Determination of network structure by extraction and random degradation, 3. Comparison with...

Makromol. Chem. 183,2237-2256 (1982) 2237

Determination of Network Structure by Extraction and Random Degradation, 3a)

Comparison with Elastometry and Equilibrium Swelling

Dedicated to Professor Dr. Drs. h.c. G. V. Schulz on his 75th birthday

Martin Hoffmann

Corporate Research Division of Bayer AG, 5090 Leverkusen, Federal Republic of Germany

(Date of receipt: June 15, 1981)

SUMMARY: Extraction and random degradation of networks make it possible to determine the molecular

massb) of the network chains, the amount of chain ends, the spatial inhomogeneity of crosslinking, the surprisingly high functionality of the crosslinks, and the efficiency and the transfer of crosslinking radicals. These results are compared with the stress-strain behaviour at low and high extensions, the stress optical coefficient, and the equilibrium swelling degree. A value of the memory term relative to that of crosslinked melts may be deduced from the swelling degree at a given modulus. With this value an “analytical” modulus may be calculated from the analytical results. It corresponds roughly to the Mooney-Rivlin modulus if one neglects the influence of functionality, but there remain deviations which suggest that another method should be developed for determining the modulus of the covalent network. The modulus measured at low extensions cannot be used to calculate the crosslink density in weakly crosslinked polymers.

1. Introduction

Complete extraction combined with random degradation of networks makes it possible to determine previously inaccessible structural details of polymer networks, as shown in Parts I and 2 of this series’,2). Since the same networks were used in experiments evaluating their stress-strain-behaviour, their stress birefringence, and their equilibrium swelling degree, it is possible to check the validity of the theories of elasticity and swelling. The analytical evaluation of the widely used stress-strain measurements is uncertain because the relationships between the structure and the constants of elastic behaviour depend on the method of measurement (e.g. strain rate) in a way which has yet to be adequately explained. Furthermore the methods of evaluation have not as yet been theoretically substantiated with any adequate degree of certainty. Thus one aspect which is still unclear3s4), is whether the dependence of the stress 0 (relative to the cross-sectional area of the stretched specimen), on the extension ratio d can be reliably evaluated according to Mooney and Rivlin5) for

a) Part 2: cf.2). b, “Relative molecular mass” (systematic IUPAC name) is shortened to “molecular mass”

throughout this paper.

0025-1 16X/82/09 2237-20/$03.00

2238 M. Hoffmann

obtaining that fraction 2C, of the modulus which is characteristic of the covalent crosslinks only:

It is also not clear whether the relationship between 2C, and the number-average molecular mass of the network chains depends on the functionality6) A,, and whether the weight fraction of the elastically inactive chain ends can be caialated from the molecular mass Mn of the uncrosslinked polymer as (a .M,,,/M,J, mcause chain scission may have occurred during the preparation of the mixture fo* crosslinking which would increase the weight fraction of chain ends, and because the factor a is questionable'). It is more correct to assume a to be 1 instead of 2, even in the absence of chain scission. This is seen easily by visualizing a distribution of n crosslinks along a chain with the molecular mass M = n Mc. It is also not clear how the memory term h2/h; can be analytically determined, i.e. how it can be established whether the network chains have the same end-to-end distance h, as that usually found with molecules of equal length in the same surroundings, or whether it is different (= h) . Furthermore, pF is a volume fraction which must not only be attributed to the presence of a filler, but may also be due to an inhomogeneity in the concentration of crosslinking units. Investigations of the stress birefringence An provide no new conclusions as to the concentration of crosslinks because An is proportional to B. The determination of the degree of crosslinking by measuring the equilibrium swelling degree q is also unreliable because, according to the most commonly used theory and to experimental results, q and 2C, are closely related, thus introducing into this method the same uncertainties as in elastometry. In addition to this, the relationship between the experimental values of q and 2C, does not quantitatively agree with the theoretical equations) so that its exponent must be fitted to experiments, leading to a semi-empirical character of the equation.

In view of this situation, it appeared necessary to develop new methods for determining the network structure. The statistical degradation of network chains and the investigation of the molecular mass distribution and of the branching of the high- molecular-mass, soluble scission products of networks, and the characterization of the extractable components proved to be particularly suitable methods's2). Their results will now be compared with the results of elastometry and swelling.

--

2. Methods used for Further Characterization of Networks

2.1. Elasticity and strain behaiiiour

A tensile strength testing machine built at Bayer makes it possible to apply uniaxial stress to the specimen in such a way that the middle of the specimen remains practically stationary and can be observed through a microscope. The machine was operated at a strain velocity of 2,5 cm/s. The specimens were stamped from sheets in the form of standard testing bars (dumbbell

Determination of Network Structure by Extraction and Random Degradation, 3 2239

shape, length of narrow section Lo = 2,8 cm, width 0,52 cm, thickness do = 0,l - 0,4 cm). The elongation (always measured during the first stretching of the sample) was calculated from the elongdtion time t (i. e. the length L of the stretched specimen which was set equal to the distance of the clamps) as L/Lo without applying corrections for relaxation, deformation of the transducer, or permanent deformation of the sample. The restoring force K was plotted with a fast-response recorder and converted, using the original cross-section Qo and A , into the stress u relative to the cross-section of the stretched specimen. This method is not suitable for an exact measurement of stress at elongations <30%. Therefore such values will not be used in the following discussion.

