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The critical phenomena of (water + AOT + decane) microemulsion with the molar ratio (ω = 30.0) of water to AOT
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J. Chem. Thermodynamics 39 (2007) 1470–1473 www.elsevier.com/locate/jct
The critical phenomena of (water + AOT + decane) microemulsion with the molar ratio (x = 30.0) of water to AOT Honglan Cai a, Xueqin An b
b,*
, Weiguo Shen
a,c
a Department of Chemistry, Lanzhou University, Lanzhou, Gansu 730000, China College of Chemistry and Environment Science, Nanjing Normal University, Nanjing, Jiangsu 210097, China c Department of Chemistry, East China University of Science and Technology, Shanghai 200237, China
Received 13 December 2006; received in revised form 2 March 2007; accepted 2 March 2007 Available online 13 March 2007
Abstract The coexistence curves of a ternary microemulsion system of {water + sodium di(2-ethylhexyl) sulfosuccinate (AOT) + n-decane} with the molar ratio (30.0) of water to AOT have been determined by measurements of refractive index at constant pressure within about 8 K from the critical temperature Tc. The critical exponent b and the critical amplitude B have been deduced from the coexistence curves. The experimental results have been analysed and compared with the system with molar ratio of 40.8 studied previously. It was found that the critical exponents b for both systems were consistent with the 3D-Ising value in a region sufficiently close to the critical temperature. The critical concentration was slightly affected by the molar ratio x, but the critical temperature significantly was raised as the molar ratio x was decreased. The volume fraction / was the better choice of the concentration variable than the effective volume fraction w and the refractive index n used for constructing the order parameter for both systems. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Microemulsion; Critical exponent; Coexistence curve; AOT
1. Introduction Three-component mixtures of water, sodium di(2-ethylhexyl) sulfosuccinnate (AOT), and n-alkane can form water-in-oil microemulsions of well-defined droplet size determined by fixing the molar ratio x of water to AOT [1,2]. For the mixture a lower consolute critical point is observed. Above critical temperature Tc, the mixture separates into two microemulsion phases of different composition but with the same ratio x [1,2]. Therefore such a microemulsion system can be regarded as a pseudobinary mixture. A controversy from experiments and theories was raised about whether these systems near critical points belonged to the 3D-Ising universality class [3,4]. Most of the experiments [5] involved measurements of critical exponents m, c and a, which characterize the divergence of the *
Corresponding author. Tel.: +86 25 83598678. E-mail address: [email protected] (X. An).
0021-9614/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jct.2007.03.003
correlation length and osmotic compressibility and the specific heat at constant pressure and critical concentration. A few experimental studies [6,7] have been carried out to determine the critical exponent b, which characterizes the shape of the coexistence curve, but the results were not precise enough to support unambiguously the Ising value or the Fisher re-normalization value. In the previous work, we studied the critical behaviour for the critical microemulsion of (water + AOT + n-decane) with the molar ratio (40.8) of water to AOT and showed that the system belongs to 3D-Ising universality class [8]. It would be interesting to examine the effect of molar ratio x on the critical behaviour. In the paper, we present precise coexistence curve measurements of a ternary microemulsion, consisting of water, n-decane, and AOT, at a constant ratio x of 30.0 and constant pressure within about 8 K temperature range above the critical temperature. The experimental results are analysed to determine the critical temperature, the critical concentration, the critical exponent and the critical ampli-
H. Cai et al. / J. Chem. Thermodynamics 39 (2007) 1470–1473
tude, and to compare with that of the system with the molar ratio (40.8) of water to AOT.
1471
minimum deviation [8,10]. A slow downward drift of the phase-separation temperature of 5 mK per day was also observed and a correction was made for each observed temperature by subtraction of the temperature shift. The apparatus used in this work was described previously [10]. 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. The accuracy in measurement of temperature difference (T Tc) was about ±0.003 K.
2. Materials and methods 2.1. Materials The AOT (mass fraction purity >0.98) surfactant was obtained from Fluka and purified according to a standard procedure [9]. The n-decane (mass fraction purity >0.99) supplied from Aldrich Chemical Co. was used without further treatment. The water was twice distilled from deionised water in our laboratory.
