nonlinear viscoelasticity of noncrystalline epdm rubber networks

Nonlinear Viscoelasticity of Noncrystalline EPDM Rubber Networks

BOUDEWIJN J. R. SCHOLTENS, DSM, Research and Patents, PO Box 18, 6160 MD Geleen, Netherlands

PAUL J. R. LEBLANS, Department of Chemistry, University of Antwerp, Antwerp, Belgium

Synopsis

An experimental study has been carried out to specify the nonlinear viscoelastic behavior of noncrystalline peroxide-cured EPDM networks covering a range in elongation ratios between one and seven. Stress-strain curves, measured at constant rates of elongation at 300, 345, and 390 K, are analyzed with two recently published methods. Time and strain effects are shown to be separable. The nonlinear strain measure, determined at various stretching rates, is studied as a function of temperature, cross-link density, and prepolymer type. These results are compared with the strain measures of two phenomenological theories, namely with the Mooney-Rivlin equation and a constitutive equation which contains the nonlinear n measure of strain, and with the predictions of the molecular theory of elasticity of polymer networks developed by Flory.

INTRODUCTION

A very important result of the molecular or statistical theories of rubber elasticity is the predicted relationship between the equilibrium stress and strain for an isothermally deformed polymer network.‘,’ However, even for noncrystalline elastomeric networks in (quasi) equilibrium, considerable deviations exist between the experimental and theoretical stress-strain curves, in particular in elongation. For decades, this behavior has been described with the phenomenological Mooney-Rivlin equation,’ however, this equation is unsatisfactory since it is not constitutive and only applicable in a limited extension range. The only molecular theory giving satisfactory results of which we are aware originates from Flory.3,4

The situation is even worse for nonequilibrium stress-strain

0 1986 by The Society of Rheology, Inc. Published by John Wiley & Sons, Inc. Journal of Rheology, 30(2), 313-335 (1986) CCC 014%6055/86/020313-23$04.00

314 SCHOLTENS AND LEBLANS

measurements. Smith proposed an empirical equation to analyze stress-strain measurements at various constant stretching rates in which the time and strain effects were separated.5 However, this approach lacks a firm theoretical basis. Tschoegl et a1.6-S derived a consistent scheme to separate time- and strain-dependent responses. However, with the nonlinear strain measure intro- duced, viz the n measure of strain, their model is capable of de- scribing stress data at moderately large deformations only, namely, at extension ratios lower than about 2.5.6-8

In the previous paper in this issue,g two methods are elaborated for analyzing stress-strain measurements of elastomeric networks at constant stretching rate. Both methods are based on a well-known single integral constitutive equation in which the time and strain effects are separated. The first, an analytical approach, results after some minor approximations in a simple equation similar to the empirical relation proposed by Smith.5 The second, a numerical approach, is exact for infinitesimal summation steps and yields results which are indistinguishable for practical summation step lengths.

In the present study these two methods are applied to stress- strain isotherms of various noncrystalline EPDM networks which have been carefully characterized in the linear region in torsion with a Rheometrics mechanical spectrometer.10 The nonlinear strain measure, SE(h), determined at various stretching rates, is studied as a function of temperature, cross-link density, and prepolymer type. The results are compared with existing molecular explanations for the Mooney-Rivlin constants.“-13 In addition, an attempt is made to describe and interpret our results on the basis of Flory’s molecular theory.3p4

EXPERIMENTAL

Materials

The elastomers are terpolymers of ethylene, propylene, and a diene monomer (EPDM). The pertinent characteristics of the three noncross-linked prepolymers for this study are given in Table I. The permanent networks, obtained as described in Ref. 10, are designated by the sample code followed by a number indi- cating the weight percentage of dicumyl peroxide used for crosslinking (0.2-1.6 wt%). More details on the preparation of these

NONLINEAR VISCOELASTICITY 315

TABLE I Pertinent Characteristics of the Noncross-linked EPDM Elastomers”

Mooney

Sample code

GPC results Composition (mol %) (kgimol)

Ethylene Diene” M,, a,

index

MLJl + 41 at 398 K

T{ W

K 64.6 2.7 45 180 48 280 N 64.8 1.1 40 90 21 280 D 58.2 0.8 50 125 28 247

“More details in Ref. 10. bInitial crystallization temperature determined by DSC, more details in Ref. 10.

networks and their linear viscoelasticity and molecular charac- terization have been published elsewhere.” Table II summarizes the various samples and measuring temperatures used in this study.

