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Determination of the Second Virial Coefficient of the Interaction between Microemulsion Droplets by Microcalorimetry Wen-Yih Chen,* Chih-Sheng Kuo, and Der-Zen Liu Department of Chemical Engineering National Central University, Chung-Li, Taiwan 320, Republic of China Received August 6, 1999. In Final Form: November 10, 1999 The study proposes a novel method of obtaining the information of second virial coefficient of interactions between microemulsion droplets by microcalorimetry. By use of a high sensitivity isothermal titration microcalorimetry to measure the dilution heat of microemulsions solution, information about the second virial coefficient of the interactions between the microemulsion droplets can be derived with the number density of microemulsion solution. The derivation is based on a hard-sphere interaction potential assumption. The variance of the derived second virial coefficients was verified by the percolation behavior of different reverse micelles solutions of dioctyl sulfosuccinate sodium salt in decane with or without solutes.
Introduction Microemulsions are thermodynamically stable mixtures of water, oil, and surfactant(s) that exhibit a rich phase behavior. The interactions between microemulsion droplets account for the concerns of the dispersity in various applications, such as a drug carrier, cosmetic applications, and as a bioreactor. For discussions of the interaction between microemulsion droplets, similar to the McMillanMayer theory1 for molecular solution, the solvent was treated as a continuum because the droplets consist of many molecules and are much larger than the solvent molecules. Although the droplets are in dynamic equilibrium between the dispersed phase and surfactants, the interaction potential appears to exhibit a distinct hardcore part with a radius. Basically, the interaction forces between droplets involve the attractive van der Waals force between surfactant interfaces and, for ionic surfactants, the electrostatic repulsive force. For a liquid film, such as the surface of a microemulsion, the replusive hydration and the entropic forces in a short-range distance contribute to the pair potential as an energy barrier.2 However, direct measurement of the short-range forces between the droplets has not been achieved successfully. Therefore, a thermodynamic aspect of the energy potential or a description of the deviation from elastic collision, such as the second virial coefficient, is needed for describing different droplet interactions. Developments in the literature regarding interactions between microemulsion droplets have been reviewed by Koper et al.3 The second virial coefficient has been experimentally determined by measuring the osmotic pressure4 and diffusion coefficient as a function of volume fraction.5 The magnitude (and sign) of the second virial coefficient was correlated with the aggregation phenomenon of microemuslion. This Letter presents the novel idea of using * To whom correspondence should be addressed: Fax: +886 3 4225258. E-mail address:
[email protected]. (1) Hill, T. L. An Introduction to Statistical Thermodynamics; Dover Publishers, Inc.: New York, 1986. (2) Israelachvilli, J. Intramolecular and Surface Forces; Academic Press Ltd.: London, 1992. (3) Koper, G. L. M.; Sager, W. F. C.; Smeets, J.; Bedeaux, D. J. Phys. Chem. 1995, 99, 13291. (4) Stauffer, D. Introduction to Percolation Theory; Taylor and Francis, Inc.: London, 1985. (5) Cassin, G.; Illy, S.; Pileni, M. P. Chem. Phys. Lett. 1994, 221, 205.