3. Results and discussion The critical volume fractions /c and the critical temperature Tc of {/(water-AOT) + (1-/) n-decane} with the molar ratio (30.0) of water to AOT were determined to be (0.099 ± 0.001) and T = 318.26 K, respectively. The critical volume fraction was very close to the value of 0.098, and the critical temperature was higher than the value of T = 310.8 K of the system with the molar ratio of 40.8 studied previously [8]. Decrease of the critical temperature by increasing x may be explained by the fact that there is an attractive interaction between dispersed droplets of microemulsion with a short-ranged, but strong potential due to the overlapping of the surfactant hydrocarbon tail. This interaction can be increased linearly with the size of droplet (namely x), which imply that the microemulsion system with the large droplets is unstable. However, when the temperature is lowered, the attraction of the background oil fluid to the surfactant orients the chain oil molecules at the globule interface and decreases the surfactant–surfactant penetration interaction between the droplets, which is beneficial to form the stable microemulsion [11,12]. The refractive indexes n of the coexisting phases in the cell were measured at various temperatures. The results are shown in figure 1a. To obtain the (T, /) coexistence curves, a series of ternary mixtures of (water + AOT + n-decane) with known volume fraction were prepared and their refractive indexes were measured in one phase at various temperatures. The
2.2. Determination of critical composition and critical temperature The critical composition /c of the mixture was approached by fixing the molar ratio of water to AOT at 30.0 and adjusting the amounts of n-decane to achieve equal volume of the two phases at the critical temperature [8]. The phase-separation temperature of the mixture with the critical composition was carefully determined 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 1 K. This may be a result of uncontrollable impurities introduced into sample during preparation or of the hydrolysis of AOT. However, the final results were no affected, because only one sample was used throughout the measurements over the whole coexistence curve and only the temperature difference (T Tc) was important in data reduction to obtain the critical parameters. Therefore the phase diagram of (T Tc) against the density variable was reproducible. 2.3. Measurements of refractive indexes The refractive indexes of the coexisting phases at various temperatures were measured according to the method of 10 8
(T–Tc)/K
6 4 2 0
a 1.392
b 1.394
1.396 1.398
n
0.0
c .1
.2
φ
.3 0.0
.2
.4
.6
.8
ψ
FIGURE 1. Plot of (a) (T Tc) against refractive indexn, (b) (T Tc) against the volume fraction /, (c) (T Tc) against the effective volume fraction w. (d) To show the coexistence curves for {/ (AOT-water) + (1 /) n-decane} with the molar ratio x = 30.0. (s) Experimental values of concentration variables (q) of the coexisting phases, (.) experimental values of diameter (qd) of the coexisting phases, (—) concentration variables (q) and (qd) of coexisting phases from calculation.
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H. Cai et al. / J. Chem. Thermodynamics 39 (2007) 1470–1473
TABLE 1 Refractive indexes n at a wavelength k = 632.8 nm of microemulsions of {/ (water-AOT) + (1 /) n-decane} at various temperatures T and volume fractions / /
T/K
n
T/K
n
T/K
n
0.0000
326.754 324.979 323.297
1.3950 1.3958 1.3966
321.696 320.149 318.563
1.3973 1.3980 1.3987
317.083 315.720
1.3994 1.4000
0.0617
315.775 314.591 313.608
1.3991 1.3996 1.4000
312.481 311.308 310.113
1.4005 1.4011 1.4016
308.913 307.735
1.4021 1.4026
0.0797
315.060 313.833 312.628
1.3991 1.3997 1.4002
311.419 310.236 309.177
1.4007 1.4012 1.4017
308.133 306.889
1.4022 1.4027
0.1198
316.445 315.225 314.049
1.3980 1.3985 1.3990
312.910 311.748 310.679
1.3995 1.4000 1.4004
309.630 308.691
1.4009 1.4013
0.1513
315.797 314.601 313.535
1.3978 1.3982 1.3987
312.424 311.370 310.356
1.3992 1.3996 1.4000
309.217 308.141
1.4005 1.4009
0.1965
316.533 315.312 314.125
1.3968 1.3973 1.3978
312.964 311.924 310.882
1.3983 1.3987 1.3991
309.916 308.952
1.3995 1.3999
0.2786
