1090 Gordon, Torkington, Ross-Murphy
Macromolecules
The Graph-like State of Matter. 9. Statistical Thermodynamics of Dilute Polymer Solutionst Manfred Gordon,* John A. Torkington, and Simon B. Ross-Murphy Institute of Polymer Science, University of Essen, Wivenhoe Park, Colchester C04 3SQ, England. Received May 23,1977
ABSTRACT: The pathway to the properties of dilute polymer solutions through the range of two-parameter theories is redirected with the aid of combinatorial algebra. The Fixman perturbational partition function for finite chains is transformed into the convergent classical Gibbs form, thus removing the restriction to the vicinity of the theta state. The Mobius inversion and inclusion/exclusion technique, whose power has recently been emphasised by G.-C. Rota, allows the classical form to be exploited, but work by several mathematicians, physicists, and chemists supplies necessary ingredients. The non-Markovian properties of Pdlya lattice walks, a problem of great mathematical beauty, enter the physical theory only a t the very end and in simplified form. The parameters in the partition function are calibrated from scattering measurements and related to theories, initiated by Dvoretsky and Erdos, of self-intersecting walks. The central limit theorem of probability theory here provides the key. Finally, the theories and experimental results on cooalent cyclization, especially by Allen, Edwards, Walsh, and Burgess, are integrated with the graph-theoretical analysis of temporary contact-pair formation. Briefly, then, for the purposes of physical as distinct from the mathematical theory, the so-called excluded volume can be “re-included” in an almost Markovian theory by the combinatorial principle of inclusion/exclusion. Further developments are likely to center on integrating into dilute solution theory, the analysis of the nature of contact potentials, pursued successfully in semidilute solutions by Huggins.
(I) Introduction Huggins’ and Flory2 offered their pioneering lattice-type theory of the thermodynamics of semidilute polymer solutions in 1941. Ever since, it has seemed important to extend the range of thermodynamic theories to the very dilute concentration range, where coils are well separated from each other. The so-called two-parameter theories3 evolved ways of dealing with configurational properties of isolated chains in solvents good, bad, or indifferent. However, a usable partition function for such systems, which would allow configurational properties to be integrated with other equilibrium properties in a general framework of statistical mechanics, has as far as we are aware eluded theoreticians. This conundrum has been dominated by the excluded volume problem, which Flory4 put on the map in the early days of polymer science. Between 1954 and 1960,progress was reviewed by polymer scientists on three occasions (Wall and Hiller: Hermans: and Casassa7;see also Stockmayel.8). Apart from citing these reviews in 1963, Hammersleyg mentioned that scores of papers had already been devoted to the excluded volume, and he added some qualitative theorems on the underlying model of nonintersecting (Pitlya) walks on lattice graphs, such as that of the cubic or hypercubic lattice. The problem is elusive, because of the non-Markovian behavior of such excursions. At the Science Research Council RencontrelO on Applications of Combinatorial Mathematics a t Aberdeen in 1975, Paul Erdos stressed the great beauty of the mathematical problem, toward whose solution several theoretical physicists had meanwhile made important contributions, for instance Edwards,” Domb,12 and de Gennes.13 As regards thermodynamic aspects, however, the perturbation treatment of volume exclusion in polymer chains, inaugurated by Teramoto14 and by Fixman,15 and extended by Yamakawa and Tanaka,16 led to divergence troubles (see IIb below) of a series of terms with alternating signs. The structure of these series, examined by heuristic methods in a previous paper of this series, leads to the conclusion that an application of combinatorial methods based on the principle of inclusion/exclusion, or, more generally, on Mobius inversion, might usefully be applied. Moreover, some t Affectionately dedicated to Professor M. L. Huggins, on the occasion of his 80th birthday.
