 K is less efficient because of the

faster fluctuating dipole fields caused by conformational motions, narrowing the

resonance lines. The mobility in sample (BD54Di46)ssp is very high especially at such

experimental conditions. The amorphous phase in the sample behaves almost as a liquid.

Too much isotropization is present to be able to distinguish between different

amorphous fractions. Figure 3.8 (see Chapter 3) shows a rather high resolution (as in

liquid like), and we could distinguish four narrow peaks corresponding to proton

resonance frequencies of aromatic (2 peaks), ester, and aliphatic 1H-atoms. In addition,

in sample (BD54Di46)ssp, the methyl group -(Me) content is relatively high as compared

to other samples. Local mobility rather then overall phase mobility may be depicted,

which results in a challenging quantification of the amorphous phase for this sample.

From a bi-exponential (LL) fit, one can determine the fraction of rigid protons

associated with the decay of the broad components (fast decay) and the fraction of more

mobile protons, associated with the decay of the narrow component. The fractions of

each component are reported in Tables 4.2.a, 4.2.b (three-phase fit) and 4.2.c (two-phase

fit).

One should note that the crystalline fraction and rigid amorphous fractions calculated by

Jansen et al. via TMDSC and reported in Chapter 3, are based on mol fractions. The

crystalline, rigid amorphous, and mobile amorphous fractions were determined by

Jansen et al., based using the following equations [1,2]:

mobile= cp / cp 0 (4.2)

heating= H melting/H 0fusion (4.3)

rigid= 1- heating + mobile (4.4)

where cp is the heat capacity increase at the half- step Tg of the non-transesterified

PBT-Dianol mixture, whereas cp0 is the heat capacity increase for 100% amorphous

PBT at the half step Tg ; H0fusion is the enthalpy of 100% crystalline PBT. The value of

cp0 for 100% amorphous PBT was assumed to be independent of the added Dianol

monomer.

Chapter 4

70

Table 4.2.a. Rigid fraction, intermediate fraction and mobile fraction obtained from static T2 experiments using a three exponentials decay fit (2D analysis).

FBD-T, total [mol%]

FDi-T, total

[mol%] Crystalline

(%) Rigid amorphous


(%)

100 0 51 22 27

96 4 74 15 11

76 24 36 31 33

54 46 22 72 2

Table 4.2.b. Rigid fraction, intermediate fraction and mobile fraction obtained from static T2 experiments using a three exponentials decay fit (1D analysis).

FBD-T, total [mol%]

FDi-T, total

[mol%] Crystalline

(%) Rigid amorphous


(%)

100 0 46 24 30

96 4 63 17 20

76 24 7 55 37

54 46 0.1 3 97

Table 4.2.c. Rigid fraction and amorphous fraction obtained from static T2 experiments using a two exponentials decay fit (2 D analysis).

FBD-T, total [mol%]

FDi-T, total [mol%]

Rigid fraction (%)

Amorphous fraction (%)

100 0 82 18

96 4 78 22 76 24 46 54

54 46 11 89

When looking at values presented in Tables 4.2, the fractions of rigid protons, and

mobile protons obtained by fitting the experimental data with the two exponential (LL)

decays (Table 4.2.c) seem to be in reasonable agreement with the summation of rigid

and amorphous fractions found by Jansen et al., also in good agreements with values

found via peak deconvolution of T1 data at short evolution (spin-lock) time presented in

Chapter 3 (Table 3.7).

For the fractions obtained using a three components fit on the T2-data from the Hahn-

echo experiments, a bad correlation has been found with Jansen’s fractions especially

Chapter 4

71

for the (BD54Di46)ssp sample. For the (BD96Di04)ssp sample the value of rigid (crystalline)

fraction is found to be quite high, probably due to the fact that this sample is much more

rigid showing a much higher Tg. Also, the intermediate (rigid amorphous) fraction is

expected to be more important in this sample; so some part of this intermediate fraction

may be seen in T2 spin-spin relaxation experiments as part of the rigid (crystalline) one.

Furthermore, the T2 values are more or less in agreement with those calculated from

pure PBT while assuming that Dianol is incorporated in all the amorphous phase. The

values are reported in Table 4.3.

Table 4.3 Percentage of the two 1H fractions present in the system obtained via calculation method based on pure PBT and assuming Dianol is incorporated in the totality of both amorphous phases.

FBD-T, total [mol%]

FDi-T, total [mol%]

Rigid fraction (%)

Amorphous fraction (%)

100 0 51 49

96 4 48 52 76 24 35 63

54 46 28 72

One should note that phase quantification has to be treated with real care.

On one hand, different methods may be used for determination of crystallinity but these

do not necessarily lead to the same crystalline fractions. Some techniques look at 1H

crystallinity such as NMR, while DSC for instance looks at crystal melts.

During the fast T2 relaxation in the solid state NMR experiments, spin diffusion cannot

happen. Therefore the choice of T2 was made to better quantify the phase fractions.

Even then the distinction between the rigid amorphous and the mobile amorphous

phases is not always straightforward. When performing T2 experiments at higher

temperature, the increased local molecular dynamics can cause increased spectroscopic

resolution, increasing the number of mobile components in the analysis. This means that

the mobile amorphous phase will show up in different components in the multi-

exponential analysis of the FID. The number of components does not reflect anymore

the number of phases present in the system. However because T2 is always decreasing

Chapter 4

72

with decreased molecular dynamics, the phase assignment to corresponding T2 values is

still more straightforward compared to T1 and T1.

The values of the total rigid fraction (sum of the crystalline and rigid amorphous phases)

obtained for the pure PBT fractions from T1, T2 (2D, 1D) and from Jansen are all in

reasonable agreements. It is only when incorporating the Dianol that the relaxation

picture starts to become more complex and ambiguous. Effectively, at higher

temperature (above Tg) Dianol contains highly mobile groups (CH3- and p-phenyl

groups). When Dianol is incorporated in the amorphous phases, these mobile groups

generally cause an increase of the T2H relaxation time.

At low concentration (BD96Di04)ssp the incorporation of Dianol induces a minor

reorganization in the amorphous phases, slightly reducing the molecular dynamics,

resulting in a decrease of their T2-values (Tables 4.1). At higher Dianol concentration

(BD76Di24)ssp , a significant increase of the T2-values of the amorphous phases is noticed

without a change of the T2-value of the crystalline phase. For the (BD54Di46)ssp sample,

even the crystalline PBT-phase is influenced by the Dianol incorporation, showing a

significant increase in molecular dynamics.

Another source of errors in the phase quantification can also be found in the use of a

CP/MAS probe for the static solid-state NMR measurements. Such a CP/MAS probe

has a rather long dead time, only allowing proper phase quantification based on T2H-

values longer than about 20 s.

For a better understanding and to lift ambiguity in quantification of the samples

(BD96Di04)ssp with the least amount of Dianol and the sample (BD54Di46)ssp with the

higher amount of Dianol, some complementary measurements have been performed on

a Varian Inova 400 spectrometer using a dedicated wide-line probe equipped with a 5

mm coil. Thanks to the shorter dead time of this wide-line probe, solid-state relaxation

experiments with Solid-Echo (SE) detection can be performed. T1, T1 (Chapter 3) and

T2H measurements using such a solid-echo detection have been performed on samples

(BD96Di04)ssp and (BD54Di46)ssp at 293 K, 353 K and at 435 K.

4.3.2. Analysis and relaxation measurements for sample (BD96Di04)ssp

For the (BD96Di04)ssp sample at room temperature, the experimental T2 data were best

fitted with a single Gaussian function. At 353 K a bi-component (GL) best fit, using a

Chapter 4

73

combination of Gaussian and exponential (Lorentzian) functions as detailed in equation

(4.5.1) results in an amount around 93% for the rigid phase and around 7% for the

mobile phase.

At 435 K, both a tri-component (GGL)-fit (equation 4.5.2) and a two-component (GL)-

fit have been performed (equation 4.5.1):

M (t) = A exp (-1/2(t/T2s ) 2) + B exp (-t/Tl ) (4.5.1)

M (t) = A exp (-1/2(t/T2s )2) + B exp (-1/2(t/T2i )

2) + C exp (-t/T2l ) (4.5.2)

The (GL)-fit as well as the (GGL)-fit result in an amount around 70% for a rigid phase

and around 30% for an amorphous phase. From the tri-component fit, this amorphous

phase seems to be composed of 17 % of the solid-amorphous inter-phase and 12 % of

the mobile-amorphous phase.

The T2H-values and the fractions of the Solid-Echo experiments at 400 MHz (Table 4.4)

fit rather well with the values from corresponding Hahn-Echo experiments at 500 MHz

(Tables 4.1 and Tables 4.2).

Table 4.4. Data for (BD96Di04)ssp extracted from a combination of Gaussian (G) and exponential (L) functions from solid-echo experiments. At RT (20 °C) only a Gaussian fit was used.

(BD96Di04)ssp Crystalline

T2s (μs) / f (%)

Rigid-Amorphous

T2 i (μs) / f (%)

Mobile-amorphous

T2l (μs) / f(%)

(T = 20 °C); G 11.9 (one phase) - -

(T= 20 °C); G after 162°C 12.1 (one phase) - -

T=353 K (80 °C) GL 13,8 (93.4) 22,3 (6.6) -

T=435 K (162 °C) GGL 16.3 (71.2) 45 (16.8) 114 (12.0)

T=435 K (162 °C) GL 16,3 (71.5) - 117 (28.5)

4.3.3. Analysis and relaxation measurements for sample (BD54Di46)ssp

For the sample (BD54Di46)ssp a Solid-Echo pulse sequence as well as a Hahn-Echo pulse

sequence were applied. One should note that phase quantification in sample

(BD54Di46)ssp may be ambiguous, as the amount of Dianol is much higher than in the

other samples, causing a much higher molecular mobility. As discussed in Chapter 3 of

Chapter 4

74

this thesis, this correlates with the low Tg (51°C) of this sample. The excess Dianol

leads on one hand to a partial dissolution of the PBT crystals and on the other hand to

inner-outer trans-esterification reactions as describes by Jansen et al. that imply a lower

average chains Mw for this copolymer [1,2].

4.3.3.1. T2H analysis with Solid-Echo detection

Different combinations of Gaussians (G) and exponentials (L) were used to fit the

experimental data for sample (BD54Di46)ssp. The results are reported in Table 4.5.a.

At room temperature (before and after cooling down from 435 K) a GL two-component

fit resulted in a rigid phase of about 80 % and a more mobile phase of about 20 %.

However the T2-values of the rigid and mobile phase are rather close to each other (13

and 20 s) implying that their molecular mobilities are similar and very slow. Being

below Tg an important part of the rigid amorphous phase is expected to be integrated in

the crystalline phase.

Table 4.5.a. Data for (BD54Di46)ssp extracted from a combination of Gaussian and exponential functions from Solid-Echo experiments (1 D analysis).

(BD54Di46)ssp Crystalline

T2s (μs) / f (%) Rigid-amorphous

T2 i (μs) / f (%) Mobile-amorphous

T2 l (μs) / f (%)

(20°C)-GL 13 (81,8) 20 (18,2)

(20°C)-after162°C-GL 13 (81,1) 24 (18,9)

T=353 K (80 °C) -GL 16 (62,4) 20 (37,6)

T=435 K (162 °C) -GL 15 (21,8) 154 (78,2)

T=435 K (162 °C) -GL-f()

104 (58,1) 3305 (41,9)

T=435 K (162 °C) -GGL-f()

26 (10,9) 127 (51,4) 3812 (37,7)

T =435 K (162 °C) -GGL-f()- Abragam

33 (14,3) 291 (47,2) 3703 (38,4)

At 353 K, slightly above Tg, a similar GL two-component fit results in comparable T2H-

values but with an increase in the more mobile fraction (from 18 to 38 %).

At 435 K a difficult GL-analysis, based on a reduced set of data points, ends up with a

fraction of 22 % for the solid phase and a T2-value around 15 s, a value typical for a

crystalline phase at temperatures sufficient above Tg. The amorphous fraction of 78 %

Chapter 4

75

has a T2H-value of 154 s showing a significantly higher molecular mobility. However

this two-component analysis produces an insufficient accurate fit. A three-component

analysis on the same single FID at this higher temperature, produced by a routine SE-

experiment, is not fitting at all. At 435 K an important part of the amorphous phase of

this (BD54Di46)ssp sample has an increased molecular mobility resulting in a

spectroscopic resolution of the chemical groups in this phase. After FT, three peaks are

resolved at 400 MHz corresponding to aromatic, ether and CH3- + CH2R groups. This

increased spectroscopic resolution causes extra oscillations on the FID-signal, which

complicates the multi-component fit of this FID. Normally in the SE-experiment, the

analysis of the FID-signal is done based on a single echo-time. The extra oscillations,

superimposed on the normal FID, make a proper multi-component analysis impossible.

A simple adaptation of the SE pulse-sequence allows a T2-determination at variable

echo-times. This allows signal area measurements, after FT and proper peak phasing, as

a function of the echo-time, avoiding the problem of chemical shift oscillations. Based

on such an adapted SE-experiment at 435 K, we performed a T2 two-component GL-

analysis and a T2 three-component GGL-analysis using the total peak area as a function

of the echo-time. For the two-component fit T2-values of 104 and 3305 s, for a rigid

and a mobile phase with 58 % and 42 % fractions are found (Table 4.5.a). The three-

component GGL-analysis however clearly produced a better fit (Figure 4.3). T2-values

of 26, 127 and 3812 s, corresponding respectively to a crystalline, a rigid amorphous

and a mobile amorphous phase with 11 %, 51 % and 38 % fractions are found (Table

4.5.1). This lower crystalline fraction is already closer to what is expected. A refinement

of the GGL-analysis can be done by introduction of the Abragam algorithm for the

second Gaussian function in the tri-component fit (see equation 4.6) [29,30]:

M (t) = A exp (-1/2(t/T2i ) 2) + B exp (-1/2(t/T2s )

2).(sin(2(t/))/ (2(t/)))

+ C exp (-t/T2l) (4.6)

The parameter introduces an oscillation, which in this case could be linked to rather

fast Me- and p-phenyl group rotations [31]. Such an analysis gives T2-values of 33, 291

and 3703 s, for the crystalline, rigid amorphous and mobile amorphous phase with 14

%, 47 % and 39 % fractions respectively (Table 4.5.a).

Chapter 4

76

It is known that a ‘90° -- 90° -- acquisition’ SE-sequence can lead to an incomplete

refocus after the echo-time, especially with samples which hold mobile phases [4]. The

second 90°-pulse does not properly refocus e.g. chemical shift differences and p-phenyl

and methyl rotations.

Figure 4.3. T2 relaxation decays curves recorded on a 400 MHz wide-line probe at 435 K with a Solid-Echo sequence for sample (BD54Di46)ssp (experimental data fitted with a tri-component GGL-fit (thick line corresponds to an Abragam fit) and a two component GL-fit (thin line)).

4.3.3.2. T2H with Hahn-Echo detection

As a comparison Hahn-Echo (HE) experiments, having a better refocusing -pulse, are

performed on the same sample at 435 K. As well chemical shift artifacts as rotation

phenomena are refocused in this case. Thanks to the shorter dead time of the dedicated

wideline probe, used on the 400 MHz system, the shorter T2-values of the very rigid

phase can be measured more accurately (more data points at short echo-times, compare

Figure 4.2.a with Figure 4.4). Also, the total peak area and the area of the individual

Chapter 4

77

peaks are used for a tri-component fit. For the resolved peaks a GGL-combination,

using equation (4.7), leads to the most optimal fit (Figure 4.4):

M (t) = A exp (-1/2(t/T2s )2) + B exp (-1/2(t/T2i )

2) + C exp (-t/T2l) (4.7)

An analysis of the total peak area resulted in T2H-values around 20, 400 and 4000 s for

the crystalline, rigid-amorphous and mobile-amorphous phases. They correspond with

the individual fractions of 26 %, 17 % and 57 %. (Table 4.5.b).

Table 4.5.b. Data for (BD54Di46)ssp extracted from a combination of Gaussian and exponential functions from Hahn-Echo experiments recorded at 435 K(1 D analysis).

Crystalline Rigid amorphous T2 i (μs)/ f (%)

Mobile amorphous T2l (μs)/ f (%)

(BD54Di46)ssp T2 s (μs) / f (%)

T =435 K (162 °C) -Total GGL

20 (25,8) 370 (17.2) 4254 (57.0)

Figure 4.4. T2 relaxation decays curves recorded on a 400 MHz wide-line probe at 435 K with a Hahn-Echo sequence for sample (BD54Di46)ssp (experimental data fitted with a tri-component GGL-fit).

Chapter 4

78

The small fractions of the crystalline phase (11 % from the SE-experiment and 26 %

from the HE-experiment) are representative for the reduction of the PBT-crystallinity

caused by the SSP-reaction with 46 % of Dianol. This is in good agreements with the

low crystalline fraction found by Jansen et al. via MDSC as well as with the rigid

fraction values found from tri- and two-component fits in Table 4.2.a (22 %), Table

4.2.b (0.1 %), and Table 4.2.c (11%) measured at 500 MHz.

4.4. Conclusion

Solid-state 1H T2 relaxation measurements allow a more straightforward analysis for the

different phases of the (BDxDiy) copolymers. As already observed in Chapter 3, the

results show a crystalline fraction that decreases with Dianol concentration.

In the rather rigid sample (BD96Di04)ssp, the crystalline phase may be over-estimated

because part of the rigid amorphous fraction (or interface) may be depicted in the

crystalline phase.

For the (BD54Di46)ssp sample, the crystalline PBT-phase is quite influenced by the Dianol

incorporation, showing a significant increase in the molecular dynamics. The pure PBT

sample shows results similar to those obtained with other techniques indicating the

presence of a “three phases” system, composed of a really rigid phase (crystalline

fraction), an intermediate phase (interfacial fraction) and a mobile amorphous phase.

When Dianol is incorporated, the three phases system holds, but the relaxation picture

and phases quantification becomes challenging, especially with this type of Diol

containing high mobile groups at temperatures high enough (well above Tg), causing

chemical shift resolution. This increased spectroscopic resolution may cause extra

oscillations on the FID-signal as observed in sample (BD54Di46)ssp, which complicates

the multi-component fit of the FID. At such high concentration of Dianol, only a

traditional 1D data analysis is realistic and the best fit was found using a combination of

Gaussian and exponential decays (GGL). The results obtained from Hahn-echo and

Solid-echo experiments were found to be in good agreements.

Chapter 4

79

4.5. Reference


[2] Jansen, M. A. G.; Goossens, J. G. P.; de Wit, G.; Bailly, C.; Koning, C. E. Anal. Chim. Acta 2006, 557, 19.