316.536 315.396 314.240
1.3956 1.3961 1.3965
312.991 311.815 310.574
1.3970 1.3974 1.3979
309.424 308.444
1.3984 1.3988
316.950 315.590 314.383
1.3943 1.3948 1.3953
313.138 312.028 310.810
1.3958 1.3962 1.3966
309.748 308.583
1.3970 1.3974
0.3531
results are listed in table 1. The refractive index of the microemulsion may be expressed as a linear function of temperature in a certain temperature range nð/; T Þ ¼ nð/; T Þ þ ðon=oT Þ/ ðT T Þ;
ð1Þ
ðon=oT Þ/ ¼ /ðonA =oT Þ þ ð1 /ÞðonB =oT Þ;
ð2Þ
nð/Þ ¼ 1:3971 0:0128/ 0:00109/2
ð4Þ
with a standard deviation of 0.0002 in refractive index. All of our experimental data used in fitting equation (4) are in the range of / < 0.354, therefore equation (4) can only be safely used over such a range. The values of refractive indices then were converted to volume fractions by equation (4). The results are shown in figure 1b. The choice of concentration variable may affect the symmetry and the size of asymptotic region of a coexistence curve. One of the choices is to define an effective volume fraction w [13] w ¼ /=½/ þ /c ð1 /Þ=ð1 /c Þ:
ð5Þ
The (T, /) was converted to (T, w) by equation (5). The results are shown in figure 1c. According to the theory of critical phenomena, the difference between concentration variables (q2 q1) can be expressed by the Wegner expression ðq2 q1 Þ ¼ Bsb þ B1 sbþD þ ;
ð6Þ
where s = (T Tc)/Tc, b and D are the critical exponents; B and B1 are the system-dependence amplitudes; q is the concentration variable, q1 and q2 are the values of q in the upper and lower coexisting phase, respectively. In the region sufficiently close to the critical temperature, the simple scaling is valid: ðq2 q1 Þ ¼ Bsb :
ð7Þ
It is well known that the region of validity of equation (7) 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 equation (7). The values of b were estimated with different values of s for n, /, and w by fitting the experimental data to equation (7). The results for / are shown in figure 2, where (T Tc)max is the cut off value for the maximum (T Tc). The values of b for the three choices of variables n, /, and w increased as the temperature range was reduced. The values of b depend on the .345
where T is chosen as T = 322.15 K, about the middle temperature of the coexistence curves shown in figure 1, and (onA/oT) and (onB/oT) represent the values of (on/oT)/ for / = 1 and 0, respectively. From the temperature dependence of refractive indexes nB of pure decane listed in table 1, the value of (o nB/oT) was calculated to be 4.55 Æ 104. Rearrangement of equations (1) and (2) yields nð/;T Þ ¼ nð/;T Þ þ ½/ðonA =oT Þ þ ð1 /ÞðonB =oT ÞðT T Þ: ð3Þ The values of n(/, T) listed in table 1 were fitted to equation (3) to obtain (onA/oT), which was 2.3 Æ 104, and the values of n(/, T). The small standard deviation of 0.0002 in refractive index indicates that equation (3) is valid. This allowed us to simplify the procedure of determination of the dependence of n on / just by fitting a polynomial form to n(/, T) for various / at T. We obtained the expression
.340 .335
β
.330 .325 .320 .315 .310 0
2
4
6
8
10
(T-Tc)max /K FIGURE 2. Plot of the values of the critical exponent against (T Tc)max for / obtained by fitting the experimental data to equation (7) (d) for volume fraction /.
H. Cai et al. / J. Chem. Thermodynamics 39 (2007) 1470–1473 TABLE 2 Critical amplitude B and B1 of three order parameters for (water + AOT + n-decane) with different molar ratios x of water to AOT in the temperature range of (T Tc) < 10 K Order parameter
B
B1
jB1/Bj
n
0.0144 ± 0.0001 0.0152 ± 0.0002 1.027 ± 0.003 1.037 ± 0.006 3.03 ± 0.01 3.22 ± 0.03
0.021 ± 0.001
1.38
0.40 ± 0.06
0.39
4.6 ± 0.3
1.43
/ w
(T Tc)max approach to the 3D-Ising value of 0.3265, and are obviously inconsistent with the Fisher renormalized value of 0.365 within experimental uncertainties in the region sufficiently close to the critical temperature. The goodness of variables used to construct the order parameters can also be tested by fitting the experimental data to equation (6) with fixed values of b = 0.3265 and D = 0.5, and comparing the significance of the first Wegner correction. The results are listed in table 2. The significance of the first Wegner correction terms may be qualitatively indicated by the ratio jB1/Bj. As can be seen in the table 2, the values of jB1/Bj for / are significantly smaller than that for n and w, which imply / is the best variable among the three choices of the variables used to define an order parameter. The results are in agreement with that of the system with the molar ratio of 40.8. It is well known that when the correct variable is used to construct an order parameter, the diameter of the coexistence curve may expressed as qd ¼ ðq2 þ q1 Þ=2 ¼ qc þ A0 s þ A1 s1a þ
