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LSU Doctoral Dissertations Graduate School

2002

Thermal rheological analysis of cure process ofepoxy prepregLiangfeng SunLouisiana State University and Agricultural and Mechanical College, lsun@lsu.edu

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Recommended CitationSun, Liangfeng, "Thermal rheological analysis of cure process of epoxy prepreg" (2002). LSU Doctoral Dissertations. 952.https://digitalcommons.lsu.edu/gradschool_dissertations/952

THERMAL RHEOLOGICAL ANALYSIS OF CURE PROCESS OF EPOXY PREPREG

A Dissertation Submitted to the Graduate Faculty of the

Louisiana State University and Agricultural and Mechanical College

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in

The Department of Chemical Engineering

by Liangfeng Sun

B.E., Shandong Institute of Light Industry, 1988 M.S., Tianjin University, 1992

M.S., Louisiana State University, 1999 May, 2002

To My Dear Wife, Xiaoqun, My Son, Kevin, and My Daughter, Lucy

ii

ACKNOWLEDGEMENTS

The author wants to express his great gratitude to his advisor, Dr. Arthur M.

Sterling and co-advisor, Dr. Su-Seng Pang, for their guidance, support, and

encouragement in this entire research work. It is their continuous support to make the

author complete his research work with little difficulty. The author would also like to

thank Dr. Ioan I. Negulescu for his assistance with differential scanning calorimetry and

thermal gravimeter measurements. Thanks give to Dr. Armando B. Corripio, Dr. Andrew

W. Maverick, Dr. Elizabeth J. Podlaha, Dr. Thomas J. Cleij, and Dr. Michael A.

Stubblefield for their kindness to serve on author’s committee and their advices and

comments.

The author thanks Dr. Patrick F. Mensah for providing part of the test material, Dr.

Jo Dweck, a visiting professor in Chemistry Department, for his discussion on the

thermal experimental data, Paul Rodriguez for his help to maintain the Rheometer, and

other staff in Chemical Engineering Department for their kind help. Special thanks give

to Dr. John R. Collier, the author’s former major advisor for master’s and doctoral

programs, and his wife, Dr. Billie J. Collier, for their guidance, support, and help in the

past years.

The author would like to express his great gratitude to his wife, Xiaoqun, his son

Kevin, his daughter Lucy, and his parents for their endless support, encouragement, and

help during his study.

iii

TABLE OF CONTENTS ACKNOWLEDGEMENTS .............................................................................................. iii LIST OF TABLES .............................................................................................................vi LIST OF FIGURES...........................................................................................................vii ABSTRACT .......................................................................................................................xi CHAPTER 1. INTRODUCTION.......................................................................................1 CHAPTER 2. BACKGROUND.........................................................................................6

2.1 Epoxy Resins.....................................................................................................6 2.2 Curing Agents....................................................................................................7 2.3 Curing Reactions .............................................................................................11 2.4 Fiber-reinforced Composites of Epoxy Resins ...............................................15 2.5 Curing Analysis Techniques ...........................................................................16

2.5.1 Differential Scanning Calorimetry Analysis .................................... 17 2.5.2 Rheological Analysis........................................................................ 17 2.5.3 Fourier Transform Infrared Spectroscopy........................................ 18 2.5.4 High Performance Liquid Chromatography .................................... 19

CHAPTER 3. MATERIAL AND MAIN EQUIPMENT.................................................20

3.1 Experimental Material .....................................................................................20 3.2 Isothermal DSC Measurements.......................................................................20 3.3 Dynamic Scanning DSC Measurements .........................................................20 3.4 Isothermal and Dynamic Scanning Rheological Measurements .....................22

CHAPTER 4. METHODOLOGY....................................................................................25

4.1 Autocatalytic Models for Isothermal Curing Process .....................................25 4.2 Autocatalytic Models for Dynamic Scanning Curing Process ........................26

4.2.1 Model Analysis Based on Kissinger and Ozawa Kinetic Approach ............................................................................. 26

4.2.2 Model Analysis Based on Borchardt and Daniels Approach .......................................................................................... 27

4.3 Rheological Models.........................................................................................28 4.3.1 Gel Time Model ............................................................................... 28 4.3.2 Viscosity Models .............................................................................. 29

4.4 Glass Transition Temperature (Tg) Models.....................................................32 CHAPTER 5. RESULTS AND DISCUSSION ...............................................................34

5.1 Isothermal DSC Cure Analysis .......................................................................34 5.1.1 Analysis of Cure Reaction Heat ....................................................... 34 5.1.2 Degree of Cure and Cure Rate ......................................................... 41 5.1.3 Cure Reaction Modeling .................................................................. 44

iv

5.2 Dynamic Scanning DSC Cure Analysis ..........................................................53 5.2.1 The Exothermal Peak Analysis of Dynamic Cure

Process.............................................................................................. 53 5.2.2 The Kinetic Analysis Based on Kissinger and Ozawa

Approach .......................................................................................... 59 5.2.3 The Kinetic Analysis Based on Borchardt and Daniels

Approach .......................................................................................... 70 5.3 Isothermal and Dynamic Scanning Rheological Cure Analysis......................76

5.3.1 Gel Time and Apparent Activation Energy (Ea) .............................. 79 5.3.2 Rheological Modeling ...................................................................... 82

5.4 Characterization of the Cure Process of Epoxy Prepreg .................................91 5.4.1 Further DSC Thermal Analysis ........................................................ 92 5.4.2 Glass Transition Temperature .......................................................... 98 5.4.3 Isothermal TTT Cure Diagram and CTT Cure

Diagram.......................................................................................... 101 5.4.4 Thermal Stability of Epoxy Prepreg............................................... 103 5.4.5 Microstructure of the Epoxy Prepreg Sample ................................ 105

CHAPTER 6. CONCLUSIONS AND FUTURE WORK .............................................107 REFERENCES................................................................................................................114 APPENDIX A. MORE ISOTHERMAL HEAT FLOW CURVES ...............................120 APPENDIX B. ADDITIONAL CURE RATE CURVES...............................................123 APPENDIX C. ADDITIONAL VISCOSITY CURVES ................................................126 VITA ...............................................................................................................................127

v

LIST OF TABLES 1. Chemical Structure of Some Epoxy Resins ............................................................ 8 2. Some Type 1 Curing Agents and Their Chemical Structures ............................... 10 3. Kinetic Parameters of the Autocatalytic Model for Isothermal Cure

Process................................................................................................................... 47 4. Values of Constant C and Critical Degree of Cure cα for Autocatalytic

Model for Isothermal Cure Process....................................................................... 51 5. Total Dynamic Cure Reaction Heat at Different Heating Rates ........................... 55 6. The Dependence of Peak Properties on the Heating Rate..................................... 59 7. Dynamic Kinetic Parameters Obtained by Method Based on Kissinger

and Ozawa Approach ............................................................................................ 66 8. Apparent Dynamic Kinetic Parameters Obtained by Method Based on

Borchardt and Daniels Approach .......................................................................... 72 9. Gel Time at Different Temperatures and the Activation Energy .......................... 80 10. Initial Viscosity η0 and Rate Constant k in Eqs. (27) and (30) of

Viscosity Models

................................................................................................... 82 11. Kinetic Parameters in Eqs. (31) and (33) of the new Viscosity Model................. 86 12. The Maximum Cure Heat at Different Isothermal Temperatures ......................... 92

vi

LIST OF FIGURES 1. A Schematic Drawing of DSC 220 from Seiko Inc. in Japan ............................... 21 2. The Bohlin VOR Rheometer and its Main Part .................................................... 23 3. Dynamic Cure From 30 to 300 oC with Heating Rate of 2 oC/min ....................... 35 4. Isothermal Cure of Epoxy Prepreg at Selected Temperatures for the

Varying Time ........................................................................................................ 36 5. Combinations of Dynamic and Isothermal Cure of Epoxy Prepreg at

Different Isothermal Temperatures ....................................................................... 39 6. Degree of Cure as a Function of Time at Different Isothermal

Temperatures ......................................................................................................... 42 7. Cure Rate as a Function of Time at Different Isothermal Temperatures .............. 43 8. The Logarithm Plot of [r/(1-α)n-k1] vs. α at 160 oC ............................................ 45 9. Comparisons of Experimental Data with the Autocatalytic Model as

Eq. (1) at the Select Isothermal Cure Temperatures.............................................. 48 10. Rate Constants k1 and k2 in Eq. (2) as a Function of Temperature........................ 49 11. Variation of Diffusion Factor with Degree of Cure at Different

Temperatures ......................................................................................................... 52 12. Heat Flow Changes at the Heating Rates of 2, 5, 10, 15, and 20 oC/min:

(a) From Right to Left by Peak; (b) From Left to Right by Peak.......................... 54 13. Non-reversal Heat Flow as a Function of Temperature for Samples

Partially Cured to, From Top to Bottom by Peak, Uncured, 170, 190, 200, 210, 220, 230, 240, and 270 oC, at a Heating Rate of 5 oC/min, Starting from Room Temperature. ....................................................................... 57

14. Plots of Logarithm Heating Rate vs. the Reciprocal of the Absolute

Exothermal Peak Temperature. The Activation Energies Were Determined by Peak Temperatures at Heating Rates of 2, 5, 10, 15, and 20 oC/min........................................................................................................ 61

15. The Iso-conversional Plots for Logarithm Heating Rate vs. Reciprocal

of the Absolute Temperature. The Apparent Activation Energies Were Determined by Iso-conversional Plots .................................................................. 63

vii

16. The Activation Energy, Calculated by the Iso-conversional Plots, as a Function of Degree of Cure................................................................................... 63

17. Degree of Cure as a Function of Temperature Calculated by Method

Based on Kissinger and Ozawa Approach and its Comparison to the Experimental Value............................................................................................... 69

18. Cure Rate as a Function of Temperature Calculated by Method Based

on Kissinger and Ozawa Approach and its Comparison to the Experimental Value............................................................................................... 71

19. Modeling of Logarithmic Cure Rate vs. the Reciprocal of the Absolute

Temperature and Comparing to the Experimental Values .................................... 73 20. Plots of Logarithm ( ))1(/()/ nmdtd ααα − vs. the Reciprocal of the

Absolute Temperature at Heating Rates of 2, 5, 10, 15, and 20 oC/min From Right to Left, Respectively. The Apparent Pre-exponents and Activation Energies Were Determined From the Intercepts and Slopes of the Linear Parts of the Curves........................................................................... 75

21. Degree of Cure as a Function of Temperature Calculated by Method

Based on Borchardt and Daniels Approach at Heating Rates of 2, 5, 10, 15, and 20 oC/min from Left to Right, Respectively and its Comparison to the Experimental Value ................................................................ 77

22. Cure Rate as a Function of Temperature Calculated by Method Base

on Borchardt and Daniels Approach at Heating Rates of 2, 5, 10, 15, and 20 oC/min From Left to Right by Peak and its Comparison to the Experimental Value............................................................................................... 77

23. Dependence of Storage Modulus G' and Loss Modulus G'' on Time at

110 oC .................................................................................................................... 79 24. Dependence of Phase Angle on Time at Isothermal Conditions........................... 81 25. Gel Time as a Function of Isothermal Cure Temperature..................................... 81 26. Initial Viscosity in Eq. (26) vs. Isothermal Cure Temperature and the

Determination of kinetics Parameters by Arrhenius Equation.............................. 84 27. Rate Constant in Eq. (26) vs. Temperature and the Determination of

Kinetic Parameters by Arrhenius Equation ........................................................... 84 28. Experimental and Calculated Viscosity Profiles at Selected Isothermal

Temperatures ......................................................................................................... 85

viii

29. Critical Time in Eq. (31) vs. Cure Temperature.................................................... 87 30. Rate Constant in Eq. (31) vs. Isothermal Cure Temperature and the

Determination of Kinetic Parameters by Arrhenius Equation............................... 87 31. The Logarithmic Plot of Viscosity vs. Temperature at Heating Rates of

2 and 5 oC/min ....................................................................................................... 88 32. Experimental and Calculated Dynamic Viscosity Profiles at a Heating

Rate of 2 oC/min. The Reaction Order n Equals 1.13 in Eq. (30). The Critical Time tc in Eq. (33) is 81.2 minutes and the Corresponding Critical Temperature is 187.4 oC .......................................................................... 90

33. Experimental and Calculated Dynamic Viscosity Profiles at a Heating

Rate of 5 oC/min. The Reaction Order n Equals 1.20 in Eq. (30). The Critical Time tc in Eq. (33) is 39.4 minutes and the Corresponding Critical Temperature is 221.8 oC........................................................................... 90

34. Generalized Isothermal Cure Phase Diagram of Time-Temperature-

Transformation ...................................................................................................... 93 35. Iso-conversional Plots of 1/T vs. Logarithmic Time for Epoxy Prepreg .............. 94 36. The Maximum Degree of Cure as a Function of Isothermal Cure

Temperature........................................................................................................... 96 37. Half-life and the Maximum Cure Time for the Isothermal Curing

Process as a Function of Isothermal Cure Temperature........................................ 96 38. The Comparison of Dynamic Cure Heats Between the Fresh and

Partially Cured Samples at a Heating Rate of 10 oC/min. The Ration of the Reaction Heat of the Cured Sample to the Reaction Heat of the Fresh Sample is 0.51 ............................................................................................. 98

39. Temperature Profile in the Modulated DSC Mode ............................................. 100 40. Reversal and Non-reversal Heat Flow for Fresh Prepreg During

Dynamic Scanning at 5 oC/min ........................................................................... 100 41. The Dependence of Tg on Degree of Cure........................................................... 101 42. The TTT Diagram of Isothermal Cure for the Epoxy Prepreg System ............... 102 43. The CTT Phase Cure Diagram for the Epoxy Prepreg System........................... 103 44. TGA Curves of Epoxy Prepreg at a Heating Rate of 5 oC/min ........................... 104

ix

45. The Comparison of the Surfaces Between The Partially (Left) and Fully (Right) Cured Samples............................................................................... 106

46. The Comparisons of the Cross Sections Between the Partially (Left)

and Fully (Right) Cured Samples ........................................................................ 106 47. The Variation of Isothermal Heat Flow With Time

(a) 110-140 oC ..................................................................................................... 120 (b) 150-180 oC ..................................................................................................... 121 (c) 190-220 oC ..................................................................................................... 122

48. Cure Rate vs. Degree of Cure

(a) 110 oC............................................................................................................. 123 (b) 130 oC ............................................................................................................ 123 (c) 150 oC............................................................................................................. 124 (d) 170 oC ............................................................................................................ 124 (e) 190 oC............................................................................................................. 125 (f) 210 oC ............................................................................................................ 125

49. The Variation of Viscosity With Time

(a) 150 oC............................................................................................................. 126 (b) 170 oC ............................................................................................................ 126

x

ABSTRACT

The cure process of epoxy prepreg used as composite pipe joints was studied by

means of differential scanning calorimetry (DSC), Bohlin Rheometer and other

techniques. Isothermal DSC measurements were conducted between 110 and 220 oC, at

10 oC intervals. The results show that the complete cure reaction could be achieved at

220 oC. The isothermal cure process was simulated with the four-parameter autocatalytic

model. Except in the late stage of cure reaction, the model agrees well with the

experimental data, especially at high temperatures. To account for the effect of diffusion

on the cure rate, a diffusion factor was introduced into the model. The modified model

greatly improved the predicated data at the late stage of cure reaction.

The dynamic cure process was different from the isothermal cure process in that it

is composed of two cure reactions. For dynamic cure process, a three-parameter

autocatalytic model was used. The parameters in the model were determined by two

methods. One was based on Kissinger and Ozawa approach. The whole curing process

was modeled with two reactions. Another method was based on Borchardt and Daniels

kinetic approach with the whole curing process modeled with one reaction. The fitted

results by the first and second methods agreed well with the experimental values in the

late and early cure stages, respectively.

Rheological properties of epoxy prepreg are closely related to the cure process.

With the development of the cure reaction, gelation occurs and epoxy prepreg becomes

difficult to process. As the temperature increases, the gel time decreases. Viscosity

profiles were described by different models. Except for the first and nth order viscosity

xi

models, new viscosity models were proposed The proposed new viscosity models are

better than the old models for both the isothermal and dynamic cure processes.

To graphically represent the phase changes of the cure process, the isothermal

cure diagrams of time-temperature-transformation (TTT) and conversion-temperature-

transformation (CTT) are constructed. Each region in TTT and CTT diagrams

corresponds with the phase state of the cure process, so the cure mechanism is clearly

shown in the diagrams.

xii

CHAPTER 1. INTRODUCTION

Epoxy resins and epoxy prepreg are important thermosetting polymers. Prepregs

are often used to produce various composite products. Prepregs are short for

preimpregnated. They refer to fiber-reinforcing materials impregnated with resin prior to

the molding process and cured by the application of heat. The reinforcing fibers may be

fiberglass, Kevlar, carbon, polyester, nylon and ceramic; but the most commonly used

one is fiberglass. High strength fibers such as, Kevlar and carbon, are used to make

composites with special properties. The polymer matrix may be either thermoset or

thermoplastic resins. Their applications as composite materials are under continuing

development in many fields. Epoxy prepreg can be molded to the desired shape according

to the needs of the final products, and cured by the application of heat. The studies on

epoxy prepreg as joints to composite pipe systems by a heat-activated method have been

reported (Stubblefield et al., 1998). We are presently working to join composite-to-

composite pipes with different joint configurations. These applications involve curing

cycles of the epoxy prepreg, in which different isothermal and dynamic curing processes

are applied. Curing cycles determine the degree of cure of epoxy prepreg and have an

important effect on the mechanical properties of the final products. Optimal curing

schedules are the keys to achieve efficiently the desired properties of the cured materials.

Although companies manufacturing the commercial epoxy prepreg materials usually

suggest curing cycles for custom applications, their curing cycles may not be the optimal

ones for special applications. In order to optimize the curing cycles for epoxy prepreg

used as composite pipe joints, it is necessary to understand the cure kinetics and

characteristics of epoxy prepreg in more detail.

1

A number of methods were used to analyze the cure kinetics and physical

properties of epoxy prepreg in this study. Differential Scanning Calorimetry (DSC)

analysis is based on the heat flow change of the epoxy prepreg sample during the cure

process. It is assumed that the heat of cure reaction equals the total area under the heat

flow-time curve. The degree of cure is proportional to the reaction heat. It was

calculated either by the residual heat or by the reaction heat at a particular time. Different

kinetic models for DSC cure analysis are available. The simpler model applied to DSC

data was the model from the mechanism of an nth order reaction. This model gave a

good fit to the experimental data only in a limited range of degree of cure (Barton and

Greenfield, 1992). More complicated models for isothermal and dynamic curing process

assumed an autocatalytic mechanism (Horie et al., 1970). The autocatalytic model may

have different forms depending on whether the value of initial cure rate is zero or not.

The kinetic parameters are obtained from the DSC analysis. At isothermal conditions,

the rate constant and reaction orders are determined at each cure temperature.

The dynamic cure kinetics is very different from the isothermal cure kinetics. The

rate constant is a function of temperature, so it will change during the dynamic heating

process. The kinetic parameters such as pre-exponential factor, activation energy, and

reaction orders are determined at each heating rate and used to predict degree of cure and

cure rate of the dynamic cure process.

Applications of epoxy prepreg require understanding its rheological properties

during the cure process, as well as the cure kinetics. Rheological analysis has been used

to study the cure process of epoxy resin (Kenny et al., 1991; Wang, 1997) and epoxy

prepreg (Simon and Gillham, 1993) and is also essential to the optimization of cure cycle.

2

Like polymers, epoxy prepreg is a viscoelastic material. During a curing process under

continuous sinusoidal stresses or strains, its viscoelastic characteristics change, which is

reflected in the variations of rheological properties such as the storage modulus G', loss

modulus G", viscosity η, and loss tangent tan δ. The G' is the elastic character of the

epoxy prepreg and reflects the energy that can be recoverable. The G" represents the

viscous part of the epoxy prepreg and reflects loss energy by dissipation. Viscosity

measures the fluidity of the epoxy resin system. Higher viscosity means the lower

fluidity of the epoxy resin systems. The loss tangent tan δ equals the ratio of the loss

modulus to storage modulus. It is used to evaluate the viscoelasticity of epoxy prepreg.

