Polymer excluded volume problem

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. ... 

There are several problems that involve single polymer chains for which an exact solution is virtually out of the question. The polymer excluded volume problem is one such case. It is then necessary to search for suitable approximation schemes for these problems. 

There have been some approaches to the problem based upon equation of motion methods. However, by far the most sophisticated approximations have involved the SCF theories. There have also been some diagrammatic approaches to the polymer excluded volume problem, but these can be shown to be intimately related to the SCF and perturbational methods. 

Edwards has also noted the strong mathematical analogies between the functional integral representation of the polymer excluded volume problem and questions associated with electronic structure in disordered systems. "" Thus a detailed discussion of this formal approach is of more than just academic interest. In the polymer case, approximation may be guided by probabilistic arguments, whereas in the disordered system analogous mathematical approximations rest upon less intuitive grounds. 

The mathematical problems associated with a fully statistical mechanical description of polymer networks (with interchain interactions possibly present) are staggering in comparison to those posed even by the polymer excluded volume problem. Hence it is expected that a formal... 

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. 

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-... 

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 ... 

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. 

In order to introduce the notation and some of the necessary concepts, as well as to motivate the introduction of the functional integral techniques, first some exact results from the configurational statistics of individual polymer chains are introduced. Functional integral techniques are then applied to these simpler problems before discussing the more difficult problems of polymer excluded volume and the description of polymers in bulk. 

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. 

It has already been noted that the configurational statistics of a single polymer chain with the long-range interactions is equivalent to the full many-body problem. Therefore, this excluded volume problem is not, in... 

As noted in the introduction, in dealing with difficult many-body problems, like the excluded volume problem, there are usually a few rather crude, but physically reasonable, kinetic-theory type arguments which often give the correct results apart from a constant factor. In the excluded volume problem, there are some smoothed density approaches where, e.g., in (6.16), the field due to the excluded volume from the polymer density... 

We have not discussed the subject of nonideal polymers in any detail apart from the excluded volume problem. Thus no mention is made of the evaluation of the potential of mean force from the monomer-solvent interaction, and subsequently the evaluation of the osmotic pressure. We refer to the treatment of Yamakawa (Ref. 5, Chapter IV) for this subject and mention only that the osmotic pressure of a polymer solution at finite concentrations is represented as a virial expansion in the polymer concentration. " The second, third, etc., virial coefficients represent the mutual interaction between two, three, etc., polymer chains in solution. Thus the functional integral techniques presented in this review should also be of use in understanding the osmotic pressure of nonideal polymer solutions. We hope that this review will stimulate such studies of this important subject. It should also be mentioned in passing that at the 0-point the second virial coefficient vanishes. In general, the osmotic pressure -n is given by the series... 

It is well-known that the Flory mean-field theory of the polymer self-excluded volume problem yields excellent results. If R is some scalar measure of the polymer chain size, such as the radius of gyration, and N is the number of monomer units in the chain, then R scales with N as... 

Edwards, S. F., Singh, P. Size of a polymer molecule in solution Part 1.-Excluded volume problem. /. Chem. Soc. - Faraday Transactions II (1979) 75, pp. 1001-1019... 

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