2554
J. Phys. Chem. B 2007, 111, 2554-2564
Nonasymptotic Critical Behavior of a Ternary Ionic System Karin I. Gutkowski, Hugo L. Bianchi, and M. Laura Japas* Unidad de ActiVidad Quı´mica, Comisio´ n Nacional de Energı´a Ato´ mica, AV. del Libertador 8250, 1429 Buenos Aires, Argentina, and Escuela de Ciencia y Tecnologı´a, UniVersidad Nacional de General San Martı´n, Martı´n de Irigoyen 3100, 1651 San Andre´ s, ProVincia de Buenos Aires, Argentina ReceiVed: October 27, 2006; In Final Form: December 28, 2006
Refractive indices n and salt concentrations ms of coexisting phases of the ternary system 1,4-dioxane + water + potassium chloride were measured along the liquid-liquid-solid coexistence curve near the liquidliquid critical end point. Refractive index measurements were carried out in the range 0.689 × 10-3 < t ) (T - Tc)/Tc < 0.118 while salt concentrations were determined for the temperature range 1.84 × 10-3 < t < 8.07 × 10-2. From these experimental results, compositions fD (mass fraction of dioxane on a salt-free basis) and densities F of coexisting phases were obtained. The shape of the coexistence curve was analyzed using alternatively n, ms, fD, and F as order parameters. In all cases, the obtained coexistence curve displays, asymptotically, Ising behavior. Outside the asymptotic critical domain, n, ms, and F show significant deviations of the effective critical exponent from its Ising value, while the concentration variable fD requires no corrections to simple scaling. On the basis of the present results, we conclude that this system shows no indication of multicritical behavior.
1. Introduction The critical behavior of ionic systems recently has been the subject of intense discussion. Experimental and theoretical studies were performed to establish the effects of long-range Coulombic interactions in the fluctuation-dominated region. The conclusion of these surveys is that asymptotically close to the critical point ionic systems display Ising behavior.1-10 However, their behavior in the nonasymptotic region remains an open issue, as for some ionic systems the crossover to mean-field behavior occurs at relatively small distances from the critical point, it is rather sharp and, in most cases, non-monotonic, all distinctive features in comparison to simple molecular systems. The origin of this peculiar nonasymptotic behavior is not clear; it has been linked to the existence of another length scale, possibly related to supramolecular structures, that couples with the correlation length of the critical fluctuations and eventually drives the system into multicritical behavior. Sharp crossover was observed in the binary ionic systems Na in liquid NH3,1,2,11 tetra-n-butylammonium picrate in 1-dodecanol, 1-tridecanol,12 and 1-tetradecanol,7 and in ternary ionic systems,8,12-14 in most cases associated with non-monotonic behavior. In particular, two ternary ionic systems received recent attention: 3-methyl pyridine + water + sodium bromide (3MPWNaBr)14-16 and 1,4-dioxane + water + potassium chloride (DWKCl).8,17 3MPWNaBr was reported to display an extraordinary behavior at high salt concentrations: almost complete mean-field behavior, strong background scattering,14-16 and an anomalous concentration dependence of the critical line.14,15 These results were interpreted in terms of the formation of ion-induced mesoscopic structures and linked to recent theoretical results18-23 predicting the existence of a structured phase (an arrangement of ions of alternating charges) in equilibrium with a disordered one, which could give rise to a tricritical point. Therefore, the ternary system 3MPWNaBr was * Author to whom correspondence should be addressed. E-mail:
[email protected].
