No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
Crystallization of polyethylene and polytetrafluoroethylene by density‐functional methods
1.See, for instance, L. Mandelkern, Crystallization of Polymers (McGraw-Hill, New York, 1964);
1.B. Wunderlich, Macromolecular Physics (Academic, New York, 1973), Vol. 1.
2.T. V. Ramakrishnan and M. Yussouff, Phys. Rev. B 19, 2775 (1979).
3.A. D. J. Haymet and D. W. Oxtoby, J. Chem. Phys. 74, 2559 (1981);
3.A. D. J. Haymet, J. Chem. Phys. 78, 4641 (1983); , J. Chem. Phys.
3.B. B. Laird, J. D. McCoy, and A. D. J. Haymet, J. Chem. Phys. 87, 5451 (1987)., J. Chem. Phys.
4.P. Tarazona, Mol. Phys. 52, 81 (1984);
4.W. A. Curtin and N. W. Ashcroft, Phys. Rev. A 32, 2909 (1985);
4.A. R. Denton and N. W. Ashcroft, Phys. Rev. A 39, 4701 (1989)., Phys. Rev. A
5.M. Baus, J. Phys. Condensed Matter 1, 3131 (1989).
6.C. Marshall, B. B. Laird, and A. D. J. Haymet, Chem. Phys. Lett. 122, 320 (1985).
7.J. D. McCoy, S. W. Rick, and A. D. J. Haymet, J. Chem. Phys. 90, 4622 (1989);
7.J. D. McCoy, S. W. Rick, and A. D. J. Haymet, J. Chem. Phys. 92, 3034 (1990); , J. Chem. Phys.
7.S. W. Rick, J. D. McCoy, and A. D. J. Haymet, J. Chem. Phys. 92, 3040 (1990)., J. Chem. Phys.
8.K. Ding, D. Chandler, S. J. Smithline, and A. D. J. Haymet, Phys. Rev. Lett. 59, 1698 (1987).
9.J. D. McCoy, K. G. Honnell, K. S. Schweizer, and J. G. Curro, Chem. Phys. Lett. 179, 374 (1991). In this earlier work we used a more primitive differential form of the flexibility correction. The numerical results are nearly identical.
10.P. J. Flory, Proc. R. Soc. London, Ser. A 234, 60 (1956).
11.For reviews, see R. Evans, in Les Houches Summer Lectures, Session XLVIII, Proceedings of the Les Houches Summer School, Session 48, edited by J. Chavrolin, J. F. Joanny, and J. Z.nn-Justin (Elsevier Science, New York, 1989);
11.A. D. J. Haymet, Annu. Rev. Phys. Chem. 38, 89 (1987);
11.M. Baus, J. Phys. Condensed Matter. 2, 2111 (1990).
11.For a recent comparison of the monatomic methods, see A. de Kuijer, W. L. Vos, J.-L. Barrat, J. P. Hansen, and J. A. Schouten, J. Chem. Phys. 93, 5187 (1990).
12.D. Chandler, J. D. McCoy, and S. J. Singer, J. Chem. Phys. 85, 5971 (1986);
12.D. Chandler, J. D. McCoy, and S. J. Singer, 85, 5977 (1986)., J. Chem. Phys.
13.J. D. McCoy, S. J. Singer, and D. Chandler, J. Chem. Phys. 87, 4953 (1987).
14.W. E. McMullen and K. F. Freed, J. Chem. Phys. 92, 1413 (1990);
14.H. Tang and K. F. Freed, J. Chem. Phys. 94, 1572 (1991)., J. Chem. Phys.
15.J. Melenkevitz and M. Muthukumar, Macromolecules 24, 4199 (1991).
16.J. V. Selinger and R. F. Bruinsma, Phys. Rev. A 43, 2910 (1991);
16.J. V. Selinger and R. F. Bruinsma, 43, 2922 (1991)., Phys. Rev. A
17.C. E. Woodward, J. Chem. Phys. 94, 3183 (1991).
18.R. P. Feynman, Statistical Mechanics: A Set of Lectures (Benjamin, Reading, 1971);
18.D. Chandler and P. G. Wolynes, J. Chem. Phys. 74, 4078 (1981).
19.Mass renormalization was discussed in the quantum DF work of Ref. 7, but without resolution.
20.K. G. Honnell, J. D. McCoy, J. G. Curro, K. S. Schweizer, A. H. Narten, and A. Habenschuss, J. Chem. Phys. 94, 4659 (1991).
