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Coagulation of Nanoparticles in Reverse Micellar Systems: A Monte Carlo Model Ravi Jain, Diwakar Shukla, and Anurag Mehra* Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400076, India Received August 25, 2005
The process of formation of nanoparticles obtained by mixing two micellized, aqueous solutions has been simulated using the Monte Carlo technique. The model includes the phenomena of finite nucleation, growth via intermicellar exchange, and coagulation of nanoparticles after their formation. Using the model, an exploratory study has been conducted to analyze whether the coagulation of nanoparticles is the reason for the formation of nanoparticles whose sizes are comparable to the size of the reverse micelles. The model explains the possible mechanism of coagulation of semiconductor nanoparticles formed within reverse micelles and its effect on the evolution of their size with time. The model is predictive in nature, and the simulation results compare well with those observed experimentally.
Introduction Nanoparticles lie between bulk and atomic dimensions and are therefore endowed with special properties, thereby making them starting material for many futuristic applications. The properties of nanoparticles depend on the crystallite sizes;1 therefore, control over the particle sizes and particle size distributions (PSDs) is highly desirable. Reverse micelles or water-in-oil microemulsions have been used over the years as a medium for the synthesis of nanoparticles because the individual micelles act as nanoreactors/precipitators. The sizes of the precipitated particles are profoundly affected by the inherent physicochemical properties of the micellar system as well as by the exchange of material between micelles. Thus, by manipulating various parameters of the reverse micellar system the size of nanoparticles can be controlled. The microemulsion mixing method involves the contacting of premicellized reactants present in two separate microemulsions. Intermicellar collisions are responsible for the occurrence of reaction and the eventual formation of particles after nucleation. A number of experimental studies have been conducted using a reverse micellar system to precipitate nanoparticles. Copper nanoparticles,2 silver and silver halides,2-4 CdS, ZnS, PbS, and Ag2S,5-8 and metal oxides such as ZnO and TiO29,10 have been prepared by such means. These studies report the effects of the water-to-surfactant molar ratio and reactant concentrations on particle sizes (and sometimes the PSDs). * Author to whom correspondences should be addressed. Email:
[email protected]. (1) Gleiter, H. Acta Mater. 2000, 48, 1. (2) Pileni, M. P. Langmuir 1997, 13, 3266. (3) Manna, A.; Imae, T.; Iida, M.; Hisamatsu, N. Langmuir 2001, 17, 6000. (4) Bagwe, R. P.; Khilar, K. C. Langmuir 1997, 13, 6432. (5) Khiew, P. S.; Radiman, S.; Huang, N.; Ahmad, M. S. J. Cryst. Growth 2003, 254, 235. (6) Hirai, T.; Sato, H.; Komasawa, I. Ind. Eng. Chem. Res. 1994, 33, 3262. (7) Suzuki, K.; Harada, M.; Shioi, A. J. Chem. Eng. Jpn. 1996, 29, 264. (8) Lianos, P.; Thomas, J. K. J. Colloid Interface Sci. 1987, 117, 505. (9) Wang, G. H.; Li, G. L. Nanostruct. Mater. 1999, 11, 663. (10) Hingorani, S.; Pillai, V.; Kumar, P.; Multani, M.; Shah, D. O. Mater. Res. Bull. 1993, 28, 1303.