For the evaluation according to Mooney and Rivlin’), u was divided by L2 - ( l / L ) and plotted versus 1/L (see Fig. 1). The left-hand section of the ordinate is 2C,, the modulus which is assuiilcd to originate from the covalent crosslinks. The right-hand section of the ordinate is 2C, + 22:. where 2C2 is presumably due to physical crosslinks, in particular to the entangle- ment of L ~ C linear macromolecules before crosslinking’). The 2C, volues found i n another lab- oratory’”’ using “equilibrium” vplues for u and L show good agreement with the values determined here, so it would appear that only the teim 2C2 is influenced by the difference in the methods of measurement as has been demonstrated‘). For this reason the values of 2C, determined here will be used for a comparison with theories giving equilibrium stress values. In view of the e r r m in the dimensions of the specimen, in 1, and in the extrapolation to 2C,, the error of 2C. is estir:ated 10 be about 5%, at very low 2C, even higher. Values of 2C, are given in Tab. 1. In Fig. 1, the Mooney-Rivlin curves turn upwards at a certain high value of’ 1 which we call L ’. This deviation from normal behaviour will be discussed in section 3.1. The values of L + depend on the type of measurement because relaxation reduces the stress as a function of 1. We assume ~.L: values of 1 + to be mcertain to ahout k 10%. Tab. 1 shows valoes of L +. Va!ues measured after swelling refer to networks minus extractable material.

Fig 1 Mcwwy-Rivlin plot of elasticity measuremetlts: ?) pdyiqoprene 3; b) 901~- cutadiene 25; c) a polysiloxane resembling 32. At high extension ratios L the cuive deviates from the straight 11-Lt and defines L +

2240 M. Hoffmann

Tab. 1. Results of elastometric and swelling experiments on networks nos. 1 - 70 (for type of polymer cf. ’) and for symbols see section 2). Values in parentheses have larger errors than usual

No. Values measured at pv f

A 7

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

0,21 0,22 0,285 0,37 0,21 0,255 0,29 0,34 0,19 0,36 0,38 0,43 0,29 0,31 0,345 0,39 0,34 0,38 0,40 0,48 0,29 0,43 0,55 0,57 0,73 0,72 0,54 0,61 0,62 0,68 0,78 0,052 0,12 0,118

(0,089) 0,095 0,122 0,115 0,123 0,11 0,16 0,175 0,17 0,175 0,20

0,195 0,225 0,30 0,345 0,163 0,20 0,27 0,327 0,163 0,35 0,395 0,46 0,285 0,297 0,38 0,41 0,36 0940 0,45 0,51 0,295 0,37 0,57 0,58

>0,73 >0,75

0,43 0,50 0,58

p 0,69 p 0,80

0,027 0,059 0,087 0,089

0,071 0,050 0,070 0,121 0,009 0,020 0,032 0,044 0,080

0,042

0,25 0,28 0,31 0,40 0,25 0,29 0,30 0,35 0,23 0,37 0,39 0,46 0,33 0,29 0,32 0,42 0,36 0,40 0,43 0,51 0,30 0,50 0,58 0,63 0,74 0,75 0964 0,70 0,72 0,71 0,80 0,095 0,125 0,155 0,170 0,155 0,175 0,165 0,175 0,121 0,254 0,265 0,258 0,26 0,28

- 0,4 - 0.3 - 0,3 - 0,4 - 0,4 - 0,4 - 0,3 -0,4 + 0,9 - 0,3 - 0,3 - 0,3 - 0,4 - 0,2 - 0.3 - 0,4 - 0,2 - 0,3 - 0,4 - 0,2 - 0,3 - 0.5 -1,1 - - -

- 0,5 + 0,s + 0,6 - -

- 0,5 - 0.6 - 0,5 - 0,5 - 0,5 - 0,5 - 0,5 - 0,5 - 0,6 090

- 0,8 - 0,6 - 0,5 - 0,04


Tab. 1. Continued

No. Values measured at pv I

o 1

2 c 1 2 c 1 + 2 c 2 A + Cn,o.ld -* dCn 104 A=1,5 du

46 0,225 47 0,21 48 0,22 49 0,0135 49a) 0,11 50 0,0155 50a) 0,105 51 0,035 51a) - 52 0,047 52a) - 53 0,039 53a) - 54 0,070 54a) - 55 0,073 55a) - 56 0,072 563 -

58 0,22 59 0,245 60 0,26 61 0,39 62 0,46 63 0,085 64 0,083 65 0,070 66 0,062 67 0,062 68 0,19 69 0,17 70 0,19