21.D. Chandler and H. C. Andersen, J. Chem. Phys. 57, 1930 (1972);
21.D. Chandler, in Studies in Statistical Mechanics VIII, edited by E. Montroll and J. Lebowitz (North-Holland, Amsterdam, 1982), p. 274.
22.D. Chandler, Y. Singh, and D. Richardson, J. Chem. Phys. 81, 1975 (1984);
22.A. Nichols III and D. Chandler, J. Chem. Phys. 84, 398 (1986)., J. Chem. Phys.
23.K. S. Schweizer and J. G. Curro, Phys. Rev. Lett. 58, 246 (1987);
23.K. G. Honnell, J. G. Curro, and K. S. Schweizer, Macromolecules 23, 3496 (1990), and references cited therein.
24.Here, the sites cannot be taken to be completely localized as in the RIS model. Instead, the degree of localization is one of the minimization parameters in the density-functional theory.
25.M. V. Volkenstein, Configurational Statistics of Polymeric Chains (Interscience, New York, 1963);
25.P. J. Flory, Statistical Mechanics of Chain Molecules (Wiley, New York, 1969).
26.(a) A. Verma, W. Murphy, and H. J. Bernstein, J. Chem. Phys. 60, 1540 (1974);
26.J. R. Durig and D. A. Compton, J. Phys. Chem. 83, 265 (1979);
26.D. A. Compton, S. Montero, and W. F. Murphy, J. Phys. Chem. 84, 3587 (1980). , J. Phys. Chem.
26.See also I. Kanesaka, R. G. Snyder, and H. L. Strauss, J. Chem. Phys. 84, 395 (1986).
26.(b) The apparent renormalization of in going from the gas to the liquid has previously been considered theoretically for short n-alkanes;
26.see L. R. Pratt, C. S. Hsu, and D. Chandler, J. Chem. Phys. 68, 4802 (1978), and references therein.
27.T. W. Bates, in Fluoropolymers, edited by L. A. Wall (Wiley, New York, 1972).
28.J.-P. Ryckaert and A. Bellemans, Faraday Discuss. Chem. Soc. 66, 95 (1978).
28.Similar values for σ are obtained from other Lennard-Jones potentials: see, for instance, W. L. Jorgensen, J. D. Madura, and C. J. Swenson, J. Am. Chem. Soc. 106, 6638 (1984). The Lennard-Jones parameters were optimized to describe n-butane behavior near room temperature.
29.J. D. Weeks, D. Chandler, and H. C. Andersen, J. Chem. Phys. 54, 5237 (1971);
29.also, see Ref. 23 for the polymeric implementation.
30.G. L. Slonimskii, A. A. Askadskii, and A. I. Kitaigorodskii, Polym. Sci. USSR 12, 556 (1970).
31.D. Y. Yoon and P. J. Flory, Macromolecules 9, 294 (1976).
32.We have also compared the calculated structure factor to scattering results for a number of n-alkanes in K. G. Honnell, J. D. McCoy, J. G. Curro, and K. S. Schweizer (unpublished). We discuss the - intermolecular potential further here.
33.J. D. McCoy, K. G. Honnell, J. G. Curro, K. S. Schweizer, and D. Honeycutt, Macromolecules (submitted, 1991).
34.R. Koyama, J. Phys. Soc. Jpn. 34, 1029 (1973).
35.Crystalline polymer chains occupy space more like cylinders than like spheres. Hence, significantly higher crystalline packing fractions are possible in polymeric systems than in atomic systems.
36.J. D. McCoy, R. McRae, and A. D. J. Haymet, Chem. Phys. Lett. 169, 549 (1990);
36.R. McRae, J. D. McCoy, and A. D. J. Haymet, J. Chem. Phys. 93, 4281 (1990).
37.P. R. Swan, J. Polym. Sci. 56, 403 (1962).
38.R. A. Orwoll and P. J. Flory, J. Am. Chem. Soc. 89, 6814 (1967);
38.P. Zoller, J. Polym. Sci. Polym. Phys. Ed. 18, 897 (1980).
39.D. T. Grubb, Macromolecules 18, 2282 (1985).
40.B. Wunderlich and G. Czornj, Macromolecules 10, 906 (1977).
41.L. Mandelkern, A. Prasad, R. G. Alamo, and G. M. Stack, Macromolecules 23, 3696 (1990).
42.L. Mandelkern, An Introduction to Macromolecules, 2nd ed. (Springer-Verlag, New York, 1983), p. 87.
43.Y. P. Khanna, G. Chomyn, R. Kumar, N. S. Murthy, K. P. O’Brien, and A. C. Reimschuessel, Macromolecules 23, 2488 (1990).