recent mathematical literature underlines the timeliness of exploring such combinatorial patterns. Gian-Carlo Rota, in particular, has been publishing a series of papers, often with distinguished co-authors, which fully justifies its running title: On the Foundations of Combinatorial Theory (some of the results are found in his recent book17).It is significant that the series started18 with the sentence: “One of the most useful principles of enumeration in discrete probability and combinatorial theory is the celebrated principle of inclusionexclusion.” The principle is then set in the wide framework of the inversion of the Mobius function of a partially ordered set as a fundamental principle of enumeration (see also ref. 19). Like the later parts of Rota’s series, his first paper is no doubt a potential gold mine for theoretical physicists and chemists. Little more than just the basic notions of these combinatorial techniques, however, are required to make the modest progress in the context of solution thermodynamics which is recorded here, namely (a) to dispose unambiguously of the question, repeatedly examined from different viewp o i n t ~ , of ~ ~the- ~source ~ of the divergence difficulties, and more importantly (b) to arrive at a usable partition function for an isolated chain in solution. Indeed, once this is done, it is not hard to disguise the connection of the problem with the classical Mobiusz5 function, by expressing the few lines of theory simply as an exercise in ordinary matrix inversion. While the combinatorics required is simple, recent work by mathematicians, physicists, and chemists has proved very beneficial in the practical implementation of our program. First, this program involved the shifting of the main burden of the physical theory from the non-Markovian to Markovian aspects of lattice walks, where the work initiated by Dvoretsky and Erdos26on self-intersecting walks, and its recent development by SpitzerZ7and Flatto,28has proved invaluable (see section IVc below). Second, the theoretical calculations and chemical experiments by Edwards, Allen, and c o - w ~ r k e r s ~ ~ have provided us very useful ingredients to reinforce the present work (see section IV4). (11) Combinatorial Relation between the Gibbs and Fixman Forms of the Partition Function The schematization of a polymer chain as a string of n beads, with pairs of beads in mutual contact to form contact pairs, is illustrated in Figure 1.The counting of the configurations available to a chain will be facilitated by the familiar
Vol. 10, No. 5, September-October 1977
Graph-like State of Matter 1091
n
a
b
C
d
a
Figure 1. The chain graph (a) corresponds to bead model (b) for a polymer chain, which here features three contact pairs (black beads). The chain graph (c) illustrates a multiple (here, a triple) intersection, corresponding to bead model (d). The walker returns twice to the bead marked 1 to form the two contact pairs ( 2 , l ) and (3,1),in accordance with variant (ii) of section IVb. In variant (iii),there would be a third contact pair (2,3).
one-to-one correspondence between chain configurations and lattice walks (Figure 2). One end bead of the chain is placed at the origin of, say, the diamond lattice, and successive beads then mark other points on the lattice visited by the “walker”. Not surprisingly, the enumeration of configurations of a chain with a given m contact pairs is easier, if no bias is allowed for or against the presence of additional contact pairs. Thus various theories, e.g., self-consistent field calculation^^^ or the Wang-Uhlenbeck theorem as generalized by Fixman,15 lead naturally to the number (or mean-square dimensions) of chains which have at least m , say, contact pairs but may have more, with an overcounting procedure explained as follows. The theories lead in the first place to the number N,, of configurations which share a t least a specified set, labeled as the jth, of m contact pairs, but may have more. Summation over all j then arrives a t a number N , which heavily overcounts any given configuration: instead of being counted once, it is counted ($) times, where $ is the number of contact pairs of the given configuration. The overcount is corrected subsequently by the terms in an alternating series (see below). The canonical Gibbs form of the partition function arises naturally as a function of the number n, of configurations which have exactly m contact pairs, where each set of exactly m contact pairs is included once in the count. The connection between Nmj and n , is furnished by a simple combinatorial lemma, which reads (for a maximum number P ( n )of contact pairs):
b
Figure 2. The transformation of an intersection in part of a walk (on a diamond lattice, not shown) in the diagram a into a contact pair in diagram b by moving the two congruent beads apart to form a contact pair in violation of the lattice condition.
where n,(n) is the number (or measure) of distinct chain configurations with exactly m contact pairs and E is the energy of a contact pair. Write
Z ( n )=
m=O
n,(n)[I - (1 - e - E / k T ) ] m
(3)
and expand in powers of (1 - exp(-EIkT)). Then
where the passage from (4) to ( 5 ) applies the lemma. Any partition function such as (2) is algebraically equivalent to a form such as (5) with alternating signs. For small E, and defining
p = EIkT
(6)
we discover the discrete form of the partition function of Fixman15
whose proof follows immediately from the definitions. Of course, if the set of configurations is replaced by any set of objects, and the set of contact pairs possessed by a configuration is replaced by any set of properties, the combinatorial interpretation of eq 1is suitably generalized. The canonical configurational partition function Z for a chain of n units (e.g., on a lattice) is written thus:
In order to put the present work in perspective with the long history of the two-parameter theories, the comparison of the two forms (eq 2 and 7 ) of the partition function is pursued under a number of headings: (2a) The passage from discrete to continuum models, which have been preferred in the past; (2b) the problem of divergence of the partition function in the continuum models based on eq 7 ; (2c) the comparison of perturbation solutions with exact solutions and the appropriate reference state about which perturbations are considered.
1092 Gordon, Torkington, Ross-Murphy
Macromolecules
(a) From Discrete to Continuum Models. The discrete lattice model leads directly by algebra from the canonical form (eq 2) to the Fixman form (eq 7), viz., to an expansion in powers of a small variable @, with coefficients which count the number of relevant configurations. The classical expansion in terms of Boltzmann weights and degeneracies (eq 2) is abandoned in ( 5 )in favor of a Taylor expansion in the variable -(1 - exp(-E/kT)), denoted as x by Fixmanlb and by Y a m a k a ~ aWe . ~ saw that in the discrete lattice treatment x is replaced (eq 5 ) by @ in the perturbation approximation ( E 2 dimensions are transient and those for
