Composite critical behavior

Yamada et al. [9,10] demonstrated that the copolymers were ferroelectric over a wide range of molar composition and that, at room temperature, they could be poled with an electric field much more readily than the PVF2 homopolymer. The main points highlighting the ferroelectric character of these materials can be summarized as follows (a) At a certain temperature, that depends on the copolymer composition, they present a solid-solid crystal phase transition. The crystalline lattice spacings change steeply near the transition point, (b) The relationship between the electric susceptibility e and temperature fits well the Curie-Weiss equation, (c) The remanent polarization of the poled samples reduces to zero at the transition temperature (Curie temperature, Tc). (d) The volume fraction of ferroelectric crystals is directly proportional to the remanent polarization, (e) The critical behavior for the dielectric relaxation is observed at Tc. 

An explanation of the observed relaxation transition of the permittivity in carbon black filled composites above the percolation threshold is again provided by percolation theory. Two different polarization mechanisms can be considered (i) polarization of the filler clusters that are assumed to be located in a non polar medium, and (ii) polarization of the polymer matrix between conducting filler clusters. Both concepts predict a critical behavior of the characteristic frequency R similar to Eq. (18). In case (i) it holds that R= , since both transitions are related to the diffusion behavior of the charge carriers on fractal clusters and are controlled by the correlation length of the clusters. Hence, R corresponds to the anomalous diffusion transition, i.e., the cross-over frequency of the conductivity as observed in Fig. 30a. In case (ii), also referred to as random resistor-capacitor model, the polarization transition is affected by the polarization behavior of the polymer matrix and it holds that [128, 136,137]... 

Binary-Liquid Option. As an alternative to this study of critical behavior in a pure fluid, one can use quite a similar technique to investigate the coexistence curve and critical point in a binary-liquid mixture. Many mixtures of organic liquids (call them A and B) exhibit an upper critical point, which is also called a consolute point. In this case, the system exists as a homogeneous one-phase solution for all compositions if Tis greater than... 

In the present article, we review recent progress in this subject area. In Sec. 2, we give a short overview on the chemical composition of the low melting salts and ILs. In Sec. 3 we address the problem of the electrolyte solution structure at conditions of low reduced temperature, where phase separations are known to occur. In Sec. 4, we consider experimental and theoretical results concerning the location of the two-phase regime in solutions of ionic fluids. In Sec. 5 we finally review theoretical and experimental results on near-critical behavior of ionic fluids. 

As the use of polymers continues to increase in multi-component and multiphase applications, such as in polymer blends and composites, the behavior of polymers at interphases and the degree of mixing polymer segments in their blends become critical issues. Understanding how the conformations and mobilities of polymer chains in these heterogeneous systems compare with those observed in their homogeneous bulk systems may provide a point of departure in the discussions of their physical characteristics. 

Note that this free energy functional is Gaussian in the Fomier coefficients of the composition, and, hence, the critical behavior still is of mean-field type. Transforming back from Fourier expansion for the spatial dependence to real space, we obtain for the free energy functional ... 

Both the statics and the collective dynamics of composition fluctuations can be described by these methods, and one can expect these schemes to capture the essential features of fluctuation effects of the field theoretical model for dense polymer blends. The pronounced effects of composition fluctuations have been illustrated by studying the formation of a microemulsion [80]. Other situations where composition fluctuations are very important and where we expect that these methods can make straightforward contributions to our understanding are, e.g., critical points of the demixing in a polymer blend, where one observes a crossover from mean field to Ising critical behavior [51,52], or random copolymers, where a fluctuation-induced microemulsion is observed [65] instead of macrophase separation which is predicted by mean-field theory [64]. 

Rubin Z, Sunshine S A, Heaney M B, Bloom I and Balberg I (1999) Critical behavior of the electrical transport properties in a tunneling-percolation system, Phys Rev B 59 12196-12199. Balberg I (2002) A comprehensive picture of the electrical phenomena in carbon black-polymer composites, Carbon 40 139-143. 

The critical behavior has been studied in details for polymer blends in a common good solvent. The composition fluctuations play an important role and the critical behavior is not of the mean field type except in the limit of... 

Monte Carlo simulations very early demonstrated the effect of thermal composition fluctuations in low molecular blends. Studies by Sariban et al. [16] exclusively found Ising critical behavior in blends of molar volume up to about 16000 cm /mol and no indications of a crossover to mean field behavior. Such a mean field crossover was later detected by Deutsch et aL [ 17] in blends with an order of magnitude larger chains. These results and the techniques of Monte Carlo simiflations have been extensively reviewed by Binder in [4]. 

Thermal fluctuations can be described within the Gaussian approximation at sufficiently high temperatures above the critical temperature. For these situations, the system fulfills the conditions of mean field approximation [9]. On the other hand, thermal composition fluctuations become strong near the critical temperature, leading to non-Hnear effects which asymptotically close to the critical temperature imply that the system obeys the universality class of 3D-Ising critical behavior. Thermal fluctuations are described by the Ginzburg-Landau Hamiltonian which is written as a fimctional of the spatially varying order parameter

[Pg.21]

Figure 5 The phase diagram for a system showing upper critical behavior. At temperature T2 mixtures are stable at all compositions whereas at Tj phase separation can occur. The origins of this phase separation are shown in the lower diagram which shows the plot of free energy of mixing against composition at Tj. A metastable region exists between the binodal and the spinodal points (< b respectively). The inset in the lower diagram shows that if composition X separates into...

In the presence of a selective solvent, ordered block copolymers form micelles that, at sufficiently high copolymer concentrations, serve to stabilize a diree-dimensional network and promote physical gelation. This study examines the steady and dynamic rheological properties of micellar solutions composed of AB diblock, ABA triblock and bidisperse mfactures of AB and ABA copolymer molecules. Of particular interest is the unexpected improvement in network development upon addition of an AB copolymer to an ABA copolymer at constant solution composition. This behavior is observed for ABA/solvent systems above and below the critical gelation concentration, and is interpreted in terms of the volume exclusion that occurs in bidisperse mixture of grafted chains. 

---