J . Phys. Chem. 1989, 93, 314-317
314
Conductlvfty of Water-In-011 Mlcroemulslons: A Quantitative Charge Fluctuatlon Model Hans-Friedrich Eicke,* Michal Borkovec, and Bikram Das-Gupta Institut fur physikalische Chemie der Universitat Basel, Kiingelbergstrasse 80, CH-4056 Basel, Switzerland (Received: November 12, 1987: In Final Form: April 4, 1988)
We present a model for the electrical conductivity of a water-in-oil microemulsion. The conductivity is explained by the migration of charged aqueous droplets in the electric field. Charged droplets are formed by spontaneous number fluctuations of the ions residing on the droplets. The magnitude of these fluctuations is directly related to the Coulomb energy that is required to charge up a droplet. The present model for the conductivity contains no adjustable parameters and is entirely consistent with experimental results on AOT-water-isooctane microemulsions.
1. Introduction Measurement of electrical conductivity is one of the key methods for the study of rnicroernulsi~ns.'-~ Microemulsions are thermodynamically stable mixtures of oil, water, and a surfactant. They are isotropic and homogeneous on a macroscopic scale but heterogeneous on a molecular scale as they are divided into water and oil domains which are separated by a surfactant monolayer.'J Their structure depends crucially on the water-to-oil ratio. Microemulsions containing a comparable amount of oil and water are usually bicontinuous-a spongelike random network. On the other hand, for high oil content we have a water-in-oil microemulsion which is a dispersion of nanometer-sized water droplets coated with a surfactant monolayer. An analogous structure is to be expected for an oil-in-water microemulsion.2~3 The conductivity of water-in-oil microemulsions shows quite a remarkable change over many orders of At higher water content (often approximately more than 20%) the conductivity is comparable to the conductivity of electrolyte solutions (10-'-1 C2-' m-I) and decreases linearly with decreasing volume fraction of watera4 At lower water content, however, the conductivity drops sharply by 3-4 orders of magnitude due to percolation transitionS (water percolation on the oil-rich side3). This sudden decrease is governed by power laws which are characterized by critical exponenh6 Below the percolation transition the conductivity of a water-in-oil microemulsion keeps decreasing with decreasing volume fraction of water but remains around 104-10-5 C2-' m-l. However, this value is still much higher than a typical conductivity of an apolar solvent ( 10-'6-10-'z C2-I m-'). This higher conductivity of a water-in-oil microemulsion has been explained by the fact that nanodroplets carry positive or negative excharges.' The migration of these charged droplets in an electric field will cause a finite conductivity (see Figure 1). From such a model it follows that upon dilution the specific conductivity of the microemulsion must remain constant. The question of the magnitude of the specific conductivity is immediately related to the average charge on a droplet. As each dropet consists typically of lo4 ionic surfactant molecules, one would
-
expect from fluctuation t h e ~ r ythat ~ . ~each droplet would carry lo2 elementary charges-the square root of the number of ions. A quick calculation shows that such a number leads by far to too large conductivities. The experimentally observed conductivity is more than 2 orders of magnitude lower. Therefore, if this physical picture is correct, there must be an opposing mechanism that drastically reduces the number of highly charged droplets. Actually, such a mechanism is well-known in ionic solvation theory. By a simple electrostatic argument Born has shown that charging up an atom or a molecule in a solvent requires a considerable amount of polarization energy.* We expect such an effect to also be important for microemulsion droplets. In this paper we exploit this idea in more detail and present a quantitative model of the conductivity of a water-in-oil microemulsion. Furthermore, we show that this model is entirely consistent with experimental results obtained on the AOTwater-isooctane system. 2. Charge Fluctuation Model Consider a nanodroplet composed of N l negatively charged surfactant molecules and Nz positively charged counterions (see Figure 1). For electroneutrality the average values are equal ( N l ) = ( N 2 ) = N . However, due to spontaneous fluctuations in these numbers, the droplet will carry an excess charge z = N2 - NI (1)
