Excluded-volume effects/interaction

Suppose we have a physical system with small rigid particles immersed in an atomic solvent. We assume that the densities of the solvent and the colloid material are roughly equal. Then the particles will not settle to the bottom of their container due to gravity. As theorists, we have to model the interactions present in the system. The obvious interaction is the excluded-volume effect caused by the finite volume of the particles. Experimental realizations are suspensions of sterically stabilized PMMA particles, (Fig. 4). Formally, the interaction potential can be written as... 

In the following paper, the possibility of equilibration of the primarily adsorbed portions of polymer was analyzed [20]. The surface coupling constant (k0) was introduced to characterize the polymer-surface interaction. The constant k0 includes an electrostatic interaction term, thus being k0 > 1 for polyelectrolytes and k0 1 for neutral polymers. It was found that, theoretically, the adsorption characteristics do not depend on the equilibration processes for k0 > 1. In contrast, for neutral polymers (k0 < 1), the difference between the equilibrium and non-equilibrium modes could be considerable. As more polymer is adsorbed, excluded-volume effects will swell out the loops of the adsorbate, so that the mutual reorientation of the polymer chains occurs. 

Equation (23) predicts a dependence of xR on M2. Experimentally, it was found that the relaxation time for flexible polymer chains in dilute solutions obeys a different scaling law, i.e. t M3/2. The Rouse model does not consider excluded volume effects or polymer-solvent interactions, it assumes a Gaussian behavior for the chain conformation even when distorted by the flow. Its domain of validity is therefore limited to modest deformations under 0-conditions. The weakest point, however, was neglecting hydrodynamic interaction which will now be discussed. 

The GvdW theory has been applied also to mixtures of Leonard-Jones fluids [15,16], The extension to mixtures is straightforward with respect to the binding energy and interaction with an external field but not quite so straightforward with respect to the excluded volume effect. The GvdW(S) theory produces for a mixture of c components an equation of state of the form... 

The range of semi-dilute network solutions is characterised by (1) polymer-polymer interactions which lead to a coil shrinkage (2) each blob acts as individual unit with both hydrodynamic and excluded volume effects and (3) for blobs in the same chain all interactions are screened out (the word blob denotes the portion of chain between two entanglements points). In this concentration range the flow characteristics and therefore also the relaxation time behaviour are not solely governed by the molar mass of the sample and its concentration, but also by the thermodynamic quality of the solvent. This leads to a shift factor, hm°d, is a function of the molar mass, concentration and solvent power. 

A similar effect may exist for hydrophobic interaction between solute and stationary phase, as one solute may adsorb more strongly to the stationary phase than another. It has also been remarked that a flexible polymer confined to a pore should be at a lower entropy than one in bulk solution, leading to exclusion in excess of that expected for a simple geometric solid.23 Even in the absence of interactions, a high concentration of a small component can lead to an excluded volume effect, since the effective volume inside the pore is reduced. 

Muthukumar and Winter [42] investigated the behavior of monodisperse polymeric fractals following Rouse chain dynamics, i.e. Gaussian chains (excluded volume fully screened) with fully screened hydrodynamic interactions. They predicted that n and d (the fractal dimension of the polymer if the excluded volume effect is fully screened) are related by... 

A problem arises, in that the strong r b dependence of My requires that close overlap of spins be prevented. Thus, even though excluded volume interactions have no effect on chain dimensions in the bulk amorphous phase, it is important in the present application to build in an excluded volume effect (simulated with appropriate hard sphere potentials), so that occasional close encounters of the RIS phantom segments do not lead to unrealistically large values of M2. 

The free energy of a polymer consists of the entropic term and the internal energy of the segmental interactions, U, which represents the excluded-volume effect, and can be expanded as a power series of the segment density p (Eq. 1) ... 

A second problem with the random walk model concerns the interaction between segments far apart along the contour of the chain but which are close together in space. This is the so-called "excluded volume" effect. The inclusion of this effect gives rise to an expansion of the chain, and in three-dimensions, 2 a, r3/5 (9), rather than the r dependence given in equation (I). 

