The excluded volume problem

P. G. De Gennes. Exponents for the excluded volume problem as derived by the Wilson method. Phys Lett 38A 339, 1972 J. des Cloiseaux. The Lagrangian theory of polymer solutions at intermediate concentrations. J Phys 26 281-291, 1975. 

H, Reiss, J. Chem. Phys., 47, 186 (1967) See also H. Yamakawa preprint On the Asymptotic Solution of the Excluded Volume Problem in a Linear Polymer Chain. ... 

Random walks on square lattices with two or more dimensions are somewhat more complicated than in one dimension, but not essentially more difficult. One easily finds, for instance, that the mean square distance after r steps is again proportional to r. However, in several dimensions it is also possible to formulate the excluded volume problem, which is the random walk with the additional stipulation that no lattice point can be occupied more than once. This model is used as a simplified description of a polymer each carbon atom can have any position in space, given only the fixed length of the links and the fact that no two carbon atoms can overlap. This problem has been the subject of extensive approximate, numerical, and asymptotic studies. They indicate that the mean square distance between the end points of a polymer of r links is proportional to r6/5 for large r. A fully satisfactory solution of the problem, however, has not been found. The difficulty is that the model is essentially non-Markovian the probability distribution of the position of the next carbon atom depends not only on the previous one or two, but on all previous positions. It can formally be treated as a Markov process by adding an infinity of variables to take the whole history into account, but that does not help in solving the problem. 

Exercise. Find the mean square distance after r steps of a random walk on a two-dimensional square lattice when U-turns are forbidden. (This is not the excluded volume problem, because each site can be visited many times, provided more than two steps intervene.)... 

The present model may impose too strong an obstruction on a real system, and it will be of great interest to know whether the idea of a higher shell substitution effect can be extended and adjusted to the excluded volume problem. Certainly, nobody can characterize the cascade theory with substitution effects as a mean field theory. 

The methods illustrated here can be used to evaluate all quantities of interest in the excluded volume problem. Another example (A ) will be considered after we derive the general Feynman rules. 

To summarize, d = 4 makes a border line, the upper critical dimensionJ for the excluded volume problem. For d < 4 both the cluster expansion and the loop expansion break down term by term in the excluded volume limit. For d i> 4 the expansions are valid, the leading n- or c-dependence of the results being trivial, however. We may state that for d > 4 the random walk model or Flory-Huggins type mean field theories catch the essential physics of the problem. As will be explained more accurately in Chap. 10 the mechanism behind this is the fact that for d > 4 two nncorrelated random walks in general do not cross. 

In addition to the aforementioned practical difficulty of the method based anA M/fy], it must be realized that its theoretical basis is also not secure. The excluded volume problem of two entangled chains, which is a fundamental part of the theory of the second virial coefficient, has not been much advanced beyond the first-order perturbation stage, and as a result the function /(a) of Eq. (10) is only imperfectly known. For these... 

Besides the earliest theories by Flory (95 ), Huggins (128") and Miller (183 ), all improved theories are concerned with the effect of ring closure of chains on the mixing entropy [Staverman (237) Guggenheim and McGlashan (7) Tompa (21) MCnster (15) and Kurata, Tamura and W atari (756)]. The problem of ring closure is essentially the excluded volume problem. 

I o some extent the excluded volume problem can be overcome if we arc able tf> calculate enough tjrders of the expansions. The power series for the end-to end distaiuie of an isolated i hain, has been calculated to sixth order, [MN84]... 

The excluded volume problem of polymer chains was taken up early in 1943 by Flory [6]. His arguments based on the chemical thermodynamics brought the conclusions (i) the existence of the Flory point ( point) where two body interactions apparently vanish, and (ii) that in non-solvent state chains behave ideally-... 