METHODS

The dynamic true stress-strain curves were measured with an Instron 1195 tensile tester equipped with a TN0 nitrogen thermostat C-C 0.1 K), and automated with an HP 85 calculator and an HP 6940 B multiprogrammer (both Hewlett Packard). Seven cross-head speeds were chosen between 5 and 500 mmlmin, at temperatures T = 300,345, and 390 K.

Rectangular samples, about 4 mm wide and 2 mm thick, were tightly fixed in the clamps, which were 50 mm apart at the start of an experiment. A homogeneous cross-sectional area of the samples ensured homogeneous elongation during the tensile experi-

TABLE II Summary of the Stress-Strain Measurements

Sample code Measuring temperature (K)

K 0.2 300, 345,390 K 0.4 300,345,390 N 0.4 300,345 N 0.8 300, 345 N 1.6 300 D 0.0 300 D 0.4 300, 345,390 D 0.8 300. 345


ments. The cross-sectional area was determined from the mass of the sample, its density, and its length. The clamps had flat jaws to prevent slippage of the cross-linked (permanent) networks. As a result of the tight clamping of the samples, which prevented sample slippage, the originally flat samples bulged slightly and ex- erted a small pressure force on the clamps. With the automated data acquisition, each 0.2 set for the first 128 stress-strain data, each 0.4 set for the next 128 data, etc., it was very easy to deter- mine by interpolation the sample length at the time of zero tensile force, usually about l-3% larger than the original distance between the clamps. This length was taken to be the original sample length. The stretching rate >; = dhldt, where h represents the elongation or stretch ratio, was calculated from this length and the cross-head speed. The experiments cover constant stretch-rate measurements at various values of i between 1.6 1O-3 and 1.6 10-l sP ‘. By interpolation the true stress-strain curves were determined at fixed values of A. These data were stored on magnetic tapes, which facilitated data manipulation.

The maximum sample length possible in the thermostat was about 37 cm, so that the maximum possible extension ratio ex- ceeded 7. However, in many experiments this maximum extension ratio was not reached due to rupture of the elongated samples near the clamps, in particular at high temperatures and low tensile velocities. Usually about 12-15 measurements were per- formed on a specific network type at a particular temperature. For each measurement a new, undeformed sample was chosen. The reproducibility of the measurements was usually better than 2 5%. The absence of sample slippage in the clamps was checked with ink marks on those parts of the sample which were just in the clamps. At the end of an experiment the homogeneity of the elongated sample was checked by inspection. All doubtful experiments were discarded.

RESULTS

Dynamic Stress-Strain Curves

The dynamic stress-strain curves were determined at seven different cross-head speeds for all the sample temperature combi- nations collected in Table II. The results were plotted as curves of


.n -I YG. ..,...., ../.. ., ,., ,,,,

1 3 5 7

Fig. 1. True tensile stress vs. elongation ratio at various stretching rates, h = dhldt, for D 0.4 at 2’ = 300 K.

the true tensile stress, crE[X(t)], vs. the elongation ratio, A. A typical example is given in Figure 1.

The characteristic shape of these curves is more pronounced in a semilogarithmic graph of the incremental stress, 8uE[X(t)]lJA, vs. A - 1, as shown in Figure 2 for a representative set of measurements. These results show that this characteristic shape is not much dependent on the stretching rate: the incremental stress-strain curves are almost parallel over the entire strain range. For A < 2, the incremental stress decreases slightly with increasing strain due to relaxation effects. The constant cross- head speed makes the strain rate decrease steadily with increasing strain. As is to be anticipated, the value of the incremental stress at constant strain increases with increasing tensile veloc- ity. For A > 2, a sharp upturn is observed, which is ascribed to non-Gaussian effects due to the limited chain extensibility. The main arguments for this explanation are that the strain at which this upturn starts is found to be almost independent of the temperature, as shown in Figure 3, while it decreases with rising cross-link density, as exemplified in Figure 4 for four networks prepared from two different prepolymers. Non-Gaussian effects are usually not expected at these relatively small stretch ratios2


3.0

2.0

1.0

cAs,[A(t)]lbh K 0.2 T=300 K i(s-') WY q 1.610 '

t

x 6.4 lo-' o 3.210 3 l 1.610 3

d

__)e (A - 1) 0.0 ( , ( , , , , , , , 1 , / , , , , , 0.10 1.0

I

l( 1

Fig. 2. Incremental stress versus h - 1 on a semilogarithmic scale at various stretching rates for K 0.2 at T = 300 K.

