EQUATION OF STATE FOR RCBBER ELASTICITY 13.5 A has been calculated' for the latter using a formula proposed by Taylor,25which takes into account the restricted rotation about the C-C bonds of the hydrocarbon chain. The radial lengths of the polyoxyethylene regions, re, calculated by subtraction of ?"h from the total radius of the hydrated micelles, are given in Table 111. The gradual increase in re with increasing temperature continues the trend observed with the micelles below the threshold temperature,' and indicates a
345 gradual extension of the polyoxyethylene chain toward its fully extended length (26 A). Acknowledgments. The author is indebted to Professor P. H. Elworthy for his advice and encouragement, and wishes to thank Air, G. Cochrane for constructing the light-scattering apparatus and Dr. J. E. Mathews for advice during its development. (26) W. Taylor, J . Chem Phys., 16, 257 (1948).
Volume Dependence of the Equation of State for Rubber Elasticity:
Poly (dimethyl siloxane)
by A. V. Tobolsky and L. H. Sperling Department of Chemistry, Princeton University, Princeton, New Jersey
(Received May 22, 1967)
The exact equation of state for rubber elasticity has been the subject of much recent work, both theoretical and experimental. In order to evaluate the volume dependence suggested in recent equations,'?6 a swelling technique has been employed. I n this manner the value of the volume dependence parameter y has been estimated to be +0.33 for poly(dimethy1 siloxane) swollen with silicone oil. The significance of previous thermoelastic work on this polymer is discussed in the light of the present result.
Introduction It has recently been suggested that the equation of state for rubber elasticity should be somewhat modified as'
Equation 1 is such that it should also be applicable to stretched swollen rubbers. In this case, the symbols of eq 1 are defined as follows: f equals force, LO equals original unstretched unswollen length at 1 atm, Vo equals original unstretched unswollen volume at 1 atm, and L and V refer to the final length and volume of the stretched swollen sample a t 1 atm. The quantity y is a new empirical parameter suggested in ref 1; if this is zero, eq 1 is the same as prior equations.2-6 The quantity C is a function of temperature but not of volume. Its molecular interpretation in terms of network parameters is well known, especially if y is zero. (See Appendix I.) Independent of detailed molecular interpretation of C, y can be regarded as a parameter which permits a volume-dependent front factor in eq
1; it was introduced on an empirical basis to be evaluated by experiment. In the experiments to be discussed below, it will be assumed that the unstretched swollen volume V' is very nearly the same as V,the stretched swollen volume, a t least when compared to the difference between Vo and V . (In deriving eq 1 it is assumed that dilational or hydrostatic pressure is applied as necessary a t all times during the stretching process, so that the stretching always occurs a t volume V . The original swelling may be regarded as similar to dilational pressure.) I n eq 1, the measurable quantities aref, LO,L, Vo, and T. As stated above, V' is assumed essentially equal to V for numerical purposes, and V' is readily mea(1) A. V. Tobolsky and M. C. Shen, J . Appl. Phys., 37, 1952 (1966). (2) P.J. Flory, Trans. Faraday SOC.,57, 829 (1961). (3) A. V. Tobolsky, D. W. Carlson, and N. Indictor, J . Polymer Sci., 54, 178 (1961). (4) (a) W.R. Krigbaum and R. J. Roe, Rubber Chem. Technol., 38, 1039 (1965);(b) G.Gee, Polymer, 1, 373 (1966). (6) L. H.Sperling and A. V. Tobolsky, J . Mucromol. Chem., 1, 799 (1966). Volume 72,Number 1 January 1068
A. V. TOBOLSKY AND L. H. SPEXLING
346
sured. It will be shown below that the quantity C is easily eliminated in the evaluation of y. (Note small corrections discussed in Appendix I.) The system poly(dimethy1 siloxane)-silicone oil was chosen for the first examination because: (1) we have previously studied the thermoelastic properties of this elastomer;6 and ( 2 ) silicone oil is an athermal,6 relatively nonvolatile swelling agent for silicone rubber. joy tV/V,).
