Relaxation mechanisms in polymeric sulfur - The Journal of Physical

Adi Eisenberg, Larry A. Teter. J. Phys. Chem. , 1967, 71 (7), pp 2332–2340. DOI: 10.1021/j100866a057. Publication Date: June 1967. ACS Legacy Archiv...
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A. EISENBERG AND L. A. TETER

2332

Relaxation Mechanisms in Polymeric Sulfurla

by A. Eisenberglb and L. A. Teter Contribution N o . 1967 f r o m the Department of Chemistry, University of California, Los Angeles, California 90024 (Received December 9, 1966)

The observation is made that polymeric sulfur above the glass transition temperature is subject to two relaxation mechanisms, a “simple” molecular flow and bond interchange. The effects of these mechanisms on the viscoelastic properties are separated quantitatively, and their relative contributions to the shear viscosity are computed. The activation energies of each of the mechanisms are also determined, and it is shown that the molecular flow is subject to a WLF type temperature dependence, while that of the interchange mechanism is of the Arrhenius type. It is tentatively suggested that the chemical relaxation mechanism is of a radical nature, with homolytic chain scission predominating.

I. Introduction It has been known for a long time that elemental sulfur can exist in a metastable “plastic” modification at room temperature, and many of the properties, including viscoelastic properties, of that modification have been studied;2 but practically no attention seems to have been paid to the relaxation and flow mechanisms in that material. The elucidation of the flow mechanisms for this polymer is, however, of great interest, because, as we shall show below, both simple diffusional motion and bond interchange are to be expected as relaxation mechanisms, and both are indeed encountered. The two relaxation mechanisms mentioned above can be expected for the following reasons. Polymeric sulfur, owing to its high molecular weight, can be expected to have a high viscositys assuming simple molecular flow to be the only mechanism. A high viscosity is indeed e n ~ o u n t e r e d indicating ,~ that this might be the relaxation mechanism. On the other hand, it is known that above l6Oo, a dynamic equilibrium exists between rings and chain^,^,^ indicating that some type of chemical bond breakage and reformation goes on in the liquid. Furthermore, Tobolsky and co-workers have shown2 that bond interchange is the only relaxation mechanism in cross-linked polysulfide rubbers a t temperatures as low as 70” or even lower in the presence of a catalyst. Thus, since elemental polymeric sulfur has a very high molecular weight and since the activation energy for the bond interchange process encountered in the polysulfide The Journal of Physieal Chemistry

rubbers is relatively low (24-36 kcal), both processes could be expected in the glass transition region and above. As has been demonstrated in a number of preceding publications from this laboratory and from the work by T o b o l ~ k y ,the ~~~ differentiation ~~ between bond interchange or molecular flow as mechanisms of stress relaxation is relatively straightforward. If polymers of various molecular weights are synthesized and the stress relaxation of the resulting materials studied, one would expect, on the basis of previous experience, to find the following situation. If the materials relax by simple molecular flow, then the maximum relaxation time (reflecting the equivalent “ultimate Maxwell element”) will vary with the molecular weight in accordance with the relationg (1) (a) Paper no. 7 in a series on the viscoelastic relaxation mechanism in inorganic polymers; based, in part, on a Ph.D. dissertation submitted by L. A. Teter to the Department of Chemistry. (b) McGill University, Montreal, Canada. (2) A. V. Tobolsky and W. J. MacKnight, “Polymeric Sulfur and Related Polymers,” Interscience Publishers, Inc., New York, N. Y.,

1965. (3) R. E. Powell and H. Eyring, J . Am. Chem. Soc., 65, 648 (1943). (4) R. F. Bacon and R. Fanelli, ibid., 65, 639 (1943). (5) (a) G. Gee, Trans. Faraday Soc., 48, 515 (1952); (b) F. Fairbrother, G. Gee, and G. T. Merrall, J . Polymer Sci., 16, 459 (1955). (6) A. V. Tobolsky and A. Eisenberg, J . Am. Chem. Soc., 81, 780 (1959). (7) A. Eisenberg and L. Teter, ibid., 87, 2108 (1965). (8) A. Eisenberg and L. Teter, Trans. SOC.Rheol., 9, 229 (1965). (9) A. V. Tobolsky and K. Murakami, J . Colloid Sci., 15, 282 (1960).

