16674
J. Phys. Chem. 1996, 100, 16674-16677
Critical Behavior of a Pseudobinary System for a Three-Component Microemulsion Xueqin An,* Jiao Feng, and Weiguo Shen Department of Chemistry, Lanzhou UniVersity, Lanzhou, Gansu 730000, People’s Republic of China ReceiVed: May 6, 1996; In Final Form: July 18, 1996X
A three-component microemulsion system can be regarded as a pseudobinary mixture at certain conditions. The coexistence curves of (T, n), (T, φ), and (T, Ψ) (n and φ are refractive index and volume fraction, respectively; Ψ is defined as Ψ ) φ/[φ + φc(1 - φ)/(1 - φc)]) for a ternary microemulsion (AOT-waterdecane) at constant pressure and a constant ratio of water to AOT have been determined within about 10 K from the critical temperature by measurements of refractive index. The critical exponent β was deduced precisely from (T, n), (T, φ), and (T, Ψ) coexistence curves within 1.5 K above Tc. These values were 0.327, 0.324, and 0.325, respectively, and were consistent with the 3D Ising exponent. The volume fraction φ is a better choice of the concentration variable than Ψ used for construction of the order parameter to fit the Ising behavior.
I. Introduction
II. Experiments
Microemulsion systems exhibit complex phase behaviors in their phase diagrams. At constant temperature and pressure, a phase diagram of a ternary mixture usually can be represented in a triangle, the corners of which represent the different components. If the temperature is also varied, an axis perpendicular to the plane of the triangle has to be introduced to depict the temperature dependence. Three-component mixtures of water, n-decane, and the surfactant sodium bis(2-ethylhexylsulfosuccinate) (AOT) can form water-in-oil (w/o) microemulsions of well-defined droplet size determined by fixing the molar ratio ω of water to AOT.1-3 For the mixture with ω ) 40.8 and a volume fraction of the droplets φc ) 0.098, one observes a lower consolute critical point. Above Tc the mixture separates into two microemulsion phases of different composition but with the same ratio ω.1,3 Such a microemulsion system can be regarded as a pseudobinary mixture. Therefore, the phase behavior can be depicted in a two-dimensional diagram with concentration of droplets along the abscissa and temperature along the ordinate. A coexistence curve of temperature T against a concentration variable, such as volume fraction φ, can then be drawn in the same way as it was done for binary mixtures.1,4 Critical microemulsion and micellar systems are of considerable interest since a controversy from experiments and theories was raised about whether these systems near the critical points belonged to the 3D Ising universality class.4,5 Most of the experiments involved measurements of critical exponents ν, γ, and R which characterize the divergence of the correlation length, the osmotic compressibility, and the specific heat at constant pressure and critical concentration.5,6 A few experimental studies7,8 have been carried out to determine the critical exponent β, which characterizes the shape of the coexistence curve, but the results were not precise enough to unambiguously support the Ising value or Fisher renormalization value. In this work, we present precise coexistence curve measurements of a ternary microemulsion, consisting of water, n-decane, and AOT, at constant pressure and a constant ratio ω of 40.8 within about 10 K temperature range above the critical temperature. The experimental results are analyzed to determine the critical exponent β and the critical amplitude B and to examine the size of asymptotical range and the anomalies of the diameters for different choices of order parameters.
Materials. The AOT surfactant was obtained from Fluka and purified according to a standard procedure.9 The n-decane (99 mass %) supplied from Aldrich Chemical Co. was used without further treatment. The water was twice distilled from deionized water in our laboratory. Preparation of Samples. The critical composition of the mixture was approached by fixing the ratio ω at 40.8 and adjusting the amounts of decane to achieve equal volumes of the two phases at critical temperature.10,11 The phase separation temperature of the mixture with the critical composition was carefully measured and taken as the critical temperature. It was observed that samples nominally of the same composition had different critical temperatures, and the difference was as much as 2 K. However, it did not affect the final results, because only one sample was used throughout the measurements of the whole coexistence curve and only temperature difference (T Tc) was important in data reductions to obtain the critical parameters. Measurements of Refractive Indices. The refractive indices were measured according to the method of minimum deviation.10-12 The apparatus used in this work was described previously.11 During measurements the temperature was constant to (0.001 K. The accuracy of measurement was about (0.01 K for temperature and (0.0001 for refractive index. A sample with the critical composition was prepared in a rectangular fluorometer cell provided with a Teflon plug. Continuous slow upward shifts of phase separation temperature near the critical point were observed. The rate of upward shifts was determined by repeating measurements of critical temperature, and a correction was made to each observed temperature by subtraction of the shift value. Several measurements with these corrections in heating and cooling runs were consistent within experimental uncertainties. This is evidence that the shift of the critical temperature does not affect the shape of the coexistence curve. The accuracy in measurement of temperature difference (T - Tc) was about (0.003 K.
