LiquidâLiquid Phase Equilibria and Interactions between Droplets in

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Liquid-liquid Phase Equilibria and Interactions between Droplets in Water-in-Oil Microemulsions Tianxiang Yin, Mingjie Wang, Xiaoyi Tao, and Weiguo Shen Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b03496 • Publication Date (Web): 27 Nov 2016 Downloaded from http://pubs.acs.org on November 28, 2016

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Liquid-liquid Phase Equilibria and Interactions between Droplets in Water-in-Oil Microemulsions Tianxiang Yin,† †

Mingjie Wang,†

Xiaoyi Tao,† Weiguo Shen†,‡,*

School of Chemistry and Molecular Engineering, East China University of Science and Technology, Shanghai 200237, China ‡

Department of Chemistry, Lanzhou University, Lanzhou, Gansu 730000, China

1

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Abstract The liquid-liquid phase equilibria of {water/sodium bis(2-ethylhexyl) sulfosuccinate (AOT)/n-decane} with the molar ratio w0 of water to AOT being 37.9 and {water/AOT/ethoxylated-2,5,8,11-tetramethyl-6-dodecyne-5,8-diol(Dynol-604)/n-decane} with w0= 37.9 and the mole fraction α of Dynol-604 in the total surfactants being 0.158 were measured in this work. From the data collected in the critical region, the critical exponent β corresponding to the width of the coexistence curve was determined, which showed good agreement with the 3D-Ising value. A thermodynamic approach based on the Carnahan-Starling-Van der Waals type equation was proposed to describe the coexistence curves and to deduce the interaction properties between droplets in the microemulsions. The interaction enthalpies were found to be positive for the studied systems, which evidenced that the entropy effect dominated the phase separations as the temperature increased. Addition of Dynol-604 into the {water/AOT/n-decane} microemulsion resulted in the decrease of the critical temperature and the interaction enthalpy. Combining the liquid-liquid equilibrium data for {water/AOT/n-decane} microemulsions with various w0 determined previously, it was shown that the interaction enthalpy decreased with w0 and tended to change its sign at low w0, which coincided with the results from the isothermal titration calorimetry investigation. All these behaviors were interpreted by the effects of entropy and enthalpy and their competition, which resulted from the release of solvent molecules entrapped in the interface of microemulsion droplets and were dependent on the rigidity of the surfactant layers and 2


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the size of the droplet. Keywords: coexistence curve; criticality of microemulsion; interaction properties of droplets; Carnahan-Starling-Van der Waals equation.

1. INTRODUCTION Thermodynamically stable nanometer-sized water-in-oil microemulsions are homogeneous and transparent which are formed spontaneously when water, non-polar component and surfactant (with or without co-surfactant) are mixed.1 In a water-in-oil microemulsion, the water cores are stabilized by surfactant interfaces to construct droplets which uniformly disperse in a non-polar solvent. Microemulsions have shown a wide range of applications in chemical reactions, drug delivery, material synthesis, food and cosmetic industry, etc.2-5 These applications are closely associated with complex phase behaviors in microemulsions, among which the equilibrium of two liquid phases with different concentrations of the droplet has attracted much attention. {Water/sodium bis(2-ethylhexyl) sulfosuccinate (AOT)/n-alkane} microemulsions are most extensively investigated ones among those widely applied microemulsions.6-10 The liquid-liquid phase equilibria of these microemulsions with different droplet concentrations in the two liquid phases commonly show lower critical solution temperatures (LCST), and their phase behaviors vary with the molar ratio of water to surfactant (w0), the chain type and the chain length (n) of the alkane.8,11-14 This was mainly attributed to the variations of droplet-droplet interactions dependent on many