3.0 boEIA(t)]lbA K0.2 i = 1.6 10 's-' T(K)

Wd 0 306 + 345 x 390

--+-(A - 1; 0.0, 1 , I I I I , I I , 8 , * , I, I I I I 0.10 1.0 )

Fig. 3. Incremental stress-strain curves as a functiond temperature for K 0.2 at k = 1.6 ~O-‘S-~.

3.0

2.0

1.0

0.0 C


@A(t)]/bA T= 300 K i = 1.6 lo-' s ' G, Wal WV 0 D 0.4 0.07

l D0.8 0.19

q K0.2 0.13 a K0.4 0.26

----+(A - 1)

1.0 10 Fig. 4. Incremental stress-strain curves for networks with various equilib-

rium moduli, G,, at i = 1.6 10e2 s -I and T = 300 K.

In our opinion it is indicative of a fraction of relatively short network chains as a result of inhomogeneities in the cross-link density.

These results are in sharp contrast with results of noncross- linked EPDM networks which exhibit strain-induced crystallization.14 At room temperature these elastomers have a similar upturn, but it disappears readily upon a slight increase in temperature, as was demonstrated in Figure 13 of Ref. 14.

Comparison of the Two Methods of Analysis

The approximate, analytical expression for the nonlinear strain measure of cured elastomers derived in the previous paper’ reads:

SE(A) = 3dA(t)llF(t) (1)

where u&(t)] represents the true tensile stress andF(t) the constant stretch rate modulus defined by?,’


F(t) = 3G, + $ i 1% 7H(~)(l - Ct”) d In 7 (2) a

where G, and H(T) are the equilibrium modulus and the relaxation time spectrum in shear, respectively, and T the relaxation time. The applicability of Eq. (1) was tested in two different ways. The first way was already proposed by Smith:5 for each stretch rate the stresses at arbitrarily selected extension ratios are determined and the times to reach these extensions. Next, these stress and time data are plotted at each selected extension ratio as log uE[h(t)] vs. log t, called the isostrain stress relaxation curves. In addition, log F(t), calculated from the linear viscoelastic characteristics in torsion,l’ . 1s plotted in this graph, as shown in Fig- ure 5 for a representative network. The resulting isostrain stress relaxation curves will be parallel with log F(t) versus log t pro- vided the data can be represented by Eq. (1). From the vertical displacement between the log u&(t)1 and log F(t) curves the value for *SE(A) can be easily obtained with Eq. (1).

For each network at a given temperature the isostrain stress relaxation curves were compared with the pertinent part of the constant stretch rate modulus curve, as exemplified in Figure 5. Most log F(t) vs. log t graphs are slightly curved, but over the limited range of the isostrain stress relaxation curves (two decades) they can be approximated very accurately by a straight line. As can be inferred from Figure 5, the slopes of these lines are smaller than - 0.1, in accordance with observations on different elastomeric networks.5,15 Similar results were obtained for the other networks and temperatures. Smith obtained straight lines as well for his isostrain stress relaxation curves.5

Within the reproducibility of the experiments, most data were situated on straight lines parallel to the pertinent part of log F(t) vs. log t. The largest deviations were observed in data of K 0.2 at 300 K (see Fig. 5) at high extensions, where the slope obtained with linear regression was slightly, but systematically smaller than the one drawn parallel to log F(t) versus log t. These minor deviations, which increased slightly with increasing cross-link density and ethylene content and decreasing temperature, are probably caused by an increase in the microparacrystallinity content of these samples with extension, as was also detected in this


K 0.2 T= 300K A=

Fig. 5. Course of the double logarithmic isostrain stress relaxation and constant stretching rate modulus curves, log u.&(t)] and log F(t), respectively, for KO.ZatT = 300K.

sample with wide-angle x-ray scattering and low-angle light scattering. l6 The same type of deviations, but much larger, was also observed in results of EPDM elastomers which crystallized upon stretching.14

The second method of testing the validity of Eq. (1) follows from a comparison of S,(X) obtained with Eq. (1) and with the numerical method, discussed in our previous paper.g A typical example is given in Figure 6. The broken curve, obtained with Eq. Cl), falls only slightly below the symbols and the drawn curve, which both pertain to data obtained with the numerical method. At A = 7 this deviation is 10% and it decreases with decreasing A. Similar results were obtained for data of different networks or temperatures, so that we conclude that S&Q for these cured elastomeric networks can be calculated with good precision with the approximate, but very simple Eq. (1).