[4] Hedesiu, C.; Demco, D. E.; Kleppinger, R.; Buda, A. A.; Blumich B.; Remerie, K.; Litvinov, V. M. Polymer 2007, 48, 763.

[5] Lequieu, W.; Van De Velde, P.; Du Prez, F. E.; Adriaensens, P.; Storme, L.; Gelan, J.; Polymer 2004, 45, 7943.

[6] Schmidt-Rohr, K.; Spiess, H. W. Multidimensional Solid-State NMR and Polymers, Academic Press, 1994.



[9] Havens, J. R.; Vanderhart, D. L. Macromolecules 1985, 18, 1663.

[10] Fedotov, V. D.; Schneider, H.; Structure and dynamics of bulk polymers by NMR-methods, Springer, 1989.

[11] Yao, Y.-F.; Graf, R.; Spiess, H. W. Macromolecules 2008, 41, 2514.

[12] Demco, D. E.; Litvinov V. M.; Rata, G.; Popescu, C.; Phan, K-H.; Schmidt, A.; Blumich, B. Macromol. Chem. Phys. 2007, 208, 2085.

[13] Hedesiu, C.; Demco, D. E.; Kleppinger, R.; Poel, G. V.; Remerie, K.; Litvinov, V. M.; Blumich, B.; Steenbakkers, R. Macromol. Mater. Eng. 2008, 293, 847.

[14] Bertmer, M.; Wang, M.; Kruger, M.; Blumich, B.; Litvinov, V. M. Chem. Mater. 2007, 19, 1089.


[16] Powles, J. G.; Strange, J. H. Proc. Phys. Soc. London 1963, 82, 6.

[17] Bloch, F.; Hansen, W. W.; Packard, M. Phys. Rev. 1946, 69, 127.

[18] Tanaka, H.; Kohrogi, F.; Suzuki, K. Eur .Polym. J. 1989, 25, 449.

[19] Bergmann, K.; Nawotki, K. Z. Polymer 1967, 219, 132.

[20] Earl, W.; Vanderhart, D. L. Macromolecules 1979, 12, 762.

[21] Dadayly, D.; Harris, R. K.; Kenwright, A. M.; Say, B. J.; Sunnectioglu, M. M. IRC Polym. Technol. 1994, 35, 4083.

[22] Tanaka, H.; Inoue,Y. Polym. Int. 1993, 31, 9.

[23] Henning, J. Concepts in Magn. Res. 1991, 3, 125.

[25] Litvinov, V. M.; Penning, J. P. Molecular Chem. Phy. 2004, 205, 1721.

Chapter 4

80

[26] Schneider, M; Gaspar, L.; Demco, D. E.; Blumich, B. J. Chem. Phys. 1999, 111, 402.

[27] Tanaka, H.; Kohrogi, F.; Suzuki, K. Eur. Polym. J. 1989, 25, 449.

[28] Kitamaru, R.; Horii, F. Adv. Polym. Sci. 1978, 26, 139.

[29] Abragam, A.; The principle of NMR. Oxford University Press, 1991.

[30] Abragam, A.; Goldman, M.; NMR: order and disorder, Oxford University Press, 1982 (1).

[31] Sergeev, N. A.; Olsezewski, M. Solid State NMR 2008, 34, 162.

Chapter 5

81

Chapter 5

Heterogeneous chain dynamics in polyester network probed by 1H static spin-spin relaxation NMR experiments

Summary

Characterization of polymer networks is a very complex task, which has captivated the

attention of polymer scientists, especially during the last two decades. To be able to

improve and optimize the macroscopic properties of cross-linked resins, we must

develop a thorough understanding at the microscopic level, including features such as

the cross-link density of a network, the number of dangling chains, loops, and eventual

polymer chains unattached to the network (commonly called the sol fraction). The

chemical structure (topology) of cross-linked networks is complicated. During the

formation of a network, many different structures may be formed, resulting in a very

heterogeneous network. Different initiation propagation termination and combination

reactions may occur, as well as side reactions with other components present in the

curing resin system. Many parameters affect the curing process, including the

temperature, the solvent, the chemical and physical properties of the (pre)-polymers,

additives and catalysts presents. The final network structure with all its variations,

heterogeneities and defects determines the properties of the network.

In this study heterogeneous chain dynamics in a cured unsaturated polyester resin (UPR-

Palatal intermediate) are investigated using 1H-NMR transverse relaxation experiments.

Different transverse 1H relaxation times T2 are reported at different temperatures. T2

measurements were performed on a cross-linked sample in the original and swollen state

at respectively 4.7 and 11.7 Tesla by means of a novel multi-variable procedure. A

spectral resolution has been implemented at high field, correlating line-shape

Chapter 5

82

information to relaxation decay. The two-dimensional data sets obtained were analyzed

in terms of two or three decaying components without any assumptions about the

underlying line shape. These results are complemented with classical two and three

component analyses based on the maximum intensity of the FID’s.

5.1. Introduction

Unsaturated polyester (UP) resins are the third-largest class of thermoset molding resins.

They constitute one of the most important matrix resins for glass fibers reinforced

composites. These materials are easy to process and are produced at low production

cost. UP-resins are produced in a two-step process: first, saturated and unsaturated

diacids or anhydrides are reacted with diols via a polycondensation reaction. Secondly,

the resulting polyester is dissolved in styrene monomer. The resin is then processed into

a three-dimensional polymer network (using a peroxide as free-radical initiator) through

a polymerization between styrene and the double bonds in the polyester chains (Figure

5.1). The cross-link density of the network will control the macroscopic properties of

the material. To tailor these properties it is important to look at the molecular structure

and the related molecular mobility of the obtained network.

Mobility in polymer networks has been extensively studied during the last two decades

by proton Nuclear Magnetic Resonance (NMR) spectroscopy, in both solid and swollen

states. NMR has proven to be a valuable tool for studying various types of polymer

networks, including epoxy resins [1], cross-linked polystyrene [2], cured EPDM [3],

cross-linked polyacrylate [4] and unsaturated polyesters [5].

Both NMR as a spectroscopic technique [5,6], and NMR as a relaxometry technique can

be performed on these types of systems; this contribution focuses on the latter

technique. The structure and the dynamics of polymer chains, largely determine the

mechanical properties of polymers. Investigation of these is thus of prime importance

and NMR is an informative non-invasive technique in that regard.

Motions in polymer materials can generally be investigated by NMR relaxometry. NMR

experiments yield information about both the local and long-range mobility of network

polymer chains. One of the most useful NMR techniques for the latter purpose is the

measurement of the transverse magnetization relaxation process, commonly referred to

Chapter 5

83

as T2 (spin-spin) relaxation measurements [7,8,9]. In the solid state, this technique

reflects slow local molecular mobility of polymers. The T2 relaxation times very often

have values smaller than a millisecond.

At such time scale, both chemical cross-links and physical entanglements affect polymer

networks chain’s motion preventing complete motional isotropy. As a result, some

residual dipolar coupling is observed. The proton T2 relaxation experiments are

governed by a de-phasing of coherence influenced by this residual dipolar interaction.

The relaxation rate depends on the strength of the interactions between the nuclear

magnetic dipoles. It has been shown that the magnitude of the residual coupling

increases with the cross-link density of the system [10,11,12,13]. When the chain

motions are less restricted, the interactions will be more averaged out, resulting in a

slower transverse relaxation.

The rate of T2 relaxation can be expressed as the sum of the individual relaxation decays

from different parts of the network without spin-diffusion. To correlate the NMR

relaxation with molecular dynamics, theoretical models are used [14].

The mathematical form of the decay function has been a subject of discussion. A simple

model based on two or more exponential decays is often used as a phenomenological

approach.

Figure 5.1. Representation of the polyester pre-polymer chain cross-linked with styrene.

The main goal of this chapter, presenting results of T2 relaxation NMR experiments on

cured unsaturated polyesters, is to help to better understand the complex chains

dynamics in chemically cross-linked polymer materials. The author tried to probe

Chapter 5

84

heterogeneity of the chain dynamics in terms of two components and three components,

one for describing the rigid part of the system due to the cross-links and the others

describing the more mobile parts.

The experimental data were then first fitted with a simple dual-exponential-decay

approach (below Tg). For a complete and precise description of the system, a spectral

resolution has been implemented, correlating line-shape information to relaxation decay.

The two-dimensional data sets obtained were analyzed in terms of two and three

decaying components without any assumption about the underlying line shape.

5.2. Experimental


The samples were supplied by DSM Resins (Zwolle, The Netherlands) in the form of a

solution of unsaturated polyester resins (UPR-Palatal intermediate) containing 36.6wt %

styrene. A catalyst (0.5% of a 1% cobalt solution) and 2% of benzoyl peroxide initiator

were added to the solution. The unsaturated polyester is formed by a poly-condensation

reaction of maleic anhydride (MA) and phthalic anhydride (OPA) with 1,2-propylene

glycol (PG) and ethylene glycol (EG). The material was degassed and transferred

between two glass-plates, cured for 24 hours at room temperature, followed by two

post-curing steps for 24 hours each at 60 oC (333 K) and 80 oC (353 K) respectively. For

the NMR experiments, the resulting networks were ground to a fine and apparently

homogeneous powder using a cryogenic grinder (Figure 5.1).

The initial composition of the material leads to the following weight fraction ratio (1;

1.5; 2; 0.5) respectively ( MA; OPA; EG; PG ) and 36% of the total mass being styrene

monomer.


The NMR experiments were performed on a DMX500 and a MSL200 Bruker

spectrometers operating at 500 MHz and 200 MHz respectively for protons. The

relaxation experiments were carried out at different temperatures on the cross-link

samples between 213 K and 506 K using a variable temperature unit with and without

Chapter 5

85

swelling in Chloroform (CDCl3) (Vs = 40-50%). The low temperature calibration was

performed using methanol. For high temperature calibration ethylene glycol was used

between room temperature (RT) and 390 K. For temperatures around 450 K and higher,

calibration was performed using the melting point of organic compounds. The samples

were equilibrated at each temperature for 20 minutes. The Hahn-Echo Pulse Sequence

(HEPS) was used to record the decay of the transverse magnetization.

A Hahn-Echo (HEPS) measurement is based on a double-pulse sequence: 90°-τ-180°-τ

-acquisition. In our experiments, the τ is varied between 0.01 and 4 ms. The second

pulse in the HEPS inverts nuclear spins and an echo signal is obtained with a maximum

at time equal to (τe) half of the 90° pulse + 2τ + the 180° pulse. By varying the pulse

spacing, intensity profiles are obtained as function of time τe. The 1H-NMR T2 HEPS

relaxation signal is acquired as a function of τe starting from the top of the echo.

5.2.3. Relaxation background

As explained in Chapter 2, mechanical properties of polymer materials are strongly

determined by their chain dynamics.

Chain dynamics in polymer systems, even in a very homogeneous material (one type of

component or phase) is extremely complex and can only be described by a continuous

distribution of correlation times. In polymer networks, as a result of diversity of motions

(Figure 5.2), different parts of the network generate different dynamics, from slow

molecular dynamics (correlation time 10-3 s) to fast molecular dynamics (correlation

time < 10–10 s). Chains dynamics in polymeric network is very often studied by spin-

spin 1H NMR relaxation analysis [1]. This relaxation is induced by the dipole-dipole

interactions between neighbouring spins and is sensitive to anisotropy of long-range

motions. In the liquid state fast isotropic tumbling of the molecules leads to motional

averaging of these dipolar interactions to zero. In the solid state, averaging leads to a

residual part q of the second moment M2 of the dipolar interactions as described in

equation (5.1). This residual coupling is a direct result of the anisotropy of molecular

motions.

Several theoretical models have been developed based on discrete correlation times in

order to fit the transverse relaxation decay. It is accepted that isotropic rotational chain

motions (q=0) exhibit an exponential decay relaxation behavior (BPP theory), whereas

Chapter 5

86

anisotropic chain motions (q≠ 0) are better described by Gaussian decay functions

(Anderson-Weiss theory).

Figure 5.2. Representation of different mobile molecular parts in polymer network: 1) sol fraction, 2) dangling chains ends; 3) inter cross-link chains.

In general, “the relaxation picture” in polymer networks (above Tg) exhibits different

molecular mobilities in the material producing different relaxation signals of different

forms and lengths. This diversity of motions (combination of isotropic and anisotropic

motions) leads to describe the relaxation with a superposition of exponential decays or

stretch exponential decay (Weybull [15]) and Gaussian decays. However, there is still

some discussion in the literature on the mathematical form of this decay.

A more general model that collects with both exponential and Gaussian is widely used

to describe the relaxation of the transverse magnetization in polymer networks [16, 17]:

CBcccA TtCTtBttqMTtAtM 222

22 expexp1expexp)( (5.1)

where M(t) represents the magnetization function, the factor A represents the fraction of

inter-cross-link chains, B corresponds to the fraction of more mobile components such

dangling chains, and C corresponds to the liquid-like fraction commonly named sol-

fraction. t is the pulse spacing in the HEPS experiment. T2A, T2B and T2C are related to

the relaxation decay, τc is the apparent correlation time of inter-cross-link orientations,

Chapter 5

87

and q is a measure of the mean residual part of the second moment M2 of the dipolar

interaction in the rigid limit (T< Tg).

5.2.4 Fitting procedure

The spin-spin 1H NMR relaxation in polymers is usually analysed by recording the

maximum of the Hahn- Echo in the time domain in function of the echo time (t) without

any spectral resolution . Hahn-Echo spectra are recorded by means of a 90°- -180°-

pulse sequence with a 90° and a 180° pulse separated by a variable time interval . The

90° pulse produces initial transverse magnetization, which difocuses during time

under the influenceof spin-spin coupling, chemical shift differences and magnetic

field inhomogeneity. The 180° pulse refocuses the effect of chemical shift and field

inhomogeneity, but leaves the spin-spin coupling unchanged. As a result an echo arises

at time τe which is attenuated only by the spin-spin relaxation. The NMR signal is

acquired as a function of time t starting from the top of the echo:

(5.2) )exp()|()(),( 000inhom0 titRGdtS ee

where Ginhom(0) reflects the inhomogeneous spectrum arising from residual anisotropy,

chemical shift differences and magnetic field inhomogeneity, and R(0| e +t) the spin-

spin relaxation decay associated with spectral frequency 0. In the usual approach, the

top of the Hahn-Echo at t = 0 is measured as a function of e without spectral

resolution. The resulting monotonous decay, a 1D data set, can be analyzed in terms of a

number of components (exponential or Gaussian). We propose to use the spectral

resolution in the full 2D data set I( e ,) after Fourier transformation to the frequency

domain (Figure 5.3, equation (5.3)).

By defining pure components to have mono-exponential relaxation and relaxation-

invariant NMR lineshapes, we impose a mathematical structure of the form:

(5.3) )/exp()(),( 2 ke

n

kke TgI

Chapter 5

88

with gk() the relaxation-invariant subspectrum of the component with relaxation time

T2k.

Figure 5.3. Representation of the data of matrix Y in a two-dimensional fitting approach.

The transition from equation (5.2) to equation (5.3) is not trivial. First, we change from

a realistic continuous description to the discrete picture that can be feasibly extracted

with numerical methods from data with significant noise without some a priori model

for Ginhom(0). Second, by assuming relaxation-invariant lineshapes gk(0) we have

tacitly separated R(0| e +t) into the two time domains according to the Laplace

relation:

)](exp[),(r)|( 00 tdtR ee (5.4)

where r(0, ) denotes the Laplace spectrum of the relaxation decay for the component

with NMR frequency 0. Third, instead of labelling the components on the basis of their

frequency 0, equation (5.3) implicitly classificies spectral components on the basis of

their relaxation rate. To clarify this point we insert equation (5.4) into equation (5.2) and

apply the Fourier transform along t and change the order of the integrals over 0 and :

(5.5) ])(/[2),(r)(]exp[),( 20

200inhom0 GddI ee

Chapter 5

89

The meaning of r(0, ) as the Laplace density function of the component with

frequency 0, has now become the spectral density function of components with

relaxation rate. Equation (5.5) also shows the Lorentzian line broadening resulting

from the relaxation during the acquisition time t. Translated into the discrete picture of

equation. (5.3), equation (5.5) defines the sub-spectrum gk() as:

(5.6) ]2)(1/[2),(r)()( 22

021

200inhom0 kkkk TTTGdg

For a mixture of components with mono-exponential relaxation and well-separated

spectral frequencies and relaxation rates it should be relatively easy to extract the

intrinsic properties from a series of Hahn-Echo spectra. In real cases, however,

components may strongly overlap in both the spectral and Laplace domain. To be able

to distinguish between the different components of the sample a data analysis approach

based on multivariate statistics has to be used. This approach uses a “Multivariate

Analysis” to estimate the line shapes of the overlapping proton resonance. There are

several methods to fit the exponential decay model, which have been used with different

degrees of success, the one used here, is based on the so called LINOL method

describes in more details in Appendix 1 & 2 (Linear-NOn-Linear fitting) [18].

5.2.5. MDSC experiments

It was shown that modulated DSC (MDSC) allows for deconvolution of signals from

amorphous and ordered phases. The method has a greater resolution then non-modulated

DSC. The amorphous phase gives rise to a reversing heat flow, which is readily

converted to specific heat capacity (Cp) and its derivative, which highlights the glass

transition temperatures.

The Tg’s and the corresponding heat capacity of the polyester network were measured

via (MDSC) using TA Q1000 DSC, equipped with an auto-sampler and a refrigerated

cooling system (RCS). The temperature calibration was obtained using Indium. The

sample was prepared in an aluminium pan (between 10 and 15 mg). An oscillating heat

flow signal with a period of 60 seconds and amplitude of 1 oC and an underlying heating

rate of 1 oC/min was used.

Chapter 5

90

5.2.6. DMTA experiments

The glass temperature and the storage modulus were measured with DMTA using a

multi-frequency sweep. Test bar sample of specific dimensions were cut out of the cured

film. Thickness of the cured bar was measured with a calibrated measuring device. The

measurements of storage modulus, loss modulus and tangent delta were carried out at a

frequency of 1 to 100 Hz. Experiments were started at 10 C up to 200 C with a ramp

speed of 5 C /min.

5.2.7. Dielectric experiments (DE)

The dielectric measurements were done using the Novocontrol alpha analyser.

Two gold-plated electrodes with standard configuration were used. The sample

thickness was set about 2 mm, and a layer of gold was applied to the material by

sputtering. The measurements were performed within a broad frequency range (0,125

Hz up to 42 MHz). Experiments were started at 10 C up to 200 C with a ramp speed

of 5 C /min.