suspected to crossover to mean-field multicritical behavior. However, later studies were unable to confirm that behavior: Measurements in the one-phase region of viscosity,24 turbidity,25 heat capacity,26 and light scattering (for both 3MP + water + NaBr27 and 3MP + heavy water + NaBr28) and of refractive indices25,29 in the two-phase region showed asymptotic Ising behavior, without indication of anomalous concentration dependence of the critical locus.24 Recently, Kostko et al.,30 with new results of light scattering, confirmed the universal Isinglike critical behavior for properly aged samples and concluded that the previous anomalous results were due to the formation of slowly decaying nonequilibrium aggregates, probably formed by hydrogen-bonding of methylpyridine and water molecules. A similar conclusion was reached by Herna´ndez et al.31 to explain the apparent disagreement between dynamic and static correlation lengths, namely, the existence of noncritical 3MPrich aggregates. The second ternary system is potassium chloride in 1,4dioxane + water mixtures (DWKCl), whose coexisting curve was investigated under salt-saturation conditions.8 The presence of the solid phase, the spectator phase,32 transforms the ternary system into a pseudo-binary system, the amount of salt in the liquid phases being controlled by the equilibrium with the noncritical solid phase. Along this path, the critical behavior is correctly represented by the universal critical exponents without renormalization by hidden variables.33 The measured property, the density of coexisting phases, was taken as the concentration variable. Asymptotically close to its lower critical end point, the system shows Ising behavior, but as the distance to the critical point is raised over t > 10-2, the effective critical exponent βeff (describing the shape of the coexistence curve) was found to increase steeply to reach values close to the corresponding mean-field exponent but without any indication of leveling off. It has been suggested17 that this fact can also be an indication of the existence of a continuous transition and a tricritical point.
10.1021/jp067069z CCC: $37.00 © 2007 American Chemical Society Published on Web 02/16/2007
Critical Behavior of a Ternary Ionic System
J. Phys. Chem. B, Vol. 111, No. 10, 2007 2555 TABLE 1: Refractive Indices of Coexisting Phases
Figure 1. Experimental setup for the determination of the refractive index: dashed line, nondeviated laser beam; solid line, deviated laser beam. The meanings of L1, L2, x1, x2, R, φm, and φ are explained in the text; see eqs 1-3.
With the aim of achieving a better understanding of the nonasymptotic critical behavior of ionic systems, we present in this paper new experimental data for the coexistence curve of the salt-saturated DWKCl system. The values of βeff have been considered the more appropriate means to study the crossover to mean-field behavior.6,7 However, in practice, this advantage is obscured by the lack of a precise identification of the order parameter with real, experimental variables and, for the nonasymptotic region, by the strong dependence of the results on the chosen variable. With this problem in mind, we have acquired new experimental information and analyzed different alternatives for the order parameter. On the basis of these results and guided by the isomorphism principle, we propose the most proper order parameter for DWKCl. When the nonasymptotic critical behavior of this ionic system is analyzed in terms of that choice, Ising behavior is obtained in an unusually extended region, up to 0.1 in reduced temperature, without indication of either mean-field critical or tricritical behavior. We present arguments for the values of βeff > 0.5 previously observed for this system. 2. Experimental Section Sample Preparation, Critical Temperature. The critical sample was prepared by mixing proper amounts of 1,4-dioxane (Merck, p.a. 99.5+%) and MilliQ deionized water (conductivity below 0.1 µS cm-1) to obtain a solution of concentration z ) nD/(nD + nW) ) 0.295, with nD and nW the amounts of 1,4dioxane and water in moles, respectively. An excess of KCl (Mallinckrodt, AR) was then added to the sample. For the refractive index measurements, the mixture was placed in a triangular prism-shaped cell (Hellma) while, for salt solubility determinations, a 0.5 L Erlenmeyer flask was employed. The sample was immersed in a water thermostat and stirred thoroughly at each temperature. The bath temperature was regulated by a Hart temperature controlling system (model 2100) with a long-term stability of 10 mK, approximately. Temperature