Modern theories do not solve analytically the excluded volume problem, instead they pursue the same problem from different angles. This is because the excluded volume problem is placed as a category of typical many body problems [36], hence an intractable one. In spite of such a situation, there is an approach to persistently seek closed solutions of the excluded volume chain. Following this trend, some empirical formulations have been put forth for the limiting case of N—>°° [37-41]. The most successful one is that of the des Cloizeaux type equation, written by the form [41] ... 

There has been a proliferation of theories of the excluded volume problem using a variety of methods to decouple the many-bodied problem. Their results can be summarized crudely by... 

In the previous chapter, we focused on the excluded volume problem. We learned to appreciate that each monomer has a certain volmne, and the monomers cannot penetrate each other. This leads to repulsion at short distances. In the case of a good solvent, repulsion is the prevaihng tendency overall, so the polymer coils swell. But what if the quality of the solvent grows worse For example, you could add some precipitant into the solvent, or change the temperature. As a result, the solvent may go through the 0 point (as we discussed in Section 8.5) and the binary interactions between the monomers will become mainly attractive. Segments will tend to stick to each other from time to time, so there will be lots of temporary couples. What will this do to the coil as a whole ... 

When in the course of generating the molecule one chooses an already occupied position, the simulation is discarded. This permits the representation of the excluded volume problem mentioned in Sect. 1.3.4. In the computation of the probability in Fig. 1.39 this termination of chains because of excluded volume is recognized by the factor Fn. Results on 1123 surviving chains, generated on a tetrahedral lattice to simulate the conformations of a carbon backbone, are also given in Fig. 1.39 [10]. These data were also shown in Fig. 1.33 with the restriction and refinements of the random flight. 

We now discuss the excluded volume problem in terms of the CERWC model when a chain comprises N segments of length A each. The repulsion potential... 

As a result of the transformations, the point will be located in the region C regardless of the initial values of Uj and 1V which predetermines the universality of the excluded volume problem the transformation R, converts all the chains into the macroscopically equivalent ones in the region C. Near the fixed point, R, can be assumed to be linearized, namely,... 

Two important consequences follow from the fact of the atomic. sccdc of length a being indi.stinguishable in experimentally observed quantities (Oono and Freed, 1981a Oono et al., 1981). First, the macroscopic properties do not depend on the existence of the natural minimum on the molecular scale a, so, they must be well-defined in the limit o —> 0. In the excluded volume problem of a molecular coil, the limit a —> 0 should be regarded as a rejection of the consideration how a segment interacts with itself (the self-excluded volume). 

Oono, Ohta, and Freed have developed a renormalization method for conformational space of Gell-Mann and Low s (195 1) type for the excluded volume problem. In a number of cases, this approach proves to be more effective than Kadanoff-Wilson s approximation (see subsection 5.1.1) it provides a higher accurau y of the calculation of such quantities as the end-to-end vector distribution function, the scattering function, the conformation of a mricromolecule as a function of polymer concentration, etc. 

One of the first polymer problems to be treated by these methods was the so-called excluded volume problem, which arises from an attempt to account for the fact that two different constituent atoms (or group of atoms) of a polymer chain cannot occupy the same region of space. The simplest approach to the analogous excluded volume problem in the case of simple fluids is van der Waals theory of fluids. In van der Waals theory an equation of state is easily written and the thermodynamic properties of the fluid are obtained from it. By contrast the polymer excluded volume problem is of the same order of mathematical difficulty and is very similar in nature to the general many-body problem. Yamakawa reviews the large number of theoretical approaches to the excluded volume problem and the resultant controversies over the properties of polymers with excluded volume. ... 

They relate stiff chains to cases of Brownian motion where the particle s position, velocity, etc., are considered. Connection is also established between stiff polymers and the quantum mechanics of systems which classically would have velocity, acceleration, etc., dependent forces. The excluded volume problem is considered in the next section, and then we discuss problems involving polymers in bulk. 

VI. THE EXCLUDED VOLUME PROBLEM THE SELF-CONSISTENT FIELD... 

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