Mutual virial coefficient

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 ... 

We now present briefly more explicit calculations of the mutual virial coefficients obtained with the use of des Cloizeaux direct renormalization method for blends of linear flexible polymers in a common good solvent, a common 0-solvent and a selective solvent and for blends of rodlike polymers and flexible polymers in a 0-solvent (marginal behavior). These calculations enable one to find (universal) prefactors relating the mutual virial coefficient to the chain volume (in Eq. 7) in the asymptotic limit. Moreover they give the corrections to the scaling behavior which explicitly depend on the interactions between unlike monomers and are actually responsible for the phase separation of flexible polymer blends in a good solvent. 

The mutual virial coefficient has also been calculated in a... 

Notice that up to this order in e. Eqs. (14) and (15) are consistent with a mutual virial coefficient varying linearly with the mass of the larger polymer B namely, = Sb Nb below, we give a sample interpretation of this result in terms of a blob model. 

In very asymmetric systems where the radius ratio o is either very large or very small, the mutual virial coefficient Gab (o equivalently the mutual interpenetration function Was) is very difficult to measure because it gives only a small contribution to the osmotic pressure or scattering intensity. In the symmetric case where the masses of the two different polymers are equal, Waa Was are of the same order of magnitude and their difference has been measured recently by lightscattering using the so-called optical 0-solvent method which gives direct access to the monomer virial coefficient difference... 

The direct renormalization method also allows the determination of the mutual virial coefficient in a common 0-solvent (g = g = 0) and a selective solvent (g = 0, g = g ). In both cases, for symmetric polymers, when the radius ratio is equal to unity we find a hard sphere interaction characterized by a dimensionless virial coefficient... 

Whatever the solvent quality, we thus find a mutual virial coefficient Gab proportional to the mass of the larger chain Nb This is a signature of the fractal nature of the polymer chains. The volume excluded by the long polymer B to the short polymer A is not proportional to the volume of the B chains. There is some interpenetration between the chains. 

All the above results for polydisperse blends are consistent with a blob model first suggested by de Gennes. In the long chain, we group the monomers into subunits with a radius equal to the radius of the smaller chains (Fig. 1). The excluded volume Gab is the total volume occupied by these blobs. Each blob having Ra° monomers the mutual virial coefficient 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... 

When both polymers are in a solvent, the excluded volume effects are absent between chains of same chemical nature, however chains of different chemical nature and equal radius interact as hard spheres, their mutual virial coefficient is proportional to their volume. The proportionality constant is universal and has been calculated as an expansion in powers of = 4-d where d is the space dimension. 

Second virial coefficient for the mutual interactions of species i and ... 

The deterioration of the solvent qnality, that is, the weakening of the attractive interactions between the polymer segments and solvent molecules, brings about the reduction in the coil size down to the state when the interaction between polymer segments and solvent molecules is the same as the mutual interaction between the polymer segments. This situation is called the theta state. Under theta conditions, the Flory-Huggins parameter % assumes a value of 0.5, the virial coefficient A2 is 0, and exponent a in the viscosity law is 0.5. Further deterioration of solvent quality leads to the collapse of coiled structure of macromolecules, to their aggregation and eventually to their precipitation, the phase separation. 

There are a number of quantitative features of Eq. (14) which are important in relation to rapid diffusional transport in binary systems. The mutual diffusion coefficient is primarily dependent on four parameters, namely the frictional coefficient 21 the virial coefficients, molecular weight of component 2 and its concentration. Therefore, for polymers for which water is a good solvent (strongly positive values of the virial coefficients), the magnitude of (D22)v and its concentration dependence will be a compromise between the increasing magnitude of with concentration and the increasing value of the virial expansion with concentration. 

It is assumed that the molecules adsorbed on the electrode surface behave as a two-dimensional gas that can be described by an equation of state. For ideal noninteracting molecules, Henry s law, Pa = R 70, is fulfilled. When the mutual interaction between adsorbed molecules is taken into account, the equation with the virial coefficient B can be presented in the form ... 