was measured using calibrated thermistors at millikelvin resolution. The thermistors’ calibrations were performed against a second reference platinum resistence thermometer (5614 PRT, Hart Scientific, NIST traceable). For each separate experiment the lower critical temperature was determined as the midpoint between the lowest temperature where a meniscus was formed and the highest temperature without phase separation, being Tc ) (311.921 ( 0.027) K for refractive index measurements and Tc ) (311.265 ( 0.036) K for solubility data. The difference in critical temperatures between both samples was assigned to impurities: Although the same quality of reactants was employed in both experiments, small traces of contaminants usually produce large changes in the critical temperature without affecting the critical exponents. Experimental Techniques. RefractiVe Index. Refractive indices were determined using the minimum deviation angle method.34,35 To allow a precise alignment of the incident beam
t 102
n1
nu
t 102
n1
nu
0.069 0.098 0.115 0.147 0.170 0.188 0.220 0.257 0.289 0.297 0.355 0.398 0.468 0.470 0.470 0.552
1.3891 1.3889 1.3887 1.3884 1.3878 1.3876 1.3876 1.3874 1.3873 1.3868 1.3871 1.3862 1.3863 1.3861 1.3864 1.3856
1.3952 1.3957 1.3953 1.3959 1.3959 1.3959 1.3961 1.3964 1.3962 1.3967 1.3967 1.3967 1.3968 1.3973 1.3970 1.3973
0.674 0.756 0.824 0.856 0.941 1.180 1.378 1.918 2.499 3.415 5.185 6.058 8.386 9.951 11.8
1.3856 1.3851 1.3850 1.3843 1.3842 1.3835 1.3833 1.3818 1.3804 1.3792 1.3769 1.3759 1.3744 1.3732 1.3722
1.3971 1.3974 1.3973 1.3978 1.3972 1.3975 1.3974 1.3973 1.3968 1.3965 1.3951 1.3938 1.3914 1.3896 1.3875
at the angle of minimum deflection, the cell was mounted on a rotary platform with fine adjustment. Platform and cell were attached to a positioning mechanism to permit their vertical displacement. In this way, the height of the cell, relative to that of the beam, was changed to probe successively upper and lower phases and to measure the reference position of the nondeflected beam (with the cell out of the laser path). The minimum deviation angle φm was determined by measuring the position of the beam at two fixed distances from the cell, L1 and L2, separate from each other by a distance d (Figure 1). The refractive index n was then obtained from
∆x ) tan(φ) d
(1)
nair sin(φ) ) nw sin(φm)
(2)
n ) nw
sin[(φm + R)/2] sin(R/2)
(3)
where ∆x ) x1 - x2 and xi is the distance from the zero point (no deviation) to the beam position after traversing the mixture, measured at position Li. φ is the measured angle, corresponding to minimum deviation, found by rotating the cuvette and related to φm by Snell’s law, eq 2, and R is the angle formed by the walls of the cell (π/3). The distance d was determined by calibration, measuring the deviation angle produced by a liquid of known refractive index (1,4-dioxane, n ) 1.4207 at 289.95 K, λ ) 632.8 nm).36 The refractive index of water nw, used as a thermostating fluid, was obtained from the literature for each temperature.37 The procedure to obtain the data was as follows: A photodetector, covered with a pinhole, was mounted in a micrometer first at position L1. A set of approximately 60 pairs of voltage-position measurements was performed for determining the whole Gaussian profile of the laser beam, and the maximum of the Gaussian curve was taken as the position of the laser beam. This procedure was done for the zero point (nondeviation, the beam passing through the water bath only) and for the deviated beam position, after it passed through the mixture. The procedure was repeated for position L2. With this information the refractive index of one of the phases was obtained. The whole operation was repeated for the other liquid phase. The refractive index was calculated using the equations given above. All measurements were performed using a 3 mW He/Ne Melles Griot laser (λ ) 632.8 nm) with an experimental accuracy of σ ) 2.5 × 10-4.
2556 J. Phys. Chem. B, Vol. 111, No. 10, 2007
Gutkowski et al. was explored as an alternative. For that purpose, refractive index data in the one-liquid-phase region were fitted to the empirical equations
n(fD,T) ) n0(fD) + nT(fD)(T - T0)
(5)
n
n0(fD) )
aif iD ∑ i)1
(6)
m
nT(fD) ) Figure 2. Results for the refractive index of the ternary KCl-saturated 1,4-dioxane + water system. In the homogeneous region, the open circles indicate experimental results, and the thin lines the values calculated using eqs 5-7 for the samples listed in Table 2. For the two-liquid-phase region, the solid circles denote experimental values; the thick line is only a guide to the eye.
