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Kinetics of the Formation of Particles in Microemulsions C. Tojo,*,† M. C. Blanco,‡ F. Rivadulla,‡ and M. A. Lo´pez-Quintela‡ Physical Chemistry Department, Faculty of Chemistry, University of Santiago de Compostela, E-15706 Santiago, Spain, and Physical Chemistry Department, Faculty of Sciences, University of Vigo, E-36200 Vigo, Spain Received August 8, 1996. In Final Form: January 9, 1997X In this paper the kinetics of the formation of nanoparticles in microemulsions is studied by Monte Carlo computer simulation. It is observed that compartmentalization of the reactants induces a separation of the nucleation and growth processes, which is more clearly observed when the concentration of reactants is relatively high. It is also observed that growth by autocatalysis and by ripening overlaps at low concentrations but occurs on different time scales at high concentration. A comparison between the results of these simulations and those from Smoluchowski’s rapid-coagulation mechanism is made, showing that this theory is adequate to model the kinetics of these reactions.
Introduction Microemulsions are currently being explored for making ultrafine particles of a variety of materials.1-18 The small size in these materials confers new properties and opens up possibilities of novel applications.1 The principal reason for using microemulsions in the preparation of ultrafine materials is that this is a powerful method for controlling particle sizes and the corresponding size distribution.19 A number of experimental investigative efforts for producing colloidal particles using this technique have been undertaken in the last 10 years.2-20 In spite of the advantages offered by the microemulsion technique, the results vary depending on the reaction conditions employed. Of course, a special requirement for a specific synthesis route is to have a good knowledge of the * To whom correspondence should be addressed. † University of Vigo. ‡ University of Santiago de Compostela. X Abstract published in Advance ACS Abstracts, March 1, 1997. (1) Fendler, J. H. Chem. Rev. 1987, 87, 877. (2) Pileni, M. P. J. Chem. Phys. 1993, 97, 6971. (3) Boutonnet, M.; Kizling, J.; Stenius P.; Maire, G. Colloids Surf. 1982, 5, 209. (4) Gobe, M.; Kon-no, K.; Kandori, K.; Kitahara, A. J. Colloid Interface Sci. 1983, 93, 293. (5) Kurihara, K.; Kizling, J.; Stenius, P.; Fendler, J. H. J. Am. Chem. Soc. 1983, 105, 2574. (6) Lo´pez-Quintela, M. A.; Rivas, J. J. Colloid Interface Sci. 1993, 158, 446. (7) Kandori, K.; Kon-No, K.; Kitahara, A. J. Colloid Interface Sci. 1988, 122, 78. (8) Lufimpadio, N.; Nagy, J. B.; Derouane, E. G. In Surfactants in Solution; Mittal, D.; Lindman, B., Eds.; Plenum Press: New York, 1983; Vol. 3, p 1483. (9) Lianos, P.; Thomas, J. K. Chem. Phys. Lett. 1986, 125, 299. (10) Petit, C.; Lixon, P.; Pileni, M. P. J. Phys. Chem. 1990, 94, 1598. (11) Petit, C.; Lixon, P.; Pileni, M. P. J. Phys. Chem. 1993, 97, 12974. (12) Motte, L.; Petit, C.; Lixon, P.; Boulanger, L.; Pileni, M. P. Langmuir 1992, 8, 1049. (13) Lo´pez-Quintela, M. A.; Quibe´n, J.; Rivas, J. In Industrial Applications of Microemulsions; Solans, C., Kunieda, H., Eds.; Surfactant Science Series; Marcel Dekker: New York, 1996; pp 247-264. (14) Towey, T. F.; Khan-Lodhi, A.; Robinson, B. H. J. Chem. Soc., Faraday Trans. 1990, 86, 3757. (15) Barnickel, P.; Wokaun, A.; Sager, W.; Eicke, H. F. J. Colloid Interface Sci. 1992, 148, 80. (16) Motte, L.; Lebrun, A.; Pileni, M. P. Prog. Colloid Polim. Sci. 1992, 89, 99. (17) Arriagada, F. J.; Osseo-Asare, K. Colloids Surf. 1992, 69, 105. (18) Khan-Lodhi, A.; Robinson, B. H.; Towey, T.; Hermann, C.; Knoche, W.; Thesing, U. In The Structure, Dynamics and Equilibrium Properties of Colloidal Systems; Bloor, D. M., Wyn-Jones, E., Eds.; NATO ASI Series C; Kluwer Academic Publishers: Dordrecht, 1990; Vol. 324, p 373. (19) Lo´pez-Quintela, M. A.; Rivas, J. Curr. Opin.Colloid Interface Sci. 1996, 1, 806. (20) Lo´pez-Quintela, M. A.; Quibe´n, J.; Rivas, J. U.S. Patent 4,983,217, 1991, and E.C. Patent 370,939, 1993.
