Virial expansion validity

Equation (3.11) is the virial expansion valid for a dilute polymer solution. One may also interpret the second order term. It describes an increase in osmotic pressure due to the contacts between the dissolved polymer molecules, which occur with a probability proportional to Cp... 

Current use of statistical thermodynamics implies that the adsorption system can be effectively separated into the gas phase and the adsorbed phase, which means that the partition function of motions normal to the surface can be represented with sufficient accuracy by that of oscillators confined to the surface. This becomes less valid, the shorter is the mean adsorption time of adatoms, i.e. the higher is the desorption temperature. Thus, near the end of the desorption experiment, especially with high heating rates, another treatment of equilibria should be used, dealing with the whole system as a single phase, the adsorbent being a boundary. This is the approach of the gas-surface virial expansion of adsorption isotherms (51, 53) or of some more general treatment of this kind. 

Let us mention first the work of Stecki who expanded Bogolubov s results in a series in A28 and who with Taylor showed that this expansion is identical to all orders in A with the generalized Boltzmann operator (85).29 Since the method is rather different from the virial expansions which we present here, we give in Appendix A.III the major thoughts of this general work valid for any concentration. 

This value of kn is actually low by an order of magnitude for dilute suspensions of charged spheres of radius Rg. This is due to the neglect of interchain correlations for c < c in the structure factor used in the derivation of Eqs. (295)-(298). If the repulsive interaction between polyelectrolyte chains dominates, as expected in salt-free solutions, the virial expansion for viscosity may be valid over considerable range of concentrations where the average distance between chains scales as. This virial series may be approxi-... 

It is generally agreed that a virial form of isotherm equation is of greater theoretical validity than the DA equation. As explained in Chapter 4, a virial equation has the advantage that since it is not based on any model it can be applied to isotherms on both non-porous and microporous adsorbents. Furthermore, unlike the DA equation, a virial expansion has the particular merit that as p — 0 it reduces to Henry s law. 

This equation may be considered to be just an extension of the virial expansion and as such is empirical. This equation has been shown to be valid up to a volume fraction of < 2 0.25 for nonpolar systems. The X parameter can be determined by measuring the activity of the solvent, Oi, in a polymer solution using the equation... 

The first term of this virial expansion [Eq. (4.67)] is linear in composition and is called the van t Hoff Law [Eq. (1-72)], which is valid for ver>--dilute solutions ------------------------------------------------------------... 

It is easily seen that this set of equations is in agreement with Eq. (19) in the limit of low densities, where only the first term in Eq. (39) is retained. But, of course, the equations are of interest only in the critical region. One remark on Eq. (39) should be made the virial expansion of is quite different from the virial expansion of G r), and probably converges to zero more quickly than G r). However, does not appear in every term of the expansion and it is not certain that the second moment of is finite at the critical point, which is a necessary condition for the validity of the Ornstein-Zernike theory. 

To proceed further, we shall use the Lie equations, and hence need an initial approximation for the excess viscosity (valid for small volume fractions) from which estimates of the infinitesimal generators can be constructed. At the present time, very little information of this type exists. Indeed, reliable values are available only for the second- and third-order coefficients V2 and 3 appearing in the viscosity virial expansion... 

In the above discussion we have assumed that the polymer molecules are free in the solution, so that they scatter light individually. When the end-functionalized polymer is added into the colloidal suspension, some of the polymer molecules adsorb on the surface of the colloidal particle, and form colloid-polymer micelles. If all the polymer molecules adsorb on the colloidal surfaces (complete adsorption), the mixture can be viewed as a single-component system consisting of only colloid-polymer micelles. In this case the standard virial expansion for the scattering light intensity is still valid, and Eq. (1) becomes ... 

Discussion of Eq. (3.21) enables some direct conclusions. First, consider the dilute case, x 1, where the virial expansion is valid. We write for Fy7(x, z)... 

This relation is very useful. First, we can employ the virial expansion Eq. (3.11) valid in the dilute range for a calculation of the osmotic modulus. The result is... 

We have seen in section II that in a good solvent A-S interactions play only a marginal role in dilute solutions and excluded volume interactions dominate. In the asymptotic limit of infinite molecular weight, polymers A and B are not distinguishable, i.e., the dimensionless mutual virial coefficient g B tends to the same universal asymptotic limit g as g and Sbb (36)). As a result, contrary to the case of a common 0-solvent or of a selective solvent, the phase separation arises from the corrections to the scaling behavior. Let us consider the symmetric caseiV = N and suppose that the solution is sufficiently dilute so that the virial expansion is valid. For such a case = 1/2 and Eq. (30) leads to a critical concentration... 

Equation (8.4.2) is valid only at infinite dilution. For finite concentrations, the use of a virial expansion of the type introduced in Eq. (8.3.22) leads to... 

This section is concerned with the semidilute and concentrated regimes in which the virial expansion is no longer valid. Our discussion is confined to 11 or SII/Sc of isotropic solutions phase equilibria including lyotropic liquid crystal formation of rigid-polymer solutions are out of the scope. 

The parameter is best obtained by fitting the equation for to the experimental heats of mixing of analogous materials as reported elsewhere It can also be obtained from any other binary quantity such as the second virial coefficient, the thermal expansion coefficient of mixture, or the volume change on mixing. is assumed to be independent of temperature but as we described in the previous section this may not be valid. At present there is no way of predicting the temperature variation and one can only use empirical expressions or assume a constant value most appropriate for the temperature range of interest. 

The validity of the virial series expansion of the osmotic equation of state becomes questionable for moderately concentrated (or semidilute ) solutions, in which the overall number density N c/wo... 

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