Another way of expressing the nonlinearity is by defining the nonlinearity or damping17 function:

h,&) = S&)ICE- ‘(A) (3)


10’

loo

lo-.

I f N 0.4,T = 300 K

I i (s-1) A 1.67 10-a x 6.67 10-a + 3.33 10-2 o 1.6710-l

+A

I 3 5 7

Fig. 6. The nonlinear strain measure for N 0.4 at 300 K. The broken curve represents the average data over all stretching rates obtained with the analytical Eq. (1). The symbols and the drawn curve pertain to data obtained at four stretching rates with the numerical method.

where CE- ’ (A) represents the difference between the ll- and 22- components of the relative Finger strain tensor:

GE-l(A) = A2 - h-l (4)

which corresponds to the strain measure predicted by the Gauss- ian statistical theories of rubber elasticity.1-4 Some representative values for &(A) based on SE(h) data determined with Eq. (1) are collected in Table III. In view of the reproducibility of the stress-strain measurements and the fact that F(t) was calculated from viscoelastic characteristics measured in torsion and ex- tended by time-temperature superposition,” the observed linearity at A = 1.25 and 1.50 is very gratifying.


TABLE III Some Values for the Damping Function /&A) = &WCs-‘(h)

Sample T Extension ratio X

code (K) 1.25 1.50 2.0 3.0 4.0 5.0 6.0 7.0

K0.2 300 1.04 0.97 0.86 0.69 0.62 0.57 0.55 0.51 345 1.12 1.07 0.98 0.88 0.78 0.68 - - 390 1.12 1.07 0.98 0.91 - - - -

K 0.4 300 1.06 0.99 0.86 0.68 0.63 0.59 0.55 - 345 1.05 0.98 0.89 0.77 0.71 - - - 390 1.05 1.01 0.89 - - - - -

N 0.4 300 1.01 0.94 0.76 0.52 0.39 0.32 0.28 0.25 345 1.05 0.95 0.79 0.56 0.45 0.39 0.35 0.32

N 0.8 300 1.05 0.97 0.80 0.57 0.48 - - - 345 1.05 0.97 0.81 - - - - -

N 1.6 300 1.05 0.98 0.82 - - - - - D O.Oa 300 0.98 0.92 0.76 0.51 0.37 0.28 0.23 - D 0.4 300 1.20 1.05 0.86 0.57 0.42 0.34 0.29 0.24

345 1.13 1.02 0.82 0.55 0.43 0.37 0.32 0.28 390 1.16 1.02 0.83 0.58 0.48 0.42 0.38 0.35

D 0.8 300 1.01 0.92 0.75 0.52 0.41 0.36 0.32 0.30 345 0.97 0.88 0.73 0.54 0.45 0.40 0.36 -

“These data were obtained with the numerical method.’

Factorability

The applicability of Eq. (11, as demonstrated in Figures 5 and 6, means that the time and strain effects are separable in these elastomeric networks up to high extensions. Indeed the indepen- dence of SE(A) of the stretching rate A, as indicated in Figure 6, implies that SE(A) is independent of time. Hence, our time- dependent data give the same strain function as equilibrium data, a consequence of this factorability already realized by Smith and Dickie.” In other words, the same nonlinear strain measure SE(A) is applicable to the relaxational and equilibrium components of the modulus. This very important conclusion is corroborated by the data in Table IV, which gives SE(A) values for three networks with different contributions of the equilibrium tensile modulus, 3G,, to F(t). For each of these networks, a comparison is made between SE(A) values determined at the highest and at the lowest stretching rate. These representative data also demonstrate no significant differences in S&A) with different relaxational contributions to F(t).