RELAXATION MECHANISMS IN POLYMERIC SULFUR

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+ CiCz +T -T T-, Tg + B log a

(ca. 0.35-cm inside diameter) and boiled under vacuum (prior to seal-off) to free it from adsorbed gases. The tubes were then heated to 300" in the furnace (subsequent to seal-off) and were maintained a t that temperature for at least 1 day. Thereupon, the samples were quickly quenched in cold alcohol (temperature ca. -25 to -28") and then brought to ca. -80" by the slow addition of crushed Dry Ice to the bath. Next, an excess of Dry Ice was added and the tops of the tubes were broken off. The tubes were then allowed to remain completely submerged in the bath for several minutes during which time the alcohol, saturated with COS,would seep between the wall of the glass tube and the sample. The tubes, one a t a time, were withdrawn from the bath and allowed to warm slightly. The pressure of the escaping C 0 2then usually caused the sample either to be extruded completely or to be forced out of the tube sufficiently so that it could be pulled out the remainder of the way with a pair of tweezers. Care had to be taken in the course of this procedure that the temperature did not rise above T,; otherwise, extensive deformation of the sample could take place. Once prepared and liberated, the samples were stored a t Dry Ice temperature until actually placed in the stress relaxation machine. 2. Arsenic-Cross-linked Polymers. In general, the arsenic trisulfide-sulfur copolymers were synthesized by the simple expedient of heating together glassy As2& obtained by the procedure to be described below and the amount of sulfur necessary to achieve the desired mole ratio of arsenic to sulfur. After seal-off under vacuum, the Pyrex glass ampoules were heated a t 450" for 1-2 days with frequent agitation of the contents. The tubes were then removed from the furnace and allowed to cool. After liberation from the tube, the samples were pulverized and sieved through a 10mesh screen. The stress relaxation samples were prepared by filling the fragmented polymer into Pyrex glass tubes ca. 0.5 cm in diameter followed by vacuum seal-off and heating a t 450" for a t least 1 day. The tubes were then withdrawn from the furnace and allowed to cool, and t,he intact sample was liberated from its tube. The sample was subsequently pressed between smoothed pieces of aluminum foil in the Carver press, the tem-

log

T~

= log A

where T,,, is the maximum relaxation time, A is a constant, C1 and C2 are the constants of the WLF equation,10 111", is the weight-average molecular weight, and B is another constant which ranges from 1 to 2 for polymers below the critical entanglement molecular weight and assumes the universal value of ca. 3.4 above the critical entanglement length (the constant A would also change accordingly). If, on the other hand, the polymer relaxes by bond interchange in the terminal flow region, then, to a very good approximation, the maximum relaxation time should 'be independent of the molecular weight but should reflect only the rate of the interchange reaction. If both bond interchange and molecular flow are occurring simultaneously with comparable relaxation times, then the situation is far more complicated;" however, one can still resolve the mechanisms to some extent and evaluate their relative contributions, for instance, to the shear viscosity. This can be done by converting stress relaxation data into compliance data and shifting the compliance curves to obtain a master curve by optimizing the overlap a t the shortest times. Following Ferry's lead12,13it is postulated that the lower envelope of points corresponds to the molecular flow (or a) mechanism, while any deviation therefrom corresponds to the bond interchange (or x ) mechanism. This will be described much more fully below and has been previously reported in a preliminary fashion." I n order to elucidate the relaxation mechanisms in polymeric sulfur, we resorted to the technique of stress relaxation and utilized, in addition to pure sulfur, a cross-linked polymer, i.e., sulfur cross-linked with As atoms allowing us to vary the molecular weight between cross-links. The reason for this choice will emerge later. Before concluding this introduction, it should be pointed out that the book by Tobolsky and MacKnight2 summarizes the pertinent preceding studies on sulfur and related polymers. Since that review is very recent, for the sake of brevity no summary of previous work is presented here. Occasional pertinent comparisons will be made in the text of the paper itself.