ð8Þ
Otherwise, the diameter shows a 2b anomaly: qd ¼ ðq2 þ q1 Þ=2 ¼ qc þ A0 s þ Cs2b þ
ð9Þ
We fitted our experimental data over the range of (T Tc) < 8 K by equations (8) and (9) in separate fitting procedures with fixed values of a = 0.11 and b = 0.3265 and obtained qc, A0, A1 and C. The characteristics of the fits are summarized in table 3. The experimental value of nc was obtained by extrapolating refractive index against temperature in the one-phase region to the critical temperTABLE 3 Parameters of equations (8) and (9) and standard deviations S in q for diameters of coexistence curves of (T, n), (T, /) and (T, w) of water + AOT + n-decane with the fixed molar ratio x = 30.0 Parameter
(T, n)
qc, qc A0 A1 S
1.3976 ± 0.0001 0.099 ± 0.001 qd = qc + A0s + A1s1a 1.3976 ± 0.0001 0.094 ± 0.001 0.073 ± 0.014 4.47 ± 0.88 0.06 ± 0.01 5.0 ± 0.6 2.4 105 1.49 103
qc A0 C S
qd = qc + A0s + Cs2b 1.3976 ± 0.0001 0.093 ± 0.001 0.142 ± 0.004 0.52 ± 0.26 0.01 ± 0.001 0.5 ± 0.1 2.5 105 1.52 103
exp
(T, /)
(T, w) 0.500 ± 0.001 0.481 ± 0.002 33.5 ± 2.8 23 ± 2 4.8 103 0.485 ± 0.002 7.7 ± 0.8 3 ± 0.2 4.5 103
1473
ature [8,10,14]. The experimental value of /c,exp was determined by the technique of ‘‘equal volumes of two coexistence phases’’. The values of /c obtained from equation (9) departed further from the /c,exp than that obtained from fitting equation (8), and the standard deviation S in fitting equation (9) for variable / is larger than that in fitting equation (8), which indicate that the (1 a) anomaly is more significant than that of 2 b. The result is consistent with that of the system with the molar ratio (40.8) of water to AOT. Combination of equations (6) and (8) yields q1 ¼ qc þ A0 s þ A1 s1a ð1=2ÞBsb ð1=2ÞB1 sbþD þ ð10Þ q2 ¼ qc þ A0 s þ A1 s
1a
b
þ ð1=2ÞBs þ ð1=2ÞB1 s
bþD
þ ð11Þ
When a, b and D were fixed at 0.11, 0.3265 and 0.5, and the values of B, B1, A0, A1 and qc were taken from tables 2 and 3, the values of q1, q2 and qd were calculated from equations (10) and (11), the results are shown as lines in figure 1. The values from the calculations are in good agreement with experimental results. Acknowledgements This work was supported by the National Nature Science Foundation of China (Nos. 20273032, 20473035, 20573056, and 20673059), the New Technique Foundation of Jiangsu Province, P.R. China (No. BG2005041) and the Special Sustentation Fund of Nanometer Technology of Shanghai City, P.R. China (No. 0652nm010). References [1] M. Kotlarchyk, S.H. Chen, J.S. Huang, M.W. Kim, Phys. Rev. A 29 (1984) 2054–2069. [2] D. Roux, A.M. Bellocq, Phys. Rev. Lett. 52 (1984) 1895–1898. [3] A. Martin, I. Lopez, F. Monroy, A.G. Casielles, F. Ortega, J. Chem. Phys. 101 (1994) 6874–6879. [4] Y. Jayalakshimi, D. Beysens, Phys. Rev. A 45 (1992) 8709–8718. [5] R. Aschauer, D. Beycens, Phys. Rev. E 47 (1993) 1850–1855. [6] R. Aschauer, D. Beycens, J. Chem. Phys. 98 (1993) 8194–8198. [7] A.M. Bellocq, D. Gazeau, J. Phys. Chem. 94 (1990) 8933–8938. [8] X.Q. An, J. Feng, W.G. Shen, J. Phys. Chem. 100 (1996) 16674–16677. [9] K.P. Johnston, G.J. McFann, R.M. Lemert, in: K.P. Johnston, J.M.L. Penninger (Eds.), Supercritical Fluid Science and Technology, ACS Symposium Series, vol. 406, American Chemical Society, Washington, DC, 1989. [10] X.Q. An, J.Y. Chen, Y.G. Huang, W.G. Shen, J. Coll. Interf. Sci. 203 (1998) 140–145. [11] P. Li, X.Q. An, W.G. Shen, Acta Phys-Chim. Sin. 17 (2001) 144–149. [12] J.S. Huang, S.A. Safran, M.W. Kim, G.S. Grest, M.J. Kotlurchyk, N. Quirke, Phys. Rev. Lett. 53 (1984) 592–595. [13] M. Corti, V. Degiorgio, M. Zulauf, Phys. Rev. Lett. 48 (1982) 1617– 1620. [14] X.Q. An, J.Y. Chen, Y.G. Huang, W.G. Shen, J. Chem. Thermdyn. 34 (2002) 1107–1116.