S0743-7463(96)00787-1 CCC: $14.00
technique so that reproducible results can be obtained. In order to gain an insight into this problem, we have carried out computer simulations on the formation of nanoparticles in microemulsions to elucidate the kinetics and mechanism of formation of these particles. This paper presents a Monte Carlo simulation study of this type of reaction showing that the compartmentalization of the reactants induces a separation of the nucleation and growth processes, which can be clearly observed when the concentration of reactants is relatively high. It is also observed that the growth by ripening and the growth by autocatalysis over an existing nucleus appear on different time scales at high concentrations. On the contrary, the time scales of both growth processes overlap when the concentration is low. Finally, a comparison between simulation results and the rapid-coagulation mechanism proposed by Smoluchowski11,14,21 is made, showing that this theory can be applied for this type of reactions at least for the experimental conditions studied here. Simulation Procedure The computer simulation of the formation of nanoparticles in microemulsions was done as follows: 1000 droplets are initially placed on a bidimensional lattice at random. A 10% portion of the space is occupied by droplets (volume fraction of droplets φ ) 10%). We use excluded volume principles; i.e., we do not allow more than one droplet to occupy a given site at any one time. For simplification, we consider only very fast chemical reactions and we assume that each collision between reactants gives rise to the formation of products. This is a valid approximation when the reaction is very quick in comparison with the interdroplet exchange rate. One half of the droplets carry c units of reactant A, and the other half carry c units of reactant B, i.e., [A] ) [B] ) c. The smallest value of c has been chosen big enough (c g 50 reactants per droplet) in order for the statistical fluctuations to be negligible. These should be taken into account by means of an appropriate statistical distribution like Poisson distribution. Reaction then proceeds allowing the droplets to perform random walks to nearest neighbor sites. Cyclic boundary conditions are enforced at the ends of the lattice. Our time unit base is 1 Monte Carlo step (mcs), which is defined as the time it takes for all droplets to move in one step into one of their nearest neighbors. All simulations were run for 10 000 mcs; in the event, all particle (21) Smoluchowski, M. Z. Phys. Chem. 1918, 92, 129.