Experimental Details, Results, and Discussion Silicone oils of two differing viscosities (50 and 2 cSt) were used to make samples with VIVOranging from 1.4 to 4.6 for the swollen polymer. The samples were swollen to equilibrium for the first observations. Subsequently, the samples were partly deswollen and the measurements were repeated. The observations were made in air, with a small amount of silicone oil placed in a standard relaxation box7 to retard sample evaporation. A highly swollen polymer in equilibrium with solvent will absorb more solvent when stretched,8 but over-all weight gains were not observed in these experiments. The unswollen polymer was also studied. All of the polymers were cut in the form of narrow rectangular strips. Stress-time curves at constant length were determined near room temperature, approximately 22”. Extensions of L/LoE 1.1were employed. I n all cases a value of equilibrium stress was obtained with the method of Chasset and Thirion.9~~0By use of this method, unswollen samples were observed for several days until the equilibrium stress was nearly attained; swelling much reduced the time needed for a reasonable approach to equilibrium. In each case, dimensions of the samples were measured and the samples were weighed before and after relaxation. This latter procedure was a precaution against slight evaporation. While weight losses were usually small, after very long times the stress of swollen samples appeared to increase slightly; this was attributed to sampIe shrinkage and was a limitation to the technique employed. The calculation of y was carried out in the following manner. We consider samples having the same value of V o and LO, though swollen to different extents. Starting with eq 1, it is convenient to introduce a new par amet er A =
J
KO ;OH]
c ---
where the quantity A. may be employed to represent the value of A for the unswollen polymer. Division of A by A. yields t’he ratio (3) The Journal of Physical Chemistry
) Figure 1. Plot of log ( & / A ) us. log ( V I V O for poly(dimethy1 siloxane) rubber swollen by silicone oil.
after noting that (VJVO)’ = 1. Equation 3 suggests that a plot of log (Ao/A) us. log (V/Vo)would yield y directly as the slope of the expected straight line. Figure 1 shows a plot of log (Ao/A)us. log (VIVO)for poly(dimethy1 siloxane) elastomer swollen with silicone oil. The slope yields y = 0.33 i 0.04. These determinations were made a t L/Lo = 1.1 and should be extended to other values of L/Lo. A nonzero value of y affects the interpretation of thermoelastic datalJ (stress-temperature measurements at constant length). This type of measurement can be related to the variation of the mean-square endto-end distance of the equivalent free chain a t volume VO,qvo, with temperature provided y is known. Forpoly(dimethylsiloxane), e’ = R d In $,,/d(l/T) can now be computed, Using data in the literature,6 E’ = -28 cal/mole. This indicates a very small energy difference between rotational isomers in the poly(dimethyl siloxane) chain, if interdependence of bond rotational states is (cautiously) neglected.11112 The value of the present experiment lies in the relative ease of evaluating y from eq 1. The change of volume with elongation or with pressure is very small,13914but with swelling the volume can be altered at will over a considerable range. There have been numerous studies on the elastic properties of swollen polymers.l5-1* However, it (6) J. E.Mark and P. J. Flory, J . Am. Chem. Soc., 86, 138 (1964). (7) A. V. Tobolsky, “Properties and Structure of Polymers,” John
Wiley and Sons, Inc., New York, N. Y.,1960,p 143. (8) T. L. Hill, “An Introduction to Statistical Thermodynamics,”’ Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960,p 414. (9) R. Chasset and P. Thirion, “Physics of Non-Crystalline Solids,” North-Holland Publishing Co., Delft, 1964, p 345. (10) L.H. Sperling and A. V. Tobolsky, to be published. (11) A. Ciferri, Trans. Faraday Soc., 57, 846,853 (1961). (12) U. Bianchi, E. Patrone, and M. Galpiaz, Makromol. Chem., 84, 230 (1965). (13) G. Gee, J. Stern, and L. R. G. Treloar, Trans. Faraday Soc., 46, 1101 (1950). (14) F. G. Hewitt and R. L. Anthony, J. Appl. Phya., 29, 1411 (1958). (15) G. Gee, Trans. Faraday SOC.,42, 584 (1946). (16) M. C. Shen and A. V . Tobolsky, J . Polymer Sci., AZ, 2513 (1964).
