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Langmuir 1999, 15, 952-956
Particle Size Distribution in the Synthesis of Nanoparticles Using Microemulsions Yongcheng Li and C.-W. Park* Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611 Received May 11, 1998. In Final Form: October 15, 1998 A Monte Carlo simulation was used to estimate the particle size distribution in the synthesis of nanoparticles using microemulsions. When two microemulsions each containing reactants were mixed together, continuous collisions of the droplets lead to a particle-forming reaction through a repeated coalescence and breakup process. The particle formation in this process typically was limited by the droplet collision, because the mixing and reaction of the reactants were fast once two droplets coalesced. The present simulation indicates that a significant number of fine particles is generated making a broad hump in the particle size distribution on the small particle side. However, the volume fraction of the fine particles is insignificant, and the particle size distribution may be considered to be a narrow one if the fine particles are neglected. This prediction appears to be consistent with experimental observations, although only a limited number of experimental data is available.
1. Introduction Water-in-oil microemulsions (i.e., reversed micellar solutions) are transparent, isotropic, and thermodynamically stable fluids in which nanometer-sized water droplets are dispersed in a continuous oil phase. The droplet sizes are usually determined by the surfactants used for the microemulsions and are known to be very uniform. Because of its size uniformity, microemulsion is considered a lucrative way to synthesize nanosize particles with a narrow size distribution. Synthesis of nanoparticles using reactions in microemulsions was first reported by Boutonnet et al.1 Since then, microemulsions have been used in the synthesis of various particles (e.g., cadmium sulfide,2 nickel, iron and cobalt,3 silica,4 aluminum hydroxide,5 molybdenum sulfide,6 copper, lead, and indium sulfides,7 silver bromide,8 Y-Ba-Cu-O and B-Pb-Sr-Ca-Cu-O ceramic superconductors9). The synthesis of nanoparticles using microemulsions may be classified into two types. The first is a single microemulsion type in which nanoparticles are produced in a microemulsion by adding a reducing or precipitating agent, in the form of gas or liquid, to a microemulsion containing the primary reactant dissolved in its aqueous droplets. The particles produced by this type of process could be much bigger than the original droplet size.10 The second is the double or multiple microemulsion type, in which two or more water-in-oil microemulsions each containing respective reactants are mixed together. The droplets of the mixed microemulsion go through numerous * To whom correspondences should be addressed. (1) Boutonnet, M.; Kizling, J.; Stenius, P.; Maire, G. Colloids Surf. 1982, 5, 209. (2) Modes, S.; Lianos, P. J. Phys. Chem. 1989, 93, 5854. (3) Nagy, J. B. Colloids Surf. 1989, 35, 201. (4) Arriagada, F. J.; Osseo-Asare, K. Colloids Surf. 1992, 69, 105. (5) Watson, D. W.; Fulton, J. L.; Smith, R. D. Mater. Lett. 1987, 6, 31. (6) Boakye, E.; Radovic, L. R.; Osseo-Asare, K. J. Colloid Interface Sci. 1994, 163, 120. (7) Dannhauser, T.; Oneil, M.; Johansson, K.; Whitten, D.; McLendon, G. J. Phys. Chem. 1986, 90, 6074. (8) Kumar, P.; Pillai, V.; Bates, S. R.; Shah, D. O. Mater. Lett. 1993, 16, 68. (9) Kumar, P.; Pillai, V.; Shah, D. O. Appl. Phys. Lett. 1993, 62, 675. (10) Arriagada, F. T.; Osseo-Asare, K. J. Colloid Interface Sci. 1995, 170, 8.
collisions and thereby the reactants are mixed and react to form solid particles as a product of the chemical reaction. Experimental data indicate that the particles formed by this type are generally smaller than the original droplets. The dispersed water droplets in a microemulsion are considered to be nearly monodispersed and spherical, and their sizes may be predicted on the basis of thermodynamic and geometric considerations.11-13 For reactions in microemulsions which involve reactants confined within the dispersed water droplets, a necessary step before the chemical reaction is the exchange of reactants after coalescence of droplets. The material exchange between droplets has been found to follow a second-order reaction mechanism, and the exchange rate constants have been investigated for various systems.14,15 If the chemical reaction itself is very fast, the kinetics of particle formation is limited by the material exchange during the collisions. For most systems, the material exchange is limited by the droplet collision frequency.16,17 In this article, a simple particle growth mechanism is assumed and the size distribution of the nanoparticles produced by a double microemulsion process are estimated using Monte Carlo simulation. 2. Particle Formation and Growth Mechanism Microemulsions are considered to be nearly monodispersed and are in a dynamic state that involves continuous collisions between droplets. Depending on the interaction potential between droplets, which is determined by the fluid (i.e., oil) and the surfactant used, some collisions result in coalescence. Once the droplets coalesce, the coalesced droplet tends to break up because it is not thermodynamically as stable as the original droplets. Thus, unless the interactions between droplets are very repulsive, the aqueous droplets in a water-in-oil micro(11) Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1981, 77, 601. (12) Oakenfull, D. G. J. Chem. Soc., Faraday Trans. 1 1980, 76, 1875. (13) Eicke, H. F.; Markovic, Z. J. Colloid Interface Sci. 1981, 79, 151. (14) Jada, A.; Lang, J.; Zana, R. J. Phys. Chem. 1989, 93, 10. (15) Suarez, M.; Levey, H.; Lang J. J. Phys. Chem. 1993, 97, 9808. (16) Gelade, E.; De Schryver, F. C. J. Photochem. 1982, 18, 223. (17) Gelade, E.; De Schryver, F. C. J. Am. Chem. Soc. 1984, 106, 5871.