3


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factors, the nature of which has been a matter of long-standing question. 9,15-20 It has been found that the droplet-droplet interaction in {water/AOT/n-alkane} microemulsions was attractive, and its strength increased with w0 and n of n-alkane.7,19,21-23 This interaction was interpreted by an attractive square-well energy potential between two droplets, which seemed to be supported by the fact that the {water/AOT/n-alkane} microemulsions had LCST and it diminished with the increase of w0 and n, because the depth of the energy potential well should increase with temperature, w0 and n.8,11-14 However, some studies showed the existence of repulsive enthalpy (or energy) interaction between the droplets,1,16,24-27 which was unfavorable to droplet aggregation and the phase separation. Therefore, drawing droplets to close together and inducing the phase separation in AOT-based microemulsions are driven by the positive entropy change due to the release of the solvent molecules confined in the surfactant tails of the droplets, thus the “energy potential” should have the Gibbs free energy character.16,28 Our recent isothermal titration microcalorimetry (ITC) studies of the {water/AOT/oil} microemulsions confirmed the importance of the entropy effect on the stability of droplets in these microemulsions, and found that droplet interaction enthalpy varied with the nature of the oil phase and the molar ratio of water to surfactant, in some cases from the positive to the negative.26,27 However, as a key feature of microemulsion systems, interactions between droplets remain a confusing and challenging issue.29 Fisher has pointed out that the critical exponents should be renormalized to characterize a binary solution when a third component is added as an impurity.30 A 4


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microemulsion may be considered as a binary solution of droplets dissolved in a solvent, or a ternary solution having renormalized critical exponents. Experimental determinations on the critical exponents corresponding to the width of coexistence curve, the correlation length and the osmotic compressibility for microemulsions have been performed by different groups,11-14,31-36 where some controversies about whether the criticality of microemulsion belongs to the 3D-Ising universality still existed. In this work, with the assumption of the droplet being stable pseudo-component and the coexisting phases possessing different droplet concentrations, we made measurements on the coexistence curves for ternary {water/AOT/n-decane} and quaternary {water/AOT/Dynol-604/n-decane} microemulsion solutions with w0=37.9, where Dynol-604 refers to ethoxylated-2,5,8,11-tetramethyl -6-dodecyne-5,8-diol and its mole fraction α in the total surfactants (i.e. the mixture of AOT and Dynol-604) is fixed at 0.158. The obtained coexistence curve data in the critical region are used to determine the critical exponent and to probe whether they are consistent with 3D-Ising values or not. Moreover, a thermodynamic approach based on the Carnahan-Starling-Van der Waals type equation is proposed to describe the coexistence curves and to deduce the interaction properties of the microemulsion droplets, which confirm the reported isothermal titration calorimetry (ITC) results and are discussed in terms of the size of droplet and the rigidity of the interface. 2. EXPERIMENTAL SECTION 2.1 Material 5


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The sodium bis(2-ethylhexyl) sulfosuccinate (AOT, >97% mass fraction) was supplied by Sigma-Aldrich Chemical Co., which was dried for two weeks under vacuum and

stored

in

a

vacuum

desiccator

over

P2O5.

The

ethoxylated-2,5,8,11-tetramethyl-6-dodecyne-5,8-diol (Dynol-604, 100% mass fraction, claimed by the supplier) was purchased from Air Products and used without further purification. The n-decane (>99% mass fraction) was from Alfa Aesar, which was dried o

and stored over 4 A molecular sieves. Water used in this work was from an ultrapure water system (DZG-303A, Leading Water Treatment Co. Ltd., Shanghai) with conductivity being less than 5.5.10-6 S.m-1. 2.2 Measurements of coexistence curves The experimental instrument was a home-built one and the detailed information of the setup and measurement procedure can be found in our previous publications.11,12 Briefly, the critical volume fraction of the droplet (surfactant + water) was approached by fixing the molar ratio w0=37.9 and adjusting the amount of n-decane to achieve the criterion of “equal volume” of the two phases by visual observation near the phase separation point. By this method, the critical volume fraction can be determined with a precision of 0.002. A sample with the critical composition then was prepared in a rectangular fluorescence cuvette and the cuvette was sealed and placed in a water bath, where the temperature was controlled by a precise temperature controller (PTC-41, Tronac Inc.) and measured by a platinum resistance thermometer connected to a Keithley 2700 digital multimeter. The standard uncertainty and the reproducibility in measurement of temperature were 0.02 K 6


