Virial coefficient mean-field

Thus one must rely on macroscopic theories and empirical adjustments for the determination of potentials of mean force. Such empirical adjustments use free energy data as solubilities, partition coefficients, virial coefficients, phase diagrams, etc., while the frictional terms are derived from diffusion coefficients and macroscopic theories for hydrodynamic interactions. In this whole field of enquiry progress is slow and much work (and thought ) will be needed in the future. 

Among other approaches, a theory for intermolecular interactions in dilute block copolymer solutions was presented by Kimura and Kurata (1981). They considered the association of diblock and triblock copolymers in solvents of varying quality. The second and third virial coefficients were determined using a mean field potential based on the segmental distribution function for a polymer chain in solution. A model for micellization of block copolymers in solution, based on the thermodynamics of associating multicomponent mixtures, was presented by Gao and Eisenberg (1993). The polydispersity of the block copolymer and its influence on micellization was a particular focus of this work. For block copolymers below the cmc, a collapsed spherical conformation was assumed. Interactions of the collapsed spheres were then described by the Hamaker equation, with an interaction energy proportional to the radius of the spheres. 

Figure 1 shows the representation of the experimental isotherm (B. G. Aristov, V. Bosacek, A. V. Kiselev, Trans. Faraday Soc. 1967 63, 2057) of xenon adsorption on partly decationized zeolite LiX-1 (the composition of this zeolite is given on p. 185) with the aid of the virial equation in the exponential form with a different number of coefficients in the series i = 1 (Henry constant), i = 2 (second virial coefficient of adsorbate in the adsorbent molecular field), i = 3, and i = 4 (coefficients determined at fixed values of the first and the second coefficients which are found by the method indicated for the adsorption of ethane, see Figure 4 on p. 41). In this case, the isotherm has an inflection point. The figure shows the role of each of these four constants in the description of this isotherm (as was also shown on Figure 3a, p. 41, for the adsorption of ethane on the same zeolite sample). The first two of these constants—Henry constant (the first virial constant) and second virial coefficient of adsorbate-adsorbate interaction in the field of the adsorbent —have definite physical meanings. 

A. V. Kiselev Zeolites are porous crystals. This means that we can find the molecular field distribution in their channels. The advantage of describing the adsorption on zeolites using the molecular theory consists in obtaining the constants which have a definite physical meaning (for example, the Henry constant and second virial coefficient). Further development of the theory needs a further improvement of the model based on the investigation of the adsorbate-zeolite systems by the use of modern physical methods. 

As X is lowered, the polymer likes the solvent more, increasing the osmotic pressure. However, the mean-field theory that is the basis of Eq. (4.71) is only valid close to the 0-temperature, where chains interpenetrate each other freely [Eq. (3.102)]. Far above the 0-temperature (in good solvent), the second virial coefficient A2 is related to chain volume [Eq. (3.104)] rather than monomer excluded volume v. Recall that the second virial coefficient can also be determined from the concentration dependence of scattering intensity [Eq. (1.91)]. 

To parameterize the polymersome model, the identification of the virial coefficients, vaij and wapv is driven by the requirement that the amphiphiles described by (10) and (12) should create a stable bilayer with given material properties. Assuming that the hydrophobic interior should be in a melt state, the coefficients vaa and Waaa are determined such that (12) enforces the A-blocks to create a melt in equilibrium with its vapor which, in a solvent free model, represents the surrounding water. It can be shown, from (12), that the equation of state of such a homogeneous melt, within mean-field approximation, is [138]... 

Figure 4. Adnesion energy (erg/cm ) for SUPC biiayers in 0.1 M salt (PBS) plus dextran polymers. Number average polymer indices - Np (number of glucose monomers). Solid and dasned curves - predictions from mean field theory with first and second virial coefficients from osmotic pressure measurements (14).

Another practical definition of 0 can be proposed. We may call 0 the temperature at which the second virial coefficient between two very large coils vanishes. Fortunately, these two definitions coincide. When we are on the dividing line, the parameter u (at the m-th iteration) gives (in dimensionless units) the virial coefficient between two subunits. Since the dividing line ends at 0, where u = 0, this coefficient vanishes when the subunits are large enough. The distinction between 0 ai 0 is essentially absent from the polymer literature (which has been written mainly on the mean field level). 

Solution activity data obtained by osmometry on dilute solutions showed that the second virial coefficient is dependent on molar mass, contradicting the Flory-Huggins theory. These problems arise from the mean-field assumption used to place the segments in the lattice. In dilute solutions, the polymer molecules are well separated and the concentration of segments is highly non-uniform. Several scaling laws were therefore developed for dilute (c < c is the polymer concentration in the solution, c is the threshold concentration for molecular overlap) and semi-dilute (c > c ) solutions. In a good solvent the threshold concentration is related to molar mass as follows ... 

The mean-field expression for the virial coefficients may be used only when different chains can interpenetrate freely. This is not the case for a polymer solution in a good solvent (Xas 1/2). The excluded volume interactions (Xas monomer virial coefficient decay to zero as an inverse power of the molecular weight of the chains. " Vfe discuss here the interpenetration of different macromolecules A and B in terms of simple scaling laws that are valid in the asymptotic limit where the molecular weights of both polymers are large. 

If the interpenetration free energy Fint is much smaller than kT, the two polymers interpenetrate almost freely. They are diaphanous to each other and a mean-field theory (Eq. 4) may be used to calculate the mutual virial coefficient Gab ... 

This result does not quite agree with experimental data as a function of molecular weight, since A2 decreases weakly at higher values of Z. The Hory-Krigbaum potential is not quite correct, and a mean field approach is not quite right. But, the major improvement in the understanding of the second osmotic virial coefficient obtained by them remains a monumental achievement. 

Logarithmic corrections to the mean-field theory of single chains are known, for example, for the finite-chain Boyle temperature Tb(N), where the second virial coefficient vanishes. The scaling of the deviation of Tb(N) from T reads [124] ... 

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