10.1021/la980550z CCC: $18.00 © 1999 American Chemical Society Published on Web 12/24/1998
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emulsion go through continuous coalescence and breakup which results in continuous exchange of the droplet contents. When two microemulsions each containing respective reactants are mixed, the contents of the aqueous droplets are mixed and redistributed very rapidly because of the collisions that involve temporary merging of the droplets into a larger droplet (fusion) and subsequent breakup of this larger droplet (fission). It was found that the mass exchange between the droplets can be extremely fast and consequently, the mass exchange is limited by the collision of the droplets.20 For a first approximation, we may assume that once two droplets are merged through a collision, the contents of the two droplets are mixed rapidly and thoroughly before breakup into two droplets of identical size. For simplicity, we may denote the contents of a droplet by an array of three numbers (or by a three-dimensional vector): the first and the second number represent the concentrations of reactants A and B, respectively, and the third number represents the volume of the solid particle present in the droplet. Then the net result of a single collision (i.e., coalescence and breakup) between two fresh droplets which contain reactants A and B, respectively, may be represented as
(1, 0, 0) + (0, 1, 0) f (0, 0, 1) + (0, 0, 0)
(1)
Here the reactant concentrations have been scaled by the initial concentrations of each reactants, before the microemulsions are mixed together. The particle volume has been scaled by the volume of the particle that is formed by the two fresh droplets each containing reactants A and B at the initial concentrations. It is assumed that one droplet can contain only one solid particle. Although we assume a simple chemical reaction in which reactants A and B make particle C as aA + bB f cC, the stoichiometric coefficients a, b, and c do not have to be all ones because the reactant concentrations and the particle volume are scaled by respective concentrations. However, we assume that the initial concentrations of the two reactants in the droplets are in stoichiometric ratio although it can be easily extended to other situations. Under the assumption that the particle formation/ growth process is limited by the droplet collision (i.e., coalescence and breakup), which is a reasonable assumption for microemulsions of viscous continuous phase, the resulting droplets (droplets 3 and 4) after a single collision between two droplets (droplets 1 and 2) can be described as
(a1, b1, c1) + (a2, b2, c2) f (a3, b3, c3) + (a4, b4, c4)
(2)
Here ai and bi (i ) 1, 2, 3, or 4) cannot be bigger than 1, and both ai and bi cannot be nonzero at the same time within a droplet because reactants A and B react rapidly. When droplets 1 and 2 coalesce to form a larger droplet, the amount of reactants A and B will be (a1 + a2) and (b1 + b2), respectively. If (a1 + a2) is bigger than (b1 + b2), reactant B will be depleted and droplets 3 and 4 will contain equal amounts of the excess reactant A (i.e., a3 ) a4 ) [(a1 + a2)-(b1 + b2)]/2 and b3 ) b4 ) 0). (18) Zana, R.; Lang, J. In Microemulsions: Structure and Dynamics; Friberg, S. E., Bothorel, P., Eds.; CRC Press: Boca Raton, FL, 1987. (19) Atik, S. S.; Thomas, J. K. Chem. Phys. Lett. 1981, 79, 351. (20) Lianos, P.; Lang, J.; Cazabat, A. M.; Zana, R. In Surfactants in Solution; Mittal, K. L., Bothorel, P., Eds.; Plenum Press: New York, 1986.