JCT 06-337
J. Chem. Thermodynamics 39 (2007) 1470–1473 www.elsevier.com/locate/jct
The critical phenomena of (water + AOT + decane) microemulsion with the molar ratio (x = 30.0) of water to AOT Honglan Cai a, Xueqin An b
b,*
, Weiguo Shen
a,c
a Department of Chemistry, Lanzhou University, Lanzhou, Gansu 730000, China College of Chemistry and Environment Science, Nanjing Normal University, Nanjing, Jiangsu 210097, China c Department of Chemistry, East China University of Science and Technology, Shanghai 200237, China
Received 13 December 2006; received in revised form 2 March 2007; accepted 2 March 2007 Available online 13 March 2007
Abstract The coexistence curves of a ternary microemulsion system of {water + sodium di(2-ethylhexyl) sulfosuccinate (AOT) + n-decane} with the molar ratio (30.0) of water to AOT have been determined by measurements of refractive index at constant pressure within about 8 K from the critical temperature Tc. The critical exponent b and the critical amplitude B have been deduced from the coexistence curves. The experimental results have been analysed and compared with the system with molar ratio of 40.8 studied previously. It was found that the critical exponents b for both systems were consistent with the 3D-Ising value in a region sufficiently close to the critical temperature. The critical concentration was slightly affected by the molar ratio x, but the critical temperature significantly was raised as the molar ratio x was decreased. The volume fraction / was the better choice of the concentration variable than the effective volume fraction w and the refractive index n used for constructing the order parameter for both systems. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Microemulsion; Critical exponent; Coexistence curve; AOT
1. Introduction Three-component mixtures of water, sodium di(2-ethylhexyl) sulfosuccinnate (AOT), and n-alkane can form water-in-oil microemulsions of well-defined droplet size determined by fixing the molar ratio x of water to AOT [1,2]. For the mixture a lower consolute critical point is observed. Above critical temperature Tc, the mixture separates into two microemulsion phases of different composition but with the same ratio x [1,2]. Therefore such a microemulsion system can be regarded as a pseudobinary mixture. A controversy from experiments and theories was raised about whether these systems near critical points belonged to the 3D-Ising universality class [3,4]. Most of the experiments [5] involved measurements of critical exponents m, c and a, which characterize the divergence of the *
Corresponding author. Tel.: +86 25 83598678. E-mail address: [email protected] (X. An).
0021-9614/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jct.2007.03.003
correlation length and osmotic compressibility and the specific heat at constant pressure and critical concentration. A few experimental studies [6,7] have been carried out to determine the critical exponent b, which characterizes the shape of the coexistence curve, but the results were not precise enough to support unambiguously the Ising value or the Fisher re-normalization value. In the previous work, we studied the critical behaviour for the critical microemulsion of (water + AOT + n-decane) with the molar ratio (40.8) of water to AOT and showed that the system belongs to 3D-Ising universality class [8]. It would be interesting to examine the effect of molar ratio x on the critical behaviour. In the paper, we present precise coexistence curve measurements of a ternary microemulsion, consisting of water, n-decane, and AOT, at a constant ratio x of 30.0 and constant pressure within about 8 K temperature range above the critical temperature. The experimental results are analysed to determine the critical temperature, the critical concentration, the critical exponent and the critical ampli-
H. Cai et al. / J. Chem. Thermodynamics 39 (2007) 1470–1473
tude, and to compare with that of the system with the molar ratio (40.8) of water to AOT.
1471
minimum deviation [8,10]. A slow downward drift of the phase-separation temperature of 5 mK per day was also observed and a correction was made for each observed temperature by subtraction of the temperature shift. The apparatus used in this work was described previously [10]. 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. The accuracy in measurement of temperature difference (T Tc) was about ±0.003 K.
2. Materials and methods 2.1. Materials The AOT (mass fraction purity >0.98) surfactant was obtained from Fluka and purified according to a standard procedure [9]. The n-decane (mass fraction purity >0.99) supplied from Aldrich Chemical Co. was used without further treatment. The water was twice distilled from deionised water in our laboratory.
