9500
Langmuir 2006, 22, 9500-9506
Articles Modeling Shell Formation in Core-Shell Nanocrystals in Reverse Micelle Systems Diwakar Shukla and Anurag Mehra* Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400076, India ReceiVed May 26, 2006. In Final Form: August 11, 2006 The mechanisms responsible for the formation of the shell in core-shell nanocrystals are ion-displacement and heterogeneous nucleation. In the ion-displacement mechanism, the shell is formed by the displacement reaction at the surface of the core nanoparticle whereas in heterogeneous nucleation the core particle induces the nucleation (or direct deposition) of shell material on its surface. The formation of core-shell nanocrystals via the post-core route has been examined in the current investigation. A purely probabilistic Monte Carlo scheme for the formation of the shell has been developed to predict the experimental results of Hota et al. (Hota, G.; Jain, S.; Khilar, K. C. Colloids Surf., A 2004, 232, 119) for the precipitation of Ag2S-coated CdS (Ag2S@CdS) nanoparticles. The simulation procedure involves two stages. In the first stage, shell formation takes place as a result of the consumption of supersaturation, ion displacement, and reaction between Ag+ and excess sulfide ions. The growth in the second stage is driven by the coagulation of nanoparticles. The results indicate that the fraction of shell deposited by the ion-displacement mechanism increases with increasing ion ratio and decreases with increasing water-to-surfactant molar ratio.
1. Introduction Core-shell nanocrystals consist of a core particle of nanodimensions that is covered by a shell of a different material, with thickness in the range of a few nanometers. Core-shell nanoparticles can be used in a variety of applications because of the enhanced properties exhibited as a result of coating with a different material. Nanosized semiconductor particles exhibit unusual optical and electronic properties on a length scale intermediate between the molecular and bulk states of matter. Core-shell nanocrystals of various materials have been synthesized using the reverse micelle route.1-4 Qi et al.3 have synthesized ZnS-coated CdS and coprecipitated ZnxCd1 - xS nanoparticles that have a tunable energy gap from single CdS nanoparticles to ZnS nanoparticles. They have characterized the core and shell particles using optical absorption and photoluminescence spectroscopy. Han et al.2 have prepared Ag2S@CdS nanoparticles using a modified reverse micelle technique and have studied their long-range absorption characteristics. Hao et al.5 have studied the optical properties of CdS-coated CdSe nanoparticles synthesized by the microemulsion method followed by refluxing in 60:1 toluene/methanol solvent, which resulted in particles with a strong narrow band-edge luminescence. The core-shell CdS@CdSe nanoparticles have also been synthesized by Xu et al.6 These coated particles showed reduced photoinstability in the luminescence spectrum. Hota et al.1 have synthesized a variety of nanocomposites including core-shell (1) Hota, G.; Jain, S.; Khilar, K. C. Colloids Surf., A 2004, 232, 119. (2) Han, M. Y.; Huang, W.; Chew, C. H.; Gan, L. M. J. Phys. Chem. B 1998, 102, 1884. (3) Qi, L.; Ma, J.; Cheng, H.; Zhao, Z. Colloids Surf., A 1996, 111, 195. (4) Loukanov, A. R.; Dushkin, C. D.; Papazova, K. I.; Kirov, A. V.; Abrashev, M. V.; Adachi, E. Colloids Surf., A 2004, 245, 9-14. (5) Hao, E.; Sun, H.; Zhou, Z.; Liu, J.; Yang, B.; Shen, J. Chem. Mater. 1999, 11, 3096. (6) Xu, L.; Huang, X.; Zhu, J.; Chen, H.; Chen, K. J. Mater. Sci. 2000, 35, 1375.