in units of the elementary charge e. Even though the "valency" of a droplet z will fluctuate in time, the conductivity of a water-in-oil microemulsion and a dilute electrolyte solution containing different ions can be evaluated in an entirely equivalent manner. This is because only the mean-square valency of the ions (or droplets) determines the conductivity. The conductivity u of a dilute electrolyte solution of different ions i with a valency zi, radius r (taken as independent of i for simplicity), and number density pi is given by8
-
(1) Eicke, H.-F. In Microemulsions;Robb, I. D., Ed.; Plenum: New York,
1982. (2) Benett, K. E.; Hatfield, J. C.; Davis, H. T.; Macosko, C. W.; Scriven, L. E. In Microemulsions; Robb, I. D., Plenum: New York, 1982. (3) Borkovec, M.; Eicke, H.-F.; Hammerich, H.; Das-Gupta, B. J . Phys. Chem. 1988, 92, 206. (4) Lagourette, B.; Peyrelasse, J.; Boned, C.; Clausse, M. Nature (London) 1979, 281, 60. Clausse, M.; Peyrelasse, J.; Heil, J.; Boned, C.; Lagourette, B. Ibid. 1981, 293, 6361. Cazabat, A,-M.; Chatenay, D.; Langevin, D.; Meunier, J. Faraday Discuss. Chem. SOC.1982, 76, 291. ( 5 ) Bhattacharya, S.; Stokes, J. P.; Kim, M. W.; Huang, J. S. Phys. Reu. Len. 1985,55,1884. Eicke, H.-F.; Hilfiker, R.;Thomas, H. Chem. Phys. Lett. 1986, 125, 295. van Dijk, M.A.; Casteleijn, G.; Joosten, J. G. H.; Levine, Y. K. J. Chem. Phys. 1986,85, 626. (6) Seaton, N. A,; Glandt, E. D. J. Phys. A: Math. Gen. 1987, 20,3029. Grest, G. S.; Webman, I.; Safran, S. A.; Bug, A. L. R. Phys. Rev. A 1986, 33, 2842. (7) Eicke, H.-F.; Denss, A. In Solution Chemistry ojSurfactants; Mittal, K. L., Ed.; Plenum: New York, 1979.
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where 7 is the viscosity of the solvent and i runs over all different ionic species in solution. In the case of microemulsion droplets it is more convenient to write eq 2 as pe2 u = -(z2)
6a7r
(3)
where p is the number of droplets per unit volume and (...) is the canonical average over all droplets. Note that due to electroneutrality ( 2 ) = 0. The quantity of interest is the mean-square charge (zz) of a droplet. It can be expressed in terms of the mean-squared fluc(8) See for example: Berry, R. S.; Rice, S . A,; Ross, J. Physical Chemistry; Wiley: New York, 1980. (9) See for example: Callen, H. B. Thermodynamics;Wiley: New York, 1960.
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The Journal of Physical Chemistry, VoZ. 93, No. 1, 1989 315
Conductivity of Water-in-Oil Microemulsions
are essentially given by the number of ions residing on the droplet. As N >> 1, the realistic case corresponds to the second limit where (z’) = l / a . This means that it is determined by the ratio of Coulomb and thermal energies. Inserting ( z z ) = 1/a into eq 3, we obtain the final result for the conductivity of a dilute microemulsion
We have replaced p in favor of the volume fraction of the droplets 4 by using the relation = 4 ~ ? p / 3 . Equation 11 predicts that the specific conductivity in a water-in-oil microemulsion u/d should be a constant and for given solvent and temperature depend on the radius of the droplets only (af3). It is interesting to note that this result is independent of the charge of the ions in question. Figure 1. Pictorial representation of aqueous microemulsion nanodroplet.
tuations of the number of ions residing on a droplet SN, = Ni (Nt) by ( 2 ’ ) = (SN2’) - 2(6N2 6N1) (SNi’) (4)
+
Such averages are related to derivatives with respect to the conjugated thermodynamic forcesg
where pj is the Chemical potential of t h e j t h component (j = 1, 2), T i s the absolute temperature, and kB is the Boltzmann constant. To evaluate ( z z ) explicitly, we need a model for the chemical potential p i of the ion i residing on a droplet. We write pi
= k(O)
+ kBT log Ni + p.I(cx)
(6)
where the first two terms on the right-hand side represent the chemical potential for an ideal solution while p p ) is the excess chemical potential
(7) We adopt a very simple model and identify the electrostatic work required to charge a droplet in the solvent with the excess Gibbs free energy, i.e.
