Synthesis of Copper Nanosize Particles in Anionic Reverse Micelles

May 15, 1995 - The size of copper nanoparticles in reverse micelles can be controlled by varying the water content of ... 5 Department of Applied Math...
0 downloads 0 Views 3MB Size
Download PDF
Langmuir 1995,11, 2385-2392

2385

Synthesis of Copper Nanosize Particles in Anionic Reverse Micelles: Effect of the Addition of a Cationic Surfactant on the Size of the Crystallites I. Lisiecki,??tM. Bjorling,'f>tL. Motte,t~~ B. Ninham,O and M. P. Pileni*$+J Laboratoire SRSI, Universitt? Pierre et Marie Curie, URA 1662, BP 52, Bat 74, 4 Place Jussieu, 75005 Paris, France, CEN Saclay, DRECAM-SCM Bat 522, 91191 Gif sur Yvette, France, and Department of Applied Mathematics, P.O. Box 4, Canberra, ACT 2601, Australia Received November 15, 1994. I n Final Form: February 22, 1995@ The size of copper nanoparticles in reverse micelles can be controlled by varying the water content of anionic reverse micelles (AOT = sodium bis(2-ethylhexyl)sulfosuccinate). In the presence of cetyltrimethylammonium chloride (CTAC), depending on the concentration, the size of the particles is strongly affected. At a low CTAC concentration ([CTAC] = 4 x M), an unexpectedly large increase in the size is observed. At the other extreme, a decrease in the size is observed at higher CTAC concentration ([CTAC] = 6.4 x M). The increase in the size at low CTAC concentration could be explained in terms of the formation of superaggregates containing most of the cationic surfactant. At higher CTAC concentrations, a random distribution of the positively charged surfactant takes place and the decrease in the size could be due to a decrease in the intermicellar attraction. Furthermore, the yield of particles drastically increases which is attributed to a change in the redox potential.

I. Introduction Since metals through their surfaces constitute a wide class of catalysts, the preparation of small monodispersed metallic particles with a large surface area is of topical importance. The preparation of such nanoparticles raises both practical and fundamental questions. At the practical end, new synthetic methods have to be devised and old ones have to be optimized. As far as the fundamental aspects are concerned, one should still answer the question, at which size does the particle loose its metallic properties? Reverse micelles are thermodynamically stable mixtures ofwater, oil, and surfactant where the water regions are separated from oil by a monolayer of surfactants. Due to the amphiphilic nature of the surfactant, numerous disordered or partially ordered phases are formed, depending on temperature and surfactant Concentration.' The ternary phase diagram sodium bis(2-ethylhexy1)sulfosuccinate (AOT)/water/isooctane shows a large zone where a reverse micellar phase (Lz)exists. In this isotropic liquid phase, the ratio of water over surfactant concentrations, (w = [H20l/[AOT]), called the water content, determines the size ofthe reverse micelles. When w values are lower than 15, the water mobility is greatly reduced (bound water). Above w = 15, the water pool radius, R,, increases linearly with the water content.2 Such a micellar system presents a percolation process when the polar volume fraction or temperature increases. The percolation threshold corresponds to the maximum of permittivity and the onset of c~nductivity.~ The effective intermicellar potential depends on temperature, solvent, and droplet size.4 This intermicellar potential can be adequately represented by a hard sphere core with very

