Critical interaction parameter

Flory-Huggins interaction parameter critical degree of polymerization (= 2/X) phase separation not observed... 

A variety of equations-of-state have been applied to supercritical fluids, ranging from simple cubic equations like the Peng-Robinson equation-of-state to the Statistical Associating Fluid Theoiy. All are able to model nonpolar systems fairly successfully, but most are increasingly chaUenged as the polarity of the components increases. The key is to calculate the solute-fluid molecular interaction parameter from the pure-component properties. Often the standard approach (i.e. corresponding states based on critical properties) is of limited accuracy due to the vastly different critical temperatures of the solutes (if known) and the solvents other properties of the solute... 

Because the appearance of a superlattice is usually well characterized qualitatively in terms of an interaction parameter w which has nothing to do, in the usual treatments, with the melting of the parent solid solution, one does not expect to find a simple relationship between the critical temperature for disordering of the superlattice, and Ts, the solidus temperature of the corresponding solid... 

Using the estimated interaction parameters phase equilibrium computations were performed. It was found that the EoS is able to represent the VL2E behavior of the methane-n-hexane system in the temperature range of 198.05 to 444.25 K reasonably well. Typical results together with the experimental data at 273.16 and 444.25 K are shown in Figures 14.14 and 14.15 respectively. However, the EoS was found to be unable to correlate the entire phase behavior in the temperature range of 195.91 K (Upper Critical Solution Temperature) and 182.46K (Lower Critical Solution Temperature). 

Prior work on the use of critical point data to estimate binary interaction parameters employed the minimization of a summation of squared differences between experimental and calculated critical temperature and/or pressure (Equation 14.39). During that minimization the EoS uses the current parameter estimates in order to compute the critical pressure and/or the critical temperature. However, the initial estimates are often away from the optimum and as a consequence, such iterative computations are difficult to converge and the overall computational requirements are significant. 

It is assumed that there are available NCP experimental binary critical point data. These data include values of the pressure, Pc, the temperature, Tc, and the mole fraction, xc, of one of the components at each of the critical points for the binary mixture. The vector k of interaction parameters is determined by fitting the EoS to the critical data. In explicit formulations the interaction parameters are obtained by the minimization of the following least squares objective function ... 

At each point on the critical locus Equations 40a and b are satisfied when the true values of the binary interaction parameters and the state variables, Tc, Pc and xc are used. As a result, following an implicit formulation, one may attempt to minimize the following residuals. 

Table 14.8 Interaction Parameter Values from Binary Critical Point Data...

Englezos, P., G. Bygrave, and N. Kalogerakis, "Interaction Parameter Estimation in Cubic Equations of State Using Binary Phase Equilibrium Critical Point Data", Ind. Eng Chem. Res.31(5), 1613-1618 (1998). 

A is a constant and p is the critical exponent which adopts values from 0.3 to 0.5. Values around p = 0.5 are observed for long-range interactions between the particles for short-range interactions (e.g. magnetic interactions) the critical exponent is closer to p 0.33. As shown in the typical curve diagram in Fig. 4.2, the order parameter experiences its most relevant changes close to the critical temperature the curve runs vertical at Tc. 

Fig. 51 Phase diagram for PS-PI diblock copolymer (Mn = 33 kg/mol, 31vol% PS) as function of temperature, T, and polymer volume fraction, cp, for solutions in dioctyl ph-thalate (DOP), di-n-butyl phthalate (DBP), diethyl phthalate (DEP) and M-tetradecane (C14). ( ) ODT (o) OOT ( ) dilute solution critical micelle temperature, cmt. Subscript 1 identifies phase as normal (PS chains reside in minor domains) subscript 2 indicates inverted phases (PS chains located in major domains). Phase boundaries are drawn as guide to eye, except for DOP in which OOT and ODT phase boundaries (solid lines) show previously determined scaling of PS-PI interaction parameter (xodt

In the most general case the Lagrangian density of a field suffers a reduction of symmetry at some critical value of an interaction parameter. Suppose that... 

This introduces two "interaction parameters" per binary pair. The pure component coefficients, a and b i, are evaluated from critical data and the acentricity, as proposed by Soave in his original paper (1). The pure component aii varies with reduced temperature so as to match vapor pressure. (Soave s recently revised expression for a (17) has not been used.)... 

The calculated critical points of the binary pairs, particularly the critical pressures, are quite sensitive to the values used for the interaction parameters in the mixing rules for a and b in the equation of state. One problem in undertaking this study is that no data are available on the critical lines of any of the binary pairs except for CO2 - H2O. Even for C02 - H2O, two sets of critical data available (18, 19) are in poor quantitative agreement, though they present the same qualitative picture of the critical phenomena. 

Most of the interaction parameters employed were taken from other studies (20, 21), and are reportedly obtained by minimizing errors in the match of phase equilibrium data. However, in (21), the SRK equation employed was slightly different from that used here. The parameters for CO2 - H2O were chosen because they had been shown to give a critical line which is qualitatively correct. The H2O - CO interaction parameter is the value given in (20) for H2S - CO. For H2O - H2, kij was taken to be -0.25 in the absence of any literature studies. 

The temperature - dependent interaction parameters were determined from 77°F to 680°F using the data of Culberson and McKetta (20) and of Sultanov et al. (18). This parameter increases with temperature and appears to converge to the value of the constant parameter used for the vapor phase as the critical temperature of water is approached. 

The data of Rebert and Hayworth (32j were used to extend the temperature - dependent interaction parameters to temperatures above the critical point of n-hexane. 

J. As with the alkane - water systems, the interaction parameters for the aqueous liquid phase were found to be temperature - dependent. However, the compositions for the benzene - rich phases could not be accurately represented using any single value for the constant interaction parameter. The calculated water mole fractions in the hydrocarbon - rich phases were always greater than the experimental values as reported by Rebert and Kay (35). The final value for the constant interaction parameter was chosen to fit the three phase locus of this system. Nevertheless, the calculated three-phase critical point was about 9°C lower than the experimental value. 

The critical data and values used for inert components were those given by Ambrose (24). The interaction parameters between the water and the inert component were found by performing a dew-point calculation as described above but with the interaction parameter k.. rather than P taken as the iteration variable. 

C 2 re needed for determlng 3 (the interaction parameter for mixed micelle formation in aqueous solution), the critical micelle... 

We can substitute our expression for the free energy of mixing in Eq. (2.78) into the stability criteria of (2.36) and (2.37) to solve for the critical interaction parameter at the onset of phase separation, Xc... 

The Flory-Huggins theory of polymer solutions has been documented elsewhere [26, 27]. The basic parameters necessary to predict polymer miscibility are the solubility parameter 6, the interaction parameter %, and the critical interaction parameter ( ) . 

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