Osmotic virial expansion

The major theories developed to predict phase separation and biomolecule partitioning in aqueous two-phase systems are mostly extensions of the widely known polymer solution theories of Flory and Huggins and the osmotic virial expansion. 

Where R is the gas constant, T the absolute temperature, and M the molecular weight of the polymer. This series is usually called the osmotic virial expansion, with A (i = 2, 3,...) being referred to as the i-th virial coefficient of the... 

Reminiscent of Eqs. (3.11) and (3.12), tliis series is called an osmotic virial expansion. Show tlrat the second osmotic virial coefficieirt B is ... 

This is known as osmotic virial expansion and is analogous to the virial expansion of the compressibility factor. At very low solute concentrations, the linear and higher-order terms in Ci on the right-hand side are negligible and eq. reverts to The coefficients B(T), C(T), and others are the osmotic virial... 

Inserting liquation P.8 in this relation, one obtains a series for c in terms of X that can be inverted to give A, as a series in c. Finally, inserting this series in the X series for K, one obtains the so-caUed osmotic virial expansion. 

In analogy to the description of polymer solutions where the theta temperature T is normally identified as an essential reference temperature [4,62,66], To can also be defined for dilute polymer blends. Since either of the two components may be the dilute species, there are two osmotic virial expansions and two theta temperatures whose SLOT expressions are [47]... 

Theta conditions in dilute polymer solutions are similar to tire state of van der Waals gases near tire Boyle temperature. At this temperature, excluded-volume effects and van der Waals attraction compensate each other, so tliat tire second virial coefficient of tire expansion of tire pressure as a function of tire concentration vanishes. On dealing witli solutions, tire quantity of interest becomes tire osmotic pressure IT ratlier tlian tire pressure. Its virial expansion may be written as... 

Coefficients in the virial expansion of the osmotic pressure as a power series in the concentration c (Chap. XII et seq.). 

Coefficients in the alternative virial expansion of the osmotic pressure (see Eqs. VII-13 and XII-76). 

This is a virial expansion form of the osmotic pressure analogous to the van der Waals fluid. Dusek and Patterson examined this equation and predicted the presence of two phases, i.e. collapsed and swollen phases. % is temperature dependent and is given by,... 

In these equations ns is the solvent refractive index, dn/dc the refractive index increment, c the polymer concentration in g/ml, T the temperature in K, R the gas constant, NA Avogadro s number, and n the osmotic pressure. Equation (B.8) follows from Eq. (B.7) by using the familiar virial expansion of the osmotic pressure... 

What does this picture predict for the scaling functions Let us first consider the osmotic pressure. Clearly for small overlap the virial expansion holds... 

Some authors [59] call the region c < c the virial regime. Probably they consider that the virial expansion for osmotic pressure would diverge as c approaches c. ... 

Section 2 brings the cluster development for the osmotic pressure. Section 3 generalizes the approach of Section 2 to distribution functions, including a new and simple derivation of the cluster expansion of the pair distribution function. Section 4 presents a new expression for the chemical potential of solvents in dilute solutions. Section 5 contains an application of our general solution theory to compact macromolecular molecules. Section 6 contains the second osmotic virial coefficient of flexible macromokcules, followed in Section 7 by concluding remarks. 

We shall now derive the virial expansion of the osmotic pressure following McMillan and Mayer and Hill (P). but simplifying the derivation. The virial expansion plays an imjwrtant role in the theory of solutions. For our purpose we introduce the grand partition function of... 

Each polymer coil in a solution contributes to viscosity. In very dilute solutions, the contribution from different coils is additive and solution viscosity r] increases above the solvent viscosity t/s linearly with polymer concentration c. The effective virial expansion for viscosity at low concentration is of the same form as Eq. (1.76) for osmotic pressure and Eq. (1.96) for light scattering ... 

This virial expansion is analogous to that used for the osmotic pressure in Chapter 1 [Eq. (1.74)] and we will see in Section 3.3.4, how the excluded volume is related to the second virial coefficient. 