44.J. H. Gibbs and E. A. DiMarzio, J. Chem. Phys. 28, 373 (1958).
45.J. F. Nagle, P. D. Gujrati, and M. Goldstein, J. Phys. Chem. 88, 4599 (1984).
46.The difference in isothermal compressibility [equivalently or between a purely hard-core system and one with soft attractions can be estimated in several ways. For small molecules an effective hard-sphere diameter can be calculated and employed with PY theory to compute the bulk compressibility. When such a calculation is compared with experiment for a wide variety of dense molecular liquids, one finds [K. S. Schweizer and D. Chandler, J. Chem. Phys. 76, 2296 (1982)]
46.that An alternative approach is to utilize the MSA (mean-spherical approximation) closure (or, more accurately, the optimized random phase approximation [H. C. Andersen and D. Chandler, J. Chem. Phys. 57, 1918 (1972)]) for fluids interacting via a Lennard-Jones (LJ) potential which leads to the relation where the subscript “HS” refers to the corresponding hard-sphere fluid, and is the integrated strength of the attractive branch of the LJ potential which can be estimated from known LJ parameters. Calculations based on the above formula also leads to a result of the form where at high density or equivalently
47.L. Verlet, Phys. Rev. 163, 201 (1968).
48.M. L. Huggins, Ann. N.Y. Acad. Sci. 43, 1 (1942). The fraction of gauche depends on lattice type. The value quoted here is for a cubic lattice.
49.Radii of gyration can be found in Polymer Handbook, 2nd ed., edited by J. Brandrup and E. H. Immergut (Wiley, New York, 1975);
49.and chain diameters, from Ref. 30.
50.The aspect ratio is for these lattices if one assumes that the only excluded volume interaction is that two adjacent bonds can not coincide.
51.A. J. Lovinger, D. Davis, F. C. Schilling, F. J. Padden, F. Bovey, and J. M. Zeigler, Macromolecules 24, 132 (1991).
52.See, for example, P. Flory, Adv. Polym. Sci. 59, 2436 (1984);
52.G. Ronca and D. Yoon, J. Chem. Phys. 76, 3295 (1982).
53.D. Frenkel, B. M. Mulder, and J. P. McTague, Phys. Rev. Lett. 52, 287 (1984);
53.D. Frenkel, J. Phys. Chem. 92, 3280 (1988).
54.Of course, for any discrete RIS-like model freezing will be predicted to occur at sufficiently low temperatures; the use of heavily coarse-grained “continuous”-type models such as the Gaussian or freely jointed chain is clearly inappropriate at low temperatures where the real polymer chain is very stiff.
55.B. J. Alder and T. E. Wainwright, J. Chem. Phys. 31, 459 (1959).
56.S. J. Singer and R. Mumaugh, J. Chem. Phys. 93, 1278 (1990).
57.S. M. Bhattacharjee, Phys. Rev. B 34, 1624 (1986).
58.The effect of pressure is not addressed in the Flory analysis; however, one can always try to incorporate pressure into the coarse-graining procedure.
59.D. S. Gaunt, J. Phys. Chem. 46, 3237 (1967);
59.J. Orban and A. Bellemans, J. Phys. Chem. 49, 363 (1968); , J. Phys. Chem.
59.L. K. Runnels, J. P. Salvant, and H. R. Streiffer, J. Phys. Chem. 52, 3252 (1970); , J. Phys. Chem.
59.for a review, see L. K. Runnels, in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic, London, 1972), p. 305.
60.J. F. Nagle, Proc. R. Soc. London, Ser. A 337, 569 (1974).
61.P. D. Gujrati, J. Phys. A 13, L437 (1980);
61.P. D. Gujrati and M. Goldstein, J. Chem. Phys. 74, 2596 (1981);
61.P. D. Gujrati, J. Stat. Phys. 28, 441 (1982);
61.J. F. Nagle, J. Stat. Phys. 38, 531 (1985); , J. Stat. Phys.
61.J. Suzuki and T. Izuyama, J. Phys. Soc. Jpn. 57, 818 (1988).
62.A. Baumgartner and D. Y. Yoon, J. Chem. Phys. 79, 521 (1983);
62.D. Y. Yoon and A. Baumgartner, Macromolecules 17, 2864 (1984);
62.R. G. Petschek, J. Chem. Phys. 81, 5210 (1984);
62.A. Baumgartner, J. Chem. Phys. 84, 1905 (1986); , J. Chem. Phys.