Tab. 1. Continued

57 0,21

0,102 0,130 0,149 0,0062 0,015 0,0104 0,025 0,008 0,013 0,017 0,030 0,021 0,040 0,034

0,052 0,070 0,051 0,070 0,110 0,148 0,209 0,032 0,185 0,29 0,092 0,092 0,068 0,058 0,049 0,205 0,18 0,186

-

0,31 0,25 0,257 0,023 0,165 0,023 0,145 0,0485 0,27 0,061 0,215 0,047 0,20 0,087

0,082 0,18 0,085 0,177 0,28 0,295 0,292 0,395 0,54 0,57

-

- - 0,076 0,069 0,071

0,18 0,195

-

- 0,3 - 0,45 - 0,3

- 0,23 - 2,8 - 0,5 - 1,5

-

( + 0.3) - 0,7 - 0,3 - 0,9 - 0,95 - 0,9

- l ,o - 0,5 - l,o - 0,5 - 0,5 - 0,4 - 0,35 - 1,4 -1,2

090 - 0,7 - 0,5 - 0,5

(0,0) - 0,3 - 0,5 - 0,6

-

-

No, ~ ~~

Values measured after swelling in toluene I

A s,

C7

2242 M. Hoffman

Tab. 1. Continued

No. Values measured after swelling in toluene f \

* U

4

4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

4,85 0,14 6,9 0,065 6,3 0,085 5,55 0,lO 5, l 0,14 7,O 0,07 4,8 0,19 4,5 0,20 4,15 0,20 5,65 0,12 5,l 0,14 4,6 0,17 4,3 0,21 5,05 0,15 4 3 0,21 4,3 0,20 3,9 0,25 5,3 0,14 5,2 0,19 4,55 0,27 4,2 0.30 3,9 0,41 3,6 0,37 435 0,21 4,05 0,28 3,85 0,30 3,7 0,36 3.45 0,37

10,2 0,011 7,2 0,020 5,2 0,044 5,O 0,044

(7,2) 0,020 5,8 0,033 6,4 0,026 5,6 0,038 4,6 -

25,l - 17,8 0,0083 14,4 0,013 13,3 0,015 8,5 0,0441 8,O 0,046 6,6 0,070 6.25 0.060

49 - 0,0039 0,003 0,0044

- -

- 15 -

- 18 ( - 16) ( - 18) - -- -

(- 15)

( - 15)

( - 12)

+ 14

I

I

I

-

- 6,O + 2,5 c 4,O

+ 5 3 + 6 + S

+ 15 + 12 + 16

-

-1,2 - 1,5 -1,5

(-795) - 4,O - 4 3 -- 5.5 - 6,5 I

- -

- 6 - 7 - 6 - 5

0 -7 -

Determination of Network Structure by Extraction and Random Degradation, 3

Tab. 1. Continued

No.

2243

Values measured after swelling in toluene I -I

A

a

- 49 a)

50 50a) 51 51 a)

52 52 a)

53 53 a)

54 54a) 55 55a) 56 56 a)

57 58 59 60 61 62 63 64 65 66 67 68 69 70

- 0,0095

0,0095

0,019

0,018

0,022

0,024

0,040

0,052 0,075 0,091 0,017 0,100 0,125

-

-

-

-

-

-

-

- - - - - - - -

a) Networks dried before investigation.

2.2. Stress birefringence

For photoelastic investigations, the specimen was as usually placed between the crossed polars of a polarizing microscope using a weak magnification (10 x 6 times). The direction of stretching formed an angle of 45' with both the analyser and the polarizer directions. The specimen remained in the focussing plane during elongation and was wider than the field of view. The intensity I , of the monochromatic light from a sodium lamp, transmitted by the unstretched specimen, was measured, using a photocell, with the polarizers being set a) parallel and b) at 90" to each other. Intensities were plotted with a strip-chart recorder. The instantaneous intensity I measured in the elongation test was divided by I, and used together with the thickness d,/A of the stretched specimen for calculating the birefringence An = ny - na (do in cm) l):

2244 M. Hoffmann

Fig. 2 shows that An is proportional to u only at small values of u. The following approximation is sufficiently accurate:

Cn,o is the photoelastic coefficient at small elongations, C, that at 1 > 1. Tab. 1 also gives the values of Cn,o and k, = dCn/du. The error in Cn,o is about +4% and that in k, = dCn/du about ?lo%.

