Thermodynamically equivalent sphere

Table 2). All the radii have a certain molar mass dependence. The magnitudes of these radii, however, can deviate strongly from each other. These differences result from the fact that they are physically differently defined. The radius of gyration, R, is solely geometrically defined the thermodynamically equivalent sphere radius, R-p is defined by the domains of interaction between two macromolecules, or in other words, on the excluded volume. The two hydrodynamic radii R and R result from the interaction of the macromolecule with the solvent (where the latter differs from R by the fact that in viscometry the particle is exposed to a shear gradient field).

The second virial coefficient has also been expressed in terms of the radius of the thermodynamic equivalent sphere ... 

Let us assume the random coil in the solution as a hard sphere of the radius / h as in the thermodynamic sphere (Figure 2.8). This hypothetical sphere is not the representative of the segment distribution, but shows the region inside the coil where the solvent flow cannot pervade. It is called the hydrodynamically equivalent sphere (Figure 2.12). Its volume is vh =4ttR /3. The radius /fn is not the same as the radius of gyration, but is... 

The crucial question is at what value of <)> is the attraction high enough to induce phase separation De Hek and Vrij (6) assume that the critical flocculation concentration is equivalent to the phase separation condition defined by the spinodal point. From the pair potential between two hard spheres in a polymer solution they calculate the second virial coefficient B2 for the particles, and derive from the spinodal condition that if B2 = 1/2 (where is the volume fraction of particles in the dispersion) phase separation occurs. For a system in thermodynamic equilibrium, two phases coexist if the chemical potential of the hard spheres is the same in the dispersion and in the floe phase (i.e., the binodal condition). 

This allows us to define a thermodynamically effective equivalent radius Rf by replacing the actual sphere radius of a hard sphere by R j< which gives... 

Having at our disposal accurate structural and thermodynamic quantities for HS fluid, the latter has been naturally considered as a RF. Although real molecules are not hard spheres, mapping their properties onto those of an equivalent HS fluid is a desirable goal and a standard procedure in the liquid-state theory, which is known as the modified hypemetted chain (MHNC) approximation. According to Rosenfeld and Ashcroft [27], it is possible to postulate that the bridge function of the actual system of density p reads... 

O is the so-called Flory s constant, a is the expansion factor of the polymer molecule, which depends from the thermodynamic quality of the solvent (a = 1 in ideal solvent), ( o> is the mean-square radius of gyration, is Avogadro s number, and is the volume of the equivalent hydrodynamic sphere. 

The OP space 91 is the manifold of all possible values of the OP that do not alter the thermodynamical potentials of the system. The energy of condensation Fcond takes a minimum value on 91. For a uniaxial nematic, the OP space is a sphere of unit radius any point on the sphere corresponds to a different orientation of the director n. Furthermore, since n = —n, any two diametrically opposite points on the sphere describe not just energetically equivalent states, but rather indistinguishable states. The unit sphere with identified antipodal points is denoted fZy, it is the OP space of a uniaxial nematic. 

Baxter (1968b) showed that the Ornstein-Zernike equation could, for some simple potentials, be written as two one-dimensional integral equations coupled by a function q(r). In the PY approximation for hard spheres, for instance, the q(r) functions are easily solved, and the direct-correlation function c(r) and the other thermodynamic properties can be obtained analytically. The pair-correlation function g(r) is derived from q(r) through numerical solution of the integral equation which governs g(r) for which a method proposed by Perram (1975) is especially useful. Baxter s method can also be used in the numerical solution of more complicated integral equations such as the hypernetted-chain (HNC) approximation in real space, avoiding the need to take Fourier transforms. An equivalent set of relations to Baxter s equations was derived earlier by Wertheim (1964). 

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