11. Experimental Techniques Since most of the techniques used here have not been described before, they will be presented in some detail. A. Preparation of Samples. 1. Polymeric Sulfur. The crystalline sulfur (Gallard, Schlesinger) of 99.9999% purity was introduced into small Pyrex tubes

(10) M.L. Williams, R. F. Landel, and J. D. Ferry, J . Am. Chem. SOC.,77, 3701 (1955). (11) A. Eisenberg, S. Saito, and L. A. Teter, J . Polyner Sci., C14,

323 (1966). (12) J. D.Ferry, W. C. Child, Jr., R. 2. Stern, D. M. Stern, M. L. Williams, and R. F. Landel, J. CoZEoid Sci., 12, 53 (1957). (13) W.C.Child, Jr., and J. D. Ferry, ibid., 12, 327,389 (1957).

Volume 71, Number 7

June 1967

A. EISENBERG AND L. A. TETER

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perature being above T,. When the sample had been pressed to B thickness of ca. 0.2 cm, it was removed from the press and ground to constant dimensions by the use of sandpaper. The final result was a bar that varied somewhat in its exact dimensions from one sample to the next. They were all cut to a suitable length with a hot razor blade. Following fabrication, the samples were all stored a t temperatures well below their T , until use. Polymers with the composition of AszSlz and higher sulfur content showed an increasing tendency toward crystallization a t elevated temperatures (above Tg but not much higher than ca. 100”) as the sulfur content increased. Above ca. loo”, the crystals in a crystallized sample would melt. B. Glass Transitions. The T i s were all measured dilatometrically using some liquid, usually a higher alcohol, of a uniform coefficient of expansion in the temperature range being measured. The Tg’s for the sulfur-arsenic system are listed in Table I. Table I Activation

Materia

AsPS~ ASPSIP ASZSIS ASZSZS sz

a

Mean

TKI

energy,

O C

koa1

69 46 32 4 - 27

28.3 37.6 40.3 35.4 33.6 36.7 f 2.1”

average deviation.

C. Preparation of As2S3.l4 A 0.3-mole sample of reagent grade As203 was dissolved in 3 1. of 3 N HC1 and heated on the steam bath with stirring while HzS gas was slowly passed into the solution. Following complete precipitation, the mixture was filtered and washed on the funnel and afterward sucked as dry as possible. The precipitate was then dissolved in ca. 0.5 1. of concentrated NH40H and kept on the steam bath for ca. 24 hr during which time a red gritty material precipitated from the original deep yellow solution. This precipitate was subsequently filtered off and dried by allowing a stream of clean dry air to pass over its surf:tce for ca. 24 hr. Following this treatment, a reddish orange powder was obtained. When heated in a test tube, the initial heating caused the material to both melt and sublime a t the same time; however, once the melting had become well advanced, little material was lost via sublimation. The Journal of Physical Chemistry

During the melting procedure, additional powdery AszS3 was added until the final volume of molten As& amounted to ca. two-thirds of the volume of the test tube. At this point, the mixture was heated to the boiling point and then poured into molds for stressrelaxation samples or else the molten material was allowed to cool in the tube for subsequent removal and pulverization ; this pulverized material was then used to prepare other arsenic-containing samples of higher sulfur content than AszS3. It must be mentioned at this point that the final composition of the cooled As2S3melt could very well deviate slightly from the stoichiometric arsenic to sulfur ratio 2:3, the most probable composition being slightly on the low sulfur side; however, since a great deal of sulfur was added to form other sulfur-arsenic samples, this possible deviation was not considered significant.