As a result of this reaction, particle growth will occur with the volume equivalent to the amount of the limiting reactant B [i.e., (b1 + b2)]. There can be three different cases for the collision; when both c1 and c2 are zero, when one of c1 and c2 is nonzero, and when both c1 and c2 are nonzero. If both c1 and c2 are zero, a new particle will be formed with a volume (b1 + b2); that is, c3 ) (b1 + b2) and c4 ) 0. Because droplets 3 and 4 are indistinguishable, it is equivalent to c3 ) 0 and c4 ) (b1 + b2). If one of c1 and c2 is nonzero (e.g., c1 is nonzero), that particle will grow by the volume (b1 + b2); that is, c3 ) c1 + (b1 + b2) and c4 ) 0. Again it is equivalent to c3 ) 0 and c4 ) c1 + (b1 + b2) because droplets 3 and 4 have no distinction. This assumption is plausible, because precipitation (or crystallization) on the surface of an existing particle should be easier than forming a new additional particle which may need a nucleating site. Finally, if both c1 and c2 are nonzero, both particles will grow sharing the volume amount (b1 + b2) in proportion to the surface area of each particle. The surface area ratio of the solid particles in droplets 1 and 2 is c12/3/c22/3. Thus, c3 ) c1 + (b1 + b2)‚[c12/3/(c12/3 + c22/3)] and c4 ) c2 + (b1 + b2)‚[c22/3/(c12/3 + c22/3)]. As pointed out previously, when the coalesced droplet breaks up into two after reaction, each droplet will contain one solid particle because we assume that one droplet can contain up to one particle. The following are a few examples of the particle formation and growth mechanism described above.
(1, 0, 0) + (0, 0, 1) f (0.5, 0, 1) + (0.5, 0, 0)
(3)
(1, 0, 0) + (0, 0, 0) f (0.5, 0, 0) + (0.5, 0, 0)
(4)
(1, 0, 0) + (0, 0.5, 1) f (0.25, 0, 1.5) + (0.25, 0, 0) (5) (0, 0.5, 1) + (0.25, 0, 1.5) f (0, 0.125, 1.108) + (0, 0.125, 1.642) (6) In this particle growth mechanism, triplet collisions (i.e., the collision of three droplets at the same time) are neglected because they have a very small probability. As the collisions proceed, the effective collisions (e.g., eqs 5 and 6) which result in new particle formation or growth will become less and less frequent, whereas the noneffective collisions (e.g., eqs 3 and 4), which do not involve the chemical reaction, will become more frequent. In the end, all collisions will become noneffective. Bigger particles can be formed when a collision occurs between droplets containing both reactant and solid particles (e.g., eqs 5 and 6). Small particles, on the other hand, are produced by collisions between the droplets containing the reactant at a low concentration. 3. Prediction of the Particle Size Distribution by a Monte Carlo Simulation The particle size distribution at different reaction stages can be predicted by a simple computer simulation. Starting from NA and NB number of fresh droplets containing reactant A(1, 0, 0) and B(0, 1, 0), respectively, a random collision is induced by randomly selecting two droplets from the pool of (NA + NB) number of droplets. The two droplets are then assumed to merge (or coalesce), react according to the mechanism described in the previous section, and break up into two droplets. Assuming that the probability of being chosen for collision is equal for all droplets, this random selection and collision process is repeated until the conversion is complete. Because two droplets merge and break into two in each collision, the total number of droplets (i.e., NA + NB) is preserved in the simulation.
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Because the three numbers attached to each droplet (i.e., the concentrations of A and B, and the particle volume) are monitored, the conversion of the reaction as well as the particle size distribution can be calculated at any stage of the collision/reaction process. Obviously, the number of droplets NA and NB should be big enough for the simulation to be statistically valid. The minimum number of droplets required for a consistent result can be obtained by comparing the particle size distribution at any value of conversion for various values of NA and NB. Once NA and NB are bigger than a certain value, further increase in NA and NB does not change the particle size distribution obtained by the simulation. It has been found that 30 000 for NA and NB is sufficiently large for consistent results. In our simulation, both NA and NB were much greater than 30 000. Figure 1 shows the particle size distribution at various values of conversion for the same value of NA and NB. The scaled size for the abscissa is the particle radius (or diameter) normalized by the size of the particle formed by the collision of two fresh reacting droplets (i.e., eq 1). The distribution density for the ordinate is calculated based on the volume and number fractions of the particles, respectively. Because the reaction process through collision is discrete, the particle size distribution remains rather discrete even when the conversion is fairly high. This discrete reaction process explains the spiky nature of the particle size distribution even when both NA and NB are very large. When the conversion is low (Figure 1a), the particle size distribution is nearly monodispersed. As the conversion increases both bigger and smaller particles are formed, and the particle size distribution becomes broader and smoother (or less spiky). At a later stage of reaction, small particle generation appears to be more significant than large particle formation. Thus, the small particle fraction increases more rapidly, which results in the existence of a broad hump on the small particle side of the particle size distribution (e.g., the dotted line of Figure 1d). Although a large number of small particles are produced, their volume fraction is not very significant. Shown in Figure 2 are the average particle size and the variance of the particle size distribution at various stages of reaction. When the average particle size is calculated based on the volume fraction, it increases only moderately with the conversion, whereas the number-average particle size decreases rather rapidly indicating the significance of fine particle generation at a later stage of reaction. Although the population (or the number) of small particles is rather high, their volume fraction is not very significant and the particle size distribution based on the volume fraction is rather narrow with a small variance. The particle size distribution broadens because of the increased number of noneffective collisions with increasing conversion which simply redistribute unreacted reactants in the droplets and eventually contribute to the particle growth and the generation of small particles. Thus, the particle size distribution will become narrower, if one of the reactants is quickly depleted before the number of noneffective collisions becomes significant. This can be achieved by making NA much bigger or much smaller than NB. In Figure 3, the particle size distributions when NA/ NB is 1.5 and 4 are given. The particle size distribution becomes narrower with increasing NA/NB. When NA/NB ) 4 the particle size distribution is nearly monodispersed with only a very small fraction of small and large particles existing. Although there have been numerous publications on the synthesis of nanoparticles using microemulsions, little information is available regarding the particle size
Li and Park
Figure 1. Particle size distribution at various conversions: (NA ) NB ) 50 000) 27.9%, (b) 71.7%, (c) 85.7%, (d) 98%. The solid and dotted lines for the particle size distribution density are based on volume and number fractions, respectively.
