Stress Relaxation in ElastomersA. M. Bueche Citation: The Journal of Chemical Physics 21, 614 (1953); doi: 10.1063/1.1698977 View online: http://dx.doi.org/10.1063/1.1698977 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/21/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Investigation of Stress Relaxation in Filled Elastomers by XPCS with Heterodyne Detection AIP Conf. Proc. 1092, 29 (2009); 10.1063/1.3086228 Infrared spectroscopy of a polyurethane elastomer under thermal stress AIP Conf. Proc. 430, 560 (1998); 10.1063/1.55727 Stress Relaxation as a Method of Analyzing Stress Growth, Stress Overshoot and SteadyState Flow ofElastomers J. Rheol. 30, 383 (1986); 10.1122/1.549854 Elastomer stress damper with constructable orifice J. Acoust. Soc. Am. 68, 1016 (1980); 10.1121/1.384877 THE RELATION BETWEEN STRESS AND STRAIN IN MULTIAXIAL DEFORMATIONS OFELASTOMERS Appl. Phys. Lett. 11, 235 (1967); 10.1063/1.1755114

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614 D. F. SMITH

ADDITIONAL COMMENTS

Unfortunately, lines suitable for resolution of Stark components and the evaluation of the dipole moment are not of sufficient intensity to observe and measure. The low intensities of the lines, however, are in accord with a dipole moment as low as the 0.554 Debye units.1

The observed structure of monomeric CIF 3 makes the existence of a dimerl ,16 somewhat perplexing.

The vibration spectrum of CIF 3 has been studied by Jones2 and others at the K-25 Laboratories and more recently at Vanderbilt University. The experimental studies of such a reactive compound have been so difficult and the spectrum so unusual that it has been impossible to assign the six fundamental vibration fre­quencies. Since there is no third group of microwave

16 H. Schmitz and H. J. Schumacher, Z. Naturforsch. 2a, 363 (1947).

THE JOURNAL OF CHEMICAL PHYSICS

lines showing spacings similar to the Cl36F 3 10.1;-1

- 21, 2;-1 observed within 500 mc of the ground state transition, it seems unlikely that any 10.1;-1- 21, 2;-1

transition of molecules in an excited state vibration lines are strong enough to be observable at - 80°C. This can be taken as evidence that there is unlikely to be a fundamental vibration as low as 200 cm-I •

ACKNOWLEDGMENT

The author is indebted to Dr. R. D. Burbank for making available to him the preliminary x-ray diffrac­tion results which greatly speeded the analysis of the spectrum; to Dr. Burbank and Dr. D. W. Magnuson for several helpful discussions; and to Mr, M. Tidwell and Mr. D. V. P. Williams, who had somewhat earlier established the optimum conditions for the observation of the CIF 3 microwave spectrum.

VOLUME 21, NUMBER 4 APRIL, 1953

Stress Relaxation in Elastomers

A. M. BUECHE

General Electric Research Laboratory, Schenectady, New York (Received November 18, 1952)

The process of chemical stress relaxation in elastomers is examined in some detail. It is predicted that in some cases it will be possible to determine the location of the chemical bond broken. The special cases of relaxation due to scission of cross links and random scission of the polymer are calculated. The shapes of the stress vs time curves are widely different.

T HE measurement of tension as a function of time in elastomers held at constant elongation has been

shown by Tobolsky to be a useful tool in investigating thermal and chemical degradation.1 By varying the temperature and the time of the experiment the visco­elastic behavior also can be studied.2 During the course of our work it was decided to investigate the possible types of stress relaxation data to be expected from thermal or chemical degradation on the basis of a few simple assumptions. In what follows it will be shown that for vulcanized elastomers it is possible in some cases to decide whether the stress relaxation is due to a random scission of polymer bonds or whether the bonds broken are of a special nature such as those introduced in the cross-linking process. In addition, of course, one may also determine rate constants and activation energies.

The tension T in an elastomer network system (at high enough temperatures and long enough times of measurement so that viscous effects may be neglected)

1 Tobolsky, Prettyman, and Dillon, J. Appl. Phys. 15, 380 (1944); Tobolsky, Metz, and Mesrobian, J. Am. Chern. Soc. 72, 1942 (1950); M. Mochulsky and A. V. Tobolsky, Ind. Eng. Chern. 40, 2155 (1948).

