Coexistence Curve for LB Mixture

For a sufficiently large fluid interaction coupling, G, the LB mixtures of components A and B phase separate into liquids having coexisting compositions $\phi _A$_A and $\phi_B$_B at equilibrium. The composition variables $\phi _A$_A and $\phi_B$_B denote the relative volume fractions of the two fluid components. These dimensionless concentration units are normalized so that $\phi_A + \phi_B = 1$_A + _B = 1. Compressibility effects on the fluid mixture can be treated through the addition of an additional vacancy component $\phi_c$_c such that _A + _B + _c = 1, as in equilibrium lattice model calculations of compressible mixtures [33], but this complication is not considered in the present paper.

Increasing the LB coupling G makes the coexisting compositions more enriched in the pure components and thus has the same qualitative effect as lowering the temperature in systems exhibiting an "upper critical temperature" type phase separation (i.e., phase separation upon cooling). The parameter G thus plays a role similar to the binary interaction parameter $\varepsilon$ in the lattice model of fluid mixtures [19,20,21,22,34],

$$\varepsilon=\varepsilon_{AB}-(\varepsilon_{AA}+\varepsilon_{BB})/2$$ (14)

where $\varepsilon_{AB}$_AB and $\varepsilon_{AA}$_AA, $\varepsilon_{BB}$_BB denote the mutual and self nearest-neighbor interactions of the fluid components. G is also analogous to the well depth parameter $\varepsilon_{LJ}$_LJ in off-lattice models of phase separation based on a Lenard-Jones or related potentials [35]. In all these models, it is the relative value of the "interaction strength", (G, $G, \varepsilon ,\varepsilon_{LJ}$ , $\varepsilon_{LJ}$_LJ ) to the temperature which is the dimensionless coupling constant defining the tendency toward phase separation. For example, the dimensionless interaction in the lattice model of fluid mixtures is conventionally defined as [19,20,21,22,34],

$$\chi = q \varepsilon/k_b T$$ (15)

where q and k_bT denotes the lattice coordination number and thermal energy respectively. The inclusion of the q factor is made to weight the number of possible nearest-neighbor interactions. In magnetic phase transitions we have the same form of dimensionless couplings as Eq. (13) where $\varepsilon$ is replaced by the "exchange interaction" J modeling the short-range magnetic interparticle interaction [18,36]. From these measures of interaction, we see that lowering the temperature has basically the same effect as increasing the interaction coupling ( $\varepsilon, \varepsilon_{LJ}$, _LJ ) for the usual case where ordering occurs upon cooling. Phase separation in the LB liquid also occurs when the temperature is lowered with G fixed and we similarly define a dimensionless coupling constant,

$$\chi_G \equiv G/k_bT$$ (16)

A reduced variable temperature $\tau$ may then be defined from the interaction coupling constant, _G ,

$$\tau_G \equiv \frac{\vert T/G-T/G_c\vert}{T/G_c}$$ (17)

for our simulation performed at fixed temperature T and variable G. For a particular fluid mixture it is natural to fix G and to vary T so that the reduced temperature variable is defined as,

$$\tau'_G \equiv \frac{\vert T/G-T_c/G\vert}{T_c/G}$$ (18)

where T_c is the critical temperature for a fixed value of G. The absolute value definition in Eqs. (17) and (18) ensures that the reduced temperature variable is positive for notational simplicity, but this requires that we must carefully distinguish between the one-phase and two-phase regions. All of the computations of the present paper are performed in the two phase region.

In Fig. 1 we present our results for the coexistence curve of a symmetric LB fluid mixture (both mass and viscosity ratios of fluid components are equal). The y-axis denotes the ratio of critical dimensionless coupling to the dimensionless coupling, G_c / $\chi _{G_c}/\chi _G$G, defined in Eq. (16) and the x-axis denotes the composition $\phi _A$_A of the A fluid. We observe the critical composition $\phi_{c,A}$_c,A of the A-component equals $\phi_c = 1/2$_c = 1/2 (a "symmetric mixture"), as required by the symmetry of exchange of the fluid components. This exchange symmetry is well known in lattice models of fluid phase separation [37,38,39,40,41]. The composition difference $\Delta \phi=\phi^{(1)}_A-\phi^{(2)}_A$ between the coexisting phases defines an order-parameter for the fluid phase separation process. The relation of $\Delta \phi$ $\Delta \phi$ to the reduced temperature is indicative of the type of critical phenomena ("universality class") under discussion. In a mean-field model of fluid phase separation $\Delta \phi$ $\Delta \phi$ is described by the general relation [17,18,34],

$$\Delta \phi = 2B \tau^\beta, \tau = (T-T_c)/T_c, T \approx T_c$$ (19)

where the order parameter exponent $\beta$ and critical amplitude B for a symmetric incompressible fluid mixture equal [34],

$$B= \sqrt{3}/2 , \beta=1/2$$ (20)