isothermal titration microcalorimetry to measure the dilution heat of the microemulsion solution, and with the statistic thermodynamics derivation and a hard-sphere assumption, information about the second virial coefficient of the interaction between droplets can be obtained. Furthermore, results of the isothermal titration microcalorimetry (ITC) method were verified by the variance of second virial coefficients with percolation temperatures of Dioctyl sulfosuccinate sodium salt (AOT) reverse micelles with or without solutes. The formation of percolating clusters by the reverse micelles can be attributed by nonelastic collisions between the droplets and can be monitored by the electric conductivity change of the solution by changing the microemulsion solution temperature. Experimental Section Microemulsion System. The experiments were performed on AOT/Water/Alkane/Solutes microemulsion systems. The microemulsions were prepared by mixing an appropriate quantity of water or aqueous buffer solution containing solutes with an AOT-in-oil solution. Subsequent microemulsions were placed in Teflon-stoppered test tubes and then left to equilibrate for 1 day at 25 °C before the conductivity and microcalorimetric experiments. Isothermal Titration Microcalorimetry (ITC). In this study, we used highly sensitive microcalorimetry which employed a Thermal Activity Monitor (Thermometric AB, Sweden), as controlled by Digitam software. A 4 mL stainless steel ampule is the microreaction system used herein as a titration mode. The heat difference was measured by high sensitivity thermopiles in an isothermal system which is stabilized at (2 × 10-4 °C. In addition, electrical calibration was required before each titration is performed. Heat of dilution of the various compositions of microemulsion systems measured by ITC was performed as follows. A 2.5 mL portion of the microemulsion solution was placed in the ampule, stirring at 60 rpm. A Hamilton microliter syringe was used to titrate a 20 µL aqueous or buffer solution into the microemulsion solution once a thermoequilibrium between the ampule and the heat sink was reached. The number of titrations was designed for the desired microemulsion solution from reverse micelles suspension to percolation solution. The enthalpy changes in the percolation process of reverse micelles were also followed by the microcalorimetry by changing the water volume faction (Φw) with the tritarion of water water into the microemulsion suspension. The water volume fraction (Φw) is defined as the ratio of the water volume to the total volume of the solution.
10.1021/la991070q CCC: $19.00 © 2000 American Chemical Society Published on Web 12/29/1999
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Figure 1. Conductivity of the reverse micelle solution verses the solution temperature of reverse micelles with different solutes (AOT/water/n-decane).
Results and Discussion In this study, the conductivity measurement is designed to detect the conductivity percolation temperature of various reverse micelle systems with or without solute. Whereas the microcalorimetric study performed herein provides interaction potential between droplets, thereby allowing the microcalorimetric results to be explained in interaction potential aspect. Figure 1 depicts the conductivity of AOT/water/n-decane reverse micelle systems with or without the presence of different solutes. The percolation temperature is defined as the peak temperature of the rate of change in the log value of the conductivity verses the temperature plot. The effects of the various solutes on the percolating cluster formation of reverse micelles have been extensively discussed and reported.6-9 With the discussion above, the dilution heat of the solution verses the concentration of reverse micelles was measured by ITC and combined with the virial equation for nonideal behavior of microemulsion solution. The present study proposes the following derivations and discussions the derivation of the second virial coefficient from dilution heat. In general, the dilution heats of the reverse micelles solution with the water volume fraction can be fitted by a function of second-order polynomial as
d(q/NkBT) d(E/NkBT) = ) b2 + b3ΦW + b4ΦW2 dΦW dΦW
(1)
where q is the heat of dilution (mJ), E is the internal energy (mJ), N is the number of particles (number of reverse micelles droplets in the solution), kB is the Boltzmann constant, T is the absolute temperature (K), FW is the water volume fraction, and b2, b3, and b4 are the fitting coefficient of a second-order polynomial function. For an open liquid system, when there is no pressure change and the dilution volume is neglected compared (6) Bommarius, A. S; Holzwarth, J. F. D; Wang, C.; Hatton, T. A. J. Phys. Chem. 1992, 94, 7232. (7) Leodidis, E. B.; Hatton, T. A. Structure and Reactivity in Reverse Micelles; Pileni, M. P., Eds.; Elsevier: Amsterdam, 1989; p 270. (8) Rodgers, M. A.; Lee, P. C. J. Phys. Chem. 1984, 88, 3480. (9) Pileni, M. P.; Zemb, T.; Petit, C. Chem. Phys. Lett. 1985, 118, 414.