3. Results and discussion The critical volume fractions /c and the critical temperature Tc of {/(water-AOT) + (1-/) n-decane} with the molar ratio (30.0) of water to AOT were determined to be (0.099 ± 0.001) and T = 318.26 K, respectively. The critical volume fraction was very close to the value of 0.098, and the critical temperature was higher than the value of T = 310.8 K of the system with the molar ratio of 40.8 studied previously [8]. Decrease of the critical temperature by increasing x may be explained by the fact that there is an attractive interaction between dispersed droplets of microemulsion with a short-ranged, but strong potential due to the overlapping of the surfactant hydrocarbon tail. This interaction can be increased linearly with the size of droplet (namely x), which imply that the microemulsion system with the large droplets is unstable. However, when the temperature is lowered, the attraction of the background oil fluid to the surfactant orients the chain oil molecules at the globule interface and decreases the surfactant–surfactant penetration interaction between the droplets, which is beneficial to form the stable microemulsion [11,12]. The refractive indexes n of the coexisting phases in the cell were measured at various temperatures. The results are shown in figure 1a. To obtain the (T, /) coexistence curves, a series of ternary mixtures of (water + AOT + n-decane) with known volume fraction were prepared and their refractive indexes were measured in one phase at various temperatures. The
2.2. Determination of critical composition and critical temperature The critical composition /c of the mixture was approached by fixing the molar ratio of water to AOT at 30.0 and adjusting the amounts of n-decane to achieve equal volume of the two phases at the critical temperature [8]. The phase-separation temperature of the mixture with the critical composition was carefully determined 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 1 K. This may be a result of uncontrollable impurities introduced into sample during preparation or of the hydrolysis of AOT. However, the final results were no affected, because only one sample was used throughout the measurements over the whole coexistence curve and only the temperature difference (T Tc) was important in data reduction to obtain the critical parameters. Therefore the phase diagram of (T Tc) against the density variable was reproducible. 2.3. Measurements of refractive indexes The refractive indexes of the coexisting phases at various temperatures were measured according to the method of 10 8
(T–Tc)/K
6 4 2 0
a 1.392
b 1.394
1.396 1.398
n
0.0
c .1
.2
φ
.3 0.0
.2
.4
.6
.8
ψ
FIGURE 1. Plot of (a) (T Tc) against refractive indexn, (b) (T Tc) against the volume fraction /, (c) (T Tc) against the effective volume fraction w. (d) To show the coexistence curves for {/ (AOT-water) + (1 /) n-decane} with the molar ratio x = 30.0. (s) Experimental values of concentration variables (q) of the coexisting phases, (.) experimental values of diameter (qd) of the coexisting phases, (—) concentration variables (q) and (qd) of coexisting phases from calculation.
1472
H. Cai et al. / J. Chem. Thermodynamics 39 (2007) 1470–1473
TABLE 1 Refractive indexes n at a wavelength k = 632.8 nm of microemulsions of {/ (water-AOT) + (1 /) n-decane} at various temperatures T and volume fractions / /
T/K
n
T/K
n
T/K
n
0.0000
326.754 324.979 323.297
1.3950 1.3958 1.3966
321.696 320.149 318.563
1.3973 1.3980 1.3987
317.083 315.720
1.3994 1.4000
0.0617
315.775 314.591 313.608
1.3991 1.3996 1.4000
312.481 311.308 310.113
1.4005 1.4011 1.4016
308.913 307.735
1.4021 1.4026
0.0797
315.060 313.833 312.628
1.3991 1.3997 1.4002
311.419 310.236 309.177
1.4007 1.4012 1.4017
308.133 306.889
1.4022 1.4027
0.1198
316.445 315.225 314.049
1.3980 1.3985 1.3990
312.910 311.748 310.679
1.3995 1.4000 1.4004
309.630 308.691
1.4009 1.4013
0.1513
315.797 314.601 313.535
1.3978 1.3982 1.3987
312.424 311.370 310.356
1.3992 1.3996 1.4000
309.217 308.141
1.4005 1.4009
0.1965
316.533 315.312 314.125
1.3968 1.3973 1.3978
312.964 311.924 310.882
1.3983 1.3987 1.3991
309.916 308.952
1.3995 1.3999
0.2786
316.536 315.396 314.240
1.3956 1.3961 1.3965