  

Figure 1: Phase diagram of LB fluid mixture. Normalized coupling $\chi _{G_c}/\chi _G$G_c / $\chi _{G_c}/\chi _G$G versus the composition $\phi _A$_A of fluid A. The solid circles represent data from the Shan-Chen model and the triangles represent data from the body-forcing model. $\chi _{G_c}/\chi _G$G_c / $\chi _{G_c}/\chi _G$G also corresponds to the temperature ratio T / T_c.

The dashed line in Fig. 1 is the predicted value. Our data are consistent with the mean-field prediction as $\tau_G \rightarrow 0$_G 0. Note that the Shan-Chen model deviates more from the mean-field prediction than does the simple body-forcing model. In general, it was found that the linear body forcing was somewhat more stable. The mean-field theory prediction (Eq. (19)) is further examined in Fig. 2 where we plot log₁₀($log (\Delta \phi)$ ) versus log₁₀ ($log (\tau_G)$_G ) for the lattice data shown in Fig. 1. It is apparent that a power law scaling of $\Delta \phi$ $\Delta \phi$ on $\tau _G$_G is observed over an appreciable temperature range. The solid line denotes the prediction of Eq. (19) with no free parameters where $\tau$ is equated with $\tau _G$_G . Note that the critical temperature is not adjustable in this comparison, in contrast to most simulations and experiments where this quantity is not known exactly. Of course, the solution of the two-dimensional Ising spin model and its lattice gas analog is an exception to this general situation [17,18]. Sengers gives an excellent review of the critical properties of fluids and fluid mixtures that provides much further information about mean-field and non-mean-field critical properties and the "crossover" between these property scaling regimes [42].

  

Figure 2: Order parameter $\Delta \phi$ versus quench depth parameter $\tau _G$_G.

Figures 1-2 not only verify that the phase separation process in LB fluids is described well by mean-field theory, but they also establish the utility of our definition of reduced temperature scale, $\tau _G$_G, which is required for other applications involving LB fluid mixtures. For example, we can quantify the quench depth of our phase separation measurements by specifying the $\tau _G$_G value. These simulations can be compared to experiments on real fluids at the corresponding $\tau$ value. Quantitative agreement with the properties of real liquids can only be expected for liquids that can be modeled by mean-field theory over a broad temperature range (see discussion below). This identification between computational and real fluids is generally restricted to a temperature range over which mean-field critical behavior is exhibited to a good approximation. Strictly speaking, no real fluids are described by mean-field critical behavior, but for many fluids the approximation should be reasonable provided $\tau$ is sufficiently far from the critical point defined by the limit, $\tau \rightarrow 0^{+}$ 0⁺. The Ginzburg criterion defines the temperature range over which mean-field theory is a reasonable approximation [33,43,44,45,46,47,48,49,50].

Real fluid mixtures are characterized by differences in the molecular shapes and volumes of the fluid molecules and asymmetries in the intermolecular interaction potentials that destroy the symmetry of exchange between the fluid components [38,39,40]. This symmetry breaking is evident in the shape of the coexistence curve. The graph of $\Delta \phi$$\Delta \phi$ versus $\tau$ becomes "skewed" so that the critical composition $\phi_c$_c no longer equals 1/2 [38,39,40]. The molecular asymmetry effect is particularly evident in polymer fluid mixtures where the ratio of the molecular weights and the backbone chain structure can be adjusted to "tune" the asymmetry of the coexistence curve [21,22,34,51]. The asymmetry becomes extreme in the case of high molecular weight polymers dissolved in a low molecular weight solvents where $\phi_{A,c}$_{A, c} of the high molecular weight component approaches zero with increasing molecular weight [21,52]. It is also possible to modify the molecular weights of a blend to achieve an almost perfect symmetry as in Fig. 1 [53]. This symmetry is not usually observed in fluid mixtures or in single component fluid phase transitions, although the degree of asymmetry is usually modest in comparison with polymer solutions.