To define alternative order parameters that eliminate the regular contribution in the analysis of critical anomalies (see below), refractive indices were measured in the one-liquid-phase region for mixtures of different dioxane/water proportions, all with an excess of KCl (saturation). The dioxane/water composition range studied in this homogeneous-liquid region spanned from the binary H2O + KCl to the binary 1,4-dioxane + KCl, at different temperatures, ranging from approximately 293 K to the phase-separation temperature. It is worth recalling that, due to the presence of pure KCl in equilibrium with the fluid, the concentration of the salt in the liquid phase is not an independent variable but is entirely determined by the temperature and the dioxane/water ratio. Salt Solubility. The mixture was stirred at each temperature, and after complete phase separation, small amounts of each phase (7 mL approximately) were withdrawn from the cell into preweighted vials. Once at room temperature, the vials were reweighted and then brought to dryness in an oven at approximately 343 K. By comparison of all of the weights, the amount of salt and solvent (1,4-dioxane + water) was obtained, and the molality mS of the salt was calculated. The overall error of the molality is estimated as 0.01 molal. 3. Results The refractive indices n of coexisting liquid phases are given in Table 1 and depicted in Figure 2. The shape of the curve, highly skewed, is a consequence of the relatively small difference between the refractive indices of dioxane and KClsaturated water: With the exception of the region within 4 K from Tc, temperature has a stronger effect on n than concentration. Due to thermal expansion, both branches of the coexistence curve display a negative slope in the T-n plane, as do pure liquids or liquid mixtures at constant compositions. Figure 2 also shows that the difference in refractive indices of coexisting phases increases with the distance to the critical point only up to approximately 320 K; for higher temperatures it clearly diminishes.38 Since the order parameter should reflect the difference between coexisting phases, playing the role of the magnetization in Ising magnets, variables other than the refractive index, but related to it, were examined as alternative order parameters. The mass fraction of dioxane on a salt-free basis
fD )
wD (wD + wW)
bif iD ∑ i)1
(7)
where T0 ) 293.15 K. The input data, refractive indices of the homogeneous-liquid phase, temperatures, and compositions, are given in Table 2. Different degrees of the polynomial representation for n0, eq 6, and for nT, eq 7, were tested. The best representation was obtained for quadratic and cubic dependence of n0 and nT on fD, respectively. The corresponding parameters ai and bi are a0 ) 1.366698, a1 ) 0.036312, a2 ) 0.015634, b0 ) 0.000038 K-1, b1 ) -0.000278 K-1, b2 ) -0.0006232 K-1, and b3 ) 0.000384 K-1. Figure 3 shows the quality of the fits for three selected concentrations (fD ) 0.352, 0.550, and 0.856) in the one-liquidphase region and in the whole range of explored temperatures. For all isopleths, deviations are within 2σ, where σ is the uncertainty of the refractive index measurement. Compositions (fD) of coexisting phases were calculated by finding the roots of the polynomial representation using temperatures and refractive indices of the coexistence curve as input data. Figure 4 shows the coexistence curve using mass fraction as a variable. In the homogeneous-liquid region, the constantfD curves look effectively vertical, and the coexistence curve seems more symmetric in terms of fD than in terms of n (see Figure 2 for comparison). As alternative order parameters, solubility and mass density were also considered. The result for the solubility of KCl in coexisting phases, expressed as molality mS (moles of KCl per kilogram of 1,4-dioxane + water), is given in Table 3 and shown in Figure 5. The coexistence curve in the mS-T coordinates shows the asymmetry characteristic of systems with a small critical concentration: The low-mS branch exhibits a smaller temperature dependence due to its closeness to the concentration physical limit, mS ) 0. The mass density of the system F was calculated at each temperature, assuming the validity of the Lorenz-Lorentz relation for a ternary mixture