312.991 311.815 310.574
1.3970 1.3974 1.3979
309.424 308.444
1.3984 1.3988
316.950 315.590 314.383
1.3943 1.3948 1.3953
313.138 312.028 310.810
1.3958 1.3962 1.3966
309.748 308.583
1.3970 1.3974
0.3531
results are listed in table 1. The refractive index of the microemulsion may be expressed as a linear function of temperature in a certain temperature range nð/; T Þ ¼ nð/; T Þ þ ðon=oT Þ/ ðT T Þ;
ð1Þ
ðon=oT Þ/ ¼ /ðonA =oT Þ þ ð1 /ÞðonB =oT Þ;
ð2Þ
nð/Þ ¼ 1:3971 0:0128/ 0:00109/2
ð4Þ
with a standard deviation of 0.0002 in refractive index. All of our experimental data used in fitting equation (4) are in the range of / < 0.354, therefore equation (4) can only be safely used over such a range. The values of refractive indices then were converted to volume fractions by equation (4). The results are shown in figure 1b. The choice of concentration variable may affect the symmetry and the size of asymptotic region of a coexistence curve. One of the choices is to define an effective volume fraction w [13] w ¼ /=½/ þ /c ð1 /Þ=ð1 /c Þ:
ð5Þ
The (T, /) was converted to (T, w) by equation (5). The results are shown in figure 1c. According to the theory of critical phenomena, the difference between concentration variables (q2 q1) can be expressed by the Wegner expression ðq2 q1 Þ ¼ Bsb þ B1 sbþD þ ;
ð6Þ
where s = (T Tc)/Tc, b and D are the critical exponents; B and B1 are the system-dependence amplitudes; q is the concentration variable, q1 and q2 are the values of q in the upper and lower coexisting phase, respectively. In the region sufficiently close to the critical temperature, the simple scaling is valid: ðq2 q1 Þ ¼ Bsb :
ð7Þ
It is well known that the region of validity of equation (7) 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 equation (7). The values of b were estimated with different values of s for n, /, and w by fitting the experimental data to equation (7). The results for / are shown in figure 2, where (T Tc)max is the cut off value for the maximum (T Tc). The values of b for the three choices of variables n, /, and w increased as the temperature range was reduced. The values of b depend on the .345
where T is chosen as T = 322.15 K, about the middle temperature of the coexistence curves shown in figure 1, and (onA/oT) and (onB/oT) represent the values of (on/oT)/ for / = 1 and 0, respectively. From the temperature dependence of refractive indexes nB of pure decane listed in table 1, the value of (o nB/oT) was calculated to be 4.55 Æ 104. Rearrangement of equations (1) and (2) yields nð/;T Þ ¼ nð/;T Þ þ ½/ðonA =oT Þ þ ð1 /ÞðonB =oT ÞðT T Þ: ð3Þ The values of n(/, T) listed in table 1 were fitted to equation (3) to obtain (onA/oT), which was 2.3 Æ 104, and the values of n(/, T). The small standard deviation of 0.0002 in refractive index indicates that equation (3) is valid. This allowed us to simplify the procedure of determination of the dependence of n on / just by fitting a polynomial form to n(/, T) for various / at T. We obtained the expression
.340 .335
β
.330 .325 .320 .315 .310 0
2
4
6
8
10
(T-Tc)max /K FIGURE 2. Plot of the values of the critical exponent against (T Tc)max for / obtained by fitting the experimental data to equation (7) (d) for volume fraction /.
H. Cai et al. / J. Chem. Thermodynamics 39 (2007) 1470–1473 TABLE 2 Critical amplitude B and B1 of three order parameters for (water + AOT + n-decane) with different molar ratios x of water to AOT in the temperature range of (T Tc) < 10 K Order parameter
B
B1
jB1/Bj
n
0.0144 ± 0.0001 0.0152 ± 0.0002 1.027 ± 0.003 1.037 ± 0.006 3.03 ± 0.01 3.22 ± 0.03
0.021 ± 0.001
1.38
0.40 ± 0.06
0.39
4.6 ± 0.3
1.43
/ w
(T Tc)max approach to the 3D-Ising value of 0.3265, and are obviously inconsistent with the Fisher renormalized value of 0.365 within experimental uncertainties in the region sufficiently close to the critical temperature. The goodness of variables used to construct the order parameters can also be tested by fitting the experimental data to equation (6) with fixed values of b = 0.3265 and D = 0.5, and comparing the significance of the first Wegner correction. The results are listed in table 2. The significance of the first Wegner correction terms may be qualitatively indicated by the ratio jB1/Bj. As can be seen in the table 2, the values of jB1/Bj for / are significantly smaller than that for n and w, which imply / is the best variable among the three choices of the variables used to define an order parameter. The results are in agreement with that of the system with the molar ratio of 40.8. It is well known that when the correct variable is used to construct an order parameter, the diameter of the coexistence curve may expressed as qd ¼ ðq2 þ q1 Þ=2 ¼ qc þ A0 s þ A1 s1a þ