TABLE IV Comparison of S,(h) Values Determined with the Numerical Method’ from

Data Measured at the Two Ultimate Stretching Rates

Sample A=3 A=5

code L,” LLax Li” Lx

K 0.4 S,(A) 6.3 6.4 15.1 15.4 3GJIW 0.66 0.51 0.68 0.53

N 0.4 SE(A) 4.8 4.5 8.7 7.9 BG&FIF(t) 0.51 0.33 0.54 0.36

D 0.4 S,(A) 4.7 4.9 8.0 8.2 BGJFIF(t) 0.39 0.19 0.42 0.22

SGJF(I) represents the relative contribution of the equilibrium modulus to F(t), i,,, = 1.6 10m3 and k,,, = 1.6 10-l s-l.

Variations of &(A.) with Temperature, Cross-Link Density, and Prepolymer Type

Whereas SE(A) is independent of time, its value generally increases with rising measuring temperature, in particular for A 2 2, as shown in Figure 7 for a typical sample. This effect can also be deduced from Table III.

The variation of SE(A) with the degree of crosslinking is less uniform. For both the N and D networks S&A) increases slightly with cross-link density for A b 4, as exemplified in Figure 8. It is interesting to note that even the uncross-linked D 0.0, which is a polymer melt, lo has a nonlinear strain function not much different from the permanent networks D 0.4 and D 0.8. In contrast, no dependence on the cross-link density is observed in K 0.2 and K 0.4, see Figure 9 and Table III.

Finally, the type of the prepolymer has a considerable effect on SE(A) as demonstrated in Figures 10 and 11 for three networks prepared with different prepolymers but of comparable cross-link density, at 300 and 345 K, respectively.

DISCUSSION

Our results show conclusively that the S,(A) curves are independent of time, so that these functions are also valid for equilibrium stress-strain measurements. Conversely, the course of S,(A)


10

10

10

0 0.4 T(K) 0 300 + 345 Y 390

i 5 i

Fig. 7. Variation of SE(h) with the measuring temperature for D 0.4, with the Finger strain measure as a reference.

may thus be compared with predictions based on equilibrium theories, as will be done below.

Another interesting phenomenon is that the shape of SE(A) does not differ greatly for uncured and cured EPDM elastomers (cf. Fig. 8). We are not aware of similar results in the literature. It seems to imply that, at least for EPDM elastomers, which are highly entangled,10*1g the nonlinear behavior of strongly relaxing melts and that of permanent networks have roughly the same molecular origin. This may be specific for EPDM elastomers due to their highly entangled state. Tschoegl et al. reported larger variations of SE(A) with cross-link density for other elastomers. 6 ~a It may however, be an intriguing point for theoreticians, who now develop theories for the nonlinear behavior for polymer


,” 1 3 5 7

Fig. 8. Variation of S,(X) with the degree of crosslinking for D 0.0,0.4 and 0.8 at 300 K; G, represent the equilibrium shear modulus,1o the Finger strain measure is given as a reference. The data of D 0.0 were obtained with the numerical method.’

melts on the one hand and permanent networks on the other hand.

The dependence of SE(L) on the experimental variables temperature and cross-link density is qualitatively in agreement with results reported by Tschoegl et a1.6-s Their n measure of strain, which is of the same form as the Seth strain measure, is for uniaxial extension defined by

S&Y) = 20X” - h-“‘2Yn (51

Tschoegl et al. consider n as an adjustable parameter or material constant, and find their equation applicable to moderately large


T 300 K + K 0.2 0.13

I a K 0.4 0.26

I

10-I -A

1 3 5 7

Fig. 9. cf. Fig. 8, but now for K 0.2 and 0.4.

deformations, when h < 2.5. 6-8 Our results cover a much wider range and cannot be described satisfactorily with this simple parametric function having such a definite shape.