10-3.q I 2 6 6 8

t i aIMPa

b z . a

- . . -\

Fig. 2. Stress-optical coefficient An/u as a function of the stress u for the polymers of Fig. 1

0,5 1,0 1.5 alMPa

2.3. Equilibrium swelling

The crosslinked specimens were swollen and thus extracted for about one month at room temperature in an excess (amounting to about 10 times the volume of the swollen specimens) of toluene. The apparatus was shaken daily. After this, the specimens were weighed in the swollen and in the dry state, and the ratio of the volumes, i.e. the degree of swelling q was calculated from these weights, taking into account the densities. Traces of moisture influence q . Therefore and because of differences of M, in the different specimens of a vulcanizate, the values of q could be reproduced to about f 3% only. Tab. 1 shows these values. According to Fig. 3 the relationship between q and 2C, is given semiempirically by Eq. (4)8), provided the specimens have been crosslinked in the molten state, which will be indicated by the subscript 0:


(4) const k

q 0 = - - - - L - (2 c, 18 (2 C1)8

With specimens which have been crosslinked in solu&on,the deviation from the normal relationship (see Eq. (4)) should indicate by how much h 2 / h ~ , , , deviates from 1 (see section 3.2).

\

-I I I I I I 1 1 1 ) I I I I I , , )

0,o 1 on1 1 2 C t l M P a

Fig. 3. Equilibrium swelling q of networks from polyisoprenes ( 0 , 0, W , 0) and polysiloxanes ( x , A) at 25 "C in toluene; 0 , W and x were crosslinked in the melt, 0 , 0 and A in solution (see Tab. 1) with pv a 0.4 ~-

3. Theoretical background of an Analytical Evaluation of Elasticity Measurements and of Equilibrium Swelling

Some of the current theories of stress strain behaviour of entropic elasticity3, lead to Eq. (1). According to other theories the modulus at low extensions characterizes the number of network chains12). In section 4.3 experimental values of 2C1 and of the modulus at low extensions will be compared with the calculated number of network chains per cm3.

3.1. Deoiations from Gaussian statistics at high elongations

The deviation from Gaussian statistics occurring at high extensions, i. e. at I + , may be due to ~rystallization'~) or, in the case of sufficiently rapid stretching, to an abnormally large elongation of the chains. A theory has already been put forward14) for this latter case using the Langevin approximation of conformational statistics.

2246 M. Hoffmann

With Gaussian statistics of the end-to-end distances of the deformed chains the following dependence of o/L on L has been derived:

(+)G = N k T ( A 2 - $) Langevin statistics leads to another dependence at higher I :

N k T 1

Z, is the number of statistical segments of a network chain of molecular mass M,. It is the number of segments which are oriented during deformation. If M, is the molecular mass of a statistical segment, then

Furthermore, Y ' is the inverse Langevin function which can be approximated by a series. Eq. (6) hence can be converted to the following form

(+)L- = N k T ( A 2 - +)[l+ *(Az + -$) + %(,I4 + A + $)] (8)

If we assume that the quotient oL/oG of an unswollen vulcanizate must be equal to 1,06 (in order to be visible outside the margins of error, so that a L + can be defined there), then the following approximation results for nonideal networks containing loops (w,) and short chain branches (wSbr):

Tab. 1 also shows values of I + . According to Fig. 4a the experiments confirm approximately Eq. (9). For polyisoprene the value of I + at M, = 1 I@ agrees roughly with that calculated from the size of the segments obtained by completely different measurements. Furthermore, no significant differences are observed between polyisoprenes with high and those with very low crystallization tendency. At the high strain rates used here, I + is not determined by crystallization, but by the abnormally high stretching of the network chains. Therefore L + is related to that part of M,, which does not belong to loops and branches. An increased number of loops and branches per network chain at high values of M, may be responsible for a smaller dependence of L + on M, than that predicted by Eq. (9). Furthermore the ratio oL/oG and Eq. (9) may be influenced by a non-uniform crosslink density. According to Fig. 4b the values of L + found after swelling specimens crosslinked in the melt or after swelling or drying samples which had been crosslinked in solution correspond roughly to the values expected for a change of the end-to-end distance h. This change should be equal to [q( l - wex)I1l3 if S (see section 4.2.1) remains constant.


100= - - 2 : c

+ - N-

s - lo=

a )

I/ I 1 1 1 1 1 I I l 1 1 l l 1

3.2. Equilibrium swelling and the memory term

Deviations from the normal dependence of q on 2C, may occur if the memory term h2/h&, deviates from 1. h,,, is the end-to-end distance of the network chains in a reference state, e.g. in a specimen crosslinked in the molten state. The influence of the memory term will now be derived theoretically.

A polymer crosslinked in the melt at the temperature of measurement so that h2/hi = 1 has @of the network chains close to the mean square end-to-end distance of the theta state G. Such a polymer swells to the equilibrium degree of swelling go, the end- to-end distance taking on the value h, which, in the case of simple (i. e. not interlaced) networks, is given by

--

--

diagrams like Fig. 1 , plotted against the

MC," is a number average value of M,,,, and Mc,deg taken from Part 2*) for polyisoprenes IR 305 ( 0 ) and polysiloxanes (A), both crosslinked in the melt.

degrees q: Networks 49 - 56 (o), 32, 34, 39 (A), 10, 42-45 (0). For extracted specimens q has been multiplied by (1 - we,). The experimental values do not follow the line, which indicates proportionality to q- j'3

molecular mass MC," of the network chains. 2-

+cr 1-

+ s' 'i

b) A&/a; (see Fig. 1 ) at different swelling 1

-i

45-

w-

Because of Flory's expansion factor aq, q,, is here dependent'on the thermodynamic quality of the swelling agent. If h, < h,, then (given otherwise equal conditions) qo should be greater, because it is only at greater values of q that the elastic recovery of the chains reaches a level which prevents further swelling.