X
Abstract published in AdVance ACS Abstracts, September 1, 1996.
S0022-3654(96)01287-7 CCC: $12.00
III. Results and Discussion The critical volume fraction of [φ(AOT-water) + (1 φ)decane] was determined to be φc ) 0.098 ( 0.001, which was consistent with that reported by Rouch et al.1 The critical temperature Tc ) 310.8, about 2 K lower than that observed by Rouch et al.1 The refractive indices n of coexisting phases in © 1996 American Chemical Society
Critical Behavior of a Three-Component Microemulsion
J. Phys. Chem., Vol. 100, No. 41, 1996 16675
TABLE 1: Coexistence Curves of (T, n), (T, O), and (T, Ψ) for Microemulsion [O(AOT-Water) + (1 - O)Decane)]a T (°C)
n1
n2
φ1
φ2
Ψ1
Ψ2
310.804 310.814 310.820 310.830 310.843 310.880 310.900 310.923 310.941 310.947 310.966 311.012 311.085 311.180 311.274 311.383 311.496 311.625 311.791 312.024 312.421 312.986 313.807 314.834 316.135 317.858 320.139
1.4002 1.4002 1.4001 1.4000 1.3999 1.3999 1.3998 1.3997 1.3996 1.3996 1.3995 1.3993 1.3991 1.3989 1.3987 1.3985 1.3983 1.3980 1.3977 1.3975 1.3970 1.3964 1.3955 1.3945 1.3936 1.3925 1.3914
1.4011 1.4011 1.4012 1.4013 1.4013 1.4013 1.4014 1.4015 1.4016 1.4016 1.4016 1.4016 1.4018 1.4019 1.4019 1.4021 1.4020 1.4021 1.4021 1.4020 1.4019 1.4018 1.4015 1.4011 1.4005 1.3998 1.3988
0.112 0.113 0.117 0.119 0.124 0.121 0.125 0.129 0.134 0.134 0.135 0.141 0.151 0.155 0.162 0.166 0.174 0.183 0.191 0.195 0.209 0.227 0.250 0.272 0.290 0.312 0.321
0.082 0.079 0.076 0.073 0.071 0.072 0.069 0.065 0.062 0.062 0.062 0.059 0.053 0.048 0.045 0.038 0.038 0.035 0.032 0.030 0.027 0.025 0.022 0.020 0.020 0.015 0.016
0.537 0.538 0.549 0.553 0.565 0.558 0.567 0.578 0.586 0.588 0.589 0.602 0.620 0.627 0.641 0.647 0.660 0.673 0.685 0.690 0.708 0.729 0.754 0.775 0.790 0.807 0.813
0.450 0.440 0.429 0.422 0.414 0.416 0.406 0.390 0.380 0.380 0.379 0.367 0.341 0.315 0.300 0.269 0.264 0.252 0.235 0.220 0.204 0.188 0.171 0.158 0.155 0.122 0.131
a Refractive indices were measured at wavelength λ ) 632.8 nm. Subscripts 1 and 2 relate to lower and upper phases, respectively.