Bagwe and Khilar have specifically studied the effect of exchange rates on the size distribution of the particles.4,11 These studies have led to speculation about the possible mechanisms involved in the formation of nanoparticles. The growth mechanism of CdS ultrafine particles in water-in-oil microemulsions has been studied by Towey et al.12 and Suzuki et al.7,13 Towey et al.12 used UV-vis spectrophotometric detection to show that in the initial period, after the mixing of reactants, the rate-determining step of the particle growth process is the coalescence between CdS-containing microemulsion droplets. Hirai et al.6 have confirmed Towey’s mechanism in which the CdS particles grow by the coagulation of the colliding particles. Suzuki et al.7 and Sato et al.14 studied the growth in later stages of mixing and concluded that the rate of coagulation decreases as the coagulating particle size decreases. Thus, the experimental work done clearly establishes the coagulation process as an important growth mechanism for nanoparticles prepared using microemulsions. Some amount of modeling work has been done and is reported in the literature. Hatton et al.15 proposed a population balance approach to model the population dynamics of reverse micellar precipitation systems. A limitation of this study was that it did not incorporate the phenomena of finite nucleation. Natarajan et al.16 also used population balances and assumed the initial reactant distribution to be Poissonian and the exchange protocol to be cooperative. These investigators added nucleation as a separate event, although the rate was assumed to be instantaneous. Ramesh et al.17 have recently proposed a population-balance-based model for predicting the particle (11) Bagwe, R. P.; Khilar, K. C. Langmuir 2000, 16, 905. (12) Towey, T. F.; Khan-Lodhi, A.; Robinson, B. H. J. Chem. Soc., Faraday Trans. 1990, 86, 3757. (13) Suzuki, K.; Mizutani, N.; Harada, M. J. Chem. Eng. Jpn. 1999, 32, 31. (14) Sato, H.; Nofumi, A.; Komasawa, I. Ind. Eng. Chem. Res. 2000, 39, 328. (15) Hatton, A. T.; Bommarius, A. S.; Holzwarth, J. F. Langmuir 1993, 9, 1241. (16) Natarajan, U.; Handique, K.; Mehra, A.; Bellare, J. R.; Khilar, K. C. Langmuir 1996, 12, 2670. (17) Ramesh, A. K.; Hota, G.; Mehra, A.; Khilar, K. C. AIChE J. 2004.
10.1021/la0523208 CCC: $30.25 © 2005 American Chemical Society Published on Web 10/22/2005
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size distribution of nanoparticles and have assumed a finite rate of nucleation. Li and Park carried out Monte Carlo simulations to construct particle size distributions for the synthesis of nanoparticles using reverse micelles.18 They assumed instantaneous nucleation and reaction in their model. They also assumed cooperative exchange of micellar contents after the fusion of two colliding micelles. Tojo et al.19 carried out Monte Carlo simulations using the random walk approach. They have shown the influence of autocatalysis and ripening on the final size distribution of the nanoparticles formed, using film flexibility as a parameter. Bandyopadhyaya et al.20 removed some of the aforementioned limitations by incorporating a finite nucleation rate in the Monte Carlo models. Sato et al.14 have proposed a kinetic model for the coagulation process on the basis of the statistical distribution of particles among the micellar droplets. The model was found to be applicable to PbS, silver halides,21 and coprecipitaed CdSZnS particles.22 However, the model is semiempirical in nature and required the estimation of model parameters by fitting the calculated results, for the overall coagulation rate constant, to the experimental results. The model also underestimates the mean particle size at high water content. The aforementioned models have been able to capture the experimentally observed trends qualitatively but have not been able to predict the sizes and particle size distribution (PSD) of nanoparticles accurately. The typical sizes of nanoparticles reported are higher than the predictions made by the models of Ramesh et al.17 and Natarajan et al.16 One of the reasons for the underprediction of such models is that they do not incorporate the coagulation of nanoparticles just after their formation. Ramesh et al.17 have demonstrated that the typical average size of nanoparticles reported by different authors should be thousand of molecules. The authors have argued that such a large number of molecules in a particle cannot be predicted by models that do not incorporate phenomena other than coalescence-decoalescence, nucleation, and collisional growth of nanoparticles. Pileni has reported that sizes of nanoparticles vary with the produced materials.2 Semiconductor nanoparticles have sizes varying from 1 to 4 nm, silver nanoparticles attain sizes up to 7 nm, and copper nanoparticles attain sizes of up to 12 nm for the same micellar system. The CdS nanoparticles sizes reported by various contributors have a size range of 2-7 nm.5,6,17 The typical micellar size in these systems is also in the range of 2-7 nm. Thus, the size of CdS and other semiconductor nanoparticles seems to be limited by the micellar core size. Silver halide nanoparticles prepared by various authors have sizes that are several times the size of the reverse micelles.4 Thus the capability of reverse micelles to limit the growth of nanoparticles varies from substance to substance, and a general model that predicts the PSD of nanoparticles should not be possible. Keeping in mind the limitations of the models available in the literature to predict nanoparticle size distributions and a need to explore the coagulation mechanism in detail, the present model attempts to study the mechanism of the formation of nanoparticles and their subsequent (18) Li, Y.; Park, C. W. Langmuir 1999, 15, 952. (19) Tojo, C.; Blanco, M. C.; Lopez-Quintela, M. A. Langmuir 1997, 13, 4527. (20) Bandyopadhyaya, R.; Kumar, R.; Gandhi, K. S.; Ramakrishna, D. Langmuir 2000, 16, 7139. (21) Sato, H.; Hirai, T.; Komasawa, I. J. Chem. Eng. Jpn. 1996, 29, 501. (22) Sato, H.; Hirai, T.; Komasawa, I. Ind. Eng. Chem. Res. 1995, 34, 2493.