The flow behavior of reacting system is closely related to the cure process. In the

early cure stage, the epoxy resin is in a liquid state. Cure reaction takes place in a

continuous liquid phase. With the advancement of the cure process, a crosslinking

reaction occurs at a critical extent of reaction. This is the onset of formation of

networking and is called the gel point (Wang and Gillham, 1993). At the gel point, epoxy

resin changes from a liquid to a rubber state. It becomes very viscous and thus difficult to

process; so the gelation has an important effect on the application process of epoxy

prepreg. Although the appearance of the gelation limits greatly the fluidity of epoxy

resins, it has little effect on the cure rate; so the gelation cannot be detected by the

analysis of cure rate, as is the case in the DSC. The gel time may be determined by a

rheological analysis of the cure process.

Besides the gel time, another important parameter is the glass transition

temperature (Tg). The Tg is the temperature region where the amorphous polymer changes

from one state to another when the temperature changes, either from glass to rubber

3

phase or from rubber to glass phase. The application temperature range of epoxy prepreg

depends on its Tg. The variation of Tg of epoxy prepreg is related to the extent of cure

reaction.

Based on the thermal and rheological analysis of the curing process, additional

information on the physical states of epoxy prepreg is obtained. Epoxy prepreg

experiences a phase transition during curing process. The construction of the phase

diagram will be beneficial to the optimization of the cure cycles.

In short, this study focuses on the thermal rheological behaviors of the epoxy

prepreg during the cure process and the optimization of the cure cycles of the epoxy

prepreg for the application as pipe joints. The specific objectives of the research are as

follows:

• To relate the heat flow or cure reaction heat to the degree of cure. The cure

process of epoxy prepreg is an exothermic process. The reaction heat released

during the cure process can be calculated from the time-dependent heat flow

curve. The relationship between the degree of cure and time will be determined.

• To related the rheological properties such as the storage and loss modules and

viscosity to the gel time and cure process. When the cure process proceeds to a

certain degree of cure, the molecular mobility of the reacting system will be

greatly limited and gelation occurs. The gel point is usually determined by the

rheological properties.

• To test the existing kinetic and viscosity models. A number of kinetic and

viscosity models have been reported recently. The first and nth order reaction

models may be used with limited accuracy. To achieve better accuracy, the

4

complicated autocatalytic reaction models will be used and compared with

experimental data.

• To propose the new models of the cure process if desirable. If the experimental

data show that the existing models are not accurate to describe the cure process,

the development of new models is necessary.

• To characterize the cure process of the epoxy prepreg. The phase transition

diagram of the cure process will be constructed and the thermal stability of the

epoxy prepreg will be analyzed.

5

CHAPTER 2. BACKGROUND

2.1 Epoxy Resins

Epoxy resins are thermosetting polymers that, before curing, have one or more

active epoxide or oxirane groups at the end(s) of the molecule and a few repeated units in

the middle of the molecule. Chemically, they can be any compounds that have one or

more 1,2-epoxy groups and can convert to thermosetting materials. Their molecular

weights can vary greatly. They exist either as liquids with lower viscosity or as solids.

Through the ring opening reaction, the active epoxide groups in the uncured epoxy can

react with many curing agents or hardeners that contain hydroxyl, carboxyl, amine, and

amino groups. Compared to other materials, epoxy resins have several unique chemical

and physical properties. Epoxy resins can be produced to have excellent chemical

resistance, excellent adhesion, good heat and electrical resistance, low shrinkage, and

good mechanical properties, such as high strength and toughness. These desirable

properties result in epoxy resins having wide markets in industry, packaging, aerospace,

construction, etc. They have found remarkable applications as bonding and adhesives,

protective coatings, electrical laminates, apparel finishes, fiber-reinforced plastics,

flooring and paving, and composite pipes. Since their first commercial production in

1940s by Devoe-Reynolds Company, the consumption of epoxy resins has grown

gradually almost every year (May, 1988). The three main manufacturers of epoxy resins

are Shell Chemical Company, Dow Chemical Company and Ciba-Geigy Plastics

Corporation. They produce most of the world’s epoxy resins. The United States and other

industrialized countries such as Japan and those in Western Europe are the main

producers and consumers of epoxy resins.

6

Since the 1930’s when the preparation of epoxy resins was patented, many types

of new epoxy resins have been developed from epoxides. Tanaka (1988) gave a complete

list of epoxides and discussed their properties and preparation. Most conventional epoxy

resins are prepared from bisphenol A and epichlorohydrin. For example, the most

commonly used epoxy resins are produced from diglycidyl ethers of bisphenol A

(DGEBA). Its properties and reaction mechanism with various curing agents have been

reported extensively (Simpson and Bidstrup, 1995; Barton et al., 1998). Other types of

epoxy resins are glycidyl ethers of novolac resins, phenoxy epoxy resins, and (cyclo)

aliphatic epoxy resins. Glycidyl ethers of novolac resins, and phenoxy epoxy resins

usually have high viscosity and better high temperatures properties while (cyclo)

aliphatic epoxy resins have low viscosity and low glass transition temperatures.

Although many accomplishments have been made in the field of epoxy resins,

researchers still make efforts to understand better their curing mechanisms, to improve

their properties, and to produce new epoxy resins.

From this literature review, it was found that the most widely used epoxy resins

were based on oxides containing aromatic groups. (Cyclo) Aliphatic epoxy resins were

seldom reported in the literature. Table 1 gives some of the commonly-used epoxy resins

and their chemical structures

2.2 Curing Agents

Curing agents play an important role in the curing process of epoxy resin because

they relate to the curing kinetics, reaction rate, gel time, degree of cure, viscosity, curing

cycle, and the final properties of the cured products. Thus many researchers have been

studying the effects of curing agents on the curing process.

7

Table 1. Chemical Structure of Some Epoxy Resins

COMPOSITION

CHEMICAL STRUCTURE

DGEBA CH CH2

n

C O

CH3

CH3

O CH2 CH

OH

CH2 O C O

CH3

CH3

CH2 CH CH2

OCH2

O

4,4′-diglycidyloxy-3,3′, 5,5′-tetramethyl biphenyl

CH2 CHO

CH2 O O

CH3

CH3 CH3

CH3

CH2 CH CH3

O

TGDDM C H2 C H C H2

O

C H2 C H

O

C H2

N NC H2

C H2 C H C H2

C H2C HO

O LCE

CH2 CHO

CH2 O CO2CO2 O CH2 CH CH2

O

POEDE CH2 CH

O

CH2 O CH2

n

CH2 O CH2 CH CH2

O

Mika and Bauer (1988) gave an overview of the epoxy curing agents and

modifiers. They discussed three main types of curing agents. The first type of curing

agents includes active hydrogen compounds and their derivatives. Compounds with

amine, amides, hydroxyl, acid or acid anhydride groups belong to this type. They usually

react with epoxy resin by polyaddition to result in an amine, ether, or ester. Aliphatic and

aromatic polyamines, polyamides, and their derivatives are the commonly used amine

type curing agents. The aliphatic amines are very reactive and have a short lifetime. Their

applications are limited because they are usually volatile, toxic or irritating to eyes and

skin and thus cause health problems. Compared to aliphatic amine, aromatic amines are

8

less reactive, less harmful to people, and need higher cure temperature and longer cure

time. Hydroxyl and anhydride curing agents are usually less reactive than amines and

require a higher cure temperature and more cure time. They have longer lifetimes.

Polyphenols are the more frequently used hydroxyl type curing agents. Polybasic acids

and acid anhydrides are the acid and anhydride type curing agents that are widely used in

the coating field. Table 2 gives a list of commonly-used type 1 curing agents and their

chemical structures.

The second type of curing agents includes the anionic and cationic initiators.

They are used to catalyze the homopolymerization of epoxy resins. Molecules, which

can provide an anion such as tertiary amine, secondary amines, and metal alkoxides are

the effective anionic initiators for epoxy resins. Molecules that can provide a cation, such

as the halides of tin, zinc, iron and the fluoroborates of these metal, are the effective

cationic initiators. The most important types of cationic initiators are the complexes of

BF3.

The third type of curing agents is called reactive crosslinkers. They usually have

higher equivalent weights and crosslink with the second hydroxyls of the epoxy resins or

by self-condensation. Examples of this type of curing agents are melamine, phenol, and

urea formaldehyde resins.

Among the three types of curing agents, compounds with active hydrogen are the

most frequently used curing agents and have gained wide commercial success. Most

anionic and cationic initiators have not been used commercially because of their long

curing cycles and other poor cured product properties. Crosslinkers are mainly used as

9

surface coatings and usually are cured at high temperatures to produce films having good

physical and chemical properties.

Table 2. Some Type 1 Curing Agents and Their Chemical Structures

COMPOUNDS

CHEMICAL STRUCTURE

DDS N H 2 S N H 2

O

O

DDM NH2 NH2 NH2

MCDEA N H2 C H2

C lC2H5

C2H5

N H2

C2H5

C2H5C l

Methylaniline NH CH3

TETA NH2 CH2CH2 2NH CH2 CH2 NH2

Phenol novolac O H

n

O H

C H 2

O H

C H 2

Phthalic anhydride

O

O

O

10

2.3 Curing Reactions

The curing reaction of epoxide is the process by which one or more kinds of

reactants, i.e., an epoxide and one or more curing agents, with or without the catalysts,

are transformed from low-molecular-weight to a highly crosslinked structure. As

mentioned earlier, the epoxy resin contains one or more 1,2-epoxide groups. Because an

oxygen atom has a high electronegativity, the chemical bonds between oxygen and

carbon atoms in the 1,2-epoxide group are the polar bonds, in which the oxygen atom

becomes partially negative, whereas the carbon atoms become partially positive. Because

the epoxide ring is strained (unstable), and polar groups (nucleophiles) can attack it, the

epoxy group is easily broken. It can react with both nucleophilic curing reagents and

electrophilic curing agents. The curing reaction is the repeated process of the ring-

opening reaction of epoxides, adding molecules and producing a higher molecular

weight and finally resulting in a three-dimensional structure. The chemical structures of

the epoxides have an important effect on the curing reactions. Tanaka and Bauer (1988)

provide more details about the relative reactivity of the various epoxides with different

curing agents and the orientation of the ring opening of epoxides. It was concluded that

the electron-withdrawing groups in the epoxides would increase the rate of reaction

when cured with nucleophilic reagents, but would decrease the rate of reaction of

epoxides when cured with electrophilic curing agents.

As discussed earlier, many curing agents may be used to react with epoxides; but

for different curing agents, there exist different mechanisms of the curing reaction. Even

for same epoxy resin systems, the cure mechanism may be different for the isothermal

and dynamic cure processes. Some of the mechanisms are presented here for reference.

11

Many polyfunctional curing agents with active hydrogen atoms such as

polyamines, polyamides, and polyphenols perform nucleophilic addition reaction with

epoxides. Tanaka and Bauer (1988) gave the following general cure reaction:

HX X CHOH

n

R CHOH

CH2 X X CH2 CHOH

R CH CH2

OR1R1 CH2

+ CH2 CH R CH CH2

OOHX XHR1

Where X represents NR2, O or S nucleophilic group or element and n is the degree of

polymerization, having a value of 0, 1, 2, …

Tanaka and Bauer (1988) discussed in detail the curing mechanisms of epoxides

with several types of curing agents. For epoxy-1-propyl phenyl ether/polyamines system,

they concluded that a primary amine would react with epoxy-1-propyl phenyl ether to

produce a secondary amine, and the secondary amine would react with the same epoxide

to produce a tertiary amine. No evidence of tertiary-catalyzed etherification between the

epoxide and the derived hydroxyl was found.

NH2 + CH2 CH RO

R1NHR1 CH2 CH

OH

R

R1 NH

R2

+ CH2 CH RO

R1 N

R2

CH2 CHOH

R

R1 N

R2

CH2 CHOH

R + CH2 CH RO

R1 NR2

CH2 OCHR

CH2 CHOH

R

On the other hand, Xu and Schlup (1998) studied the curing mechanism of epoxy

resin/amine system by near-infrared spectroscopy and derived the following equation of

curing reaction:

12

R1 NH CH2 CH R + CH2 CH R

OOH

R1 N CH2 2

OH

CH R

They pointed out that the etherification during the epoxy resin cure was significant only

at certain reaction conditions such as at a high curing temperature and for only some

epoxy resin/amine systems. For the tetraglycidyl 4,4’-diaminodiphenylmethane and

methylaniline system, they found that the etherification reaction during cure is more

significant. The main curing reactions, similar to the above equations, were also used by

other researchers in the different epoxy resin/amine systems (Gonis, 1999; Laza et al.,

1998). Grenier-Loustalot and Grenier (1992) presented the following reaction

mechanisms when they studied the curing process of epoxy resin/amine system in the

presence of glass and carbon fibers:

C H 2 C H C H 2

O

C H 2 C H C H 2

O

N C H 2

C H 2

NC H 2 C H

C H C H 3

C H 2+

O

O

N H 2 S N H 2

O

O

D D S

CH2 CH CH2

O

CH2 CH CH2

O

N CH2

CH2

NCH2 CH

CH CH3

CH2

O

DDS

OH

N SO2 NH2

TGDDM

CH2 CH CH2

CH2 CH CH2

O

N CH2

CH2

NCH2 CH

CH CH3

CH2

O

OH

N SO2 NH2

TGDDMOH

NHSO2NH2

DDS

They studied the crosslinking mechanisms and kinetics of carbon and glass fiber

reinforced epoxy resin by several methods and found that the presence of the carbon and

glass fibers does not change these reaction mechanisms.

13

Arturo et al. (1998) discussed the possible influence of other products, i.e. the

products presented or obtained during the course of reaction on the reaction rate in the

epoxy resin/amine systems. They proposed the following reactions:

A 1 + E A 2O H

k 1 A 2 + E A 3O H

k 2

A 1 + E A 2

k 1

A 2 + E A 3

A 1 + E A 2A 1

A 2 + E A 3A 1

A 1 + E A 2A 2

A 2 + E A 3A 2

A 1 + E A 2A 3

A 2 + E A 3A 3

I I I k 2III

k 1I V

k 1II

k 1I

k 2I

k 2I I

k 2I V

where E, OH, A1, A2, A3 are the epoxy, hydroxyl, primary, secondary, and tertiary

amines, respectively. By these mechanisms, they studied the reactivities of cis and trans

isomers of 1,2-DCH with a bisphenol A epoxy resin. Like the reaction of epoxy

resin/amine system, the reaction of the epoxy resin/polyphenol system may be initiated

by the attack of the nucleophilic hydroxyl group to the partially positive carbon atom.

The curing reactions of epoxy resins with other nucleophilic curing agents such

as polyamides, polyureas, polyurethanes, polyisocyanates, polyhydric alcohols, and

polycarboxylic acids were seldom reported in the literatures. Their reaction mechanisms

were discussed by Tanaka and Bauer (1988).

Unlike mechanisms of polyaddition, the stepwise polymerization of epoxy resin

is initiated by anionic and cationic reagents. Anionic polymerization of epoxides may be

induced by initiators such as metal hydroxides and secondary and tertiary amines.

Cationic polymerization may be induced by using Lewis acids as initiators. Many

inorganic halids could be used as cationic initiators. Tanaka and Bauer also discussed the

14

mechanisms of anionic and cationic polymerization of epoxy resin. They pointed out that

the products from anionic and cationic polymerization with monoepoxides have

relatively low molecular weights. One important factor of polymerization is

stoichiometry. It has effects on the viscosity and the gel time of the epoxy resin system

(Hadad, 1988).

Many cure reactions of epoxy resins have been discussed. The most frequently

studied and important cure reactions are epoxide-amine reaction, epoxide-

phenol/anhydride reaction, epoxide-epoxide reaction.

2.4 Fiber-reinforced Composites of Epoxy Resins

Composites are usually called reinforced plastics. They are composed of a

continuous reinforcing fiber and a polymer matrix. Thermoset resins, such as epoxy

resins, have high heat resistance and great structural durability. While reinforcing fibers

can provide strength and stiffness, epoxy resin can give the desired shape or structure to

the composite and distribute the stress between the enforced fibers. The combination of

high strength of the reinforcing fiber and the epoxy matrix gives composites many

desirable chemical and physical properties. They have high strength and can meet

specific strength requirements in engineering applications. In rigidity, they have high

stiffness and meet the low deformation requirement under high load. In weight, they are

light materials. Compared to steel and aluminum, they have the highest ratio of strength

to weight, which is one of the important requirements in the aerospace industry.

Chemically, they are stable and have excellent resistance against corrosion and wearing.

Economically, they are cost-effective materials and easy to use. Because of these unique

chemical and physical properties, the fiber-reinforced epoxy composites find applications

15

in many areas such as spacecraft, aircraft, marine vessels, corrosion-resistance chemical

tanks, electrical equipment, vehicles, reinforced epoxy pipes, construction materials, and

tooling.

Many composite products are manufactured from different prepregs. Prepregs are

produced according to their end uses and applications. Several factors such as epoxy

resin, curing agent, reinforced fiber and its orientation, and volume fraction of fibers or

epoxy resin content in the fiber/resin prepreg have important effects on the properties of

prepregs and their final products. The analysis of the cure process on epoxy prepregs

provides a solid foundation for their applications. For one application, epoxy prepregs

can be used to join composite pipes. Stubblefield et al. (1998) developed a heat-activated

method to join composite pipes with epoxy prepregs. In their study, several ply epoxy

prepregs were wrapped around the ends of the two composite pipes. A shrink tape was

applied and placed over the prepreg laminate. When heat was applied to the shrink tape

and prepregs, the prepregs were cured and the tape shrank and compressed the prepregs

to improve adhesion between the thermal coupling and composite pipe. This is a cost-

effective method for joining composite-to-composite pipe. Joining composite-to-

composite pipes with a Tee configuration, using epoxy prepreg fabric, is presently under

study.

2.5 Curing Analysis Techniques

Several techniques have been used to study the curing process of epoxy resins or

epoxy prepreg. They can be classified as chemical and/or physical methods. Fourier

transform infrared spectroscopy (FTIR), nuclear magnetic resonance (NMR), high

performance liquid chromatography (HPLC), infrared spectroscopy and radiochemical

16

methods are traditionally considered as chemical-based approaches. Differential

scanning calorimetry (DSC), thermal scanning rheometry (TSR), thermomechanical

analyzer (TMA) and dynamical mechanical analyzer (DMA) are considered as physical

methods. These chemical and physical methods are sometimes used together to analyze

the curing mechanism and determine kinetic parameters for models.

2.5.1 Differential Scanning Calorimetry Analysis

DSC is a quantitative differential thermal analysis technique (Hatakeyama and

Quinn, 1994). During measurement with DSC, the temperature difference between the

sample and reference is measured as a function of temperature or time. The temperature

difference is considered to be proportional to the heat flux change.

In the study of curing kinetics of epoxy resins, it is assumed that the degree of

reaction (cure) can be related to the heat of reaction. Both isothermal and dynamic

methods can be adopted to determine the kinetic parameters with DSC. For the

isothermal method, the sample is quickly heated to the preset temperature. The system is

kept at that temperature and the instrument records the change of heat flux as a function

of time. For the dynamic method, the heat flux is recorded when the sample is scanned at

a constant heating rate from low temperature to high temperature. The area under the

heat flux curve and above baseline is calculated as the heat of reaction.

2.5.2 Rheological Analysis

Epoxy resins exhibit both viscous and elastic properties. During the curing

process, their viscosity increases quickly in the gel region. The viscosity can be related to

degree of cure. Rheological equipment can be used to measure effectively the epoxy

17

resin properties. The currently used rheometer is the Bohlin VOR Rheometer. Detailed

information on this rheometer will be provided later.

Hinrichs (1983) discussed the requirements for a rheological instrument to

measure successfully the rheological properties of the polymer. The 50mm parallel plates

are beneficial for low viscosity and the 25mm parallel plates are good for high viscosity.

The viscosity of the polymer materials can be measured in the continuous rotation or

oscillation mode.