5.3.1. Network heterogeneity and temperature effect on T2 measurement

5.3.1.1. The experimental overall lineshapes and Hahn-Echo decays (both as 1D

data sets) as a function of temperature

Figures 5.4.a and 5.4.b show the experimental overall line shapes, and Hahn-Echo

decays (both as 1D data-set) as a function of temperature demonstrating the problem of

solving complex polymer dynamics in an unambiguous way. Figure 5.4.a depicts the

influence of temperature on the NMR line shape. Passing the glass transition

temperature (Tg = 80 C), the mobility in the polymer network increases. It is shown

that at higher temperatures (above Tg) more degree of freedom is observed; the motions

are rapid enough and dipole interactions start to average out leading to a narrowing of

the NMR lines. In addition, chemical resolution is improved. For this polymeric system,

Chapter 5

91

the 1H NMR transverse relaxation decay results are presented in Figure 5.4.b. These fits

were obtained with a two exponential decay model, suggesting two types of mobility in

the sample. Fast decays are recorded at low temperature (300 K), while at higher

temperatures (409 K-506 K) slower decays are observed. At higher temperatures (above

Tg) a two exponential decay model seems not to be sufficient anymore.

Figure 5.4.a. (left) NMR line recorded at 2= 16 s ( corresponding to the first point of the exponential decay). Figure 5.4.b. (right) Hahn-echo decays as a function of temperature .

Generally, in chemically cross-link polymeric network, the strong static proton dipolar

interaction makes the spin-spin relaxation process dominant in the relaxation. However,

if the experimental temperature is well above the Tg, the dipole-dipole interactions

become weaker and the relaxation will be dominated by local conformational motions,

narrowing the resonance lines. This will be reflected by much higher relaxation time

values. The extracted values of relaxation times T2 are presented in Tables 5.

It has been already mentioned earlier that the decay of the transverse magnetization, in a

high cross-link density network but also in a rigid matrix (e.g. crystalline phase), is

generally described by a Gaussian function at very short Hahn-Echo times (10 to 30

microseconds), especially at temperatures below the Tg. For the experimental data

below the Tg additional fitting experiments were performed and the data were best fitted

with a three component GGL- model (Figure 5.5).

Chapter 5

92

Table 5.1. T2 relaxation times obtained with 2 exponential decay fit for the cross-linked polyester resins recorded at 500MHz.

T(K) 2K T2s (μs) T2l (μs)

298 9 64

362 12 101

393 26 270

441 51 563

464 84 910

506 109 1083

Table 5.2. T2 relaxation times obtained with three exponential decay fit for the cross-linked polyester resin recorded at 500MHz (2D fit analyses).

T(K) 2K T2s (μs) T2m (μs) T2l (ms)

298 9 64

362 11 65 0.7

393 19 112 0.6

441 33 198 1.8

464 48 284 4

506 62 338 7

Table 5.3. T2 relaxation times obtained with GGL fit for the cross-linked polyester resin recorded at 500MHz (1D fit analyses).

T(K) 2K T2s (μs) T2m (μs) T2l (μs)

298 10 23 155

362 10.7 31.9 140

393 10.2 39 276

It is generally admitted, that for highly cross-linked polymeric networks below the glass

transition temperature (Tg), the broad component should depict a T2s relaxation time in

the order of s. In the present case (Tables 5), the T2s relaxation data show T2 relaxation

times for the broad component varying from 10 to about 100 s as a function of

increased temperature. The increase of the short relaxation time value (T2s) above Tg is

an argument for the absence of crystallinity as well as in favor of network chains

heterogeneity.

Chapter 5

93

Figure 5.5. Three component LLL, GLL and GGL fittings with GGL the most optimal fit. This results for the sol fraction 1) T2l= 276 s with 43,7% fraction estimste for the dangling chains ends 2) T2i=39 s with 27 % estimated for the inter cross-link chains 3) 10.2 μs with 29.4% (at 393 K calibrated temperature).

For all temperatures above the glass transition, the best fit was found using three

component decays following a LLL-best fit (Table 5.2), suggesting that the system at

higher temperature is best described by three types of 1H mobility (Figure 5.2); (1)

protons that are part of the rigid cross-link chain segments which show a fast relaxation

and a T2s in the order of tens of microseconds; (2) protons that are part of network

defects, such as dangling chain ends T2i and (3) protons from sol fractions with groups

that depict a much longer relaxation with a T2l-value of a few hundreds of microseconds

to the ms. This multi-relaxation behavior confirms network heterogeneity. In addition to

NMR, Dynamic Mechanical Thermal Analysis (DMTA), Dielectric Spectroscopy (DES)

and Modulated Differential Scanning Calorimetry (MDSC) experiments have been

performed. In DMTA results, as seen in Figure 5.6, two types of relaxation can be

observed. In terms of mechanical behavior, transitions occur in such polymer materials

when the energy thresholds for certain molecular motions are exceeded. This increased

Chapter 5

94

molecular mobility, caused by the increase in temperature, leads to an overall softening

of the material. This process usually manifests itself as a drop in the storage modulus

and a peak in the loss modulus. In the polyester network, two peaks are observed; one

peak generally assigned to long range chain molecular motions (so called relaxation,

associated with the glass transition), and a secondary relaxation also known as local

mode relaxation, or relaxation, due to small-scale motions (side chain motions and/or

rotations of terminal groups).

Figure 5.6. DMTA loss modulus (multi-frequency sweep) recorded for the cross-linked polyester.

The dielectric spectroscopy (DES) data presented in Figure 5.7 show a very broad Tg

regime between 50 and 100 C. Beyond that temperature range, the dielectric constant

shows an additional relaxation mechanism at the lowest frequencies. Often, this kind of

relaxation mechanism is indicative of certain heterogeneity of the material (presence of

at least two phases with different electrical conductivity).

Chapter 5

95

Additional modulated MDSC experiments have also been performed and two-glass

transitions temperatures were depicted (Tg = 60 C; Tg = 89 C) as seen in Figure 5.8.

All these complementary information and techniques probe and confirm the network

chains heterogeneity.

Figure 5.7. Dielectric measurements. Dielectric constant versus temperature at different frequencies (0.125 Hz, 0.5 Hz, 1 Hz, 256 Hz, 4 KHz, 33 KHz, 2 MHz, 41 MHz; ranging from low frequencies to high frequencies, following the arrow).

In solid-state NMR relaxation experiments, models for the curve shape are still under

debate. In general, physical interpretation of these results is very complex, especially

when fitting two or more relaxation components with weakly separated characteristic

times to a monotonous decay. One way to overcome these ambiguities and to be able to

understand a little further chain dynamics in polyester networks is to combine spectral

and relaxation resolution by correlating line shape information and Hahn-Echo T2H-

relaxation experiments (see Appendix 1 & 2 for more detail).

Chapter 5

96

Figure 5.8. MDSC heating run of the polyester network (top, reversing heat flow vs. T(oC); endo up). Bottom, Reversing Cp vs. T( oC).

Chapter 5

97

5.3.1.2. Combined spectral and relaxation resolution in the 2D-data sets

In Figures 5.9, different experimental Hahn-Echo spectra just above the glass transition

temperature, are represented in function of (half Hahn-Echo time) at respectively 16

μs, 210 μs and 2 ms. An additional line shape is represented and obtained by a

subjective subtraction of the first two Hahn-Echo spectra, obtained at 16 μs and 210 μs.

Some adjustment of the scaling parameter of the two spectra has been done in such a

way that smoothing of the line was obtained. From this two-dimensional approach (2D),

different information could be gained.

First of all, the spectra recorded at 16 μs, 210 μs, and 2 ms show variations in the line

shape, which is already narrowing at 200 μs. This seems to be in agreement with the

relaxation data obtained in Table 5.1, suggesting a T2 relaxation for the broad

component around 100 μs at high temperature. Furthermore, one could observe that on

the spectrum recorded at 2 ms three peaks could easily be assigned. The first two

resonating around respectively 1.3 ppm (-CH3 from PG and CH2- and CH- from PS) and

3.5 ppm (-0CH2 and -0CH from EG and PG) could be assigned to aliphatic protons and

the one at 7 ppm to aromatics. This seems to indicate that the 1H of the polyester

backbone chains as well as the polystyrene chains show the same type of mobility. The

third peak corresponding to the aromatic protons could be attributed both to the

aromatic protons present in the polyester backbone and to the protons present in the

polystyrene cross-link junctions. Also the relative peak intensities of the broad

component at 16 s (at such temperature) closely resemble the envelope of the three-

peak pattern of the mobile component at 2 ms.

This observation could already suggest that the chemical composition of the rigid and

the mobile phase should be relatively close. This can be confirmed by the calculated

proton mol fractions of the aromatic, glycolic and aliphatic protons based on the initial

sample composition (see Appendix 3 of the thesis). The respective calculated values of

33.7, 40 and 26.4 %, correlate quite well with the estimated peak area of the spectrum at

2 ms in Figure 5.9.a recorded at 385 K.

Chapter 5

98

Figure 5.9.a. (left) NMR lines recorded at 2 = 16s, 0.2ms, 2ms (corresponding to the first point of the exponential decay). Figure 5.9.b. (right) Hahn Echo decay recorded at 385 K corresponding to the point selected on the line (at various intensities).

Figure 5.9.b, presents the spectrally resolved T2-behavior. One could see that the

aromatic, glycolic and aliphatic protons depict a similar slower T2 relaxation compared

to the T2 evolution corresponding to the traces and at the left and right side of the

broad signal of the more rigid phase. This observation confirms that the chemical

composition of the mobile and the immobile components in the polymeric system is

closely the same.

5.3.1.3. Multivariate algorithm ("model free", without lineshape assumptions)

Applying the Multivariate Algorithm approach described in the Appendix 1 & 2 of this

thesis, so called "model free" approach, (without line shape assumptions) should help to

probe more objective information.

Figure 5.10.a, shows modeled peaks (at 407 K) obtained for the broad component

SA(W) and the mobile one SB(W). The line shape of the broad component SA(W) seems

to perfectly corresponds to the one in Figure 5.9.a obtained by a subjective subtraction

of the first two Hahn-Echo spectra at respectively 16 μs and 200 μs. Figure 5.10.b shows

the corresponding simulated relaxation decays that assumes that for SB(w), all the

protons (aliphatic, glycolic and aromatics) have the same time decay within the mobile

Chapter 5

99

component. This is in good agreement with what is observed in Figure 5.9.b. The two

approaches (1D and 2D) lead to the same results.

Figure 5.10.a. (left) modelled peaks obtained for the broad and the mobile component obtained at temperature 407K (left). Figure 5.10.b. (right) T2 simulated plots obtained for the broad and the mobile component t(right).

There is an easy way to mathematically check the correspondence between the broad

and narrow line shape by artificially broadening the narrow component. Ideally one

could use the difference in T2 values, because the homogeneous line width is inversely

related to the T2 value. The precise procedure would be to transform the narrow-

component line shape back into the time domain (inverse FT), then multiply with an

exponential decay with decay rate (1/T2s - 1/T2l), and finally transform back into the

frequency domain (FT). Performing such mathematical transformation confirms that the

line shape of the broad component (at such temperature) corresponds to the envelope of

the three-peak pattern of the mobile components.

5.3.1.4. Relaxation in the swollen network

The influence of temperature on T2 relaxation measurements and molecular mobility of

non-swollen cross-linked polyesters resins samples are shown in Tables 5.1, 5.2 & 5.3

and in Figure 5.11.a. The relaxation times of the broad and the narrow components

increase with temperature. It is well known that when temperature increases, the

molecular mobility of network chains is gradually decoupled from that of network

defects. This effect is strengthened at temperatures above the glass transition and results

Chapter 5

100

in a major distinction in the relaxation behavior. The slight increase of the short

component is often attributed to disentanglements of physical cross-links but also to the

increase in frequency of ‘large spatial scale’ motions. The slight decrease in the

strength of inter-chain proton dipole-dipole interactions has also an influence. At some

point, T2s should reach a maximum value (Figure 5.11.a).

Figure 5.11.a. T2 vs. temperature recorded at 500 MHz; correspond to the T2s of the

rigid (short) component, correspond to the T2l of the mobile (long) component.

In general, this is better observed in swollen samples as can be seen in Table 5.4 and

Figure 5.11.b. The physical constraints imposed by the cross-links lead to an

independence of T2 at high temperature (well above Tg) and a plateau is observed (T2pl).

A maximum value of T2s is reached. This value is in the order of magnitude of what is

generally observed for very rigid non-swollen networks around 50 s (Table 5.2).

One could precise that this maximum T2s value is generally used to estimate the cross-

link density of a network by relating the T2pl to the number of statistical segments

between network junctions (equation 5.7). The number of statistical segment between

cross-links can then be used to estimate the average molar mass between cross-links Mc

(equation 5.8) as derived by Gotlib et al. [10]:

Chapter 5

101

Figure 5.11.b. T2s relaxation times for the rigid component at 500 MHz for cross-linked polyester resin swollen in CDCl3.

Z = (T2pl) / [k (T2

rl) ] (5.7 )

Mc = Z c M ru / N (5.8)

where k is the theoretical coefficient, which depends on the angle between segment axis

and the inter-nuclear vector for the nearest nuclear spins. If the spin-spin connecting

vector and the segment direction are perpendicular to each other, then k= 3/10; if they

are parallel k=3/5 [18]. T2rl is admitted to be the T2-value below the glass transition

temperature of the rigid lattice whereas T2pl is the T2- plateau value well above the glass

transition temperature. The maximum of error in the determination of the cross-linked

density by this theory is generally estimated at 20-35% [10]. The biggest assumption is

made in k, and c (representing the number of backbone bonds in one KUHN segment).

It is admitted to be equal to 2 for freely rotating chain and around 10 for polystyrene in

cyclohexane.

Chapter 5

102

In the present unsaturated polyester network, the average molecular mass between

polystyrene (PS) cross-links was estimated for (Mru / N = 52), k= 3/10 and c = 10. The

obtained average molecular mass between PS cross-links (Mc ≂ 482 g/mol) corresponds

to an estimated length of 4.6 styrene monomer units between cross-links.

The mean length between the cross-links in the polyester/PS network can also be

estimated starting from the monomer fractions (see simulation in Appendix 3). From

the calculated proton mol fractions, a mean length of 5.28 styrene monomer units is

estimated for the PS cross-links, supposing a complete reaction of the MA-units. The

estimated mean length between the cross-points in the polyester backbone is 5.79

monomer units. This is rather short and close to the value of 5.28 styrene units for the

PS cross-links. From these short cross-link lengths we do not expect crystallization to

occur. The values obtained through the simulation in Appendix 3 and the one calculated

from Gotlib’s approximations are very close.

In the non-swollen sample, the increase of the long component T2l is much more

pronounced and should keep on increasing with temperature, as it is generally originated

from the relaxation of network defects such as dangling chains end, free chains that are

not attached to the network, chain loops and also to the relaxation of side chain groups

such as -Me groups or (-Ǿ) rings.

Table 5.4 presents T2l from 220 K and up for the swollen sample. A constant value of

about 5 ms is depicted from 300 K to 400 K. Surprisingly, higher values are observed at

low temperature (below 300 K). This effect is generally observed when non-

homogeneous swelling of the network at low temperature occurs or when specific

solvent interactions with the polymer chains are present. A reduction of molecular

mobility is in this case probably a consequence of solvent induced organization [19].

5.3.2 Quantification of rigid and mobile components

Since the integral of the NMR peak is directly proportional to the number of protons in

the material, the relative overall quantities of the network attributed to the rigid

component and the more mobile ones can be calculated in a straightforward way.

Chapter 5

103

Table 5.4. T2 relaxation times obtained with three exponential decay fit for the swollen cross-link polyester resin recorded at 500 MHz.

T(K) 2K T2s (μs) T2i (μs) T2l (ms)

223 11 105 17

253 16 156 12

270 20 192 9

298 26 204 2

320 34 281 5.6

351 42 316 4

385 34 280 5

It is interesting to look at the overall fraction of components in function of temperature.

Tables 5.5 and 5.6 depict the changes in relative fraction of components as a function of

temperature obtained from experiments probed at 500 MHz in the original sample (non

swollen) and in the swollen sample. The main component in the system is undoubtedly

the rigid one representing at room temperature about (80%) of the total system. As

expected from such polymeric system, at high temperature, the amount of rigid material

is decreasing (Table 5.5). Consequently the amount of mobile component in the network

increases. The fraction of component in the non-swollen system seems to be

temperature dependent. This dependence would be surprising if we considered the

system as an ‘ideal’ rigid chemically cross-linked network described by only two

components. This deviation from the ideal network and the temperature dependence on

the fractions of components suggest that the chemical network seems to hold not only

rigid cross-links segments but also physical entanglements.

Also, one can see that the two-component model is too simple to describe this type of

material. It assumes that all the protons have the same time decay and the same NMR

peak shape within the same component (mobile or rigid), which is just not realistic.

With such an approach, small relaxation differences between different protons

belonging to the same component are not considered. This assumption does not

constitute a big drawback at lower temperatures, where the low mobility of the system

yields a broad NMR peak for each component. In this case, the peak shape and the

relaxation decay are dominated by residual anisotropy probed in a sub-millisecond time

and a simple two-component model is enough to describe the system.

Chapter 5

104

Table 5.5. Fractions of components obtained with two (298 K ) and three exponential decay fit for the non-swollen cross-link polyester resin recorded at 500 MHz.

T(K) 2K Rigid fraction

(%) Intermediate fraction

(%) Mobile fraction

(%)

298 91 9

362 78 21 1

393 64 28 8

441 61 33 6

464 57 37 5

506 54 42 4

Table 5.6. Fractions of components obtained with three exponential decay fit for the swollen cross-link polyester resin recorded at 500 MHz.

T(K) 2K Rigid fraction

(%) Intermediate fraction

(%) Mobile fraction

(%)

223 80 12 8

253 72 20 8

298 60 30 10

320 61 33 6

351 59 35 6

385 61 32 7

At higher temperatures however a different panorama is observed. When the

temperature increases well above Tg, the main source of local field seems to be residual

susceptibility and chemical shift rather then residual dipolar field. In that case, mobility

keeps on increasing with temperature resulting in a drastic reduction of the NMR

linewidth. Therefore, the differences among the different protons present within the

same component become more evident, since the peak shape is becoming influenced not

only by the mobility of the system but also by the chemical shift. At 450 K and 506 K,

the intensity of the Me-signal of the propylene glycol (PG) has an increased value at 2=

4 ms, due to its fast conformational rotation, compared to e.g. the aromatic signal of PS

and OPA (shorter T2). This makes the differences in relaxation decay of the different

protons of the same phase more significant. In other words, above a certain temperature

(generally the glass transition temperature), the chemical shift becomes more relevant;

Chapter 5

105

(b)(a)

(Hz)

Inte

nsi

ty (

a.u

.)