If a solution containing the volume fraction pv of polymer is crosslinked, the network chain ends are separated by the distance h,; in a thermodynamically non-

b)

2248 M. Hoffmann

ideal swelling agent this distance is greater than h,,, = h,,,=, by the expansion factor a,. If the assumptions leading to Eq. (10) are valid, the degree of swelling q of the specimen which already contains chains with a > 1 is therefore smaller by the factor a-3 than that of a specimen crosslinked to the same value of M, in the melt.

q = - 90

a3

Consequently, if a polymer crosslinked in a solution of a volume fraction of polymer pV is dried, then the end-to-end distance of the network chains must be reduced from h, to a smaller value. If, as hitherto assumed, no further rearrangements occur in the network, then a distance h,,, must be assumed at v, = 1 . This distance can be calculated from lo, as follows:

(+$ = Pv

If h, is equal to the normal end-to-end distance of a network chain of molecular mass M,, therefore, h,,, is much smaller than h,. As a consequence of this, the swelling capacity q,+, of the dried polymer crosslinked in solution at pV is much greater than qo, viz. that of a polymer crosslinked to the same M, at pV = 1 . From Eqs. (11) and (12) it follows that'"

_ _ In the dried polymer, which has been crosslinked at rp,, the value of h2/hi has thus

been reduced. Using Eq. (12) it is true to say that for h, = h, - a,:

The consequence of this is, that, in accordance with Eq. (l), the modulus 2C1 decreases to a value

Thus, if the experimental values of q,+ are plotted versus (2C1),-, (Fig. 3), the data point representing a polymer crosslinked in solution is situated above the curve determined with melt-crosslinked polymers. In accordance with Eq. (15) the swelling of specimens with (2C1),,1 would be greater than that of specimens with (2C1), by the factor l/(v,,2/3 * if q were calculated from (2C1),+1 with the assumption that Eq. (4) is valid. In reality, it is larger by the factor 1/(pv *a:) in accordance with Eq. (13). The value measured for q,-* is therefore higher than the corresponding value of the normal curve (at the same (2C1),+,) by a factor q/qnorm:


The smaller M,, the more the factor a: approaches 1. When M = 104, this factor is about 1,22 with polyisoprenes, 1,37 with polybutadienes and 1,03 with polysiloxanes (all in toluene). When M = 2.104, it is 1,41 with polyisoprenes, and when M = 4,4 * 103, it is about 1 ,O.

Using Eq. (14), Eq. (16) may be written in a form showing the memory term to be a function of q/qnorm at constant 2C1 :

- 213 - (4/9)b

Eq.' (17) has been derived under the assumption that the factor S (see section 4.3) remains constant during swelling.

4. Discussion and Comparison with Results of Extraction and Degradation

4.1. Chain end correction term, filler term, functionality term

4.1.1. Chain end correction term

As polyisoprene IR 305 originally does not contain appreciable amounts of molecules with long chain branches, Mc,ex corresponding to Mc,deg, we assume that w,, and Mc,ex give correct values of we leading to the chain end correction term, 1 - M,/Mn in Tab. 2.

4.1.2. Influence of network inhomogeneity on 2C1 (filler-term)

The term (1 + 2,5 v ) ~ . . .) in Eq. (1) may be used to calculate the influence of network inhomogeneity on the stress at a given extension, if we insert wSl (see Part 1')) instead of q+. This yields too large effects, if the regions of anomalous high crosslinking density are not much harder than the matrix. As they have a large swelling degree (see')), their influence on 2C, is small and may be approximated by:

With the values of wSl from Tab. 3 in Part 22) the values of (1 + 2,5 v ) ~ . . .) given in Tab. 2 are calculated. They do not deviate strongly from 1 ,O. However, Eq. (18) does not apply in the case of networks where the strongly crosslinked regions touch each other and form the matrix. Furthermore Eq. (18) cannot be used to calculate the influence of an arrangement of polyfunctional crosslinks, leading to a chain with the length corresponding to the kinetic chain length of the polymerization of double bonds. These chains do not have a high extensibility and may form a network with a higher modulus than that calculated from Mc,n.

