Composition fluctuations correlation

The partial structure factors for binary (Bhatia and Thorton, 1970) and multicomponent (Bhatia and Ratti, 1977) liquids have been expressed in terms of fluctuation correlation factors, which at zero wave number are related to the thermodynamic properties. An associated solution model in the limits of nearly complete association or nearly complete dissociation has been used to illustrate the composition dependence of the composition-fluctuation factor at zero wave number, Scc(0). For a binary liquid this is inversely proportional to the second derivative of the Gibbs energy of mixing with respect to atom fraction. 

The derivation of the entropy in terms of the long-wavelength composition fluctuations seems to need no generalization to time-dependent temperature variations. However, the composition correlations which appear in the entropy expression will not respond fully and in phase with a rapid temperature variation, and a computation of this response is now necessary for a computation of the dynamic heat capacity. It will be supposed, as in the theory of the viscosity, that the radial distribution function satisfies the generalized diffusion equation... 

To calculate the correlation functions for composition fluctuations in a binary system, consider the Ising Hamiltonian of Eq. (2.17) with an additional term A,-5,- where A,- is a local field. Show that the correlation function... 

Use the continuum description of the binary system described by the free energy of Eq. (2.28) to derive an expression for the correlation function for composition fluctuations. To do this, treat Eq. (2.28) as a new effective Hamiltonian. Examine the properties of the correlation function near the critical point and compare with the results of the lattice-gas model of Eq. (2.22). Which fluctuations does the continuum model best describe ... 

Figure 5.6. Schematic plots of the polymer segment concentration as a function of the distance in bulk polymer solutions. In (a) the concentration of chains is low enough that on average different polymer chains do not overlap. The segment concentration has large spatial fluctuations this is a dilute solution. In (b) chains start to overlap, but there are still strong composition fluctuations imposed by the connectivity of the chains characterised by a correlation length This is the so-called semi-dilute concentration regime. In (c) the solution is concentrated and there are no concentration fluctuations on length scales larger than the monomer size.

In the vicinity of the critical point of a binary mixture one observes universal behavior, which mirrors the divergence of the correlation length of composition fluctuations. The universal behavior does not depend on the details of the system but only on the dimensionahty of space and the type of order parameter. Therefore, binary polymer blends fall into the same imiversality class as mixtures of small molecules, metalHc alloys, or the three-dimensional Ising model, hi the vicinity of the critical point, Xc = 2 for a symmetric blend [ 14], the difference of the composition of the two coexisting phases—the order parameter m—vanishes like m - XcN), where the critical exponent... 

We have demonstrated this for one specific case of an incompressible blend and suspect that it may be a featiue of incompressible blends in general. The observation that the fluctuations in W and o) are not correlated with each other is presumably related to the fact that the (vanishing) density fluctuations do not influence the composition fluctuations. If that is true, we can conclude that field-theoretic Monte Carlo can be used to study fluctuations in polymer mixtures in the limits of high and low compressibiUties. Whether it can also be applied at intermediate compressibilities will have to be explored in the future. 

The influence of the chain connectivity on the dynamics of the composition fluctuations does not only influence two-point correlation functions like the global structure factor but it is also visible in the time evolution of composition profiles in the vicinity of a surface [109]. 

Fig. 4. Depolarized intensity correlation functions for the (PS/PI) polymer blend of Figure 3. The relaxation mode that is shown in the VH scattering geometry is due to double scattering induced by composition fluctuations. A 353 K (VH) 343 K (VH) o333K(VH).

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