The Direct Lattice Sum. Dispersion forces between two atoms can be described by a potential function expressed in terms containing inverse powers of the internuclear separations, s. The simplest function of this sort includes a potential energy of attraction proportional to the inverse sixth power of the separation and a repulsion that is zero at distances of separation greater than a particular value se and infinite at separations less than sc. This is the so-called hard sphere or van der Waals model. Such an approximate potential function can be improved in two respects. Investigations of the second virial coefficient have revealed that the potential energy of repulsion is best described as proportional to the inverse twelfth power of the separation and the term in sr9, which accounts for the greater part of the total attraction potential, due to the attraction of mutually induced dipoles, should have added to it the dipole-quadrupole and quadrupole-quadru-pole attractions, expressed as terms in sr8 and s-10, respectively. The complete potential function for the forces between two atoms is, therefore ... 

The limiting slope of the zero concentration line of the plot of Kc/Ro against sin 6/2 (Fig. 3-4) gives (l6n /3X Mw)f g. The mutual intercept of the zero concentration and zero angle lines gives, and the limiting slope of the zero angle line can be used to obtain the second virial coefficient as indicated by Eq. (3-20). 

The independence of and of T must be expected in those regions of the phase diagram, where the attraction between the rods is negligible in comparison with their mutual repulsion. Since the high-temperature corridor of the phase diagram is situated at i 1/p I, i.e. in the region of validity of the second virial approximation, this statement can be reformulated as follows the values of and do not depend on temperature if the contribution of the attraction to the second virial coefficient, Ba, is much less than that of the repulsion, B,. 

The fact that the second virial coefficient of polymer chains becomes negative in worse than 0-solvents is indicative of mutual segmental attraction (Flory, 1953). Moreover, as discussed in Chapter 8, when sterically stabilized particles coagulate under the action of van der Waals forces, an entirely different pattern of flocculation behaviour ensues. Specifically, flocculation occurs in dispersion media that are better solvents than 0-solvents, i.e. when the segments are undoubtedly mutually repulsive. 

According to Fig. 7.3.1 the isotherm slopes are approximately equal in formamide and methanol whereas the slope for the aqueous system is considerably larger. The mutual repulsion of adsorbed anions is therefore evidently stronger in methanol and formamide than in water. The interaction parameter is also found to depend strongly on the anion for a given solvent. For example in the formamide system the second virial coefficient (which is directly related to the interaction parameter) for adsorption of 1 ions is 310 A /ion compared with 2000 A /ion for Cl ions. Thus the simple adsorption model of point charges undergoing lateral coulombic repulsion represents a considerable oversimplification in non-aqueous solutions as in aqueous solutions. Studies of adsorption of halide anions from mixed electrolyte solutions in formamide and methanoF reveal complex behaviour which cannot be explained in terms of a simple model. 

The second virial coefficient depends on the excluded volume u. The macromolecules arrange themselves with little mutual interference since the total excluded volume Ni u is much smaller than the total volume V. The total number of possible ways of arranging these Ni macromolecules is calculated from the partition function O,... 

We have not discussed the subject of nonideal polymers in any detail apart from the excluded volume problem. Thus no mention is made of the evaluation of the potential of mean force from the monomer-solvent interaction, and subsequently the evaluation of the osmotic pressure. We refer to the treatment of Yamakawa (Ref. 5, Chapter IV) for this subject and mention only that the osmotic pressure of a polymer solution at finite concentrations is represented as a virial expansion in the polymer concentration. " The second, third, etc., virial coefficients represent the mutual interaction between two, three, etc., polymer chains in solution. Thus the functional integral techniques presented in this review should also be of use in understanding the osmotic pressure of nonideal polymer solutions. We hope that this review will stimulate such studies of this important subject. It should also be mentioned in passing that at the 0-point the second virial coefficient vanishes. In general, the osmotic pressure -n is given by the series... 

Keesom, W. M. (1915) The second virial coefficient for rigid spherical molecules whose mutual attraction is equivalent to that of a quadruplet placed at its center Proc. Roy. Acad. Sci., Amsterdam, 18, 636-646. 

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