Order parameter miscibility

The development of the miscibility gap for W < 0 and the antiphases ( Tjeq) for W > 0 have entirely different kinetic implications. For decomposition, mass flux is necessary for the evolution of two phases with differing compositions. Furthermore, interfaces between these two phases necessarily develop. The evolution of ordered phases from disordered phases (i.e., the onset of nonzero structural order parameters) can occur with no mass flux macroscopic diffusion is not necessary. Because the 77+q-phase is thermodynamically equivalent to the 7/iq-phase, the development of 77+q-phase in one material location is simultaneous with the evolution of r lq-phase at another location. The impingement of these two phases creates an antiphase domain boundary. These interfaces are regions of local heterogeneity and increase the free energy above the homogeneous value given by Eq. 17.14. The kinetic implications of macroscopic diffusion and of the development of interfaces are treated in Chapter 18. 

The two coexisting phases are characterized by the two values of their order parameter, piiq and Pvap- If one prepares a macroscopic system of volume V, at a fixed density, < P < Pwqi inside the miscibility gap, it will phase separate into two macroscopic domains in which the densities attain their coexistence values. The volumes of the domains, Viiq and Kap, are dictated by the lever rule ... 

Using multicanonical simulation techniques we can sample all configurations in the pertinent interval of density (order parameter) and determine their free energy. It is important to note that the simulation samples all states at a fixed order parameter with the Boltzmann weight of the canonical ensemble. What are the typical configurations that a finite system of volume V adopts inside the miscibility gap in the canonical ensemble [48-55] ... 

A study of the miscibility of liquid crystals is of great importance from two points of view [31]. First, mixtures manifest a variety of phases separated by phase transition lines. Varying the composition of a mixture, we can study the interaction of different structural modes (the interaction of order parameters), investigate various pretransitional phenomena [32], etc. Mixing an unknown substance with a compound having well-defined phases we can also identify the structure of the unknown substance. On the other hand, mixtures are extremely important from the technological point of view. The best liquid crystalline materials for displays are, as a rule, multicomponent mixtures with a wide temperature range of operation. Here, the problem is to compose a thermodynamically stable eutectic mixture. 

Lohse et al. have summarized the results of recent work in this area [21]. The focus of the work is obtaining the interaction parameter x of the Hory-Huggins-Stavermann equation for the free energy of mixing per unit volume for a polymer blend. For two polymers to be miscible, the interaction parameter has to be very small, of the order of 0.01. The interaction density coefficient X = ( y/y)R7 , a more relevant term, is directly measured by SANS using random phase approximation study. It may be related to the square of the Hildebrand solubility parameter (d) difference which is an established criterion for polymer-polymer miscibility ... 

Suppose now that there is chosen as the third component (component 1) a monomeric substance in which each of the polymer components (2 and 3) is separately miscible in all proportions in the absence of the other. In order that this condition may be fulfilled, both X12 and xi3 are required to be less than one-half. Aside from this stipulation the actual values of these parameters are of minor importance only hence we may let Xi2 = xi3- As before we take X2 = Xz = x,... 

For simplicity and in order to avoid potential misrepresentation of the experimental equilibrium surface, we recommend the use of 2-D interpolation. Extrapolation of the experimental data should generally be avoided. It should be kept in mind that, if prediction of complete miscibility is demanded from the EoS at conditions where no data points are available, a strong prior is imposed on the parameter estimation from a Bayesian point of view. 

The utility of the DSC for studying polymer-polymer miscibility has been demonstrated for poly(vinyl chloride)/nitrile rubber polyfvinyl methyl ether)/poly-styrene and poly(2,6-dimethyl 1,4-diphenylene oxide)/poly(styrene-co-chloro-styrene)It has also been particularly useful for measuring the melting point depressions of crystalline polymers in blends Mn order to calculate the interaction parameter as will be discussed later. 

It was found that the solubility of C02 in component 7 (oil) could be used in much the same way as the mixing parameter to vary the relative rate of movement of C02 and oil, and thus control the extent of enhanced oil recovery in order to match the simulated oil recovery to the laboratory model results. Desirably, the solubility should fall in a narrow range in such matching, so that the value of solubility so obtained could be used for predictive purposes, just as a narrow range of mixing parameter is found to hold in matching laboratory and field miscible floods. 

Effect of Unlike-Pair Interactions on Phase Behavior. No adjustment of the unlike-pair interaction parameter was necessary for this system to obtain agreement between experimental data and simulation results (this is, however, also true of the cubic equation-of-state that reproduces the properties of this system with an interaction parameter interesting question that is ideally suited for study by simulation is the relationship between observed macroscopic phase equilibrium behavior and the intermolecular interactions in a model system. Acetone and carbon dioxide are mutually miscible above a pressure of approximately 80 bar at this temperature. Many systems of interest for supercritical extraction processes are immiscible up to much higher pressures. In order to investigate the transition to an immiscible system as a function of the strength of the intermolecular forces, we performed a series of calculations with lower strengths of the unlike-pair interactions. Values of - 0.90, 0.80, 0.70 were investigated. 

PMMA is typical of many polymer pairs, for which the parameter is positive and of order 0.01, making only low molar mass polymers form miscible blends. PVME/PS, PS/PPO, and PS/TMPC have a strongly negative x parameter over a wide range of temperatures (of order — 0.01) but since >0 and Bblends phase separate on heating. PEO/ PMMA, PP/hhPP and PlB/hhPP, all represent blends with very weak interactions between components (x = 0). 

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