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coagulation. The model has been used to predict the time evolution of nanoparticles. The investigations in the present study are done for CdS nanoparticles because there is strong experimental evidence of their coagulation and data required for simulation are available. Though the predictions using the model are done for CdS nanoparticles, the approach and framework used for describing the coagulation process can be applied to other semiconductor particles with adjustments in the values of materialdependent simulation parameters. (e.g., β, n*, A, Ksp; see below for more details.) To the best of our knowledge, this is the first nonempirical model proposed for the coagulation process. Model Formulation The system considered here is the reaction between A and B that produces C. The reactants are solubilized in two separate microemulsions that are then mixed together. Micelles are in a state of random Brownian motion, which leads to collisions between them. Only some collisions, which occur between sufficiently energetic micelles, result in the formation of a dimer. Reaction occurs when micelles containing A and B undergo such fusion, and this is followed by fission of the fused mass back to two micelles. instantaneous
finite nucleation rate
A(l) + B(l) 98 C(l) 98 C(s) V (1) Assumptions. The following assumptions have been made in developing the model. (1) The initial reactant distribution in the micelles is assumed to be known. A Poisson distribution of the reactant has been assumed. This is because of the low mean occupancies of reactants in micelles and the rapid rate of collision of micelles causing a redispersion of their contents. Low reactant concentrations result in fractional mean occupancies of micelles, leading to a situation where many micelles are empty. The distribution of reactant molecules in the reverse micelles is given by
pk )
kkave-kavg k!
(2)
where pk is the probability of a micelle containing k molecules of the reactant and kav is the average number of molecules in a single micelle. (2) The collisions are assumed to occur only because of the Brownian motion of the micelles, and only binary collisions are important. Also, it is assumed that the amount of material in the micelle does not affect its velocity (i.e., a heavily packed micelle is as “mobile” as an empty micelle). (3) Nucleation and intermicellar exchange proceed at a finite rate. However, the reaction is assumed to be instantaneous. (4) Particle growth in a micelle is assumed to be instantaneous and may occur via two mechanisms. One of the ways in which a particle can grow is by picking up available molecules of product C. The other way in which a particle can grow is by coagulation with other particles. Hence, if two micelles containing a particle each coalesce, then the particles coagulate to form a single large nanoparticle that is contained in one of the daughter micelle formed after decoalescence of the dimer, thereby producing an empty micelle. This has been schematically represented in Figure 1.
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Figure 1. Growth of nanoparticles via two routes: (a) growth of particles by picking liquid C molecules and (b) coagulation of two nanoparticles. The pink matter represents C molecules in the liquid phase, and the violet matter represent nucleated nanoparticles.