Another phenomenological, but in this case nonconstitutive, equation which is widely used is that of Mooney-Rivlin.2’11 - 13,1g This equation is not factorable into a strain and a time function. The nonlinear strain measure corresponding to it is dependent on the ratio of its two constants, namely C2/C1, according to the following equation:

SE(h) = [

1 + h-‘C2ICl 1 + C&l I

CE- l(A) (6)

Because C&/C1 is in general time dependent,‘2’1g S,(A) determined with Eq. (6) is a function of time as well. For all these reasons it is


lo+

100

lo-

4 G&IPa

I

T ~300 K + K 0.2 0.13 h n N 0.4 0.13

o D 0.8 0.19

-A

3 5 7

Fig. 10. Variation of SE(A) with the type of prepolymer for three networks with roughly equal equilibrium modulus at 300 K. Tbe Finger strain measure is given as a reference.

not surprising that these constants C1 and C2 lack a clear molecular explanation.2,11-‘3,‘g Figure 12 shows how SE(A) varies with Cz/Ci. Our results cannot be fitted satisfactorily with Eq. (6) because they all demonstrate a relatively high.linearity of SE(A) for A i 2.5. In addition, we do not observe a dependence of SE(A) on time (cf. Fig. 6). However, the observed dependence of SE(A) on the cross-link density agrees qualitatively with numerous studies which used Eq. (6): the deviations from linear behavior decrease with increasing cross-link density. “Jo

Boyer and Miller discussed three possible molecular origins of this nonlinearity. l3 One of their speculations, namely strain- induced crystallization must be rejected for the present elastomers on the basis of experimental evidence? even at an exten-


1

Fig. 11.

-A

3 5 7

cf. Fig. 10, but now at 345 K.

sion ratio as high as 10 the sample with the highest ethylene content, K, did not show the wide-angle x-ray diffraction spots characteristic of strain-induced crystallization.‘4 Besides, in our opinion, strain-induced crystallization can never be responsible for the kind of deviations observed, because it will work in the opposite direction. Another, related, speculative cause, namely intermolecular order must also be rejected for these elastomers. The present samples indeed show some intermolecular order,10,16 but the most ordered sample, K, is the one with the smallest deviations from C, -‘(A) (cf. Figs. 10 and 11) and thus with the lowest CzICi (cf. Fig. 12). Their last speculation, namely chain entanglements, is treated quantitatively and in great detail by Flory in his molecular theory.3p4

In the theory of Flory (and Erman) the restrictions imposed on the mobility of the chains by the extensive interpenetration of


Fig. 12. Variation of SEoI) with C2/C1 as determined with Eq. (6).

neighboring, but topologically remote chains are completely con- centrated in the junctions by laying restraints on their fluctuations. These restraints diminish with elongation, which causes a gradual transition from the affne theory to the phantom theory, respectively. 3,4 Thus the junctions, rather than the chains between junctions, are considered to be affected by entanglements. Entanglements are certainly not envisaged as discrete loci which can be equated to a set of enumerable cross-links. In its most recent version this theory implies the following equation for the nonlinear strain measure in uniaxial extension:

(7)

where fJfph is a complicated function of h containing the parameters K and c.4 The parameter K specifies the severity of the entan-


TABLE V Summary of Parameters to Describe Experimental S,(A) Curves

Sample Temperature code (K) K 5 4s

K 0.2

K 0.4

N 0.4

N 0.8

N 1.6 D 0.0 D 0.4

D 0.8

300 20 0 1.8 345 20 0 0.8 390 20 0 0.7 300 20 0 1.7 345 20 0 1.0 390 20 0 0.9 300 10 0 6.0 345 10 0 4.0 300 10 0 3.5 345 10 0 3.5 300 20 0 3.0 300 10 0 9.0 300 10 0 6.0 345 10 0 5.0 390 10 0 3.3 300 10 0 4.9 345 10 0 3.9

glement constraints relative to those imposed by the phantom network; for K = 0 the phantom theory applies without any en- tanglement contribution. In the limit of very large K the entang- lement constraints are maximal, corresponding to a total sup- pression of junction fluctuations. The parameter 5 describes departures from affine transformation of the a priori probability function of these fluctuations.4 In addition, fclfph contains the network characteristics CL, corresponding to the number of active junctions, and 5, being the cycle rank.3,4,20 For perfect networks free of defects CL/( equals 2/(+ - 21, where 4 is the junction functionality.314 For real tetrafunctional networks not far beyond the gel point the effective functionality is close to three,‘l so that p,l( for our networks is expected to range from 2 to 1.