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and 0.002 K, respectively. The phase separation temperature was searched by changing the temperature of the water bath. After each change of the temperature, the sample was shaken rigorously and then kept undisturbed in the water bath at least one hour to observe whether the phase separation occurred or not. The phase separation temperature was taken as the critical solution temperature with a precision of 0.002 K. After the critical solution temperature was measured, the water bath temperature was increased step by step. At each temperature we took sufficient time to ensure the phase equilibrium of the system before the refractive indices of two coexisting phases were measured by the “minimum deviations angle” technique.11,12 The standard uncertainty in refractive index measurement was estimated to be 0.0001. All the experiments were carried out at the pressure in the sealed sample cell, where the vapor pressure depended on the temperature. However, the effect of such change in pressure on liquid-liquid equilibrium was negligible. 3. RESULTS AND DISCUSSIONS 3.1 Critical behaviors of microemulsions In this work, we defined the droplet constructed by the water and the surfactant as the pseudo second component denoted by subscript 2, which mixed with a non-polar solvent to form a pseudo binary microemulsion solution. The critical volume fractions φ2,c of the droplet and the critical temperatures Tc were determined to be φ2,c=0.096, Tc=311.256K for {(1-φ2) n-decane + φ2 (AOT + water)}with w0= 37.9; and φ2,c=0.104, Tc=306.680K for {(1-φ2) n-decane + φ2 (AOT/Dynol-604 + water)} with w0=37.9 and 7


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α=0.158; respectively. It should be noted that although the standard uncertainty in measurement of Tc for the critical samples prepared by the same batch of chemicals from the same source was about 0.05 K, we determined T and Tc with a reproducibility of ±0.002 K for a certain prepared and sealed sample with a critical composition. Therefore, the standard uncertainty of Tc − T

was ±0.003 K in determination of a whole

coexistence curve, which is critical important for the study of the critical phenomena. The critical temperatures Tc and the critical volume fractions φ2,c for {(1-φ2) n-decane + φ2 (AOT + water)} with various w0 were studied previously,8,13,14,37 together with those obtained in this work, are summarized in Figure 1. It may be seen from Figure 1 that the critical temperatures decreases as w0 increases, while the critical volume fraction keeps approximately unchanged.

Figure 1. Dependences of (a) critical temperature Tc (circles) and (b) critical volume fraction φ2,c of the droplet (triangles) on molar ratio of water to surfactant w0 for {(1-φ2) n-decane + φ2 (AOT + water)} and {(1-φ2) n-decane + φ2 (AOT/Dynol-604 + water)} with Dynol-604 mole fraction α in the total surfactants being 0.158 (filled

8


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symbols). The pentagrams symbols refer to the corresponding theoretical values calculated by eq 20 and eq 25 for Tc and φ2,c, respectively. The measured refractive indices n for each coexisting phase of the studied microemulsion systems at various temperatures are listed in columns 2, 3, 7, and 8 of Table S1 (supporting information). Due to the fact that only negligibly small critical anomaly of the refractive index is presented near the critical point;38 in a certain temperature range, the refractive index n of a microemulsion solution can be expressed as an analytic function of volume fraction φ2 and T, which may be expressed as: n(T , φ2 ) = n(T 0 , φ2 ) + R (φ2 )(T − T 0 ) R(φ2 ) = (1 − φ2 ) R1 + φ2 R2

(1) (2)

where T0 is the temperature close to the upper boundary of the measured coexistence curve; R(φ2) is the derivative of n with respect to T for a particular volume fraction φ2, R1 and R2 are the values of R(φ2) for φ2=0 and φ2=1, respectively. Refractive indices n of pure n-decane at various temperatures were measured and are listed in Table S2 (supporting information), from which the value of R1=−0.000465 K-1 was obtained by a linear fit with eq 1. The values of refractive indices n(T , φ2 ) at different volume fractions and temperatures were also measured and are listed in Table S2 (supporting information), which were fitted with eq 1 to obtain the values of n(T 0 , φ2 ) and R(φ2) at various volume fractions with the standard deviations of fits being less than 0.0002. Then the value of R2 was determined by fitting these R(φ2) values to eq 2, which gave R2=−0.000154 K-1 for {(1-φ2) n-decane + φ2 (AOT + water)} and R2=−0.000120 K-1 for 9


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{(1-φ2) n-decane + φ2 (AOT/Dynol-604 + water)}, respectively. The values of n(T0, φ2) were further fitted to the polynomials to obtain n (T 0 = 317.121K , ϕ2 ) = 1.3995 − 0.0213ϕ2 + 0.0014ϕ22

(3)

for {(1-φ2) n-decane + φ2 (AOT + water)} and n (T 0 = 314.688 K , ϕ2 ) = 1.4004 − 0.0239ϕ2 + 0.0062ϕ 22

(4)

for {(1-φ2) n-decane + φ2 (AOT/Dynol-604 + water)} with standard deviations of the fits being about 0.0002. By means of the Newton’s iteration method, the measured refractive indices were converted to the volume fractions by simultaneously solving eqs 1, 2, and 3 (or 4). The coexistence curves are shown in Figure 2 as the plots of temperature against refractive index (T, n) and temperature against volume fraction of the droplet (T, φ2).