(5) Nucleation occurs when the number of product molecules (C) in a micelle exceeds the critical nucleation number, kc. Furthermore, the number of nucleation events in a micelle is limited to one, and the transfer of a nucleus to another micelle containing a nucleus (during the fusion process) does not occur. Therefore, one micelle can contain only one particle. (6) A random exchange style has been assumed. A random redistribution or exchange style implies that the total number of C molecules in the dimer formed is randomly distributed among the two micelles that are formed after fission of the dimer. This supposition, in turn, is based on the assumption that there is no significant force of attraction or repulsion among the micellar contents. (7) The particles are assumed to be spherical in shape. Khiew et al.5 have reported that CdS and PbS nanoparticles have spherical morphologies based on the study of the TEM images of these nanoparticles. Fusion-Fission Rules. (1) The fusion between two micelles, containing different reactants in each, results in complete mixing of the solutes followed by the redistribution of the contents according to the chosen exchange protocol. (2) The fusion between a nucleated and a nonnucleated but nonempty micelle results in the transfer of the entire
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amount of solute C into the micelle containing the particle, leading to growth of the existing particle and the formation of an empty micelle. (3) The autocatalysis effect has been considered for collisions involving nucleated micelles. This effect states that the reaction and therefore the product formation occur preferentially in the micelle containing a particle or, if both contain particles, then in the micelle that contains the larger particle. This effect assumes that the reaction is likely to occur on the surface of the particle. (4) The fusion between two micelles containing particles leads to coagulation of the two nanoparticles to form a large nanoparticle contained in one of the daughter micelles and an empty micelle. Coagulation of Nanoparticles. As has been outlined in the preceding section, the coagulation of nanoparticles has been incorporated into the model. The coagulation of nanoparticles is driven by intermicellar collisions. Hence, the coagulation of nanoparticles should go on indefinitely untill only one large particle is left in the entire population of micelles. However, this has not been observed in experimental studies. In most of the studies related to the growth of CdS nanoparticles, the particles grow to a maximum size corresponding to that of the micellar core.6,7,14 Hence, in addition to the rules outlined above, there must be a constraint on the sizes that nanoparticles could attain. Also, it is desirable to understand the mechanism of coagulation in order to predict the size distribution of nanoparticles accurately. The coagulation of nanoparticles could be determined by the degree of “dryness” or “wetness” of the particles. If the particles are dry, it would imply that the collision between them would not lead to a large particle because the colliding particles would not be able to stick to each other after the collision. If the particles are wet, they can easily stick to each other after colliding. The degree of dryness and wetness should be a time-dependent property (i.e., a freshly nucleated particle would be wetter than another particle that had been made some time ago). Although this chain of argument seems to have a physical basis, it would be very difficult to quantify the degree of dryness or wetness of particles. Also, there could be complexities such as in the situation when dry and wet particles collide with each other. Hence, a model based on these lines would be a complex one. Many experimental studies have demonstrated that the water-to-surfactant molar ratio (R) determines the size of the nanoparticles formed.2,4,17 The value of R determines the size of the micelles, which in turn determines the size of the nanoparticles formed. This points toward the possibility that micellar size could limit the growth of CdS nanoparticles. However, the micelles act as nanoreactors and aid in dividing the reactants among themselves in very minute amounts so that eventually a large number of nanosized particles are formed. The micellar structure is not rigid and stable but is in a state of continuous change. Theoretically, it cannot stop a nanoparticle from growing, and hence the particle should eventually be able to become larger than the reverse micelle itself. However, it has been seen that the average nanoparticle size becomes constant after some time. Various theories have been proposed for a constraint on the growth of nanoparticles. Israelachvili has proposed an explanation for the constrained growth of particles due to coagulation.23 According to the author, an exponentially repulsive hydration force of decay length of less (23) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1985.