2.5.3 Fourier Transform Infrared Spectroscopy

The main component of most commercial FTIR is an interferometer (Smith,

1996). A typical Michelson interferometer consists of four arms. Arm one contains a

source of infrared radiation. Arm two and arm three contain a fixed mirror and a moving

mirror, respectively. Arm four is open. A beamsplitter is placed at the intersection of

the four arms. By transmitting half the infrared light that impinges upon it and reflecting

another half of it, the beamsplitter divides infrared light into two light beams. The

transmitted light beam strikes the fixed mirror while the reflected light beam strikes the

moving mirror. After reflecting off their fixed and moving mirrors respectively, the two

light beams with the optical path difference recombine at the interferometer, leave the

interferometer, interact with the sample, and hit the detector. To identify a sample with

FTIR, the solid sample should be ground into a powder mixed with KBr powder to

prepare the very thin film pallet. Each active group of the component in the sample has

its own characteristic absorbance or transmittance peak to the light. The concentration of

each component can be calculated from the area under its peak.

18

2.5.4 High Performance Liquid Chromatography

HPLC usually consists of a high-pressure pump, a solvent reservoir, a packed

column, an injection unit, a detector and a displaying monitor (Lindsay, 1987). The

pump provides the necessary pressure to move the mobile phase through the packed

column. The injection unit is used to inject the sample into the packed column. The

detector, such as a uv-absorbance detector, monitors the mobile phase from the packed

column. The individual component information is displayed on the monitor. To identify

a sample with HPLC, a sample solution is prepared first by extracting the weighed

prepreg with the appropriate organic solvent, such as tetrahydrofuran. The extractable

resin is fully soluble. Because the different components in the sample will have different

retention times in the column and thus will have different, unique peaks, the

concentration of each component can be calculated from the area(s) under its peak(s).

In this chapter, the background of the research is briefly discussed. This includes

the epoxy resin, the curing agents, the cure reaction, epoxy composites, and techniques

used in the cure analysis. The details of the experimental set up and analysis instruments

are given next.

19

CHAPTER 3. MATERIAL AND MAIN EQUIPMENT

3.1 Experimental Material

The commercial prepreg used as sample in this study is Hexcel 8552

carbon/epoxy prepreg, which is an amine cured epoxy resin system. It was supplied by

Hexcel Corporation. The carbon synthetic fiber areal weight in the prepreg tape was 137

g/m2. The content of the multifunctional epoxy resins and the curing agent of the prepreg

was 33 % by weight.

3.2 Isothermal DSC Measurements

A Differential Scanning Calorimeter from Seiko Inc. in Japan was used to

measure the heat flow change during the isothermal curing process. The structure of a

Seiko DSC instrument design is shown in Fig. 1 (Negulescu, 1997). It consists mainly of

a sample holder and a reference holder, temperature controller, and a heating block. To

prepare the samples, the prepreg was cut into very small pieces, put into an aluminum

pan, and then the pan was sealed. By the recommendation of the Seiko instrument, the

amount of sample was in the range of 7-10 mg. A larger sample size could increase the

signal of the heat flow, but it would increase the temperature gradient inside the sample.

The reference was an empty aluminum pan with cover. The purging gas was nitrogen.

The flux of nitrogen was set to 100 ml/min. The sampling time was set to 6 s for all the

isothermal runs in the temperature range between 110 and 200 oC. The sampling time for

other runs was set to 3 s.

3.3 Dynamic Scanning DSC Measurements

The measurements of heat flow of the samples during dynamic scanning

processes were conducted using the DSC 2920 from TA Instruments Inc. in USA. The

20

Lid

Insulatedsurroundings

Microfurnaces Microfurnaces

Controller andprogrammer

PRPS TS TR

T(t) --program

P -- Control

Sample Reference

Fig. 1 A Schematic Drawing of DSC 220 from Seiko Inc. in Japan

21

instrument could be run in the conventional DSC mode and modulated DSC mode. In the

conventional DSC mode, the dynamic scanning process was conducted at a simple

constant heating rate. In the modulated DSC mode, a sinusoidal temperature profile was

overlaid on a primary temperature ramp, which changed at a constant heating rate. The

net effect was that the actual heating rate was sometimes greater than and sometimes less

than the constant heating rate. By the modulated DSC mode, the total heat flow signal

could be separated into reversal heat flow and non-reversal or kinetic heat flow. The

same prepreg provided samples for both the isothermal DSC measurements and the

dynamic scanning DSC measurements. Following the recommendation from TA

Instruments, the size of the sample for the kinetic study was in the range of 5 to 10 mg.

Considering the content of the resin in the prepreg, the size of the sample was usually 9 to

10 mg. Both the sample container and the reference were aluminum pans. Nitrogen was

used as the purging gas. The sampling time was set to 0.2 second per point.

3.4 Isothermal and Dynamic Scanning Rheological Measurements

The isothermal and dynamic rheological measurements were conducted by a

Bohlin VOR Rheometer, as shown in Fig. 2 (Seyfzadeh, 1999). The lower plate was

driven by a DC motor while the upper plate was kept stationary. Through the upper plate,

torque bar, and a lever arm, the response of the sample to the deformation caused by the

lower plate was transmitted to the sensor in the LVDT torque measurement system to

produce the electrical signal for the computer analysis. A temperature control unit

maintains the sample at a fixed temperature. It is a fully computer controlled device with

a well-defined menu system. The output data are viewed on a monitor in graphical and

table form during the measuring time.

22

Torque bar

Air bearing

LVDT torque measurement

DC motor with tacho servo

Gear boxesClutch

Position servo actuator

Sample cellTemperature sensorHot air inlet

Hot air outlet

Fig. 2 The Bohlin VOR Rheometer and its Main Part

23

There are several modes to be selected for the rheological measurements in this

rheometer. The parallel plates of 25 mm in diameter under the oscillation mode with 300

gcm torque bar were chosen in this study. The size of one layer of prepreg sample was

25.4 mm (1") in diameter and about 0.15 mm in thickness. Because one layer of epoxy

prepreg is very thin, seven layers of the prepreg about 1 mm in thickness were used as the

sample to improve the accuracy of the measurements. Preliminary tests were conducted

to determine the optimum instrumental parameters such as the angular frequency, gap

between two plates, and strain. The optimized experimental conditions are the maximum

strain of 0.18, a gap of about 1mm, and a frequency of 0.2 Hz. During experiments, the

strain was set to be automatically adjustable to ensure the right torque range. For

isothermal measurements, the sample chamber was preheated to the desired temperature

and stabilized at that temperature for half hour. The sample was put into the chamber and

measurement was started. For dynamic measurements, the samples were put into the

chamber at the room temperature and heated at a certain rate.

This chapter discussed the experimental material, instruments, and measurement

procedures and sampling. The models for the cure analysis are discussed next.

24

CHAPTER 4. METHODOLOGY

4.1 Autocatalytic Models for Isothermal Curing Process

The autocatalytic model for an isothermal cure reaction is as follows (Horie,

1970):

nmkkdtdr )1)(( 21 ααα

−+== (1)

where r and dtdα is the cure rate, α is the degree of cure, m and n are the orders of cure

reaction, k1 and k2 are the rate constants.

The cure rate including the diffusion factor introduced by Chern and Poehlein

(1987) is expressed as followed:

nmkkfdtd

))()(( 121 αααα−= + (2)

where )(αf is the diffusion factor. It has the following form:

)exp(11)(

)( cCfαα

α−+

= (3)

where and C cα are the two empirical constants which are temperature dependant. cα is

called the critical degree of cure.

Combining Eqs. (2) and (3), Eq. (1) with diffusion control can be re-expressed as

follows:

nm

ckk

Cdtd

))((exp1

1 1)( 21)(

ααααα

−= +−+

(4)

25

4.2 Autocatalytic Models for Dynamic Scanning Curing Process

4.2.1 Model Analysis Based on Kissinger and Ozawa Kinetic Approach

The kinetic model for a dynamic curing process with a constant heating rate can

be expressed in the following form (Gonis et al., 1999):

)()( αα gTkdtd = (5)

where is the rate constant (which depends on the temperature T ), and )(Tk )(αg is a

function of α only. It may have different forms, depending on the cure mechanism. The

rate constant can be further expressed by an Arrhenius equation, )(Tk

RTEa

AeTk −=)( (6)

where A is the pre-exponential factor, Ea is the activation energy, R is the gas constant,

and T is the absolute temperature. Substituting Eq. (6) into Eq. (5) yields

)(αα gAedtd RT

E a−= (7)

For a dynamic curing process with constant heating rate, the temperature

increases with the increment of cure time t. The relationship between dtdα and dT

dα can be

expressed as,

dTd

dtdT

dtd αα )(= (8)

where dtdT is the constant heating rate.

Substituting Eq. (8) into Eq. (7) and rearranging yields

RTE a

egdTdAdt

dT −−= )( (1) αα (9)

Taking the logarithm on both sides of Eq. (9) yields,

26

TREg

dTdA

dtdT a 1)()(ln)ln(lnln )( −++−= αα (10)

For the autocatalytic model with the initial cure rate of zero, the term )(αg may

have the following form (Yang et al., 1999),

nmg )1()( ααα −= (11)

Substituting Eq. (11) into Eq. (10) yields

TRE

dTdAdt

dT anm 1)())1(ln()ln(lnln )( −+−+−= ααα (12)

The pre-exponential factor A and activation energy E can be determined by the

Kissinger (1957) and Ozawa (1970) kinetic approach.

a

4.2.2 Model Analysis Based on Borchardt and Daniels Approach

Substituting Eq. (11) into Eq. (7) yields the autocatalytic model for the dynamic

curing process,

nmRTE a

Aedtd )1( ααα −=

− (13)

By taking the logarithm of both sides of Eq. (13), a linear expression for the

logarithm of cure rate can be obtained,

TREnmAdt

d a 1)()1ln(lnlnln )( −+−++= ααα (14)

Parameters A, m, n, and Ea can be obtained by multiple linear regression, in which

the dependent variable is )n( dtdl α , and the independent variables are αln , )1ln( α− , and

T1 .

27

4.3 Rheological Models

4.3.1 Gel Time Model

Gel time, which was detected by the rheological measurement, varies with the

isothermal cure rate of reaction. By integrating the Eq. (5) from zero time to gel time t ,

the relationship between and cure rate is obtained,

gel

gelt

αα

αd

gTkt gel

gel ∫⋅=0 )(

1)(

1 (15)

where gelα is the degree of cure at gel time.

Substitute Eq. (6) into Eq. (15) and take logarithm on both sides to get the

relationship between the gel time and isothermal cure temperature,

TR

EdgA

t agel

gel 1)(

11ln)ln(0

0

⋅+

= ∫

αα

α (16)

According to Flory’s expression (1953), the degree of cure gelα at gel time

depends on the functionalities of the epoxy systems only. So it can be considered a

constant for a given epoxy systems regardless the cure temperature.

By considering the first term on the left side of Eq. (16) as a constant c, a linear

relationship of ln( vs. )geltT1 is obtained and Eq. (16) can be rewritten as:

TRE

ct agel

1)ln( ⋅+= (17)

From Eq. (17), the apparent activation energy can be calculated from the slope of

the curve of vs. )ln( geltT1 .

28

4.3.2 Viscosity Models

During the curing process, the viscosity of the sample is not only a function of

curing temperature, but also a function of degree of cure. The variation of viscosity is the

result of the combination of physical and chemical processes and can be empirically

expressed as (Ampudia et al., 1997):

)()( αζψη T= (18)

where )(Tψ is a function of curing temperature only; )(αζ is a function of degree of

cure.

Terms )(Tψ and )(αζ can be empirically expressed with the simple form

respectively:

0)( ηψ =T and α

αζ−

=1

1)( (19)

where η0 is the initial viscosity which is a constant at isothermal cure conditions; α is the

degree of cure.

Substituting Eq. (19) into Eq. (18) to get the relationship of viscosity vs. the

degree of cure,

αηη

−=

11

0 (20)

For dynamic scanning cure process, the initial viscosity η0 depends on cure

temperature and can be further expressed in Arrhenius equation,

RTE

eAη

ηη =0 (21)

where Aη and Eη are the initial viscosity at ∞=T and the viscous flow activation energy,

respectively.

29

The degree of cure α in Eq. (20) is a function of cure time. Depending on the

cure kinetics, the relationship of α vs. time t may have different forms. For the first order

reaction, it can be expressed as:

)1( αα−= k

dtd

(22)

Eq. (22) was solved to get Eq. (23),

∫=−

tkdt

e 0

11

α (23)

Substitute Eq. (23) into Eq. (20) to get:

∫=tkdt

e 00ηη (24)

Substitute Eq. (6) into (24) and take logarithm on both sides to obtain:

∫−

++=t

RTE

k dteARTE

Ak

0lnln η

ηη (25)

where Ak and Ek are the apparent rate constant at ∞=T and the kinetic activation energy,

respectively.

For the first order reaction with the isothermal cure process, temperature T and

rate constant k are constant. Eqs. (24) and (25) become:

kte0ηη = (26)

RTE

k

k

etARTE

A−

++= ηηη lnln (27)

Eqs. (25) and (27) are the empirical four-parameter model of viscosity introduced

by Roller (1975). According to Eq. (26), η0 and k can be obtained from plot of ηln vs. t

30

at each temperature. The values of Aη, Eη, Ak, and Ek are determined by Eqs. (6) and (21)

respectively.

For the nth order reaction, it can be expressed as:

nkdtd )1( αα

−= (28)

Eq. (28) was solved to get Eq. (29),

11

0 ))1(1(1

1 −∫−+=−

ntkdtnα

(29)

Substitute Eq. (29) into Eq. (20) and take logarithm on both sides to get:

))1(1ln(1

1lnln0∫

−−+

−++=

tRTE

k dteAnnRT

EA

kη

ηη (30)

This is the empirical five-parameter model of viscosity for the nth (n≠1) order reaction

introduced by Dusi (1983).

A new viscosity model for an isothermal cure process of epoxy prepreg is

proposed:

∞−∞ +

+−

= ηηηη )(0

1 cttke (31)

where η∞ is final viscosity; k is the rate constant of cure reaction; and tc is the critical

time which follows Arrhenius behavior, i.e.

RTE

tc

t

eAt = (32)

Eq. (31) is proposed based on a Boltzmann function to produce a sigmoidal curve,

which the viscosity profile for the isothermal cure process seems to follow, especially in

the gel region. Eq. (31) is just a fitting function which is based on the mathematical

knowledge instead of the rheological theory. It has a similar form as a Boltzmann

31

function, but each parameter in Eq. (31) has its own physical meaning. The parameters in

Eq. (31) are determined by the multiple non-linear regression method.

For dynamic cure process, Eq. (31) becomes:

∞−

∞ +∫∫+

−=

−−η

ηη

η

η

)(001

dteAdteA

RTE

ctRTE

kt

RTE

k

kk

e

eA (33)

The values of Aη and Eη are determined from the linear part of viscosity profile in

the dynamic cure process.

4.4 Glass Transition Temperature (Tg) Models

The Tg increases with the increment of degree of cure. There exists a unique

relationship between the Tg and degree of cure regardless of the cure temperature and

cure path (Wisanrakkit and Gillham, 1990; Karkanas and Partridge, 2000; Wingard,

2000). Based on corresponding state principles, a theoretical equation relating Tg to the

degree of cure was introduced by DiBenedetto (1987) and reported earlier by Nielsen

(1969). The DiBenedetto equation cannot be applied to some polymer systems, especially

for highly cross-linked systems (Boey and Qiang, 2000; Adabbo and William, 1982).

Pascault and Williams (1990) introduced the maximum glass transition temperature T

of the system (attained when α = 1 in the DiBenedetto equation) and further extended the

DiBenedetto equation as follows:

∞g

λααλαα

+−

+−= ∞

)1()1( 0 gg

g

TTT (34)

where Tg0 is the glass transition temperature when α = 0, and λ = Fx/Fm, is an adjustable

structure-dependent parameter between 0 and 1, Fx/Fm is the ratio of segmental mobilities.

32

Eq. (34) can be applied to polymer systems at higher reaction extents (Simon and

Gillham1993).

On the other hand, Venditt and Gillham introduced another equation (Venditti and

Gillham, 1997) used for thermosetting polymers by the Couchman procedure (Couchman

and Karasz, 1978), which was based on the entropy consideration of the mixed system,

))1(

)ln()ln()1(exp(

0

00

αα

αα

p

p

gp

pg

g

cc

Tcc

TT

∆

∆+−

∆

∆+−

=∞

∞∞

(35)

where and ∆ are the heat capacity differences between the liquid/rubbery state

and glassy state for the uncured and fully cured systems, respectively.

0pc∆ ∞pc

In this chapter, a number of models of the cure process are presented. These

models are very useful to the understanding of the cure process. The details of the

thermal and rheological analysis of epoxy prepreg will be discussed next.

33

CHAPTER 5. RESULTS AND DISCUSSION

5.1 Isothermal DSC Cure Analysis

During the isothermal curing measurements, the variation of the heat flow or

enthalpy of the epoxy prepreg sample is caused by the cure reaction. The instrument

records the heat flow change with respect to the cure time based on the sample size. For

the purpose of comparison, the heat flow for each run has been normalized to one gram

of sample.

5.1.1 Analysis of Cure Reaction Heat

Both dynamic and isothermal measurements were done to obtain more

information about the curing process. For dynamic measurements, the sample was

scanned from -5 to 300 oC. Fig. 3 gives the heat flow change at the heating rate of 2

oC/min in the temperature range from 30 to 300 oC. It shows the difference between the

first and second run, as well as heat flow changes with temperature. The sample for the

second run was from the sample in the first run after it was cured. It was indicated that

the appreciable cure reaction took place at about 110 oC. The heat capacity of the

uncured sample is larger that that of the cured sample, thus the heat flow for the first run

is lower than that of the second run before the onset temperature of the cure reaction.

With the information from the dynamic cure, a series of isothermal measurements

were performed, starting from 110 oC. Two runs were made on each sample. To achieve

almost constant heat flow in the late cure stage, the measurement time was set long

enough, from 100 minutes at 220 oC to 720 minutes at 110 oC. After the first run, the

sample was left in the cell of the instrument to cool naturally to room temperature. The

second run at each temperature was repeated with the same procedure and cure condition

34

0 40 80 120 160 200 240 280 320-0.08

-0.04

0.00

0.04

0.08

0.12

110 oC

DSC-1st run DSC-2nd run

Hea

t flo

w

(W/g

) E

xoth

erm

Temperature ( oC)

Fig. 3 Dynamic Cure From 30 to 300 oC with Heating Rate of 2 oC/min

35

0 50 100 150 200 250 300-0.09

-0.06

-0.03

0.00

0.03

0.06

0.09

0.12

T=160 oC

Hea

t flo

w (

W/g

) E

xoth

erm

Time (min)

DSC-1st run DSC-2nd run

(a) 160 oC for 260 Minutes

0 15 30 45 60 75 90 105-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7 DSC- 1st run DSC- 2nd run

T=220 oC

Hea

t flo

w

(W/g

) E

xoth

erm

Time (min)

(b) 220 oC for 100 Minutes

Fig. 4 Isothermal Cure of Epoxy Prepreg at Selected Temperatures for the Varying Time

36

as the first run. The heat flow was plotted against cure time for the first and second runs

at each cure temperature. As shown in Fig. 4 for the isothermal cure processes at 120,

160, and 220 oC (heat flow curves at other isothermal temperatures are given in Appendix

A), each curve for the first run exhibits a peak and finally becomes horizontal while the

curve for the second run does not. Similar dynamic behavior between the two samples

was seen in Fig. 3. Except for the start up of the isothermal measurement, the curve for

the second run is almost horizontal over the whole cure time. This indicates that the

isothermal cure reaction is almost complete after the first run at each cure temperature.

The curve for the second run can be used as the baseline to the first run to integrate the

area under the peak of the first run. This area is thought to be the isothermal cure reaction

heat.

It must be mentioned that the curve for the second run in Fig. 4, at each

temperature, has been shifted along the heat flow axis and used as the base line for the

purpose of the calculation of the cure reaction heat only. Usually the curve for the

second run needs to be shifted to have the same heat flow as the first runs at the end of

their cure reactions. The experimental points in the last 10 minutes for both runs were

averaged and the difference of the average values was calculated. The curve for the

second run was then shifted up or down, depending whether the average value in the last

10 minutes for the second run is greater or less than that for the first run. The magnitude

of the shift at each temperature is different, ranging from –1.2 × 10-3 to 1.5 × 10-3 W/g or

from –2.99 to 5.99% of the value to be shifted. Reasons to shift the curve for the second

run are as follows:

37

• At the end of the first run, the curve becomes almost horizontal; this means that

the curing reaction is almost complete at the end of the first run at that

temperature.

