The Excluded Volume

In order to obtain the maximum Z( Na ), we minimize Equation 2.19 with respect to the orientation distribution function while retaining the normalization condition 2.20. The Euler-Lagrange equation is 

When the molecules are brought to the distance of separation a, the combined concentration of polymer segments belonging to both molecules in the corresponding volume element 8V is 

Letting Vi represent the volume of a solvent molecule, the corresponding numbers of solvent molecules are 

Then from Eq. (40) the free energy of mixing segments and solvent in the initial state in which the two volume elements are well separated is 

Similarly for the volume element when the molecules overlap 

In order to render the expression for d AFa) in a usable form, it remains to evaluate pk and pi. We have already pointed out that the average segment density of a molecule will be greatest at the center of gravity and that it will decrease smoothly as the distance 5 (Fig. 114,a) from the center is increased. While the distribution will not be exactly a Gaussian function of s, it may be so represented without introducing an appreciable error in our final result, which can be shown to be insensitive to the exact form assumed for the radial dependence of the segment density. Hence we may let 

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A number of refinements and applications are in the literature. Corrections may be made for discreteness of charge [36] or the excluded volume of the hydrated ions [19, 37]. The effects of surface roughness on the electrical double layer have been treated by several groups [38-41] by means of perturbative expansions and numerical analysis. Several geometries have been treated, including two eccentric spheres such as found in encapsulated proteins or drugs [42], and biconcave disks with elastic membranes to model red blood cells [43]. The double-layer repulsion between two spheres has been a topic of much attention due to its importance in colloidal stability. A new numeri-... 

Flere b corresponds to the repulsive part of the potential, which is equivalent to the excluded volume due to the finite atomic size, and a/v corresponds to the attractive part of the potential. The van der Waals equation... 

The nth virial coefficient = < is independent of the temperature. It is tempting to assume that the pressure of hard spheres in tln-ee dimensions is given by a similar expression, with d replaced by the excluded volume b, but this is clearly an approximation as shown by our previous discussion of the virial series for hard spheres. This is the excluded volume correction used in van der Waals equation, which is discussed next. Other ID models have been solved exactly in [14, 15 and 16]. ... 

We have seen that the DFI theory in the limiting case neglects excluded volume effects in fact the excluded volume of the centra ion can be introduced into the theory as explained after A2.4.48. If the radius of the ions is taken as a for all ions, we have, in first order. 

The simplest extension to the DH equation that does at least allow the qualitative trends at higher concentrations to be examined is to treat the excluded volume rationally. This model, in which the ion of charge z-Cq is given an ionic radius d- is temied the primitive model. If we assume an essentially spherical equation for the u. . 

That state of affairs in which the poorness of the solvent exactly compensates for the excluded volume effect is called a 0 condition or Flory condition, after... 

It is convenient to begin by backtracking to a discussion of AS for an athermal mixture. We shall consider a dilute solution containing N2 solute molecules, each of which has an excluded volume u. The excluded volume of a particle is that volume for which the center of mass of a second particle is excluded from entering. Although we assume no specific geometry for the molecules at this time, Fig. 8.10 shows how the excluded volume is defined for two spheres of radius a. The two spheres are in surface contact when their centers are separated by a distance 2a. The excluded volume for the pair has the volume (4/3)7r(2a), or eight times the volume of one sphere. This volume is indicated by the broken line in Fig. 8.10. Since this volume is associated with the interaction of two spheres, the excluded volume per sphere is... 

Regardless of the particle geometry, the excluded volume exceeds the actual volume of the molecules by a factor which depends on the shape of the particles. 

Equation (8.97) shows that the second virial coefficient is a measure of the excluded volume of the solute according to the model we have considered. From the assumption that solute molecules come into surface contact in defining the excluded volume, it is apparent that this concept is easier to apply to, say, compact protein molecules in which hydrogen bonding and disulfide bridges maintain the tertiary structure (see Sec. 1.4) than to random coils. We shall return to the latter presently, but for now let us consider the application of Eq. (8.97) to a globular protein. This is the objective of the following example. 