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micelles in the population being considered, Nmic is the number density of micelles, and q is the Brownian collision rate. For equal-sized entities, q has been derived by Smoluchowski.26
q)
Figure 2. Schematic illustration of the origins of restrictions of nanoparticle growth. (a) Micelles that are fully occupied cannot fuse with another micelle. (b) When the coagulated particle size is larger than the size of the micelle, the particles cannot coagulate.
than 1 nm is observed between hydrophilic surfaces when in neutral or alkaline concentrated electrolyte solutions. Another possible explanation is that when the nanoparticle size approaches the size of the micelle they might get stabilized because of the absorption of the surfactant molecules on the particles. This has been experimentally illustrated by Kandori et al.24 for the case of CaCO3 nanoparticles. The AOT reverse micelles stabilize colloids in organic solvents,25 although their capability to limit growth could vary from substance to substance. Experimental data shows that the size range of CdS nanoparticles is the same as the size range of micelle sizes for various R values. This indicates that in the case of CdS nanoparticles the AOT molecules stabilize the particles effectively and prevent further growth. Hence, if a nanoparticle has reached the micellar core size, it is “shielded” from other nanoparticles because of the absorption of the surfactant molecules onto it, rendering it ineffective for further coagulation. When two micelles containing CdS nanoparticles of sizes such that their coagulation would result in the formation of a particle that is larger than the core size collide, then coagulation is not allowed to take place. Coagulation in this case would not be possible because both particles would not be able to come together on one side of the dimer before its fission takes place. Keeping these in mind, a constraint on the growth of nanoparticles via coagulation has been put forth: (1) As soon as a micelle is fully occupied, it becomes ineffective and hence is unavailable for further growth. (2) If two micelles containing particles of X and Y molecules respectively fuse together and X + Y is more than the number of molecules required to fill a micelle fully, then their coagulation is not allowed, and the fusionfission becomes redundant. This restriction provides a logical way to end the simulations. The restriction mechanisms for particle coagulation are depicted in Figure 2. There are only two basic events in the model, namely, fusion-fission of micelles and nucleation. Coalescence. The frequency of coalescence is given by
fc )
(21)βqN
micN
(3)
where β is the coalescence efficiency (i.e., the fraction of collisions between micelles that actually results in the exchange of micellar contents), N is the number of (24) Kandori, F.; Konno, K.; Kithara, A. J. Colloid Interface Sci. 1998, 122, 78. (25) Cooper, W. D.; Wright, P. J. Chem. Soc., Faraday Trans. 1 1974, 70, 858.
8kbT 3η
(4)
where kb is the Boltzmann’s constant and η is the viscosity of the medium. Nucleation. Homogeneous nucleation has been assumed for determining the nucleation rate in a micelle because the size of the micelles is very small. The nucleation rate, ki, in a nonnucleated micelle containing i product species (i > n*) is given by27,28
{
0
ki )
[
iA exp -
16πσ3V2met
]
3(kBT)2(ln λ)2
if i < n* if i g n*
}
(5)
where A is the preexponential factor, σ is the interfacial tension between the particle and water, Vm is the volume of one metal atom, λ is the computed supersaturation based on the solubility product of the precipitated species and number of such molecules in the micelle, and n* is the critical nucleation number. The overall nucleation frequency is given by the sum of the nucleation rates in all of the micelles as N
fn )
kn(i) ∑ i)1
(6)
where kn(i) denotes the nucleation rate in the ith micelle. Choice of Events. The Monte Carlo methodology makes use of random numbers coupled with the probabilities of events in order to make a choice as to whether an event is to be executed. For the system under consideration, first the probabilities of all events are calculated as follows
f t ) fc + fn
(7)
pc )
fc ft
(8)
pn )
fn ft
(9)
where fc and fn are the frequencies of coalescence and nucleation and pc and pn are their probabilities, respectively. A probabilistically random choice involves the generation of a random number and determining whether it falls within the limits set for the probability of a specific event to occur. The most probable event has the largest range of values that the random number can fall into (for the event to be executed). In this manner, the randomness of the system is retained in the model, and the most probable event still has the highest probability of being chosen. Choice of Micelles. After the event has been chosen, the micelles that have to undergo that event are to be (26) Smoluchowski, M. V. Z. Phys. 1916, 17, 585. (27) Banyopadhyaya, R.; Kumar, R.; Gandhi, K. S.; Ramakrishna, D. Langmuir 1997, 13, 3610. (28) Adamson, A. Physical Chemistry of Surfaces; John Wiley and Sons: New York, 1990.