[
]
FWRW FDRD FSRS n2 - 1 4 ) πFNA + + 2 MW MD MS n +2 3
(8)
where n is the refractive index, Mi and Ri are the molar mass and polarizability (at λ ) 632.8 nm) of component i, and Fi is its mass fraction in the ternary system. In contrast to fD, Fi ) wi/(wS + wD + wW) takes into account all three components. The Fi values were calculated by combining the values of fD and mS measured along the coexistence curve. The polarizabilities of water and KCl, RW and RS in eq 8, were obtained by fitting the Lorenz-Lorentz relation for isothermal (298 K), subsaturated n-mS data for the binary mixture, taken from the literature.39 The polarizability of pure 1,4-dioxane, RD, was calculated with the same equation for a one-component system with n ) 1.4207 and F ) 1.0366.36 4. Analysis
(4)
Theoretical Background. Close to the critical point, the differences in order parameters of upper (dioxane-rich) and
Critical Behavior of a Ternary Ionic System
J. Phys. Chem. B, Vol. 111, No. 10, 2007 2557
TABLE 2: Refractive Index in the One Phase Region as a Function of the Mass Fraction of 1,4-Dioxane (fD) and Temperature n
fD
T (K)
n
fD
T (K)
1.3666 1.3675 1.3679 1.3679 1.3680 1.3814 1.3802 1.3792 1.3778 1.3806 1.3798 1.3790 1.3785 1.3782 1.3777 1.3778 1.3779 1.3922 1.3916 1.3911 1.3908 1.3903 1.3899 1.3893 1.3890 1.3885 1.3878 1.3874 1.3872 1.3977 1.3955 1.3945 1.3931 1.3922 1.3969 1.3968 1.3968 1.3964 1.3964 1.3950 1.3948 1.3949 1.3945 1.3936 1.3940 1.3939 1.3934 1.3928 1.3922 1.3923 1.3925 1.4040 1.4022 1.4003 1.3985
0 0 0 0 0 0.352 0.352 0.352 0.352 0.355 0.355 0.355 0.355 0.355 0.355 0.355 0.355 0.550 0.550 0.550 0.550 0.550 0.550 0.550 0.550 0.550 0.550 0.550 0.550 0.660 0.660 0.660 0.660 0.660 0.666 0.666 0.666 0.666 0.666 0.666 0.666 0.666 0.666 0.666 0.666 0.666 0.666 0.666 0.666 0.666 0.666 0.766 0.766 0.766 0.766
293.562 305.879 317.915 325.044 333.130 293.558 302.957 312.846 323.022 298.206 307.840 312.846 317.870 320.446 323.025 323.955 324.042 289.403 292.198 293.552 295.872 298.201 300.565 301.040 302.956 305.397 307.844 310.330 311.838 293.558 298.207 302.966 307.867 309.860 295.386 295.404 295.865 297.263 298.201 301.021 302.956 302.960 303.936 305.853 305.876 305.881 307.846 309.338 310.810 310.832 310.838 293.558 298.201 302.956 307.844
1.3968 1.4055 1.4039 1.4019 1.4001 1.4008 1.3988 1.3982 1.4085 1.4085 1.4074 1.4068 1.4060 1.4057 1.4045 1.4033 1.4009 1.4005 1.3988 1.3977 1.3968 1.3970 1.3972 1.4096 1.4057 1.4044 1.4041 1.4022 1.4013 1.4021 1.4000 1.3978 1.4120 1.4096 1.4075 1.4055 1.4037 1.4009 1.3990 1.4104 1.4096 1.4086 1.4064 1.4064 1.4058 1.4040 1.4019 1.3991 1.3987 1.3974 1.4203 1.4179 1.4139 1.4094 1.3993
0.766 0.801 0.801 0.801 0.801 0.801 0.801 0.801 0.858 0.858 0.858 0.858 0.858 0.858 0.858 0.858 0.858 0.858 0.858 0.858 0.858 0.858 0.858 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.901 0.901 0.901 0.901 0.901 0.901 0.901 0.910 0.910 0.910 0.910 0.910 0.910 0.910 0.910 0.910 0.910 0.910 1 1 1 1 1
311.848 293.557 298.213 302.968 307.832 307.846 310.840 312.344 295.415 295.422 298.122 300.103 301.050 302.965 305.405 307.855 312.846 314.859 317.907 320.946 321.940 323.010 323.015 293.554 302.960 306.870 307.850 312.565 312.835 313.063 317.940 320.990 293.569 298.201 302.966 307.857 312.852 317.914 323.020 298.202 300.101 302.968 307.855 307.862 308.810 312.865 317.930 323.040 324.010 328.090 289.045 293.554 302.960 312.846 333.140
lower (water-rich) phases ∆M and the diameter M of the coexistence curves are described by40-42
∆M ) |Mu - M1| ) Btβ(1 + B1t∆ + ‚‚‚)
(9)
and
M )
Mu + M1 ) Mc + Dt + D1-Rt(1-R) + D2βt2β 2
(10)
where M denotes a concentration variable, either n, fD, mS, or F, t is the reduced critical temperature t ) (T - Tc)/Tc, and β
Figure 3. Deviation plot for the polynomial representation of the refractive index as a function of T and fD (see eq 5). For the sake of clarity, only three isopleths are represented; their fD values are 0.352, 0.550, and 0.858 for solid circles, solid triangles, and open squares, respectively. All data correspond to the KCl-saturated homogeneousliquid phase.