2250 M. Hoffmann

Tab. 2. Values of correction terms for the calculation of the modulus of elasticity of networks nos. 1 - 70 (for meaning of symbols see sections 2 and 4). Values in parentheses have larger errors than usual

- M , , 1 + 2,5 pF.. .') 2 Corrected

1 - - analytical 1 - A h2 V)p,eff No. -

P V hnom Mn,n f v modulus 2

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

0,95 0,95 0,95 0,95 0,94 0,95 0,95 0,95 0,96 0,98 0,98 0,97 0,98 0,98 0,98 0,98 0,98 0,98 0,98 0,98 0,97 0,97 0,98 0,98 0,98 0,98 0,98 0,99 0,99 0,99 0,99 0,89 0,91 0,95 0,94 0,94 0,94 0.95 0,95 0,95 0,49 0,79 0,84 0,88 0,93 0,93 0,93

0,90 0,91 0,93 0,93 0,88 0,91 0,92 0,93 0,90 0,95 0,96 0,96 0,92 0,93 0,94 0,94 0,94 0,95 0,95 0,94 0,93 0,88 0,92 0,94 0,96 0,97 0,92 0,96 0,97 0,97 0,97 0,79 0,82 0,90 0,90 0,87 0,87 0,88 0,89 0,89

0,72 0,76 0,80 0,85 0,85 0,91

0,4

0,187 0,240 0,290 0,375 0,177 0,202 0,238 0,306 0,117 0,348 0,488 0,626 0,245 0,308 0,425 0,633 0,478 0,652 0,696 0,782 0,413 0,178 0,259 0,430 0,660 0,811 0,234 0,372 0,576 0,735 1,145 0,0187 0,0292 0,0523 0,0724 0,0254 0,0348 0,0323 0,0361 0,170 0,00416

(0,0121) 0.0204 0,0423 0,0949 0,0781 0,1544


Tab. 2. Continued -

Mc,n 1 + 2,5 pF.. . a) 2 Corrected 1-- 1 - - analytical

No. & h 2 -

9 v h n o m M"Jl f v modulus 2

48 0,93 49 0,70 50 0,77 51 0,65 52 0,91 53 0,93 54 0,93 55 0,94 56 0,91 57 0,91 58 0,93 59 0,94 60 0,64 61 0,93 62 0,96 63 0,95 64 0,95 65 0,93 66 0,93 67 0,93 68 0,96 69 0,96 70 0,96

a) See Eq. (1).

(1 7 0 ) - - - - - - - - 1,02 1,00 0,98 0,94 1 ,oo 1 9 0 0 0,92 0,83 0,81 0,82 0,79 1,01 0.99 0,98

0,91 0,76 0,82

( O S 5 )

0,86 0,84 0,93 0,89 0,83 0,85 0.87

(035) 0,82 0,88

(0~4)

- - - - - - - -

0,191 0 , m 0,0149 0,0039 0,0115 0,040 0,0335 0,116 0,115 0,106 0,152 0,194 0,0192 0,140 0,237 - - - - - - - -

4.1.3. Term correcting the modulus because of the functionality

Values off, have been determined in Part 22). A further method for the analytical determination of the functionality uses the semiempirical swelling equation (Eq. (9)) and Eq. (59) of Part 1 assuming wsbr = 0 and a given value off,:

Fig. 5 shows that Eq. (19) is experimentally verified for poly(dimethylsi1oxanes) in toluene. In case of networks 68 to 70 the [q]& values are too low, indicating the influence of the molecules with low M, and high f, which retarded dissolution of the network, giving smaller [q]&. A similar correlation as Eq. (19) can also be established between [q]; and 2C,. The analytical values off, have been used to calculate the values of the correction term 1 - (2/fv) shown in Tab. 2.

2252 M. Hoffmann

- I : A

10 I " ' I ' " ' I 2 3 5 10

Q

Fig. 5. degradation products (at t + ) of polysiloxanes as a function of q, the equilibrium degree of swelling before degradation: Networks with functionality f, ie. 4 (0 , 32 - 37, 40) and with higher functionalities ( A , 38, 39, 68 - 70)

Intrinsic viscosity [q ]& of

4.1.4. Effective polymer concentration in the network

The factor (1 - wex) reduces the polymer concentration which prevailed during crosslinking to that which has to be used in Eq. (I), i.e. to 9p,eff (see Tab. 2).

4.2. The memory term and the factor S characterizing the complexity of networks

4.2.1. Factor S

The assumption which leads to Eqs. (10) - (15), namely that no further rearrangements take place in the network during changes of 9, appears questionable when one considers real networks, where several simple networks are interlaced, leading to S3 = h: . (2N/f,) S= 1. If such networks are made to swell, the swelling could be brought about by reducing S without the chains having to expand. During such a swelling, however, conformational rearrangements compatible with a given end-to- end distance h are impeded, because the crosslinks must be arranged in a different way with respect to their neighbours. It is in any case to be expected that, in real networks, hq will grow more slowly with q than it is shown in Eq. (lo), and that q/qnorm is smaller than predicted by Eq. (16). For the same reason (2C,), q1'3 could be smaller than (2C1), . This is confirmed by the numerical values given in Tab. 1 for equilibrium swelling and for the modulus (2C,), of the swollen specimens. Similar results have been reported for other networks9). According to Eq. (1) the modulus is proportional to (hz/h,2). In swelling to q the value of hi//$" is expected to increase proportional to q$3 where qeff is that part of q which does not arise from a variation of S. We therefore find qeff /q from the following equation:

