Flexible chain molecules excluded volume

Firstly, for rigid-chain polymas the excluded volume effects in thermodynamically good solvents (mainly determining the conformation and size of a flexible-chain molecule in solution are negligible. This has been confirmed both experimentally and theoretically ... 

This kind of perfect flexibility means that C3 may lie anywhere on the surface of the sphere. According to the model, it is not even excluded from Cj. This model of a perfectly flexible chain is not a realistic representation of an actual polymer molecule. The latter is subject to fixed bond angles and experiences some degree of hindrance to rotation around bonds. We shall consider the effect of these constraints, as well as the effect of solvent-polymer interactions, after we explore the properties of the perfectly flexible chain. Even in this revised model, we shall not correct for the volume excluded by the polymer chain itself. 

In an ensemble of flexible polymer chains, the instantaneous separation of two segments i and j varies from one molecule to another. Ensemble averages such as required in Eq. 2 are obtained by specifying W(r-jj), the probability that segments i and j are separated by ry. In an elastomeric rubber which is not so highly swollen that excluded volume interactions become important, and which is not too greatly deformed, W(r- j) takes a particularly simple form... 

The nematic phase (N) is the least ordered, and hence the most fluid liquid crystal phase. The order in this type of LC phases is based on a rigid and anisometric (in most cases rod-shaped or disc-shaped) molecular architecture. Such molecules tend to minimize the excluded volume between them, and this leads to long range orientational order. For rod-like molecules the ratio between molecular length and its broadness determines the stability of the nematic phase with respect to the isotropic liquid state and the stability rises with increase of this ratio. In most cases the rigid cores are combined with flexible chains, typically alkyl chains, which hinder crystallization and in this way retain fluidity despite of the onset of order. 

Figure 3.3 illustrates the idea of excluded volume. It shows two protein molecules as two adjacent spheres of the same radius R. Because molecules are not penetrable by each other, the volume of a solution occupied by a macromolecule is not accessible to other macromolecules. A minimal distance between two adjacent spherical molecules of a globular protein equals the sum of their radii, or the diameter of one of them. This means that around each protein molecule there is an excluded volume U), which is 8-fold larger than that of protein molecule itself and is not accessible for centres of other protein molecules. The excluded volume is still larger for non-spherical macromolecules and depends on the flexibility of the macromolecular chain, and its configurational, rotational, vibrational properties and hydration (Tanford 1961). 

If we consider the size of a polymer molecule, assuming that it consists of a freely rotating chain, with no constiaints on either angle or rotation or of which regions of space may be occupied, we arrive at the so-called unperturbed dimension, written (r), /. Such an approach fails to take account of the fact that real molecules are not completely flexible, or that the volume element occupied by one segment is excluded to another segment, i.e. in terms of the lattice model of a polymer solution, no lattice site may be occupied twice. Real molecides are thus bigger than the unperturbed dimension, which may be expressed mathematically... 

Chain stiffness prevents the monomer units of a polymer molecule from having contact with one another. Thus, we may expect that there exists for a semi-flexible polymer a certain contour length Lc below which the excluded-volume effect on the chain dimensions disappears (in a statisitical sense). This prediction is borne out by the data of Figure 5-3, which show that (5 )/M follows the curve for an unperturbed wormlike chain until Mw reaches 3 x 10 (L 4 X 10 ). Obviously, Lc ought to be larger for a stiffer polymer, i.e., one with larger persistence length q. 

Here, we are concerned with long flexible polymer chains dissolved in a good solvent. A good solvent is defined as one, in which it is energetically more favoiurable for the monomers of the polymer to be surrounded by molecules of the solvent than by other monomers. As a consequence, there exists around each monomer a region (the excluded volume) in which the chance of finding another monomer is very small. It turns out, that such diluted polymer solutions have physical properties that are independent of most of the details of the chemical microstructure of the chains and of the solvent. 

Monte Carlo simulations with lattice models of chain dynamics have been concerned with segmental and with whole molecule relaxation. In each case the autocorrelation functions depend upon details of the lattice model. A cubic-lattice model, with random stifle bead motions has been used to study the relaxation of seven characteristics of instantaneous aspherical shape. Three decay functions are observed for iarge molecules with and without excluded volume considerations. The large difference in relaxation times and in their chain length dependence obtained here, and in earlier studies on including the excluded volume effect, is attributed to restrictions associated with the single bead mechanism, a view justified by simulations which obtained more flexible chains by... 

The Flory-Huggins theory uses the lattice model to arrange the polymer chains and solvents. We have looked at the lattice chain model in Section 1.4 for an excluded-volume chain. Figure 2.1 shows a two-dimensional version of the lattice model. The system consists of si,e sites. Each site can be occupied by either a monomer of the polymer or a solvent molecule (the monomer and the solvent molecule occupies the same volume). Double occupancy and vacancy are not allowed. A hnear polymer chain occupies N sites on a string of N-l bonds. There is no preference in the direction the next bond takes when a polymer chain is laid onto the lattice sites (flexible). Polymer chains consisting of N monomers are laid onto empty sites one by one until there are a total tip chains. Then, the unoccupied sites are filled with solvent molecules. The volume fraction of the polymer is related to rip by... 

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