represent the values of (∂n/∂T) for φ ) 1 and 0, respectively. From the temperature dependence of refractive indices of pure decane listed in Table 2, the value of (∂nB/∂T) was calculated to be -4.6 × 10-4 K-1. Rearrangement of eqs 1 and 3 yields
n(φ, T) ) n(φ, T0) + [φ(∂nA/∂T) + (1 - φ)(∂nB/∂T)]∆T (4) The values of n(φ, T) listed in Table 1 were fitted to eq 4 to obtain (∂nA/∂T), which was -1.5 × 10-4 K-1, and the values of n(φ, T0). The small standard deviation of 0.0002 in refractive index indicates that eq 4 is valid. This allowed us to simplify the procedure of determination of the dependence of n on φ just by fitting a polynomial form to n(φ, T0) for various φ at T0. We obtained the expression
n(φ, T0) ) 1.4015 - 0.0302φ + 0.0235φ2 - 0.0301φ3
(5)
with a standard deviation 0.498 is unreliable. The values of refractive indices then were converted to volume fractions by calculating n(φ, T0) through eq 4 and iteratively solving eq 5. The results are listed in columns 4 and 5 of Table 1 and shown in Figure 1b. The choice of the concentration variable may affect the symmetry and the size of the asymptotic region of a coexistence curve. One of the choices is to define an effective volume fraction Ψ:4,14
Ψ ) φ/[φ + R(1 - φ)]
(6)
R ) φc/(1 - φc)
(7)
We converted (T, φ) to (T, Ψ) by eqs 6 and 7. The results are listed in columns 6 and 7 in Table 1 and shown in Figure 1c. The difference of concentration variables (F2 - F1) may be expressed by the Wegner expression
(F2 - F1) ) Bτβ + B1τβ + ∆ + . . . Figure 1. Coexistence curve of (T, n), (T, φ), and (T, Ψ) for [φ(AOTwater) + (1 - φ)n-decane]: (a) temperature T vs refractive index n; (b) temperature T vs volume fraction φ; (c) temperature T vs effective volume fraction Ψ. (9) Diameter of the coexistence curve.
the cell were measured at various temperatures. The results are listed in columns 2 and 3 of Table 1 and shown in Figure 1a. To obtain the (T, φ) coexistence curve, a series of ternary mixtures of water-AOT-decane with known volume fractions were prepared, and their refractive indices in the one-phase region at various temperatures were measured. The results are listed in Table 2. With the assumption that no significant critical anomaly is present in the refractive index,13 the refractive index of the microemulsion may be expressed as a linear function of temperature in a certain temperature T range:
n(φ, T) ) n(φ, T0) + (∂n/∂T)∆T
(1)
∆T ) T - T 0
(2)
(∂n/∂T) ) φ(∂nA/∂T) + (1 - φ)(∂nB/∂T)
(3)
T0 is chosen as 315.15 K (about the middle temperature of the coexistence curve determined in this work), (∂n/∂T) is the derivative of n with respect to T, and (∂nA/∂T) and (∂nB/∂T)
(8)
where τ ) (T - Tc )/Tc, b, and ∆ are critical exponents, F is the concentration variable or the order parameter, and F2 and F1 are the values of F in the upper and lower coexisting phases. In the region sufficiently close to the critical temperature, the simple scaling is valid:
(F2 - F1) ) Bτβ
(9)
It is well-known that the region of validity of eq 9 is affected by the choices of the variables. A wrong choice of the variable may cause a significant reduction of the region of validity of eq 9. We estimated the value of b in different τ regions for n, φ, and Ψ by fitting the experimental data to eq 9. The results are shown in Figure 2, where (T - Tc)max is the cutoff value for maximum (T - Tc). The values of b depend on (T - Tc)max, and the errors in determination of β increase as (T - Tc) is narrowed. When all experimental data were used to fit eq 9, the values of β were obtained to be 0.325 ( 0.002, 0.310 ( 0.003, and 0.270 ( 0.017 for φ, n, and Ψ, which are significantly inconsistent with the Fisher renormalized value of 0.365. However, the value of 0.325 ( 0.002 for φ is consistent with the 3D Ising value of 0.3265 ( 0.001. Within the range of 1.5 K from the critical temperature, the average values of β are 0.327 ( 0.003, 0.324 ( 0.004, and 0.325 ( 0.004 for φ, n, and Ψ, respectively. They all approach the 3D Ising value within experimental uncertainties.
16676 J. Phys. Chem., Vol. 100, No. 41, 1996
An et al.