• For the second run, the sample has already been cured. The cured sample at the

end of the first run should be the same as the sample in the second run. So they

should have the same heat capacity and heat flow so long as the temperature

profiles are the same.

• If there is a difference of heat flow between the end of the first run and second run,

it should result from the different initial conditions, such as the initial heat flow to

start the run. It is difficult to control all conditions such that the same initial heat

flow value will occur even if all of the operating procedures at the start of the first

and second run are practically the same.

• The cure heat is calculated by integrating the area between the peak of the heat

flow curve and its baseline. So the absolute value of the first run has no effect on

the calculation of the cure heat.

At different cure temperatures, the isothermal cure heat is different. Its value

increases with the increment of temperature. When the cure temperature was raised to

220 oC, the cure heat was 177.5 J/g. This value is thought to be the total reaction heat of

isothermal cure because it is very close to the isothermal cure heat of 177.1 J/g at 210 oC,

which means that no additional cure heat was released and the cure reaction was

complete at 220 oC. This is supported by the experimental results from a combination of

dynamic and isothermal cure discussed next. All of the other values for reaction heats

cured isothermally below 220 oC were considered as the partial isothermal reaction heats.

38

0 150 300 450 600 750-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

residue heat=18.6 (J/g)

DSC-1 st run baseline

Hea

t flo

w (

W/g

) Ex

othe

rm

Time (min)

0

40

80

120

160

200

240

280

320

Te

mpe

ratu

re (o C

)

temperature

(a) Cure from 30 to 180 oC (Heating), Holding 80 Minutes, 180 to 30 oC (Cooling), and 30 to 300 oC (Re-heating)

0 150 300 450 600 750-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

residue heat=12.3 (J/g)

DSC-- 1st run baseline

Hea

t flo

w (W

/g)

Exot

herm

Time (min)

0

40

80

120

160

200

240

280

320

Tem

pera

ture

(o C)

temperature

(b) Cure from 30 to 190 oC (Heating), holding 80 Minutes, 190 to 30 oC (cooling), and 30 to 300 oC (Re-heating)

0 150 300 450 600 750-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

residue heat=6.1 (J/g)

DSC-- 1 st run baseline

Hea

t flo

w (

W/g

) E

xoth

erm

Time (min)

0

40

80

120

160

200

240

280

320

Tem

pera

ture

(o C)

temperature

(c) Cure from 30 to 200 oC (Heating), Holding 80 Minutes, 200 to 30 oC (Cooling), and 30 to 300 oC (Re-heating)

0 150 300 450 600 750-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14 DSC-- 1st run

Hea

t flo

w (

W/g

) E

xoth

erm

Time (min)

0

40

80

120

160

200

240

280

320

residue heat=0

Te

mpe

ratu

re (o C

)

temperature

(d) Cure from 30 to 220 oC (Heating), Holding 80 Minutes, 220 to 30 oC (Cooling), and 30 to 300 oC (Re-heating)

Fig. 5 Combinations of Dynamic and Isothermal Cure of Epoxy Prepreg at Different Isothermal Temperatures

39

The experimental result obtained by the isothermal cure was confirmed by the

combination of dynamic and isothermal cure. Several measurements with the

combination of dynamic and isothermal cure were done. The sample was first heated to

the desired temperature at the rate of 1 oC/min, held at that temperature for 80 minutes to

allow the sample to achieve complete isothermal cure, then cooled down to 20 oC, and

finally reheated to 300 oC with the same heating rate. Fig. 5 gives the heat flow change

with time. Each one has four or five positive peaks and three negative peaks. The first

positive peak is from the normal cure reaction, which is an exothermic peak. The fifth

positive peak, if it exists, comes from the residual cure reaction, which is also an

exothermic peak. The other positive and negative peaks result from the sudden changes

of the heating rate, which consequently cause the sudden changes of the heat flow. So

they are not actually the exothermic or endothermic peaks. The residual reaction heat was

calculated by integrating the area under the peak with the linear baseline between the

onset and ending temperatures of polymerization. The sample after isothermal cure at 180

oC has the residual reaction heat of 18.6 J/g. This value is very close to 18.1 J/g, the

difference of isothermal reaction heat at 180 and 220 oC. The residual reaction heats after

cure at 190 and 200 oC are 12.3 and 6.1 J/g, respectively. The differences of isothermal

reaction heat at 190 and 200 oC to that at 220 oC are 10.4 and 5.3 J/g, respectively. After

isothermal cure at 220 oC, the residual reaction heat is zero. This proves that the complete

isothermal cure reaction could be achieved at 220 oC and that the cure reaction heat

obtained by the above method is reliable.

40

5.1.2 Degree of Cure and Cure Rate

The curing process is an exothermic reaction. The cumulative heat generated

during the process of reaction is usually related to the degree of cure. It is assumed that

the degree of cure is proportional to the reaction heat. One commonly used method

assumes that the degree of cure equals the ratio of the difference between total and

residual reaction heat to the total reaction heat. The sample is prepared and partially

cured for a series of times at a certain temperature before measurement. Its residual heats

are then measured. This method takes much time to prepare the partially cured sample.

In our experiments, the sample used is fresh and uncured. Its reaction heat at each

sampling time is determined by integrating the curve of heat flow from the beginning to

the determined time, so the degree of cure can be directly calculated from the partial

reaction heat. It is expressed as follows (Morancho and Salla, 1999):

totalt

HH

∆∆=α (36)

where α is the degree of cure, tH∆ is the partial heat of reaction at time t, and ∆Htotal is

the total heat of reaction. As discussed earlier, the total reaction heat of the studied

sample is 177.5 J/g.

Once the partial reaction heats at each sampling time and temperature have been

measured, the degree of cure can be easily calculated by Eq. (36). Its relationship to cure

time at the temperature range from 110 to 220oC is shown in Fig. 6. Compared to the

value of 1 at 220 oC, the final degree of cure at 110 oC is only about 0.64. The time

needed to reach the final degree of cure is also much different, depending on the

isothermal cure temperature.

41

0 100 200 300 400 500 600 700 8000.0

0.2

0.4

0.6

0.8

1.0

160 oC 150 oC 140 oC 130 oC 120 oC 110 oC

Deg

ree

of c

ure

(α)

Time (min)

(a) 110-160 oC

0 25 50 75 100 125 150 175 2000.0

0.2

0.4

0.6

0.8

1.0

220 oC 210 oC 200 oC 190 oC 180 oC 170 oC

Deg

ree

of c

ure

(α)

Time (min)

(b) 170-220 oC

Fig. 6 Degree of Cure as a Function of Time at Different Isothermal Temperatures

42

0 100 200 300 400 500 600 700 8000.0

5.0x10-5

1.0x10-4

1.5x10-4

2.0x10-4

2.5x10-4

3.0x10-4

3.5x10-4

4.0x10-4

Cur

e ra

te (

s-1)

Time (min)

160 oC 150 oC 140 oC 130 oC 120 oC 110 oC

(a) 110-160 oC

0 10 20 30 40 50 60 70 800.0

4.0x10-4

8.0x10-4

1.2x10-3

1.6x10-3

2.0x10-3

2.4x10-3

2.8x10-3

3.2x10-3

3.6x10-3

220 oC 210 oC 200 oC 190 oC 180 oC 170 oC

Cur

e ra

te

(s-1)

Time (min)

(b) 170-220 oC

Fig. 7 Cure Rate as a Function of Time at Different Isothermal Temperatures

43

The cure rate at each sampling time and temperature can be calculated by

differentiating the degree of cure to time. The changes of cure rate with time at each

isothermal temperature from 110 to 220oC are shown in Fig. 7. In the early stages of cure

reaction, the cure rate at a higher temperature is faster than that at a lower temperature;

but in the late stages, the cure rate is slower at the higher cure temperature. Note that for

the lower temperature range of 110 to 160 oC, the slope of the cure rate curve is negative

near time zero. This agrees with the autocatalytic model to be discussed next when

reaction order is greater than 1. m

5.1.3 Cure Reaction Modeling

The autocatalytic model for the isothermal cure process as Eq. (1) has been given

previously. Different methods to calculate kinetic parameters in Eq. (1), page 25 have

been reported (Han et al., 1998; Barton et al., 1992; Yang et al., 1999).

Equation (1) can be differentiated with respect to α, to give Eq. (37):

)]()1([)1( 21)1(

2)1( mmn kknmkd

dr ααααα +−−−= −− (37)

If m > 1and α = 0, we have

1nkddr −=α (38)

The rate constant k1 can be obtained by extrapolating the curve of cure rate vs.

degree of cure and its value equals the cure rate at 0=α . From Eq. (38), we know that

the value of nk1 equals the negative derivative of cure rate to degree of cure when degree

of cure is zero. So the reaction order n can be easily calculated. Theoretically, if the cure

reaction order m is greater than one, the derivatives of cure rate to degree of cure should

be negative. Our experimental data indicate that in the low temperature ranges from 110

44

to 160 oC, the derivatives of cure rate to degree of cure are indeed negative when the

degree of cure nears zero.

Once the values of k1 and n are known, Eq. (1) can be expressed as follows:

mn kkr αα 21 ])1([ =−− (39)

By Eq. (39), if we make a log-log plot of ])1( 1k[ rn −−α vs. degree of cure, the rate

constant k2 and reaction order m can be easily obtained.

With the above method, we analyzed the experimental data at 160 oC. The rate

constant k1 equals 3.40 × 10-4 ± 2.1 × 10-6 s-1. The derivative of cure rate to degree of cure

at α=0 is –8.20 × 10-4 ± 2.6 × 10-6 s-1. The value of the cure reaction order n was then

calculated. Its value is 2.41 ± 0.02. The relationship between ])1( 1k[ rn −−α and degree of

cure α, and their linear fitting are given in Fig. 8.

-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0-11

-10

-9

-8

-7

-6

-5

m=1.46ln(k2)=-5.94

ln[r/

(1-α

)n -k1]

ln α

Fig. 8 The Logarithm Plot of [r/(1-α)n-k1] vs. α at 160 oC

45

The calculated values for rate constant k2 and reaction order m are 2.63 × 10-3 ±

6.59 × 10-6 s-1 and 1.46 ± 0.002, respectively. It was found that the values of the four

parameters obtained in this method are close to the ones obtained through nonlinear least

square regressions discussed next.

Another easier and efficient way to analyze the data is by the nonlinear

regressions of the experimental data. In our data analysis, we employed Origin software

to do nonlinear least squares curve fitting to the experimental data. To obtain successfully

the four parameters in the autocatalytic model, the selection of initial values for the

parameters and ranges of experimental data is very important. During the process of

nonlinear regressions, the sum of the squares of the derivations of the theoretical values

from the experimental values, which is called χ2, decreases and the parameters change.

The regression stops when χ2 is minimum. The parameters thus obtained achieve the best

values for the model. The values for the rate constants and reaction orders are listed in

Table 3. The fitting curves and experimental data at some temperatures are provided in

Fig. 9 (More curves are given in Appendix B). Except for data in the late stage of cure

reactions, the fitting curves agree well with the experimental data.

As seen in Table 3, the cure reaction orders m and n decrease with the increment

of temperature. The rate constants k1 and k2 increase with the increment of temperature

and follow the Arrhenius law as Eq. (6). By Eq. (6), the plots of ln and vs. 1/T

with their linear regression curves, shown in Fig. 10, are provided. From the intercepts

and slopes of the regression curves, the pre exponential factors A

1k 2ln k

1 and A2, and activation

energies Ea1 and Ea2 can be determined. Their values are also given in Table 3. The cure

temperature has more effect on rate constant k1 than k2.

46

Table 3. Kinetic Parameters of the Autocatalytic Model for Isothermal Cure Process

Temp. (oC)

k1 (×10-4

s-1)

SE (×10-7

s-1)

k2 (×10-3

s-1)

SE (×10 –5

s-1)

m SE (×10-2)

n SE (×10-2)

110 0.36 1 1.22 2 1.98 0.73 3.97 1.3

120 0.55 2 1.40 3 1.87 1.01 3.60 1.69

130 0.85 5 1.76 4 1.77 1.38 3.34 2.00

140 1.40 7 2.04 5 1.68 1.39 3.02 2.02

150 2.02 0 2.51 4 1.53 0.72 2.82 1.23

160 3.45 0 2.77 4 1.50 0.60 2.45 1.11

170 4.47 0 3.11 6 1.25 0.85 2.25 1.23

180 6.27 0 3.58 7 1.16 0.86 2.08 1.40

190 9.03 0 3.76 9 0.99 1.05 1.82 1.61

200 10.3 0 3.88 4 0.66 0.53 1.68 0.58

210 14.3 0 4.09 2 0.52 0.26 1.51 0.28

220 19.4 0 5.14 16 0.47 1.18 1.47 1.70 Ea1

(KJ/mol) SE

(KJ/mol) A1

(s-1) SE (s-1)

Ea2 (KJ/mol)

SE (KJ/mol)

A2 (s-1)

SE (s-1)

57.3 1.05 2416 714.1 19.6 0.884 0.60 0.149

47

0.0 0.2 0.4 0.6 0.8 1.00.0

1.0x10-5

2.0x10-5

3.0x10-5

4.0x10-5

5.0x10-5

6.0x10-5

7.0x10-5

T=120 oC

experimental model without diffusion model with diffusion

Cur

e ra

te (

s-1)

Degree of cure (α)

(a) 120 oC

0.0 0.2 0.4 0.6 0.8 1.00.0

2.0x10-5

4.0x10-5

6.0x10-5

8.0x10-5

1.0x10-4

1.2x10-4

1.4x10-4

1.6x10-4

T=140 oC

experimental model without diffusion model with diffision

Cur

e ra

te

(s-1)

Degree of cure (α)

(b) 140 oC

0.0 0.2 0.4 0.6 0.8 1.00.0

5.0x10-5

1.0x10-4

1.5x10-4

2.0x10-4

2.5x10-4

3.0x10-4

3.5x10-4

4.0x10-4

T=160 oC

experimental model without diffusion model with diffusion

Cur

e ra

te (

s-1)

Degree of cure (α)

(c) 160 oC

0.0 0.2 0.4 0.6 0.8 1.00.0

1.0x10-4

2.0x10-4

3.0x10-4

4.0x10-4

5.0x10-4

6.0x10-4

7.0x10-4

8.0x10-4

T=180 oC

experimental model without diffusion model with diffusion

Cur

e ra

te (

s-1)

Degree of cure (α)

(a) 180 oC

0.0 0.2 0.4 0.6 0.8 1.00.0

3.0x10-4

6.0x10-4

9.0x10-4

1.2x10-3

1.5x10-3

1.8x10-3

T=200 oC

experimental model without diffusion model with diffusion

Cur

e ra

te

(s-1)

Degree of cure (α)

(e) 200 oC

0.0 0.2 0.4 0.6 0.8 1.00.0

5.0x10-4

1.0x10-3

1.5x10-3

2.0x10-3

2.5x10-3

3.0x10-3

3.5x10-3

T=220 oC

experimental model without diffusion

Cur

e ra

te

(s-1)

Degree of cure (α)

(f) 220 oC

Fig. 9 Comparisons of Experimental Data with the Autocatalytic Model as Eq. (1) at the Select Isothermal Cure Temperatures

48

2.0x10-3 2.2x10-3 2.4x10-3 2.6x10-3 2.8x10-3-11

-10

-9

-8

-7

-6

-5

-Ea1/R=-6896.75ln(A1)=7.79ln

(k1)

1/T (K-1)

(a) ln (k1) vs. 1/T

2.0x10-3 2.2x10-3 2.4x10-3 2.6x10-3 2.8x10-3-7.0

-6.5

-6.0

-5.5

-5.0

-Ea2/R=-2355.68ln(A2)=-0.51

ln(k

2)

1/T (K-1)

(b) ln (k2) vs. 1/T

Fig. 10 Rate Constants k1 and k2 in Eq. (2) as a Function of Temperature

49

In the late stage of the curing process, the sample approaches the solid state. The

movement of the reacting groups and the products is greatly limited and thus the rate of

reaction is not controlled by the chemical kinetics, but by the diffusion of the reacting

groups and products. From Fig. 9, it is noticed that there exist large derivations of

theoretical values from experimental data in the late cure stage, especially at low

temperatures. To account for the effect of the diffusion on the rate of cure reaction, the

diffusion factor should be included in the kinetic model, as shown in Eq. (2) or (4) (page

25).

There are six parameters in Eq. (4). In the data analysis with Eq. (4), rate

constants k1 and k2, and reaction orders m and n have already been determined. To

determine the remaining two constants C and αc, we again use the same software to do

the nonlinear least square regressions. During the regression process, χ2 values decrease

and C and αc are varied until χ2 converges to a minimum value. The regression curves

with diffusion factor at different temperatures are also provided in Fig. 9 for comparisons.

It can be observed that the model with a diffusion factor has very good agreement with

the experimental data.

The values for constant C and critical degree of cure αc at different temperatures

are listed in Table 4. The critical values for degree of cure increase with the increment of

temperature. Fig. 11 gives the relationship between the diffusion factor and degree of

cure at different temperatures. It can be shown that the diffusion factor remains almost

unchanged with values close to 1 in the early cure stage. But in the late cure stage, it

decreases greatly until it approaches zero.

50

Table 4. Values of Constant C and Critical Degree of Cure cα for Autocatalytic Model for Isothermal Cure Process

Temperature

( oC) C SE α c SE

(× 10-4) 110 50.0 0.167 0.5980 0.8

120 41.4 0.139 0.6247 0.9

130 57.7 0.359 0.6641 1.4

140 38.2 0.180 0.6783 1.6

150 49.7 0.396 0.7276 2.1

160 42.3 0.314 0.7637 2.3

170 43.1 0.350 0.8039 2.4

180 39.3 0.380 0.8485 2.8

190 50.1 0.821 0.9070 3.7

200 75.7 3.548 0.9523 6.7

51

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Diff

usio

n fa

ctor

(f)

Degree of cure (α)

110 oC 120 oC 130 oC 140 oC 150 oC 160 oC 170 oC 180 oC 190 oC 200 oC

Fig. 11 Variation of Diffusion Factor with Degree of Cure at Different Temperatures

52

5.2 Dynamic Scanning DSC Cure Analysis

The dynamic cure kinetics is different from isothermal cure kinetics. The number

of peaks and /or shoulders in the isothermal and dynamic scanning DSC thermograms

may be different (Kenny et al., 1991). As shown in Fig. 4, there was only a peak in the

isothermal DSC thermograms. In the dynamic scanning DSC study discussed next, a peak

and a shoulder appeared in DSC thermograms. The kinetic parameters obtained from an

isothermal cure studies are not good in predicting the dynamic cure behavior (Karkanas

and Partridge, 2000). The best way to understand the dynamic curing process is through

the dynamic curing experimental data.

In the dynamic measurements, signals recorded by the instrument were based on

the sample size. For the heat flow to be discussed next, normalization is also to one gram.

5.2.1 The Exothermal Peak Analysis of Dynamic Cure Process

The dynamic DSC measurements were conducted at the heating rates of 2, 5, 10,

15, and 20 oC/min, with the temperature ranging from zero to 345 oC. The heat flow

changes measured by the conventional DSC mode during the heating process are shown

in Fig. 12. Fig. 12 (a), which shows the heat flow changes versus the time, was used to

calculate the cure reaction heat by the method of integration. A baseline under the peak

of each heat flow curve was required in order to determine the cure reaction heat. Two

types of baselines were generally used: the straight baseline and sigmoidal baseline. The

straight baseline was based on the assumption that the heat capacity changed linearly

with the temperature. The sigmoidal baseline considered the possible change of heat

capacity during the transition of the dynamic scanning. Dupuy et al. (2000) discussed in

detail the influence of the baseline on the calculation of the cure reaction heat. In this

53

0 20 40 60 80 100 120 140-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

20 oC/min

15 oC/min10 oC/min

5 oC/minheating rate=2 oC/min

Hea

t flo

w (W

/g)

exo

ther

m

Time (min)

(a) Heat Flow vs. Time

0 50 100 150 200 250 300 350-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

20 oC/min

15 oC/min

10 oC/min5 oC/min

heating rate=2 oC/min

Hea

t flo

w (W

/g)

exot

herm

Temperature (oC)

(b) Heat Flow vs. Temperature

Fig. 12 Heat Flow Changes at the Heating Rates of 2, 5, 10, 15, and 20 oC/min: (a) From Right to Left by Peak; (b) From Left to Right by Peak

54

study, a straight baseline was used to integrate the peak of heat flow with respect to time

to give the reaction heat. Table 5 lists the values of the total reaction heat at each heating

rate for the uncured samples. Little difference between the total reaction heats was

observed at the different heating rates. The cure reaction at each heating rate could be

considered complete.