III. Results and Interpretations A. Multiple-Mechanism Relaxation and Separation of the Mechanisms. Although the criteria for the occurrence of simultaneous multiple-mechanism relaxation have been discussed before,” they will be reviewed here briefly because of their importance. These criteria are given in following paragraphs. (1) The timetemperature superposition principle15*16is not applicable, for instance, to stress-relaxation data obtained on a sample of the material at different temperatures. I n some regions of temperature or modulus, time-temperature superposition may very well apply, but its inapplicability in others demonstrates the presence of additional mechanisms besides simple molecular flow (the a mechanism) ; the material must, of course, be in the region of linear viscoelasticity. The failure of timetemperature superposition manifests itself also in a change in the distribution of relaxation times. In the case of sulfur this is particularly evident in the rubbery modulus region, where a dip appears in the distribution and deepens with increasing temperature, indicating that the rubbery plateau lengthens and that the relaxation times associated with the two mechanisms move farther and farther apart. This was discussed briefly elsewhere. l1 (2) A log-log plot of the maximum relaxation time T~ against the shift factors a , is nonlinear. It should be recalled that the shift factor is the horizontal shift (14) J. W. Mellor, “A Comprehensive Treatise on Inorganic Theoretical Chemistry,” Vol. IX, Longmans, Green, and Co., London, 1929, p 273. (15) A. V. Tobolsky, “Properties and Structure of Polymers,” John Wiley and Sons, Inc., New York, N. Y., 1960. (16) J. D.Ferry, “Viscoelastic Properties of Polymers,” John Wiley and Sons, Inc., New York, N. Y.,1961.

RELAXATION MECHANISMS IN POLYMERIC SULFUR

needed to bring into coincidence the stress-relaxation curves obtained a t various temperature^'^^'^ with the curve obtained a t T = T, used as the reference, while the maximum relaxation timee refers to the “equivalent ultimate Maxwell Element” and is obtained from a plot of log E,(t) (the time-dependent Young’s modulus in stress relaxation) against time on a linear scale. If only one mechanism were observed, then the kinetic factors governing both of these values should be identical. If, on the other hand, two mechanisms of unequal activation energy are encountered, then the short-time relaxation behavior (reflected in uT) should be governed by one set of kinetic parameters, while the maximum relaxation time is governed by another set, leading to a nonlinear relationship. The slope of the log-log plot should be related to the ratio of the activation energies of the two processes. An example of such a plot is shown in Figure 1 for As2Sl2; the points are experimental while the line is calculated from the ratio of the activation energies, the method of computation of which will be described in a subsequent section. As was mentioned in the Introduction, the relative contributions to the relaxation behavior by each of the mechanisms can be determined by the following procedure. l 1 First, the stress-relaxation data are converted to creep compliance data [ J ( t ) ]by the method suggested by Tobolsky and Ak10nis.l~ The curves are then subjected to the usual shifting procedure, so as to maximize the overlap in the short-time region of any one curve with the master curve. Since the creep compliance for multiple mechanism relaxations is in some ways analogous to the reciprocal resistance of a system of resistors connected in parallel [for which 1/R = (1/R1) (l/Rz), quite analogous to J ( t ) = Jl(t) + J 2 ( t ) ] ,the lower envelope of points (obtained from the short-time segments by simple shifting) is subtracted from the individual curves obtained a t various temperatures and the difference is ascribed to the second mechanism, the lower envelope of points being representative of the first mechanism. An example of this procedure is shown in Figure 2 for As2S16. B. The Activation Energy of the Two Processes. Since the identification of the mechanisms is based to a very large extent on the activation energies, the estimation of these will now be described in detail. As was pointed out before, time-temperature superposition has been found to be valid above a certain modulus region (107-10s dynes/cm2 depending on the crosslink density). The shift factors in the region of applicability for all of the materials are of the WLF form,lo as shown in Figure 3, the constants of the WLF equation

+

2335

logaT =

T - T, c1Cz + T - Tg

being C1 = 14.1 and C2 = 47.4. From these, by the usual differentiation of log uT with respect to 1/T, the activation energies for the first mechanism can be calculated.

0‘

Or

-

- 3 €

1

2

3

4

5

6

log r,,,(sec)

Figure 1. The maximum relaxation time the shift factor ( a ~for ) As&: points, experimental (46-110’); line, calculated.

us.