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size distributions became stable after about 2000-8000 mcs, depending on the synthesis variables. Microemulsions are used as microreactors because they can exchange the content of their water pools by a collision process. The rate of communication between droplets is very fast,22 and it is assumed that this exchange can only take place when an energetic collision between two droplets is able to establish a water channel between the droplets,22 forming a “transient droplet dimer”.23 This process is energetically unfavorable because it implies a change in the curvature of the surfactant film.13,23 Therefore, the rate constant of this process is not diffusioncontrolled, and only a very small fraction of the total collisions (1 in 1000 for AOT surfactant films) between droplets leads to reactant exchange.22 In most cases droplet communication is the rate-determining step in particle formation.14 To take this into account in our simulation, we consider that when two droplets step onto contiguous sites they can establish a water channel forming a transient dimer.23 At this stage, water cores can interchange their contents. Depending on the nature of the species inside the droplet, the interchange criteria will be different: 1. The interchange of reactants in absence of nuclei leads to two possible situations: 1.1. If both droplets carry the same reactant, one atom from the droplet containing a higher number of reactants will go to the droplet containing a lower number of reactants (concentration gradient). 1.2. When the two droplets carry different reactants, the chemical reaction gives rise to the formation of the nuclei (A + B f P). Assuming again a concentration gradient, this first nucleus will locate inside the droplet carrying a lower number of reactants. 2. The interchange of reactants in the presence of nuclei presents a more complicated situation, because some reactions are catalyzed by the existing nuclei. For this kind of reaction, the exchange of reactants between two droplets containing nuclei and reactants will be in such a way that we favor the reaction on the existing nucleus (if only one of the two water pools carry a nucleus), or we favor the reaction on the bigger nucleus (if both water pools carry them). Therefore, we consider that a larger nucleus has a greater probability of playing as a catalyst because of its larger surface. This procedure may be called autocatalysis. 3. The interchange of nuclei becomes important as the reaction takes place, because more and more droplets carry P units of aggregations (nuclei). At this stage, collisions between two droplets both containing nuclei are the most probable. The interchange of nuclei and/or reactants during the same collision is allowed. To simulate the interchange of nuclei between droplets, some aspects have to be taken into account: 3.1. The interchange becomes more difficult as the nuclei grow, as was stated above. The flexibility of the surfactant film around the droplets governs the ease with which channels communicating colliding droplets can form and also the size of these channels. To study the influence of the film flexibility we have introduced the variable f, defined as the maximum size of the nuclei which can be interchanged: particles with more than f units were not allowed to transfer from one droplet to another. In this (22) Fletcher, P. D. I.; Robinson, B. H.; Bermejo-Barrera, F.; Oakenfull, D. G. In Microemulsions; Robb, I. D., Ed.; Plenum Press: New York, 1982; p 221. (23) Zana, R.; Lang, J. In Microemulsions; Structure and Dynamics; Frieberg, S. E., Bothorel, P., Eds.; CRC Press: Boca Raton, FL, 1987; p 153.
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way, a highly flexible surfactant film will allow the interchange of larger nuclei than a rigid surfactant film. 3.2. Another aspect to take into account is droplet size. As it is well-known, water is readily solubilized in the polar core, forming a so-called “water pool”, defined by the water-surfactant molar ratio. On the basis of simple geometrical arguments, the size of the droplet is expected to be proportional to the water-surfactant molar ratio. To take into account the influence of droplet size, we have introduced in the simulation a parameter q which restricts the maximum number of products P (and therefore the maximum particle size) which can be carried by a droplet. It should be noted that, as it was experimentally observed,8,24-27 the final size of the particles may be slightly bigger or smaller than the droplet size, depending on the flexibility of the surfactant film and/or the surfactant adsorption. In this paper, we restrict our study to the case of very big droplet sizes (q ) 5000 units per droplet), so that this variable does not affect our results. 3.3. It is well established that larger particles grow as smaller ones disappear. This process is known as Ostwald ripening. The mechanism of ripening relies on mass transport from small particles to larger particles. So, if one droplet containing a nucleus with an agglomeration number i (Pi) collides with another droplet containing a nucleus with higher agglomeration number (Pj), the smaller nucleus can be interchanged from the initial droplet to the droplet carrying the larger nucleus, giving rise to a bigger particle (Pi + Pj f Pi+j), provided the film flexibility and droplet size allow this interchange. This process can be called growth by ripening. As time goes some water pools will contain large particles, but other droplets will be empty. When one droplet becomes empty, it is removed from the system to get a less time-consuming run. We have monitored the evolution of particle size distribution as a function of time, for different values of the synthesis variables. In this study, we restrict our attention to autocatalytic reactions. Results and Discussion I. Study of the Nucleation and Growth Processes. Figure 1 shows the time evolution of particles for two different particle sizes. Different particle sizes mean different numbers of P units constituting a nucleus and correspond to different observation windows. Curves a in Figure 1 represent particles with a small number of units, which simulate the nuclei, and curves b correspond to particles with a large number of units, which represent the approximate final size of the particles. Curves a in Figure 1 clearly show the existence of two well-defined processes associated with the nucleation and growth of the particles. It is consistent with the mechanism proposed by Lo´pez-Quintela et al.6 deduced for Fe particles obtained in Aerosol OT (AOT) microemulsions using a stopped-flow technique and measuring the time-resolved small angle X-ray scattering (SAXS) with synchrotron radiation. Nucleation implies an increase in the number of scattering centers (number of particles) for a given observation window, and therefore, it gives an increase in the scattered intensity. On the contrary, the growth of particles is associated with a decrease of the scattered (24) Monnoyer, Ph.; Fonseca, A.; Nagy, J. B. Colloids Surf. 1995, 100, 233. (25) Ravet, I.; Lufimpadio, N.; Gourgue, A.; Nagy, J. B. Acta Chim. Hung. 1985, 119, 155. (26) Boutonnet, M.; Kizling, J.; Touronde, R.; Marie, G.; Stenius, P. Appl. Catal. 1986, 20, 163. (27) Nagy, J.; Gourgue, A.; Derouane, E. G. Stud. Surf. Sci. Catal. 1991, 16, 193.