(c)

(b)(a)

(Hz)

Inte

nsi

ty (

a.u

.)

(c)

Figure 5.12. Error computed as the difference between experimental and modeled data of T2 experiments recorded at 500 MHz (resp. at 297 K (a), at 393 K (b) and 463 K (c)).

(d) (e)

(f)

(d) (e)

(f)

Figure 5.13. Error computed as the difference between experimental and modeled data of T2 experiments recorded at 200 MHz (resp. at 297 K (d), at 393 K (e) and 463 K (f))

Chapter 5

106

the overlap between the NMR lines is reduced. The failure of a too simple (two-

components) model becomes then more evident. This certainly biases the solution: the

higher the temperature, the higher the error as can be observed in Figures 5.12 and 5.13.

In Figure 5.12 and Figure 5.13, the error computed as the difference between

experimental and modelled data of T2 experiments recorded at 500 MHZ and 200 MHz

is represented. One could see that the lower the field, the smaller is the error, also the

higher the temperature, the more systematic becomes the error.

The differences in fraction of the components could have another origin as well, related

to changes in the composition of the material itself. Effectively, after performing the

measurements up to 500 K, cooled down the sample to 297 K and measured again, a

change in the shape of the relaxation decay was observed and then different T2s and T2l

values were obtained (see Figure 5.14). It seems that the system becomes slightly more

rigid. This may suggests that some changes occurred in the material.

Figure 5.14. Post-curing effect at room temperature (+ correspond to the decay recorded at 500 MHz at room temperature prior to temperature treatment. correspond to the decay recorded at room temperature after temperature treatment).

Chapter 5

107

This could be attributed to different reasons such as, ageing of the system, post-curing

of the system, or, rearrangements occurring in term of phase separation. Effectively,

above the Tg the mobility of the system increases and the system can vitrify some more

on that same time scale.

5.4. Conclusion

The 1H spin-spin NMR relaxation data show a strong heterogeneous dynamic behavior.

The data obtained via other characterization techniques such as DMTA, DES and

MDSC confirmed this result. A combination of gaussian and exponential decays were

required to fit the experimental data. A simple dual exponential decay fit generally used

in affine homogeneous networks with low cross-link density was just not realistic.

Three types of 1H mobility were depicted above Tg, and combining high spectral

resolution NMR experiments to 1H-NMR relaxation experiments (2D data analysis)

helped for a better understanding of the network chain dynamics.

Finally, the average molecular weight between cross-links was estimated and found to

be relatively low (only a few polystyrene monomer units). However, the use of models

to estimate the average molecular weight between cross-links implies to take the

quantitative aspect of that work with some care.

Chapter 5

108

5.5. References

[1] Litvinov, V. M.; De Prajna, P. Spectroscopy of Rubbery Materials, RAPRA Technology 2002.

[2] O’Donnell, J. H.; Whittaker, A. K. Polymer 1992, 33, 62.




[6] Graf, R.; Demco, D. E.; Hafne, S. H.; Spiess, W. Solid State Nuclear Magnetic Resonance 1998, 12, 139.

[7] Heuert, U.; Knorgen, M.; Menge, H.; Scheler, G.; Schneider, H. Polymer Bulletin 1996, 37, 489.

[8] Kuhn, W.; Barth, P.; Denner, P.; Muller, R. Solid State Nuclear Magnetic Resonance 1996, 6, 295.

[9] Fisher, E.; Grindberg, F.; Kimmich, R. Journal of Chem. Phys. 1998, 109, 846.

[10] Gotlib, Y.; Lifshitz, M. I.; Shevelen, V. A.; Lishanskij, I. S.; Balanina, I. V. Polym. Sci. 1976, 491.

[11] Simon, G.; Schneider, H.; Hausler, K. G. Prog. Colloid Polym. Sci. 1988, 78, 30.

[12] Simon, G.; Schneider, H. Polym. Bull. 1989, 21, 475.

[13] Fedotov, V. D.; Tshernov, V. M.; Wolfson, S. I. Polym. Sci. 1978, 260.

[14] Brereton, M. G. Macromolecules 1990, 23, 1119.


[16] Litvinov V. M.; Dias, A. A. Macromolecules 1999, 32, 3624.

[17] Simon, G.; Baumann, K.; Gronski, W. Macromolecules 1992, 25, 3624.

[18] Ruytinx, B.; Berghmans, H.; Adriaensens, P.; Gelan, J. Macromolecules 2001, 3, 552.

[19] Menge, M.; Hotopf, S.; Heuert, U.; Schneider, H. Polymer 2000, 41, 3019.

[20] Orza, R. A.; Magusin, P. C. M. M.; Litvinov, V. M.; van Duin, M.; Michels, M. A. J. Macro. Symp. 2007, 40, 899.

Chapter 6

partly reproduced from: Ziari, M.; Asselen Van, O. L. J.; Jansen, M. A. G.; Goossens, J. G. P.; Schoenmakers P. J. Macromolecular Symposia 2008, 265, 290.

109

Chapter 6

A FTIR study on the solid-state copolymerization of bis(2-

hydroxyethyl)terephthalate and poly(butylene terephthalate) and the

resulting copolymers

Summary

The aim of this work was to study the solid-state copolymerization (SSP) of bis(2-

hydroxyethyl)terephthalate (BHET) with poly(butylene terephthalate) (PBT) by FTIR

spectroscopy. The development of the chemical microstructure during the SSP-reaction

was examined as a function of the BHET content, showing the different regimes.

Depending on the ratio of PBT to BHET, a competition exists between annealing of

PBT, incorporation of BHET in amorphous PBT and BHET homopolymerization.

The thermal behaviour of the resulting copolymers was also investigated using infra-red

dynamic spectra. For low BHET-concentrations, only crystallization of PBT-sequences

was observed, while for high BHET-concentrations, only crystallization of PET-

sequences was detectable with a cross-over behaviour for intermediate concentrations.

Chapter 6

110

6.1. Introduction

Poly(butylene terephthalate) (PBT) and poly(ethylene terephthalate) (PET) are two

semi-crystalline polymers that are used in many engineering applications. The main

advantage of PBT is its high crystallization rate, making it suitable for injection molding

applications with short molding cycle times [1]. PET has a considerably lower

crystallization rate and is therefore mainly used for fiber applications and packaging [2].

The glass transition temperature of PBT is rather low compared to PET (Tg 45 °C for

PBT and Tg 80 °C for PET). To obtain a Tg higher than 45 °C, PBT and PET can be

reactively blended in the melt. The concomitant ester-interchange reactions occurring in

the melt first result in the formation of block copolymers, but as the reaction proceeds,

random PBT-PET copolymers are obtained [3-5]. These random PBT-PET copolymers

have a Tg in between that of the PBT and PET homopolymers. However, the shorter and

more irregular homopolymer sequences of these random copolymers consequently lead

to a lower melting temperature, crystallization rate and crystallinity with respect to pure

PBT. It is therefore desirable to synthesize PBT-PET copolymers having a Tg higher

than the PBT homopolymer, but with a crystallization behavior comparable to that of

PBT. Therefore, a copolymerization method should be used that enables to retain large

crystallizable homopolymer PBT blocks with a non-random chemical microstructure.

In previous work [6], PBT-PET copolymers were made by copolymerization in the melt

(MP) and the solid state (SSP) of bis(2-hydroxyethyl)terephthalate (BHET) with PBT.

The chemical microstructure of the synthesized PBT-PET copolymers was examined by 13C-NMR spectroscopy sequence distribution analysis. As expected, the chemical

microstructure of the PBT-PET copolymers obtained by MP was fully random.

However, when BHET was incorporated into PBT via SSP, a non-random blocky

chemical microstructure was obtained. When the fraction of BHET used for

incorporation in PBT was increased, the blocky character of the chemical microstructure

became more pronounced, suggesting that BHET may react by self-condensation to

form homopolymer PET blocks [7, 8]. The 13C-NMR sequence distribution analysis also

showed that transesterification reactions occurred between BHET monomer and PBT. It

was not clear whether the formed homopolymer PET blocks are present as a separate

phase or whether these blocks reside in the amorphous phase of PBT.

Chapter 6

111

In this chapter, the incorporation of BHET into PBT via SSP is studied in more detail.

The kinetics of the incorporation via SSP is studied under isothermal conditions by

using FTIR spectroscopy. During the SSP process, a competition between annealing and

dissolution of the PBT-crystals, incorporation of BHET in the amorphous PBT and

BHET homopolymerization exists, of which the relative rates depend on the ratio of

PBT/BHET. In a previous study, the miscibility of the BHET with PBT and the thermal

properties of the resultant BDxEGy-copolymers obtained via SSP were investigated by

using DSC [9]. It was observed that BHET is partially miscible in the amorphous PBT,

depending on the PBT/BHET ratio. The crystallization behaviour of the resultant

BDxEGy copolymers, where BDx represents the initial mol% PBT and EGy the mol% of

hydroxyl end groups per mole BHET, was studied by DSC. It was observed that when

more BHET is incorporated the onset of crystallization shifts to lower temperatures,

while the crystallization exotherm becomes broader. However, the (BD30EG70)ssp

copolymer displayed a much sharper crystallization exotherm with a slightly higher

onset. A similar observation was made by Misra et al. [10]. However, DSC lacks the

possibility to investigate the crystallization behaviour of the individual components

when the crystallization exotherms display a large overlap. In this respect, temperature-

dependent FTIR spectroscopy is much more discriminating by following absorption

bands, which can either be assigned to the crystalline or amorphous phase of PBT or

PET.

The SSP kinetics were investigated under isothermal conditions using two

compositions. In that way, the development of the chemical microstructure during the

SSP reaction could be examined together with the miscibility of BHET and PBT.

Temperature-dependent experiments were performed on three different BDxEGy

copolymers and compared to PBT using the dynamic infra-red spectra obtained through

the 2D-IR correlation algorithm developed by Noda et al. [11, 12].

6.2. Experimental

The BDEG copolymers were obtained by incorporation of BHET into PBT by using

SSP. At first, the kinetics of the SSP reactions was investigated with attenuated total

reflection (ATR)-FTIR spectroscopy. A BioRad FTS3000 spectrometer equipped with a

Chapter 6

112

MCT detector was used to record the spectra. The samples were pressed onto the

diamond crystal of a Speak Golden Gate ATR accessory. Next, the sample was heated

to 120 C and the initial spectrum was recorded. Upon starting the experiment, the

sample was heated to 180 C using a heating rate of 30 C/min. Spectra were recorded

during the reaction with a time interval of 60 seconds using a spectral resolution of 4

cm-1, co-adding 62 scans. At 180 C, the sample consists of a semi-crystalline fraction

of PBT and a liquid fraction of BHET. During the initial stage of the SSP-reaction, the

fraction of the liquid BHET in contact with the crystal increases, while the fraction of

PBT decreases. It becomes then very difficult to monitor the reaction quantitatively.

Therefore, it was decided to perform all experiments in the transmission mode. Then, a

more or less constant cross-section of the sample is monitored. These experiments were

performed using a BioRad UMA500 IR microscope coupled to a BioRad FTS6000

spectrometer. A Linkam TMS600 hot stage was used for temperature control. To obtain

a thin sample, the powder samples were cold pressed by a diamond anvil cell. Next, the

sample was placed in between two zinc selenide (ZnSe)-windows and transferred to the

Linkam hot stage. Spectra were recorded with a resolution of 2 cm-1 co-adding 250

scans. The crystallization behaviour of three BDEG copolymers were also followed by

using the same set-up as described above with a cooling rate of 10 °C/min. The samples,

however, were not covered with a second ZnSe window. For these experiments, spectra

were recorded using a resolution of 2 cm-1 co-adding 100 scans. The 2D-Pocha

software, developed at the Kwansei Gakuin University (Japan), was used for the data

treatment. The spectrum at 180 °C was taken as the mean spectrum for the dynamic

spectra calculations.

6.3. Theory

Two-dimensional infra-red (2D-IR) spectroscopy is an analytical technique based on

time-resolved detection of IR absorption to look at molecular interactions. In 2D-IR, a

spectrum is obtained as a function of two independent wave numbers. The general

experimental approach used in 2D correlation is based on the detection of dynamic

variations of spectroscopic signals induced by an external perturbation.

Chapter 6

113

The effect of such induced-perturbations leads to changes of the local molecular

environment and will be manifested by time-dependent fluctuations of various spectra,

yielding so-called dynamic-spectra. Mathematical manipulation of these spectra will

give a useful set of 2D-IR spectra.

There are different types of external perturbations that could be used; the type of

information that can be obtained from the dynamic spectra will depend on the type of

perturbation. A more detailed information on the formalism can be found in NODA

feature article [11]. This approach is used to lift many analytical ambiguities. For

instance, it helps to simplify and understand complex spectra containing many

overlapped peaks. It can also be used to enhance the spectral resolution by spreading the

absorption peaks into a second dimension as much as it could help to identify different

intramolecular interactions through selective correlation of IR absorption bands. In this

chapter, thermal excitation has been used. Temperature-dependent experiments were

performed on three different BDxEGy copolymers and compared to PBT. Hence, the

microstructures of the copolymers could be investigated.

6.4.. Results and discussion

6.4.1. Assignment of the absorption bands of the pure components

Table 6.1 shows the assignment of most relevant absorption bands of PBT, PET and

BHET. For crystalline PBT, a band at 1454 cm-1 (CH2 bending) is observed, that for

amorphous PBT shifts to 1468 cm-1 together with absorption bands at 1320 cm-1 (ring

ester in plane) and 1389 cm-1 (CH2 wagging). Crystalline PET shows characteristic

absorption bands at 1471 cm-1 (CH2 bending), at 1342 cm-1 (CH2 wagging) and at 990

cm-1, of which the latter band is only observed if chain folding occurs. Typical bands of

amorphous PET can be observed at 1371 cm-1 (CH2 wagging), 1458 cm-1 (CH2 bending).

For BHET, a lot of absorption bands overlap with PBT or PET. To follow the

copolymerization, the absorption band at 3550 cm-1 (OH stretching) can be used.

Chapter 6

114

Table 6.1. Assignment of the absorption bands of the pure components [13].

PBT ( cm-1)

PET ( cm-1)

BHET ( cm-1)

amorphous crystalline amorphous crystalline

OH stretching 3550

CH2 bending 1468 1458/1454 1458/1440 1471 1453

Ring-in-plane-def 1408 1409 1409 1409 1407

Ring CH in-plane def 1380

CH2 wagging 1389 1389 1371 1342 1372

Ring CCH 1320 1315

Chain folding 990

6.4.2. Kinetics of the SSP reaction studied by FTIR spectroscopy

Since the spectra of PBT, BHET and PET show overlapping absorption bands, it was

decided to focus on the four following absorption bands: the crystalline PET absorption

band at 1342 cm-1, the absorption band at 1371 cm-1 of amorphous PET or BHET, the

crystalline PBT band at 1454 cm-1 and the OH-stretch vibration band of BHET at 3550

cm-1 at starting time (t = 0)..

The SSP-kinetics at 180 C was followed for the (BD70EG30)feed and (BD30EG70)feed

systems. Figures 6.1a and 6.1b show the individual spectra before and after the SSP-

reactions, while the kinetic results are shown in Figures 6.2a and 6.2b, respectively. The

disappearance of the absorption band at 3550 cm-1 shows that in both cases the BHET is

converted. For the (BD70EG30)feed system (Figure 6.2a), the intensity decreases gradually

and the behaviour is typical for a series of two irreversible consecutive first-order

reactions. This was also observed for the incorporation of 2,2-bis[4-(2-

hydroxyethoxy)phenyl]-propane (Dianol 220) [14]. The (BD30EG70)feed system (Figure

6.2b) shows a different behaviour. After a similar gradual decrease of the intensity up to

approx. 20 min, the reaction accelerates and finally reaches a plateau level after approx.

50 min. In both figures, the absorption band of the PET CH2 wagging mode in the

crystalline state is also depicted, although the band position changes from at 1340 to

1342 cm-1 on going from (BD70EG30)feed to (BD30EG70)feed. Cole et al. studied the

microstructure of PET homopolymer using reflection and transmission FTIR

Chapter 6

(a)(a)

(b)(b)

Figure 6.1. Spectra before (-) and after (..) the SSP-reaction: (a) (BD70EG30)feed and (b) (BD30EG70)feed.

115

Chapter 6

116

spectroscopy and showed that the ethylene glycol segments for PET in the crystalline

phase are always in the mutual trans orientation (T), while in the amorphous phase a

mutual gauche orientation predominantly exists (G) [15]. They also observed that

different conformation states can also exist for the C-O bond of the glycol group, which

can likewise exist in a trans (t) or gauche (g) conformation as well as the terephthalic

acid segment. If the latter is in a non-trans conformation and the ethylene glycol

segment is in a trans (T) conformation, a “rigid” amorphous band will be observed at

1340-1338 cm-1. X-ray diffraction measurements showed that all three groups adopt the

trans conformation (T, t, Tb) in the crystalline phase. Then, an absorption band at 1342

cm-1 of the glycol C-C unit will appear. The change in peak position is a strong

indication that for the (BD70EG30)feed system the BHET is incorporated in the PBT

chain, while for the (BD30EG70)feed system separate PET crystals can be formed. The

difference in behaviour is also found for the 1371 cm-1 band. For the (BD30EG70)feed

system, the same intensity profile is found as for the OH-stretching vibration band of

BHET (3550 cm-1), but (BD70EG30)feed system remains at the same absorption level for

the first 30 min and then starts to decrease up to approx. 40 min followed by a plateau.

This might indicate that during the first 30 min, the decrease of the intensity due to the

conversion of BHET is compensated by the formation of amorphous PET, which also

has an absorption band at that wavenumber.

For the (BD30EG70)feed system, the crystalline PBT band at 1454 cm-1 decreases,

probably because of the decrease of the BHET that also has an absorption band at 1454

cm-1, although this decrease may also be attributed to partial dissolution of the PBT-

crystals that may occur because of the high mole fraction of BHET acting as a solvent

for PBT. On the contrary, for the (BD70EG30)feed system, the band at 1454 cm-1

increases. This behaviour can only be explained by the occurrence of annealing and

perfectioning of the PBT-crystals due to the high mobility during the SSP-reaction.

Chapter 6

117

(a)(a)

(b)(b)

Figure 6.2. Relative intensities of the indicated absorption bands of crystalline PET, amorphous PET, PBT and BHET during SSP of: (a) (BD70EG30)feed and (b) (BD30EG70)feed.