( T ~ )

The determination of the activation energy for the second mechanism is based on the following factors. It is clear from Figure 2 that the deviations from superposition are pronounced, the deviations appearing even larger 01: a stress-relaxation plot. This has been taken to mean that at long times, the second mechanism becomes predominant, in those temperature regions in which it is not obscured by the first mechanism, i e . , a t low temperatures (see Figures 6 and 7). Thus, if we determine the maximum relaxation time in the usual manner, the value should reflect within experimental error only the kinetic factors due to the second mechanism. Therefore, if one plots log rm us. 1/T, the slope should reflect the activation energy of the second mechanism. Plots for all the samples of log rm us. 1000/T are shown in Figure 4, and the activation energies calculated from them are listed in Table I, indicating that the activation energies for the second mechanism for all the samples are close. (17) A. V. Tobolsky and J. J. Aklonis, O.N.R. Technical Report

RLT 88, Princeton, N. J., 1965.

Volums 71,Number 7 June 1967

A. EISENBERG AND L. A. TBTNR

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TIME (SeC)

Figure 2. Separation of the CY and x mechannisms in AsaSls. The smooth curved h e is the lower envelope of points (points omitted for the sake of clarity). Temperatures are indicated near the experimental points. Reference temperature is Tg,32'. Plot of log J ( t ) us. log t: A, 35.8';

U, 46.6';

0, 40.0';

0,50.0';

e, 55.4'.

1000 / T (*K-')

Lower lines represent contribution of J,.

Figure 4. Log n,, U8. 1000/T for all samples: points, experimental: 0, &SO; 8 , AszS~e;8,ASSIS; 0, AsaS2~; 0, S,. The l i e s are the "best" straight lines through the points, the activation energies W i g given in Table I.

0

IO

20

30

40

50

60

T-Tg ("CI

Figure 3. Shift factors (log U T ) us. T - TBfor arsenic-sulfur copolymers: points, experimental: 0,Ask&; @, As~Sl~ls; 0, ASZSZ~; line, calculated from WLF equation and CI= 14.1 and CZ= 47.4.

There is one other method of determining the activation energy for the second mechanism. From the compliance plots due to the second mechanism, the viscosity due to that mechanism can be computed. As can be seen in Figure 2, the plots are linear with a The Journal of Physical C h h t w

slope of 1, indicating that only irrecoverable compliance is involved here, i e . , a pure viscosity. The viscosity is obtained from the simple relation r] = t/J(t). If the master curve has been plotted for T = T,as the reference temperature, then the viscosities thus calculated have to be divided by the shift factors due to the first mechanism; ;.e., the logarithmic shift factor has to be subtracted from the logarithm of the viscosity.ll-l* It should be recalled that r] = 7mG so here again a plot of log 7m vs. 1/T would give the activation energy. However, since G does not vary appreciably with temperature over the narrow temperature range accessible, a plot of log 7 us. 1/T yields the correct value of the activation energy. This plot is presented for in Figure 5. For this material, a value of 39.3 kcal/mole was obtained which, within experimental error, is identical with the value calculated from T ~ . rm is known as a function of temperature from the plot in Figure 4 (the expression for the smoothline was used here), and log UT is known from the WLF expression. A simple plot of one variable against the other yields Figure 1, the slope of which, incidentally, yields the ratio of the activation energies of the various processes; the curvature indicates the variation of the activation energy for one of the processes. C. Identzjkation cf the Mechanisms. The identi-