Table 5. Total Dynamic Cure Reaction Heat at Different Heating Rates

Heating rate (oC/min)

2 5 10 15 20 averageHeat of cure reaction

(J/g) 184.8 183.4 184.2 183.9 183.7 184.0

SE 0.2 0.9 1.6 2.2 0.9 1.2

Unlike the isothermal curing process controlled by diffusion (Nunez et al., 2000),

the diffusion control had no effect on the dynamic curing process at the studied heating

rates. The averaged dynamic cure reaction heat was 184.0 ± 0.48 J/g. Once the reaction

heat at each temperature and the total reaction heat were determined, the degree of cure α

at each time or temperature could be calculated by Eq. (36). In Eq. (36), ∆Ht represents

the dynamic reaction heat at time t and ∆Htotal represents the total reaction heat at a

certain heating rate for the dynamic cure process. By differentiating the degree of cure α

with respect to time, the relationship of the cure rate vs. time or temperature was

determined. These data would be used as the source data to simulate the dynamic curing

process.

Fig. 12 (b) shows the heat flow changes versus temperature. It is seen that the

start and ending points shifted to the higher temperatures at the higher heating rate.

55

Accordingly, the maximum heat flow and the exothermal peak temperature also increased.

At each heating rate, the curve exhibited a peak and a shoulder. This suggested that:

• The curing process was composed of two cure reactions with the different

activation energies.

• The peak, which occurred at lower temperature, was caused by Reaction 1.

• The shoulder, which occurred at higher temperature, was caused by Reaction 2.

• Reaction 2 was more sensitive to temperature than Reaction 1. As heating rate

increases, the rate for Reaction 2 increased faster than the rate for Reaction 1.

• Reactions 1 and 2 occur simultaneously. The relative contributions of these two

reactions to the total reaction vary with the cure time in a given temperature range.

• The mechanisms of Reactions 1 and 2 may or may not the same. Further analysis

of these two reactions is required to better understand their cure mechanisms.

Further experiments with the partially cured samples in the modulated DSC mode

supported the above conclusions. In the modulated DSC mode, the non-reversal heat

flow that is caused by the cure reaction only was separated from the general heat flow of

the curing process. The partially cured samples were prepared in a specially designed

oven. The uncured sample was put into the oven and heated from room temperature to the

desired temperature at a constant heating rate of 5 oC/min, with nitrogen as the purging

gas. After the oven reached the desired temperature, the sample was quickly moved into

the refrigerator and cooled in a sealed jar. The samples were partially cured to 170, 190,

200, 210, 220, 230, 240, and 270 oC, separately.

The non-reversal heat flows versus temperature are shown in Fig. 13. The values

on the left of each corresponding peak are the exothermal peak temperature, and in

56

-100 -50 0 50 100 150 200 250 300 350-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

204.9 (170, 7%)

205.3 (fresh, 0%)

204.6 (190, 17%)

209.3 (200, 29%)

232.1 (210, 45%)

234.9 (220, 72%)236.3 (230, 78%)236.2 (240, 86%)

peak temperatures (oC)

Non

-rev

ersa

l hea

t flo

w (W

/g)

Exo

ther

m

Temperature (oC)

Fig. 13 Non-reversal Heat Flow as a Function of Temperature for Samples Partially Cured to, From Top to Bottom by Peak, Uncured, 170, 190, 200, 210, 220, 230, 240, and 270 oC, at a Heating Rate of 5 oC/min, Starting from Room Temperature.

57

parentheses, the cure temperature and degree of cure. The sample cured to 270 oC had no

peak. This indicated that the sample was completely cured when cured to 270 oC. For all

other samples, their exothermal peak temperatures could be divided into two categories:

those around the lower temperature of 205 oC (the first peak) and those around the higher

temperature of 236 oC (the second peak). The regions around the first peak were

dominated by Reaction 1. All of the shoulders caused by Reaction 2 occurred after the

peaks. Sample cured up to 190 oC (17% cured) showed no significant change of the

exothermal peak temperature, so the effect of the second reaction to the first peak was

small. However, when the sample was cured to 200 oC (29% cured), its peak temperature

was shifted to 209 oC, so the effect of the second reaction on the first peak was apparent.

The regions around the second peaks were dominated by Reaction 2. When the sample

was cured to 210 oC (45% cured), the first peak disappeared. Instead, the shoulder

occurred before the second peak, which began at 232 oC. It was similar for the sample

cured to 220 oC (72% cured). The second peak temperatures for both samples were

affected by Reaction 1. For the samples cured to 230 (78% cured) and 240 oC (86%

cured), the second peak temperatures showed very little difference, so the effect of the

first reaction to the second peak temperature could be neglected.

As shown in Fig. 12 (b), all the first exothermal peak temperatures at different

heating rates were apparent, so it was not difficult to determine the first exothermal peak

temperature at each heating rate. However, the second peak temperature, which was

caused by Reaction 2, was not apparent. Because of the effect of the first cure reaction, it

occurred at the shoulder of the heat flow curve. In such a case, the second exothermal

peak temperature was determined as the temperature at the turning point in the shoulder

58

region. It was reported that other techniques, such as the Gaussian distribution, were used

for the peak separation (Karkanas and Partridge, 2000). The values for the first and

second exothermal peak temperatures are provided in Table 6.

Table 6. The Dependence of Peak Properties on the Heating Rate

Heating rate (oC/min)

2 5 10 15 20

Tp (oC) 181.03 206.06 227.21 240.14 250.35

αp 0.4167 0.4602 0.4832 0.5022 0.5151 Peak 1 pdT

d )( α (1/K)

0.0163 0.0152 0.0145 0.0146 0.0149

Tp (oC) 218.75 235.31 248.63 256.98 262.95

αp 0.8642 0.8283 0.7696 0.7345 0.6971 Peak 2 (Shoulder)

pdTd )( α

(1/K) 0.0075 0.0097 0.0117 0.0128 0.0137

5.2.2 The Kinetic Analysis Based on Kissinger and Ozawa Approach

For the dynamic curing process, the cure rate was not only a function of degree of

cure, but also a function of temperature. At different heating rates, the curing processes

took place in different temperature ranges. The curing process was greatly affected by the

heating rate. As discussed earlier, the exothermal peak temperature increased with the

increment of heating rate. Eq. (12) can be used to describe the relationship between the

heating rate and exothermic peak temperature for the autocatalytic cure process with the

cure rate of zero at the beginning of cure reaction, which was based on Kissinger (1957)

and Ozawa (1970) approach. The pre-exponential factor A in Eq. (12) changed with the

heating rate, so the average value of the pre-exponential factors at the different heating

rates was used instead. Theoretically, the derivative of cure rate with respect to

59

temperature equals zero at the peak temperature, so the derivative of degree of cure with

respect to temperature )dTd( α at the peak temperature should be constant, regardless of the

heating rate. The term ln( at the peak temperature changed with the heating

rate, but compared to term

))1( nm α−α

Aln , its value was very small. Based on the above

considerations, the general linear form between heating rate and the reversal of the peak

temperature is

ln )( dtdT =

Ac ln(ln −=

pdTd )α

)1)((p

a

TREc −+ (40)

where Tp is the absolute temperature at the exothermic peak; )RE a−( is the slope of the

curve; c is the intercept and

np

mppdT

d )1(ln) ααα −+ (41)

where A is the average value of the pre-exponential factors at the five heating rates. The

terms pdTd )( α and αp are the derivative of degree of cure to temperature and degree of cure

at the exothermic peak, respectively. Their values are also shown in Table 6. It was

noticed that the difference of ( at the different heating rates was small. For peak 1,

the degree of cure αp at the peak temperature increased with the increment of heating rate.

For peak 2, it was the opposite. The logarithm plots of heating rate to the reciprocal of the

absolute peak temperature are given in Fig. 14. It is shown that there exists a very good

linear relationship between heating rate and the reversal of the exothermic peak

temperature.

The values for intercepts and activation energy calculated from slopes of the two

peaks are given in Table 7, page 66, where c1 and Ea1 are the intercept and activation

60

0.0018 0.0019 0.0020 0.0021 0.0022 0.0023-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5 peak 1 peak 2

ln(h

eatin

g ra

te (K

/s) )

1/T (1/K)

Fig. 14 Plots of Logarithm Heating Rate vs. the Reciprocal of the Absolute Exothermal Peak Temperature. The Activation Energies Were Determined by Peak Temperatures at Heating Rates of 2, 5, 10, 15, and 20 oC/min

energy for peak 1 (Reaction 1) and c2 and Ea2 are the intercept and activation energy for

peak 2 (Reaction 2). The activation energies for Reactions 1 and 2 were 65.84 and

114.87 KJ/mol, respectively. The activation energy for Reaction 2 was much higher than

that for Reaction 1. Therefore, Reaction 2 was more sensitive to the temperature than

Reaction 1. At a higher heating rate, the cure reactions shifted to a higher cure

temperature range. At peak 1, which was caused mainly by Reaction 1, the total degree of

cure at the higher heating rate increased due to the larger contribution from Reaction 2 at

the higher cure temperature range. At peak 2, which was caused mainly by Reaction 2,

the total degree of cure at the higher heating rate decreased due to the smaller

contribution from Reaction 1 at the higher cure temperature range.

Based on Eq. (10), a series of iso-conversional plots could be obtained. In this

case, each plot has the same degree of cure. At the different heating rates, the temperature

61

required to achieve the same degree of cure is different. It increased with the increment of

heating rate. At each iso-conversional curve, the apparent activation energy Ea should be

constant. The iso-conversional plots of the logarithmic heating rate versus the reciprocal

of the absolute temperature are shown in Fig. 15. For all the iso-conversional curves, a

good linear relationship is observed between the logarithm heating rate and the reciprocal

of the absolute temperature. The apparent activation energy at each degree of cure is

calculated from its slope in the iso-conversional curve and plotted in Fig. 16. The iso-

conversional plots helped to understand the details of the curing process. As shown in

Fig. 16, the apparent activation energy increased with the increment of degree of cure. At

the lower degree of cure of 0.05, the apparent activation energy is close to the activation

energy of the first reaction. At the higher degree of cure of 0.95, the apparent activation

energy is close to the activation energy of the second reaction. So at the early stage

of the cure reaction, the curing process was dominated by the first reaction, while at the

later stage of cure reaction, the curing process was dominated by the second reaction.

This assumption was further supported by the calculated results discussed next.

Equation (41) can be rearranged to obtain an expression for the average pre-

exponential factor A ,

n

p

m

p

pc

dTde

A)1(

)(

αα

α

−= (42)

Having obtained the average pre-exponential factor A and the activation energy,

we must now determine the actual pre-exponential factor and orders of cure reactions at

each specific heating rate. To reflect the change of the pre-exponential factor A with the

heating rate, the average pre-exponential factor was modified by introducing a new

62

0.0016 0.0018 0.0020 0.0022 0.0024 0.0026-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5degree of cure(%)

5 10 20 40 60 80 90 95 linear fit

ln (h

eatin

g ra

te (K

/s))

1/T (1/k)

Fig. 15 The Iso-conversional Plots for Logarithm Heating Rate vs. Reciprocal of the Absolute Temperature. The Apparent Activation Energies Were Determined by Iso-conversional Plots

0 20 40 60 80 10060

70

80

90

100

110

120

E a (kJ

/mol

)

Degree of cure α (%)

Fig. 16 The Activation Energy, Calculated by the Iso-conversional Plots, as a Function of Degree of Cure

63

parameter Ar to obtain an expression for the pre-exponential factor A at each heating rate.

If we let AAAr = , then Eq. (42) can be written as

n

p

m

p

pc

rdTde

AA)1(

)(

αα

α

−⋅= (43)

where Ar was the correction factor of the specific pre-exponential factor to the average

pre-exponential factor, which varied with the heating rate. The regression processes

indicated that the introduction of Ar could greatly improve the fitting results.

Substituting Eq. (11) and (43) into Eq. (7) and rearranging, the final expression

for the dynamic cure rate is

np

mp

nmRTE

pc

r

a

edTdeAdt

d)1(

)1()(αααααα

−−⋅⋅⋅=

−⋅ (44)

In Eq. (44), the dependent variable is the cure rate dtdα and the independent

variables are cure temperature T and degree of cure α. The correction factor Ar and the

orders of cure reaction m and n were determined by a multiple nonlinear least square

regression method, which is based on the Levenberg-Marquardt algorithm. The multiple

nonlinear least square regression process is the same as the single nonlinear least square

regression process, which was talked previously.

As discussed earlier, two cure reactions occurred during the dynamic curing

process. The contributions of these two reactions to the total reaction were not the same

during the whole curing process. In the early cure stage, Reaction 1 dominated the curing

process. In the later curing process, Reaction 2 dominated the curing process. The details

of these two reactions can be learned from the curve fitting results. To achieve better

nonlinear fit results, the proper range of cure rate should be selected as the source data.

64

Another important thing is to decide which reaction to fit first. One option was to select

the early stage of cure rate as the source data to fit Reaction 1 first, then use the

difference between total cure rate and fitting rate from Reaction 1 as the source data to fit

Reaction 2. Another option was to select the late stage of cure rate as the source data to

fit Reaction 2 first, then used the difference between total cure rate and fitting rate from

Reaction 2 as the source data to fit Reaction 1. It was found that the second option

achieved the better fitting results because it had a smaller residual error. Therefore, the

second option was used and the range of cure rate was selected staring from about 80 %

degree of cure and ending with 100 % degree of cure. The values for the correction

factor Ar and orders of cure reaction m and n were obtained through multiple nonlinear

regressions. During the regression process for Reaction 2, it was found that reaction order

decreased to a very small value, about 102m -17, so its value was set to zero. The values

of Ar, m, and n for Reactions 1 and 2 are listed in Table 7, which shows the data at

different heating rates along with standard errors. In Table 7 and the flowing equations,

please note that A1, Ea1 refer as the pre-exponential factor and the activation energy for

Reaction 1, respectively and A2, Ea2 refer as the pre-exponential factor and the activation

energy for Reaction 2, respectively.

Reaction 1 exhibited the behavior of the autocatalytic reaction. The average

values for m and at the studied heating rates were 0.50 and 2.60, respectively. With a

value of zero for m

1 1n

2, Reaction 2 became the simple nth order reaction. So Reactions 1 and

2 have the different cure mechanisms. The average value of reaction order for Reaction 2

at the studied heating rates was 0.74. For both reactions, it was shown that the orders of

reactions changed little with the heating rate, so the effect of heating rate on the reaction

65

Table 7. Dynamic Kinetic Parameters Obtained by Method Based on Kissinger and Ozawa Approach

Heating rate (oC/min)

2 5 10 15 20

Ar1 0.868 0.698 0.554 0.448 0.373

SE 0.001 0.002 0.001 0.001 0.001 A1

(×104 s-1) 10.839 10.004 8.217 7.282 6.648

SE 0.514 0.474 0.389 0.345 0.315

c1 14.046

SE 0.0474

Ea1 65.84 (KJ/mol)

SE 0.194 (KJ/mol)

m1 0.546 0.542 0.491 0.464 0.474

SE 0.004 0.006 0.006 0.007 0.007

n1 2.461 2.582 2.630 2.665 2.675

Rea

ctio

n 1

SE 0.007 0.011 0.013 0.015 0.017

Ar2 0.763 0.894 0.870 0.849 0.829

SE 0.002 0.003 0.003 0.003 0.003 A2

(×109 s-1) 1.447 1.712 1.542 1.480 1.401

SE 0.151 0.179 0.161 0.155 0.146

c2 24.676

SE 0.105

Ea2 114.77 (KJ/mol)

SE 0.45 (KJ/mol)

m2 0 0 0 0 0

n2 0.789 0.754 0.724 0.721 0.718

Rea

ctio

n 2

SE 0.002 0.003 0.004 0.004 0.003

66

orders was not significant. The averaged total reaction order of these two reactions was

about 1.9 for all of the heating rates. This value was close to the reaction order of 2,

which was assumed the value of the total order of the cure reaction (Yang et al., 1999;

Dupuy et al., 2000).

As seen in Table 7, all values for Ar at the studied heating rates were less than 1,

in the range of 0.868-0.373. Now that the values for αp, pdTd )( α (Table 6), c (intercept in

Fig. 14), Ar, m, and n (by nonlinear regressions) are known, the pre-exponential factor A

at each heating rate can be calculated by Eq. (43). Its value is listed in Table 7. Unlike

the orders of reaction, the change of the pre-exponential factor with the heating rate for

Reaction 1 was apparent. As the heating rate increased, the pre-exponential factor A1

decreased. This implies that the kinetic rate constant at the same temperature for Reaction

1 decreased with the increment of the heating rate. Therefore, as the heating rate

increased, the exothermal peak temperature caused by Reaction 1 shifted to the higher

cure temperature. The pre-exponential factor for Reaction 2 showed little change with

the heating rate, so the effect of heating rate on the kinetic rate constant in Reaction 2 was

small.

Having obtained the kinetic parameters for these two reactions, we could calculate

the values for degree of cure and cure rate for each reaction by solving the differential

equations. According to Eq. (4), the cure rates dt

d 1α of Reactions 1 and dt

d 2α of Reaction

2 are the function of the total degree of cure α, which equals the sum of the degree of

cure α1 by Reaction 1 and degree of cure α2 by Reaction 2,

21 ααα += (45)

67

For each reaction, substituting Eqs. (8), (11), and (45) into Eq. (7) and rearranging we

obtain

111

)1()()( 2121111 nmRT

E a

eAdtdT

dTd ααααα

−−+=−− (46)

222

)1()()( 2121212 nmRT

E a

eAdtdT

dTd ααααα

−−+=−− (47)

where A1, Ea1, m1 and n1 are the pre-exponential factor, activation energy, and reaction

orders by Reaction 1, respectively, and A2, Ea2, m2 and n2 are the pre-exponential factor,

activation energy, and reaction orders by Reaction 2, respectively.

Equations (46) and (47) comprise a system of nonlinear ordinary differential

equations, where the dependent variables are the degree of cure α1 and α2, and the

independent variable is temperature T. There is no known analytic solution to the above

system of equations. Matlab software was used to find the numerical solution. The solver

used was ode45, which is based on the Runge-Kutta (4,5) algorithm. The calculated

results for degree of cure α1, α2 and α at each heating rate are plotted in Fig. 17.

At different heating rates, the final contributions of reactions 1 and 2 to the total

reaction were different. At a 2 oC/min heating rate, the final degrees of cure for Reaction

1 and 2 were 0.70 and 0.30, respectively, while at a 20 oC/min heating rate, the final

degrees of cure for Reaction 1 and 2 were 0.38 and 0.62, respectively. As heating rate

increased, the final degree of cure from Reaction 1 decreased. Except in the early cure

stage, the calculated total degree of cure agreed well with the experimental data at all the

heating rates.

