Dimensionless virial coefficients

Figure 4. Dimensionless virial coefficient of the interaction, calculated as a function of the electrolyte concentration for various p/e) ratios, compared with the experimental results of ref 22 (circles). The crosses represent the experimental results when the partial dissociation of the sodium acetate is taken into account.

The expressions for the virial coefficients may be written also in dimensionless form. Defining dimensionless virial coefficients by... 

The dimensionless virial coefficients are determined from the expansion where g is the dimensionless second virial coefficient... 

In the asymptotic limit where the radii of both chains are infinite (the radius ratio o = Ra Rb being kept constant) the dimensionless virial coefficients Baa Sbb Sab finite universal value. Starting from... 

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... 

In a dilute solution in a common good solvent for both blocks, the interactions between. different copolymers may be studied using the same direct renormalization procedures as the interactions between two homopolymers A and B equivalent to the two blocks.As for blends, in the asymptotic limit of infinite molecular masses, the chemical difference between the two blocks is irrelevant and the dimensionless virial coefficient gc between block copolymers defined by Eq. (10) is equal to the same value g as for homopolymers. The interactions which may provoke the formation of mesophases are here again due to the corrections to the scaling behavior ... 

It turns out that of = 3 is very special, because it is the upper critical dimension for tricritical behavior. This is the deep reason underlying the fact that polymers at the theta point in = 3 are quasi-ideal (i.e., have size exponent = j and have all dimensionless virial coefficients vanishing in the limit of infinite chain length). In dimension rf < 3, polymers at the theta point are not quasi-ideal." ... 

The osmotic second virial coefficient A2 is another interesting solution property, whose value should be zero at the theta point. It can be directly related with the molecular second virial coefficient, expressed as B2=A2M /N2 (in volume units). For an EV chain in a good solvent, the second virial coefficient should be proportional to the chain volume and therefore scales proportionally to the cube of the mean size [ 16]. It can, therefore, be expressed in terms of a dimensionless interpenetration factor that is defined as... 

Here D is the translational diffusion coefficient of the solute molecule at C —> 0 with C the mass concentration of the solute, kd the diffusion second virial coefficient, f a dimensionless parameter depending on polymer chain structure and solvent, and the mean square radius of gyration of the polymer chain. Hence, for C and q small enough, Eq. (2.3) may be approximated by... 

For the values of the parameters employed (a relatively large Hamaker constant), the potential barrier is only a few kT or less hence, the apoferritin should coagulate at almost all the concentrations studied. Since experiment shows that the proteins did not coagulate, another repulsion should be present, at least al low separation distances. This repulsion, while essential for the stability of the system, did not affect much, because of its short range, the behavior of the second virial coefficient. In the calculation of the second virial coefficient, it was assumed that the distance of closest approach between apoferritin proteins cannot be less than 8 A. This value leads to a dimensionless second virial coefficient for the hard spheres repulsion of 4.8 instead of 4. 

Hill osmotic virial coefficient for components i and j, dimensionless. 

We also use physical dimensionless parameters, g and h which are respectively related to the second and to the third virial coefficient. More precisely, we write... 

For d — 3 the three-body interaction is just marginal. This is not the case for d < 3 (and in particular for d = 2) since, then, the dimensionless parameter y = cS3 t becomes infinite when S - oo. In this case, a system with two-body and three-body interactions is tricritical when the second virial coefficient A2 vanishes (or z = 0). Then, for an isolated chain, the dependence of R2 with respect to S (for large S) is characterized by an exponent v, (v, > 1/2)... 

Dimensionless form of equation-of-state parameter b Second virial coefficient (m /mol)... 

For uncharged coronal blocks A, the short-ranged (van der Waals) interactions between monomer units are described in terms of a virial expansion. The latter accounts for the monomer-monomer binary (pair) interactions, with second virial coefficient VaO, or the ternary interactions with third virial coefficient waa . We assume that the monomer unit length, a, is the same for both blocks A and B. In the following, we use a as a unit length to make all lengths dimensionless and eliminate a in further equations. We also assume that the (dimensionless) second virial coefficient Va > 0 and that the third virial coefficient Wa 1. 

As the above expressions for Uk meike clear, the reduced virial coefficients are dimensionless, and therefore independent of the sphere diameter a. To simplify the following discussion, we will therefore set <7=1. With this choice, the product of /-functions in the denominator is nonzero in the region where r/ < 1 for all / n. The maximum volume assumed by a parallelotope within this region is just 1. (If the edges incident with vertex n are used to define the parallelotope, it is just a unit cube for other choices of reference vertex the parallelotope is skewed, but still has unit voltune.) For the numerator, the product of /- functions is nonzero when rim < 1 for all /m n. The maximum parallelotope volume within this region can be determined simply by maximizing the expression for the parallelotope volume, subject to the... 

A useful empirical correlation for the second virial coefficient is based on the Pitzer method. In this method, the dimensionless ratio, BPc/RT, is expressed in the form... 

The dimensionless second virial coefficient in the two-parameter theory (Equation 3.1 147)... 

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). 

Figure 3.1 The dimensionless second virial coefficient Sip llRT ) of argon as a function of reduced temperature TjT, where p is the critical pressure and 7 is the critical temperature. Values computed from the equation of state of Tegeler et al The insert shows the same data on an inverse temperature scale.

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