We saw in Chap. 1 that the random coil is characterized by a spherical domain for which the radius of gyration is a convenient size measure. As a tentative approach to extending the excluded volume concept to random coils, therefore, we write for the volume of the coil domain (subscript d) = (4/3) n r, and combining this result with Eq. (8.90), we obtain... 

The above argument shows that complete overlap of coil domains is improbable for large n and hence gives plausibility to the excluded volume concept as applied to random coils. More importantly, however, it introduces the notion that coil interpenetration must be discussed in terms of probability. For hard spheres the probability of interpenetration is zero, but for random coils the boundaries of the domain are softer and the probability for interpenetration must be analyzed in more detail. One method for doing this will be discussed in the next section. Before turning to this, however, we note that the Flory-Huggins theory can also be used to yield a value for the second virial coefficient. 

We begin our attempt to reconcile these two expressions for the excluded volume of the random coil by reviewing some ideas about random coils from Chap. 1 ... 

The full Flory-Krigbaum theory results in the following expression for the excluded volume ... 

If we disregard the term in brackets in Eq. (8.112)-which we shall call f(y) and which is close to unity for values of x that are not too different from 1/2-this expression gives the same result for the excluded volume of the coil as given by Eq. (8.104) from a comparison of Eqs. (8.97) and (8.103). We may regard f(y) as a correction factor which is required for Eq. (8.104) to be valid as the difference 1/2 - x increases. In the next section we shall discuss the implications of Eq. (8.112) in greater detail. 

Our primary interest in the Flory-Krigbaum theory is in the conclusion that the second virial coefficient and the excluded volume depend on solvent-solute interactions and not exclusively on the size of the polymer molecule itself. It is entirely reasonable that this should be the case in light of the discussion in Sec. 1.11 on the expansion or contraction of the coil depending on the solvent. The present discussion incorporates these ideas into a consideration of solution nonideality. 

It is interesting to note that for a van der Waals gas, the second virial coefficient equals b - a/RT, and this equals zero at the Boyle temperature. This shows that the excluded volume (the van der Waals b term) and the intermolecular attractions (the a term) cancel out at the Boyle temperature. This kind of compensation is also typical of 0 conditions. 

Of interest is the manner in which cavities of the appropriate size are introduced into ion-selective membranes. These membranes typically consist of highly plasticized poly(vinyl chloride) (see Membrane technology). Plasticizers (qv) are organic solvents such as phthalates, sebacates, trimelLitates, and organic phosphates of various kinds, and cavities may simply be the excluded volumes maintained by these solvent molecules themselves. More often, however, neutral carrier molecules (20) are added to the membrane. These molecules are shaped like donuts and have holes that have the same sizes as the ions of interest, eg, valinomycin [2001-95-8] C H QN O g, and nonactin [6833-84-7] have wrap around stmctures like methyl monensin... 

Here,. Ai(X) is the partial SASA of atom i (which depends on the solute configuration X), and Yi is an atomic free energy per unit area associated with atom i. We refer to those models as full SASA. Because it is so simple, this approach is widely used in computations on biomolecules [96-98]. Variations of the solvent-exposed area models are the shell model of Scheraga [99,100], the excluded-volume model of Colonna-Cesari and Sander [101,102], and the Gaussian model of Lazaridis and Karplus [103]. Full SASA models have been used for investigating the thermal denaturation of proteins [103] and to examine protein-protein association [104]. 

Figure 4 Excluded volume for the Di agonist pharmacophore. The mesh volume shown by the black lines is a cross section of the excluded volume representing the receptor binding pocket. Dihydrexidine (see text) is shown in the receptor pocket. The gray mesh represents the receptor essential volume of inactive analogs. The hydroxyl binding, amine binding, and accessory regions are labeled, as is the steric occlusion region.

FIG. 2 Geometry for a simulation of a cylindrical pore in contact with a bulk-like aqueous phase. A, V, P, and W denote the aqueous phase, the excluded volume, the pore wall, and the confining walls, respectively. 