Figure 4. Representation of the coexistence curve using the 1,4-dioxane mass fraction (salt-free basis) fD as an order parameter. Values of fD were obtained from refractive index measurements by finding the roots of eqs 5-7: solid circles, experimental values for coexisting liquid phases and the diameter; solid lines, compositions of coexisting liquid phases and diameter as obtained from the fit to eqs 9 and 10, with coefficients given in Tables 4 (fit I) and 5 (fit C); open circles, results for the homogeneous liquid. The vertical dashed lines show the experimental values of fD in samples listed in Table 2.
TABLE 3: Solubility of KCl in Coexisting Phases mS (mol kg-1) t 102
upper phase
lower phase
0.184 0.408 0.638 1.01 1.01 1.51 1.51 2.02 2.53 2.53 4.05 6.09 8.07
0.236 0.184 0.140 0.110 0.109 0.099 0.099 0.068 0.068 0.067 0.031 0.026 0.028
0.884 1.092 1.198 1.397 1.392 1.631 1.660 1.798 2.031 2.008 2.486 3.117 3.602
) 0.326, ∆ ) 0.51, and R ) 0.11 are the critical exponents of the three-dimensional Ising universality class.43 The second term on the right-hand side of eq 10 is the regular rectilinear dependence, the third arises from the mixing of physical fields into the scaling fields, and the last term takes into account the effect on the diameter of an incorrect choice of order parameter44 or, as discussed recently,45 is a direct consequence of complete scaling. The quantities B and B1, the leading amplitude for the difference in order parameters and the first amplitude for its Wegner expansion, are system-dependent, as well as all D coefficients in eq 10.
2558 J. Phys. Chem. B, Vol. 111, No. 10, 2007
Gutkowski et al. dent) concentration variable. In general, for a mixture of three components with mole fractions x1 ) (1 - x2 - x3), x2, and x3, the new concentrations z and x are defined through
x1 ) (1 - z)(1 - x) x2 ) z(1 - x) x3 ) x
(11)
By virtue of this definition, the new concentration z is just the mole fraction of component 2 on a basis free of component 3, z ) n2/(n2 + n1). In terms of z and x, the molar Gibbs energy for a ternary system is expressed as
G(p,T,x2,x3) ) µ1 + µ21z + (µ31 - zµ21)x Figure 5. Concentration of KCl along the solid-liquid-liquid coexistence curve and its diameter. The concentration of KCl (solute) is expressed in moles of KCl per kilogram of solvents (molality). The circles represent the experimental results while the lines indicate the molality of coexisting phases (solid line) and the diameter (dashed line), according to the results shown in Tables 4 and 5.