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selected. In the case of coalescence, two random integers between 1 and N are generated, corresponding to each micelle. If the values of both of the random numbers lie in the specified range for collision to occur, then the micellar contents are redistributed according to the chosen protocol. The choice of whether a micelle will undergo nucleation is made by generating a random number and comparing it with the probabilities of nucleation for each micelle. Hence, the following method is used to choose the micelle. j
pjn )
kn(i) ∑ i)1 fn
(10)
If a generated random number between 0 and 1 lies between pjn and pj+1 n , then the jth micelle is chosen for nucleation. To evaluate the time elapsed between two events, the concept of the quiescence interval is used.29 This method ensures that the mean rate of the process is conserved while events occur randomly. The quiescence time interval is calculated using the equation
ln(1 - u1) TQ ) ft
(11)
where u1 is a random number between 0 and 1. Results and Discussion The nanoparticles that are formed grow by two different mechanisms (viz., the growth of particles by picking nonnucleated product molecules and the growth due to the coagulation of nanoparticles). The initial growth of nanoparticles is primarily due to picking up nonnucleated product molecules. This is because the probability of collision between two micelles, each containing a particle, is low. When all of the reactant has been exhausted and all C(l) molecules have been consumed to form particles, coagulation is the only mechanism that allows the particles to grow further. The simulation has been carried out for a large population size of 108 micelles to minimize the statistical error. Simulation results for the time evolution of the average size of nanoparticles has been shown in Figure 2 and has been discussed in the following subsection. Time Evolution of the Average Size of Nanoparticles. Figure 3 shows the growth of the average size of CdS nanoparticles with time and its comparison with the results obtained experimentally.6 The parameters used for the simulation are listed in Table 1. The simulations have been performed for a water-to-surfactant molar ratio (R) of 5. The mean occupancy of reactants in the micelles has been calculated using the experimental description of Hirai et al.6 The value of the collision efficiency (β) has been calculated on the basis of the experimental observation by Hirai et al.6 that the intermicellar exchange rate is 5.6 × 107 M-1 s-1. The time for the exhaustion of reactants in the system and 100% conversion to solid nanoparticles is only about 0.01 s. The mean aggregation number (MAN) of molecules in the nanoparticles has been calculated to be 7. This translates to a nanoparticle size of only 1 nm. Thereafter, nanoparticles grow because of coagulation with other nanoparticles. (29) Shah, B. H.; Ramakrishna, D.; Borwanker, J. D. AIChE J. 1977, 23, 897.
Figure 3. Time evolution of the average size of CdS nanoparticles. Comparison of experimental data6 and model predictions. Table 1. Parameters Used for Generating the PSD of CdS Nanoparticles for the Experimental Setup of Hirai et al.6 parameter
value
source
β θ η B A dm Nmic Ksp Vmic n* R
0.01 1.097 × 10-17 0.001 kg m-1 s-1 600 278.42 s-1 5.45 nm 5.58 × 1023 m-3 3.6 × 10-23 mol2 m-6 4 × 10-26 m3 2 6
Bandyopadhyaya et al. (2000) Smoluchowski (1916) Bandyopadhyaya et al. (2000) Bandyopadhyaya et al. (2000) Bandyopadhyaya et al. (2000) Hirai et al. (1994) Hirai et al. (1994) Bandyopadhyaya et al. (2000) Lianos and Thomas (1987) Bandyopadhyaya et al. (2000) Hirai et al. (1994)
Coagulation Kinetics. The coagulation of nanoparticles proceeds rapidly just after their formation as is clear from Figure 3. After approximately 1 s, the rate of growth of the average particle size decreases drastically, and eventually the average particle diameter plateaus. The initial phase of rapid growth has been termed the Smoluchowski growth by Suzuki et al.7 Figure 3 illustrates the comparison with the results obtained by Hirai et al.6 and the computed results. The trend obtained for the increase in average particle diameter is qualitatively similar to that obtained by the model developed. The particle diameters predicted by the model are also quantitatively close to the experimental data reported but are slightly higher. The final average particle size as measured by Hirai et al.6 is