ð8Þ
Otherwise, the diameter shows a 2b anomaly: qd ¼ ðq2 þ q1 Þ=2 ¼ qc þ A0 s þ Cs2b þ
ð9Þ
We fitted our experimental data over the range of (T Tc) < 8 K by equations (8) and (9) in separate fitting procedures with fixed values of a = 0.11 and b = 0.3265 and obtained qc, A0, A1 and C. The characteristics of the fits are summarized in table 3. The experimental value of nc was obtained by extrapolating refractive index against temperature in the one-phase region to the critical temperTABLE 3 Parameters of equations (8) and (9) and standard deviations S in q for diameters of coexistence curves of (T, n), (T, /) and (T, w) of water + AOT + n-decane with the fixed molar ratio x = 30.0 Parameter
(T, n)
qc, qc A0 A1 S
1.3976 ± 0.0001 0.099 ± 0.001 qd = qc + A0s + A1s1a 1.3976 ± 0.0001 0.094 ± 0.001 0.073 ± 0.014 4.47 ± 0.88 0.06 ± 0.01 5.0 ± 0.6 2.4 105 1.49 103
qc A0 C S
qd = qc + A0s + Cs2b 1.3976 ± 0.0001 0.093 ± 0.001 0.142 ± 0.004 0.52 ± 0.26 0.01 ± 0.001 0.5 ± 0.1 2.5 105 1.52 103
exp
(T, /)
(T, w) 0.500 ± 0.001 0.481 ± 0.002 33.5 ± 2.8 23 ± 2 4.8 103 0.485 ± 0.002 7.7 ± 0.8 3 ± 0.2 4.5 103
1473
ature [8,10,14]. The experimental value of /c,exp was determined by the technique of ‘‘equal volumes of two coexistence phases’’. The values of /c obtained from equation (9) departed further from the /c,exp than that obtained from fitting equation (8), and the standard deviation S in fitting equation (9) for variable / is larger than that in fitting equation (8), which indicate that the (1 a) anomaly is more significant than that of 2 b. The result is consistent with that of the system with the molar ratio (40.8) of water to AOT. Combination of equations (6) and (8) yields q1 ¼ qc þ A0 s þ A1 s1a ð1=2ÞBsb ð1=2ÞB1 sbþD þ ð10Þ q2 ¼ qc þ A0 s þ A1 s
1a
b
þ ð1=2ÞBs þ ð1=2ÞB1 s
bþD
þ ð11Þ
When a, b and D were fixed at 0.11, 0.3265 and 0.5, and the values of B, B1, A0, A1 and qc were taken from tables 2 and 3, the values of q1, q2 and qd were calculated from equations (10) and (11), the results are shown as lines in figure 1. The values from the calculations are in good agreement with experimental results. Acknowledgements This work was supported by the National Nature Science Foundation of China (Nos. 20273032, 20473035, 20573056, and 20673059), the New Technique Foundation of Jiangsu Province, P.R. China (No. BG2005041) and the Special Sustentation Fund of Nanometer Technology of Shanghai City, P.R. China (No. 0652nm010). References [1] M. Kotlarchyk, S.H. Chen, J.S. Huang, M.W. Kim, Phys. Rev. A 29 (1984) 2054–2069. [2] D. Roux, A.M. Bellocq, Phys. Rev. Lett. 52 (1984) 1895–1898. [3] A. Martin, I. Lopez, F. Monroy, A.G. Casielles, F. Ortega, J. Chem. Phys. 101 (1994) 6874–6879. [4] Y. Jayalakshimi, D. Beysens, Phys. Rev. A 45 (1992) 8709–8718. [5] R. Aschauer, D. Beycens, Phys. Rev. E 47 (1993) 1850–1855. [6] R. Aschauer, D. Beycens, J. Chem. Phys. 98 (1993) 8194–8198. [7] A.M. Bellocq, D. Gazeau, J. Phys. Chem. 94 (1990) 8933–8938. [8] X.Q. An, J. Feng, W.G. Shen, J. Phys. Chem. 100 (1996) 16674–16677. [9] K.P. Johnston, G.J. McFann, R.M. Lemert, in: K.P. Johnston, J.M.L. Penninger (Eds.), Supercritical Fluid Science and Technology, ACS Symposium Series, vol. 406, American Chemical Society, Washington, DC, 1989. [10] X.Q. An, J.Y. Chen, Y.G. Huang, W.G. Shen, J. Coll. Interf. Sci. 203 (1998) 140–145. [11] P. Li, X.Q. An, W.G. Shen, Acta Phys-Chim. Sin. 17 (2001) 144–149. [12] J.S. Huang, S.A. Safran, M.W. Kim, G.S. Grest, M.J. Kotlurchyk, N. Quirke, Phys. Rev. Lett. 53 (1984) 592–595. [13] M. Corti, V. Degiorgio, M. Zulauf, Phys. Rev. Lett. 48 (1982) 1617– 1620. [14] X.Q. An, J.Y. Chen, Y.G. Huang, W.G. Shen, J. Chem. Thermdyn. 34 (2002) 1107–1116.
JCT 06-337