Chapter 6

118

6.4.3. Temperature dependent experiments

Changes in the FTIR-spectra were monitored as a function of temperature. Figure 6.3

shows the dynamic spectra of BDEG copolymers with different PBT/BHET ratios

cooled from the melt in a temperature range from 240 to 170 C. For the PBT

homopolymer, positive bands at 1454, 1386 and 1320 cm-1 develop during cooling.

These bands can be assigned to crystalline PBT (CH2 bending and wagging modes).

Also the positive band at 1408 cm-1 can be assigned to amorphous PBT (See Table 1).

In the spectra of the (BD70EG30)ssp system, beside the positive bands of crystalline PBT,

also a positive band is observed at 1338 cm-1, which can be assigned to CH2 wagging of

an ethylene segment (trans conformation) with the terephthalate segment in a non-trans

configuration. This is an indication that ethylene segments are incorporated into the

PBT chain.

In the spectra corresponding to sample (BD50EG50)ssp the positive bands of PBT are

weaker and positive bands at 1471 and 1342 cm-1 corresponding to crystalline PET

appear. Negative bands can be observed at 1315 and 1371 cm-1, which can be assigned

to amorphous PET, as well as a band at 1465 cm-1, which can be assigned to amorphous

PBT. These spectral features clearly indicate that PET crystals are formed. Furthermore,

the crystalline PBT and PET bands appear almost at the same time. The limited

miscibility of BHET with PBT promotes the formation of separate PET-crystals.

However, no visible absorption band at 990 cm-1 related to PET chain folding is found,

suggesting that the crystalline PET chain segments are not long enough.

In the spectra corresponding to the (BD30EG70)ssp system, weak spectral features of

crystalline PBT are present together with strong positive and negative features of PET.

If we have a closer look at the bands at 1320 and 1454 cm-1 and the band at 1342 cm-1, it

can be noticed that PET crystallizes much earlier, indicating that PBT and PET

crystallize independently and are mostly like in a separate phase. This is confirmed by

the presence of an absorption band at 990 cm-1, that can be assigned to chain folding of

PET.

These observations are in line with the previously reported DSC-results. When more

BHET is incorporated, the onset of crystallization shifts to lower temperatures, while the

crystallization exotherm becomes broader, mainly because the PBT crystallization is

Chapter 6

119

disturbed by the presence of short PET sequences in PBT. However, the exotherm of the

(BD30EG70)ssp copolymer, which is much sharper with a slightly higher onset, is due to

crystallization of PET, while at lower temperatures only a small PBT crystalline fraction

is formed. Most likely, because of the limited miscibility of BHET in the amorphous

phase of PBT and the concomitant formation of PET homopolymer by self-

condensation, the large interfacial area leads to a strong nucleation effect.

Figure 6.3. Dynamic spectra as a function of temperature: 210-170 °C with 10 °C steps: (a) PBT, b) BD70EG30, (c) BD50EG50, and (d) BD30EG70.

Chapter 6

120

Figure 6.4 shows the crystalline PBT absorption bands at 1454 cm-1, and the crystalline

PET absorption bands at (1342 cm-1) as function of temperature after cooling the

samples from 240 C to 110 C. One could observe that crystalline PBT bands decrease

with decreasing fraction of PBT in the (BDxEGy) mixtures in accordance to earlier

results.

In addition, both samples (BD30EG70)ssp and (BD50EG50)ssp show a strong intensity for

the PET crystalline absorption bands at 1342 cm-1.

-1

0

1

2

3

4

100 120 140 160 180 200 220 240

Temperature (oC)

Abs

orba

nce

(a.u

.)

PBT, 1458 cm-1

80% PBT, 1454 cm-1

80% PBT, 1339 cm-1

70%PBT, 1454 cm-1

70% PBT, 1339 cm-1

50% PBT, 1454 cm-1

50% PBT, 1342 cm-1

30%PBT, 1458 cm-1

30% PBT, 1342 cm-1

Figure 6.4. Absorbance of crystalline PBT absorption bands at 1454 cm-1 and crystalline PET absorption bands at 1342 cm-1 as function of temperature (cooling from 240 C to 110 C).

From the intensity of the crystalline PBT absorption bands at 1454 cm-1 and after

normalization (using the band at 1410 cm-1 present in PBT and PET crystals), the

crystallinity of PBT in each sample was estimated. These calculations were based on the

crystallinity of pure PBT (54%). The results are reported below in Table 6.2.

Chapter 6

121

Table 6.2. Estimated values (%) of crystallinity in the sample based on pure PBT crystallinity

Samples (BDxEGy)ssp Crystallinity

(%)

Pure PBT 54

(BD80EG20) 35.5

(BD70EG30) 37

(BD50EG50) 20

The crystallinity values are in good agreement with the earlier study by Jansen et al.

using DSC,for the samples with high amount of PBT, where no separate PET crystals

are expected. For the sample (BD50EG50) a big discrepancy is observed with the DSC

results, and shows a much lower crystallinity. One can notice that the crystallinity

determined via DSC is based on the amount of heat needed to break the totality of

crystals order, whereas the crystallinity determined with FTIR spectroscopy is based on

the intensity of the crystalline PBT absorption band. The much lower value obtained via

IR , for the sample (BD50EG50), may be a net indication of the presence of additional

PET crystals in this sample that are not taken into account in this estimation.

6.5. Conclusions

FTIR spectroscopy in transmission can be used to follow the kinetics of BHET-

incorporation in PBT during SSP and to study the microstructure of the resulting BDEG

copolymer. These results show that depending on the ratio of PBT/BHET, a competition

exists between annealing of PBT, incorporation of BHET in amorphous PBT and BHET

homopolymerization. For the (BD70EG30)feed system, BHET is incorporated in PBT

forming a non-random copolymer, while for high BHET concentrations a separate PET

phase is found.

The dynamic spectra measured during cooling from the melt clearly showed that for low

and intermediate BHET-concentrations, crystallization of PBT is responsible for the

crystallization exotherm, as observed by DSC, while for high BHET-concentrations the

DSC crystallization exotherm is caused by crystallization of PET.

Chapter 6

122

Furthermore, the crystallinity of PBT in each sample was estimated. For samples

containing high concentration of PBT, the results for the crystallinity were found to be

in good agreement with the values measured by DSC. Nevertheless, for the sample with

lower amount of PBT, (BD50EG50)ssp, a significant discrepancy among the results was

found between the two techniques. The much lower crystallinity value obtained via

FTIR for this sample indicates that there are additional crystals of PET. Indeed, the

crystallinity determined with FTIR spectroscopy takes only into account the crystalline

PBT absorption band.

Chapter 6

123

6.6. References

[1] Radusch, H. J.; Handbook of Thermoplastic Polyesters, S. Fakirov Ed. 2002.

[2] Gupta, V. B.; Bashir, Z. Handbook of Thermoplastic Polyesters, S. Fakirov, Ed. 2002.


[4] Jacques, B.; Devaux, J.; Legras, R.; Nield, E. J. Polym. Sci., Polym. Chem. 1996, 34, 1189.

[5] Kim, J. H.; Lyoo, W. S.; Ha, W. S. J. Appl. Polym. Sci. 2001, 82, 159.

[6] Jansen, M. A. G.; Goossens, J. G. P.; de Wit, G.; Bailly, C.; Koning, C. E. Anal. Chem. Acta 2006, 557, 19.

[7] Tomita, K. Polymer 1973, 14, 50.

[8] Lin, C. C.; Baliga, S. J. Appl. Polym. Sci. 1986, 31, 2483.

[9] Jansen, M. A. G.; Wu, L. H.; Goossens, J. G. P.; de Wit, G.; Bailly, C.; Koning, C. E. J. Pol. Sci., Pol. Chem. 2007, 45, 882.

[10] Misra, A.; Garg, S. N. J. Pol. Sci., Pol. Phys. 1986, 24, 983.

[11] Noda, I. Applied Spectroscopy 1990, 4, 44.

[12] Harrington, P.; Urbas, A.; Tandler, P. J. Chemometrics and Intelligent Laboratory Systems 2000, 50, 149.

[13] Cole, K. C.; Ajji, A.; Pellerin, E. Macromolecules 2002, 35, 770.


[15] Cole, K. C.; Ajji, A.; Pellerin, E. Macromol. Symp. 2002, 184, 1.

Appendix 1

partly reproduced from: Vivo-Tryols, G.; Ziari, M.; Magusin, P. C. M. M.; Schoenmakers P. J. Analytica Chimica Acta 2009, 37, 641.

125

Appendix 1

Effect of initial estimates and constrains selection in Multivariate Curve

Resolution – Alternating Least Squares. Application to low-resolution NMR data

Summary

A comprehensive study of the applicability of multivariate curve resolution (MCR) methods

to series of T2-relaxation filtered 1H NMR spectra of a crosslinked polymer network is

presented. A collection of Hahn-Echo NMR spectra is obtained at different echo times,

yielding two-way data. In this study the applicability of two different types of orthogonal

projection approach (OPA1 and OPA2) (column-wise and row-wise) were tested. Four

different strategies of alternating least squares methods were also examined (ALS1, ALS2,

ALS3 and ALS4). These strategies differed on the order of measurement for which the

constraints were applied in the final output, and the way in which SSR was calculated to

monitor for convergence. In the spectral order of measurement, a non-negativity constraint

was imposed, whereas in the time order of measurement, the signal was forced to follow an

exponential decay. This yielded up to eight MCR configurations, giving different results.

For solid-state NMR, the dissimilarity in NMR profiles is significantly lower than the

dissimilarity in signal decays, and therefore OPA2 performed better. A final output with a

constrained solution in relaxation-time was preferred (instead of a constrained solution in

NMR spectra) for practical purposes. Differences between the solutions given from the two

ALS configurations can be interpreted as a sign of lack of fit.

Appendix 1

126

1. Introduction

Multivariate analysis is a modern approach to interpret complex experimental data. The

underlying idea is that a coupled analysis of several data sets together (i.e., the information

coming from multiple channels at the same time) yields significant advantages over

analysing every data set separately. This has been demonstrated in many cases. The more

complex the instruments, the greater the advantages of a multivariate approach.

One factor that affects the selection of the most appropriate method(s) is the order of the

experimental data sets. According to the terminology introduced by Kovalski [1],

instruments can be classified according to the order of the tensor that represents the data

obtained from a single experiment. Zeroth-order instruments yield a scalar (i.e. a zeroth-

order tensor), first-order instruments yield a vector (first-order tensor), second-order

instruments yield a matrix (second-order tensor), etc. Among second-order instruments, a

distinction must be made between bilinear and non-bilinear data. This determines the type

of algorithms that can be used for data treatment. If the data is bilinear, the D matrix

representing the data of the second-order instrument can be decomposed as follows:

εCSD (1)

where C is a matrix having in each column the signal of one compound in the first order of

measurement and S is a matrix with in each row the signal of one compound in the second

order of measurement and represents the noise of the data. Chemically speaking, Eq. (1)

means that the number of sources of variance equals the number of chemical components in

the studied system [2]. For example, for a mixture of r compounds in high-pressure liquid

chromatography (HPLC) with ultraviolet (UV) detection, D would be the two-dimensional

spectrochromatogram of UV spectra (horizontal) as a function of elution time (vertical). The

matrix C contains the r elution profiles (column-wise) and the matrix S contains the r UV

spectra (row-wise) of the r pure compounds. In practice, the pure-component features may

not be known in advance. By means of multivariate analysis, the signals of the individual

(overlapping) compounds can be separated. Even in cases of severe co-elution, the peaks of

each compound can be quantified, provided that the signals (spectra) of the compounds

obtained by the (multichannel) detector are sufficiently different.

Appendix 1

127

Basically, the aim of multivariate-analysis algorithms applied to second-order instruments

yielding bilinear data is to solve Eq. (1), removing the noise of the signal and obtaining

both C and S. These two matrices contain all the individual signals of each pure compound

in the two orders of measurement. Eq. (1) can be solved by several methods, including

Alternating Least Squares [3] or Heuristic Evolving Latent Projections (HELP) [4].

Alternatively, if three-way data is available (which implies access to several experiments

performed in a second-order instrument) Generalised Rank-Annihilation Method (GRAM)

[5] or PArallel Rank-FACtor analysis (PARAFAC) [6] constitute an alternative that

indirectly solves Eq. (1). The common denominator in these techniques is to detect the

different sources of variance in the raw-data matrix (D) and to assign each one to a specific

compound. From among this family of techniques, ALS has become popular. This is

probably due to its simplicity –in programming terms– and to the fact that only one sample

is needed. The latter aspect helps to avoid the need for data alignment. Data alignment

strategies are normally a must when multivariate techniques are applied to the

chromatographic field. This requirement normally arises when more than one

chromatographic separation is simultaneously subjected to multivariate analysis. In ALS, if

more than one sample is available, the data from different experiments can be adjoined

forming an enriched two-way data matrix, avoiding the need for peak alignment.

Two problems must be solved when applying the ALS method to second-order instruments.

First, ALS needs to start with an initial guess for either the C or the S matrix. This is

normally tackled by performing several peak-purity assays on the original data matrix, for

example using Orthogonal-Projection Approach (OPA) [7] or SIMPLe-to-use Interactive

Self-modelling Mixture Analysis (SIMPLISMA) [8]. OPA constitutes an excellent

approach, yielding reasonably good initial guesses for C or S matrices. A second problem to

overcome is the so-called rotational ambiguity [9]. This refers to the fact that multiple

matrix pairs C, S mathematically fulfil Eq. (1), while only one of these represents the true

physical solution. In ALS, this problem is partially solved by imposing constraints that

modify the C and S solutions so as to yield profiles that are chemically meaningful in both

orders of measurement. However, depending on the problem, some rotational ambiguity

may remain even with imposed constraints.

OPA-ALS has been applied successfully to second-order instruments involving a separation

with a multi-channel detector such as HPLC-DAD [10] or CE-DAD [11]. In these cases,

Appendix 1

128

OPA was applied to provide an initial estimate of the spectrum of each compound (S

matrix). Instead, OPA may also be applied to obtain initial estimates of the individual

chromatographic peak profiles (C matrix). In HPLC-DAD the former is preferred, because

the selectivity is higher in the chromatographic domain than in the (UV) spectral domain.

The applications of ALS go far beyond hyphenated chromatographic techniques. The

method has been used, for example, for near-infrared-spectroscopy (NIR) data [12], Raman

spectroscopic images [13], kinetic-reaction data obtained with NMR [14], or Diffusion-

ordered spectroscopy (DOSY-NMR) data [15]. In each application, the constraints applied

to C and S profiles are different, based on (chemical) knowledge in each field.

In this work we apply OPA-ALS to a series of Hahn-Echo 1H-NMR spectra as a function of

the echo time for a cross-linked unsaturated polyester resin. Series of Hahn-Echo spectra are

obtained at various echo times. This yields a second-order data set, in which one of the

orders of the measurement is the conventional NMR spectrum (frequency or chemical-shift

axis) and the other the decay as a function of the Hahn-Echo time. The same kind of data is

obtained in DOSY NMR [15]. The entire signal is often analyzed in terms of a few

components, assigned to different phases or fractions in the sample polymers [16-18]. Each

"component" yields a signal of different shape in the NMR spectrum (first order of

measurement) and signal-decay curve (second order of measurement). The signal decay for

each component is assumed to be exponential. There are various reasons to treat the

outcome of such analyses with care, especially when it comes to the physical interpretation

of the results. First, according to general relaxation theory, exponential relaxation is only

expected for rotational chain motions, which are isotropic and fast at a miliseconds

timescale. Chain motions in rubbery polymers are anisotropic and therefore expected to

produce non-exponential decays [19, 20]. Models for describing the curve shape are still

under debate [21]. Second, chain dynamics are highly complex – even within a single

polymer phase or fraction – and they may be more-realistically described with a continuous

distribution of correlation times. In practice, decay curves are often fitted as a sum of

exponential relaxation curves (mono-, bi- and tri-exponential relaxation decays). The

adequate precision obtained with such empirical models may result from the combined

effects of motional anisotropy and physical heterogeneity.

There are important differences with previously published applications of OPA-ALS that

render the present case more difficult. Firstly, if we try to distinguish between fractions of a

Appendix 1

129

polymer that are chemically similar, but physically different (e.g. crystalline and amorphous

regions in a polymer), then we may expect that the NMR spectra are highly correlated. The

peaks should be located at the same frequencies and only differ in their widths. This calls

for caution: some steps in the ALS algorithm may yield highly unstable results, since the S

matrix (containing NMR spectra) will be nearly singular. As it has been discussed in other

areas (like imaging spectroscopy [22]), it is not a priori clear whether the OPA procedure

should be performed in such a way as to obtain initial estimates of signal decay curves

(according to ref. [15]) or to provide initial estimates of NMR spectra. Secondly, the

constraints that can be imposed to the C (signal decays) and S (NMR spectra) matrices are

of different nature. As mentioned above, it is assumed that the signal decay of both

components should show some kind of exponential behaviour. The possibility of fitting

each column of the C matrix to an exponential model can be imposed as a constraint within

the ALS procedure. This has been called in the literature as hard-soft modelling [23].

In this article we aim to describe and assess the different configurations of the OPA-ALS

method as applied to solid-state NMR data. The different possibilities of applying OPA

(obtaining initial estimates for NMR peaks or for exponential decays) are critically

evaluated. Also, different configurations of the ALS method can be used. Some of the

results obtained are significantly different when considering the different configurations.

2. Theory

2.1. Hahn-Echo decay in solid-state 1H-NMR of a polymer network.

A T2 Hahn-Echo Pulse Sequence (HEPS) relaxation experiment in 1H-NMR can be

classified as a second-order experiment, yielding a second-order tensor of data (i.e. a

matrix). The first order of measurement is the NMR-frequency axis expressed in relative or

absolute frequency units (ppm or kHz), whereas the second order is related to the relaxation

time (measured in time units). Following the terminology in Eq. (1), D contains the

experimental data obtained from a T2 HEPS experiment. D is n × m sized, n being the

number of relaxation times assayed, and m the number of frequency points collected in the

NMR spectrum. An arbitrary element of this matrix, di,j, collects the signal observed at the

ith pulse spacing (t) and the jth frequency. Accordingly, a column in the D matrix, dj, is a

Appendix 1

130

column vector (nx1 sized) representing the signal decay (intensity measured at different t

values) at the jth frequency (or chemical shift). Similarly, a row in the D matrix, di, is a row

vector (1xm sized) showing an NMR peak profile (intensity at different frequencies) at the

ith pulse spacing.