Figure 2. Coexistence curves of temperature against refractive index (T, n) and temperature against volume fraction (T, φ2) for (a) {(1-φ2) n-decane + φ2 (AOT + water)} with w0= 37.9 and (b) {(1-φ2) n-decane + φ2 (AOT/Dynol-604 + water)} with w0= 37.9 and α=0.158. The width of a coexistence curve Z cxc can be expressed in the following way:39

10


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Z cxc ≡

where τ =

T − Tc Tc

Z+ −Z− ≈ Bˆ 0Zτ β (1 + Bˆ1Zτ ∆ ) 2

(5)

is the reduced temperature; β and ∆ are universal critical exponents;40

Z is the physical density, such as the refractive index n, volume fraction ϕ2 etc,. The second term in eq 5 represents the correction-to-scaling contribution which possibly becomes significant in a wider temperature range from the critical point and indicates a crossover from the critical region to the non-critical region; hence it is also required to be examined.

The widths of the coexistence curve for n and φ2 in the temperature range T − Tc < 1K were fitted to eq 5 with neglecting the correction-to-scaling term Bˆ1Zτ ∆ to

get the values of Bˆ 0Z and β, which are listed in Table 1. The fitting results are shown in Figure 3, indicating good agreements with the experimental data. As shown in Table 1, in the temperature range T − Tc < 1K , the values of β for {(1-φ2) n-decane + φ2 (AOT + water)} and {(1-φ2) n-decane + φ2 (AOT/Dynol-604 + water)} both coincide well with the theoretical one (0.326) of the 3D-Ising universality40 which was derived for one-component systems and extended to binary mixtures by the principle of isomorphism,39 and significantly deviate from the Fisher renormalization value (0.367). It may be attributed to the fact that the surfactant as the third component in the microemulsion does not act as the impurity but joins the formation of the droplet with the water. It has been commonly accepted that the nature of the droplets remains essentially

11


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constant for these microemulsion systems under our experimental conditions of measurement of the coexistence curve;6,33 hence they can be treated as pseudo-binary solutions and the 3D-Ising universality instead of the fisher renormalization is valid. It also has been confirmed by different independent experiments for similar microemulsion systems.11-14,33,35 Moreover, in a wider temperature range of T − Tc < 6K , we fitted the experimental data of Zcxc to eq 5 with the values of β and ∆ being fixed at theoretical values 0.326 and 0.50 respectively to obtain the values of Bˆ 0Z and Bˆ1Z , which are listed in columns 4 and 5 of Table 1. From Table 1, it may be seen that the differences between the values of Bˆ 0Z from the two fits described above and the correction-to-scaling term are not significant, which implies the critical singularity being applicable for the coexistence curves in the temperature range of about 6 K from the critical point. Therefore, it raises a question whether any classical solution models based on the mean field theory can reasonably well describe the microemulsion coexistence curves and deduce out interaction information in microeulsions, as it has been done commonly for the ordinary binary solutions; although strictly speaking those mean field treatments and the corresponding deduced interaction properties are only approximate because of the neglect of the critical anomaly in the near critical region.

.

Table 1. Values of Critical Amplitudes Bˆ0Z and Bˆ1Z , Critical Exponent β in eq 5 for Coexistence Curves of (T, n) and (T, φ2) for {(1-φ2) n-decane + φ2 (AOT + water)} with w0= 37.9 and {(1-φ2) n-decane + φ2 (AOT/Dynol-604 + water)} with w0= 37.9 and α=0.158. 12


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T − Tc < 1K

Order parameter

Bˆ0Z

T − Tc < 6K

β

Bˆ0Z

Bˆ1Z

{(1-φ2) n-decane + φ2 (AOT + water)}

n

0.0121 ±0.0005

0.327 ± 0.006

0.0123 ± 0.0001

-0.50 ± 0.08

φ2

0.57± 0.02

0.329± 0.006

0.571 ± 0.004

-0.23± 0.08

{(1-φ2) n-decane + φ2 (AOT/Dynol-604 + water)}

n

0.0124 ± 0.0004

0.326 ± 0.005

0.0128 ± 0.0001

-0.74± 0.09

φ2

0.51 ± 0.02

0.327 ± 0.005

0.517 ± 0.005

-0.27 ± 0.10

Figure 3. Log-log plots of φ2,cxc against τ for (a) {(1-φ2) n-decane + φ2 (AOT + water)} with w0= 37.9 and (b) {(1-φ2) n-decane + φ2 (AOT/Dynol-604 + water)} with w0= 37.9 and α=0.158 in the temperature range T − Tc < 1K . The solid lines are fitting results by eq 5. 3.2 A classical thermodynamic approach to describe the microemulsion coexistence curves In a pseudo binary microemulsion solution, the chemical potentials of the solvent 13