This expression of osmotic pressure can be written in the form of the virial expansion in terms of number density of A monomers c = 4>jb [see Eq. (3.8)]... 

Recall the mean-field virial expansion for the osmotic pressure of polymer solutions discussed in Section 4.5.1 [Eq. (4.67)]. 

The semidilute osmotic pressure has a stronger concentration dependence than predicted by the mean-field virial expansion [Eq. (5.41)] -------... 

The mean-field prediction for the osmotic pressure [Eq. (5.40)] in 0-sol-vents is the virial expansion with vanishing excluded volume (v = 0) ... 

Our purpose in this section is to derive a set of useful expressions for the chemical potentials starting with the principles of statistical mechanics. The expressions we shall obtain take the form of virial expansions similar to those of the Edmond and Ogston (6) but having a very different theoretical basis. Our model parameters are isobaric-isothermal virial coefficients which are about an order of magnitude smaller than the osmotic virial coefficients in the Edmond and Ogston model. We shall develop the theory neglecting the effect of polydispersity because we empirically did not find this to be very important at the level of accuracy commonly attainable in experimental phase diagrams for these systems. 

At this point we note that Equation 13 is the McMillan-Mayer (16) expansion for the osmotic compressibility factor which is fundamentally different from the analogous expansion that was obtained from the formalism of Hill (Equation 10). We also identify B as a McMillan-Mayer osmotic virial coefficient. 

A theoretical treatment of aqueous two-phase extraction at the isoelectric point is presented. We extend the constant pressure solution theory of Hill to the prediction of the chemical potential of a species in a system containing soivent, two polymers and protein. The theory leads to an osmotic virial-type expansion and gives a fundamentai interpretation of the osmotic viriai coefficients in terms of forces between species. The expansion is identical to the Edmunds-Ogston-type expression oniy when certain assumptions are made — one of which is that the solvent is non-interacting. The coefficients are calculated using simple excluded volume models for polymer-protein interactions and are then inserted into the expansion to predict isoelectric partition coefficients. The results are compared with trends observed experimentally for protein partition coefficients as functions of protein and polymer molecular weights. 

The theory leads to an osmotic virial type expansion and gives a fundamental interpretation of the coefficients appeEiring in this expansion in terms of forces between the species. The expansion reduces to the Edmunds-Ogston expression only when certain assumptions are made -namely that the fluids are incompressible and that the solvent is... 

Equations 13 and 14 have the same functional form as that postulated by Edmonds and Ogston (3) and later generalized by King et aL (4). The significance of the work presented here is that it enables us to give a fundamental interpretation of the coefficients and the reference potential in terms of forces between the species. It also allows us to relate these coefficients to the virial coefficients which appear in the McMillan-Meyer virial expansion (12) of the osmotic pressure. In the equations of Ogston and of King et the coefficients are set equal to the virial coefficients of the McMillan-Meyer virial expansion, but, as we shall see, these coefficients are equivalent only when certain assumptions are made. 

Generally speaking, positive values of A2 mean net repulsion between the particles, while negative values of A2 correspond to attraction. Eor more detailed analysis of the values of the second osmotic virial coefficient, the use of other definitions of the particle concentration is more convenient. The common virial expansion"... 

In practice, the most important set of thermodynamic variables is of course T, P, pA, employed in (6.34). However, relation (6.33) is also useful and has enjoyed considerable attention in osmotic experiments where pB is kept constant. This set of variables provides relations which bear a remarkable analogy to the virial expansion of various quantities of real gases. We demonstrate this point by extracting the first-order expansion of the osmotic pressure n in the solute density pA. This can be obtained by the use of the thermodynamic relation... 

The virial expansion of the osmotic pressure, although formally exact, is not very useful beyond the first-order correction to the DI limiting case. Higher-order correction terms involve higher-order potentials of mean force about which very little is known. 

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