-- --


1 - (qeff/q) is the relative part of q which is caused by a variation of S. An experimental determination of this quantity via Eq. (20) has to take into account that the values of q in Tab. 1 refer to networks which were extracted. Since the removal of extractable material does not influence hPv, these values have to be corrected for their use in Eq. (20) by multiplying with (1 - we,). No correction has to be applied to the moduli. The ratio 2C1,,/(2C1,,) varies with q-0,43 and not with q-0,33 as expected from a simple theoretical approach. This is due to the difference in Sq and S,. We calculate qeff/q = l/qoJ5 and at q = 5 , a value of (qeff/q) = 0,79. This is not confirmed by the dependence of I: / I ; on q in Fig. 4b; I: / I ; equals hqv/hq and has a value of about l/q0943 which is even smaller than expected. So we can draw the conclusion that only a small part of q results from a change of S with q and that the change of S is connected with conformational restrictions reducing I + .

4.2.2. Memory term

In accordance with Fig. 3, the degrees of swelling and the moduli of elasticity of the specimens crosslinked in the melt follow the semiempirical correlation very well (see Eq. (4)). Because of the good agreement observed with so many specimens one may conclude that h2/h,2 has the same value 1,0 for all these specimens. This corresponds to results from other experiment~’~). Values of h2/hi are given in Tab. 2; the error is estimated to be about f 3%. With some polysiloxane networks one finds deviations from the usual correlation between qo and (2C1),. Evaluated with Eq. (16) and a, = 1 these deviations lead to the conclusion that these samples were crosslinked in solution (volume fraction of polymer p, < 1). In the case of networks 54 to 57, the calculated values of pv show good agreement with those provided by the manufacturer after this analysis was carried out.

A deviation from theory may result from the fact that q depends on the mass of the elastically effective part of the network chains, i. e. on M,(1 - w, - w,,,,), whilst 2C1 depends on the molecular mass M, of the total polymer. Therefore the elastically effective part of the network chain is elongated by swelling more strongly than expected from M,, and any deviation from Eq. (4) may be explained by a different amount of loops or short branches. Such unknown elements of the network structure prevent a quantitative discussion of the experimentally detected deviations from Eq. (4). Nevertheless the value of q at a given 2C1 indicates roughly whether h deviates much or not at all from its normal value h,,,,.

- _ --

4.3. Experimental examination of current theories of rubber elasticity and equilibrium swelling

Mean values of McTn which had been determined by extraction and degradation are used to calculate the modulus of elasticity p p R T/M,,, and to correct this value by means of the correction terms discussed in sections 4.1 and 4.2, i. e. the relative value

2254 M. Hoffmann

_ _ h2/h&,, of the memory term, the chain end correction term (1 - we), and the filler term arising from an inhomogeneity of the crosslinking density. The correction term 1 - (2/f,) has not been applied. Values of this “analytical” modulus are also shown in Tab. 2. If one compares them with the values of the modulus 0/[d2 - (l/d)] at I = 1,5, it can be easily seen that the modulus at low extensions measured at high strain rates cannot be used to calculate the number of network chains in weakly crosslinked polymers. Fig. 6 compares the Mooney-Rivlin value of the modulus, i. e. 2C1, with the analytical modulus. 2C1 is not equal to the analytical modulus and increases less than proportional to it, though it has the correct order of magnitude.

/ I I I I 1 I

00 OJ 1

Fig. 6. Modulus 2C, as a function of the “analytical” modulus (see text) for crosslinked melts of polyisoprenes (o), polybutadienes (A) and polysiloxanes (A), and for polyisoprenes crosslinked in solutions with the volume fraction of polymer 8, = 0,5 (0). The inclination of line 2 is 0,8 and not 1 ,O. Line 3 representing values for polymers crosslinked in solutions and line 1 representing values for polybutadienes do not coincide with line 2 as would be expected theoretically

This result will not be improved if we apply the correction term 1 - (2/f,) to the analytical modulus. Furthermore, at a given value of the analytical modulus, the value of 2C, seems to depend on the chemical nature of the polymer: With polybutadienes it is about 40% higher than with polyisoprenes. The high value of 2C, in the said case may arise from an anomalous structure of these networks (see section 4.1) or from a wrong method of evaluating the modulus. The semiempirical Mooney-Rivlin


extrapolation to 2C, might have to be substituted by a better method of extrapolation, which has been suggested also for other reasonsI4). An extrapolation using a plot with a linear l / I Y 3 scale as abscissa (instead of a scale linear in l / I ) reduces the modulus found for low degrees of crosslinking and for high 2C2 to more resonable values”.