RELAXATION MECHANISMS IN POLYMERIC SULFUR

fication of the mechanism is based on the following factors. (1) The shift f.ctors corresponding to the first mechanism are of the WLF form. (2) The shift factors for the first mechanism are very much in line with those obtained for polymeric selenium,' which has been shown to relax by simple molecular flow (the a mechanism). (3) The activation energy for the second mechanism does not, vary with temperature; ie., it is of the Arrhenius type. (4) The compliance resulting from the second mechanism is strictly irreversible, as shown by linearity of the difference plots and the slope of 1. For these re:tsons, the first mechanism is identified as the a mechanism, while the second is assigned to bond interchange and is called the x mechanism owing t o its chemical nature. The linearity of the log 7mus. 1000/T plot is believed to be particularly significant and to indicate that the size of the reacting group does not change with temperature, in contrast to molecular motion of the type responsible for the a mechanism, which is governed by the WLF relation. D. Relative Contributions to the Viscosity by Each of the Mechanism. The method of determining the viscosity due to the x mechanism (7,) was described in section B. The calculation of the viscosity due to the a mechanism now remains to be described. It should be recalled, however, that even the highest temperature experiments may still contain a contribution of the x mechanism and that therefore the lowest envelope of points a t the longest times may not be a true reflection of the complete a mechanism; i.e., we might have to go to still higher temperatures. I n order to obtain the a viscosity by one method, the lowest envelope of points on the compliance curve is recalculated back into a modulus curve, and the distribution of relaxation times is computed. The viscosity is then calculated from the equations: 7(t) = JH(7)d(7), where v(t) is the tensile viscosity and H ( T ) d(7) is the distribution of relaxation times, and 9 = ~ ( ~ ) / 3To . obtain the total viscosity for any one temperature, the above procedure is followed except that here the real relaxation master curve for the particular temperature is utilized rather than the master curve corresponding to the lowest envelope on the compliance plot. It should be stressed that in this procedure, the a-viscosity curve is a result of a recalculation of modulus into compliance, subsequent shifting to obtain a master curve, a recalculation back to modulus, computation of the distribution of relaxation times, and, finally, graphic integration. The results, therefore are not expected to be highly accurate. Figures

2337

;"'F

/

I

3.00

I

I

3.10 I O O O I T (OK-')

3.20

I 3.30

Figure 5. Log 7, us. 1000/T for AszSU.: points, experimental; line indicates activation energy of 39.3 kcal.

6 and 7 show these results for both pure sulfur and the AszS16 sample. In both of these cases the curve crosses the q X curve above Tg, the a viscosity predominating a t higher temperatures. The total viscosity, as might be expected, is lower than that due to either of the two mechanisms alone, quite analogous to the case of the two resistors connected in parallel. It should be noted that in these two materials a t the highest accessible temperatures (ca. 50" above T, for AsI2Sl6and 70" above T, for pure sulfur) the extrapolated chemical viscosity is a t least an order of magnitude greater than the total observed viscosity. Thus, the observed viscosity in those temperature regions is almost entirely a reflection of the a process. The total viscosity and the a viscosity in those temperature regions are identical, and the a viscosity as a function of temperature can be obtained simply by an application of the WLF equation to the observed viscosity a t the highest temperatures. The viscosity-temperature plots for the a mechanism obtained by these two procedures agree within experimental error. If, however, a t the highest accessible temperatures 7obsd and q, do not diverge appreciably (which is the case for As2S12), then the viscosity data obtained Volume 71. Number 7 June 1967

A. EISENBERG AND L. A. TETER

2338

c

\

Figure 6. The total viscosity (0)and the calculated viscosities due to the CY (- . - .) and x (- - - -) mechanisms for pure sulfur.

a t the highest temperature cannot be ascribed to qa only. As might be expected, the relation 1 1 _ -- _

rl

71,

+ -1

0

IO

20

30

40

50

8

T-Tg C'( 1

Figure 7. The total viscosity (0)and the calculated viscosities d u e t o t h e a ( - . - . ) a n d x ( - - - - ) mechanisms. The solid line is obtained from the relation 1/q = ( l / q x ) (l/qa).

+

rlx

has been found applicable to all these systems; the total viscosity value for As2Sl6calculated in this manner is also shown in Figure 7. In spite of the fact that the x viscosity is usually the lower one a t T,,the mechanism of the glass transition is undoubtedly short-range motion of chain segments rather than bond interchange. It should be recalled that the extrapolated CY viscosity represents a cooperative movement of the entire chain (as evidenced, for instance, by the molecular weight dependence of the viscosity of normal polymers) rather than that of small segments, but that the latter are of critical importance to the glass transition. , E. Speculations on the Detailed Interchange Mechanism. Three items need to be considered in this section. The first concerns itself with the problem of the identity of the species subject to bond interchange in pure sulfur on the one hand and in the As crosspolymers on the other, the second inquires into whether the interchange reaction proceeds by a free-radical or an ionic mechanism, while the third describes a mechanistic scheme for the interchange reaction. The first problem is easy to settle. Owing to the The Journal of Physical Chemistry