short range attractive taih5r6 All collisions between two reverse micelles are not purely elastic, and some of them lead to a n exchange p r o c e s ~ . ~ ~ ~ In the present paper, it is shown that the size of the metallic copper particles can be controlled by the micellar water content. Addition of a low concentration of cetyltrimethylammonium chloride (CTAC) (less than one per micelle) induces an unexpectedly strong increase in the size. At the other extreme, a decrease in the size is observed at higher concentrations. Furthermore, the yield of particles drastically increases with CTAC addition. 11. Experimental Section Isooctane,sodium bis(2-ethylhexyl)sulfosuccinate(AOT),and hydrazine are obtained from Sigma, Fluka, and Kodak respectively. They were used without further purification. Absorption spectra are recorded on Perkin-Elmer Lambda 5 and Hewlett Packard HP 8452A spectrophotometers. Electron micrographs and diffraction patterns are obtained using a Jeol electron microscope (Model Jem.100CX.2). A drop of solution containing the colloidal particles is evaporated on a copper grid. To eliminate partially the AOT surfactant, the grid is washed before its examination. Size distributionsare achieved with about 1000 particles, and the measures of particle size are the means of the distribution. The quasi-elastic light scattering (QELS)is performed at 20 "C using an argon ion laser beam (514.5A) focused at the center of a cylindrical sample cell (diameter 8 mm). The fluctuations of the scattered intensity are measured at 45" from the direction of the incident beam, and the autocorrelationfunctionis obtained with a 136-channelBrookhaven 2130 AT digital correlator. All samples are filtered and centrifuged to eliminate dust.

111. Synthesis and Characterization The preparation of colloidal copper particles is achieved by mixing the reverse micellar system with a n aqueous ( 5 ) Bothorel, P.; Roux, D.;Lemaire, B. J.Phys. Chem. 1983,87,1023.

Universitk Pierre et Marie Curie. * CEN Saclay. 5 Department of Applied Mathematics. Abstract published in Advance A C S Abstracts, May 15, 1995. (1)Structure and reactiuity in reuerse micelles; Pileni, M. P., Ed.; Elsevier: Amsterdam, 1989. +

@

(2) Pileni, M. P.; Zemb, T.; Petit, C. Chem.Phys. Lett. 1985,118,414. (3)Van Dijk, M.A. Phys. Reu. Lett. 1986,55,1003. (4)Huang, J. S.J . Chem. Phys. 1985,82,480.

(6)Safran, S.A,; Kim, M. W.; Grest, G.; Kotlarchyk, M.; Quirke, N.; Huang, J. S. Phys. Rev. Lett. 1984,53,592. (7) Shepherd,J. C. W.; Steinmann,A.;Eicke, H. F. J . CoZZoidInterface Sci. 1976,56,168. (8)Toprakcioglu, C.; Dore, J. C.; Chieux, P.;Robinson, B. H. J . Chem. SOC.,Faraday Trans. 1 1984,80,413. (9)Felderhof, B. U. J . Phys. A: Math. Gen. 1978,11, 929. (10)Pileni, M. P.; Zemb, T.; Petit, C. Chem. Phys. Lett. 1986,118, 414. (11)Brochette, P. These de 1'Universite Paris 6,1987.

0743-7463/95/2411-2385$09.00/00 1995 American Chemical Society

Lisiecki et al.

2386 Langmuir, Vol. 11, No. 7, 1995

solution of hydrazine. The AOT reverse micellar solution consisted of a mixture of copper bis(2-ethylhexy1)sulfosM) and uccinate surfactant, Cu(AOT12, ([Cu(AOT)21= sodium bis(2-ethylhexy1)sulfosuccinate surfactant, NaAOT M) in isooctane. Hydrazine is added ([NaAOTl = 8 x to the micellar solution, and the reduction takes place ( [N2H4]= 3 x M). During the reaction, the samples are kept under a shelter of oxygen and characterization of the micellar solution is achieved 5 h after the hydrazine addition. The particles are characterized by UV-visible absorption spectroscopy, transmission electron microscopy, and electron diffraction. From absorption spectroscopy it has been demonstrated that colloidal copper particles are characterized by a plasmon peak centered a t 557 nm. The intensity of this peak depends on the particle ~ i z e . ' ~For J ~ a given average size, the change in the absorption spectrum indicates a change in the concentration of particles. From the data presented below, it seems obvious that, in the absence of CTAC, the overall copper ions present in solution are not reduced. So it is impossible to give a n exact value of the production yield of particles. Only a relative yield can be given from the change of the optical density of particles having the same average size. In the case of oxide formation a strong absorption a t 800 nm is observed. This does not occur when pure metallic particles are formed. Transmission electron microscopy showed that the particles synthesized are spherical with diameters between 2 and 12 nm. High-resolution electron microscop of the particles showed a n interreticular distance of 2 This value is in good agreement with that calculated from the lattice of metallic copper. Electron diffraction showed concentric circles characteristic of a face-centered-cubic phase with a lattice dimension equal to 3.61 A. The distance between the highest density reticular plane (111) can be calculated (3.61(311"= 2.09 A). The presence of oxide could be detected by electron diffraction.