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(n-decane in this work) and the solute (microemulsion droplet) µ1 and µ2 may be expressed by

µ1 = µ10 ( T , P ) + RT ln( x1γ 1 )

(6)

µ2 = µ2* ( T , P ) + RT ln( x2γ 2 )

(7)

where subscripts 1 and 2 refer to the solvent and the solute respectively; γ is the activity coefficient; x is mole fraction. µ1 and µ2 depend not only on temperature T and pressure P but also on the concentration of the solution. Since γ 1 → 1 as x1 → 1 , and

γ 2 → 1 as x2 → 0 ; hence µ10 (T , P ) is the chemical potential of the pure solvent and µ2* (T , P ) is the chemical potential of the solute in a hypothetical state (i.e. x2=1, while the concentration of the droplet in the solution is infinite dilution). The potential differences ∆µ1 and ∆µ2 are defined as ∆µ1 = µ1 − µ10 (T , P )

(8)

∆µ2 = µ2 − µ2* (T , P )

(9)

∆µ1 may be expressed as ∆µ1 = −πν 1 = −πν 2

ν1 ν2

(10)

where v1 and v2 are the molar volumes of the solvent and the solute respectively; π is the osmotic pressure. The water-in-oil microemulsion droplets are commonly modeled as spheres with the brush-like surfactant interface surrounded by the hydrophobic solvent molecules, which interact with each others. According to this structure model for the microemulsion

14


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droplets, the interaction between droplets may be simply considered as the sum of two contributions: one is the contribution of the hard sphere excluded volume; the other incorporates all non-hard sphere interactions. The first contribution has been well described by the Carnahan-Starling approach,41 and the second one was represented by a simple term of λϕ22 by Vrij et. al.42 Combination of above two contributions yields a so called Carnahan-Starling-Van der Waals equation of state, which , has been successfully used to describe the dependence of the osmotic pressure on the volume fraction φ2 of the droplet in microemulsion systems:23,42,43 πν 2 RT

= ϕ2

1 + ϕ2 + ϕ22 − ϕ23

(1 − ϕ2 )

3

− λϕ22

(11)

where λ is the interaction parameter. Combination of eq 10 and eq 11 gives: ∆µ1 = − RT

 ν 1  1 + ϕ 2 + ϕ 22 − ϕ 23 2 − λϕ  ϕ2 2  3  ν 2  (1 − ϕ2 ) 

(12)

According to the Gibbs-Duhem equation at constant temperature and pressure, we have d ( ∆µ 2 ) = −

x1 d ( ∆µ1 ) x2

(13)

The ratio of x1 x2 can be expressed by: x1 νϕ1 ν 1 ν 2 1 − ϕ 2 = = x2 νϕ 2 ν 2 ν 1 ϕ2

(14)

where v is the molar volume of the solution. Combination of eqs 12, 13, and 14 gives: d ( ∆µ 2 ) = RT

1 − ϕ2

ϕ2

 1+ϕ + ϕ 2 −ϕ3  2 2 2 d  ϕ2 − λϕ 22  3   (1 − ϕ2 )  

(15)

Integrating eq 15 and assuming λ being independent on φ2, we have

15


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 7 + ϕ 2 − 2ϕ 3  2 2 2 ϕ λ ϕ ∆µ2 = RT  + + − + I ln 1  ( ) 2 2 2  2 (1 − ϕ2 ) 

(16)

where I is the integration constant. Combing eqs 7, 9 and 16 and using the condition of ideal dilute solution: γ 2 → 1 as ϕ2 → 0 , hence ∆µ2 = ln x2 , we have 7  ∆µ2 = ln x2 = RT  + ln ϕ2 + λ + I  = 0 2 

(17)

which results in 7  ϕ I = −  + ln 2 + λ  . x2 2 

(18)