CdS@Ag2S nanoparticles. The methodology used is the postcore method, wherein the core nanoparticles are synthesized using the reverse micelle method. This is followed by the addition of another micelle solution containing the ions required for shell formation. Another popular method of core-shell nanoparticle synthesis is the partial microemulsion method wherein the core particles are synthesized by the direct addition of a solution of one of the core-forming ions to a micelle solution containing the counterions. The shell is deposited on the core nanocrystals by a similar process. Loukanov et al.4 have used the partial microemulsion route to synthesize core-shell semiconductor nanoparticles and have done experiments to investigate the dependence of shell thickness on the water-to-surfactant molar ratio. It may be noted that apart from Hota et al.1 that the experimental papers mentioned above do not report particle sizes and their distributions. For example, Qi et al.3 have reported only the absorption spectrum of the particles, whereas Loukanov et al.4 have provided only the average particle sizes and not the distributions. Various models have been proposed to investigate the precipitation of nanoparticles in reverse micelles.7-12 Li and Park7 proposed a purely probabilistic Monte Carlo model for the precipitation of nanoparticles in two-microemulsion systems. Singh et al.11 have proposed a more computationally efficient Monte Carlo scheme for similar systems. Ramesh et al.12 have proposed a stochastic population balance model to study the influence of system parameters on the nanoparticle sizes obtained. (7) Li, Y.; Park, C. W. Langmuir 1999, 15, 952. (8) Jain, R.; Shukla, D.; Mehra, A. Langmuir 2005, 21, 11528-33. (9) Shukla, D.; Mehra, A. Nanotechnology 2006, 17, 261-267. (10) Bandyopadhyaya, R.; Kumar, R.; Gandhi, K. S.; Ramakrishna, D. Langmuir 2000, 16, 7139. (11) Singh, R.; Durairaj, M. R.; Kumar, S. Langmuir 2003, 19, 6317. (12) Kumar, A.; Hota, G.; Mehra, A.; Khilar, K. AIChE J. 2004, 50, 15561567.
10.1021/la061499z CCC: $33.50 © 2006 American Chemical Society Published on Web 10/12/2006
Modeling Shell Formation in ReVerse Micelle Systems
Langmuir, Vol. 22, No. 23, 2006 9501
Figure 1. Schematic of the formation of a Ag2S shell in Ag2S@CdS core-shell nanocrystals via the ion-displacement mechanism.
Jain et al.8 and Shukla and Mehra9 have studied the effect of the coagulation of nanoparticles on the size and PSDs of nanoparticles. Although there are many models for the precipitation of nanoparticles in reverse micelles, there are very few investigations that have attempted to understand the mechanism of the formation of complex nanostructures such as core-shell nanoparticles, nanodisks, and so forth. Jain et al.13 have modeled the process of the formation of core-shell nanoparticles in reverse micelles. Their model deals with shell formation via an ion-displacement mechanism and predicts the experimental results of Han et al.2 In the current model, we investigate an alternative mechanism for shell formation in core-shell nanoparticles and analyze the conditions under which different mechanisms may operate and compete with each other.
2. Model Formulation The mechanism by which the shell is formed over the core nanoparticles is not well understood. This is partially because of the lack of adequate characterization tools for core-shell nanocrystals. It is not possible to directly observe the phenomena taking place inside the micelle that leads to the formation of shell. Zhou et al.14 have put forward the ion displacement mechanism for the formation of core-shell nanocrystals. Because of the addition of only cations (Ag+) to the microemulsion containing core CdS nanoparticles, the reaction that can lead to the formation of shell is the displacement reaction at the core surface. The procedure of ion displacement can be shown by the following equation
2Ag+ + CdS(s) f Cd2 + + Ag2S(s)
(1)
The above reaction has a favorable Gibbs free energy change; therefore, the ion displacement can take place. Han et al.2 have also suggested the ion-displacement mechanism for the formation of the shell over the core nanoparticles. The schematic of the ion-displacement mechanism is shown in Figure 1. In some cases, the free-energy change may not be favorable. For example, in the experimental results of Qi et al.3 the CdS-coated Zns nanoparticles cannot be formed via the ion-displacement mechanism as the displacement reaction is not feasible. In such a case, the only possible mechanism that can lead to shell formation is the direct deposition of shell-forming material on the surface of core particles. Hota et al.1 have suggested that heterogeneous nucleation and growth occurs that results in the formation of a shell over the core nanoparticles. The schematic of the heterogeneous nucleation and growth mechanism is shown in (13) Jain, R.; Shukla, D.; Mehra, A. Ind. Eng. Chem. Res. 2006, 45, 22492254. (14) Zhou, H. S.; Han, M.; Dong, Z.; White, T. J.; Knoll, W. J. Lumin. 1996, 70, 21.
Figure 2. Schematic of the formation of the Ag2S shell in Ag2S@CdS core-shell nanocrystals via the heterogeneous nucleation mechanism. Two types of particles can form in this case depending on whether a core particle is present in one of the reactant-containing (colliding) micelles.