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Figure 1. Time evolution of droplets carrying particles. A (c ) 500): Curve a represents the number of droplets containing particles of sizes between 100 and 150 P units. Curve b plots the number of droplets containing the biggest particles (sizes between 500 and 550 P units). B (c ) 100): Curve a represents the number of droplets containing particles with sizes between 50 and 100 P units. Curve b plots the number of droplets containing the biggest particles (sizes between 100 and 150 P units). In both figures an autocatalytic reaction is simulated, using a surfactant film flexibility f ) 5 (nuclei of less than 5 units can be interchanged), and a large value of droplet size (q ) 5000).
Figure 2. Time course of absorbance at λ ) 260 nm during the formation of particles in microemulsions. Microemulsion: 16.54% Brij 30/2.47% water, R ) 3.01, [PtIV] ) 0.01 M, [hydrazine] ) 0.05 M.
intensity, because the observation window corresponds to the diffraction of smaller particles which are disappearing during the growth process. The presence of this maximum (although not so well-defined) has also been spectrophotometrically detected by Towey et al.14 for the formation of CdS in AOT microemulsions. The sharpness of this maximum depends on concentration, as one can observe in Figure 1 (compare the smeared maximum in Figure 1B with the well-defined one in Figure 1A). Curves b in Figure 1 represent the growth of final largest particles. A similar behavior can be observed in the growth of Pt particles in water/Brij30/n-heptane microemulsions (Figure 2). An initial decreasing of the absorbance corresponding to the nucleation can be observed, because the chosen wavelength corresponds to the absorption band of the reactants. After that, there is an increasing of the absorbance due to the growth process, because in the experimental conditions nucleation and growth are two separated processes. To explain the different behaviors in Figure 1, we can assume18 that the rates of nucleation and growth are dependent on the reactant concentration. The reaction
rate is proportional to the concentration. So a high concentration means that the reactants will react rapidly resulting in the generation of a large number of seed nuclei. After that, the growth of the particles begins. Conversely, if the concentration is low, the reaction will be slower and some particles may grow whereas the reaction is still taking place. Therefore, nucleation and growth take place on a different time scale when the concentration is high, and these processes occur almost simultaneously at low concentrations. This fact can be observed on Figures 3 and 4, which show the particle size distribution corresponding to different times during the course of the reaction for the synthesis variables used in parts A and B of Figure 1, respectively. These figures include four histograms: histogram A represents the beginning of the reaction (10-15% conversion); histograms B and C correspond to 25 and 50% conversion, respectively; and the last histogram in each figure corresponds to the equilibrium distribution of particle sizes (100%). By observing both figures, one can observe that, at the beginning of the reaction (histograms A) a large quantity of droplets carrying very small particles appear. These particles will grow both because of the autocatalytic effect of an existing nucleus and by interchange of nuclei between droplets (“Ostwald ripening”). When the reaction has reached about 25% conversion, a small number of larger particles appears at low concentrations. The mean size of these particles is almost equal to the mean size of the particles in the final distribution (compare histograms B and D in Figure 4). One can conclude that, at low concentrations, large particles appear early in the reaction, when the nucleation process is still taking place. In contrast, at high concentrations (c ) 500), when the reaction has reached 25% conversion, most of the particles are much smaller than the final mean size (compare histograms B and D in Figure 3). This means that a great quantity of seed nuclei have been formed while the growth process has not yet begun, i.e., there is no overlapping of nucleation and growth. In this case the mean size of the largest particles increases as time goes on. This explains the results shown in Figure 1: at high concentration, nucleation and growth appear clearly separated; conversely, at low concentration, both processes are closer.