K

IV. Results 1. Determination of the Droplet Size andMicellar Interactions in the Absence and in the Presence of CTAC. For dilute solutions, the average diffusion coefficient D, obtained from QELS, depends linearly on the volume fraction of the particles, 4:

D

= D&1+ a+)

where DOis the diffusion coefficient a t infinite dilution and a is the second vinal coefficient. The latter is the sum of the well-known second osmotic virial coefficient and other terms which take into account hydrodynamical effect^.^ Assuming that the particles are spherical, the hydrodynamical radius, Rh, is given by the StokesEinstein expression:

Do= kTI6qRh where k, T , and 7 are the Boltzmann constant, the temperature, and the shear viscosity of the solvent, respectively. The hydrodynamics radii, Rh,determined by QELS a t various water contents (Table 11, are in good agreement with those estimated from a geometrical model assuming the size of the spherical droplet is controlled by the volume of water molecules and the surface per head polar group of the surfactant.1° Taking into account the length of the alkyl chains of the AOT (0.8 nm), the polar radii (12)Lisiecki, I.; Pileni, M. P. J. Am. Chem. SOC.1993,115,3887. (13) Lisiecki, I.; Pileni, M. P. Submitted for publication.

Table 1. Evolution of the Hydrodynamic Radii Obtained from QELS (Rh) and from Calculation (Rh(cdc), nm) and the Corresponding Second Virial Coefficients (aand with the Micellar Water Content (w)in the Absence and in the Presence of CTAC [CTACHAOTI = 0 [CTACl/[AOTl= 0,2% [CTACl/[AOTl = 10% W

Rh

Rh(cal)

U

3 1.97 2.34 -3.7 5 2.24 2.66 -7.9 10 3.00 3.47 -5.7

Rh

a

1.93 2.55 3.31

-3.1 -4.9 -3.8

Rh

a

2.01 -0.20 2.27 -0.86 3.00 -0.35

a(ca~)

-1.41 -3.81 -1.70

determined by QELS (Table 1)are also in good agreement with those obtained previously by small-angle X-ray scattering (SAXS).l1 Addition of CTAC to 0.1 M AOT reverse micelles, a t a given water content, leaves the hydrodynamics radius approximately unchanged within experimental error (Table 1))whereas an increase in the second virial coefficient is obtained. The increase in the second virial coefficient (Table 1)is strongly dependent on the CTAC concentration and indicates a decrease in the intermicellar attraction. 2. Synthesis of Copper Metallic Particles Obtained in the Absence and in the Presence of CTAC. In previous papers,12J3it was demonstrated that addition of a 3 x M aqueous solution of hydrazine to a reverse M of NaAOT, micellar solution formed by 8 x M of Cu(AOT)2,water, and isooctane as the bulk solvent, induces the formation of nanosize copper metallic particles. Figure 1shows the electron microscopy pattern obtained when the copper particles are made in reverse micelles a t various water content. An increase in the average diameter of the copper particles from 2 to 12 nm (Figure 2) is observed with increasing water content. The particle size does not notably change for water contents greater than w = 10. In the presence of CTAC, as in the absence, an increase in the particle size with increasing water content is observed. However, the magnitude of the change depends on the CTAC concentration and the water content (Figures 3 and 4). At low water content (w = 2) no change neither in the production yield nor in the average size (the diameter remains equal to 2 nm) is observed. Addition of 4 x M CTAC to 0.1 M of AOT reverse micelles induces changes which depend on the water content (Figure 3): At water contents greater than 2, drastic changes in the particle size and in the production yield are observed. At a water content equal to 3, addition of 4 x lo-* M CTAC induces a n increase in the particle size, from 5 to 9 nm, and of the production yield. At w = 10 and above the nanosize particles remain unchanged and equal to 12 nm but show a large increase in the M of CTAC production yield. Thus, addition of 4 x induces a faster increase in the size of the particles with the water content than that observed in the absence of CTAC (Figure 2). The particles reach the same limiting size (12 nm), and the production yield strongly increases. Upon addition of 6.4 x M CTAC to 0.1 M AOT reverse micelles, the particle size increases with the water content, as observed in the absence of CTAC. However, for a given water content, the particle sizes are smaller than that observed in the absence of CTAC and the limiting size is reached more slowly (Figure 2). Large copper metallic particles (greater than 10 nm) are characterized by an absorption spectrum with a maximum centered a t 566 nm which is due to a plasmon peak.13 The optical density recorded a t 566 nm is directly proportional to the number of particles formed. Figure 5 shows the increase in the 566-nm optical density with CTAC concentration a t a high water content (w = 201,