EQUATION OF STATEFOR RUBBER ELASTICITY
347
appears to be difficult to make a quantitative analysis of y from this literature. The volume dependence of the front factor in eq 1 has been introduced in the form (Vo/V)y. This particular form makes calculation of many of the thermoelastic properties relatively simple. Although y has been found to be a constant for poly(dimethy1 siloxane) elastomer, there is no guarantee that the volume dependence of the front factor will be equally simple for other polymers. An elegant extension of the volume dependent front factor was given by Ciferri, Smith, and Bashaw,Ig who pointed out that y can be expressed in a more general manner. They wrote the equation of state in the form
network chain in the unswollen polymer may be considered as having 8 solvent unperturbed dimensions, whereas a network chain in the polymer highly swollen by oligomer may be considered to be in a nearly athermal solvent . 2 0 ~2 1 In solution, the mean-square end-to-end distance of polymer chains compared to the unperturbed distance (in a 8 solvent) is given by20
r2/ro2= cy2 (9) where cy is the swelling or expansion factor. If the solvent is a low molecular weight species of the polymer, the following equation has been suggestedz1 for dilute solutions as -
cy’
=
KZ”’/X,
(10)
where all terms except B have the same meaning as in eq 1. The quantity B was assumed to depend upon both V and 1,. Applying the well-known thermoelastic relation, it was shown that
where K 1, Z is the degree of polymerization of the linear polymer, and X , is that of the solvent. It would be very interesting if an equation for cy could be developed for concentrated solutions and swollen networks. Such a relation could then be applied to eq
where f e is the energetic component of the stress and = (dV/bT)p,v/V. Combining eq 4 and 5 yields
Acknowledgments. We wish to acknowledge the support of the Office of Naval Research. We also wish to thank Dr. A. C. Martellock of the General Electric Company, Waterford, N. Y., for the preparation of the poly(dimethy1 siloxane) elastomer?
(4)
p
& f
=
-T
[
In
(flT) 2)T
]P.L
-
[y7 pT
-
+
(-bb InIn BV)
pT
(6)
L,T
Appendix I Further treatment of the parameter C is desirable here from a theoretical point of view, and also to show how to handle the data if VOand LOare not exactly the same in all samples. A particular molecular interpretation of the parameter C gives the following’s6
Comparison with ref 1 shows
(7) where y need no longer be constant. If the Mooney equation is applicable, Ciferri, Smith, and Bashaw have also derived a relation among y and the Mooney constants CIand CZ. A Molecular Interpretation of y. We present here an interpretation of y which emphasizes polymer-solvent interactions during swelling. Equation 1 may be written in a more detailed molecular form, applicable to networks under pressure (plus or minus), or to swollen networki3.’I6 f =
(L)(-)[z L E]
N0kT -rizv0 LO rf2vo
(8)
L2 Vo
The quantity ~ v o /which ~ v appears in eq 8 is the only term that .relates to y. It is interesting to consider that this quantity may be a function of volume because of the varying solvent-polymer interactions at different degrees of swell. It has been suggested that a
where N o equals the number of network chains in the equals Boltzmann’s constant times temperasample, LTture, and rrZvoequals the mean-square end-to-end distance of the actual network chains at volume Vo. Lo and rfZy,were defined in the text. Slight variations in LOand V Ooccurred. Noting that V ois proportional to No
-
-
C=
constant X VokT r&, -
_ I
Lo
rf2vo
(A2)
(17) A. Oplatka and A. Katchalsky, Makromol. Chem., 92, 251 (1966). (18) P. J. Flory, Chem. Rev., 35, 51 (1944). (19) A. Ciferri, K. J. Smith, Jr., and R. N. Bashaw, private communication. (20) P. J. Flory, “Principles of Polymer Chemistry,” Cornel1 University Press, Ithaca, N. Y.,1953. (21) P. J. Flory, J . Chem. Phys., 17, 303 (1949). (22) NOTEADDEDIN PROOF. Utiliaing eq 26’ of ref 21, we have d e rived a value af y = 0.40 for a special case. Volume 74,Number 1 January 1088
H.B. PALMER, J. LAHAYE, AND K. C. Hou
348
If Loand V oare constant, the value of C appearing in eq 2 and 3 is a constant. Slight variations in LOand V O
affect C as shown in eq A2. This variation of accounted for in eq 2 and 3.