Taking the approximation that the molar volume of the solution may be replaced by that of the solvent in a sufficiently diluted solution, we have

into eq 16 with

ϕ2 x2

being replaced by

ϕ2 x2

=

ν2 . Substitution of eq 18 ν1

ν2 yields: ν1

 7ϕ − 3ϕ 2 − ϕ 3  ν1 2 2 ϕ λϕ ϕ ∆µ2 = RT  2 + + + − ln ln 2 ( )  2 2 2 2 ν2  (1 − ϕ2 ) 

(19)

In eq 19, the last term represents the interaction contribution to ∆µ2 , which relates to the interactions in the microemulsions. The equilibrium of two microemulsion phases denoted by “+” and “–” respectively at a certain temperature and a constant pressure requires ∆µ1+ = ∆µ1− and ∆µ2+ = ∆µ2− . With these conditions and a simple temperature dependence of λ

λ = a+b T

(20)

the data of ϕ2+ and ϕ 2− at various temperatures listed in Table S1 (supporting information) for the {(1-φ2) n-decane + φ2 (AOT + water)} with w0=37.9 and {(1-φ2) 16


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n-decane + φ2 (AOT/Dynol-604 + water)} with w0=37.9 and α=0.158 were fitted by using eqs 12 and 19 simultaneously to obtain the optimal values of parameters a and b in eq 20. We also fitted the experimental data of coexistence curves for {(1-φ2) n-decane + φ2 (AOT + water)} with various w0 we reported previously8,13,14,37 to calculate the corresponding values a and b. The fitting results of the coexistence curves are compared with the experimental data in Figure 4, showing good agreements with each others, which evidences that this classical thermodynamic model fairly well describes the microemulsion coexistence curves even though the non-classical 3D-Ising criticality is applicable for them. All the values of a, b, and the interaction parameter at critical point

λc = a + b Tc together with the experimental values of Tc are listed in Table 2 and the dependences of a, b, and λc on w0 are shown in Figure 5. It can be seen from Figure 5 that the values of a and b vary approximately linearly with w0, while λc keeps almost unchanged.

Figure 4. Comparisons of the fitting results (solid lines) calculated by eqs 12 and 19 with the experimental data for {(1-φ2) n-decane + φ2 (AOT + water)} with various w0 17


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(circles) and {(1-φ2) n-decane + φ2 (AOT/Dynol-604 + water)} with w0= 37.9 and

α=0.158 (triangles). Table 2. Critical Temperatures Tc, Fitted Values of Parameters a and b and Calculated Values of λc , ∆Hi and ∆Si for {(1-φ2) n-decane + φ2 (AOT + water)} with Various w0 and {(1-φ2) n-decane + φ2 (AOT/Dynol-604 + water)} with w0= 37.9 and α=0.158.

b w0

Tc/K

a /1000 K

λc

∆H i

∆S i

/ kJ.mol-1

/ kJ.mol-1.K-1

{(1-φ2) n-decane + φ2 (AOT + water)} 25.1

322.315

64 ± 3

-16.9 ± 1.0

11.6±0.1

281 ± 16

1.06 ± 0.05

30

318.262

134 ± 8

-38.8 ± 2.4

12.1±0.4

646 ± 41

2.23 ± 0.13

35

312.892

196 ± 20

-57.5 ± 6.3

12.2±0.2

956 ± 104

3.3 ± 0.3

37.9

311.256

272 ± 25

-80.8 ± 7.7

12.4±0.3

1343 ± 130

4.5 ± 0.4

45.2

305.379

368 ± 27

-108.8 ± 8.3

11.7±0.2

1809 ± 140

6.1 ± 0.4

{(1-φ2) n-decane + φ2 (AOT/Dynol-604 + water)} 37.9

306.680

150 ± 7

-42.4 ± 2.2

11.7±0.2

705 ± 36

2.5 ± 0.1

18


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Figure 5. Plots of a, b, λc and ∆Hi as functions of w0 for {(1-φ2) n-decane + φ2 (AOT + water)} with various w0 (unfilled symbols) and {(1-φ2) n-decane + φ2 (AOT/Dynol-604 + water)} with w0= 37.9 and α=0.158 (filled symbols). Lines are used for guidance. At the critical solution point, the first and the second derivatives of ∆µi (i=1 or 2) with respect to φ2 approach to zero  ∂∆µi  =0    ∂ϕ 2 Tc ,ϕ2 ,c

(21)