62.R. Boyd, Macromolecules 19, 1128 (1986);
62.A. Kolinski, J. Skolnick, and R. Yaris, Macromolecules 19, 2560 (1986); , Macromolecules
62.S. C. Mathur and W. L. Mattice, Macromolecules 21, 1354 (1988); , Macromolecules
62.W. G. Madden, J. Chem. Phys. 88, 3934 (1988);
62.G. F. Tuthill and Z. Sui, J. Chem. Phys. 88, 8000 (1988); , J. Chem. Phys.
62.A. Kolinski, K. Kurcinski, and A. Orszagh, Acta Phys. Pol. A 75, 879 (1989);
62.A. L. Rodrigues, H.-P. Wittmann, and K. Binder, Macromolecules 23, 4327 (1990).
63.Notice that we are not investigating the Flory lattice model—we are working in the continuum. It is also important to bear in mind that DF theory does not contain a separability assumption.
64.The expansion is technically a functional Taylor series but can be thought of as a multidimensional Taylor series with the average particle number at each point in space as the variables.
65.Technically, if the free-energy functional were to be evaluated for any bulk density between the equilibrium liquid and solid states, it would be constant. This is because the bulk density is the average over all possible configurations, and in this region, to good approximation, there are two equally stable states: the liquid and the solid. By constraining the bulk density to be a particular value one is simply determining what fraction of time the system will spend as liquid and what fraction as solid. Because the free-energy is the same for both liquid and solid, it will be the same for any combination of the two. This would make density functional theory difficult to apply, and, instead, one constrains the systems to be either always liquid or always solid. In this case, a bulk density between liquid and solid would indicate either a distorted liquid or solid. As a consequence, a distinct double well structure is developed, and it is relatively simple to determine the coexisting states. A good discussion of this is to be found in Chapter 4 of Quantum Many-Particle Systems by J. W. Negele and H. Orland (Addison-Wesley, New York, 1988).
66.L. D. Landau and E. M. Lifshitz, Statistical Physics (Pergamon, New York, 1980), pt. 1.
67.We use the term “an” rather than “the” because, although uniquely defined (essentially) for a monatomic system, the term “ideal system,” as will be discussed shortly, is not at all unique for polyatomic and especially not for polymeric systems. We define, as best we can, an ideal system as one which is similar to the real system but is mathematically tractable. Also, notice that although an “ideal gas” constitutes an “ideal system” this is not necessary, and tractability is the only requirement. The importance of the ideal system was discussed in detail in Ref. 7.
68.Of course, the more realistic the ideal system, the more difficult the evaluation of will be. By starting with relatively unrealistic ideal systems and stepping through increasingly realistic systems, the exact solution can be converged upon.
69.R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford, New York, 1989).
70.The Coulombic term in TF theory can be thought of as arising from a Debye-Huckel approximation of the direct correlation function term.
71.Closures for liquid-state theory can be found by minimizing in the presence of a particle-generating field. For the molecular case, one can imagine using a molecule-generating field and an ideal system which retains all bonds. One would expect this to be a very accurate closure because the ideal correction to the external field only compensates for packing between molecules. If the “mixture-of-sites” ideal system is used, the ideal external field also corrects for bonding between the sites. Since the ideal system now consists of separate sites, a molecule-generating field seems inappropriate: a site-generating field is more in the spirit of the approximations. This is exactly what the highly successful RISM liquid-state theory uses: a “mixture-of-sites” ideal system and a site-generating field. Bonding constraints are enforced by—and only by—the ideal external field. We suggest that it is important for the bonding to be treated the same in both the ideal system and the generating field. If, as is suggested in Ref. 12, bonds were explicitly included in the ideal system, the generating field should also have explicit bonds.
72.Notice that the ideal-gas equation of state is seen in
73.See, for instance, J. Wilks and D. S. Betts, An Introduction to Liquid Helium, 2nd ed. (Oxford University, London, 1987);
73.S. W. Rick, D. L. Lynch, and J. D. Doll, J. Chem. Phys. (submitted).
74.O. Olabisi and R. Simha, Macromolecules 8, 206 (1975).
75.D. C. Bassett, in Developments in Crystalline Polymers—1, edited by D. C. Bassett (Applied Science, London, 1982).
76.H. W. Starkweather, Jr., P. Zoller, G. A. Jones, and A. J. Vega, J. Polym. Sci. Polym. Phys. Ed. 20, 751 (1982).
77.P. R. Swan, J. Polym. Sci. 56, 403 (1962).
Article metrics loading...
Full text loading...
Most read this month
Most cited this month