The effect of the solvent is a complex one, but can again be taken care of empirically. It is possible to define a solvent (a theta, 6, solvent) whereby the net effect of an unfavourable segment-solvent interaction is to reduce the dimensions of the chain so as to exactly compensate for the excluded volume effect. 

Should the macromolecules interact with each other, then d In yildcj does not vanish. In actual experience, its value is almost always positive, largely because of excluded volume effects. Then, c,[0 In Jt/dc,] will then increase in magnitude as Cj increases and r decreases. Thus, the downward curvature shown in curve B of Figure 21.3 is typical of nonideal behavior. 

If the backbone as well as the side chains consist of flexible units, the molecular conformation arises out of the competition of the entropic elasticity of the confined side chains and the backbone [ 153 -155]. In this case, coiling of the side chains can occur only at the expense of the stretching of the backbone. In addition to the excluded volume effects, short range enthalpic interactions may become important. This is particularly the case for densely substituted monoden-dron jacketed polymers, where the molecular conformation can be controlled by the optimum assembly of the dendrons [22-26,156]. If the brush contains io-nizable groups, the conformation and flexibility may be additionally affected by Coulomb forces depending on the ionic strength of the solvent [79,80]. 

The geometrical structure of the chain in Fig. 2 is determined entirely by 6, (j), and B. The model only describes the linker geometry and does not account for excluded volume effects and other forms of nucleosome-nucleosome interaction it assumes that the core particles are point-like and that they are located at the joints of the linkers, which are straight rods. 

It is generally recognized that the average dimensions of a macromolecule are dependent on two factors one related to short distance interactions, those to which conformational analysis is applied, and one concerned with long distance interactions that are often considered in terms of excluded volume effects or covolume effects. One may, therefore, put ... 

If interactions between parts of the molecule separated by many links (the excluded volume effect ) is absent, so that the chains obey random-flight statistics, takes its unperturbed value, (s ). Theoretical calculations of the dimensions of branched molecules usually assume random flight chains, and values of the mean-square radius so obtained are estimates of . 

The size of a polymer molecule in solution is influenced by both the excluded volume effect and thermodynamic interactions between polymer segments and the solvent, so that in general =t= . The Flory (/S) expansion factor a is introduced to express this effect, by writing ... 

In Flory s theory (/< ), a polymer-solvent system is characterized by a temperature 0 at which (i) excluded-volume effects are just balanced by polymer-solvent interactions, so that os=l, (ii) the second virial coefficient is zero, irrespective of the MW of the polymer, and (iii) the polymer, of infinite molecular weight, is just completely miscible with the solvent The fundamental definition of the temperature is a macroscopic one, namely that for T near 0 the excess chemical potential of the solvent in a solution of polymer volume fraction v2 is of the form (18) ... 

Here a0 is a constant called the effective bond length of the chain, and as(z) is a dimensionless quantity called the linear expansion factor of the chain. The latter depends on long-range interactions between pairs of monomer units and chain length through the so-called excluded-volume parameter z. For details of these quantities characterizing the dimensions of random-coil polymers, the reader is referred to a recently published book by Yamakawa (40). At this place we simply note that as tends to unity in the absence of excluded-volume effect. 

In order that the contribution from random-coil sequences might be more correctly evaluated, Miller and Flory (43) carried out a calculation, taking into account the hindrance to internal rotation of bonds, van der Waals interactions of non-bonded atoms, dipole-dipole interactions between atomic groups, and so forth. However, no excluded-volume effect was allowed for. Actually, they... 

The excluded volume effect is associated with a reduction in the mixing entropy of the system. The resulting steric5 interactions contribute... 

Thermodynamically unfavourable interactions are ubiquitous in mixed biopolymer systems. As described in chapters 3 and 5, they arise mainly from excluded volume effects — the physical volume of one biopolymer molecule is inaccessible to other biopolymer molecules — and also from electrostatic repulsion between like-charged groups on different molecules (Ogston, 1970 Nagasawa and Takahashi, 1972 Tanford, 1961). 

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