 ∂ 2 ∆µi  =0  2   ∂ϕ2 T ,ϕ

(22)

c

2,c

For i=1, substitution of eq 12 into eqs 21 and 22 gives

λc =

2 3 4 1 + 4ϕ2,c + 4ϕ2,c − 4ϕ2,c + ϕ2,c

λc =

2ϕ2,c (1 − ϕ2,c )

4

(23)

2 4 + 10ϕ 2,c − 2ϕ 2,c

(1 − ϕ2,c )

5

(24)

where the subscript “c” represents the critical state. Combination of eqs 23 and 24 yields 2 3 4 5 1 − 5ϕ 2,c − 20ϕ 2,c − 4ϕ 2,c + 5ϕ 2,c − ϕ 2,c =0

(25)

19


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Numerically solving eq 25, we obtained ϕ2,c =0.13±0.01, which was further substituted into eq 23 or 24 to get λc =10.7±0.8. It is somewhat smaller than the experimental values of λc listed in Table 2. When eq 19 was substituted into eqs 21 and 22 for i=2, the same results were achieved. Using the value of λc =10.7±0.8 and the values of a and b listed in Table 2, we calculated the theoretical values of Tc and their uncertainties by eq 20. The theoretical values of ϕ2,c and Tc are compared with experimental ones in Figure 1, where larger values ϕ2,c and somewhat systematic smaller values of Tc predicted by the theory as compared with those from the experiments are illustrated. These discrepancies including that of λc indicate that although the classical theory can describe the coexistence curves fairly good, they failed to predict the critical properties precisely. It is not surprising since the mean-field theory such as the Carnahan-Starling-van der waals equation does not consider the critical anomaly resulted from the critical fluctuation in the critical region.

3.3 Thermodynamic properties of interactions between microemulsion droplets According to the Gibbs-Helmhotz equation, we have

H 2 − H 2* 1 ∂ ( ∆µ2 T ) =− 2 RT R ∂T

(26)

Substitution of eqs 19 and 20 to eq 26 gives H2 − H2* = Rb(ϕ22 − 2ϕ2 )

(27)

Neglecting the high order term leads to H 2 − H 2* ≈ −2 Rbϕ2

(28)

20


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where H 2 − H 2* refers to the enthalpy change per mole droplet for concentrating its infinite solution to a given volume fraction φ2. Define ∆H i = −2 Rb as the interaction enthalpy between microemulsion droplets which is the enthalpy change for concentrating 1 mole droplet in its infinite solution to the hypothetical pure droplet state (i.e. φ2=1, while the concentration of the droplet in the solution is so dilute that only two-body interaction exists). The values of ∆Hi are listed in column 6 of Table 2 and the variation of ∆Hi with w0 is shown in Figure 5. All the interaction enthalpies ∆Hi for {water/AOT/n-decane} microemulsions with various w0 listed in Table 2 are positive, suggesting repulsive enthalpy interactions between microemulsion droplets. It may be clearly seen from Figure 5 that the repulsive interaction enthalpy ∆Hi almost linearly increases with w0. According to this linear relation indicating in Figure 5, it is reasonably speculated that the repulsive interaction (i.e. ∆H i > 0 ) will change into the attractive interaction (i.e. ∆H i < 0 ) at low w0. These results coincide qualitatively well with those we reported previously obtained by means of isothermal titration calorimetry,27 which can be interpreted as follows. The overlapping of the surfactant tails between two neighboring droplets as they close to each other is accompanied by release of the solvent molecules entrapped in the surfactant interface. The former contributes negatively to ∆Hi ; while the latter contributes positively to ∆Hi . The competition between these two opposite contributions decides the sign of ∆Hi . Since more solvent molecules may be entrapped in the surfactant layer of the droplet with

21


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large size (i.e. with large value of w0), the positive contribution to ∆Hi is dominate in the region of large w0; however as w0 decreases, the interface becomes more rigid and less solvent molecules are entrapped in the interface, thus the negative contribution plays more important role and finally changes ∆Hi from the repulsive to the attractive. The interaction parameter λ includes enthalpy contribution b and the entropy contribution a. The later results from the increase of the freedom of the solvent molecules after they are released from the tails of the surfactants in the droplet interface when two droplets close to each other. By defining the interaction entropy between microemulsion droplets ∆Si = 2Ra , which is the entropy change for concentrating 1 mole droplet in its infinite solution to the hypothetical pure droplet state, the interaction Gibbs free energy