The breaking of the particle exchange symmetry arising from differences in molecular shape, rigidity, mass and other molecular parameters is difficult to describe in a mesoscale fluid model of phase separation. We can obtain a simple model of this symmetry breaking phenomenon, however, by considering the idealized Flory-Huggins (FH) mean-field model of polymer blend phase separation [21] which accounts minimally for the molecular mass asymmetry of the fluid components (Actually the model accounts for a volume asymmetry since this incompressible polymer blend model assumes all lattice sites are occupied and have equal density). Notably the FH model completely ignores polymer topology, monomer asymmetry, polydispersity in the size and monomer-monomer interactions and other factors that surely influence polymer blend stability, but the mass ratio in the FH model does provide a parameter that allows the asymmetry of the coexistence curve to be "tuned" to fit observations on real blends. (The recently developed lattice cluster mean-field theory generalizes the FH model by incorporating leading order correlations associated with chemical connectivity and monomer structure [33].) We first consider the case where the particle masses are "asymmetric" in the LB fluid model in the same spirit of approximation. Figure 3 shows the coexistence curve for a LB fluid mixture having a mass ratio _M = M_A / M_B = 3 where the concentration difference $\Delta \phi$ between the coexisting phases is given in number density concentration units rather than the volume fraction units of Fig. 1. We examine the scaling of $\Delta \phi$ on the quench depth parameter $\tau _G$_G in Fig. 4 where we find a mean-field scaling exponent 1/2 as in Figs. 1 and 2 and a shift of the critical coupling to the value G_c = 0.0135. The asymmetry of the coexistence curve is quantified by calculating the dependence of the average composition $\bar{\phi_A} = (\phi^{(1)}_A+\phi^{(2)}_A)/2$ in the coexistence curve shown in Fig. 3 where $\phi_A^{(1)}$ and $\phi_A^{(2)}$ are compositions of the coexisting phases. According to the "law of rectilinear diameter" of Cailletet and Mathias [54], $\bar{\phi}$ is linear function of $\tau$. This linearity is found to a good approximation in the asymmetric fluid phase separation coexistence curve shown in Fig. 3. The average composition $\phi _A$_A is shown in Fig. 5 where the line denotes the rectilinear diameter fit,

$$\bar{\phi_A} = \phi_{A,c} +A\tau_G$$ (21)

where A is a constant, A = 0.11 and $\bar{\phi_A}$ and $\phi_{A,c}$_{A, c} are given in number fraction units. This type of plot is an effective way to determine the critical composition of an asymmetric fluid mixture [55]. Fluctuation corrections to mean-field theory can lead to deviations from Eq. (21) in real fluids that are important near the critical point [56].

  

Figure 3: Phase diagram of an asymmetric LB fluid mixture.

  

Figure 4: Order parameter $\Delta \phi$ vs quench depth parameter $\tau _G$_G for an asymmetric LB mixture.

We can also break the symmetry of interparticle exchange and thus distort the shape of the phase boundary by varying the relative volumes of the fluid particles (see Appendix B). In Fig. 6 we show the phase boundary calculated for a range of values of $\lambda$, the ratio of the volumes of components A and B, $\lambda = V_A /V_B$ = V_A / V_B. We assume spherically shaped particles so that $\lambda$ scales as the cube of the ratio of the particle radii, $\lambda = (R_A/R_B)^3$ = R_A / R_B) ³. By convention we take the A fluid component to have the largest molecular volume. Increasing the particle size asymmetry strongly increases the asymmetry of the phase boundary, in qualitative agreement with the Flory-Huggins theory of polymer phase separation [21,22,34]. Note that the data in Fig. 6 is given in volume fraction units. The simple Flory mean-field treatment of the Flory-Huggins lattice model indicates that _c = 1 / [1 + $\phi_c = 1 / [1 + \lambda^{\frac{1}{2}}]$ ½], where $\lambda$ is the relative chain molecular volume (see below). The true critical composition seems to be approximated reasonably well by a similar expression _c = 1 / [1 + $\phi_c = 1 / [1 + \lambda]$], and the arrows in the figure show the result in comparison with the data. We note that the phase diagrams of micelle and protein solutions where there is a large asymmetry in the size of the phase separating species, tend to be asymmetric as in Fig. 6 [57].

In applications of the LB model to real measurements we can phenomenologically adjust the relative mass (and thus the critical composition in Figs. 3 and 4) or relative particle volume $\lambda$ and identify the LB order parameter variable [$\phi _A$_A (number density) or volume fraction units, respectively] with the experimentally determined order parameter concentration unit. While this is generally an approximation, we expect it to provide a reasonable mimic of the critical properties of asymmetric fluid mixtures as in previous experience with the FH model [58].

  

Figure 5: Rectilinear diameter = (_A + $\phi =(\phi _A+\phi _B)/2$_B ) /2 versus quench depth $\tau _G$_G .

  

Figure 6: Influence of particle size on phase boundary asymmetry.
$\lambda =1.$ = 1.0 (filled circles),4.63 (filled squares), and 125. (open circles). Dashed lines are included to help guide the eye.