Now that we have curves of degree of cure α1, α2, and α versus the temperature,

the dependence of dα1/dt, dα2/dt, and dα/dt on temperature is easily obtained by

68

100 150 200 250 300 350

0.0

0.2

0.4

0.6

0.8

1.0 experimentalcalculation:

total reaction reaction 1 reaction 2

Deg

ree

of c

ure

α

Temperature (oc)

(a) Heating Rate: 2 oC/min

100 150 200 250 300 350

0.0

0.2

0.4

0.6

0.8

1.0 experimentalcalculation:

total reaction reaction 1 reaction 2

Deg

ree

of c

ure

α

Temperature (oC)

(b) Heating Rate: 5 oC/min

100 150 200 250 300 350

0.0

0.2

0.4

0.6

0.8

1.0 experimentalcalculation:

total reaction reaction 1 reaction 2

Deg

ree

of c

ure

α

Temperature (oC)

(c) Heating Rate: 10 oC/min

100 150 200 250 300 350

0.0

0.2

0.4

0.6

0.8

1.0 experimentalcalculation:

total reaction reaction 1 reaction 2

Deg

ree

of c

ure

α

Temperature (oC)

(d) Heating Rate: 15 oC/min

100 150 200 250 300 350

0.0

0.2

0.4

0.6

0.8

1.0 experimentalcalculation:

total reaction reaction 1 reaction 2

Deg

ree

of c

ure

α

Temperature (oC) (e) Heating Rate: 20 oC/min

Fig. 17 Degree of Cure as a Function of Temperature Calculated by Method Based on Kissinger and Ozawa Approach and its Comparison to the Experimental Value

69

differentiating the degree of cure α1, α2, and α with respect to temperature T. By Eq. (8),

the cure rates dα1/dt, dα2/dt, and dα/dt versus temperature T can be calculated. The plots

are given in Fig. 18. It is shown that the maximum cure rates for reactions 1 and 2

changed with heating rate. At a 2 oC/min heating rate, the maximum cure rate of Reaction

1 was much higher than that of Reaction 2, while at a 20 oC/min heating rate, the

maximum cure rate of Reaction 1 was much lower than that of Reaction 2. Increasing the

heating rate increases the maximum cure rates for both reactions, but the maximum cure

rate of Reaction 2 increased much faster than that of Reaction 1. The curve for the

calculated total cure rate successfully predicted a peak and a shoulder in the cure process.

5.2.3 The Kinetic Analysis Based on Borchardt and Daniels Approach

In the previous section, the curing process was modeled with two autocatalytic

cure reactions. The pre-exponent factor and activation energy for each reaction were

determined first by the characteristics of the peaks at the different heating rates. The cure

reaction orders were then determined by the multiple regressions. The whole modeling

process was a little complicated. In this section, the whole curing process is studied in

term of one autocatalytic cure reaction. The pre-exponent factor and activation energy

for the reaction are now called the apparent pre-exponent factor and activation energy.

Contrary to the previous method, all the four parameters can be determined

simultaneously. Eq. (13) is the autocatalytic model that can be used for the kinetic

analysis based on Borchardt and Daniels approach (1956).

In principle, Eq. (13) could be solved by multiple nonlinear regressions. Because

the cure rate was an exponential function of the reciprocal of the absolute temperature, it

was difficult to get a good solution. The fitting results showed that there exist large

70

100 150 200 250 300 350

0

1x10-4

2x10-4

3x10-4

4x10-4

5x10-4

6x10-4

7x10-4

experimentalcalculation:

total reaction reaction 1 reaction 2

Cur

e ra

te (1

/s)

Temperature (oC) (a) Heating Rate: 2 oC/min

100 150 200 250 300 350

0.0

2.0x10-4

4.0x10-4

6.0x10-4

8.0x10-4

1.0x10-3

1.2x10-3

1.4x10-3

experimentalcalculation:

total reaction reaction 1 reaction 2

Cur

e ra

te (1

/s)

Temperature (oC)

(b) Heating Rate: 5 oC/min

100 150 200 250 300 350

0.0

5.0x10-4

1.0x10-3

1.5x10-3

2.0x10-3

2.5x10-3

3.0x10-3

experimentalcalculation:

total reaction reaction 1 reaction 2

Cur

e ra

te (1

/s)

Temperature (oC)

(c) Heating Rate: 10 oC/min

100 150 200 250 300 350

0.000

0.001

0.002

0.003

0.004 experimental

calculation: total reaction reaction 1 reaction 2

Cur

e ra

te (1

/s)

Temperature (oC) (d) Heating Rate: 15 oC/min

100 150 200 250 300 350

0

1x10-3

2x10-3

3x10-3

4x10-3

5x10-3

6x10-3

experimentalcalculation:

total reaction reaction 1 reaction 2

Cur

e ra

te (1

/s)

Temperature (oC) (e) Heating Rate: 20 oC/min

Fig. 18 Cure Rate as a Function of Temperature Calculated by Method Based on Kissinger and Ozawa Approach and its Comparison to the Experimental Value

71

errors for the kinetic parameters obtained by such a method, so the linear expression for

the logarithmic cure rate as Eq. (14) was used instead.

The plots of the logarithm cure rate versus the reciprocal of the absolute

temperature at the heating rates of 2, 5, 10, 15, 20 oC/min are given in Fig. 19. As shown

in Fig. 19, only a part of the curve exhibited a linear behavior. During the fitting process,

the linear part with the negative slope was selected as the source data. The reaction orders

m and n obtained by the multiple linear regressions are listed in Table 8. The multiple

linear fitting curves are also given in Fig. 19. As seen in Fig. 19, a large difference

between the experimental and fitted value occurred at a late stage of the curing process

because of the complexity of the actual dynamic cure reaction.

Table 8. Apparent Dynamic Kinetic Parameters Obtained by Method Based on Borchardt and Daniels Approach

Heating rate (oC/min)

2 5 10 15 20

A (×105 s-1) 1.780 2.144 2.170 3.031 4.298

SE 0.012 0.013 0.021 0.015 0.017

Ea (KJ/mol) 69.93 71.52 71.82 73.46 74.99

SE 0.023 0.021 0.039 0.019 0.016

m 0.281 0.216 0.217 0.202 0.220

SE 0.005 0.004 0.012 0.004 0.005

n 1.499 1.238 1.338 1.267 1.247

SE 0.013 0.013 0.019 0.007 0.009

In Table 8, it was noticed that the variation of m and n with the heating rate was

small. Compared to the average values of 0.25 and 1.67 for m and n previously obtained,

the average values for m and n by this method were 0.23 and 1.32, respectively. The

72

1.6x10-3 1.8x10-3 2.0x10-3 2.2x10-3 2.4x10-3 2.6x10-3 2.8x10-3-18

-16

-14

-12

-10

-8

-6

-4 experimental model

ln(c

ure

rate

(1/s

))

1/T (1/K) (a) Heating Rate: 2 oC/min

1.6x10-3 1.8x10-3 2.0x10-3 2.2x10-3 2.4x10-3 2.6x10-3 2.8x10-3-18

-16

-14

-12

-10

-8

-6

-4 experimental model

ln(c

ure

rate

(1/s

))

1/T (1/K)

(b) Heating Rate: 5 oC/min

1.6x10-3 1.8x10-3 2.0x10-3 2.2x10-3 2.4x10-3 2.6x10-3-14

-12

-10

-8

-6

-4 experimental model

ln(c

ure

rate

(1/s

))

1/T (1/K)

(c) Heating Rate: 10 oC/min

1.6x10-3 1.8x10-3 2.0x10-3 2.2x10-3 2.4x10-3 2.6x10-3-14

-12

-10

-8

-6

-4 experimental model

ln(c

ure

rate

(1/s

))

1/T (1/K)

(d) Heating Rate: 15 oC/min

1.6x10-3 1.8x10-3 2.0x10-3 2.2x10-3 2.4x10-3 2.6x10-3-14

-12

-10

-8

-6

-4 experimental model

ln(c

ure

rate

(1/s

))

1/T (1/K) (e) Heating Rate: 20 oC/min

Fig. 19 Modeling of Logarithmic Cure Rate vs. the Reciprocal of the Absolute Temperature and Comparing to the Experimental Values

73

values for reaction orders m and n were determined with small standard errors. However,

the apparent pre-exponential factor A and activation energy Ea had large standard errors

(not shown in Table 8). Especially for A, the standard error was in the range of 10-30%,

i.e. 10-30 % of A. Therefore, it is necessary to re-determine the apparent pre-exponential

factor A and activation energy Ea to reduce the magnitudes of the standard errors.

Once the orders of cure reaction m and n are determined, the apparent pre-

exponential factor A and activation energy Ea can be determined with small standard

errors by the Barett method (Riccardi et al., 1984). Equation (13) can be rearranged to

yield

RTE

nm

a

Aedtd

−=

− )1(

)(

αα

α

(48)

If there is a linear relationship between the logarithm of ))1(() nmdtd αα( α − and

T1 , the apparent pre-exponential factor A and activation energy Ea could be determined

from the intercept and slope of line obtained from Eq. (48). The plots of

)))1(()(ln( nmdtd ααα − versus

T1 are shown in Fig. 20. It is noticed that a main portion of

the curve exhibits a linear behavior. By fitting to the linear part of the curve, the apparent

pre-exponential factor A and activation energy Ea were determined. Their values, with

standard errors, are given in Table 8. Both the apparent pre-exponential factor A and

activation energy Ea were increased with the increment of the heating rates. It was

noticed that the standard errors for A and Ea were greatly reduced. Therefore, values for A

and Ea were more reliable. The values for the apparent pre-exponential factor A and the

apparent activation energy Ea were greater than that of the first reaction, but less than that

74

of the second reaction, which was discussed in the method based on the Kissinger and

Ozawa approach.

0.0016 0.0018 0.0020 0.0022 0.0024 0.0026 0.0028-12

-10

-8

-6

-4

-2

0heating rate (oC/min)

2 5 10 15 20

ln((

dα/d

t)/(α

m(1

-α)n )

(1/s

))

1/T (1/K)

Fig. 20 Plots of Logarithm ( vs. the Reciprocal of the Absolute Temperature at Heating Rates of 2, 5, 10, 15, and 20

))1(/()/ nmdtd ααα −oC/min From Right to Left,

Respectively. The Apparent Pre-exponents and Activation Energies Were Determined From the Intercepts and Slopes of the Linear Parts of the Curves

With all of the kinetic parameters in Eq. (13) known, the degree of cure can be

calculated by solving the differential equation. Substitution of Eq. (8) into Eq. (13) and

rearranging yields the ordinary differential equation,

nmRTE a

AedtdT

dTd )1()( 1 ααα

−=−− (49)

where the dependent variable is degree of cure α, and the independent variable is the

absolute temperature T.

Equation (49) has no known analytical solution. Matlab software was used again

to find the numerical solution. The calculated results at heating rates of 2, 5, 10, 15, 20

75

oC/min are plotted in Fig. 21. At up to 0.8 of degree of cure, the calculated results

offered very good agreement with the experimental value. It was also noticed that the

curves shifted to higher temperature ranges with the increment of heating rate. By

differentiating the degree of cure α with respect to the absolute temperature T, the cure

rate dtdα could be obtained by Eq. (8). The plots of calculated cure rate dt

dα versus the

absolute temperature are given in Fig. 22. At each heating rate, the calculated bell-shape

curve successfully predicted the peak of cure rate, especially at higher heating rates, but

failed to show the appearance of the shoulder as indicated by the experimental value.

Therefore, this method is more appropriate to model the simple curing process involved

one cure reaction.

5.3 Isothermal and Dynamic Scanning Rheological Cure Analysis

Rheological analysis of physical properties such as storage modulus, viscosity,

and gel time provides additional information about the curing process of epoxy prepreg.

Many rheological models have been developed to predict viscosity profiles of the cure

process. The first and nth order viscosity models, to be discussed later, express viscosity

as an exponential function of the cure time. Parameters in these models are easily

determined by the rheological experimental data only. The first and nth viscosity models

have been frequently used in the rheological analysis of the cure process (Dusi et al.,

1983; Theriault et al., 1999; Wang et al., 1997). These models do not incorporate the

effect of gelation on the viscosity and the predication accuracy is not good. The modified

Williams-Landel-Ferry (WFL) models for viscosity (Tajima and Crozier, 1983; Mijovic

and Lee, 1989) describe viscosity as the function of both cure temperature and glass

transition temperature. These models have been extensively used and they were reported

76

100 150 200 250 300 350

0.0

0.2

0.4

0.6

0.8

1.0

20 oC/min

15 oC/min

10 oC/min

5 oC/min

heating rate=2 oC/min

experimental calculation

Deg

ree

of c

ure

α

Temperature (oC)

Fig. 21 Degree of Cure as a Function of Temperature Calculated by Method Based on Borchardt and Daniels Approach at Heating Rates of 2, 5, 10, 15, and 20 oC/min from Left to Right, Respectively and its Comparison to the Experimental Value

100 150 200 250 300 350

0.0

1.0x10-3

2.0x10-3

3.0x10-3

4.0x10-3

5.0x10-3

6.0x10-3

2 oC/min

5 oC/min

10 oC/min

15 oC/min

heating rate=20 oC/min experimental calculation

Cur

e ra

te (1

/s)

Temperature (oC)

Fig. 22 Cure Rate as a Function of Temperature Calculated by Method Based on Borchardt and Daniels Approach at Heating Rates of 2, 5, 10, 15, and 20 oC/min From Left to Right by Peak and its Comparison to the Experimental Value

77

to achieve good accuracy (Karkanas and Partrige, 2000). When applying the WFL

models, one needs to know the relationship between the glass transition temperature and

cure time, which can be determined by thermal analysis. A percolation model for

viscosity (Serrano et al., 1990) expresses the variation of viscosity with degree of cure

by a power law. By introducing the degree of critical reaction into the model, the gel

effect on the cure process was taken into account. It was reported that the percolation

model fit the experimental data quite well (Serrano et al., 1990). For the application of

the percolation model, a kinetics model is necessary in order to determine the relationship

between the degree of cure and time. The characteristics of other viscosity models for

cure applications were discussed by Halley and Mackay (1996).

During the rheological measurements of the cure process, a sinusoidal shear strain

or stress is applied to the prepreg sample between two parallel plates and the response is

monitored and finally recorded as the complex modulus G*, storage modulus G', loss

modulus G", viscosity η, and phase angle δ in degrees. A series of isothermal rheological

measurements were carried out from 110 to 180 oC at 10 oC intervals, as well as dynamic

rheological measurements. Fig. 23 shows the typical variations of storage modulus G'

and loss modulus G" with time at 110 oC. During the initial cure period, the loss modulus

is greater than storage modulus. The viscous character of the epoxy prepreg dominates its

elasticity. In the final cure stage, the storage modulus is much greater than loss modulus.

The epoxy prepreg becomes mostly an elastic solid. Under isothermal conditions, the

storage modulus increases with the advancement of the cure process, and it finally

reaches a plateau value. In contrast, the loss modulus increases due to the increment of

the molecule weight through polymerization, reaches a maximum, and then decreases

78

100 200 300 400 500102

103

104

105

106

107

G' G''

G' o

r G'' (

Pas

)

Time (minute)

Fig. 23 Dependence of Storage Modulus G' and Loss Modulus G'' on Time at 110 oC

rapidly due to the solidification of the epoxy system when it approaches the gelled glass

region.

5.3.1 Gel Time and Apparent Activation Energy (Ea)

The gel point of the cure process is closely related to rheological properties. It

indicates the beginning of crosslinking for the cure reaction, where the resin system

changes from a liquid to a rubber state. The gel time can be determined according to

different criteria (Gillham and Benci, 1974; Chambon and Winter, 1987). The commonly

used criteria for gel time are as follows:

• Criterion 1, the gel time is determined from the crossing point between the base

line and the tangent drawn from the turning point of G' curve (Ampudia et al.,

1997; Laza et al. 1998).

79

• Criterion 2, the gel time is thought as time where tan δ equals 1, or G' and G"

curves crossover (Tung and Dynes, 1982; Laza et al. 1998).

• Criterion 3, the gel time is taken as the point where tan δ is independent of

frequency (Scanlan and Winter, 1991; Raghavan et al., 1996).

• Criterion 4, the gel time is the time required for viscosity to reach a very large

value or tends to infinity (Mijovic and Kenny, 1993).

In this study, the determination of gel time was based on the second criterion.

The profiles of phase angle δ (in degrees) at different isothermal cure temperatures are

given in Fig. 24. The values for gel time, determined from Fig. 24 by criterion 2, are

listed in Table 9. As the isothermal temperature increases, the gel time decreases. The

relationship between gel time and temperature is analyzed by cure kinetics. The kinetic

model as Eq. (5) is used for the gelation analysis. Eq. (17) shows the relationship

between the gel time and isothermal cure temperature. According to Eq. (17), the semi-

logarithmic plot of gel time vs. the reciprocal of the absolute temperature is drawn in Fig.

25. A linear fit of the experimental data gives a value for the apparent activation energy

of 63.0 KJ/mol.

Table 9. Gel Time at Different Temperatures and the Activation Energy

Temperature (oC) 110 120 130 140 150 160 170 180

tgel (s) 14630 10260 6672 4244 2498 1645 1079 760

Ea (KJ/mol) 63.0

SE (KJ/mol) 1.59

80

0 100 200 300 400 5000

10

20

30

40

50

60

70

80

90 110 oC 120 oC 130 oC 140 oC 150 oC 160 oC 170 oC 180 oC

Pha

se a

ngle

δ (d

egre

e)

Time (minute)

Fig. 24 Dependence of Phase Angle on Time at Isothermal Conditions

0.0020 0.0022 0.0024 0.0026 0.00286

7

8

9

10 experimental linear fit

ln(t ge

l) (s

ec)

1/T (1/K)

Fig. 25 Gel Time as a Function of Isothermal Cure Temperature

81

5.3.2 Rheological Modeling

Different viscosity models have been discussed previously. These models are

now verified with the experimental viscosity data. By Eq. (26), the initial viscosity η0

and rate constant k at each temperature are calculated from the intercept and slope,

respectively of the linear part of the curve of logarithmic viscosity vs. time before the gel

point. Their values at each isothermal temperature are listed in Table 10.

Table 10. Initial Viscosity η0 and Rate Constant k in Eqs. (27) and (30) of the Viscosity Models

Temperature (oC) ln(η0) (Pa⋅s) SE (ln Pa⋅s) k (s-1) SE (s-1)

110 6.1959 0.0380 3.739 x 10-4 3.75 x 10-6

120 5.8725 0.0278 5.900 x 10-4 4.14 x 10-6

130 5.0064 0.0508 9.306 x 10-4 1.22 x 10-5

140 4.8302 0.0801 1.57 x 10-3 2.85 x 10-5

150 4.2855 0.1313 2.60 x 10-3 7.23 x 10-5

160 4.0307 0.0899 4.44 x 10-3 7.63 x 10-5

170 3.3986 0.1949 6.71 x 10-3 2.44 x 10-4

180 3.2307 0.0786 1.061 x 10-2 1.2 x 10-4 Pre-exponential

factor Aη = 1.8274 × 10-6 (Pa⋅s) Ak=1.306 × 106 (s-1)

SE 1.3197 × 10-6 (Pa⋅s) 5.182 × 105 (s-1) Activation energy

(KJ/mol) Eη = 63.0 Ek=70.0

SE (KJ/mol) 3.16 1.45

As temperature increases, the rate constant increases while the initial viscosity

decreases. Their relationship with temperature obeys the Arrhenius law as Eqs. (6) and

(21). The plots of logarithmic initial viscosity and rate constant vs. the reciprocal of

82

temperature, including their linear fits, are shown in Figs. 26 and 27, respectively. The

values of Aη, Eη, Ak, and Ek are determined from the linear fits and are listed in Table 10.

The calculated viscosity, by Eq. (26), and experimental data at some temperatures are

shown in Fig. 28 (More curves are given in Appendix C). Large discrepancies occur in

the gel region.

In the viscosity model presented as Eq. (26), the relationship between viscosity

and time at isothermal conditions is a pure exponential type. It does not take into account

the effect of gelation on the viscosity. This viscosity model is not able to predict

accurately the viscosity change. First, it fails to predict the occurrence of the gel point,

where the viscosity rises sharply. Second, the predicted final viscosity goes to infinity.

So a large error exists in the later cure stage. Because of the limitation of the empirical

model of viscosity, a new model of viscosity for isothermal cure process of epoxy resin

systems is proposed and used to fit the experimental viscosity. As seen in Eq. (31), the

proposed new viscosity model introduces two additional parameters, the critical time tc

and final viscosity η∞. All the parameters η0, η∞, tc and k in Eq. (31) are determined at

the same time by fitting experimental viscosity with respect to time by nonlinear least

square approach. The fitted curves are also shown in Fig. 28. The predicted viscosities

have very good agreement with the experimental data, even in the gel region. It seems

clear that the viscosity profile at each temperature can be better described by the

proposed viscosity model.

