Critical binary mixture

Let us examine the critical dynamics near the bulk spinodal point in isotropic gels, where K + in = A(T — Ts) is very small, Ts being the so-called spinodal temperature [4,51,83-85]. Here, the linear theory indicates that the conventional diffusion constant D = (K + / )/ is proportional to T — Ts. Tanaka proposed that the density fluctuations should be collectively convected by the fluid velocity field as in near-critical binary mixtures and are governed by the renormalized diffusion constant (Kawasaki s formula) [84],... 

In ref. ( °) the applications of the analogous dependences (eq. (1) and (2)) for the pressure evolution of the glass temperature and the melting temperature in supercooled liquids were shown. It is noteworthy that both alcohols and water are important technological agents, also used as additives to the CO2 basic critical system. For the discussed case of binary mixtures of limited miscibility the critical behavior is the inherent feature of the system containing water and alcohol or nitrobenzene or nitrotoluene and alkanes, even under atmospheric pressure. When critical binary mixtures are considered as the base for the SCF technologies, no additional component is needed. 

Analysis of the slow forward relaxation (12) reveals [63] that the associated Kerr constant follows a power law, i.e., B is proportional to the distance in temperature from Tc as (1 — Tc/Ty", with

static electrical birefringence is in accord with the droplet model [66,67] of critical binary mixtures. The central idea of the droplet model is that the electric field distorts (orients or vectorially amplifies) the spontaneous critical concentration fluctuations. The resulting anisotropic fluctuations then play the role of nonspherical particles in ordinary electrical birefringence. The magnitude of the concentration fluctuations rapidly increases as T,. is approached. 

Onuki, A. (1986) Shear flow problems in critical binary mixtures. Physica A... 

Boots has applied the theory of multiple scattering to critical binary mixtures and concludes that it is essentially repeated single scattering so that the general structure of scattering formulae is unaltered. Cohen has considered in detail the theoretical relationships between thermodynamic parameters for polymer solutions and the intensities of Rayleigh and Brillouin scattering peaks. 

Povodyreveta/. (1997) have developedasix-term Landau expansion crossover scaling model to describe the thermodynamic properties of near-critical binary mixtures, based on the same model for pure fluids and the isomorphism principle of the critical phenomena. The model describes densities and concentrations at vapor-liquid equflibrium and isochoric heat capacities in the one-phase region. The description shows crossover from asymptotic Ising-hke critical behavior to classical (mean-field) behavior. This model was applied to aqueous solutions of sodium chloride. 

Kiselev, S. B. Kulikov, V D. (1994) Crossover behaviour of the transport coefficients of critical binary mixtures. Int. J. Thermophys., 15,283-308. 

II.3. Critical binary mixture (Dp= 2.5) In contact with a solid wall, a binary mixture has a concentration Cs at the wall different from the bulk concentration C. If the mixture is near critical, this modification is not restricted to the interface, but extends over a length... 

Figure 15 Schematic representation of hierarchical structures developed in critical binary mixtures of A and B molecules (A/B) in a phase-separation process of the late stage SD. Note that the two components here have the dynamic symmetry (i.e., nearly equal mobilities) and equal volume fraction, (a) to (c) refers to (1) global, (2) interface, and (3) interphase structure and (4) local structure, respectively, where r, Am, Rm, h, int, fr, and Rg refer to the length scale of observation, the characteristic length of the phase-separating domain structures, the scattering mean radius of curvature, the thickness of the diffuse boundary (interphase), the thermal correlation length within the interphase, the thermal correlation length within the phase-separated domains, and the radius of gyration of polymers, respectively. From Hashimoto, T. J. Polym. Sci., Part B Polym. Phys. 2004, 42, 3207-3262.= ...

Figure A2.5.11. Typical pressure-temperature phase diagrams for a two-component fluid system. The fiill curves are vapour pressure lines for the pure fluids, ending at critical points. The dotted curves are critical lines, while the dashed curves are tliree-phase lines. The dashed horizontal lines are not part of the phase diagram, but indicate constant-pressure paths for the T, x) diagrams in figure A2.5.12. All but the type VI diagrams are predicted by the van der Waals equation for binary mixtures. Adapted from figures in [3].

Figure A2.5.28. The coexistence curve and the heat capacity of the binary mixture 3-methylpentane + nitroethane. The circles are the experimental points, and the lines are calculated from the two-tenn crossover model. Reproduced from [28], 2000 Supercritical Fluids—Fundamentals and Applications ed E Kiran, P G Debenedetti and C J Peters (Dordrecht Kluwer) Anisimov M A and Sengers J V Critical and crossover phenomena in fluids and fluid mixtures, p 16, figure 3, by kind pemiission from Kluwer Academic Publishers.

Figure A2.5.31. Calculated TIT, 0 2 phase diagram in the vicmity of the tricritical point for binary mixtures of ethane n = 2) witii a higher hydrocarbon of contmuous n. The system is in a sealed tube at fixed tricritical density and composition. The tricritical point is at the confluence of the four lines. Because of the fixing of the density and the composition, the system does not pass tiirough critical end points if the critical end-point lines were shown, the three-phase region would be larger. An experiment increasing the temperature in a closed tube would be represented by a vertical line on this diagram. Reproduced from [40], figure 8, by pennission of the American Institute of Physics.