Copper Nanosize Particles in Anionic Reverse Micelles

Langmuir, Vol. 11, No. 7,1995 2387

Figure 1. Electron microscopy patterns of copper metallic particles synthesized in M Cu(AOT)2/8 x M NaAOT/water/ isooctane reverse micelles at various water contents. [N2H4]= 3 x M. that is to say for a large particle size (12nm). The large particles, a strong interaction between the particles and increase in the copper particle production yield, with CTAC the silicium cell leads to the formation of a copper mirror concentration, is confirmed by the electron microscopy on the cell walls. The mirror remains stable as long as patterns given in Figure 6. Because of the high yield of oxygen does not penetrate into the cell.

Lisiecki et al.

2388 Langmuir, Vol. 11, No. 7, 1995

0

5

10

15

20

25

w=[H,O]/[AOT]

Figure 2. Variation of the diameter of copper particles M Cu(AOT)2/8 x M NaAOT/water/ synthesized in isooctane reversemicelles with the water contentsin the absence and in the presence of various CTAC concentrations. Similar behavior is observed at lower water contents. This can be observed from the electron microscopypatterns in Figures 3 and 4. However, it cannot be demonstrated from the change of the 566-nm optical density because the plasmon peak disappears progressively with a decrease in the particle size.12

V. Discussion The QELS measurements show a n abrupt increase (to less negative values) of the apparent vinal coefficient by CTAC addition in AOT reverse micelles. This effect is observed even at low CTAC concentration ([CTAC] = 4 x M). This increase in the apparent virial coefficient depends on CTAC concentration (Table 1).This indicates a n apparent weakening of the attraction force, or an enhanced repulsion between micelles with increasing CTAC concentration. However, the diffusion coefficient extrapolated to the infinite dilution, DO, is consistent with that of AOT micelles alone. This indicates that the micellar radius is approximately unchanged by CTAC addition. (A slight increase within the experimental error is observed.) Some of the decreased attractive force between the micelles can be understood in terms of a n enhanced steric repulsion between micelles. The CTAC hydrocarbon tail is composed of sixteen carbon atoms, whereas the longest part of the AOT tail only consists of six. For each CTAC molecule residing in a n AOT micelle, a tail of ten carbons is therefore dangling outside the steric range of the AOT tails. This causes a local steric repulsion and perturbs the formation of dimers and multimers caused by the attraction between micelles. This perturbation can be interpreted as a decrease in the average attraction. With assumption that a random distribution of CTAC into spherical AOT micelles and with use of a simple geometrical model, the decrease in the virial coefficient, a, can be calculated (see Appendix A). The results show that approximately half of the decrease at high CTAC concentration can be accounted for (Table 1). However, when CTAC is added, we also expect a decrease in the size of the aqueous core, and consequently of the interdroplet attraction, as the total surface (given by the surfactant concentration) increases for a fixed volume of water. This predicted decrease in the droplet core size is not in conflict with the QELS measurements. The longer tails of CTAC ought to increase the hydrodynamics radius of the droplets, thus counteracting the decrease in the core size. The end result of this competition is not easy to quantify. As a matter of fact, the total surface depends nontrivially on the effects of electrostatic screening in the mixed surfactant monolayer, and the increase in the hydrodynamics radius is difficult to predict. The results presented above confirm that the size of the produced metallic copper particles is controlled by the

