L. MANDELKERN, J. G. FATOU, AR‘D C. HOWARD
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The Nucleation of Long-chain Molecules1
by L. Mandelkern, J. G. Fatou, and C. Howard Department of Chemistry and Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida (Received J u n e 10,1964)
A theory pertinent to the nucleation of long-chain molecules o f finite molecular weight has been developed. Although, for very high molecular weights, the free energy of nucleus formation and the critical dinlensions of nuclei assume the same values as in the usually employed infinite molecular weight approximation, in the molecular weight range of usual interest, significant differences are observed as a function of molecular weight and of temperature. I n particular, it is found that the critical number of repeating units per chain, that are required to form a nucleus, does not become infinite at the melting temperature. Rloreover, when the molecular nature of the nucleus and mature crystallite differ, it becomes theoretically possible, depending on the situation, for the homogeneous steady-state nucleation rate to vanish either below or above the equilibrium melting temperature of the system.
Introduction The important role played by nucleation processes in the liquid-crystal transformation of long-chain molecules has been strongly emphasized in the past. Nucleation theory has been important in analyzing and understanding t h e kinetics of isothermal crystallization from the melt arid the temperature coefficient of this process.2ajb It also appears very likely that the observed dependence of morphology and crystalline texture on the crystallization temperature steins from considerations of the dimensions of critical size nuclei and their free energy of f0rination.~-5 In treating the nucleation of long-chain molecules, the nucleus has been taken to have an asymmetric geometry with different arrangements of the polymer chains within the nucleus being assumed. 116,7 However, the assumption has always been made, explicitly or otherwise, that one is dealing with a chain of infinite niolecular weight. I n this approxiniation, the equations which describe the critical condition for nucleation (irrespective of detailed model) have the same analytical form as the equations derived for a collection of mononieric molecules with a similar geometric arra,ngement in the nucleus.8 I n the present paper, we re-examine the nucleation theory pertinent to cha,ins of large but finite molecular length. By avoiding the infinite niolecular weight approxiination we observe significant departures from monomer theory in the molecular weight range of usual The Journal of Physical Chemistry
interest. Furthermore, a connection can be made between the nucleation of long-chain molecules of precisely the same chain length, wherein the formation of molecular crystals is theoretically possible under certain circumstance^,^ and the case where the required uniformity of chain length cannot be attained, even for a molecular weight fraction. Our discussion will be limited to three-dimensional homogeneous nucleation since we shall be concerned solely with the effect of the finite chain length. The results obtained can be easily extended to other types of nucleation when pertinent. Results and Discussion I n developing a nucleation theory appropriate to (1) This work was supported by a grant from the U. S. Army Research Office (Durham) and a contx-act with the Division of Biology and Medicine, Atomic Energy Commission. (2) (a) L. Mandelkern, F. A. Quinn, Jr., and P. J. Flory, J . A p p l . P h y s . , 25, 830 (1954); (b) L. Mandelkern, “Growth and Perfection of Crystals,” R. H . Doremus, B. W. Roberts, and D. Turnbull, Ed., John Wiley and Sons, Inc., New York, N . Y., 1958. (3) P. J. Flory and A. D. McIntyre. J . Polymer Sci., 18, 592 (1955). (4) L. Mandelkern, ibid., 47, 494 (1960). (5) J. D . Hoffman and J. J. Weeks, J . Res. -Vatl. B u r . Std., A66, 13 (1962).
(6) L. Mandelkern, J . A p p l . Phys., 26, 443 (1955). (7) J. D. Hoffman and J. I. Lauritzen, J . Res. A-atl. B u r . S t d . , A64, 73 (1960); A65, 297 (1961). ( 8 ) L. *Mandelkern, “Crystallization of Polymers,” McGraw-Hill Book Co., New York, N. Y., 1964, p. 248. (9) P. J. Flory and A. Vrij, J . Am. Chem. Soc., 85, 3548 (1963).
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NUCLEATIOS OF LONG-CHAIN MOLECULES
chain molecules we shall adopt the same classical procedures that have been used for monomeric systems. The expression for the free energy of forming a m a l l crystallite or nucleus from the melt is composed of two types of terms. One term is the bulk free energy of fusion which characterizes the crystalline phase of finite size. Of opposite sign are terms which represent the excess surface free energies contributed by the interfaces present. It is thus tacitly assumed that the nucleus is sufficiently large so that its interior is hoinogeneous and that the exterior surfaces (real or eff ective) can be defined. The resulting free energy surface is then examined, by standard methods, for the conditions of stability of nuclei and of mature crystallites. We need make no assumptions at present in regard to the molecular nature of the mature crystallites that are eventually evolved. With the above stipulations, the free energy change in forming a cylindrically arranged crystalline array of { units long and p sequences in cross section for a pure system containing N polymer molecules each comprised of x repeating units is given by6)lO
AF
= 2g1/zp1/z[~u -
pRT In D - {pAfu -
stipulations that equilibrium with respect to the crystallite size and amount of crystallinity prevails. The inherent assumption has been made that each sequence in the ordered array which comprises the nucleus is denoted by a different chain. Since we shall only be concerned with the development of small amounts of crystallinity, the first term of the series expansion of In [l - { p / x N ] can be used so that
The surface described by eq. 2 possesses neither a maximum nor minimum but does contain a saddle point. The coordinates of the saddle point are defined by ( b A F / b p ) , and ( b A F / b { ) , = 0, which also prescribe the dimensions of a nucleus of critical size. These dimensions are given by the relations
and RT
2 where uu is the lateral interfacial free energy per molecule and D is defined as exp 1 - 2oJRT ] , where ue is the excess interfacial free energy per repeating unit as it emerges from the crystal face normal to the chain direction. The free energy of fusion per repeating unit for a chain of infinite molecular weight is given by Afu. The first two terms in eq. 1 represent the positive contribution to the total free energy change of the interfaces present. The third term represents the bulk free energy of fusion for the { p units involved in the traiisformation, behaving as though they were part of an infinite molecular weighit chain. The latter two terms result from the finite length of the chains. The first of these terms expresses the entropy gain which results from the increased volume available to the ends of the molecule after melting. The last term results from the fact that only a portion of the chain units of a given molecule are involved in the nucleus. It represents the entropy gain that arises from the number of different ways a sequence of { units can be located in a chain x units long with the terminal units being excluded from the sequence in q u e ~ t i o n . ~The ~ ’ ~inclusion of this later term is of prime importance to the present discussion, for it introduces the eFi’ect of a finite chain. Equation 1 can be recognized as the net free energy of fusion, previously calculated by F’lory,’o but devoid of the
[Afu - 2
+ x - f * +.,1
2ue - RT In
(” -
+
’)
(4)
where the asterisks denote the critical dimensions a t the saddle point. The value of the free energy change a t the saddle point, obtained by substituting eq. 3 and 4 into eq. 2, is found to be
Equation 5 is identical with that obtained for the infinite molecular weight polymer, as well as for a nucleus composed of monomers arranged in a similar geometric array. If x is very large and l*
