Composition fluctuations correlation length

In spite of the constant density of the gel, the friction of the poly(N-isopropylacrylamide) gel reversibly decreases by three orders of magnitude and appears to diminish as the gel approaches a certain temperature. This phenomenon should be universal and may be observed in any gel under optimal experimental conditions of the solvent composition and the temperature because the unique parameter describing the friction is the correlation length which tends to diverge in the vicinity of the volume phase transition point of gels. The exponent v for the correlation length obtained from the frictional experiment is far from the theoretical value. It will, therefore, be important to study a poly(N-isopropylacrylamide) gel prepared at the critical isochore where the frictional property of gel may be governed by the critical density fluctuations of the gel. 

The above equation provides a basis for correlating the temperature dependence of a transport coefficient such as mass diffusivity in the supercritical region. The effects of composition, solute, and solvent characteristics can also be introduced into the correlations via and A which are system-dependent amplitudes. However, a rigorous ftest of the applicability of equation 5 requires independent measurements of the decay rate of the order-parameter fluctuations, the correlation length, and the viscosity. 

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... 

The data appear to obey a scaling law, but the physical interpretation is not completely clear. Carmona, Prudhon, and Barreau [10 had suggested that the relative fluctuations in resistivity in similar composites are due to fluctuations in the correlation length. A random-walk argument in three dimensions predicts the relative fluctuations in resistivity for a statistical ensemble of cubic samples of side L containing a percolative network with a correlation length of is given by [14 ... 

As shown in Fig. 4.18, confinement of the liquid mixture leads to a strong depression of the critical temperatiire T, in accordance with earlier studies [84, 91, 113, 114]. This critical point shift can be attributed to the cutoff of the range of concentration fluctuations when the correlation length becomes equal to the pore width. Confinement also leads to a shift of the critical point to a more water-rich composition (i.e., in the direction of the component, which is preferred by the walls). This finding is consistent with results of earlier studies [93, 115] and can be rationalized by the fact that the pore liquid is highly inhomogeneoas. 

Fig. 4 Structure factor of binary blend dPB/PS in Zimm representation at different temperatures (top) and pressures (middle), and polymer concentrations (bottom), with the other parameters constant. From the fitted straight lines the susceptibility and correlation length is evaluated. The increase of scattering is caused by stronger thermal composition fluctuations when approaching the critical point...

Equation 12 can also be considered as an Ornstein-Zernicke equation describing the degree of thermal composition fluctuations of correlation length The correlation length is evaluated from V 2 S(0) and becomes infinite at the critical point as described by the scaling law The... 

Linear and non-linear effects of thermal composition fluctuations become visible in a scattering experiment. Within the mean field and Ising regimes the susceptibility S(0) and the correlation length are described by simple scaling laws as functions of the reduced temperature r according to Cr ) with the critical amplitudes C( o) and the critical exponents y(v). The critical exponents y(v) are known to be equal to y = 1 and 1.239 0.003 and V = 0.5 and 0.634 0.001 in the mean field and Ising cases, respectively [66]. The mean field case has already been discussed in the context with Eq. 11. 

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