Next: Interfacial Tension Up: Coexistence Curve for LB Previous: Coexistence Curve for LB

Interfacial Composition Profile and Correlation Length

The interface between phase separated liquids becomes diffuse near a critical point where the interfacial tension becomes relatively low. The width of this interface can be quantified through the determination of the composition interfacial profile $\phi _A(z)$A(z) which measures the local composition along a coordinate, z, normal to an interface between the coexisting phases.

An interface in near critical fluid mixtures can be probed by optical reflectivity [59,60,61] or ellipsometry [62,63,64] to determine its width, but accurate measurements of composition gradients are the interface across usually difficult. Direct measurement of $\phi _A(z)$A(z) has recently become possible in thin films by neutron reflection [65,66], but the broadening of these profiles by capillary waves and surface wetting effects complicates the interpretation of those measurements so that it is hard to quantitatively evaluate theory in this important area [67, 68,69,70]. The LB model allows the determination of $\phi _A(z)$A(z) for an ideal mean-field theory fluid. Some insight into the fluctuation contribution to $\phi _A(z)$A(z) can be obtained by comparing these calculations to Monte Carlo calculations of $\phi _A(z)$A(z) [69,70]. An important property that derives from the determination of $\phi _A(z)$A(z) is the correlation length
$\xi^-$- in the two-phase region which governs the average width of the interface (see below). This definition of the correlation length is more involved in asymmetric fluid mixtures since the composition profile, $\phi _A(z)$A(z), is asymmetric about the center (z = 0) of the fluid interface.


Fig. 7 shows an equilibrium interfacial composition profile $\phi _A(z)$A(z) for a symmetric LB mixture having a quench depth in the two-phase region, $\tau_G=0.1$G = 0.1. The numerically determined profile $\phi _A(z)$A(z) is fit well by the mean-field theory prediction [11,63,70,71,72],


\begin{displaymath}\phi_A(z)=\bar{\phi} +(\Delta \phi/2)\tanh(z/w) \end{displaymath} (22)

  
\begin{figure} \begin{center} \vspace{2. cm} \special{psfile=fig7.ps angle=-90 h... ...t=113 voffset=50 vscale=40 hscale= 40} \end{center}\vspace{5.0 cm}\end{figure}

Figure 7: Interface composition profile $\phi _A$A for a quench depth $\tau _G=0.08$G = 0.08.
z is in the direction normal to the center of the fluid interface. Solid line is fit to Eq. 22.


for all $\tau _G$G considered in our paper. The dependence of the interfacial width w on reduced temperature $\tau _G$G is indicated in Fig. 8. The mean-field correlation length $\xi^-$- of the fluid mixture in the two-phase mixture is related to w by [64,70,72],


\begin{displaymath}2\xi^{-} \equiv w \end{displaymath} (23)

so that the determination $\phi _A(z)$A(z) affords a means of determining the basic property $\xi^-$-. A fit to the w data nearest the critical point in Fig. 8 gives,


\begin{displaymath}\xi^{-}=(0.96\pm 0.05)\tau^{-\nu}_G \end{displaymath} (24)

where the mean-field value of the critical exponent $\nu=1/2$ = ½ is assumed. Far away from the critical point, where $\tau_G \sim O(1) $G ~ O(1), the correlation length becomes comparable to the lattice spacing, as in Monte Carlo simulation of phase separation in small molecule liquids [70]. The lattice spacing in the LB model should be interpreted as being comparable to the average range of the interparticle interaction potential. This scale is typically comparable to the average molecular dimensions of the molecule involved [73] and for polymers this scale can be fairly large [74]. Particle clustering can also increase the magnitude of this scale in small molecule liquids [75].


  
\begin{figure} \begin{center} \vspace{2. cm} \special{psfile=fig8.ps angle=-90 h... ...et=113 voffset=50 vscale=40 hscale=40} \end{center}\vspace{5.0 cm}\end{figure}

Figure 8: Interfacial width, w as a function of the reduced interaction $\tau _G$G.


In a mean-field model of phase-separation the correlation length $ \xi $- has the same singular dependence on reduced temperature, $\tau _G$G, in the one and two-phase regions ($\xi^+$+ and $\xi^-$-, respectively) [76],


\begin{displaymath}\xi=\xi^-_o \tau^{-1/2}_G, \xi=\xi^+_o \tau^{-1/2}_G, \end{displaymath} (25)

but the correlation length amplitudes, 0+, $\xi^+_o,\xi^-_o$0-, are related by a constant ratio in mean field theory [76],

\begin{displaymath}\xi^+_o / \xi^-_o=2^{1/2}, \end{displaymath} (26)

This "universal" ratio is closer to 2 in real fluid mixtures exhibiting Ising-type criticality [77,78]. The discrepancy between Eq. (26) and measurement is illustrative of the large property changes that critical fluctuations can induce. Moreover, the critical temperature, in three dimensions, can be shifted from its mean-field value by as much as 25% by fluctuations, so that the interaction parameters must be treated as phenomenological parameters in comparison to experiments in order to "absorb" these discrepancies [34].

The interfacial composition profile $\phi _A(z)$A(z) of asymmetric fluid mixtures composed of mixtures of dissimilar molecules is also asymmetric [79]. We briefly illustrate this effect in Fig. 9 for a fluid having a mass ratio $\delta _M=3$M = 3 and a quench depth $\tau_G=0.1$G = 0.1. Again, we present our concentration data for the asymmetric fluid in terms of number density units to simply realize the effect of fluid asymmetry [80]. While the equilibrium profile may appear to be like the tanh profile [Eq. (22)] found in the symmetric case, we could not obtain a good fit to this function. A good empirical description of this profile is found by first taking a derivative of the profile in Fig. 9 to find its inflection point and by then fitting to a tanh profile on each side of the inflection point [65,81]. Fig. 9 shows a fit to the data where the characteristic widths are wL=5.75 , wR=5.0 correspond to the left and right sides of the inflection point. The average $\bar{w}=(w_L + w_R)/2$ provides a good measure of the average interfacial width. We plan to discuss the properties of the LB model with a molecular volume asymmetry further in a separate publication so that this case is not discussed here. In all the discussion below we restrict ourselves to $\lambda = 1$ = 1.


  
\begin{figure} \begin{center} \vspace{2. cm} \special{psfile=fig9.ps angle=-90 hoffset=113 voffset=50 vscale=40 hscale=40} \end{center}\vspace{5. cm}\end{figure}

Figure 9: The interfacial composition (number density) profile $\phi _A(z)$A (z) of an
asymmetric fluid mixture having a mass ratio, $\delta _M=3$M = 3, and quench depth, $\tau _G=0.08$G = 0.08.

It has sometimes been reported that two correlation lengths exist in the two-phase region of fluid mixtures having asymmetric coexistence curves [75]. These measurements are made by performing light or neutron scattering on coexisting phases in macroscopically phase separated samples. The scale of the composition fluctuations appear to occur at different scales in the $\phi _A$A -rich and $\phi _A$A -poor coexisting phases [75]. Apparently, the measurement process can give rise to unequal weighting in the different phases to the two sides of the $\phi _A(z)$A(z) profile, leading to different $\xi^-$- estimates. Although such measurements provide some insight into the asymmetry of the $\phi _A(z)$A(z) profile, they should not be interpreted as implying the existence of two distinct correlation lengths in the two phase region.

Next: Interfacial Tension Up: Coexistence Curve for LB Previous: Coexistence Curve for LB