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Figure 3. Size distribution of nanoparticles during the course of the reaction at high concentration (c ) 500 reactants per droplet) and low value of the film flexibility (f ) 5, nuclei of less than 5 units can be interchanged between droplets). The last histogram corresponds to the equilibrium distribution, and histograms A, B, and C correspond to 10, 25, and 50% conversion, respectively.
A second point of interest is the achievement, at the end of the reaction, of unimodal distribution of particle sizes at low concentrations and a bimodal distribution at high concentrations. The existence of these unimodal and bimodal types of size distributions has been experimentally observed.15,24,28 As an example, Figure 5 shows the experimental histogram obtained from electron micrographs of Pt particles prepared in the conditions used in Figure 2 at two different concentrations. At low concentrations one can observe a transient b-type distribution at early times (see histogram B in Figure 4), which tends to a u-type distribution as time goes on. However, at high concentrations (Figure 3) the bimodality is more important as the time increases. Arguments similar to those explained previously can be used to explain this phenomena. As times goes on, growth becomes more important for all values of concentration. Taking into account that the surfactant film used in these simulations is not very flexible (f ) 5), the growth has to be mainly due to the formation of P units on an existing nuclei (autocatalysis with surface effect). These bigger particles give rise to the second maximum of the bimodal distribution. The first maximum is accounted for by the smallest particles, which cannot be interchanged because of the low value of film flexibility and are not favored by the autocatalysis. At low concentrations this first maximum does not appear, (28) Rivadulla, F. Master Thesis, University of Santiago de Compostela, 1996.
because the nuclei are basically growing from the beginning. Because of their very small size, the remaining seed nuclei (first stacked bar) can be interchanged despite the low value of the film flexibility. For a better understanding of the role of the surfactant film flexibility, Figure 6 shows the behavior of the size histograms as a function of time at high concentrations and high values of film flexibility (compare Figures 3 and 6). The main feature is the transition from a bimodal distribution at f ) 5 to a unimodal distribution at f ) 30. Taking into account that both figures represent an autocatalytic reaction, the difference between them is due to the growth by interchange. A high value of f means that the smallest particles can be interchanged and adsorbed by other droplets containing bigger nuclei, and therefore a u-type distribution is obtained. In both cases the concentration is high so that nucleation and growth occur at different time scales (large particles do not appear at short times). It can be seen that a narrower particle size distribution is obtained using a rigid film. This observation agrees with experimental results.11,17 As an example, Pileni et al.11 found a similar behavior studying silver nanoclusters in AOT reverse micelles using two different alkanes (isooctane and cyclohexane), i.e., different values of film flexibility. It is interesting to point out that a high polydispersity is obtained as film flexibility increases for any time during the course of the reaction.
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Figure 4. Size distribution of nanoparticles during the course of the reaction at low concentration (c ) 100 reactants per droplet) and low value of the film flexibility (f ) 5, nuclei of less than 5 units can be interchanged between droplets). The last histogram corresponds to the equilibrium distribution, and histograms A, B, and C correspond to 10, 25, and 50% conversion, respectively.
Figure 5. Influence of concentration on the size distribution of platinum particles. Microemulsion: 16.54% Brij 30/2.47% water, R ) [H2O]/[Brij 30] ) 3.01, [N2H4] ) 0.125 M. (A) [PtIV] ) 0.05 M; (B) [PtIV] ) 0.20 M.