distribution. Thus, detailed comparison between the experimental results and the present simulation is difficult with the exception of a few cases. In the experiment of
Particle Size Distribution Using Microemulsions
Figure 2. (a) Average particle size and (b) variance of the particle size distribution. The solid and dotted lines for the particle size and variance are based on volume and number fractions, respectively.
Pillai,21 Fe(OH)3 particles were synthesized with water in octane microemulsions with cetyltrimethylammonium bromide and 1-butanol, in which the two reactants Fe(NO3)3 and NH3H2O were dissolved in water. The average droplet size of the microemulsion measured by the quasielastic light-scattering (QELS) method was reported to be 30 nm. Because the concentration of Fe(NO3)3 in the droplets was 0.05 M, the particle size corresponding to the scaled size of 1 was estimated to be about 3.5 nm. He reported that most of the particles ranged from 3 to 7 nm, which appears to be in reasonable agreement with the prediction of the present study if fine particles are neglected. Many experimental results have shown a particle size distribution which is a skewed bell shape with a small hump below the most probable size.22 This observation is again consistent with the prediction of the present simulation. In experiments, the efficiency of the particle recovery increases with the particle size and it is difficult to separate the particles smaller than a few nanometers from the liquid phases. Thus, the reports of experimental studies may tend to neglect the presence of particles smaller than a few nanometers. The present study does not provide any information about the time rate of the particle formation reaction. However, if the estimates of the number of droplet collisions per unit volume and the collision time are (21) Pillai, V. K. Ph.D. Dissertation, University of Florida, 1995. (22) Tanori, J.; Pileni, M. P. Langmuir 1997, 13, 639.
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Figure 3. Dependence of particle size distribution on the number ratio of reactant droplets. (a) NA/NB ) 3/2; (b) NA/NB ) 4/1.
available, the conversion versus time relation can be obtained by the simulation. The droplet collision frequency depends on numerous factors such as the size and the number concentration of the droplets, liquid viscosity, and the interaction potential between droplets. Once this information is available, the collision frequency may be estimated by a kinetic theory,23,24 although a major difficulty still exists in obtaining the interaction potential between droplets. 4. Summary When two microemulsions each containing reactants are brought together, continuous collisions of the droplets leads to a mixing of the droplet contents. Consequently, a chemical reaction occurs forming a solid product within the microemulsion droplets. If the mixing and reaction time is much lesser than the diffusion time of the droplets, the particle formation is limited by the droplet collision and a simple numerical scheme can be applied to estimate the particle size distribution. The present simulation, which assumes that the particle formation is limited by the droplet collision, predicts the generation of a significant number of small particles making a broad hump in the particle size distribution. However, the volume fraction of the fine particles is insignificant, and the particle size distribution can be considered to be a narrow one if the fine particles are neglected. This prediction appears to be (23) Debye, P. Trans. Electrochem. Soc. 1942, 82, 265. (24) Jordan, P. C. Chemical Kinetics and Transport; Plenum Press: New York, 1979.
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consistent with experimental observations although only a limited number of experimental data is available. Acknowledgment. The authors acknowledge the financial support of the Engineering Research Center (ERC) for Particle Science and Technology at the Uni-
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versity of Florida, the National Science Foundation grant EEC-94-02989, and the Industrial Partners of the ERC. LA980550Z