where z is given by eq 1, eo is the dielectric permittivity of the vacuum, and t is the dielectric constant of the solvent. The excess Gibbs free energy (eq 8) is also the basis of Born’s theory of ionic solvation. Now we can calculate (z2) explicitly. Using eq 6, 7, and 8, we first evaluate the 2 X 2 matrix with the elements (dpi/dNj)Nk+hT and find
*
-a
1/N2
+
CY
(9)
We have introduced the abbreviation a = e2/(4?rkBTeoer).The derivatives (dN,/dp,),, required in eq 5 can be obtained most easily by noting the fact that the matrices with the elements (dpt/dN’)Nh and (dN,/dpj)P,,,bT are related by simple matrix inversion? fnverting the matrix in eq 9 and using eq 5 and 4, we obtain 2N (z2) = (10) 1 2Na
+
There are two limiting cases to consider. For a > 1 limit of eq 10 is important. Finally, one should mention that the present model might shed some light on conduction processes in nonaqueous colloidal solutionsZoand water-in-oil emulsions.21
Acknowledgment. We thank U. Hofmeier and D. Spielman for help with the experimental work and C. Quellet for useful comments on the manuscript. This research has been supported by the Swiss National Science Foundation. (18) Maitra, A. N.; Dinah, S.;Varshey, M. Colloids Surf. 1987,24, 119. Eicke, H.-F. Pure Appl. Chem. 1980, 52, 1349. (19) See for example: Davies, C. W. Ion Association; Buttenvorths: London, 1962. (20) Novotny, V. Colloids Surf.1986, 21, 219. (21) Becher, P. Emulsions: Theory and Practice; Reinhold: New York, 1965.
(16) Fletcher, P. D. I.; Howe, A. M.; Robinson, B. H. J. Chem. SOC., Faraday Trans. 1 1987,83,985. (17) Chen, S.H. Annu. Rev.Phys. Chem. 1986,37, 351. Eicke, H.-F.; Rehak, J. Helv. Chim. Acta 1976, 59, 2883.
Adsorption of Nitroxide Solutions on X Zeolite. 3. Electron Spin Resonance and Electron Spin Echo Modulation Study of Deuteriated Methanol Solutions M. Romanelli, M. F. Ottaviani, G. Martini,* Department of Chemistry, University of Florence, 501 21 Firenze, Italy
and L. Kevan* Department of Chemistry, University of Houston, Houston, Texas 77004 (Received: September 24, 1987; In Final Form: June 2, 1988)
Electron spin resonance (ESR) and electron spin echo (ESE) spectroscopies are used to study the properties of mono- and tetradeuteriomethanol solutions of neutral and negative nitroxides adsorbed on synthetic X-type zeolite. A comparison is made with the same nitroxide unadsorbed solutions: ESR spectra as a function of temperature enable a study of the dynamics of both the liquids and the probes, the location of the probes inside the faujasite cavity of the zeolite, and the dependence of surface effects on the probe mobility. Computer analysis of the modulation of ESE patterns of the nitroxides in solutions confirmed the formation of a "complex" between methanol and the nitroxides, in which the distance between the radical unpaired electron and the deuterium nuclei of the alcohol function was -2.5-2.7 A. After adsorption on zeolite, this complex formation was prevented, and deuterium nuclei resided at distances greater than 3.0 A. The reasons for these findings are discussed, and the present results are compared with those previously obtained in ethanol solutions.
Introduction parts 1 and 2 of this series continuous-wave electron spin (ESR) and pulsed electron spin echo (ESE) spectroscopies were to study motion, localization, and features of nitroxides in deuteriated ethanols in solution and adsorbed onto synthetic X
In this work we extend studies by ESR and ESE to negative and neutral nitroxides in alcohol solutions without &hydrogens, that is, methanol, both in bulk solution and in the adsorbed state on zeolite. As was previously due to the sensitivity of the echo modulation to the distance and number of nuclei surrounding the paramagnetic centers,35 mono- and perdeuteriated methanols
(1) Mazzoleni, M. F.; Ottaviani, M. F.; Romanelli, M.; Martini, G. J . Phys. Chem. 1988, 92, 1953. (2) Romanelli, M.; Martini, G.; Kevan, L.J. Phys. Chem. 1988,92, 1958.
(3) Kevan, L. Time Domain Electron Spin Resonance; Kevan, L., Schwartz, R. N., Eds.; Wiley-Interscience: New York, 1979; Chapter 8. (4) Dikanov, S . A.; Tsvetkov, Yu. D.Zh. Strukt. Khim. 1979, 22, 984. (5) Mims, V. B.; Peisach, J.; Davis, J. J . Chem. Phys. 1977, 66, 5536.
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