II. Study of the Ostwald Ripening. The simulation procedure can help us to determine if the growth takes place via reaction on an existing nucleus or via the Ostwald ripening. Both processes can or cannot occur simultaneously, depending on their relative rates, which are related to the reactant concentrations. This can be clearly
observed in Figure 7, which shows the number of particles having less than 50 units versus time, for two different values of concentration. It can be observed that a shoulder appears at high values of concentration, which clearly separates two regimes during the course of the formation of the particles. The first one corresponds mainly to the
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Figure 6. Size distribution of nanoparticles during the course of the reaction at high concentration (c ) 500) and high value of the film flexibility (f ) 30). The last histogram corresponds to the equilibrium distribution and histograms A, B, and C correspond to 10, 25, and 50% conversion, respectively.
Figure 7. Time evolution of droplets carrying particles of sizes smaller than 50 units, for different values of the surfactant film flexibility. Part A shows the results at high concentration (c ) 500), and part B shows results at low concentration (c ) 100).
reaction (nucleation and growth by autocatalysis), and the second one corresponds to the growth by ripening. This fact can be understood on the basis of the influence of the concentration on the reaction rate: high concentration values lead to the quick formation of a large number of seed nuclei, because the reaction is fast. These nuclei
will grow, but both processes, growth by autocatalysis and growth by ripening, will occur at different time scales. This phenomenon is reflected by the two regimes observed at high concentrations in Figure 7A. It can be observed that nuclei disappear more quickly as the film flexibility decreases, because a rigid film (small value of f) does not
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allow an interchange of nuclei, so the system contains a higher number of catalysts. Likewise, the fact that at high flexibility nuclei disappear slowly confirm that the growth is mainly controlled by autocatalysis at this stage. By observing Figure 7A, one can also observe that the most important contribution to growth is due to ripening when the film is very flexible (compare c and r in Figure 7A), whereas autocatalysis is the most important contribution for films with low flexibility (compare c′ and r′). The existence of these two growth mechanisms can also be experimentally observed in the formation of platinum particles as can be seen in Figure 2. When the reaction is slow (low value of c), growth by reaction and ripening takes place at the same time. In this situation, they cannot be distinguished, and the shoulder does not appear as shown in Figure 7B (c ) 100). This is also reflected in the growth of the largest particles, which are mainly formed by ripening. So in Figure 1A (high concentration) one can observe that a long time-lag is observed until the largest particles start to appear (time lag ) 2000 mcs for c ) 500). However, when reaction and ripening overlap (low concentration), the time lag is short; i.e., the largest particles appear almost from the beginning of the reaction (time lag ) 250 mcs for c ) 100). This is an important result, because in many cases experimental techniques allow only observation of the largest particles; therefore from the time-lags one can distinguish for a given film flexibility the competition between the process of Ostwald ripening and the growth by autocatalysis. It is interesting to point out that the existence of this “induction period” for the formation of the final particles can be experimentally observed in Figure 2 and has also been observed by other authors.14 III. Kinetic Model. The time lag corresponding to the appearance of the maximum shown in Figure 1 has been used by different authors11,14 to estimate the agglomeration number of the particles, assuming a Smoluchowski rapidcoagulation mechanism and taking into account a droplet communication-controlled reaction scheme.29 This model can be described as follows:30 After nucleation, the growth phase starts with a suspension of monomers of concentration c0. These monomers undergo Brownian motion until they combine. It can be shown30 that the concentration of a particular monomer composed by n atoms passes through a maximum at the time max
t
) (n - 1)τ/2
(1)
where τ is the time at which the total concentration of particles is half the original value. By considering that the rate constant for droplet communication is proportional to the time τ, Towey et al.14 obtained a good qualitative agreement between the model and the experimental results for the formation of CdS particles in AOT microemulsions. In order to compare our simulation results with those predicted from the Smoluchowski theory, we have obtained from our simulation data values of tmax for each agglomeration number n using different values of concentration. By plotting tmax versus n we have verified eq 1 and obtained the time τ. As an example, Figure 8 shows the behavior for c ) 1000 reactants per droplet and f ) 5. One can observe a good linear relationship between tmax and n as expected from eq 1. Table 1 shows the τ values obtained from the slope, using different values of concentration and film flexibility. As expected, the slope is practically independent on concentration. However it (29) Fletcher, D. I.; Howe, A. M.; Robinson, B. H. J. Chem. Soc., Faraday Trans. 1987, 83, 985. (30) Nielsen, A. E. Kinetics of Precipitation; Pergamon Press: Oxford, 1964.