108

similarity of the activation energies for all of the materials from pure sulfur to AstSs, it seems highly probable that S-S bonds are interchanging in all cases. In pure sulfur nothing else is possible, while the probability that an S-As bond interchange reaction should have the same activation energy as an S-S interchange is exceedingly small. The distinction between an ionic and free-radical mechanism is somewhat more difficult. In view of two lines of evidence to be described below, it is highly probable that a free-radical reaction is involved. The evidence follows. (1) Tobolsky, et al.,ls investigated the stress relaxation of cross-linked polysulfide rubbers cured with various recipes, some of which most certainly introduced ionic groups onto the end of some sulfur terminated chains, while one (2,4-toluene diisocyanate plus N-methyl-2-pyrrolidone) did not. It was found that all of the cures which introduced ionic groups yielded relatively low activation energies, ranging around 25 kcal, but the one that introduced no ionic groups yielded an activation energy of 36.6 kcal. While originally this was ascribed to a possible cleavage of

2339

RELAXATION MECHANISMS IN POLYMERIC SULFUR

the isourethane linkage, it seems probable now that a free-radical S-8 interchange predominated in that sample. Additicjnal confirmation comes from the work described below. (2) Recently, Kende, Pickering, and Tobolsky18 investigated the thermal decomposition of dimethyl tetrasulfide by following the disappearance of previously incorporated Banfield free radical. The rate constant for the disappearance reaction yielded an activation energy of 36.6 kcal. Since it is extremely difficult, if not impossible, to postulate an ionic mechanism for the splitting of the dimethyl tetrasulfide and the subsequent reaction with the stable organic free radical, the authors postulated that the decomposition of dimethyl tetrasulfide is a free-radical reaction. It seems to us that, most probably, this is also the case in the bond interchange of pure polymeric sulfur and of sulfur cross-linked with arsenic, since the activation energies are closer to 36 than 25 kcal. With regard to the detailed mechanistic steps, the following sequence of reactions seems reasonable

-s*

-ki

-s-s-

+ se

2-s.

-ss,*

(0 (2)

In this scheme, reaction 4 is an exact reversal of reaction 2, involving a chain end reacting with a ring to prolong the chain or a chain end “biting off” an eightmembered ring and regenerating a free radical. It is clear that rings of various sizes may be present in the sulfur, but this is of little consequence to the present argument since neither reaction 2 nor reaction 4 nor reactions involving small rings of other sizes contribute to stress relaxation. Only reaction 1, which presumably has an activation energy of 36.6 kcal, and reaction 3 are active in stress relaxation, reaction 5 being merely a recombination reaction, which, while possibly introducing stress a t a microscopic level, does not do so macroscopically since the microscopic stresses are introduced in all directions with equal probability. Finally, it should be clear that, in a strict sense, reaction 1 represents the initiation of a kinetic chain and reaction 3 is one of the possibly many identical propagation steps, reaction (5) being the chain termination reaction, both in the sense of the “kinetic chain” and the polymeric chain.