water content.12 At low water contents, the particle size increases rapidly with increasing water content and finally levels off to a limiting size (Figure 2). One reason for this influence is the changing state of the water molecules. At a low water content, most water molecules are required for hydration and are bound to the interface. With increasing water content the volume-to-surface ratio increases (linearly for spheres) and bound and bulk water molecules begin to coexist. The water structure could affect the particle synthesis by changing the coppersulfonate head group interactions, the number of hydrated copper ions, and the redox potential of copper ions and hydrazine. It is reasonable to assume that hydrazine also is strongly affected by the microenvironment since it behaves similarly to water.14J5 An additional consequence of the increasing volume-to-surface ratio is that the aggregation number, and thus the number of copper ions per micelle, increases quadratically with w. Increasing water content also gives rise to an increase in the intermicellar attraction.16 Since a n increased attraction would lead to an increase in the average number of nearest neighbors, this effect would give a higher rate of mass transport, assuming that the rate of the exchange mechanism is otherwise unchanged. Addition of CTAC to AOT reverse micelles induces large changes in the size and in the production yield of the copper metallic particles except at a very low water content (w = 2), where no changes in the size and in the yield are observed. At w = 2, there is not enough water to hydrate the reactants and CTAC does not drastically change the rate of the chemical reactions. From a slightly higher water content, w = 3, and more, the result of the particle synthesis depends on the CTAC concentration added. At relatively “high”CTAC concentration ([CTACI = 6.4 x M), the size of the particles are smaller than that observed in the absence of CTAC. This effect could be attributed to a change in the intermicellar potential between droplets which is directly related to the intermicellar exchange process. As a matter of fact, it has been demonstrated that a decrease in the intermicellar attractions brought about by replacing isooctane by cyclohexane1’or by increasing the polar volume fractionla leads to a decrease in the size of the copper metallic particles.12 At a low CTAC concentration ([CTACI = 4 x M), a CTAC/AOT ratio of less than 1 CTAC molecule per micelle, the size of the copper particles is vastly enhanced and augments faster with increasing water content than in the absence of CTAC. To explain such behaviors, several parameters have to be taken into account. At a water content of w = 3, CTAC induces a change in the size of the particles (from 5 to 9 nm), keeping the same low polydispersity as is obtained in the absence of CTAC. This large change in size cannot be explained by a random distribution of CTAC molecules (one in every fifth micelle). To rationalize these data, it seems to be reasonable to assume an adsorption and cooperative association of CTAC molecules into a few mixed micelles which would induce a n increase in the size of a few droplets in which the chemical reaction would take place. From the value of the size of the copper particles obtained in the presence and in the absence of (14)Lumry, R.; Battistel, E.; Jolicoeur, C. Faraday Symp. Chem. SOC.1982, 17, 93.

(15) Ramada, M.S.;Evans, D. E.; Lumry, R. J . Phys. Chem. 1983,

- ----

87. . , AFiRA .

(16) Robertus, C.;Philipse, W.; Joosten,J.;Levine,Y. J . Chem.Phys.

1989,90,4482.

(17) Towey, T. F.;Khan-Lodl, A.; Robinson, B. H. J . Chem. Soc., Faraday Trans.2 1990,86,3757. (18) Pitre, F.;Regnaut, C.; Pileni, M.P. Langmuir 1993, 9,2857.