When applying the HEPS technique to a polymer network, different parts of the network

may produce different relaxation decays and NMR spectra. All these different molecular

parts (or “components”) are characterized, on the one hand, by different mobilities (different

relaxation behaviour) and, on the other, by different chemical shifts (different NMR peaks

or peak shapes). A single component of the network will give rise to a unique NMR peak

and relaxation behaviour, both profiles being independent of each other. In other words, the

relaxation behaviour of a given component is independent to the frequency at which it is

measured and the NMR spectrum has the same shape at any relaxation time (see Appendix

for a more-detailed justification of this assumption). This implies, strictly speaking, that the

D matrix is bilinear, so we can assume the model described in Eq. (1). For p components

present, Eq. (1) hence adopts the form:

(2) εDεCSεscDp

1kkk

ˆ

where ck is a column vector (n×1 sized) of the signal decay for component k, and sk the row

vector (1×m sized) containing the NMR spectrum for component k, and a matrix (n×m

sized) collecting the error of the model. Therefore, C is a matrix of relaxation decays (n×p

sized) formed adjoining all the ck elements. Similarly, S is a matrix of NMR spectra (p×m

sized) formed adjoining all the sk elements. The hat symbol (“^”) is used to indicate the part

of the experimental matrix that is modelled: . CSD ˆ

1H NMR Hahn-Echo decays of cross-linked elastomers can often be empirically described

in terms of two or three exponentials [16-18]. The fastest decay component typically arises

from short network chains whose mobility is strongly restricted by crosslinks at both sides.

The slower decay component(s) are due to longer network chains, so-called “dangling ends”

connected to the network at just one end of the chain, and a sol fraction of uncured material.

In practice it is often assumed that every decay component adopts an exponential form in

the relaxation decay. Therefore, for a two-component model (p=2) the ith-row and jth-

column element of the D matrix , is defined as: ˆjid ,

ˆ

Appendix 1

131

biai Ttj

Ttjji ebead 22

,ˆ (3)

where the terms aj and bj are the respective intensities of components A and B at the jth

value of the chemical shift along the relative frequency axis, ti is the ith echo time in the

HEPS experiment, and T2a and T2b are the respective transversal-relaxation times of the two

components. If the last equation holds, matrix C is defined as follows:

bnan

ba

ba

TtTt

TtTt

TtTt

22

2222

2121

expexp

......

expexp

expexp

C (4)

and matrix S is defined as:

(5)

m

m

bbb

aaa

...

...

21

21S

where a1, a2, ... am and b1, b2, ... bm are defined in Eq. (3).

2.3. The OPA-ALS method

2.3.1. ALS1, ALS2, ALS3 and ALS4

The task of solving Eq. (1) consists of finding the C and S pair of matrices that minimise the

differences between D and CS. This implies forcing the unmodelled variance () to be

minimal. Mathematically speaking, this can be formulated as minimising the Sum of

Squares of Residuals:

(6)

m

j

n

ijiSSR

1 1

2,

22 εCSD

Unfortunately, there is more than one pair of CS matrices that make SSR minimal. For any

solution C, S = C0, S0 of Eq. 1, there is a continuous family of mathematically correct

solutions C0 T-1, T S0, where T is any invertible (pp) matrix. It is obvious that the

problem has an infinite number of solutions, because an infinite number of T matrices exist.

This is called rotational ambiguity.

Additional constraints on C and S can offer a way out of this rotational ambiguity. The ALS

algorithm is constructed in such a way that a solution of T is varied in order to direct the C

Appendix 1

132

and S matrices to a final result that meets certain (minimal) requirements for a physically

meaningful solution. Specific constraints are imposed on C and S (e.g. unimodality or non-

negativity [9]) depending on the nature of the problem. In the problem presented here,

different constraints are applied to C and S. As explained before (see section 2.1. and

Attachment), it is assumed that the columns in the C matrix are described by exponential

decays [24]. An additional reasonable assumption is that the NMR spectra in matrix S are

positive across the whole frequency range, so that a non-negativity constraint is applied to

S. In the ALS algorithm these constraints are used to direct the process to the true or, at

least, a physically meaningful solution.

However, ALS can be applied in two different ways, denoted as ALS1 and ALS2 in Figure

A1.1. Antecedents of these two configurations can be found in ref. [25]. In ALS1 we start

with an approximation of the C matrix, C0, in which the subindex 0 denotes that this is the

0th iteration. In a second step, constraints are applied to C0. The points in each column of the

C0 matrix are fitted to an exponential decay. This can be done for each column

independently by linear regression, after calculating the logarithm of the values. The fitted

points (once retransformed back to the exponential form) constitute the constrained matrix,

, where the superindex c indicates that the constraints have been applied. In the linear

regression, not only is obtained, but also the values of T2a and T2b (cf. Eq. (3), Figure

A1.1). If is known, an estimate of S0 can be obtained by minimizing SSR:

c0C

c0C

c0C

DCS

c00 (8)

where denotes the Moore-Penrose generalised inverse of . In a subsequent step,

constraints are applied over S0 (non-negativity) zeroing those values from S0 that are

negative. This yields a constrained matrix of NMR peak profiles, . In a way similar to

Eq. (8), a new estimate for C (C1) can be found if is known:

c0C

c0C

Sc0

c0S

c01 SDC (9)

All these operations constitute a closed loop, so the whole process can be repeated using

subsequnt estimates of C (Ck) until convergence is reached (see Figure A1.1). The

convergence is checked by monitoring SSR, which is calculated in this case by substituting

Appendix 1

133

C1 and in Eq. (6). When SSR starts to increase or no longer decreases significantly, the

process is stopped. Note that when the convergence is reached after q iterations, a matrix of

unconstrained signal decays (Cq+1) and constrained NMR peaks ( ) is obtained.

c0S

cqS

A second way in which the ALS algorithm can be applied is denoted by ALS2 in Figure

A1.1. We now start with an approximation of the S matrix, S0. From S0 a constrained

matrix, , is obtained after applying constraints (non-negativity). Eq. (9) is then used to

find the unconstrained C matrix, which we call C0 instead of C1 in this case. This matrix is

converted into after applying constraints (fitting exponential decays to the columns of

C0 and obtaining T2a and T2b). Eq. (8) then yields to a new solution of matrix S, S1. The

process can be repeated until convergence is reached. In this case SSR is calculated using S1

and . A matrix of unconstrained NMR peaks ( ) and a matrix of constrained (fitted)

signal decays ( ) are obtained after q iterations. Note that ALS1 and ALS2 do not differ

only in the initial guesses (column-wise or row-wise), but also in the quantity used to

monitor convergence (SSR is calculated differently) and type of outputs given

(unconstrained or constrained).

c0S

c0C

cqC

c0C 1qS

A deeper examination of the algorithm's architecture reveals the existence of two extra

alternatives, both based on the way in which SSR is calculated. One can keep in memory

the regressed (unconstrained) solutions of Sq and Cq throughout the loop, and use their

product to calculate and monitor SSR. The logical output of the algorithm is then a

completely unconstrained solution. We have labelled this procedure as ALS3 (see Figure

A1.1). An alternative consists of keeping in memory the constrained solutions, and

that have been obtained during a single loop, and use them to calculate SSR. The output of

the algorithm, which will be called ALS4, is then a constrained collection of exponential

decays and NMR profiles.

cqS

cqC

2.3.2. OPA1 and OPA2

From the last section, it is obvious that an initial estimate of the C or the S matrix is needed

to start the ALS1 or the ALS2 algorithm, respectively. This can be solved in several ways,

the most popular of which is the Orthogonal-Projection Approach (OPA [7]). Only a brief

description of the OPA method is given here. More details can be found elsewhere [7].

Appendix 1

134

Figure A1.1. Flow diagram of the algorithms OPA1-ALS1,OPA1-ALS2, OPA2-ALS1 and OPA2-ALS2.

Appendix 1

135

Let us suppose that we want to obtain an initial estimate of the C matrix (signal decays).

The OPA method is based on finding those columns in the D matrix that give the most

dissimilar signal decays. For that purpose, the dissimilarity at each chemical shift, dj, is

calculated:

jT

jjd XXdet (10)

where the Xj matrix is defined by adjoining a set of reference-signal decays, R, with the

signal decay at the jth chemical shift:

jdRX (11)

where dj is the vector containing the jth column of D. In the first iteration, the R matrix

contains the mean signal decay (taken across all the signal decays of D). Once the chemical

shift with the maximum dissimilarity has been located (the ath row in D), the corresponding

signal decay at this particular chemical shift, da, replaces the mean in R: . After this

first iteration, the dissimilarity is calculated again across all signal decays and the chemical

shift with maximum dissimilarity is located (the bth row in D). The R matrix then

incorporates db, increasing its number of columns:

adR

bdRR . This operation is repeated

until the number of columns of the R matrix equals the number of expected components

(which is two in our example). Once the process has ended, the resulting R matrix is taken

as the initial estimate of the C matrix.

However, the OPA method can also be applied to obtain an estimate of the S matrix. In this

case, Eq. (11) takes the form:

TidRX (12)

where diT is the transposed vector containing the ith row of D. In the same way, in the first

iteration the R matrix contains the transposed mean NMR spectrum (taken across all the

NMR spectra contained in D). In this case, the method is based on finding those time delays

that yield the most dissimilar NMR spectra. In each iteration, the most dissimilar transposed

NMR spectrum is incorporated in the R matrix as a new column. When the number of

columns of R equals the number of expected components (two in our case), the transpose of

R is taken as the initial estimate of S. We will define OPA1 and OPA2 as the methods

yielding initial estimates of S and C, respectively.

Appendix 1

136

other order of measurement. When a

At this point, applying OPA1 before ALS1 or OPA2 before ALS2 are straightforward

ms OPA1-ALS1 and OPA2-ALS2. However, two other

The samples were supplied by DSM Resins (Zwolle, The Netherlands) in the form of

ter resins in 33 wt % of styrene. A catalyst (0.5% of a 1%

phthalic anhydride with 1,2-propylene glycol and ethylene glycol. The material was brought

The OPA method is usually applied to the order of measurement with the largest selectivity,

subsequently comparing dissimilarities in the

chromatographic technique is involved (e.g., HPLC-DAD), the selectivity is larger in the

chromatographic direction (chromatograms) than in the spectral direction (spectra). It is

typically easier to find elution zones in which only one component is eluting, than to find

wavelengths at which only one component is absorbing. Consequently, the OPA method is

normally applied to obtain estimates of the spectra of the components, rather than of the

chromatographic peak profiles. However, when the instrument does not involve a

chromatographic separation (as in the example studied in this paper), it is not clear which

order of measurement yields the maximum selectivity. So far, to the authors’ knowledge, no

studies have been performed to compare the results of processes starting with OPA-based

estimates of C or S in NMR techniques.

2.3.3. Eight ALS configurations

options. We will call these algorith

combinations are also possible. After applying OPA1 to obtain an estimate of C, Eq. (8) can

be applied (without applying constraints to C) to obtain an estimate of S and the ALS2

configuration can subsequently be applied (OPA1-ALS2). In a similar way, OPA2-ALS1

can be considered by applying Eq. (9) to matrix S to obtain an initial estimate of C, before

starting ALS1. Finally, ALS3 and ALS4 can also be applied with both OPA1 and OPA2. In

total, eight configurations are possible, namely: OPA1-ALS1; OPA1-ALS2; OPA1-ALS3;

OPA1-ALS4; OPA2-ALS1; OPA2-ALS2; OPA2-ALS3; OPA2-ALS4;

3. Experimental

3.1. Sample preparation

solutions of unsaturated-polyes

cobalt solution) and 2% of benzoyl peroxide initiator (2%) were added to the solution. The

polyester chains had been formed by a polycondensation reaction of maleic anhydride and

Appendix 1

137

00 spectrometer equipped with

a 7-mm Magic Angle Spinning (MAS) probe. To prevent unwanted sample rotation caused

ting, an empty sample holder was loaded on top of the one

AB 7 (The Mathworks, Natick, MA, USA), were

used for data treatment.

4.1. Comparison of OPA1 and OPA2 to obtain initial estimates

are presented in Figure A1.2 and Figure A1.3),

respectively, and correspond to the HEPS-NMR experiments at 505 K. Results obtained at

raphs a-c (in

between two glass plates, degassed and then cured for 24 hours at room temperature.

Thereafter, two post-curing step were performed during 24 hours each at 600C and 800C,

respectively. For the NMR experiments, the resulting networks were ground to a fine and

apparently homogeneous powder using a cryogenic grinder.

3.2. NMR experiments

The NMR experiments were performed on a Bruker DMX5

by the N2 flow used for hea

containing the sample material. The relaxation experiments reported in this article were

carried out at 300, 441, 465 and 505 (calibrated temperature scale) on the cross-linked

samples. The Hahn-Echo pulse 90º--180- sequence (HEPS) with 180-pulse duration 10

s and echo times 2 between 0.004 and 4 ms (see section 2.1.) was used to record the

decay of the transverse magnetization.

3.3. Software

Home-built routines, written in MATL

4. Results and discussion

The results of OPA1 and OPA2

other temperatures were similar and these are not presented here for brevity. G

both figures) depict the R matrix when zero (part a), one (part b) and two (part c)

components are considered (see figure captions for a more detailed explanation). As can be

seen, OPA2 yields better initial estimates when compared with OPA1. This will be

confirmed numerically in the next section. At this point, the result can be tested qualitatively

by comparing the vector of dissimilarities (part f) in both figures. In Figure A1.2f two bands

of high dissimilarity are clearly observed at both sides of the center of the NMR spectrum.

Appendix 1

138

mpares the results in terms of the Square Sum of Residuals -SSR; Equation

(6)- obtained when applying the eight configurations to data obtained at four different

n the

nt. ALS1 yields unconstrained relaxation decays and constrained NMR

This means that the initial (signal decay) estimates (matrix R, Figure A1.2c) are not

sufficient to explain all the variance of the signal. In other words, when the columns of the

D matrix cannot be explained as a linear combination of the vectors contained in R, the

determinant of Eq. (10) increases. This is because the most dissimilar signal decay is always

found around the NMR peak maxima (Figures A1.2d and A1.2f). In contrast, Figure A1.3f

does not show equally clear (highly dissimilar) zones as Figure A1.2f. Although one

dissimilar zone is found around 0.01 ms, the general trend is noisier than Figure A1.2f. This

is because the most dissimilar regions in parts 2d and 2e are found at different time decays,

yielding thus less-correlated NMR spectra (Figure A1.2c). The differences in quality of the

initial estimates will be even clearer in the next section, in which the numerical results will

be compared.

4.2. Comparison of the results obtained with the eight configurations

Table A1.1 co

temperatures. The results that were qualitatively evaluated in the previous section o

performance of OPA1 and OPA2 are confirmed here quantitatively. For the same kind of

output, OPA2 always gives rise to lower SSR values. On the other hand, OPA1 caused most

of the divergence situations. Therefore, it can be concluded that the initial estimates

obtained by evaluating the dissimilarities between signal decays (i.e. OPA1) give rise to a

local minimum.

The results comparing the different ALS versions (using the same input) can be interpreted

considering the fact that outputs obtained from ALS1, ALS2, ALS3 and ALS4 algorithms

are not equivale

spectra, while ALS2 yields constrained relaxation decays and unconstrained NMR

spectraALS3 yields both unconstrained results, whereas in ALS4 both orders of

measurement are constrained. Any constraint implies an increase in SSR (the function has

less flexibility to adapt to the experimental data). Fitting an exponential decay (hard-

modelling) implies a stricter constraint than applying non-negativity (soft-modelling).

Obviously, the stricter the constraints are, the higher the SSR values will be. Thus, lower

values of SSR are expected when using ALS1 compared to the results obtained with ALS2.

Appendix 1

139

Figu a through through e the diss an be followed o calculate the id in this

cay is located. T idering th

d and e are repr ese two com

res A1.2. Illustration of the OPA1 method for the NMR data at T = 505 K. Parts c depict initial guesses (R matrix of signal decays), whereas in parts dimilarity curves are represented (NMR spectra). The OPA procedure c from left to right and from top to bottom. Part a depicts the mean decay used t

dissimilarity curve (part d). The maximum of the curve is overlagraph (dashed line), which indicates the frequency at which the highest dissimilar de

his decay is depicted in part b. Part e depicts the dissimilarity curve conse decay represented in part b as initial guess. The maximum of the dissimilarity is again

overlaid as a dashed line. The two extracted decays of maximum dissimilarity in parts esented together in part c (solid and dashed lines). The dissimilarity with th

ponents is depicted in f. All y-axes are relative.

Appendix 1

140

ive.

Figure A1.3. Illustration of the OPA2 procedure for the NMR data at T = 505 K. Similarly to Fig. A.2, parts a through c depict initial guesses (NMR spectra), whereas in parts d through e the dissimilarity curves are represented (decays). The procedure of OPA can be followed from left to right and from top to bottom. Part a depicts the mean spectrum used to calculate the (dis-)similarity curve (part d). The maximum of the curve is overlaid in this graph (dashed line), indicating the time decay at which the most dissimilar spectrum is located. This spectrum is depicted in part b. Part e depicts the dissimilarity curve considering the spectrum represented in part b as initial guess. The maximum of the dissimilarity is again overlaid as a dashed line. The two extracted spectra of maximum dissimilarity in parts d and e are represented together in part c (solid and dashed lines). The dissimilarity with these two components is depicted in part f. All y-axes are relat

Appendix 1

141

Also, ALS3 yields normally the lowest SSR value (when compared with other algorithm

configurations using the same input), whereas ALS4 gives normally the highest. This is

clear from Table A1.1 with OPA2. Exceptions to this rule are found in case of divergence,

due to the different ways in which SSR is calculated to monitor convergence.

Method Temperature, K OPA1-ALS1 OPA1-ALS2 OPA2-ALS1 OPA2-ALS2

300 1.06 7.55 0.47 1.91

441 0.18 1.09 0.17 2.22

465 5.60 10.75 0.32 3.39

505 1.50 2.60 0.16 0.86

Figure A1.4 depicts the graphical results (both NMR profiles and exponential decays)

obtained with ALS1 and ALS2 (starting with both OPA1 and OPA2) at the highest

temperature (505 K). Results obtained with ALS3 and ALS4 are not included for brevity. It

can be observed that the final constrained NMR spectra, , once introduced in Eq. (9),

yield a collection of signal decays, , that do not meet the constraints. This can be seen

clearly in Figure A1.4a and Figure A1.4e, in which th put from OPA1-ALS1 is

depicted. One of the unconstrained signal decays is by no means an exponential decay. A

similar effect – be it to a le hen using the OPA2-ALS1 configuration

to

the signal decays obtained with ALS1, whereas the NMR profiles are similar to the NMR

Table A1.1. SSR (×1017) obtained with the eight configurations explained in the text.

cfS

e out

fC

sser extent – is found w

(Figure A1.4c and A1.4g). Similarly, but less clearly, the final constrained signal decays,

cfC , once introduced in Eq. (9), can yield a collection of NMR spectra, fS , that

occasionally do not meet the constrains. This effect is clearly noticeable in the results

obtained using the OPA1-ALS2 configuration (Figure A1.4b and Figure A1.4f). Forcing the

signal decays to be exponential yields negative regions in one of the NMR spectra. Starting

with more correct initial candidates (OPA2) greatly alleviates this problem. Negative parts

in the NMR spectra in Figure A1.4d are almost unnoticeable.

