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Statistical Thermodynamics of Rubber Elasticity
1.F. T. Wall and P. J. Flory, J. Chem. Phys. 19, 1435 (1951).
2.H. M. James and E. Guth, J. Chem. Phys. 11, 455 (1943).
3.H. M. James and E. Guth, J. Chem. Phys. 15, 669 (1947).
4.H. M. James and E. Guth, J. Polymer Sci. 4, 153 (1949).
5.In reference 1, Wall and Flory have attributed to us ideas that we have not proposed, and have described parts of our theory in a way that appears to us to be quite misleading. We shall not here cite and answer all these points in detail, hoping that persons interested in our theory will read our own presentation of it.
6.Neglect of these interferences is perhaps the most unsatisfactory feature of our model, since they actually reduce by a large factor the number of configurations accessible to the network. In Appendix A of reference 4 we have given our reasons for believing that in soft rubber‐like materials, not too highly extended, these interferences reduce the number of configurations by a constant factor, and change the computed entropy only by an unimportant constant. On the other hand, the Gaussian network model is quite unsuited to any attempt to calculate the internal energy U, which enters our theory of ideal rubber‐like materials as an unspecified function of volume and temperature, independent of the form of the material, as with a liquid.
7.F. T. Wall, J. Chem. Phys. 10, 485 (1942).
8.Kuhn, Pasternak, and Kuhn, Helv. Chim. Acta 30, 1705 (1947).
9.P. J. Flory and J. Rehner, Jr., J. Chem. Phys. 11, 512 (1943).
10.P. J. Flory and J. Rehner, Jr., J. Chem. Phys. 11, 521 (1943).
11.Isihara, Hashitsuma, and Tatibana, J. Chem. Phys. 19, 1508 (1951).
12.Ming Chen Wang and E. Guth, J. Chem. Phys. 20, 1144 (1952).
13.This is essentially the method used in our calculation of the increase in rigidity during cure, in reference 3.
14.Compare, for instance, the calculations of the pressure exerted by a classical ideal gas, based on consideration of (a) the actual motions of the gas molecules in a particular case, (b) all possible motions of the gas molecules, taken with appropriate weights, and (c) the change of entropy of the gas with volume.
15.F. T. Wall, J. Chem. Phys. 10, 132 (1942).
16.F. T. Wall, J. Chem. Phys. 11, 527 (1943).
17.Our result can be written down immediately as a special case of general results obtained by a quite different method: H. M. James, J. Chem. Phys. 15, 651 (1947).
18.Reference 17, Eq. (7.5).
19.The difference between distributions of instantaneous extensions and of time‐average extensions is well illustrated by the linear network considered in this section. The instantaneous extensions have a displaced Gaussian distribution, but the time‐average extensions all have the same value,
20.More precisely, each chain is to be treated as able to assume a number of configurations proportional to
21.See, for instance, reference 2, Sec. 6.
22.See for instance, J. E. Mayer and M. G. Mayer, Statistical Mechanics (John Wiley and Sons, Inc., New York, 1940) pp. 78–86.
23.P. J. Flory, J. Chem. Phys. 18, 108 (1950).
24.In Sec. VI we shall apply Flory’s method of calculation to obtain this result. As discussed there, we do not believe Flory’s calculations are correct for functionalities greater than 2.
25.Our N replaces ν of Flory’s notation. There is another term in the entropy of swelling where n is the number of solvent molecules) that is not included by Wall and Flory or in the present discussion.
26.There appears to be some confusion in regard to this term in the paper of Wall and Flory. They refer to reference 23 in justifying introduction of the additional term in the entropy. This would correspond to assuming yet they describe the term as arising from the “requirement that the selected units meet in pairs.” Since there are N chains, there will be 2N chain ends, and one would expect the requirement that these meet in pairs to introduce a term in the entropy; Wall and Flory’s formula would then be identical with our own.
27.W. Kuhn and F. Grün, J. Polymer Sci. 1, 183 (1946).
28.W. Kuhn and H. Kuhn, Helv. Chim. Acta 29, 1615 (1946).
29.Kuhn, Pasternak, and Kuhn, Helv. Chim. Acta 30, 1705 (1947).
30.In this calculation performs a double function, as the range used in defining the and as a range proportional to the range of extensions possible for an individual chain. It is the second of these roles that fixes the variation of with α.
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