2 Bischoff, Catsiff, and Tobolsky, J. Am. Chern. Soc. 74, 3378 (1952).

can be written as3

T=RTv",(OI), (1)

where R is the gas constant, T is the absolute tempera­ture, v is the number of moles of elastically active chains between cross links in a unit volume of polymer, and ",(01) is a function of the elongation which describes the shape of the stress-strain curve, Even though there is some doubt as to the accuracy with which this equation represents the data on elastomers, the approxi­mation has been shown to be reasonably good provided it is applied at low elongations, and provided one lets v include the number of chains formed because of accidental intertwining of one chain with another. It, at least, serves the purpose of pointing out that the tension is proportional to the number of chains intro­duced by chemical cross links and the effect of tempera­ture.

In the following we will assume that Eq. (1) rep­resents the facts. This implies that the experiments with which the results are to be compared are done in such a way as to eliminate viscous effects and that, if thermal or chemical bond scission were eliminated, T

would be constant with time provided that T and 01 S See, for example, P. J. Flory, Chern. Revs. 35, 51 (1944).

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STRESS RELAXATION IN ELASTOMERS 615

are held constant. To investigate chemical relaxation then we must focus our attention on II.

Two processes of stress relaxation will be of interest. In both cases we shall assume, as is generally found at low elongations, that the relative rate of relaxation is independent of the stress on the sample. l The first will be the one where the stress is relieved by the reaction of some chemical agent (held at a constant concentra­tion) with the cross links.

We shall define a cross link by its action. It will be a structural unit which has the ability to join polymer molecules together in such a way as to require two cross links to form two polymer molecules into a closed ring. Such a ring will in general have branches, since in a random cross-linking process of a high polymer the probability of the ends being incorporated into the ring is very small. Thus, a cross link can be visualized as the reaction product between two trifunctional units each of which is incorporated in a polymer chain.

Flory3 has pointed out that, if X 0 such cross links are added to a unit volume containing No polymer mole­cules, the number of elastically active chains 110 is

Jlo=2Xo-2No. (2)

He has shown this relation to hold provided that Xo is at least I.S No. If we let Z designate the degree of polymerization of the No molecules, the average length Z. of the polymer chains between cross links is

Zc=NoZ/2X o. (3)

Since the cross linking does not all occur at the ends of the initial polymer molecules there will be some dan­gling chain ends. These are not elastically active. The average length of the ends is Zc. The fraction of the sample which is elastically active is then

(4)

Now if we assume that the cross links are broken according to the law

1 dX ---=K, (5)

X dt

where K is proportional to the rate constant given by the theory of absolute reaction rates, X becomes a function of time t, given by

X=Xo exp( -Kt). (6)

The subscript refers to the value at t=O. This rate law has been chosen to demonstrate the principles involved .. Analogous expressions could be developed for other rate laws. Combining Eqs. (1), (2), and (6) we find that

r=2RT[Xo exp( -Kt)-No}p(a). (7)

Thus, in general, a plot of log r vs t will not give a straight line even though Tcp(a) is held constant. If

however, the number of cross links is large enough so that No may be neglected with respect to Xo, the linear approximation will be good for small values of Kt. Theoretically, the deviation from linearity could be used to determine Xo/No. This would presumably require high precision measurements; the simple relationship given by Eq. (7) would not be expected to hold when X 0 and No are of the same order of magnitude because of the approximation used in getting Eq. (2). For the same reason Eq. (7) would not be expected to hold for large values of Kt. In many practical cases, however, the contribution of No will be small and one would expect, if the mechanism is the one postulated, to get a linear relationship between logr and t.

So far we have limited the process causing the relaxa­tion to a destruction of very special and highly localized bonds. There remains the possibility that any bond along the network chain could be destroyed. In a rubber cross linked by a random process one would expect a distribution of lengths of network chains. Since the reaction causing stress relaxation could occur at any bond along the chain, it seems probable that long chains will be destroyed at a higher rate than short ones. The observed rate of decrease of tension will then be due to the sum of the rates for the chains of different lengths.