The regressed values of critical time tc and rate constant k in Eq. (31) at each

temperature are listed in Table 11. The variation in critical time with respect to

temperature is the same as one observed in gel time and can also be described by an

83

0.0020 0.0022 0.0024 0.0026 0.00282

3

4

5

6

7 linear fit

ln (η

0) (P

as)

1/T (1/K)

Fig. 26 Initial Viscosity in Eq. (26) vs. Isothermal Cure Temperature and the Determination of Kinetics Parameters by Arrhenius Equation

0.0020 0.0022 0.0024 0.0026 0.0028-9

-8

-7

-6

-5

-4 linear fit

ln (k

) (1/

s)

1/T (1/K)

Fig. 27 Rate Constant in Eq. (26) vs. Temperature and the Determination of Kinetic Parameters by Arrhenius Equation

84

0 50 100 150 200 250 300

0.0

4.0x104

8.0x104

1.2x105

1.6x105

2.0x105

2.4x105

2.8x105

experimental Eq. (26) Eq. (31)

Vis

cosi

ty (P

a.s)

Time (minute) (a) 110 oC

0 50 100 150 200 250

0.0

4.0x104

8.0x104

1.2x105

1.6x105

2.0x105

2.4x105

2.8x105

experimental Eq. (26) Eq. (31)

Vis

cosi

ty (P

a.s)

Time (minute) (b) 120 oC

0 20 40 60 80 100 120 140

0.0

4.0x104

8.0x104

1.2x105

1.6x105

2.0x105

2.4x105

experimental Eq. (26) Eq. (31)

Vis

cosi

ty (P

a.s)

Time (minute) (c) 130 oC

0 10 20 30 40 50 60 70 80 90 10

0.0

4.0x104

8.0x104

1.2x105

1.6x105

2.0x105

2.4x105

0

experimental Eq. (26) Eq. (31)

Vis

cosi

ty (P

a.s)

Time (minute) (d) 140 oC

0 5 10 15 20 25 30 35 40

0.0

3.0x104

6.0x104

9.0x104

1.2x105

1.5x105

1.8x105

experimental Eq. (26) Eq. (31)

Visc

osity

(Pa.

s)

Time (minute) (e) 160 oC

0 5 10 15 20

0.0

3.0x104

6.0x104

9.0x104

1.2x105

1.5x105

1.8x105

experimental Eq. (26) Eq. (31)

Visc

osity

(Pa.

s)

Time (minute) (f) 180 oC

Fig. 28 Experimental and Calculated Viscosity Profiles at Selected Isothermal Temperatures

85

Table 11. Kinetic Parameters in Eqs. (31) and (33) of the new Viscosity Model Temperature (oC) tc (s) SE (s) k (s-1) SE (s-1)

110 15519.5 49.5 6.0518 × 10-4 4.29 × 10-6

120 10024.2 26.0 9.5540 × 10-4 1.02 × 10-5

130 6710.5 30.6 1.734 × 10-3 3.41 × 10-5

140 4107.2 14.7 2.888 × 10-3 6.00 × 10-5

150 2594.1 11.1 4.623 × 10-3 8.72 × 10-5

160 1742.0 11.8 7.045 × 10-3 1.27 × 10-4

170 1133.2 3.3 1.257 × 10-2 1.4 × 10-4

180 786.7 5.7 1.370 × 10-2 3.7 × 10-4 Pre-exponential

factor (s-1) At = 5.3887 × 10-5 Ak=1.3157 × 106

SE 1.4546 × 10-5 7.395 × 105 Activation energy

(KJ/mol) Et = 62.3 Ek=67.9

SE (KJ/mol) 0.96 2.20

Arrhenius law as Eq. (32). As seen in Fig. 29, there is a very linear relationship between

the logarithmic critical time and the reciprocal of absolute temperature. The rate constant

in Eq. (31) is larger than that in Eq. (26) for the same temperature and also obeys an

Arrhenius equation as a function of temperature. The relationship of ln vs. kT1 and the

linear fit are given in Fig. 30. The fitted values of pre-exponential factor and activation

energies are listed in Table 11. The kinetic activation energy obtained from rate constant

in Eq. (31) is 67.9 KJ/mol, which is closed to the value of 70.0 KJ/mol obtained from the

rate constant in Eq. (26). It is also interesting to note that the activation energies obtained

from gel time in Eq. (17), the initial viscosity in Eq. (21), and the critical time in Eq. (32)

are close to each other, with the values of 63.0, 63.0, and 62.3 KJ/mol, respectively.

86

0.0020 0.0022 0.0024 0.0026 0.00286

7

8

9

10

11 linear fit

ln(t c)

(s)

1/T (1/K)

Fig. 29 Critical Time in Eq. (31) vs. Cure Temperature

0.0020 0.0022 0.0024 0.0026 0.0028-9

-8

-7

-6

-5

-4

-3 linear fit

ln (k

) (1/

s)

1/T (1/K)

Fig. 30 Rate Constant in Eq. (31) vs. Isothermal Cure Temperature and the Determination of Kinetic Parameters by Arrhenius Equation

87

The viscosity profile in the dynamic cure process is different from that in the

isothermal cure process. As temperature increases, the viscosity decreases, reaches a

minimum, and then increases greatly. Fig. 31 gives the logarithmic plots of dynamic

viscosity vs. temperature at heating rates of 2 and 5 oC/min.

20 50 100 300102

103

104

105

106

heating rate 2 (oC/min) 5 (oC/min)

113.9 oC

188.2 oC

106.9 oC159.0 oC

Vis

cosi

ty (P

a.s)

Temperature (0C)

Fig. 31 The Logarithmic Plot of Viscosity vs. Temperature at Heating Rates of 2 and 5 oC/min

At each heating rate, the viscosity curve shows the onset temperature of cure

reaction and the minimum viscosity temperature. As the heating rate increases, the onset

temperature of the cure reaction increases from 106.9 to 113.9 oC, and the minimum

viscosity temperature increases from 159.0 to 188.2 oC. The values of onset temperature

of cure reaction at heating rates of 2 and 5 oC/min are close to those obtained by DSC

measurements, which occur at 110 and 114.5 oC, respectively. It is also worth noting that

the minimum viscosity decreases with the increment of heating rate. As it nears the

minimum viscosity, the epoxy resin system has the greatest flow ability. But the cure

88

reaction rate does not necessarily achieve the fastest rate at the minimum viscosity

temperature. In fact, the DSC kinetic study showed that the maximum cure rate

temperatures at the heating rates of 2 and 5 oC/min are 181 and 206 oC, respectively. So

they are larger than the minimum viscosity temperatures at the same heating rates.

The minimum viscosity has an important effect on the application of epoxy

prepreg. If the minimum viscosity is too low, it may cause the epoxy resin to distribute

unevenly among reinforced fibers before it is fully cured.

The model parameters obtained from isothermal conditions are useful for the

study of variation of viscosity in the dynamic cure process. The variation of viscosity

during the dynamic cure process can be described by the dynamic models previous

discussed, as seen in Eqs. (25), (30) and (33). The kinetic parameters Ak and Ek in the

dynamic models have been determined from isothermal rate constants. To improve the

fitting results, the viscous flow parameters Aη and Eη in the dynamic models were

determined by fitting the linear part of the plot of viscosity vs. time before the beginning

of the cure reaction under dynamic conditions. The calculated values for Aη and Eη were

8.20 × 10-5 Pas and 54.7 KJ/mol, respectively. The viscous flow activation energy

determined from dynamic cure process is lower than that determined from isothermal

conditions. The integrals in the dynamic models were determined numerically using the

trapezoidal rule. This was done with Origin software. The reaction order n in Eq. (30)

and the critical time tc in Eq. (33) were determined by a nonlinear least square regression

method. The viscosities calculated by Eqs. (25), (30) and (33) and their comparisons

with experimental data at heating rates of 2 and 5 oC/min are shown in Figs. 32 and 33,

respectively. It is clear that the nth order reaction model for the dynamic cure process is

89

0 10 20 30 40 50 60 70 80 90 100

0.0

5.0x104

1.0x105

1.5x105

2.0x105

2.5x105

Tem

pera

ture

(o C)

viscosity Eq. (25) Eq. (30) Eq. (33)

Vis

cosi

ty (P

a.s)

Time (minute)

0

50

100

150

200

250

temperature

Fig. 32 Experimental and Calculated Dynamic Viscosity Profiles at a Heating Rate of 2 oC/min. The Reaction Order n Equals 1.13 in Eq. (30). The Critical Time tc in Eq. (33) is 81.2 minutes and the Corresponding Critical Temperature is 187.4 oC

0 5 10 15 20 25 30 35 40 45 50

0.0

5.0x104

1.0x105

1.5x105

2.0x105

2.5x105

3.0x105

viscosity Eq. (25) Eq. (30) Eq. (33)

Vis

cosi

ty (P

a.s)

Time (minute)

0

50

100

150

200

250

Tem

pera

ture

(o C)

temperature

Fig. 33 Experimental and Calculated Dynamic Viscosity Profiles at a Heating Rate of 5 oC/min. The Reaction Order n Equals 1.20 in Eq. (30). The Critical Time tc in Eq. (33) is 39.4 minutes and the Corresponding Critical Temperature is 221.8 oC

90

better than the first order reaction model for the dynamic cure process and that the new

model based on the Boltzmann function is the best among the three models for dynamic

cure process.

5.4 Characterization of the Cure Process of Epoxy Prepreg

The characteristics of the cure process of epoxy prepreg can be better understood

by the combination of the DSC thermal analysis, the rheological analysis, and the

analyses from other techniques. The variation of the isothermal maximum degree of cure

indicates the effect of the diffusion control of the cure process, as seen from iso-

conversional plots. The maximum cure time and half-life of the isothermal cure process

are very useful to design cure cycles. The phenomena of both gelation and vitrification

may occur in the cure process of epoxy prepreg. Depending on the epoxy prepreg

systems, gelation occurs at a special conversion point (Barton et al., 1992; Flory, 1953),

where the epoxy resin changes from a liquid state to a rubber state. The phenomenon of

vitrification of the cure process is closely related to Tg. When Tg is lower than the cure

temperature, the cure process is controlled by kinetics. However, Tg increases from Tg0

with the advancement of cure reaction. When it reaches the cure temperature,

vitrification occurs (where the epoxy resin changes into the glassy state.) The cure

reaction rate becomes slower after vitrification and eventually stops. When the cure

temperature increases, a new reaction may be triggered and cause the possible increment

of the maximum degree of cure.

Gelation and vitrification of the cure process cause the phase transition of the

epoxy prepreg system. The phase transitions can be described by a Time-Temperature-

Transformation (TTT) isothermal cure diagram (Enns and Gillham, 1983; Wisanrakkit et

91

al., 1990) and a Conversion-Temperature-Transformation (CTT) diagram (Riccardiet al.,

1990; Wang and Gillham, 1993). A typical TTT isothermal cure diagram for

thermosetting polymers is given in Fig. 34 (Wisanrakkit et al., 1990). It is divided into

several regions, which correspond to the different physical states of the polymer being

cured: sol glass, liquid, sol/gel rubber, sol/gel glass, gelled rubber, gelled glass, and char.

The diagram shows clearly the possible change of the phase state during the isothermal

cure process.

5.4.1 Further DSC Thermal Analysis

As discussed earlier, the variation of the isothermal heat flow from the epoxy

prepreg sample is caused by the cure reaction, which is an exothermal process. The cure

reaction heat can be calculated from the isothermal DSC curves. The values for the

isothermal cure reaction heat are listed in the Table 12. As shown in Table 12, the

maximum isothermal cure heat at each temperature is different. It increases with the

increment of temperature. The total cure heat could be achieved at 220 oC, with the value

of 177.5 J/g.

Table 12. The Maximum Cure Heat at Different Isothermal Temperatures

Temperature (oC) 110 120 130 140 150 160 170 180 190 200 210 220

Maximum heat (J/g) 113.8 119.8 125.5 131.2 137.6 145.5 152.1 159.1 166.3 172.6 177.4 177.5

The transient degree of cure at each isothermal temperature has been previously

calculated. For the same degree of cure, the required cure time changes with the different

isothermal temperatures. The iso-conversional plots of 1/T vs. log under various

constant degrees of cure are shown in Fig. 35. For curves with degrees of cure from 0.1

to 0.6, an almost linear relationship between the reciprocal of the absolute cure

t

92

Full cure

Fig. 34 Generalized Isothermal Cure Phase Diagram of Time-Temperature-Transformation

93

101 102 103 104 1051.8

2.0

2.2

2.4

2.6

2.8degree of cure

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1/T

x103 (

1/K

)

Time (second)

Fig. 35 Iso-conversional Plots of 1/T vs. Logarithmic Time for Epoxy Prepreg

94

temperature and logarithmic cure time is observed over the isothermal temperature range

of 110-220 oC. This linear relationship means that the curing process in this region is

controlled by the curing kinetics. For curves with degrees of cure from 0.7 to 0.9, a

nonlinear relationship is observed. With the increment of degree of cure, the nonlinear

part of the curve shifts to a higher isothermal cure temperature. This means that the

curing process in this region is not controlled by kinetics, but rather by diffusion. The

region controlled by diffusion shifts to a higher temperature at the higher degree of cure.

Dusi et al (1985) suggested that this region is the onset of vitrification. The details of

phase transition of the cure process are discussed later.

From the maximum cure heat at each isothermal temperature, the corresponding

maximum degree of cure can be calculated (Prime, 1981). Its relationship to the

isothermal cure temperature is shown in Fig. 36. Note that the maximum degree of cure

increases linearly with the increment of the cure temperature until it rises to 1, the degree

of cure for the fully cured epoxy prepreg sample. This is similar to the results reported by

Kenny et al. (1991) and Lee and Wei (2000). At lower isothermal cure temperatures, the

cure reaction stops at a lower degree of cure because of diffusion control. Increasing the

cure temperature shifts the diffusion control to the kinetic control and the cure reaction

stops at a higher degree of cure, so the maximum degree of cure increases at the higher

temperatures.

Corresponding to the maximum degree of cure at each isothermal cure

temperature, there exists a maximum cure time. The values for the maximum cure time

at different isothermal cure temperatures are very different. The relationship between the

maximum cure time and the isothermal cure temperature is given in Fig. 37. Half-life is

95

100 120 140 160 180 200 220 2400.5

0.6

0.7

0.8

0.9

1.0

Max

imum

deg

ree

of c

ure

Temperature (oC)

Fig. 36 The Maximum Degree of Cure as a Function of Isothermal Cure Temperature

100 120 140 160 180 200 220 2400

100

200

300

400

500

600

700

800 half life maximum cure time

142 oC60 minutes

Cur

e tim

e (m

inut

e)

Temperature (oC)

Fig. 37 Half-life and the Maximum Cure Time for the Isothermal Curing Process as a Function of Isothermal Cure Temperature

96

another important parameter for the isothermal cure process. It is the time required to

reach 50 percent conversion at a certain cure temperature. The relationship between the

half-life and isothermal cure temperature is also given in Fig. 37. It is actually an iso-

conversional plot of cure time vs. temperature at degree of cure of 0.5. By comparison,

it is clear that half-life is much smaller than the maximum cure time at the same

isothermal temperature. This is not difficult to understand. At the same temperatures,

the cure rate in the later cure stage becomes much smaller. So it needs much more time in

the late cure stage than in the early cure stage to increase the same value of degree of cure.

The curves suggest that both the maximum cure time and half-life decay exponentially

with respect to the isothermal cure temperature. From Fig. 37, the 60-minute half-life

temperature was determined, which was 142 oC.

Half-life temperature is very useful to verify DSC thermal cure results. The

following experiments were designed based on the conventional 60-minute half-life

temperature. An uncured sample was scanned with DSC at a heating rate of 10 oC/min.

Its heat flow and dynamic cure heat are shown in Fig. 38. Another uncured sample was

aged at 142 oC for 60 minutes and then quickly moved from the instrument chamber to a

container held below 0 oC. After the aged sample became cooled and the instrument was

ready, the aged sampled was scanned with DSC, at the same heating rate of 10 oC/min.

The heat flow and the residual heat of the partially aged sample are also shown in Fig. 38.

The dynamic cure heat of the partially aged sample should be close to half of the dynamic

cure heat of the uncured sample. The data in Fig. 38 shows that the ratio of the cure heat

for the sample cured at 142 oC for 60 minutes to the cure heat of the uncured sample is

97

0.51, which is close to 0.5. So the experimental results discussed earlier are also

conformed to be reasonable by the half-life temperature.

50 100 150 200 250 300 350-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

base line

after cure at 142 oC for60 minutes

before cure

reaction heat=94.6 J/g

reaction heat=185.1 J/g

Hea

t flo

w (W

/g)

Exo

ther

m

Temperature (oC)

Fig. 38 The Comparison of Dynamic Cure Heats Between the Fresh and Partially Cured Samples at a Heating Rate of 10 oC/min. The Ration of the Reaction Heat of the Cured Sample to the Reaction Heat of the Fresh Sample is 0.51

5.4.2 Glass Transition Temperature

The Tg increases with degree of cure. Epoxy prepreg samples with different

degrees of cure were prepared in a specially designed oven by curing the samples from

room temperature to a series of designated temperatures at a heating rate of 5 oC/min and

under constant flow of nitrogen. The measurements of Tg for all of the prepared samples

were conducted through the modulated DSC mode. The underling-heating rate was set to

5 oC/min with a temperature amplitude of 2 oC. The actual modulated temperature

changes at a rate greater or less than the underling-heating rate. The details of the

98

temperature profiles are shown in Fig. 39. Through the modulated DSC mode, the total

heat flow of the sample could be divided into the reversal heat flow and the non-reversal

heat flow. The reversal heat flow corresponds to a phase transition into a glass state,

which can be used to detect Tg. The non-reversal heat flow is caused by the exothermal

reaction of the process. The exothermal reaction heat can be thought of as the residual

heat of as the sample. From the curve of the non-reversal heat flow versus the cure time,

the heat of the exothermal cure reaction was determined by drawing a straight base line

between the onset and end point of the peak and integrating the peak curve based on the

base line. The value of the integrated area represented the residual heat. From the

residual heat, the degree of cure of the measured sample could be calculated (Prime,

1981). Fig. 40 gives reversal and non-reversal heat flow changes with respect to

temperature for the uncured prepreg sample. From the curve of reversal heat flow versus

temperature, the detected Tg for the uncured sample was 1.39 oC, which was taken as the

inflection point in the glass transition region. The T’gs and residual heats for the other

samples were determined in the same way as above. The Tg for the fully cured epoxy

prepreg is 231.5 oC. The relationship of Tg with the degree of cure is given in Fig. 41; a

nonlinear relationship between the Tg and degree of cure is observed. As discussed

earlier, it was assumed that Tg depends only on the degree of cure, regardless of the cure

cycle or cure history. The Tg can be thought of as the index of the degree of cure for the

epoxy prepreg system (Wang, 1997; Wisanrakkit and Gillham, 1990). Eqs. (34) and (35)

were used separately to fit the experimental Tg versus degree of cure. The model results

from the two equations are also given in Fig. 41. It is shown that both equations fit the

experimental data very well.

99

-50 0 50 100 150 200 250 300 350 400-50

0

50

100

150

200

250

300

350

400inside

outside

Mod

ulat

ed te

mpe

ratu

re (o C

)

Temperature (oC)

-30

-20

-10

0

10

20

30

D

eriv

. Mod

ulat

ed te

mpe

ratu

re (o C

/min

)

-2

-1

0

1

2

Tem

pera

ture

am

plitu

de (o C

)

Fig. 39 Temperature Profile in the Modulated DSC Mode

-100 -50 0 50 100 150 200 250 300 350-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

baseline

residual heat=183.6 (J/g)

115.9 oC268.88 oC

1.39 oC (I)N

onre

vers

al H

eat f

low

(W/g

) E

xoth

erm

Rev

ersa

l hea

t flo

w (W

/g)

Exo

ther

m

Temperature (oC)

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Fig. 40 Reversal and Non-reversal Heat Flow for Fresh Prepreg During Dynamic Scanning at 5 oC/min

100

0.0 0.2 0.4 0.6 0.8 1.0

0

50

100

150

200

250

experimental eq. (34) eq. (35)

Gla

ss tr

ansi

tion

tem

pera

ture

(o C)

Degree of cure

Fig. 41 The Dependence of Tg on Degree of Cure

From a nonlinear, least squares, regression analysis, the structure-related

parameter λ and 0p

p

cc

∆

∆ ∞ were determined. The calculated values for λ and 0p

p

cc

∆

∆ ∞ were

0.64 and 0.87, respectively. Materials with different structures may have different values

of λ. Previously reported values of λ and 0p

p

cc

∆

∆ ∞ for epoxy resin systems ranged from

0.16 to 0.69 (Boey and Qiang, 2000; Simon and Gillham, 1993; Venditti and Gillham,

1997). The curvature of the curve for Tg versus degree of cure is affected by the value of

λ (Boey and Qiang, 2000; Pascault and Williams, 1990). A smaller value of λ will result

in a large curvature. This means that Tg increases faster at the higher degree of cure.