∆Gi may be expressed by ∆Gi = ∆Hi − T ∆Si = −2RT λ

(29)

The values of ∆Si are listed in Table 2. It is indicated by Table 2 that both the entropy and enthalpy contributions increase with w0 and compensate each others, which leads to a constant value of the interaction parameter λc at the critical solution point, but small increase in ∆Gc and decrease in Tc, hence promotion of the phase separation as increase of w0. For each of microemulsions with a fixed w0, ∆H i = −2 Rb is unchanged; thus as temperature increases, ∆Gi decreases according to eq. 29; therefore, the entropy effect promotes the occurrence of the condensed phases and dominates the temperature induced phase separation, resulting in the lower critical solution point in the microemulsions. The experimental results listed in Table 2 show that after Dynol-604 is added into 22


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{water/AOT/n-decane} microemulsion with w0= 37.9, the values of Tc, ∆Hi , ∆Si , a and the absolute values of b all significantly decrease. It may be attributed to the fact that addition of Dynol-604 into this microemulsion strengthens the interfacial rigidity of the microemulsion droplet because the added Dynol-604 shields the electrostatic repulsion between AOT head groups. This shield effect makes it difficult for solvent molecules to enter into the surfactant layer and hence reduces both the interaction enthalpy ∆Hi and the interaction entropy ∆Si from the release of the entrapped solvent molecules. Equation 29 may be rewritten at the critical solution point as Tc =

∆Hi ∆Si − 2 Rλc

(30)

with λc being unchanged after addition of Dynol-604. Eq 30 indicates that decrease of

∆Hi reduces Tc, while decrease of ∆Si increases Tc. The opposite contributions of enthalpy and entropy effects on the critical temperature compensate and compete with each other to result in the reduction of the critical solution temperature and hence the increase of ∆Gc , which indicates that the enthalpy effect is dominant on the reduction of Tc and promotion of the liquid-liquid phase separation of the microemulsion after Dynol-604 was added into the {water/AOT/n-decane} microemulsion.

4. CONCLUSION To sum up, we measured the liquid-liquid coexistence curves of {water/AOT/ n-decane}

with

the

molar

ratio

of

water

to

AOT

w0

being

37.9

and

{water/AOT/Dynol-604/n-decane} with w0 being 37.9 and the mole fraction α of

23


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Dynol-604 in the total surfactants being 0.158. By analyzing the experimental data in the critical region, the values of the critical exponent β corresponding to the width of the coexistence curve were determined and showed agreements with the 3D-Ising prediction. We proposed a classical thermodynamic approach based on the Carnahan-Starling-Van der Waals type equation and found that it well described the liquid-liquid phase equilibria of these systems, even though the non-classical 3D Ising criticality was applicable for them. The interactions between droplets in microemulsions were deduced from analysis of the liquid-liquid equilibrium data through the thermodynamic model and their dependences on w0 were discussed. Particularly, we found that the interaction enthalpies between droplets were all positive for the studied systems, which evidenced that the entropy effect dominated the phase separations as the temperature increased. The interaction enthalpy was found to decrease with w0, which coincided qualitatively well with the reported results from measurements of isothermal titration calorimetry. These behaviors were interpreted by the competition of the entropy and enthalpy effects, all resulting from the release of solvent molecules entrapped in the surfactant tails. The comparison of the liquid-liquid phase equilibrium behaviors of {water/AOT/ n-decane} microemulsions with and without addition of Dynol-604 showed the differences in various interactions between the corresponding microemulsion droplets, which was attributed to the fact that the non-ionic Dynol-604 can shield the electrostatic repulsion between AOT head groups, thus reduce the rigidity of the interfaces. This work provides a new approach to investigate the phase behaviors and the interactions between droplets 24


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in microemulsions from the liquid-liquid phase equilibrium data.

Supporting information

Refractive indices and coexistence curves for {(1-φ2) n-decane + φ2 (AOT + water)} with w0=37.9 and {(1-φ2) n-decane + φ2 (AOT/Dynol-604 + water)} with w0=37.9 and

α=0.158. Refractive indices n for pure n-decane.

Corresponding Authors

* E-mail: [email protected] (W.G. Shen).

Funding Sources This work was supported by the National Natural Science Foundation of China (Projects 21373085, 21303055 and 21403067) and the Fundamental Research Funds for the Central Universities (No. WJ1516001).

Notes The authors declare no competing financial interest.

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TOC Graphic

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