We have attempted to fit our experimental s,(A) curves with Eq. (7) using K, 5, and l.~/[ as adjustable parameters. In all cases it was possible to find a set of parameter values for a best tit, as illustrated in Figure 13 for some representative samples. In view of the accuracy of the measurements the agreement is very gratifying. The values required for ~1s are, however, in general

332

10'

10"

10

I T 300K + K 0.2 A NO.4 0 DO.0

J 1 3 5 7

Fig. 13. Experimental S,(h) data (symbols) compared with theoretical predictions using Eq. (7) and the parameter values collected in Table V (drawn curves).

SCHOLTENS AND LEBLANS

much higher than expected. We can only explain this phenomenon by arguing that also many two-functional junctions are ac- tively constrained. In other words, the restriction of the junction mobility alone is not satisfactory to explain the results on these EPDM elastomers. It seems that the number of possible chain configurations must be limited as well. This explanation seems almost equivalent to an additional contribution of trapped entanglements to the small strain equilibrium modulus. The fact that the degree of nonlinearity of the elastomeric melt D 0.0, containing no junctions, does not differ much from that of the cross-linked networks D 0.4 and D 0.8 is in accordance with the conclusion stated above that the restriction of possible chain configurations must play an important role as well. Moreover, it is puzzling that the nonlinear strain measure for D 0.0, an elas-


IO'

100

lo-

i

1

:I

1 l- 1

----ci 3 5 7

\

Fig. 14. The nonlinear strain measure calculated with Eq. (7) and Flory’s theory4 for various values of k/e, with K = 10 and L = 0.

tomeric melt, can be described so satisfactorily with an equation designed for permanent networks in equilibrium. This may re- quire a reconsideration of the molecular origin of nonlinearity, which is at the moment regarded to be different for polymer melts” and networks.3,4

It is interesting to note that with Flory’s theory a reasonable explanation can be found for the more linear behavior of the K networks (cf. Figs. 10 and 11). In an earlier study, one of the authors concluded that due to the high diene content of this sample, functionalities higher than four were probably obtained.” This results in a lower value of p/c and, with Flory’s theory, in a more linear behavior, as demonstrated in Figure 14. Another experimental observation which can be explained at least qualitatively with this theory is the increase in linearity with increas-


ing degree of crosslinking (cf. Fig. 8). According to Dusek’l this causes an increase in the average functionality, so that k/.( decreases and S,(A) becomes more linear (cf. Fig. 14). For the K 0.2 network the average functionality is probably already so highlo that a further increase hardly affects p./(. The general increase in linearity with rising temperature (cf. Fig. 7) might be ascribed to the presence of fewer constraints on the chain configurations at higher temperatures in the small strain range due to the higher Brownian motions of the chains. As a result (fJf& = r would decrease with rising temperature, so that S&I) approaches C,-‘(A) more closely. Thus, several experimental observations may be explained qualitatively with this theory.

CONCLUSIONS

Two recently developed methods for analyzing stress-strain measurements of cross-linked elastomers yield good results when applied to constant stretch rate experiments with EPDM elastomers. Within the inaccuracy of the measurements and the analytical method the agreement is satisfactory. The nonlinear strain measure SE(h) obtained with these methods is independent of time or stretching rate, but increases with temperature. An interesting feature of the present results is that SE(h) is not greatly different for noncross-linked EPDM melts and their corresponding permanent networks.

Two speculative explanations from the literature for the molecular origin of these effects, namely strain-induced crystallization or intermolecular order,13 could be rejected for the present elastomers. Flory’s recent molecular theory of rubber elasticity,3,4 which is based on constraints on the junction fluctuations, could be successfully used to describe the data. The values of F/[ required to obtain the best fit imply that these junctions are not the only restricted loci. For EPDM elastomers there seems to be an additional contribution of restricted chains as well. The dependence of SE(A) on temperature, degree of crosslinking, and prepolymer type can be explained satisfactorily with this theory.

The authors are grateful to N. J. M. H. Oversier, J. C. Joosten, and E. Peters for their assistance in automating the Instron tensile tester and the data acquisition and manipulation.


References

1. P. J. Flory, Princ$es of Polymer Chemistry, Cornell University Press, Ithaca, NY (1953).

2. L. R. G. Treloar, The Physics ofRubberEZasticity, 3rd ed., Clarendon Press, Oxford (1975).

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Received June 18, 1985 Accepted November 7, 1985

nonlinear viscoelasticity of noncrystalline epdm rubber networks

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