AVAILABLE METHODS OF ESTIMATING THE MOST PROBABLE CONFIGURATIONS OF SIMPLE MODELS OF A MACROMOLECULE BY H. N. V. TEMPERLEY Atomic Weapons Research Establishment, Aldermaston, Berkshire Received 23rd January, 1958 A summary is given of the present theoretical position for a linear polymer, and some new results for a branched polymer are published for the first time. The latter problem appears to be simpler in some respects. In a long hydrocarbon chain, the lengths of C-C links are distributed about a mean value, and successive links are inclined to one another at nearly the tetra- hedral angle, so that there are, effectively, up to three distinct ways of adding one more carbon atom to a given linear chain. The mathematical methods of dealing with such a molecule have to take account of several quite distinct facts.(a) There may be short-range correlations between the directions of successive links, e.g. it is known that the configuration of a hydrocarbon chain in which three successive links lie in a plane is energetically more favourable than any other. (b) Certain configurations are sterically impossible. (c) In many cases, the possibility of branching cannot be neglected. (d) There are interactions between the atoms in neighbouring molecules. For comparison with experiment we need various averages over a typical molecule, such as the number of configurations possible and the effective size of a molecule of given length. If we could neglect (b)-(d), the problem becomes one of “ random-walk ” type. It is fairly easy, in any given case, to take account of correlations between successive links, it being only necessary to replace the dgebraic variable occurring in the generating function for the random-walk problem by a matrix of fairly low order.lp2 Thus, restriction (a) does not materially complicate the problem.The effect of restriction (b) is dficult to calculate properly, but can be said to be essentially soluble if we merely want to investigate the number of configurations of the molecule. The result seems to be that the effect of (b) shows itself as a reduction of the “ entropy per link ”, in other words, the number of permissible ways of adding a new link to a long chain is significantly less than it is for a short one. (This fact vitiated many of the early treatments that attempted to apply perturbation methods to unrestricted random-walk theory.A reduction in the entropy per link means that “ almost all ” the configurations permitted by the unrestricted theory are ruled out.) Many workers have introduced the concept of an “effective length of segment” which implies that the actual chain, with effects (a) and (b) taken into account, should behave in a way very similar to a completely random chain with a smaller number of longer links. (For example, it is often useful to think of a long hydrocarbon molecule with only a small fraction of the linkages departing from the trans configuration. Thus, we have a number of “ plane-zig-zag ” segments connected by gauche linkages.) A recent analysis of a wide variety of experimental data 3 did seem to show that this concept was a physically valid one, the effective length being of the order of 10-30 C-C linkages for a hydrocarbon chain and 2-5 monomers for a rubber 92H .N. V. TEMPERLEY 93 chain. There was a little evidence that the effect of (d) could be more important in solids than in liquids or solutions. THE MONTROLL MODEL According to this model 4 the monomer is taken of invariable length, and suc- cessive links are constrained to lie along the nearest-neighbour bonds of some simple lattice, all configurations containing closed rings being deleted. The lattice corresponding most closely to the hydrocarbon chain is the diamond lattice, but the correspondence is by no means perfect. One is therefore led to ask whether results are sensitive to the lattice type, or to the number of dimensions (in view of the fact that the simple random walk is sensitive to the number of dimensions).Another question is whether the form of the results is sensitive to the “ excluded volume ”, e.g. what difference does it make if we delete configura- tions in which two non-neighbours in the chain approach within one lattice distance, instead of coinciding ? Montroll4 showed that the correct consequence of his model would be a progressive reduction in the entropy per link, as loops of larger and larger sizes were successively eliminated. This conclusion has been confirmed by Wall and others,s who build up non-crossing chains by means of a machine programme of “ random addition of links ”, the build-up of any chain being stopped, and a new chain started, as soon as the first loop occurs.It was found that the toraZ reduc- tion in “ entropy per link” quickly