Figure 2. Dilution enthalpy of the reverse micelle solution with various water volume fractions. Each peak indicates a titration of 20 µL volume of water into the reverse micelle solution (AOT/water/n-decane with 30 mM CuCl2).
with the total system solution volume, the dilution heat observed then is equal to the internal energy change of the system. Furthermore, the internal energy changes of the system with the number density of the reverse micelles in the solution can be expressed by the virial equation as
E
)
NkBT
3 2
∞
-T
1 dBi+1
∑ i)1 i
Fi
dT
(2)
where F ) FW/Vd (Vd is the volume of the reverse micelle droplet) and Bi is the virial coefficient. Comparing eq 1 with eq 2 reveals that the coefficient of the polynomial fitting of the experimental dilution heat data can be affiliated with the virial coefficients of eq 2 by the following
Vdb2 ) -T(dB2/dT) Vd2b3 ) -T(dB3/dT)
(3)
From statistic thermodynamics, the second virial coefficient can be represented by the interaction potential energy function U(r) as
B2(T) ) -2π
[ (
) ]
U(r)
∫0∞ exp - kBT
- 1 r2 dr
(4)
If a square-well attractive potential energy function U(r) is selected and plugged into eq 4, eq 3 can be declared as follows
Vdb2 ) -T(dB2/dT) ) -B0(λ3 - 1)e/kBT(/kBT) < 0 (5) where
B0 ) (16/3)πRHS3 λ ) (2RHS + σ)/2RHS and σ and denote the width and depth of the square-well attractive potential function, respectively. Analyzing eq 5 reveals, qualitatively, that a negative b2 value indicates an attractive interaction and a more negative value of b2 (higher values of and σ) declares that a nonelastic collision is more likely to happen between
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percolation from a thermodynamic perspective but also suggest that a simple dilution heat can be used to inquire the interaction potential information between reverse micelle or, in general, between colloids. The results can also provide, possibly, more detailed discussions of the colloid interaction mechanism. Conclusion
Figure 3. A second-order polynomial fitting of the dilution enthalpy of the reverse micelle solution with water volume fractions (AOT/water/n-decane with 30 mM CuCl2). Table 1. b2 Value and the Conductivity Percolation Temperatures of Various Reverse Micelle Solution Systems reverse micelle solution
b2 value
Tp (°C)
AOT/decane/[CuCl2] ) 30 mM AOT/decane/[NaCl] ) 30 mM AOT/decane/water AOT/decane/[Trp] ) 20[RM]
-2.178 × 10-20 -5.202 × 10-20 -1.164 × 10-19 -1.304 × 10-19
30.3 24.3 14.1 13.2
the reverse micelle droplets. Considering the percolation behavior from the perspective of an interaction potential, the more attractive interaction potential (higher values of and σ) would result in a system of reverse micelles solution easier to form percolation (lower TP value). Therefore, if two-body interaction is considered only, a larger negative value of a second virial coefficient indicates a lower Tp value of the reverse micelle solution. The above attempt was demonstrated as follows: The dilution heat of the reverse micelle solution with various water volume fractions of (AOT/water/n-decane with 30 mM CuCl2) was plotted in Figure 2, as an example. The values of heat generated in Figure 2 with the water volume fraction are plotted in Figure 3 with a second-order polynomial fitting. The b2 values of the fitting equation were obtained for various systems and are listed in Table 1 with the TP values from the conductivity measurements. In summary, the b2 value from a thermodynamics aspect derivation of a dilution can be qualitatively well correlated with the Tp from a conductivity percolation measurement. The above results not only can be used to describe the conductivity
This study, we have developed a microcalorimetry method of obtaining the second virial coefficient of the interaction potential between microemulsion droplets, and the validity of the coefficient was examined by the temperature of a percolating cluster forming various reverse micelles systems with different solutes. This novel method should be able to serve as one of the methods for determining the stability of the microemuslion and also should be helpful in understanding the interaction mechanism of microemulsion droplet systems. Glossary W0 Tp Φw Vd q E N T RHS σ B b
ratio of water to surfactant molar concentrations percolation temperature (°C) aqueous volume fraction of the microemulsion system volume of reverse micelle droplet heat of dilution (mJ) internal energy (mJ) number of particles absolute temperature (K) radius of a hard sphere width of the square-well potential function depth of the square-well potential function virial coefficient coefficient of the fitting polynomial of the dilution heat of a reverse micelle solution
Acknowledgment. The authors thank the National Science Council of the Republic of China for financial support of this research under Grant No. NSC87-2214E008-011. Helpful discussions on the interaction potential measurement by ITC with Professor Heng-Kwong Tsao are also acknowledged. LA991070Q