If the rate of reaction 3 is significant with regard to reaction 1, i.e., if sulfur relaxes by a true free-radical chain reaction, then the activation energy should be lower than the observed value of 36.6 kcal which applies to the initiation reaction only, the actual value depending on the values of the activation energies for the propagation and termination steps. In a parallel study of hydrogen- or bromine-terminated sulfur p ~ l y r n e r s ’we ~ have observed that the activation energy of the interchange process decreases with increasing concentration of H or Br, reaching values of ca. 24 and 19 kcal for the hydrogen- and bromine-terminated systems, respectively. It is interesting to note that the values for the activation energy for the interchange reaction in polysulfide rubbers range from 19.4 to 36.6 kcal, the highest value having been obtained for a diisocyanate cure (36.6 kcal), the others ranging around 25 kcal; the low values of the activation energy in the polysulfides were ascribed to the presence of an ionic mechanism, whereas the high value of 36 kcal found in our work seems to be due to a freeradical mechanism, since it is exceedingly similar to the value found by Kende, Pickering, and Tobolsky for the activation energy for the decomposition of dimethyl tetrasulfide and also for the diisocyanatecured polysulfide rubber. It seems reasonable to suppose that the decrease of the activation energy due to the addition of hydrogen and bromine is most probably due to increasing importance of an anionic mechanism. Similarly the wide variations encountered in the activation energies for the polysulfide rubbers might also be explained on the basis of changes of the relative importance of a free-radical and an ionic mechanism. In summary, we can only say that the bond interchange in sulfur and sulfur-arsenic polymers proceeds most probably by a free-radical mechanism in which the initiation reaction seems to be most important; Le., the kinetic chain length is very low. This, however, is only a very preliminary conclusion, and much further work remains to be done. F . Arsenic as a Cross-Link for Sulfur. There are two possible approaches to the evaluation of the cross-linking efficiency of arsenic in polymeric sulfur ; one is based on the assumption that each arsenic atom forms a trifunctional cross-link; the rubbery modulus of the material should thus be a simple function of the arsenic concentration. The second approach starts with the statistical theory of rubber elasticity20 and (18) I. Kende, T. L. Pickering, and A. V. Tobolsky, J . Am. Chen. SOC.,87, 5582 (1965). (19) A. Eisenberg and L. Teter, to be published.

Volume 7 1 , Number 7 June 1967

A. EISENBERG AND L. A. TETER

2340

Table I1

Composition

AS& ASZSI~ AS2S25

S atoms between cross-links calcd from stoichiometry

(from column 2)

P

OK

(from column 3)

4 5 8

128 160 256

2.2 2.2 2.1

319 305 277

1 . 4 X 100 1 . 0 x 100 8 . 7 X lo*

Mc(caicd)

Ecalcdra

TE,

Eexptl

iMc(exptl)*

x x x

1800 2400 5100

1 7 3

108 107 107

Mc(caIcd)/ Mc(expt1)

a E = 3pRZ‘/M,; p = density, weighted average of densities of S and As&; M , = molecular weight between cross-links. equation footnote a using Eexpt1.

attempts to predict the concentration of effective cross-links in the polymer. Utilizing the first method, the average molecular weight between cross-links can be calculated (assuming only simple stoichiometry), and thus also the modulus. The results are shown in Table 11, the resulting values being labeled with the subscript calcd. The second method allows the calculation of some type of effective chain length between cross-links, and these values are also listed in Table I1 being labeled with the subscript exptl. It is seen that the effective M , is 14-20 times larger than that calculated on the basis of simple stoichiometry, indicating that a substantial fraction of the As atoms is not effective as cross-links. There are several possible reasons for this, the most probable being the formation of some types of rings or loops along the chain in which case the As would not act as a cross-link but would merely tend to change the structure of the chain. G. The Rate Constant for the Interchange Reactiun. In a study of the viscoelastic properties of cross-linked polyethylene tetrasulfide polymersz1Tobolsky, Beevers, and Owen suggest the expression for the rate constant as a function of temperature

The Journal

of

Physical Chemistry

14 15 20

From

k = 5.86 X 10l2exp(-25,900/RT) the rate constant being related to the relaxation time by the relation rm = l/km, where m is the number of interchangeable bonds between cross-links. Since we are not sure of the structure of the As-cross-linked polymers, we applied the same reasoning to the pure sulfur sample, the value of m being simply the number of S-S bonds between entanglements tabulated as Mc(exptl) in Table I1 as calculated from the theory of rubber elasticity. The results are

k = 1.74 X 10’’ exp(-33,400/RT)

It is interesting to note that a t 17” the lines interesect, but owing to the different nature of the material and possibly also of the mechanism, the intersection is probably accidental. Acknowledgment. The financial assistance of the Office of Naval Research is gratefully acknowledged. (20) L. R. G. Treloar, “The Physics of Rubber Elasticity,” 2nd ed, Clarendon Press, Oxford, 1958. (21) A. V. Tobolsky, R. B. Beevere, and G . D. T. Owen, J . Colloid Sci., 18, 359 (1963).