Figure 2. Hence, to form the Ag2S@CdS nanocrystal, excess sulfide ions are used while forming the core CdS nanoparticles. When the micellized AgNO3 solution is added, a Ag2S shell is formed upon reaction with the excess sulfide ions present in the micelles. The following reaction leads to formation of Ag2S
2Ag+ + S2- f Ag2S(s)
(2)
Furthermore, the ion-displacement mechanism would lead to the deposition of only a monolayer of shell whereas Hota et al.1 and Loukanov et al.4 have prepared core-shell nanoparticles with shell thicknesses larger than the monolayer of the shellforming compound. Therefore, a combined mechanism that applies the ideas of heterogeneous nucleation and the iondisplacement mechanism is proposed in the present work to analyze the experimental results of Hota et al.1 2.1. Ion-Displacement Mechanism. The model for the iondisplacement mechanism proposed by Jain et al.13 assumes that when the outer surface of the core particle is fully covered by Ag2S ions no further displacement can take place. A new Ag+ ion would be able to displace a Cd2 + ion via the displacement reaction, subject to the availability of Cd2 + ions on the surface of the core-shell crystal. The displacement reaction has been assumed to be instantaneous. In order that ns, the number of shell particles, covers the surface of the core formed by nc nanoparticles, the projected area of the shell molecules on the core surface must be greater than or equal to the surface area of the core. The following expression gives the number of shell molecules (ns) required to completely form a monolayer of shell on the surface of core nanoparticle containing nc molecules.
ns (nc)
2/3
(
g2
3Mc
)
2/3
πFcNArs
3
(3)
In the present model, we use a value of 5 for the ratio ns/nc2/3 based on the typical values of the constants appearing on the right-hand side of the above equation.13 2.2. Heterogeneous Nucleation Mechanism. During shell formation, the core nanoparticle acts as the center for heterogeneous nucleation and subsequent deposition of shell-forming molecules. The shell molecules are assumed to precipitate instantly on the core surface on contact. The phenomenon of finite homogeneous nucleation also occurs and leads to the formation of nanoparticles of only shell-forming material. In a simplistic model for the deposition of shell material on the core
9502 Langmuir, Vol. 22, No. 23, 2006
Shukla and Mehra
Figure 3. Core-shell nanoparticle with a shell-deposition mechanism such that (a) the outer layers can be filled before the inner layer is complete. (b) The outer layers are not permitted to fill until the inner layer is complete.
surface, the deposited layers of shell can form only when the layers below it are completely filled as illustrated in Figure 3a. Every molecule of shell deposited on the core particle surface blocks the core surface molecules available for the displacement reaction. The liquid product molecules can precipitate either on the core surface or on the surface of already deposited partial layers of shell. If we take this fact into account, then the resulting structure would contain partial layers of shell as illustrated in Figure 3b. Such deposition of partial layers of shell is important when both the ion-displacement and heterogeneous nucleation mechanism are active simultaneously. Consider the case of a molecule that comes in contact with a core-shell nanoparticle with N partial layers of shell. Precipitation of the molecule occurs on a layer in proportion to the exposed surface area of that layer. The layers are numbered starting from the core surface outward. Let ai, bi be the number of shell molecules present in the layer and the maximum number of shell molecules required for complete deposition of the ith layer, respectively. The probability of deposition of a molecule on the first layer is given by
p1 ) 1 -
a1 b1
(4)
The probability of deposition on the second layer will be
p2 )
(
)
a1 a2b1 a1 a2 1) b1 b2a1 b1 b2
(5)
The analogy can be extended to the ith layer as
pi )
ai - 1 ai bi - 1 bi
(6)
These equations for the calculation of probability are valid only under the assumption that the (i + 1)th layer deposits on the surface of ith layer. This constraint is expressed in the form of an inequality,
1g
ai aN a1 g ‚‚‚ g g ‚‚‚ g g 0 b1 bi bN
(7)
The sum Σipi is always equal to 1. This implies that (i + 1)th layer cannot form as a bridge connecting two noncontiguous regions of the ith layer. The probability for deposition on a completely filled layer is zero. A layer can be complete only when the layers below it are completely filled.