TABLE 2: Refractive Indices at Wavelength λ ) 632.8 nm for [O(AOT-Water) + (1 - O)Decane] at Various Temperatures T φ
T/K
n
T/K
n
T/K
n
0.000
308.286 310.297 311.293 300.200 301.850 299.388 300.735 301.526 303.420 294.838 296.559 300.093 302.351 300.213 302.073 300.662 301.264 302.048 301.014 302.541 302.159 303.703 300.241 302.095 300.755 302.454 300.720 302.474 297.142 301.232 298.429 300.361
1.4047 1.4039 1.4034 1.4076 1.4069 1.4074 1.4069 1.4057 1.4048 1.4078 1.4070 1.4051 1.4041 1.4044 1.4036 1.4038 1.4036 1.4031 1.4031 1.4025 1.4026 1.4020 1.4032 1.4024 1.4023 1.4016 1.4017 1.4010 1.3979 1.3965 1.3938 1.3932
312.309 313.286 314.288 303.380 305.041 302.455 304.281 305.342 307.255 298.235 299.770 303.887 305.421 303.916 305.770 302.383 303.717 303.776 304.460 305.756 305.179 306.601 303.659 305.439 304.013 305.719 304.142 305.639 304.050 305.682 302.637 304.525
1.4029 1.4023 1.4019 1.4061 1.4055 1.4060 1.4052 1.4040 1.4032 1.4063 1.4057 1.4035 1.4029 1.4029 1.4022 1.4032 1.4027 1.4025 1.4017 1.4011 1.4014 1.4009 1.4017 1.4010 1.4011 1.4003 1.4004 1.3997 1.3955 1.3949 1.3924 1.3918
315.291 317.298 319.200 306.944 308.750 306.233 308.112 308.962 310.430 302.087 304.312 307.062 308.780 307.813
1.4016 1.4006 1.3997 1.4046 1.4037 1.4044 1.4036 1.4023 1.4017 1.4046 1.4036 1.4021 1.4013 1.4012
304.920 305.880 306.264 306.921 308.723 308.092
1.4021 1.4016 1.4015 1.4007 1.3999 1.4003
307.156 309.427 307.617 309.527 307.648 309.589 307.758 309.229 306.297
1.4003 1.3994 1.3995 1.3988 1.3990 1.3981 1.3942 1.3936 1.3912
0.0199 0.0393 0.0650 0.0837 0.1017 0.1215 0.1295 0.1486 0.1500 0.1698 0.1904 0.2153 0.3782 0.4980
T/K
n
310.414
1.4025
306.066 307.677
1.4029 1.4022
307.695 307.874
1.4010 1.4008
309.450
1.3997
311.741
1.3972
311.012
1.3929
B|. The value of this ratio for φ is significantly smaller than that for n and Ψ, which implies that φ is the best variable among the three choices of the variables used to defined an order parameter. This is in contradiction with that reported by Martin et al.4 It is well-known that when a good variable is used to construct an order parameter, the diameter of the coexistence curve may be expressed as
Fd ) (F2 + F1)/2 ) Fc + A0τ + A1τ1- R + . . . (10a) otherwise, the diameter shows a 2β anomaly:
Fd ) (F2 + F1)/2 ) Fc + A0τ + Cτ2β + . . . Figure 2. Values of the critical exponent β in the difference range of (T - Tc) for n, φ, and Ψ obtained by fitting the experimental data to eq 9: (b) refractive index; (9) volume fraction; (2) effective volume fraction.
TABLE 3: Paramters of Equation 8 for Coexistence Curves of (T, n), (T, O), and (T, Ψ) for [O(AOT-Water) + (1 O)Decane) order parameter n φ Ψ
B 0.0266 ( 0.0003 0.0304 ( 0.0002 1.025 ( 0.004 1.068 ( 0.008 2.50 ( 0.10 3.59 ( 0.15
B1
|B1/B|
-0.037 ( 0.002
1.22
-0.41 ( 0.07
0.38
-10.5 ( 1.3
2.9
The goodness of variables used to construct the order parameters can also be tested by fitting the experimental data to eq 8 with fixed values of β ) 0.3265 and ∆ ) 0.5. The results are listed in Table 3. The significance of Wegner correction terms may be qualitatively indicated by the ratio |B1/
(10b)
We fitted our experimental data to eqs 10a and 10b in separate fitting procedures with fixed values of a ) 0.11 and β ) 0.3265. we obtained Fc, A0, A1, and C. The characteristics of the fits are summarized in Table 4. The experimental value of nc was obtained by extrapolating refractive index against temperature in the one-phase region to the critical temperature.11 The uncertainties of optimal parameters reported in Table 4 include no systematic ones contributed by converting n to φ and φ to Ψ. These uncertainties in φc and Ψc were estimated to be (0.003 and (0.007, respectively. Therefore, the values φc and nc are consistent with the experimental results. This is evidence that no significant