Copper Nanosize Particles in Anionic Reverse Micelles

Lungmuir, Vol. 11, No. 7,1995 2389

I

I

Figure 3. Electron microscopy patterns of copper metallic particles synthesized in Cu(AOT)flaAOT/water/isooctanereverse micelles at various water contents: (A) [NaAOTl = 8 x M, and [N2H41= 3 x M; (B)[NaAOTl= M, [Cu(AOT)2]= 8x M, [Cu(AOThI = M, [N2H41= 3 x M, and [CTAC] = 4 x M. CTAC, the average number of CTAC associated per micelle can be estimated. As a matter of fact, in the absence of CTAC, the 9-nm-diameter particles have been obtained for a water content equal to 5. So the few micelles containing CTAC could have a water pool radius corresponding to a water content equal to 5 instead of 3. Then the size of the "factory" is close to 1.5-nm diameter instead of 0.9 nm which increases the volume of the microreactor by a factor of 5. This small number of mixed micelles acting as relatively large factories and the required degree of cooperative adsorption of CTAC may be calculated using the same type of geometric arguments presented above. Consider the thought experiment of cooperative addition of CTAC into a fraction of the otherwise unchanged pure AOT micelles. Equilibriumof this new system may be restored by redistributing water and AOT between the mixed and pure micelles:

(i)First the redistribution of water only is considered. It is obvious that the mixed micelles will steal the water from the pure micelles simply because of its greater concentration of salt (the surfactant may be regarded as a univalent salt). For a total concentration of 0.4 mM CTAC and 0.1 M AOT, assuming a similar hydration requirement for CTAC and AOT, v d estimating the CTAC head polar group area to be 30 A2, four CTAC per AOT in every thousand micelles is required to give a factory with a diameter of 1.5 nm (W * 5, Appendix B). This estimate is on the large side. The hydration requirement in the mixed micelles is probably increased, compared to AOT, due to the sodium and chloride counterions released from the interface. (ii) If we also allow AOT to redistribute, it will also prefer the mixed micelles by virtue of the hydrophobic counterions provided by CTA+and the subsequententropic gain in the release of their respective counterions. This

2390 Langmuir, Vol. 11,No.7,1995

Lisiecki et al.

w=2

Figure 4. Electron microscopy patterns of copper metallic particles synthesized in Cu(AOT)flaAOT/water/isooctanereverse M; (B)[NaAOTl= 8 x micelles at various water contents: (A) [NaAOTl = 8 x M, [Cu(AOT)2]= M, [N2H4]= 3 x M, [Cu(AOT)21= M, [NzH41 = 3 x M, and [CTAC] = 6.4 x M.

0

1

2

3

4

5

6

7

[CTAC].1O3

Figure 5. Evolution of the optical density at 566 nm with CTAC concentration UI = 20.

effect would also lower the required XL. Thus, the stipulated increase in the size of the factory is not that improbable. For the pure micelles the size will tend to

decrease due to the loss of water and to increase due to the loss of AOT. A tentative explanation of the increase in the particle production could be as follows: a change of the monolayer flexibility, caused by the incorporated CTAC molecules, could increase the number of efficient collisions and allow association of pure AOT micelles with the factory that would not otherwise occur. Once catalysis is complete in the factory, the removal of Cu2+ions in the water pool causes rearrangement. The synthesized Cu particles are coated by AOT alone and provided by exchange with the excess AOT micelles, and the factory is reconstituted to repeat the process. It is the local environment of associated CTAC head groups in the factory that enhances the nucleation process. The increase in the particle production can be attributed to the fact that the Cu2+ions are less associated with the surfactant interface in the presence of CTA+,which may

Copper Nanosize Particles in Anionic Reverse Micelles

Langmuir, Vol. 11, No. 7, 1995 2391

that of an "enzyme". This effect is observed for a given CTAC concentration and disappears a t relatively high CTAC concentration. The system provided by AOT reverse micelles with a small admixture of cationic surfactant thus appears to exhibit a wide potential for a broad range of nanotechnological applications.