Similar effects are observed with ALS3 and ALS4. With ALS3, signal decays are similar

Appendix 1

142

straints, since

both signal decays and NMR profiles are always constrained. As the results obtained with

ALS3 and ALS4 can be deduced from ALS1 and ALS2, ALS3 and ALS4 configurations

tw results obtained from ALS1, ALS2, ALS3 and ALS4 are

rpreted as a la b that nen ation

too simple plain the obtained in HEPS-NMR exper s on polymer

rks [21]. H r, includin component e model led to divergence, due to

ilities in th putation of the Moore-Penrose pseudoinverse (8) and (9)-.

bly, the pea iles in NM d the signal decays were too simi r the different

omponents and the and matrices were nearly singular. Another source of a lack of fit

dis

on the scale of the signal.

profiles obtained with ALS2. Therefore, with OPA1-ALS3, both profiles (signal decays and

NMR spectra) are away from meeting the constraints. With OPA2-ALS3, the problem is

alleviated for non-negativity applied to the NMR profiles. Inspecting the results with ALS4

gives no information about how difficult is for this system to meet the con

will not be studied further.

Differences be

inte

een the

ck of fit. It has een suggested the two-compo t model of Equ

(3) is to ex results iment

netwo oweve g more s in th

instab e com -Eqs.

Proba k prof R an lar fo

S Cc

can be that the signal decays do not follow an exponential behaviour. It has been reported in

the literature [21] that the signal decay in HEPS-NMR is a combined function of Gaussian

and exponential components.

4.3. Comparing dissimilarities in both orders of measurement

A more detailed study into why the OPA2 configuration performs better than the OPA1 in

providing initial estimates is shown in Table A1.2. In this table the dissimilarities between

the deconvolved signal decay profiles and the NMR spectra profiles are compared. The

objective of this comparison is to check which order of measurement features the higher

similarity, in order to study quantitatively which order of measurement is most

appropriate for applying the OPA method. For this purpose the true solutions of C and S

should be compared. The true NMR spectra of the two components are unknown, so the

best solutions (those obtained with OPA2 configuration) have been used for the

computations. In this particular case, calculation of the dissimilarity using Eq. (10) is not

recommended, since the value of the measurement depends

Appendix 1

143

Figures A1.4. Results of a two-component decomposition after applying the four algorithms described in this article to the NMR experiments at T = 505 K. Parts a through e represent NMR peak profiles, whereas parts f through j depict time decays. Algorithms: a,e OPA1-ALS1; b,f OPA1-ALS2; c,g OPA2-ALS1; d,h OPA2-ALS2.

Appendix 1

144

ormalised dissimilarity measurements are more suitable in this case. Instead, we have

calculated the dissimilarity according to ref. [26], as the sine of the angle between the two

vectors (representing the NMR spectra or the signal decays). The resulting values are

normalized. A value of 1 implies that the profiles completely different, whereas 0 implies

that the profiles are fully equivalent.

In studies involving second-order NMR data (such as DOSY-NMR [15]), in which a series

of NMR spectra are obtained applying different time delays, a good initial guess for ALS is

found by checking those frequencies of the NMR spectrum that yield the most dissimilar

time decay. This means that the OPA1 configuration is used and that OPA2 is normally not

applied. In second-order NMR spectroscopy, more selectivity is expected in the NMR

spectra rather than in the time delay. Due to the exponential nature of the signal decays, it is

easier to find frequencies containing a single (pure) compound than specific time delays

yielding NMR spectra of a pure component. However, solid-state NMR yields a different

perspective than liquid NMR. When applying Hahn-Echo techniques on cross-linked

polymer networks, regions of selectivity (pure or dominant components) are hard to find in

either order of measurement. If only two components are considered, the maxima in the

NMR spectra of both components are found at the same frequency; only differences in band

broadening can be found. This makes the dissimilarity in NMR profiles in this particular

case always significantly lower than the dissimilarity in signal decays, independent of the

ALS method applied and on the temperature (see Table A1.2). This explains why in this

particular case OPA2 performs better.

Table A1.2. Dissimilarity obtained with the deconvolved profiles (P and D matrices) of OPA2-based methods. The dissimilarity is calculated according to ref. [23].

NMR profiles Signal decay profiles

N

Temperature,

K OPA2-ALS1 OPA2-ALS2 OPA2-ALS1 OPA2-ALS2

300 0.535 0.536 0.834 0.760

441 0.413 0.480 0.716 0.628

465 0.391 0.426 0.638 0.595

505 0.337 0.407 0.636 0.562

Appendix 1

145

each specific chemical or physical component yields a

single source of variance it is possible to separate the contributions of each component to

onstrained signal decays and NMR spectra). The

ombination of all these configurations gives rise to eight different ways of applying OPA-

As for the signals obtained from solid-state HEPS-NMR experiments on polymer networks,

a d to perform better ions. This can be

lained from a ity ec sign e only

ences in p e (and not in peak positi ed i MR spectra

corresponding to the different components of the network. A way of testing which OPA

a ch performs consists of comparing the dissimilarity of either

spectra or ignal deca rder of m ent yieldin ost dissimilar

indicates t itable O nfiguratio s been dem ed that signal

4. Conclusions

Multivariate Curve Resolution (MCR) methods are designed to interpret data from second-

order instruments. These methods are based on the separation of the sources of variance that

are contained in the output matrix. If

the raw experimental data. Solid-state HEPS-NMR experiments on polymer networks yield

second-order data, which can be analysed using MCR methods. If the presence of two

components is assumed, the variance produced by each component can be determined, and

the contribution of each component to the total signal can be established. Orthogonal-

Projection Approach in combination with Alternating Least Squares (OPA-ALS) belongs to

the family of MCR methods. It has been applied in this work to analyse and separate the

contributions of two components present in polyester polymer networks. Each component is

attributed to a part of the polymer with a different mobility.

The OPA method can be applied in two ways, i.e. by evaluating the dissimilarities in signal

decays at each frequency in the NMR spectra (OPA1) or by evaluating the dissimilarities in

NMR spectra at each time delay (OPA2). The ALS method can also be applied in four ways

(ALS1, ALS2, ALS3 and ALS4), depending on (i) which order of measurement is used as

initial estimate (signal decays or NMR profiles) (ii) which profiles (unconstrained or

constrained) are used to calculate SSR to monitor convergence and (iii) the output produced

by the algorithm (constrained or unc

c

ALS. All of these have been compared in this work, yielding different results.

OPA2 configur

exp

tions were foun than OPA1 configurat

lower selectiv in the NMR sp tra than in the al decays, sinc

differ eak shap on) are expect n the N

pproa best the deconvolved

NMR of the s ys: the o easurem g the m

value he most su PA co n. It ha onstrat

Appendix 1

146

milar than NMR spectra, so that OPA2 is recommended instead of

OPA1.

decays are more dissi

Differences between the outputs obtained from the different ALS configurations and the

same OPA strategy have to be interpreted with caution. As the different configurations force

the output to be more or less constrained, the sum of squared residuals (SSR) obtained with

this output cannot be compared directly. On the other hand, significant differences were

observed between the retrieved spectra and signal decays from the different ALS

configurations. This can be interpreted either as a lack of fit or rotational ambiguity. Among

all configurations, OPA2-ALS2 performed the best and its use is recommended for the

analysis of solid-state HEPS-NMR data obtained on polymer networks.

Appendix 1

147

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[13] Wang, J. H. ; Hopke, P. K. ; Hancewicz, T. M. ; Zhang, S.L. Anal. Chim. Acta 2003, 476, 93.

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[1

Appendix 2

149

Appendix 2

Combining linear and non-linear regression algorithms as an alternative

for multivariate curve resolution problems of low selectivity

1. Introduction

Consider solving the following equation for bilinear data:

(1) εYεPDY ˆ

where Y stands for the experimental data generated by a second-order instrument [1], P and

D are the matrices containing (column-wise and row-wise, respectively) the profiles of each

chemical component in the first- and second-order of measurement, is de unmodelled

variance and by definition PDY ˆ . If the instrument is, for example, a HPLC-DAD, then

Y is the spectrochromatogram of a mixture, and P and D contain the chromatographic peak

profiles and the spectra of each chemical component, respectively.

As normally Y is (experimentally) known, the chemical problem consists of retrieving P

and D, i.e., the profiles of each chemical component in both orders of measurement. This

can be done in different ways [2], most of them based on finding the least-squares solution

that minimizes . Unfortunately, there is not a unique PD pair of matrices that minimizes .

This problem has been formulated as "rotational ambiguity" [3] and arises from considering

a rotation matrix T (T'T=I) that:

TDPT'PT (2)

where Pn=PT' and Dn=TD can be considered as a new pair of PD matrices that satisfy Eq.

(1). As an infinite number of T matrices exist, an infinite number of solutions exist for Eq.

(1).

Appendix 2

150

There are two ways to constrain the number of solutions of Eq. (1). The first one involves

considering more than one experiment (i.e., more than one Y matrix), building up three-way

data. The most popular methods of this family are, among others, PARAFAC [4] or GRAM

[5]. An alternative, when three-way data is not available -or its use is not convenient-, is to

make use of chemical knowledge. From the chemical point of view, only certain P and D

profiles make sense. For example, chromatographic peak profiles (in P matrix) should be

unimodal and non-negative, and spectra (in D matrix) should be non-negative. These

constrains can be imposed in P and D to restrict the number of solutions. The family of

techniques that impose constraints on P and D based on chemical knowledge was

introduced some decades ago by Tauler et. al., and -having many variants- has been named

as Multivariate Curve Resolution (MCR).

Two kinds of constraints can be distinguished in MCR methods. The first has been named

soft modelling [6], and does not fit the data to a mathematical equation, but corrects the P

and D matrices to meet the constraints. A typical example in the field of HPLC-DAD is to

correct for the negative parts of all spectra contained in D matrix, zeroing the values that are

below zero. An alternative (called hard modelling) is to fit the data to a mathematical

equation. Following the example before, the chromatographic profiles contained in P could

be fitted to a peak model.

The present work is related to this second sub-family of techniques. Traditionally, hard

modelling is applied as an extra constrain inside the MCR algorithm, so the fitting of the

mathematical equation is nested as an extra operation inside the MCR process. This

alternative was studied in a previous article [7]. It was found that several configurations of

the algorithm were possible, and all of them were studied. Another alternative is presented

and explored in this work. It consists of fitting the model parameters with a non-linear

algorithm (which gives an estimation of P, for example), and nesting a linear fitting (to

calculate D in this case) inside. Both methods seem to be complementary, having

advantages and disadvantages. All of them are analyzed in this work.

The example used to illustrate the algorithm performance is a solid-state Hahn-echo NMR

data of a polymer network. With this technique, changing the delay time between NMR

excitation and signal measurement, second-order data is obtained: one of the order of

measurement accounts for conventional chemical shifts (NMR spectrum) and the other for

Appendix 2

151

the signal decay. It can be assumed that the signal decay follows an exponential behavior,

which brings the possibility of fitting each column of the P matrix (each component of the

polymer network) to an exponential model (hard modeling). A direct outcome of the

algorithm is an estimation of the relaxation time of the components of the polymer network,

which is highly valuable to characterize the polymer topology.

2. Experimental

2.1. Sample preparation

The samples were supplied by DSM Resins (Zwolle, The Netherlands) in the form of

solutions of unsaturated-polyester resins in 33 wt % of styrene. A catalyst (0.5% of a

1% cobalt solution) and 2% of benzoyl peroxide initiator (2%) were added to the

solution. The polyester chains had been formed by a polycondensation reaction of

maleic anhydride and phthalic anhydride with 1-2 propylene glycol and ethylene glycol.

The material was brought between two glass plates, degassed and then cured for 24

hours at room temperature. Thereafter, two post-curing step were performed during 24

hours each at 600C and 800C, respectively. For the NMR experiments, the resulting

networks were ground to a fine and apparently homogeneous powder using a cryogenic

grinder.

2.2. NMR experiments

The NMR experiments were performed on a Bruker (Karsruhe, Germany) NMR

spectrometer of 500 MHz. The relaxation experiments were carried out at different

temperatures (from 213 K to 450 K) on the cross-linked samples. The Hahn-echo pulse

sequence (HEPS) measurement consists of a double-pulse sequence: 90°x−tHe –

180°x−tHe—acquisition, where tHe is varied between 1 and 4 ms. The second pulse in the

HEPS inverts nuclear spins of mobile molecules only, and an echo signal is formed with

a maximum at time t =2 tHe after the first pulse. By varying the pulse spacing in the

HEPS (t), information about intensity profiles is obtained as a function of time t. A

single t2 HEPS relaxation experiment in 1H-NMR yields thus two-way data. The first

order (measured in frequency units) accounts for conventional chemical shifts, whereas

Appendix 2

152

the second is related to the relaxation time (measured in time units). For consistency

with section 1, we will call Y to the experimental data-matrix obtained from a single

NMR experiment.

2.3. Solid-state NMR of a polymer network

When applying the experiment described in the previous section to a polymer network, the

relaxation signals in polymer networks will look more like different molecular parts that

produce separate relaxation signals of different form. All these different molecular parts

(or “components”) are characterized, on one hand, by different mobilities (relaxation

behaviour), and, on the other, by different chemical shifts (different NMR peaks). A

component of the molecule will give rise to a unique NMR peak and relaxation

behaviour, being both profiles independent one to each other. If this happens it is said

that the matrix Y is bilinear, so Eq. (1) holds. In Eq. (1), P is a matrix of relaxation

decays (n×p sized). Similarly, D is a matrix of NMR peaks (p×m sized). In this context,

each k “component” corresponds to each part of the molecule producing a different

relaxation decay and NMR peak. According to Litvinov’s approach [ref.], we assume

that only two components are present, so p = 2: one component (component A) refers to

the cross-links chains of the network, whereas the second component (component B)

corresponds to the fraction of more mobile components [8]. In general it is also assumed

that both components adopt an exponential form in its relaxation decay. Therefore the

ith-row and jth-column element of the matrix -see Eq. (1)-, , is defined as: Y jiy ,ˆ

biai Ttj

Ttjji ebeay 22

,ˆ (3)

where the terms aj and bj are the signals at a certain j value of the chemical shift

corresponding to components A and B, respectively; ti is the ith pulse spacing in the

HEPS experiment; and T2a and T2b are constants related to the half-time relaxation

decay for each component.

2.4. The OPA-ALS method

In a previous article the performance of several configurations of OPA-ALS algorithms was

studied [7]. Only a brief description is given here (for full details see the previous

reference). The OPA-ALS algorithm consists of two parts, namely (i) the OPA method [9]

Appendix 2

153

(retrieves initial estimates of the P or D matrix) and (ii) the ALS algorithm (which finds a

solution of Eq. (1), starting from the initial estimates found by OPA). Two versions of the

OPA method were proposed [7] (OPA1 and OPA2), depending on how the Y matrix is

analysed (column-wise or row-wise) giving an estimation of P or D in each case. The ALS

algorithm can be designed also in two ways. Taking into account that P and D matrices are

forced to meet certain constraints, ALS can give two kinds of answers: unconstrained P and

constrained D (i.e., ALS1 algorithm) or constrained P and unconstrained D (i.e., ALS2

algorithm) [7]. Therefore, four combinations of OPA-ALS algorithm exist: OPA1-ALS1,

OPA2-ALS1, OPA1-ALS2, OPA2-ALS2.

2.5. The non-linear algorithm

This algorithm arises from the idea that the (constrained) matrix P -as described in Eq.

(3)- can be described using a limited number of parameters. Taking into account that the

exponential decay of each component can be described with only one parameter, in the

example shown only two values (T2a and T2b) are needed to describe P, since only two

components are assumed. The method consists of applying a non-linear procedure to

minimise SSR. In this work the quadratically convergent Powell method [10] was used.

Starting with an initial set of (T2a and T2b) parameters, the Powell algorithm works by

varying the initial parameter estimates, evaluating the quality of the fitting simulating a

trial-and-error procedure. Each time that the quality of the fitting for a set of model

parameters is requested by the Powell routine, the SSR is evaluated -Eq. (3)-. This

implies several steps. First of all the P matrix is calculated (provided that the

parameters of the relaxation decays are supposed). In a second step, the D matrix

(containing the NMR peak profiles) is computed using Eq. (8). Once D is known, SSR

is calculated. It should be noted that Eq. (8) implies in fact a linear regression, which is

nested into a general non-linear fitting. The algorithm ends when the Powell method

does not find any significant decrease in SSR.

Appendix 2

154

3. Results and discussion

3.1 Comparison of OPA-ALS methods with non-linear method

There exist two relevant differences between the family of OPA-ALS methods

explained in ref. [7] and the non-linear method: ALS-based methods can be designed to

apply constrains in both P and D matrices, whereas in the non-linear algorithm, only

constraints on P or on D are possible. The non-linear method was designed in this

particular case to apply constraints on P, but to modify the algorithm in order to apply

constraints in D instead is straightforward. Another important difference refers to the

type of constraints that can be applied: whereas hard- and soft-modelling can be applied

to ALS, only hard-modelling is possible with the non-linear algorithm. Indeed, the

constraints applied in P (or D) matrices should consist of fitting a parametric model:

designing the non-linear algorithm to apply soft-modelling (i.e. constraints as non-

negativity or unimodality) is conceptually impossible, since the non-linear algorithm

(Powell in this case) can run only fitting parametric models. Therefore, OPA-ALS is

more flexible –in terms of constraints– than the non-linear algorithm. In this particular

case the non-linear algorithm is however specially suited: hard-modelling is naturally

applied (an exponential decay is fitted to P), and –as will be seen– to apply constraints

on D is not mandatory.

In order to evaluate properly how the non-linear algorithm and the OPA-ALS methods

compete, in a first experiment all OPA-ALS algorithms were run without applying any

constraint in D. The influence of applying non-negativity to the D matrix will be

discussed later. The results of the five methods (OPA1-ALS1, OPA1-ALS2, OPA2-

ALS1, OPA2-ALS2 and non-linear algorithm) are presented in Tables A2.1, A2.2 and

A2.3, respectively. As can be seen attending to the residuals, the OPA1-ALS2 (and in a

lesser extent OPA2-ALS2) method yields the lowest residual. This is not surprising,

since one should take into account that the ALS2 method yields unconstrained P and

constrained D. Moreover, in this particular case no constraints were applied to D,

therefore the results given by ALS2 methods are completely unconstrained. The more

the constrained the solution is, the higher the SSR will be, since the modeled function

has less flexibility to adapt to the experimental data. Therefore, the comparison of ALS2

with ALS1 or non-linear algorithm is not fair.