Since, in contrast to experiment, the simulation knows in detail what the connectivity looks hke, how long the strands are, and how the network loops are distributed, one can attribute this behavior to the non-crossability of the chains. Actually, one can even go further by allowing the chains to cross each other but still keep the excluded volume. Such a technical trick, which is only possible in simulations, allows one to isolate the effect of entanglement and non-crossability in such a case. As one would expect, if one allows chains to cross through each other one recovers the so-called phantom network result. 

In Eq. (15) the second term reflects the gain in entropy when a chain breaks so that the two new ends can explore a volume Entropy is increased because the excluded volume repulsion on scales less than is reduced by breaking the chain this effect is accounted for by the additional exponent 9 = y — )/v where 7 > 1 is a standard critical exponent, the value of 7 being larger in 2 dimensions than in 3 dimensions 72 = 43/32 1.34, 73j 1.17. In MFA 7 = 1, = 0, and Eq. (15) simplifies to Eq. (9), where correlations, brought about by mutual avoidance of chains, i.e., excluded volume, are ignored. 

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. 

Let us consider a simple self-avoiding walk (SAW) on a lattice. The net interaction of solvent-solvent, chain-solvent and chain-chain is summarized in the excluded volume between the monomers. The empty lattice sites then represent the solvent. In order to fulfill the excluded volume requirement each lattice site can be occupied only once. Since this is the only requirement, each available conformation of an A-step walk has the same probability. If we fix the first step, then each new step is taken with probability q— 1), where q is the coordination number of the lattice ( = 4 for a square lattice, = 6 for a simple cubic lattice, etc.). 

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... 

Universal SEC calibration reflects differences in the excluded volume of polymer molecules with identical molecular weight caused by varying coil conformation, coil geometry, and interactive propenies. Intrinsic viscosity, in the notation of Staudinger/ Mark/Houwink power law ([77]=fC.M ), summarizes these phenom-... 

Some GPC analysts use totally excluded, rather than totally permeated, flow markers to make flow rate corrections. Most of the previously mentioned requirements for totally permeated flow marker selection still are requirements for a totally excluded flow marker. Coelution effects can often be avoided in this approach. It must be pointed out that species eluting at the excluded volume of a column set are not immune to adsorption problems and may even have variability issues arising from viscosity effects of these necessarily higher molecular weight species from the column. 

The simplest situation is the symmetrical one (NA = NB), with the solvent equally good for both blocks. We imagine that the excluded volume interactions of A and B are stronger than the A-B repulsive interactions so that the overall structure of the layer is like that of a single component in other words, both components are equally stretched. The issue is whether or not they are homogeneously mixed with one another in the monolayer. This is essentially a two-dimensional random mixing process. In that spirit, we write the free energy... 

The quantity b has the dimension of a volume and is known as the excluded volume or the binary cluster integral. The mean force potential is a function of temperature (principally as a result of the soft interactions). For a given solvent or mixture of solvents, there exists a temperature (called the 0-temperature or Te) where the solvent is just poor enough so that the polymer feels an effective repulsion toward the solvent molecules and yet, good enough to balance the expansion of the coil caused by the excluded volume of the polymer chain. Under this condition of perfect balance, all the binary cluster integrals are equal to zero and the chain behaves like an ideal chain. 

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

Unfortunately, exclusion chromatography has some inherent disadvantages that make its selection as the separation method of choice a little difficult. Although the separation is based on molecular size, which might be considered an ideal rationale, the total separation must be contained in the pore volume of the stationary phase. That is to say all the solutes must be eluted between the excluded volume and the dead volume, which is approximately half the column dead volume. In a 25 cm long, 4.6 mm i.d. column packed with silica gel, this means that all the solutes must be eluted in about 2 ml of mobile phase. It follows, that to achieve a reasonable separation of a multi-component mixture, the peaks must be very narrow and each occupy only a few microliters of mobile phase. Scott and Kucera (9) constructed a column 14 meters long and 1 mm i.d. packed with 5ja... 

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