Appendix A. Random Distribution of CTAC The steric repulsion of the dangling CTAC tails may be quantitatively linked to a decrease in the intermicellar attraction (as observed by QELS) by using a crude geometricmodel and some results from Baxter's adhesive hard sphere model (AHS).19 In particular, the average number of nearest neighbors (2)in AHS is given by20

8 where Q, is the hard sphere volume fraction and iz is related to the adhesive well depth (7-l) by21

Note that iz is zero for nonadhesive spheres, so that 2 is in fact the average number of nearest neighbors in excess of a reference hard sphere system. The reverse transformation from A to t i d 9

Figure 6. Electron microscopy pattern of copper metallic particles obtained in the absence (A)and in the presence of 6.4 x M CTAC (B)at w = 20.

The relation between the virial coefficient a measured in QELS and z-l can be shown to bel8

a = 1.56 - 1.022-'

Figure 7. Schematic model of an AOT reverse micelle.

change the redox potential of the copper ions. With increasingwater content, more bulk water is also available for partitioning of Cu2+ions.

VI. Conclusions In the present paper, it has been shown that the size of metallic copper particles can be substantially changed. Thisis probably due to the changesin the redox potentials, the intermicellar interactions, the various interactions between the reagents and the interface, as well as in the microenvironment. It is especially intriguing to find that the size of the particles increases a t a low CTAC concentration, whereas it decreases a t a high CTAC concentration. Too much CTAC seems to "poison" the system. Addition of a low amount of CTAC to AOT reverse micelles induces large changes in the size and in the yield of copper particles. This has been explained in term of the appearance of a factory which plays a similar role as

(A4)

It is the link between iz and a (or vice versa), provided by eqs A2-A4, that is exploited in what follows. The perturbation of 2 caused by the CTAC tails will affect both iz and Q, in eq Al. Consider first the effect on A, when the AOT reverse micelles are treated as adhesive hard spheres with radiusRAm. For simplicity, we assume 2 to be determined by the probability to form dimers, i.e., that the attraction is weak and/or the system is dilute. The local steric repulsion from the CTAC tails perturb the formation of dimers. If the range of the attractive force is shorter than the repulsive range of the CTAC tail, then dimer formation is in fact excluded from a surface swept by the dangling tail. The resulting decrease in the dimer formation may be interpreted as a decrease in the average attraction. Noting that the Kuhn segment of polymethylene is approximately ten carbons, the length of the tail may be approximated by the Kuhn length I, 1.25 nm.22 Furthermore, the tails are flexible enough to release any angular constraint on the tail-end position. A crude, quantitative estimate of the repulsive range of a CTAC tail is therefore the sphere of radius I, spanned by the tail end. The maximum surface lost per CTAC (ACTA&i.e., where dimer formation is excluded, can then be obtained by simple geometry to be (see Figure 7)

In the absence of CTAC, the dimer probability is propor(19) Baxter, R. J. J. Chem.Phys. 1968,49 (6),2770. (20) Chiew, Y. C.; Glandt, E. D. J . Phys. A: Math. Gen. 1983, 16, 2599. (21) Barboy, B. J. Chem.Phys. 1974,61 (81,3194. (22) Cantor, C. R.; Schimmel, P. R. Biophysical Chemistry. Part 3, The Behaviour of Biological Macromolecules;Freeman: San Francisco, 1980.