C can be
On the Kinetics and Mechanism of the Thermal Decomposition of
Methane in a Flow System’ by H. B. Palmer, J. Lahaye, and K. C. Hou Department of Fuel Science, The Pennsylvania State University, Univereity Parlc, Pennsylvania (Received July $1, 1067)
16802
The rate of thermal decomposition of methane has been studied in a flow system at temperatures from 1323 to 1523’K. The experimental data are generally in close agreement with a recent, similar study by Eisenberg and Bliss. However, a different interpretation is given to the results. I n particular, it is concluded that nucleation of carbon in the gas phase causes the decomposition to accelerate because of heterogeneous decomposition of methane on the nuclei. The conclusion is supported experimentally. A qualitative model of the pyrolysis process is presented and discussed.
Introduction The kinetics and mechanism of the homogeneous thermal decomposition of methane should be completely understood by now, but they are not. Decomposition in more or less conventional static and flow systems has been studied for many years. Kramer and Happe12 have reviewed a number of those studies. Results of three of them are included in Figure 1, an Arrhenius plot of some reported first-order rate constants. The line through the results from flow and static systems is expressed by log Ic(sec-l) = 13.0
- 18.6 X
10*/T
(1)
corresponding to an activation energy of 85 kcal. Shock tube dataJ8v4when combined with the results of measurements of the rate of carbon film formation from pyrolyzing methaneJ6yield a quite different result log k(sec-’) = 14.6 - 22.5 X 10a/T
(2)
corresponding to an activation energy of 103 kcal. This line is also shown in Figure 1, as is the result from rapid compression experiments reported recently by Kondratiev,%which agrees well with eq 2. In the hope of understanding the discrepancy between these expressions, we have carried out some new measurements of the thermal decomposition rate using a conventional flow system. Since doing so, we have become aware of a recent and detailed study of EisenThe Journal of Physical ChemMtry
berg and Bliss7 (hereafter referred to as (EB)) in which a similar flow reactor was used. We shall refer to their work throughout this paper.
Experimental Section The experimental system has been described elsewhere.s-10 It is basically a hot porcelain tube of 5-mm i.d., having a long plateau in its temperature profile. Methane at concentrations in the neighborhood of 10% by volume is carried into the reactor in a stream of helium a t a total pressure of about 740 torr. Products are analyzed by gas chromatography using a 5-ft silica gel column at 50”. (1) Work supported in part by a grant from the J. M. Huber Corp. (2) L. Kramer and J. Happel, “The Chemistry of Petroleum Hydro-
carbons,” Vol. 11, B. T. Brooks, 8. S. Kurtz, C. E. Boord, and L. Sohmerling, Ed., Reinhold Publishing Corp., New York, N. Y., 1955, Chapter 25, p 71. (3) (a) G. B. Skinner and R. A. Ruehrwein, J. Phys. Chem., 63, 1736 (1959); (b) H. 8. Gliok, “Seventh Symposium (International) on Combustion,” Butterworth and Co. Ltd., London, 1959, p 98. (4) V. Kevorkian, C. E. Heath, and M. Boudart, J. Phye. Chem., 64, 964 (1960). (5) H. B. Palmer and T. J. Hirt, ibid., 67,709 (1963). (6) V. N. Kondratiev, “Tenth Symposium (International) on Combustion,” The Combustion Institute, Pittsburgh, Pa., 1965, p 319. (7) B. Eisenberg and H. Bliss, Chem. Eng. Progr., Symp. Ser., 63,
No. 72, 3 (1967). (8) H. B. Palmer and F. L. Dormish, J. Phys. Chem., 68, 1563 (1964). (9) K. C. Hou and H. B. Palmer, ibid., 69, 858 (1965). (10) K. C. Hou and H. B. Palmer, ibid., 69, 863 (1965).