critical anomaly is present in refractive indices and that the refractive indices were properly converted to volume fraction φ. The values of Ψc obtained from fitting eqs 10a and 10b were 0.487 and 0.490, and both significantly departed from the “experimental” value of 0.5 determined by eqs 6 and 7. This is evidence that Ψ is not a good variable for constructing an order parameter. Although eqs 10a and 10b omit the higher Wegner correction terms and artificially separate the effects of terms 2β and (1 - a), comparing the standard deviations S and Fc in fitting eqs 10a and 10b may give some indication of the
Critical Behavior of a Three-Component Microemulsion TABLE 4: Parameters of Equations 10a and 10b and Standard Deviations S in G for Diameters of Coexistence Curves of (T, n), (T, O), and (T, Ψ) of [O(AOT-Water) + (1 - O)Decane]a parameter Fc,expt
(T, n) 1.4007 ( 0.0001
(T, Ψ)
(T, φ) 0.098 ( 0.001
0.500
Fc A0 A1 S
Fd ) Fc + A0τ + A1τ1-R 1.4007 ( 0.0001 0.095 ( 0.001 0.487 ( 0.001 0.11 ( 0.02 -9.2 ( 0.7 23 ( 2 -0.21 ( 0.01 8.0 ( 0.4 -16 ( 2 0.0001 0.0019 0.0066
Fc A0 C S
Fd ) Fc + A0τ + Cτ 1.4007 ( 0.0001 0.094 ( 0.001 -0.114 ( 0.006 -0.4 ( 0.2 -0.024 ( 0.002 0.91 ( 0.07 0.0001 0.0024 2β
0.490 ( 0.001 6.5 ( 0.6 -2.0 ( 0.2 0.0056
aF c,expt is the critical value of the order parameter determined by the techniques described in the text.
significance of terms 2β and (1 - a). When Ψ was used to construct an order parameter, the standard deviation in fitting eq 10a was 0.0066, larger than that of 0.0056 in fitting eq 10b, and the value of Ψc obtained from fitting eq 10a departed further from 0.5 than that obtained from fitting eq 10b. This indicates that the choice of Ψ more likely leads to the 2β anomaly. In contrast to the variable Ψ, the results for φ listed in Table 4 indicate that the diameter of the coexistence curve supports a (1 - a) anomaly. The following conclusions can be drawn: (1) The present results confirm that the pseudobinary mixture of the microemulsion we studied belongs to the 3D Ising universality class. In a region sufficiently close to the critical
J. Phys. Chem., Vol. 100, No. 41, 1996 16677 temperature, the shapes of coexistence curves may be characterized by simple scaling with the 3D Ising value of critical exponent. (2) Our results show that φ is a better choice of the concentration variable than Ψ for constructing an order parameter. Acknowledgment. This work was supported by the State Education Committee, People’s Republic of China. References and Notes (1) Rouch, J.; Safouane, A.; Tartaglia, P.; Chen, S. H. Progr. Colloid Polym. Sci. 1989, 79, 279. (2) Kotlarchyk, M.; Chen, S. H.; Huang, J. S.; Kim, M. W. A. Phys. ReV. A 1983, 28, 508; 1984, 29, 2054. (3) Belloq, A. M.; Roux, D. Phys. ReV. Lett. 1984, 52, 1895. (4) Martin, A.; Lopez, I.; Monroy, F.; Casielles, A. G.; Ortega, F. J. Chem. Phys. 1994, 101, 6874. (5) See, e.g.: Jayalakshmi, Y.; Beysens, D. Phys. ReV. A. 1992, 45, 8079. (6) See, e.g.: Aschauer, R.; Beysens, D. Phys. ReV. E. 1993, 47, 1850. (7) Aschauer, R.; Baysens, D. J. Chem. Phys. 1993, 98 (10), 8194. (8) Honorat, P.; Roux, K.; Belloq, A. M. J. Phys. Lett. 1984, 45, 961. (9) Johnston, K. P.; McFann, G. J.; Lemert, R. M. In Supercritical Fluid Science and Technology; Johnston, K. P., Penninger, J. M., Eds.; ACS Symposium Series 406; American Chemical Society: Washington, DC, 1989. (10) Shen, W.; Smith, G. R.; Knobler, C. M.; Scott, R. L. J. Phys. Chem. 1991, 95, 3376. (11) An, X.; Shen, W.; Wang, H.; Zheng, G. J. Chem. Thermodyn. 1993, 25, 1373. (12) Levelt Sengers, J. M. H. Experimental Thermodynamics; Le Neindre, B., Bodar, Bl., Eds.; Butterworth: London, 1975; Vol. 2, p 657. (13) Sengers, J. V.; Bedeaux, D.; Mazur, P.; Breer, S. C. Physics 1980, 104A, 573. (14) Sanchez, I. C. J. Appl. Phys. 1985, 58 (8), 287.
JP961287I