The elastic behaviour at high extensions seems to follow roughly the theoretical predictions. This indicates statistical coiling of the molecules even at high extensions. But it can be seen from Fig. 4a that the specimens with high Mc,n have smaller values of A+ than expected from the theoretical dependence of I + on and from the values of I + at smaller Mc,n. This might be due to a content of loops which increases with Mc,n. Polybutadienes seem to have values of I + which are related to the strongly crosslinked regions forming the matrix.

The stress-optical coefficients do not depend on other structural elements than those influencing the stress 0. In networks which contain solvents they sometimes exhibit different values than in the melt, because the solvent changes the internal field. The constant k, of Eq. (3) increases with decreasing polymer concentration and is roughly proportional to q- ’ . The dependence of the equilibrium value q of the swelling degree on the modulus 2C, does not follow the theoretical prediction.

This deviation from theory will be increased if one compares q with the “analytical” modulus. It seems to arise from a change of the expansion coefficients with q and from an influence of the factor S (see section 4.4) on the energy of mixing.

OJ5-

8 OJ- I c .- $OW

2 4 6 LP contents in wt.-%

Fig. 7. a) Modulus 2C, as a function of lauroyl peroxide (LP) contents for melt- crosslinked ( 0 ) and solution- crosslinked (0); pv = 0,5) polyisoprene IR 305. b) “Analytical modulus” of the same polymer samples as a function of the peroxide contents; 0, 0 : same meaning as in a)

2 4 6 L P contents in wt . -%

2256 M. Hoffmann

4.4. Structure of solution-crosslinked polymers

The modulus 2C1 of a gel obtained by crosslinking a polymer in solution is nearly proportional to the peroxide concentration used for crosslinking (see Fig. 7). At a given concentration of the peroxide the value of 2C1 depends on the volume concentration pv of polymer which prevailed at crosslinking, 2C1 being nearly proportional to pv. This cannot be explained under the assumptions that the radical efficiency is near 1 ,O and that no radical transfer from the polymer molecule to the alkyl groups of the initiator or to the other polymer molecules occurs. But at a given peroxide concentration the values of the “analytical” modulus do not seem to depend on the polymer concentration. Though the errors of the values are rather large, we may conclude that in solution-crosslinked gels some factor in 2C,, probably the factor h2/hz, has a lower value than 1, i. e. about 0,6. This is confirmed by the values of A: /A; in Fig. 4b, where the values are smaller than expected and indicate that the value of A; is too high. At pv, h2/hi must therefore be lower than 1,0, i.e. G O , ~ . This may be due to some orientation of the molecules perpendicular to the direction of extension which originates from shrinking of the volume during evaporation of solvent and may not be fully relaxed. It is also not clear, why the swelling degrees of polymers crosslinked at pv = 0,26 are much smaller than expected.

--

--

I would like to thank Dr. Kromer and Dr. Bareiss who have carefully read the manuscript and helped to avoid misleading formulations.

’) M. Hoffmann, Makromol. Chem. 183, 2191 (1982) 2, M. Hoffmann, Makromol. Chem. 183, 2213 (1982) 3, P. J. Flory, J. Chem. Phys. 66, 5720 (1977); P. J. Flory, Polymer 20, 1317 (1979) 4, M. Hoffmann, Prog. Colloid. Polym. Sci. 66, 73 (1979) 9 M. Mooney, J. Appl. Phys. 11, 582 (1940); R. S. Rivlin, D. W. Saunders, Philos. Trans. R.

SOC. London, Ser. A: 243, 251 (1951) 6, J. A. Duiser, A. J. Staverman, “Physics of Non-crystalline Solids”, J. A. Prins, ed.,

North-Holland Publ., Amsterdam 1965; W. W. Graessley, Macromolecules 8, 186, 865 (1975); P. J . Flory, Proc. R. Soc. London, Ser. A: 351, 351 (1976)

’) A. V. Tobolsky, “Properties and Structure of Polymers”, Wiley, New York 1960; E. Helfland, A. Tonelli, Macromolecules 8,248 (1975); M. T. Shaw, J. Chem. Phys. 54,2172 (1971)

*) P. J. Flory, J. Rehner Jr., J. Chem. Phys. 11, 521 (1943); P. J . Flory, J. Chem. Phys. 18, 108 (1950)

9, M. Hoffmann, Kolloid-Z. 250, 197 (1972) lo) W. Oppermann, G. Rehage, 26th IUPAC International Symposium on Macromolecules,

1 1 ) C. Burri, “Das Polarisationsmikroskop”, BirkhBuser, Base1 1950 12) T. K. Su, J. E. Mark, Macromolecules 10, 120 (1977) 1 3 ) W. Kuhn, F. Griin, Kolloid-Z. 101, 248 (1942); H. M. James, E. Guth, J. Chem. Phys. 11,

14) W. Oppermann, thesis, Universit2t Clausthal, Germany (1980)

Mainz 1979, preprint E4-1, vol. 111, p. 1360

455 (1943)

M. Beltzung, C. Picot, P. Rempp, J. Herz, Macromolecules, in press

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