Figure A3.3.2 A schematic phase diagram for a typical binary mixture showmg stable, unstable and metastable regions according to a van der Waals mean field description. The coexistence curve (outer curve) and the spinodal curve (iimer curve) meet at the (upper) critical pomt. A critical quench corresponds to a sudden decrease in temperature along a constant order parameter (concentration) path passing through the critical point. Other constant order parameter paths ending within tire coexistence curve are called off-critical quenches.

The Class I binary diagram is the simplest case (see Fig. 6a). The P—T diagram consists of a vapor—pressure curve (soHd line) for each pure component, ending at the pure component critical point. The loci of critical points for the binary mixtures (shown by the dashed curve) are continuous from the critical point of component one, C , to the critical point of component two,Cp . Additional binary mixtures that exhibit Class I behavior are CO2—/ -hexane and CO2—benzene. More compHcated behavior exists for other classes, including the appearance of upper critical solution temperature (UCST) lines, two-phase (Hquid—Hquid) immiscihility lines, and even three-phase (Hquid—Hquid—gas) immiscihility lines. More complete discussions are available (1,4,22). Additional simple binary system examples for Class III include CO2—hexadecane and CO2—H2O Class IV, CO2—nitrobenzene Class V, ethane—/ -propanol and Class VI, H2O—/ -butanol. 

Concentrations of moderator at or above that which causes the surface of a stationary phase to be completely covered can only govern the interactions that take place in the mobile phase. It follows that retention can be modified by using different mixtures of solvents as the mobile phase, or in GC by using mixed stationary phases. The theory behind solute retention by mixed stationary phases was first examined by Purnell and, at the time, his discoveries were met with considerable criticism and disbelief. Purnell et al. [5], Laub and Purnell [6] and Laub [7], examined the effect of mixed phases on solute retention and concluded that, for a wide range of binary mixtures, the corrected retention volume of a solute was linearly related to the volume fraction of either one of the two phases. This was quite an unexpected relationship, as at that time it was tentatively (although not rationally) assumed that the retention volume would be some form of the exponent of the stationary phase composition. It was also found that certain mixtures did not obey this rule and these will be discussed later. In terms of an expression for solute retention, the results of Purnell and his co-workers can be given as follows,... 

Since the boiling point properties of the components in the mixture being separated are so critical to the distillation process, the vapor-liquid equilibrium (VLE) relationship is of importance. Specifically, it is the VLE data for a mixture which establishes the required height of a column for a desired degree of separation. Constant pressure VLE data is derived from boiling point diagrams, from which a VLE curve can be constructed like the one illustrated in Figure 9 for a binary mixture. The VLE plot shown expresses the bubble-point and the dew-point of a binary mixture at constant pressure. The curve is called the equilibrium line, and it describes the compositions of the liquid and vapor in equilibrium at a constant pressure condition. 

The difficulties encountered in the Chao-Seader correlation can, at least in part, be overcome by the somewhat different formulation recently developed by Chueh (C2, C3). In Chueh s equations, the partial molar volumes in the liquid phase are functions of composition and temperature, as indicated in Section IV further, the unsymmetric convention is used for the normalization of activity coefficients, thereby avoiding all arbitrary extrapolations to find the properties of hypothetical states finally, a flexible two-parameter model is used for describing the effect of composition and temperature on liquid-phase activity coefficients. The flexibility of the model necessarily requires some binary data over a range of composition and temperature to obtain the desired accuracy, especially in the critical region, more binary data are required for Chueh s method than for that of Chao and Seader (Cl). Fortunately, reliable data for high-pressure equilibria are now available for a variety of binary mixtures of nonpolar fluids, mostly hydrocarbons. Chueh s method, therefore, is primarily applicable to equilibrium problems encountered in the petroleum, natural-gas, and related industries. 

Equations (76) and (77) contain two constants, A and B, which, for any binary pair, are functions of temperature only. These equations appear to be satisfactory for accurately representing activity coefficients of nonpolar binary mixtures from the dilute region up to the critical composition. As examples, Figs. 12 and 13 present typical results of data reduction for two systems in these calculations, the reference pressure Pr was set equal to zero. 

We consider a binary liquid mixture of components 1 and 3 to be consistent with our previous notation, we reserve the subscript 2 for the gaseous component. Components 1 and 3 are completely miscible at room temperature the (upper) critical solution temperature Tc is far below room temperature, as indicated by the lower curve in Fig. 27. Suppose now that we dissolve a small amount of component 2 in the binary mixture what happens to the critical solution temperature This question was considered by Prigogine (P14), who assumed that for any binary pair which can be formed from the three components 1, 2 and 3, the excess Gibbs energy (symmetric convention) is given by... 

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 ... 

In contrast to the Gibbs ensemble discussed later in this chapter, a number of simulations are required per coexistence point, but the number can be quite small, especially for vapor-liquid equilibrium calculations away from the critical point. For example, for a one-component system near the triple point, the density of the dense liquid can be obtained from a single NPT simulation at zero pressure. The chemical potential of the liquid, in turn, determines the density of the (near-ideal) vapor phase so that only one simulation is required. The method has been extended to mixtures [12, 13]. Significantly lower statistical uncertainties were obtained in [13] compared to earlier Gibbs ensemble calculations of the same Lennard-Jones binary mixtures, but the NPT + test particle method calculations were based on longer simulations. 

Before describing variations in the critical points in the four-component water-gas shift mixture it is instructive to examine the critical points in the various binary mixtures. There are six binary pairs to consider. 

It is most significant that certain binary mixtures have no critical point. With this in mind it is not surprising that certain four-component mixtures have no critical point. 

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