TABLE 4: Results for the Fit of the Difference of Order Parameter ∆M in Coexisting Phases According to Eq 9a variable fD z n (t < 10-2) mS ln mS
fit
B
I II I II
1.368 ( 0.007 1.366 ( 0.014 1.250 ( 0.008 1.334 ( 0.014 0.0615 ( 0.0004 4.465 ( 0.040 11.08 ( 0.13 10.50 ( 0.31
I II
The n data were fitted for t < 10 critical analysis. a
B1
-2
0.0132 ( 0.082 -0.434 ( 0.083 2.95 ( 0.21 0.5 ( 2.6
χ2 0.72 0.75 2.3 0.61 1.0 1.6 0.91 0.59
(12)
and the changes in G are related to the changes in its independent variables (at constant pressure) by
dG ) µ21(1 - x) dz + (µ31 - zµ21) dx - S dT
(13)
with S the molar entropy of the mixture, T the temperature, µ21 ) µ2 - µ1, and µ31 ) µ3 - µ1. The isomorphic potential for the ternary system can thus be constructed by applying the following Legendre transformation of the Gibbs energy
G f G - x(µ31 - zµ21)
(14)
By application of the transformation to eq 12, the expression for the new potential, represented here by Ω, is
Ω ) µ1 + zµ21
(15)
to avoid regular effects in the
As mentioned in the Introduction, the selection of an appropriate variable to play the role of order parameter M is a key issue in the interpretation of the coexistence curve data. From a theoretical point of view, this choice is guided by the properties of the lattice-gas model, the equivalent to the Ising model of magnets. For a one-component system, magnetization and the magnetic field (conjugate variables) are replaced by density and chemical potential, and the associated thermodynamic potential is the Helmholtz energy density A ) A/V. In mixtures, the analysis of the critical behavior is advised by the principle of isomorphism.46 In particular for binary liquids with liquid-liquid demixing, the thermodynamic potential isomorphic with A of pure systems is G, the Gibbs energy, and order parameter and ordering field are associated with concentration and chemical potential difference µ21 ) µ2 - µ1, respectively. For most liquid mixtures, volume fraction is a better choice than mole fraction as a concentration variable,47-49 especially for asymmetric mixtures, having a critical mole fraction far from x ) 0.5. As for A(T,F), G depends on only one density variable, namely, x, as required in an isomorphic representation. For ternary incompressible systems, the thermodynamic potential isomorphic with G(T,x) of a binary system, with only one independent density variable, can be obtained from a Legendre transformation of the Gibbs potential G(T,x2,x3) to convert one of the densities into a field variable. The new potential can be defined by choosing as the new field either the chemical potential of one of the components or the difference in chemical potentials of two of them. As in this paper we describe results near a critical end point where the noncritical spectator phase is a pure component, our obvious selection of independent variable is the chemical potential of that component. To obtain the isomorphic potential, first new concentration variables are defined to eliminate the (dependent) variable associated with the salt concentration from the other (indepen-
and the differential of Ω becomes
dΩ(T,z,µ3) ) µ21 dz -
x S dµ dT (16) (1 - x) 3 (1 - x)
with T, z, and µ3 the natural variables of Ω, the new isomorphic potential. In particular, if the spectator phase has a single component (labeled 3), then its chemical potential is a function only of T (at constant pressure); thus
µ3 )
µ/σ 3
dµ3 dµ/σ 3 ) ) -S/σ and 3 dT dT
(17)
/σ where µ/σ 3 and S3 denote the chemical potential and molar entropy of component 3 in that pure spectator phase, respectively. In that case, eq 16 simplifies into
dΩ(T,z,µ3 ) µ/σ 3 ) ) µ21 dz -
S - xS/σ 3 (1 - x)
dT
(18)
Note that Ω, eq 15, and dΩ, eq 18, have, respectively, the same form as G and dG for binary systems: Along this path the system can be regarded as a pseudo-binary system. Moreover, according to its definition, the concentration variable z does not take into account the presence of component 3, which becomes a hidden component. Thus, thermodynamically speaking, the role of component 3 is that of an impurity (although a very concentrated one) whose concentration is constrained by the phase equilibrium with the pure spectator phase. Although it is clear that thermodynamic arguments alone are insufficient to decide which property should be chosen as the order parameter,44 the expression of Ω suggests z or the related concentration fD as good candidates. Analysis of the Data. The order parameter and the diameter were fitted separately performing weighted least-squares fits to eqs 9 and 10 to optimize the goodness of the fits χ2. The
Critical Behavior of a Ternary Ionic System
J. Phys. Chem. B, Vol. 111, No. 10, 2007 2559
TABLE 5: Results for the Fit of the Diameter According to Eq 10a variable fD
z
n (t < 10-2) mS ln mS
a
fit A B C D E F A B C D E F A B C D E F A B C
The n data are analyzed for t