3. Simulation Procedure Simulations based on the proposed model have been performed to study the formation of semiconductor core-shell nanocrystals (Ag2S@CdS). Quantitative data pertaining to experiments reported in the literature are limited. Han et al.2 reported data on Ag2S-coated CdS nanocrystal formation with no excess sulfide ions added at the core-formation stage. The mechanism that operates under these condition is the ion-displacement mechanism. Hota et al.1 have synthesized Ag2S-coated CdS nanocrystals with excess sulfide ions added at the core-formation stage, so both the ion-displacement and heterogeneous nucleation mechanisms are operational during shell formation. Qi et al.3 have synthesized CdS-coated ZnS nanoparticles in which the ion-displacement reaction is not feasible so that only heterogeneous nucleation contributes to shell formation. These three experimental scenarios represent three possible cases that can occur during the shellformation stage, depending upon the feasibility of the displacement reaction and the availability of excess sulfide ions during shell formation. Qi et al.3 have reported only the absorption spectrum of core-shell nanoparticles so that there is no direct data for the case where only homogeneous nucleation works. The algorithm used for present simulations is based on the Monte Carlo schemes of Bandyopadhyaya et al.10 and Shukla and Mehra.9 The simulation is done in two stages. The first stage deals with mixing the microemulsion containing shell-forming ions and the subsequent complete precipitation of Ag2S. This stage is completed within a few milliseconds. During this stage, the coagulation of core-shell nanoparticles is not allowed because the time scale for the coagulation for these particles will be much higher than the order of milliseconds because of the large particle sizes. The second stage deals only with the coagulation of nanoparticles. The following coagulation event can take place between the particles: 1. The coagulation of two Ag2S particles leads to the formation of larger nanoparticles, with the number of molecules equal to the sum of the number of Ag2S molecules contained in the coagulating particles. 2. The coagulation of two core-shell nanoparticles (Ag2S@ CdS) leads to the formation of a composite particle resulting in the loss of the core-shell character of the coagulating particles. 3. The coagulation of a core-shell nanoparticle and a Ag2S particle leads to the deposition of the Ag2S nanoparticle on the surface of the shell present around the core. During the first stage of simulation, coagulation events of type 2 will not take place because the frequency of coagulation, which decreases with increasing particle size, is very small because of the large sizes of these particles. In the following
Modeling Shell Formation in ReVerse Micelle Systems
Langmuir, Vol. 22, No. 23, 2006 9503
paragraphs, we present the important features of our purely probabilistic Monte Carlo scheme: 1. An initial distribution of core nanoparticles (CdS) is assumed. The particle size distribution (PSD) is taken from the experimental results (or can be obtained using the simulations illustrated by Jain et al.13). The total number of micelles that contain core particles is calculated using the initial mean occupancy of the core-forming reactants. These micelles also contain the excess sulfide ions required for the precipitation of Ag2S. The number of Ag+-containing micelles added to the existing solution is half the total number of micelles associated with core particles. Such a recipe has been followed by the Hota et al.1 The initial distribution of Ag+ and S2- is assumed to be Poissonian. 2. The fusion and fission events now take place between all of the micelles in the solution. The following fusion-fission rules govern the exchange of material between the micelles in addition to the rules listed by Jain et al.13 (a) The fusion between two micelles containing particles leads to the coagulation of the two nanoparticles to form a large nanoparticle contained in one of the daughter micelle. The coagulation of two Ag2S particles and the coagulation of a core-shell nanoparticle with a Ag2S particle are allowed in the first stage. (b) If the micelle containing a core particle and a S2- ions fuses with another micelle containing Ag+ ions, then the reaction between Ag+ and S2- ions is given precedence over ion displacement. If all of the sulfide ions are consumed and the resulting Ag2S molecules are deposited on the core particle surface, then the remaining Ag+ ions are used for ion displacement. 3. The deposition of Ag2S(l) on the surface of a core particle takes place according to the probability of deposition on each shell layer. The probabilities are calculated using eq 6. A random k-1 k-1 number is generated, and if it lies between (∑i)1 pi, ∑i)1 pi], then the molecule is deposited on the kth layer. The molecules are deposited one-by-one and probabilities are updated after the deposition of each molecule. 