approximately 3.5 nm, whereas the model prediction is about 4.2 nm. More complex models that incorporate size-dependent collision frequencies and areas can possibly improve the agreement between model and experimental results. Monte Carlo models without the coagulation of nanoparticles20 underpredict the mean particle sizes. The particle diameter predicted by the model without coagulation is about 1.2 nm, which is much lower than the experimental result. Therefore, the present Monte Carlo model with the coagulation process explains the large sizes obtained through the microemulsion-mediated synthesis of nanoparticles. There are several factors causing the decrease in the rate of growth of the particle size with time. First, because rapid coagulation takes place just after the formation of nanoparticles, the rate of production of empty micelles is also very high. Hence, as coagulation progresses, the population of empty micelles also increases, thereby causing a reduction in the probability of collision between
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(2) The variation of average nanoparticle size with time as predicted by the model has been compared with the experimental values reported in the literature. The trend and average size predictions are in close agreement with the reported values. (3) The close agreement in the nature of the graph for coagulation kinetics leads to the deduction that the growth of nanoparticles in reverse micellar systems due to coagulation is limited by the absorption of surfactant molecules, when the nanoparticle size becomes the same as the micelle containing it. This puts a limit on the final average nanoparticle size obtained. Notation
Figure 4. Time evolution of the average size of CdS nanoparticles. Comparison with and without the restriction imposed by the absorption of surfactant molecules on the nanoparticle when the nanoparticle becomes as large as the micelle that contains it.
two micelles that have particles. Hirai et al.6 have experimentally shown that after 100 s only a few particlecontaining micelles are left in a population of 1017 micelles. This reduction in the number of particle-containing micelles leads to a reduced rate of growth of the average particle size. Second, we consider the absorption of surfactant molecules onto the surface of the nanoparticles as they approach the size of the micelle containing them. As more and more micelles get fully packed with particle molecules, many collisions become ineffective because the nanoparticles are “shielded” by the surfactant molecules. These factors leading to a decreased rate of increase in the average particle size have been compared in Figure 4. Without putting a constraint on the sizes that nanoparticles can attain, we see that the rate of increase in the average particle size decreases. This is because of the reduced probability of a collision between particlecontaining micelles with time. Conclusions (1) A model has been developed on the basis of the Monte Carlo methodology to predict the average size of nanoparticles synthesized via the two-microemulsion route. The model includes the phenomena of finite nucleation and coagulation of nanoparticles after their formation.
A ) preexponential factor of nucleation rate in micelles, s-1 B ) exponential term in nucleation rate expression E[A] ) Average number of A molecules in a micelle E[B] ) Average number of B molecules in a micelle fc ) total fission-fusion or coalescence frequency of all micelles, s-1 fn ) total nucleation frequency of all micelles, s-1 fr ) total reaction frequency of all micelles, s-1 ft ) total frequency of all events, s-1 Ksp ) solubility product of the precipitate, mol2 m-6 kav ) average occupancy in the reverse micellar system kn ) nucleation rate constant, s-1 MAN ) mean aggregation number NA ) Avogadro’s number ) 6.023 × 1023, mol-1 Nmic ) number density of micelles, m-3 n* ) critical nucleation number p ) probability of an event in a micelle qm ) Brownian collision frequency between micelles, m3 s-1 R ) water-to-surfactant molar ratio T ) temperature on the absolute scale, K TQ ) quiescence time interval, s Vm ) volume of a precipitate molecule, m3 Vmic ) volume of the micellar core, m3 Greek Letters β ) coalescence efficiency η ) viscosity of the continuous phase, Pa s λ ) supersaturation in a micelle F ) density of the precipitate, kg m-3 σ ) interfacial tension between nucleus and micellar core liquid, Pa m-1 LA0523208