Figure 8. Time needed to attain a maximum concentration of particles, tmax, versus the agglomeration number n. Synthesis variables: concentration c ) 1000, film flexibility f ) 5.
Figure 9. Time evolution of droplets carrying particles of sizes between 100 and 150 units, for different values of the film flexibility and concentration c ) 500 reactants per droplet. Table 1. τ Values Obtained from Equation 1 for Different Concentrations (c) and Film Flexibilities (f) c
τ (f ) 5)
τ (f ) 30)
100 200 500 800 1000
3.0 2.5 2.7 2.8 2.8
1.3 1.7 2.5 2.2 2.3
can be observed that as the film flexibility is increased, the mean value of τ becomes smaller; i.e., the maximum is reached at short times for flexible films. This implies that if the concentration is kept constant when studying a given particle size (i.e., a given number of monomers n) evolving with time, it is expected that the maximum corresponding to tmax will appear earlier as the surfactant film becomes more flexible. Figure 9 plots the time course for the number of droplets containing nuclei with 100 < n < 150 monomers, for different values of the surfactant film flexibility, and c ) constant ) 500 reactants per droplet. It is observed that tmax becomes smaller as the film flexibility increases, because in effect, easier interdroplet communication occurs. The fact that we observe that τ diminishes as f increases can be explained by assuming that the interdroplet communication is the ratedetermining step for the particle formation in micro-
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emulsions.14,22 Our results therefore agree with Towey’s experimental results14 in the sense that τ can be related to the film flexibility, so that a rigid film would be associated with a slow interdroplet communication rate. Although in our simulation we do not explicitly indicate that the rate constant for droplet communication depends on the flexibility of the surfactant film, the fact that final large particles are favored by ripening at high f values implies that the formation rate of these particles by this mechanism will be quicker as f increases, because the effective rate constant for droplet communication is greater. We can also apply this model to deduce the final size of the particle from the experimental results shown in Figure 5. The interdroplet communication constant (kex) using Brij as surfactant is about 1/104 times the interdroplet communication constant for AOT.28 Therefore kex ≈ 106 M-1 s-1/104 ) 102 M-1 s-1. According to Robinson and co-workers:22 τ-1 ) kexc0 ≈ 102(5 × 10-2) ) 5 s-1. Therefore, from eq 1 and using the experimental value of tmax ) 2500 s, one can obtain n ≈ 104 atoms. Assuming a spherical shape, the final radius of the platinum particles should be about 3.5 nm, which agrees very well with the size determined by transmission electron microscopy measurements (r ) 4 nm).
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Conclusions According on the simulation results, one can conclude that nanoparticle formation in microemulsions occurs in three different steps (nucleation, growth by autocatalysis, and growth by ripening), which can or cannot overlap depending mainly on concentration. The surfactant film flexibility affects the kinetics of the reaction as rigid films can be associated with a slow interdroplet communication rate, when the interdroplet communication is the ratedetermining step in particle formation in microemulsions.14,22 It is also observed that the time lag of the maximum can be used to identify the kind of growth which dominates in a particular reaction. Finally, it has been shown that the Smoluchowski model is a useful tool which can be applied for the study of formation of nanoparticles in microemulsions. Comparison with experimental results show sthe applicability of the developed computer simulation model. Acknowledgment. The authors are grateful for the financial support of the Regional Government of Galicia (Projects XUGA9320903B95 and XUGA34701A95). LA9607870
