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Grüneisen Numbers for Polymeric Solids
1.Yasaku Wada, Conference on Relaxation Phenomena in Polymeric Systems (sponsored by the Institute of Physical and Chemical Research of Tokyo), Tokyo, Japan, September, 1966.
2.R. E. Barker, Jr., J. Appl. Phys. 34, 107 (1963).
3.T. M. Birshtein and O. B. Ptitsyn, Conformations of Macromolecules (Interscience Publishers, Inc., New York, 1966), pp. 1–5.
4.J. H. Gibbs and E. A. Di Marzio, J. Chem. Phys. 25, 185 (1956);
4.J. H. Gibbs and E. A. Di Marzio, J. Chem. Phys. 28, 373 (1958); , J. Chem. Phys.
4.J. H. Gibbs and E. A. Di Marzio, J. Chem. Phys. 43, 139 (1965)., J. Chem. Phys.
5.R. E. Robertson, J. Chem. Phys. 44, 3950 (1966);
5.C. W. Joynson, J. Appl. Phys. 37, 3969 (1966).
6.D. D. Fitts, Ann. Rev. Phys. Chem. 17, 59 (1966).
7.T. H. K. Barron, Phil. Mag. (London) 46, 720 (1955).
8.A. Schauer, Can. J. Phys. 42, 1857 (1964).
9.E. Grüneisen, Geiger‐Scheel Handbuch der Physick, F. Henning, Ed. (Verlag Julius Springer, Berlin, 1962), Vol. X.
10.C. Kittel, Introduction to Solid State Physics (John Wiley & Sons, Inc., New York, 1956), 2nd ed.
11.Y. K. Huang, J. Chem. Phys. 45, 1979 (1966).
12.G. Allen, G. Gee, and G. J. Wilson, Polymer 1, 456 (1960)
12.and G. Allen, G. Gee, and G. J. Wilson, 1, 467 (1960). , Polymer
12.U. Bianchi, G. Agabio, and A. Turturro, J. Phys. Chem. 69, 4392 (1965).
12.U. Bianchi, G. Agabio, and A. Turturro, Since and we have so The Hildebrand [Phys. Rev. 34, 984 (1929)] definition of “free volume” is where is the volume at which
13.This relation is called the Debye equation of state when is the Debye lattice energy, i.e., where and There is confusion over the definition of the Debye function the present usage corresponds to that of Ref. 9 and to the tabulated values of M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, Inc., New York, 1965), p. 998, although they did not explicity use the symbol, In this notation, the energy of a one‐dimensional lattice is The “Debye function” that Wunderlich (Ref. 16) has tabulated is where The Debye function of Tarasov (Ref. 21) is .
14.A. W. Lawson, J. Phys. Chem. Solids, 3, 250 (1957);
14.see also R. E. Howard and A. B. Lidiard, Rept. Progr. Phys. 27, 179 (1964).
15.F. E. Karasz, H. E. Bair, and J. M. O’Reilly, J. Phys. Chem. 69, 2657 (1965).
16.B. Wunderlich, J. Chem. Phys. 37, 1203 (1962);
16.B. Wunderlich, 37, 1207 (1962), , J. Chem. Phys.
16.and B. Wunderlich, J. Polymer Sci. C1, 41 (1963).
17.Wunderlich (Ref. 16) has estimated that for PE, the betweenchain/intrachain energy ratio is For PE, Wada calculated the elastic constants ratio
18.Wavenumbers in are often used to specify the lattice frequencies but since refers to the number of vibrations in a time interval of 33.3364 psec the actual wavenumbers for ultrasonic or phonon waves are given by
19.S. C. Saxena and C. M. Kachhava, Am. J. Phys. 34, 704 (1966).
20.A. A. Maradudin, E. W. Montroll, and G. H. Weiss, Solid State Physics, F. Seitz and D. Turnbull, Eds. (Academic Press Inc., New York, 1963), p. 44 ff.
21.V. V. Tarasov, Israel Program for Scientific Translations (Distributed by Oldbourne Press, London, 1963), p. 41. Also see numerous references therein to Tarasov’s important contributions to the problems of heat capacities in polymers.
22.J. C. Slater, Phys. Rev. 57, 744 (1940).
23.C. Weir, J. Res. Natl. Bur. Std. 46, 207 (1951);
23.C. Weir, 50, 311 (1953); , J. Res. Natl. Bur. Stand.
23.and C. Weir, 53, 245 (1954)., J. Res. Natl. Bur. Stand.
24.The widely discussed “melting equations” of E. A. Kraut and G. C. Kennedy [Phys. Rev. Letters 16, 608 (1966)]
24.can be put into a convenient quasi‐universal form (for metals) by using Eq. (19b) and the results of Vaidya and Gopal [Phys. Rev. Letters, 17, 635 (1966)].
24.We obtain , where is the melting point at p.
25.W. C. Overton, Jr., J. Chem. Phys. 37, 116 (1962), has found, for metals, that by applying thermodynamic rules to a relation similar to Eq. (18). He shows how to obtain from ultrasonic data.
26.V. S. Nanda and R. Simha, J. Chem. Phys. 41, 1884 (1964).
27.R. E. Hanneman and H. C. Gatos, J. Appl. Phys. 36, 1794 (1965).
28.When the periodic table is used to try to connect related sequences of elements in Fig. 5 it is found that there are loops and complicated crossovers, especially near the bottom of the plot.
29.J. C. Slater, Chemical Physics (McGraw‐Hill Book Co., Inc., New York, 1939), p. 220.
30.C. M. Kachhava and S. C. Saxena, Indian J. Phys. 40, 273 (1966).
31.P. J. Flory, R. A. Orwoll, and A. Vrij, J. Am. Chem. Soc. 86, 3507 (1964);
31.P. J. Flory, J. Am. Chem. Soc. 87, 1833 (1965);
31.and A. Abe and P. J. Flory, J. Am. Chem. Soc. 87, 1838 (1965).
32.E. A. Moelwyn‐Hughes, Physical Chemistry (Pergamon Press, Inc., New York, 1961), 2nd ed., pp. 306, 319–328. Many specific exceptions to Eq. (33) exist, e.g., the diatomics: and NO; the exponents are (instead of 6) and
33.E. A. Power, W. J. Meath, and J. O. Hirschfelder, Phys. Rev. Letters 17, 799 (1966).
34.A. G. DeRocco and W. G. Hoover, Proc. Natl. Acad. Sci. USA 46, 1057 (1960).
35.A. Peterlin, E. W. Fischer, and C. Reinhold, J. Chem. Phys. 7, 1403 (1962).
36.L. Salem, Nature 193, 476 (1962)
36.and L. Salem, J. Chem. Phys. 37, 2100 (1962).
37.R. Zwanzig, J. Chem. Phys. 39, 2251 (1963).
38.R. L. McCullough and J. J. Hermans, J. Chem. Phys. 45, 1941 (1966).
39.The extra term in the general form of Eq. (51) is due to the relations ; ; when Eq. (51) results.
40.The value of Karasz et al. for is used. The are in rough accord with the energy based on Wada’s boundary frequency.
41.J. I. Zaken, R. Simha, and H. C. Hershey, J. Appl. Polymer Sci. 10, 1455 (1966).
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