2392 Langmuir, Vol. 11, No. 7, 1995

Lisiecki et al.

tional to the square of the AOT micellar surface area because all connection points on both spheres in a dimer are equally probable. Since we assume random distribution of CTAC (in this model), the same surface area, i.e., the same number of connection points, is lost on both micelles. Consequently, the decrease in the dimer probability is proportional to the square of the lost surface area per micelle, and the new parameter A in the presence of CTAC can be expressed as

where the subscript 0 denotes the unperturbed system, and nCTAC is the number of CTAC molecules per micelle. It is obvious that eq A6 is valid only if the micellar surface area exceeds the lost surface area, i.e., when nCTAC is small. EquationA6 shows that the steric repulsion ofthe dangling tails decreases the attraction parameter A, both by perturbing the dimer formation and by increasing the volume fraction. In order to quantify the above result nCTAC, RAOT, @o, and @ in eqA6 must be related to the system parameters, e.g. the AOT concentration [AOTI, the relative water content w = [HzOl/[[AOTl,and the relative CTAC content x = [CTACY[[AOTl,whereas A0 must be measured by QELS or some other technique. Since these parameters depend more or less on the size ofthe reverse micelles, aconvenient starting point is to evaluate the micelle radii. We distinguish between four radii (see Figure 71, Rw,R,, RHS, and R h , which are the water core, the polar core (including the surfactant head groups),the hard core (including steric repulsion from the tails), and the hydrodynamic (including bound solvent) radii, respectively. Assuming spherical micelles, the water core radius is easily obtained by simple geometric arguments.1° With the assumption of a n as average aggregation number ofAOT per micelle (Nagg), well as a random distribution of CTAC so that

(A71

~ C T A C= XNagg

the volume of the water core is

V = WVpag, = (4/3)nRW3 where V, is the volume of a water molecule, and the corresponding surface area is

S = (XAOT

+ x&TAc)Nagg

4d3,

(A9)

where ZAOTand &TAC are the head group areas for AOT and CTAC, respectively. Eliminating the unknown Nagg from eqs A7 and A8, the water core radius becomes

R, =

3Ww

+

~ A O T X~CTAC

(A101

be expressed as

R, = R,

+ 6,

(A131

where z stands for any of the subscripts AOT, HS, or h, and 6 , is the corresponding increase of the radius. To model the dependence of the apparent hard sphere radius in the presence of CTAC, we may simply scale it with the lost area from above to give

The hard sphere volume fraction then becomes

where N Ais Avogadro's number and RHSis assumed to be in angstroms. Even though the above model does not purport to give exact quantitative predictions, it should nevertheless give physically reasonable results. The above crude model is expected to be more realistic for large water contents. For small w ,the parameter values seem to change more rapidly with w. Since our interest is a rough estimate of the steric effect of CTAC, a fine tuning of the parameters for each w is not necessary. (Such a fine tuning would probably increase the calculated effects.) The values of the model = 490 A2, parameters used in the calculations ?re ACTAC Vw = 30 A3,ZAOT= 60 A2,&TAC = 30 A2,6, = 0.4 nm, ~ A O T = 0.6 nm, and ~ C T A C= 1.8 nm. These values are rough estimates and may in fact not be constant as other parameters change.

Appendix B. Nonrandom Distribution of CTAC Here, we suppose that CTAC distributes unevenly among the AOT micelles and consider the simplest case where only redistribution of water is allowed. Thus, in the above notation Naggis constant for all micelles. Furthermore, we assume that the hydration requirement for AOT and CTAC is similar and remains constant. Let the mixed micelles contain XL CTAC per AOT, i.e. XL = [CTACI/(~L[AOTI) where f~ is the fraction of AOT forming mixed micelles, then the above assumptions are mathematically summarized as

WL = WS(l

+

XL)

(B1)

where WL and W Sare the relative water contents in the mixed and pure micelles, respectively. Thus the mixed micelles will be somewhat larger than the pure AOT micelles. From conservation of water we find that

Using the latter two equations, Naggis given by and the size of the water core of the mixed micelles is given by The polar core radius includes the electron rich head group region. For simplicity, the head group region may be considered of constant thickness, and the polar core radius becomes

R, = R,

+ 6,

(A12)

The hard core and the hydrodynamic radii may similarly

RL =

3WLVw

+

~ A O T XL~CTAC

in analogy with eq A10. The parameter values used are found in Appendix A. LA940910S