approached a limiting value. A similar conclusion was reached by Hammersley and Morton 6 for the diamond lattice, based on hand computation of a few very long chains. Temperley 1 has given reasons for thinking that “ the limiting entropy per link ” should be closely related to the Curie temperature for the corresponding Ising lattice, and the numerical agreement is indeed fairly good. Although no analytic solution is yet available for any of these models, even in two dimensions, it does now seem that we know enough about them to estimate the limiting “ entropy per link ” or “ effective number of choices ” in any reason- able case.The position about the other interesting properties, such as the effective size of a molecule of known total length, is much less satisfactory. We do not yet know, for example, in what circumstances the effective radius varies as N4 (as it would for a completely random walk), or as some different power of N, and the answer does not depend in any clear-cut way on the number of dimensions, but also seems to depend on the type of lattice, and to be a function of the excluded volume.5 Furthermore, there are indications that the limiting behaviour is ap- proached extremely slowly in three dimensions. (In number theory, one often meets situations in which the asymptotic behaviour is not realized until some slowly varying function like log (log N ) becomes large, but this type of thing does not seem to have been previously met in statistical mechanics.) One observation of Wall and others,s that the probability of formation of a ring is, for almost all lattices, inversely proportional to the square of the size of the ring, seems to indicate that the generating functions for the Ising lattice par- tition function and for the restricted random walk have similar analytic behaviour, which is reasonable (cp.Temperley 1 for a discussion of the analytic feature probably responsible for this behaviour). BRANCHED CONFIGURATIONS The Montroll model can also give a representation of a polymer in which unlimited branching is allowed, that is to say, we take our lattice, and select certain lines from it in such a way that they are all connected, but closed loops are absent; in other words, we form a Cayley tree, which is, according to the Montroll model, a possible configuration of a branched polymer, steric effects being allowed for.For simplicity, consider the plane square lattice (though some4 - 1 . . . . . . . . - 1 . . . . - I . . . . - 1 - 1 4 - 1 . . . . . . - 1 . . . . - 1 . . . . 0 , 0 - 1 4 - l . . . - 1 . . . . - 1 . . . . 0 A ( 3 ~ 2 0 ) ~ ~ ~ (1) where A is a constant of the order of unity. This result holds provided only that both M and N are large, independently of their ratio. Thus (2) gives us the number of ways in which a branched polymer containing MN points can be exactly “ folded up ” to form an M x N rectangle. Each choice of M and N corresponds to a diflerent set of possible configurations of the branched polymer, so that the total number of such configurations would be obtained by deriving an expression similar to (2) for all possible closed domains containing MN lattice points.The number of such domains can be estimated asymptotically by putting x = 1 in expression (1 1) of ref. (l), that is, we want the coefficient of zMN in z(l - 2 ) 3 1 - 5z + 722 - 423’ which is of the order of 3MN. Actually (3) enumerates precisely the closed domains of a particular type that can be formed on the square lattice, but it is possible to show that the total number of distinct domains is not greatly in excess of this. We can take a final step by saying that (2) does not depend very much on the shape of the domain, but mainly on the total number of lattice points, and it is known that the properties of the Ising model are not very sensitive to boundary con- ditions.7 Thus (2) is probably nearly right for any shape of domain, so that our estimate of the number of configurations of a branched-chain polymer is of the order of a factor of 9-10 per monomer (multiplying (2) by the result deduced from (3)). Thus, we have the strange situation that the polymer with branching is an analytically simpler problem than the linear polymer. The higher-order minors of (1) seem to be related to problems of solution and degradation of branched polymers, but we shall not pursue this matter further here. 1 Temperley, Physic. Rev., 1956, 103, 1. 2 Gillis, Proc. Cambr. Phil. SOC., 1955, 51, 639. 3 Temperley, J. Res. Nut. Bur. Stand., 1956, 56, 55. 4 Montroll, J. Chem. Physics, 1950, 18, 734. 5 Wall et al., J . Chem. Physics, 1954, 22, 1036; 1955, 23, 913 and 2314; 1957, 26, 6 Hammersley and Morton, J. Roy. Statist. SOC., 1954, 16, 23. 7 Temperley, Proc. Physic. SOC., 1957, 70, 192. 1742; 27, 186.