Appendix 2

155

Table A2.1. SSR (×1017) obtained with the five algorithms explained in the text. No constraint was applied to D.

Temperature OPA1-ALS1

OPA1-ALS2

OPA2-ALS1

OPA2-ALS2 non-linear

1 1.91 0.46 7.67 0.47 1.31

2 2.06 0.17 9.29 0.18 0.88

3 2.92 0.32 23.83 0.36 1.54

4 0.66 0.16 5.78 0.17 0.45


OPA1-ALS2

OPA2-ALS1


Table A2.2. T2a values obtained with the five algorithms explained in the text. No constraint was applied to D.

1 0.011 0.011 0.075 0.312 0.010

2 0.044 0.033 0.464 0.495 0.051

3 0.076 0.054 0.670 0.725 0.084

4 0.109 0.073 0.687 0.743 0.109

Table A2.3. T2b values obtained with the five algorithms explained in the text. No constraint was applied to D.


OPA1-ALS2

OPA2-ALS1


1 0.482 0.482 0.015 0.040 0.064

2 0.899 0.922 0.202 0.154 0.564

3 1.392 1.338 0.417 0.314 0.910

4 1.667 1.581 0.420 0.322 1.084

Appendix 2

156

On the contrary, ALS1 and the non-linear method can be compared properly, since both

yield to constrained P and unconstrained D. Among these three algorithms, non-linear

algorithm generated the lowest SSR. Thus, having the possibility of hard modeling, the

combination of linear and non-linear fitting performs better than ALS-based methods. It

seems that, when the non-linear algorithm governs the algorithm architecture, the model

parameters (T2a and T2b) are varied in a more smart way to get a better solution.

The solutions of the five algorithms are presented graphically in Figure A2.1 for the

highest temperature. As can be seen, only OPA1 algorithms and non-linear algorithm

give reasonable results on NMR spectra. As it was discussed before, OPA2 method

failed to yield reasonably good initial estimates, and a subsequent ALS method could

not conduct the solution towards the correct one (independently if it is ALS1 or ALS2).

In these cases, the experimental decays found were too correlated, and, therefore P was

nearly singular. This caused a significant variance inflation when P is inverted -Eq. (8)-

carrying out a significant error in D (NMR spectra). On the other hand, ALS2 methods

do not result in pure exponential decays (P is unconstrained at the end of one iteration),

so both T2a and T2b obtained do not correspond with the final result. Therefore, only

OPA1-ALS1 is competitive with non-linear, performing the former worse than the

latter.

The only inconvenience of LINOL (compared to ALS-based methods) is the

computation time. Mean computation times for the four temperatures were 10 s (OPA1-

ALS1) 2 s (OPA1-ALS2) 1 s (OPA2-ALS1) 1 s (OPA2-ALS2) and 98 s (LINOL).

OPA-based algorithms terminated after two iterations.

One can think that operating the ALS-based algorithms with constrained D (applying

non-negativity to the solutions) can conduct the ALS algorithm to a better solution.

Table A2.4 presents the SSR obtained with the four algorithms. The quality of the

results was indeed increased in OPA2-ALS1 algorithm, but however the results were in

general still the worse: applying the non-negativity constraint could not solve the

problem of bad initial estimates. The rest of the four algorithms yielded a higher SSR

when constraints are applied. As commented, the more the constrained the solution is,

the higher the SSR will be. This does not happen in OPA2-ALS1, since the (bad) initial

estimates are improved by means of imposing constraints in D. In the case of OPA2-

Appendix 2

157

Figure A2.1. Results of a two-component decomposition after applying the five algorithms described in this article to the NMR experiments at T = 505 K. Parts a-e represent NMR peak profiles, whereas parts f-j depict time decays. Algorithms: a,f OPA1-ALS1; b,g OPA1-ALS2; c,h OPA2-ALS1; d,i OPA2-ALS2; e,j non-linear algorithm.

Appendix 2

158

ALS2 this last effect is less noticeable since the SSR are computed in the final result

(with constrained D), which certainly increases the residuals. Computation times

increased in relation with unconstrained D: 24 s (OPA1-ALS1) 8 s (OPA2-ALS1) 2 s

(OPA2-ALS2) and 10 s (OPA2-ALS2).

Table A2.4. SSR values obtained with the four ALS algorithms. Non-negativity constraint was applied to D.


OPA1-ALS2

OPA2-ALS1

OPA2-ALS2

1 1.91 0.47 7.55 1.06

2 2.22 0.17 1.09 0.18

3 3.39 0.32 10.75 5.60

4 0.86 0.16 2.60 1.50

4. Conclusions

MCR methods are a family of techniques aimed to separate the different sources of

variance of bilinear data. This can be used to separate the (partially overlapped) signals

obtained with second-order instruments. Either soft- or hard-modelling can be applied to

MCR methods. When hard-modelling is applied, a mathematical model is applied inside

the MCR algorithm to fit the data (as an extra constrain). The application of hard-

modelling opens the possibility to re-design the MCR algorithm with a different

architecture. Instead of fitting the model inside the MCR algorithm, the MCR algorithm

is included inside a non-linear fitting method. The non-linear method finds the

parameters of the mathematical model, and a linear regression (half of a loop of the

MCR algorithm) is applied each time the function is evaluated.

The method has been tested with solid-state Hahn-Echo NMR data from polymer

networks. This kind of experiments produces two-way data, which can be treated as

bilinear. The goal of the experiment was to separate the contributions of different parts

of a polymer network. In this particular example, the non-linear method performed

Appendix 2

159

better than the MCR algorithm. The method might be applied to other examples of hard-

modelling of bilinear data.

Appendix 2

160

References

[1] Booksh, K. S.; Kowalski, B. R. K. Anal. Chem. 1994, 66, 782.

[2] Cuesta-Sánchez, F.; Toft, J.; van den Bogaert, B.; Massart, D. L. Anal. Chem. 1996, 68, 79.

[3] de Juan, A.; Tauler, R. Anal. Chim. Acta 2003, 500, 195.

[4] Faber, N. M.; Bro, R.; Hopke, P. K. Chemom. Intel. Lab. Syst. 2003, 65, 119.

[5] Erickson, B. C.; Pell, R. J.; Kowalski, B. R. Talanta 1991, 38, 1459.

[6] de Juan, A.; Maeder, M.; Martínez, M.; Tauler, R. Chemom. Intel. Lab. Syst. 2000, 54, 123.

[7] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in C, Second Edition, Cambridge University Press 1992.

Appendix 3

161

Appendix 3

Calculation of the (mean) number of monomer units between two cross-linked

junctions

To better interpret the NMR spectral data and the T2H-relaxation measurements we

propose a simulation, based on the assumption of a total (100%) conversion of the

network (full cross-linking). Starting with the relative weight of the monomer units and

their molecular weights, mol percents of these monomers in the unsaturated polyester

network (UP) were obtained (see Table A3.1).

Figure A3.1. Chemical structure of UP network.

Due to the excess of styrene to MA one could expect, in the ideal cross-link reaction,

that all the double bonds of MA have reacted to form the polyester/PS chemical cross-

linked network (Figure A3.1).

Appendix 3

162

Table A3.1. mol % of monomer in the (UP) network

Monomer unit OPA PG MA PS EG Sum

MG of units 148 76 98 104 62

Weight-fraction of 5 1.5 0.5 1.0 2 5

Weight-% 19.2 6.4 12.8 36 25.6 100

n mol in 100 gr net 0.130 0.084 0.131 0.346 0.413

From this assumption, the estimated mean number of styrene units per MA units can be

estimated, as well as the mean number of polyether length per MA-unit:

- PS/MA=0.346/0.131 = 2.64. (1)

- (PG+EG) / (MA) = (0.084+0.413) / (0.131) =3.79. (2)

One may note that (1) can also correspond to the number of PS dangling ends.

Furthermore, the mean number of monomer units in the polyester between two MA-

cross-links can also be derived:

(OPA+PG+MA+EG) / (MA) = (0.130+0.084+0.131+0.413) / (0.131) = 5.79 (3)

No crystallinity in this chemical network should be expected. The mean lengths of the

cross-links are too short. The PS cross-linked junctions correspond to (2*2.64) = 5.28

monomer units and the mean length between the cross points of the polyester backbone

corresponds to 5.79 (3) monomer units as mentioned above. No long linear chains that

can crystallize are present. The extra methyl group of the PG-units is also in favor of

the amorphous phase.

Above Tg (e.g. at 450 K) we expect a comparable molecular mobility for the polyester

backbone and for the PS cross-links, but a different mobility (faster) for the dangling

ends.

The proton spectra above Tg most probably will show the mobile dangling ends,

composed of aromatic signals of PS and OPA around 7 ppm, of ending polyether signals

Appendix 3

163

around 4 ppm and between 1 and 2 ppm the Me-signals from PG together with CH2-

signals from PS-ends.

Table A3.2. Estimation of proton mol % of full cross-linked polyester / polystyrene network

Type of

protons

shifts

in ppm structure mols * n protons

n mol

protons

proton

mol %

Aromatic +/- 7 PS + OPA .346*5+.130*4 2.25 38.5

Glycolic +/- 4 PG + EG .084*3+.413*4 1.90 32.5

Aliphatic 1 to 2 PG + PS + MA .084*3+.346*3+.131*3 1.69 28.9

Acknowledgments

165

Acknowledgements

With pleasure, I come to the acknowledgements.

This thesis carries only my name as author but is the achievement of my own effort

together with several contributions. I am deeply indebted to many people who shared

my Dutch experience so far and made this PhD successful.

I would like to dedicate the first tribute to my promotor Prof.dr. J. Gelan: without your

help, availability, and guidance, this PhD could not have been completed. Your trust

together with the many discussions we had in Hasselt, strengthened my determination: I

owe you a big thank you.

I also would like to express my sincere appreciation to my copromotor Prof.dr. P.J.

Lemstra. You gave me the opportunity to join your research group SKT to carry out my

PhD work. Thank you for your frank approach, and for your always smart and

appropriate observations.

I wish to extend my thanks to the members of the Committee for reading my thesis and

providing constructive comments. My gratitude goes especially to Prof. J. Amouroux:

you accepted me in your prestigious department at Ecole de Chimie Paris and you

considerably helped to shape my career. Your devotion to science and students made

you a reference to many. Also, special thanks to Prof. Hans Geerlings: Thank you

Professor for your availability to discuss, and willingness to help and share your

knowledge. Last but not least I would like to thank Prof.dr. Adriaensens, Dr.ir.

Goossens, and Dr.ir. Magusin for sharing their advice and comments.

I am grateful to all the members of the SKT department. From all of you, I had

something to learn. Thank you for creating such an inspiring working atmosphere, and

for being always so supportive and helpful. In particular, many thanks to Otto van

Asselen for helping me with the FTIR experiment, also for his special kindness and

availability. A big thank to Jules H & K, Martijn, Lijing, Thierry, Joost, Blanca, Casper,

a special thanks to Elly and Tran for helping arranging the defense; also thanks to Bob,

Ineke, Said, and finally to my dearest Cees and Irina. I am also extremely thankful to

Acknowledgments

166

Jacques Schievink from VSSD, the Lucky Luke of the printer: without your flexibility I

would have never made it on time.

Like all experimental projects, this project required much cooperation, with different

Universities and Institutes. Thank you Brahim from Molecular Heterogeneous Catalysis

group in TUe (SKA): your help with the NMR experiments was very much appreciated.

My gratitude goes also to all former and actual members of the Polymer Analysis Group

in Amsterdam (UvA). I would like to particularly thank Prof. Schoenmakers, Wijbren

for your constant and invaluable help in the laboratory, Wim, Philippo, Mauro and also

you, Gabriel Vivo, for your amazing skills and for helping me with the data treatment

part. Next, I wish to record my very sincere and grateful thanks to Prof. B. Jerome from

the FNWI-UvA: you have been so supportive and have always provided very strong

advice. A Woman in Science, and an example to me! I would also like to express my

acknowledgements to the DPI (Dutch Polymer Institute) for financing this PhD project;

to Y. Krager and P. Steeman from DSM, for providing me with samples and for helping

me with the dielectric spectroscopy experiments (DES). Furthermore, thank you to J.

Bloomberg from Shell, for making available the cryogenic grinder.

I experienced the same friendly atmosphere during my work at the Chemistry

Department of Hasselt University (Diepenbeek campus): I offer my most sincere thanks

to the members of the NMR team for their help and guidance.

I am also extremely grateful to Shell for given me the opportunity to finish my thesis

while working for the Company. I am particularly in debt to Rini Reynhout: thank you

Rini for your endless support and precious advice, I will carry them with me forever;

also many thanks to Gerard de Nazelle: your passionate dedication to science and your

determination served to strengthen my own determination to finish. Your support was

crucial in completing this manuscript.

I would also like to express appreciation to Sander van Bavel: thank you Sander for

being such a good friend and for sharing your own PhD experience to help me progress.

I cannot forget Hilbrand Klaver: thanks especially for your patience and bravery to deal

with my high level of stress while sharing the same office. I cannot forget to thank Wim

Wieldraaijer, Frank Niele, for being so inspiring; Michael, Heiko, Alfred, Zakaria,

Patrice, Wouter, Minke, Jan, Joost, Graham and Shanjoy for the many advices. Thank

Acknowledgments

167

you as well to Hans Gosselink, Herman Kuipers, Herman van Wechem and Hans

Stapersma for their flexibility and support.

This thesis meant also for me the opportunity to discover The Netherlands and enjoy the

Dutch way of living (how many bicycles did I buy and lose in Amsterdam?…). I am so

thankful to all of you, who have been part of my Dutch life, and have filled it with so

many discoveries and adventures, have made it full of great fun and so rich. The list is

very long but I cannot forget Jasper, Robin, Manu, Elio, Luca, Fatima, Javier,

Alexander, Caroline and Stephanie (my Charlie’s angels), Alessandra and Luc, Viney,

Nawar (Noa), Marina, Joost, Marcelino Dorothee, Erika, Estella, Cedric and Karima,

Philippe, Lorn and Alain, Nizar, Karim, Chafik, Kosta, Hugo, Leena, Maikel, Sylvia,

Alex, Lokash, Sebastien, Pilar, Sander and my dear Hawaiian friend, Noah Johnson.

I am particularly grateful to Herman, Suzanne, Maaike, Marie Roelofsen and Bastiaan

who always make me feel at home despite the cultural differences and the Dutch

weather; thanks to Mehdi: “ton soutien et ta patience m’ ont aidée dans des moments

difficiles”; a special thank you as well to Basilius: My dear friend, thank you for the

fantastic years we had, and all your support! I will never forget ‘’The Beach’’. Thanks

to my friend Susan Behr: your advices were very precious and helped me progress. And

last but not least, thank you to my two paranimfen, Christien Oele and Valentine

Kreykamp. Thank you ladies it is such an honor to have you as my paranimfen, thanks

for being such great friends.

And further South, there you were all my faithful friends in France and in Algeria,

always close to my heart. Again the list is too long but special thanks to Sabry: c’ est toi

qui m’a fait découvrir ce pays et qui m’a aidé à m’y installer. Je n’oublierai jamais! Un

immense merci à oncle Rabah et ma tante Claude, ainsi qu’à Cherifa et Moh, sans

oublier mes chères et tendres tantes Zina et Louisa, mon oncle Mahdi et Danielle,

‘’tonton’’ Youcef, Lilia, Jelel, Selim, Karim, Réda et Mehdi, Setti, Akli, Baya, Tarik,

Mina, Fayna, Hayet, Hocine, Selma, Amar, Bachira, Leila, Hadjira, Nadra, Nezim, Ines,

Meriem K.

This thesis with the accompanying hectic work is now coming to an end: it has been a

large part of my life, during which I have always hoped to one day complete this

manuscript. Yet, three of my closest friends have shown me a much deeper meaning of

Acknowledgments

168

hope, something which echoes Vaclav Havel’s words: “hope is not the conviction that

something will turn out well…” [my thesis for instance…] “… but the certainty that

something makes sense, regardless of how it turns out”. Laurus, Manola, Farah, my

dearest friends: it is your lives, your examples beyond your deaths, which help me make

sense of our world every day – and make me hold on firmly to hope. I offer you this

thesis, with all my hope and efforts.

And finally, last but by every means first, how will I ever be able to tell you all my

thanks, to you my parents, and to you, my brother. You have tirelessly provided me with

the moral support, your encouragement and understanding and above all, your

unconditional love. Un spécial merci à Maman: cette thèse, je te la dois! Merci pour

avoir toujours cru en moi!

Anybody who has been seriously engaged in scientific work of any kind

realizes that over the entrance to the gates of the temple of science are written the

words: ‘Ye must have faith’.

Max Planck.

Curriculum Vitae

169

Curriculum Vitae

The author of this thesis Maya Ziari was born in Algiers, Algeria. After obtaining her

‘baccalaureat’ in Mathematics and Physics in Nice (France), she completed a bachelor

degree (B.Sc) in Physical Chemistry (University Paris V), followed by a master degree

(M.Sc) in Chemical Engineering in 1998 (University Paris VI-ENSCP) under the

supervision of Prof. J. Amouroux within the Plasma Process department. During her

master, she carried out a traineeship at the Institute for Fundamental Electronics in

Orsay (University Paris XI).

At the beginning of 2000 she started working as a Research Associate at the University

of Amsterdam (UvA) in The Netherlands, on a project granted by Dow Chemicals to

investigate Ultra-High-Molecular-Mass (UHMM) polymers using field flow

fractionation (FFF) and light scattering techniques (LALS/MALS).

In 2002, she started a PhD at Eindhoven University of Technology (TU/e) in the

Polymer Technology Department under the supervision of Prof.dr. P. J. Lemstra and

Prof.dr. J. Gelan. During her PhD study, the author also obtained the diploma of the

‘Register Polymerkundige (RPK)’ organized by the ‘National Dutch Graduate School of

Polymer Science and Technology’ (PTN, Polymer Technologie Nederland).

In 2006, she joined in Amsterdam the Department of Innovation and Research (GSIR)

in Shell Global Solutions International BV, as a Research Associate on a contractor

basis. In 2009, she joined Shell (NAM) on a permanent contract and was transferred to a

new role as Appraisal Engineer in the Department of Exploration and Production.

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