Consider that at time t=O we have lIiO chains of length Zi. Then by Eqs. (2) and (4)

LZilliO=NoZ(1-2Zc/Z) =2(Xo-No)Zc. (8) i

Now, if we make the reasonable assumption that the rate at which chains of length Zi are broken is propor­tional to Z, and that a rate law of the type used in Eq. (5) is correct, we can write

11= Lllio exp( -kZit). i

(9)

Before we can proceed we need to know the relation­ship between lIiO and Z,. For a process of random cross linking, we can estimate this relationship in the follow­ing way.

Consider the system of No primary molecules of length Z joined by Xo cross links. There would then be N oZ chain elements. Referring to Eq. (4), we realize that the total number of these chain elements that are elastically active is

( 2Zc) NoZ 1-Z .

Of the latter, 2(Xo- No) are trifunctional elements that are involved in cross links in excess of those needed to tie all the molecules together and thus contribute to the elastic properties. The fraction p of the elastically active elements that are trifunctional is

p=2Xo/NoZ=1/Zc. (10)

If we assume that these are distributed at random, it can

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616 A. M. BUECHE

0.01

0001 O)------;';;-----.;!--;;---~~---L 4.0

KIOR hi

FIG. 1. Plot of 6rX1Q3/RT.,.(a) vs Kt or kt describing stress relaxation due to disruption of the cross links, Eq. (7), or due to random chain scission, Eq. (14).

be shown4 that the number of chains of length Zi is

vio=2(Xo-No)[p(1- p)ZH], (11)

where a grouping having two trifunctional elements side by side is defined as a chain of Z,= 1. Substituting this into Eq. (9), we have

'" v=2(Xo-No)PL [(1-p) ZH exp(-kZit)]. (12) Z,-1

This sum can be evaluated to give

v=2(Xo-No)p[exp(kt)-1+ p]-l, (13)

and using Eq. (1)

T= 2RT(Xo-No) p[exp(kt)-l+ p]-1'P(a). (14)

For small values of kt the exponential term may be

• The method is essentially the same as that used by P. J. Flory, J. Am. Chern. Soc. 58, 1877 (1936).

expanded giving

[ (kt)2 (kt) 3 ]-1

T=2RT(Xo-No)p p+kt+-+-+··· 'P(a). 2! 3!

(15)

Thus ·from a plot of 1/ T VS t it should be possible by inclusion of the term in (kt)2 to get p and k. From these and the value of T at t=O the entire curve could be constructed. For large values of kt a plot of 10gT vs t should be linear with a slope of - k. In Fig. 1 are plotted the curves from Eqs. (14) and (7) assuming an original polymer of molecular weight 6X loo, a density of unity, a degree of polymerization of 6X lOS, and SO cross links per original molecule. As can be seen, the curves are quite different for the two mechanisms of stress re­laxation.

We have assumed in obtaining Eq. (14) that all chains react with rates proportional to their lengths. This approximation may fail in the case of very short chains. For an example of an effect of this sort let us assume that chains of lengths up to and including Z i = j - 1 do not react at all and that chains with lengths Z,";?j react with rates (Zi- j-l)k. The expression corresponding to Eq. (12) becomes

v=2(Xo-NO)p{ r:1

(1_p)Z,-1 Z .... 1

+ EP- p)ZH exp[ -k(Z.- j+1)t]}. (16)

Upon integration this will yield one term independent of time and another term that depends on time. A variety of stress relaxation curves can be obtained, depending on the value of j used. Thus the above expressions are sensitive to the relative chemical behavior of the short and long chains. There is also some doubt as to the effect very short chains exert on the elastic properties of the sample. In view of these unknowns it seems use­less to generalize further at this time.

There is still another possibility for the analysis of stress relaxation data. Presumably, by the correct mathematical procedure, Eq. (9) could be inverted and the number of chains of length Zi could be computed. The methods of handling such a problem are known.6

ACKNOWLEDGMENT

The author wishes to thank Dr. B. H. Zimm for his valuable suggestions in preparing this manuscript.

5 E. Blade and G. E. Kimball, J. Chern. Phys. 18, 626 (1950).

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