5.4.3 Isothermal TTT Cure Diagram and CTT Cure Diagram

The relationship between degree of cure and cure time at each isothermal cure

temperature for the epoxy prepreg has been established. As discussed earlier, the Tg is a

101

function of degree of cure only and is independent of cure temperature. So the

relationship between Tg and the isothermal cure time can be determined. This means that

at each isothermal cure temperature, the cure time required for Tg to rise to the cure

temperature can be obtained. This relationship can be represented by vitrification curve.

Rheological properties of the cure process of epoxy prepreg were studied by means of a

Bohlin rheometer. The gel time has been previously determined-it was the time required

to achieve equal storage and loss moduli. The relationship between isothermal cure

temperature and gel time is represented by the gelation curve. By plotting the

vitrification and gelation curves in the same graph, the isothermal TTT cure diagram of

epoxy prepreg is constructed. It is shown in Fig. 42. Depending on the isothermal cure

path with respect to temperature and time, the cure process is controlled by kinetics in the

liquid and rubber regions and is controlled by diffusion in the glass region.

10 100 500100

120

140

160

180

200

220

240

gelation vitrification

Gelled Glass regionSol/gel rubber region

Liquid regionTem

pera

ture

(o C)

Gel or vitrification time (min)

Fig. 42 The TTT Diagram of Isothermal Cure for the Epoxy Prepreg System

102

Similar to the isothermal TTT cure diagram, a CTT cure diagram is constructed

and given in Fig. 43. The calculated degree of cure at gel time changes in a range of

0.49-0.53 in the studied isothermal cure temperatures range. The average value for

degree of cure at gel time is 0.51. This value is used to draw a straight gelation line in

Fig. 43. The vitrification curve is obtained from the relationship between the Tg and

degree of cure. As shown in Fig. 43, the glass transition temperature at the gel point gelTg

is determined. Its value is 93.4 oC. Depending on the cure temperature and degree of

cure of the epoxy prepreg being cured, it is clearly known from the CTT diagram whether

the cure process of the epoxy prepreg is kinetically or diffusion controlled.

0.0 0.2 0.4 0.6 0.8 1.0

0

50

100

150

200

250

vitrifi

catio

n

Gel

atio

n

0.51393.4 oC

Sol/gel rubberregion

Liquid region

Sol glassregion

Gelled glassregion

gelTg

Tg0

Tg∞

Tem

pera

ture

(o C)

Degree of cure

Fig. 43 The CTT Phase Cure Diagram for the Epoxy Prepreg System

5.4.4 Thermal Stability of Epoxy Prepreg

The thermal stability of an epoxy prepreg has important effects on the curing

process and the mechanical properties of the cured materials. TGA is capable of

103

detecting very small weight loss owing to the volatilization or degradation of the prepreg.

An uncured, 15.6880 mg, prepreg sample and another fully cured, 11.2710 mg, prepreg

sample were scanned with TGA from room temperature to 600 oC, at a heating rate of 5

oC/min. Fig. 44 shows the variations of the residual weight and its derivative with

respect to the temperature for both samples.

As seen in Fig.44, the two TGA curves exhibit a similar profile, meaning that

both uncured and fully cured epoxy prepreg samples follow a similar mechanism of

degradation. Over a wide range temperature, from room temperature to 307 oC, the

uncured sample was almost stable. The onset temperature of the degradation is about 307

oC. The weight loss for the sample increased greatly above this temperature. The

maximum rate of weight loss occurs at 334 oC. At the temperature of 580 oC, the weight

loss for each sample is up to 23 %.

0 100 200 300 400 500 60075

80

85

90

95

100

uncured fully cured

580oC77.8%

Der

ivat

ive

wei

ght (

%/o C

)

Wei

ght (

%)

Temperature (oC)

0

1

2

3

4

307 oC98.79 %

334 oC

Fig. 44 TGA Curves of Epoxy Prepreg at a Heating Rate of 5 oC/min

104

5.4.5 Microstructure of the Epoxy Prepreg Sample

The microstructure of the epoxy prepreg sample is associated with the degree of

cure. Two samples were prepared. One was partially cured with a degree of cure of 0.17.

Another sample was a fully-cured epoxy prepreg. Figs. 45 and 46 show the micrographs

of the surface and a cross section for these two samples, with the micrographs of the

partially cured sample on the left. As shown in Fig. 45, it is clear that the partially cured

sample shows more and bigger gaps between fibers. From the point view of the surface

structures of the samples, it seems that fully cured sample is better than the partially

cured sample. From Fig. 46, it seems that the fully cured sample is denser than the

partially cured sample. This means that the shrinkage of the fully cured sample is higher

than that of the partially cured sample and consequently can achieve better bonding

property between prepreg layers in the application of composite joints.

In this chapter, the kinetic and viscosity modeling of the curing process of the

epoxy prepreg were explored and developed and the characteristics of the cure process

were discussed. The kinetic and physical parameters were determined. It is noted that the

samples used in this study were the commercial epoxy prepreg, which chemical

structures were not known in detail. For other epoxy prepreg systems, their thermal

rheological properties can be analyzed using the same techniques, models and methods

through limited experiments, depending on its application temperature range.

105

Fig. 45 The Comparison of the Surfaces Between The Partially (Left) and Fully (Right) Cured Samples

Fig. 46 The Comparisons of the Cross Sections Between the Partially (Left) and Fully (Right) Cured Samples

106

CHAPTER 6. CONCLUSIONS AND FUTURE WORK

The study of the cure process is essential to optimize the cure cycle of epoxy

prepreg in its applications as joints for composite pipe systems. The isothermal DSC

measurement of epoxy prepreg is very useful in elucidating the curing process and in

determining the kinetic parameters for the model of the isothermal cure process. The

determination of cure reaction heat under isothermal conditions can be improved by

introducing the heat flow curve for the second DSC run as the baseline to integrate the

heat flow for the first DSC run. The isothermal cure reaction heat increases with the

increment of cure temperature. The maximum reaction heat of isothermal cure can be

achieved at 220 oC. The results from the isothermal cure process were partially

supported by the data from the combination of dynamic and isothermal measurements.

The degree of cure at isothermal cure temperature below 220 oC is less than 1. In the

earlier stage of the isothermal cure reaction, the cure rate at the higher temperatures is

faster than the cure rate at the lower temperatures, while in the late cure stage, the cure

rate is lower at the higher temperatures. The relationship between cure rate and degree of

cure was simulated by the autocatalytic four-parameter model. Except in the late cure

stage, the model can provide a good predication in the large range of experimental data,

especially at a high isothermal cure temperature. The kinetic rate constants k and

increase with the increment of cure temperature while the orders of reaction m and n

decrease. In the late stage of cure reaction, the effect of diffusion on the cure rate is

apparent, especially at low isothermal cure temperatures. The modified autocatalytic

four-parameter model, including the diffusion factor, was used to simulate the

1

2k

107

experimental data again. The simulated results with the modified model were greatly

improved in the late cure stage.

The dynamic DSC thermograms provide additional information about the curing

process. The cure reaction heats at studied heating rates showed no significant difference.

Unlike the isothermal cure process, the whole dynamic curing process was composed of

two cure reactions. In the modeling method based on the Kissinger and Ozawa approach,

the obtained activation energy and pre-exponential factor of the first reaction were much

smaller than that of the second reaction. The reaction orders between Reaction 1 and

Reaction 2 were also very different. While the first reaction exhibited the behavior of an

autocatalytic reaction, the second reaction followed the nth order reaction model. At

different heating rates, the variation of the reaction orders was very small, but the pre-

exponential factor A decreased with the increment of the heating rate. The calculated

results indicated that at the early stage, Reaction 1 contributed more than Reaction 2 to

the total reaction. The final contribution of Reaction 2 to the total reaction increased with

the increment of heating rate. Except at the early cure stage, the calculated total degree

of cure gave good prediction of the experimental value. The calculated total cure rate

successfully predicted the appearance of a peak and a shoulder in the dynamic curing

process. In the method based on the Borchardt and Daniels kinetic approach, the

determined apparent pre-exponential factor and activation energy increased with the

increment of the heating rate. The effect of the heating rate on the reaction orders was

not significant. The calculated degree of cure agreed well with the experimental data in

the early stage of curing process, but had a large deviance in the later stage at all of the

heating rates. The calculated cure rates exhibited bell-shape curves and failed to predict

1

108

the appearance of the shoulders, so this method is more appropriate to model only the

simple one reaction curing process.

More information is obtained by rheological analysis of the curing process.

Rheological properties such as storage modulus, loss modulus, and viscosity are closely

related to the cure temperature and time. When gelation occurs, the epoxy resin system

changes from liquid to rubber state. Gel time can be determined by different criteria

based on the variations of storage modulus, viscosity, and loss tangent. As isothermal

temperature increases, the gel time of cure process decreases. The relationship of gel time

vs. temperature follows the Arrhenius law and thus the apparent activation energy can be

obtained. During the curing process, the variation of viscosity vs. time is predictable by

several viscosity models. Under isothermal conditions, the empirical first order reaction

viscosity model has a larger deviation from the experimental data. The proposed new

viscosity based on the Boltzmann function agrees very well with the experimental data.

The critical time in the new model decreases with the increment of the isothermal

temperature and the relationship can be described by an Arrhenius equation. The

activation energies determined by the gel time, initial viscosity, and critical time are close

each other. At the same temperatures, the kinetic rate constant in the first reaction

viscosity model is in the same range as one in the new viscosity model. The calculated

kinetic activation energies from kinetic rate constants in both old and new models are

very close. For the dynamic curing process, the empirical nth order reaction viscosity

model is better than the first order reaction model, but still less superior than the new

model. The proposed new viscosity model improves the fitting results. As the heating

rate increases, the minimum viscosity decreases and the temperature corresponding to the

109

minimum viscosity increases. At the temperature corresponding to the minimum

viscosity, the cure rate is not necessarily the fastest.

The results from DSC and rheological analyses and other techniques help to

understand the characteristics of cure process of epoxy prepreg. The limited cure reaction

heat at the lower isothermal temperature indicated that diffusion control was involved in

the isothermal cure process when cure temperature is low. The iso-conversional plots

reveal that the cure process is controlled by kinetics for degree of cure up to 60% in the

studied temperature ranges. At the degree of cure of 70% and above, the curing process

at low temperatures is controlled by physical diffusion. A linear relationship was

observed for the maximum degree of cure and its corresponding isothermal cure

temperature. The half-life is much smaller than the maximum cure time at the same

isothermal cure temperature. Both half-life and the maximum cure time decay

exponentially with the increment of cure temperature. The glass transition temperature

increases nonlinearly with the increment of degree of cure. The Tgs for the uncured and

fully cured epoxy prepregs are 1.4 and 231.0 oC, respectively. The unique relationship of

Tg versus degree of cure can be modeled by the modified Dibenedetto equation and the

new equation proposed by Venditt and Gillhamm. Both equations agree well with the

experimental Tg. The isothermal TTT and CTT cure diagrams are very useful to the

design of cure cycle. The appearance of gelation and vitrification closely relates to the

phase transition. The different phase states are clearly shown in the different regions of

the TTT and CTT diagrams. The TTT and CTT diagrams are capable of predicting when

and where the cure process is kinetically controlled and controlled by diffusion. The

TGA analysis indicates that both uncured and fully cured epoxy prepregs follow the

110

similar mechanism of decomposition. At a heating rate of 5 oC/min, the beginning of the

degradation of the epoxy prepreg occurs at about 307 oC. At about 334 oC, the epoxy

prepreg achieves the maximum rate of weight loss. The final weight loss was up to 23%.

The micrograph study shows that the strength of the epoxy prepreg can be improved by

increasing the degree of cure.

In conclusion, this study has systematically investigated the thermal and

rheological properties of the cure process of epoxy prepreg used as composite pipe joints.

The main contributions of this research work are as follows:

• Some existing models used to describe the cure process of epoxy prepreg were

tested and conformed. The autocatalytic four-parameter model with diffusion

control proved to be the best models for the isothermal cure process of epoxy

prepreg. The calculated values by this model agree very well with the DSC data in

the entire cure process. In the analysis of the glass transition temperature, only

two reported glass transition temperature models fit well with experimental data.

In the rheological of cure process, it was found that the traditional viscosity

models produce large errors in the later cure stage.

• New efficient methods to determine parameters in the models were explored and

developed. The parameters in the kinetic models, which are usually assumed to be

a first or nth order reaction, or an autocatalytic model with an assumed value of

total reaction order, are traditionally determined by simple linear regression. In

the new method, the determinations of the parameters employ more statistical

knowledge multiple and non-linear regression analyses. The new methods can

111

be applied to the complicated models and greatly improved the prediction

accuracy of the models.

• New viscosity models for the isothermal and dynamic cure process of epoxy

prepreg were proposed. The new viscosity models are based the mathematical

knowledge, rather than rheological theory, to produce a sigmoidal curve. All the

parameters in the new viscosity models have their own physical meaning. By

introducing the critical time, the proposed viscosity models are much superior

than the traditional viscosity models in the rheological analysis of cure process,

especially in the later cure stages.

With the knowledge from the cure analysis of epoxy prepreg, the future work will

focus on the application of epoxy prepreg to composite pipe joints. Composite piping

systems have achieved applications in industry because of their unique properties. In

their application, however, reliable and economic joints are required. Heat activated

coupling joints for pipe systems are under study. The pipe joints are wrapped with

prepreg and, when heat is applied, the prepreg cures to seal the joints.

Several factors affect the quality of the heat activated coupling joints. Related to

composite pipe joints, this study may be extended as follows:

• Effect of environmental conditions on the cure process. In this study, all the

measurements by DSC were conducted in a special environmental condition, i.e.

the samples were exposed to the continuous flow nitrogen. This is the

conventional research method in thermal analysis. However, the application

conditions of epoxy prepreg are very different—it is usually exposed to the

normal atmosphere with oxygen and hydrogen. So further experiments and

112

analyses in air are needed to find the possible effect of oxygen on the cure

reactions.

• Effect of the cure cycle on the pipe joints. Optimized cure cycles applied to

different temperature range may be obtained from this study. Their effect on the

coupling joints should be considered. An efficient cure process should involve an

optimal cure cycle. The optimal cure cycles of epoxy prepreg in the lower

temperature ranges need longer cure time because of the slow cure rates. On the

other hand, the optimal cure cycles of epoxy prepreg in the higher temperature

ranges need short cure time, but more energy.

• Effect of other factors on the pipe joints. This includes the temperature

distribution among layers, thickness and length of the prepreg applied. This work

needs the help of researchers in the field of mechanical engineering. To achieve

good tenacity, several or many layers of epoxy prepreg are usually wrapped

around pipes. The layer exposed to heating source usually has a higher

temperature. Higher temperature results in faster cure reactions and produces

more cure heat. Thus a large temperature difference among layers may exist. The

effect of temperature distribution on the internal tense should be investigated.

113

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119

APPENDIX A. MORE ISOTHERMAL HEAT FLOW CURVES

0 8000 16000 24000 32000 40000 48000-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0 8000 16000 24000 32000 40000 48000-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0 8000 16000 24000 32000 40000 48000-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0 8000 16000 24000 32000 40000 48000-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

t=110 oCt=120 oC

t=130 oC

Hea

t flo

w (W

/g)

Time (second)

t=140 oC

(a) 110-140 oC

Fig. 47 (a) The Variation of Isothermal Heat Flow With Time

120

0 4000 8000 12000 16000 20000-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0 4000 8000 12000 16000 20000-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0 4000 8000 12000 16000 20000-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0 4000 8000 12000 16000 20000-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

t=150 oC

Hea

t flo

w (W

/g)

t=160 oC

t=170 oC

Time (second)

t=180 oC

Fig. 47 (b) 150-180 oC

121

0 2000 4000 6000 8000 10000-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2000 4000 6000 8000 10000-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2000 4000 6000 8000 10000-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2000 4000 6000 8000 10000-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t=190 oC

Hea

t flo

w (W

/g)

t=200 oC

t=210 oC

Time (second)

t=220 oC

Fig. 47 (c) 190-220 oC

122

APPENDIX B. ADDITIONAL CURE RATE CURVES

0.0 0.2 0.4 0.6 0.8 1.00.0

1.0x10-5

2.0x10-5

3.0x10-5

4.0x10-5

5.0x10-5

experimental model without diffusion model with diffusion

Cur

e ra

te (

s-1)

Degree of cure (α)

(a) 110 oC

Fig. 48 (a) Cure Rate vs. Degree of Cure

0.0 0.2 0.4 0.6 0.8 1.00.0

2.0x10-5

4.0x10-5

6.0x10-5

8.0x10-5

1.0x10-4

1.2x10-4

experimental model without diffusion model with diffusion

Cur

e ra

te (s

-1)

Degree of cure (α)

(b) 130 oC

Fig. 48 (b)

123

0.0 0.2 0.4 0.6 0.8 1.00.0

4.0x10-5

8.0x10-5

1.2x10-4

1.6x10-4

2.0x10-4

2.4x10-4

2.8x10-4

experimental model without diffusion model with diffusion

Cur

e ra

te (s

-1)

Degree of cure (α)

(c) 150 oC

Fig. 48 (c)

0.0 0.2 0.4 0.6 0.8 1.00.0

1.0x10-4

2.0x10-4

3.0x10-4

4.0x10-4

5.0x10-4

6.0x10-4

experimental model without diffusion model with diffusion

Cur

e ra

te (s

-1)

Degree of cure (α)

(d) 170 oC

Fig. 48 (d)

124

0.0 0.2 0.4 0.6 0.8 1.00.0

2.0x10-4

4.0x10-4

6.0x10-4

8.0x10-4

1.0x10-3

1.2x10-3

experimental model without diffusion model with diffusion

Cur

e ra

te (s

-1)

Degree of cure (α)

(e) 190 oC

Fig. 48 (e)

0.0 0.2 0.4 0.6 0.8 1.00.0

4.0x10-4

8.0x10-4

1.2x10-3

1.6x10-3

2.0x10-3

2.4x10-3

2.8x10-3

experimental model without diffusion

cure

rate

(s-1)

Degree of cure (α)

(f) 210 oC

Fig. 48 (f)

125

APPENDIX C. ADDITIONAL VISCOSITY CURVES

0 10 20 30 40 50 60

0.0

2.0x104

4.0x104

6.0x104

8.0x104

1.0x105

1.2x105

1.4x105

experimental model by Eq. (15) model by Eq. (20)

Vis

cosi

ty (P

as)

Time (minute)

(a) 150 oC

Fig. 49 (a) The Variation of Viscosity With Time

0 5 10 15 20 25 30

0.0

2.0x104

4.0x104

6.0x104

8.0x104

1.0x105

1.2x105

1.4x105

experimental model by Eq. (15) model by Eq. (20)

Vis

cosi

ty (P

as)

Time (minute)

(b) 170 oC

Fig. 49 (b)

126

127

VITA

Liangfeng Sun was born in Shandong, People’s Republic of China, on November

27, 1964. He received a bachelor of engineering degree in pulp and paper engineering in

1988 from Shandong Institute of Light Industry. In August 1989 he entered the master’s

program at Tianjin University and graduated with a master of science degree in chemical

engineering in 1992. Then he worked as a research associate at Tianjin University. He

received admission to the doctoral program in the Department of Chemical Engineering

at Louisiana State University and resumed his advanced education at Louisiana State

University in August 1996. He passed the doctoral qualifier exam in chemical

engineering in May 1997. In May 1999, He received a master of science in chemical

engineering from Louisiana State University. At the same, he also entered the master’s

program in the Department of Computer Science at Louisiana State University and

passed the comprehensive exam for master’s students in September 2000. He received

the degree of Doctor of Philosophy in chemical engineering in May 2002.