3.1. Stage 1. Each micelle is assigned six integers, which includes the number of molecules of A, number of molecules of B, number of liquid product molecules C(l), number of nucleated product molecules C(s), number of molecules forming the core nanoparticle, and number of molecules that can be displaced via ion displacement. The last two numbers are required only for the micelles that contain the core nanoparticles. Each core nanoparticle also has an associated set of integers that include the number of molecules present in each layer of the shell deposited on the core surface. The algorithm used for simulation is outlined below. 1. The initial micelle profile is generated using the mean occupancy of reactants and the experimental particle size distribution of core nanoparticles. 2. The two basic events in the model are fusion-fission (or coalescence-decoalescence) of micelles and nucleation. The event is decided on the basis of the probability of occurrence of that particular event. It is assumed that the probability of occurrence of the event is directly proportional to the frequency of that event. 3. The frequency of coalescence is given by
1 fcoal ) βqNmicN 2
()
(8)
where β is the coalescence efficiency, N is the number of micelles in the population being considered, Nmic is the number density of micelles, and q is the Brownian collision rate. For equally
sized entities, q has been derived by Smoluchowski16
q)
8kBT 3η
(9)
where kB is the Boltzmann constant and η is the viscosity of the medium. 4. 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 expression that gives the rate kn(i) in a nonnucleated micelle containing i product species (i > n*) is taken from our previous papers dealing with nanoparticle formation in reverse micelles.9,13 5. These individual frequencies are normalized by the sum of the frequencies of both the events. The events is then decided using a random number lying between [0, 1). 6. After the event has been chosen, the micelles that have to undergo that event are to be selected. In the case of coalescence, two random integers between 1 and N are generated, corresponding to each micelle. On collision, the fusion-fission rules outlined in the section on the simulation procedure are used to obtain the contents of the daughter micelles. For nucleation, a random number is generated and is compared with the probabilities of nucleation for each micelle. If a generated random number between 0 and 1 lies between pin and pj+1 n , then the jth micelle is chosen for nucleation. The probability pjn is given by j
pjn )
kn(i) ∑ i)1 (10)
fn
After the successful completion of the event, the micelle involved in the event is updated, and the process is repeated until complete conversion is achieved. 3.2. Stage II. In this stage, the coagulation of nanoparticles is considered. All the three types of coagulation events are feasible in this stage. There are only two types of micelles left in the systemsmicelles containing particles and empty micelles. The coagulation event is the only feasible event that can change the micelle profile. Every coagulation event generates an empty micelle; therefore, the number of micelles containing particles decrease with time. This leads to inaccuracy in prediction if the initial population of micelles is small. Therefore, the micelle population is duplicated 105 times to maintain the accuracy of prediction. Micelles are classified into bins according to the diameter of the nanoparticle contained in them. The diameter of a coreshell particle is calculated by adding the volumes of all of the molecules forming both shell and core. The frequency of coagulation between micelles belonging to group i and those belonging to group j is calculated as
)
(11)
βq Nci (Nci - 1)Nmic exp(-βd dip) 2N
(12)
(
βq Nci Ncj Nmic 2dip d jp exp -βd i f ) N dp + djp c ij
fijc )
(15) Jain, R.; Mehra, A. Langmuir 2004, 20, 6507. (16) Smoluchowski, M. V. Z. Phys. 1916, 17, 585. (17) Adamson, A. Physical Chemistry of Surfaces; John Wiley and Sons: NewYork, 1990.
9504 Langmuir, Vol. 22, No. 23, 2006
Shukla and Mehra
Table 1. Parameters Used for Simulating the Formation of Ag2S@CdS Core-Shell Nanocrystals1 parameter
value
β q η B A Nmic Ksp n* R
0.01 1.097 × 10-17 0.001 kg m-1 s-1 740 278.42 s-1 4.46 × 10 21 m-3 6 × 10-50 mol2 m-6 2 10
4. Results and Discussion
The bins that will be involved in the coagulation event are chosen depending on the probability of coagulation between the particles contained in them. A uniform random number u2 lying between (0, 1] is generated, and the index i that satisfies the inequality i
∑ k)1
i+1
pc(k) < u1 e
pc(k) ∑ k)1
(13)
determines the bins involved in the event. The probabilities pc(k) are calculated as follows
(
)
f cij i(i - 1) p +j ) c 2 f c
(14)
where i ∈{I, 1 e i e D}, j ∈{I, j e i}∀i, and f c is the total coagulation frequency, calculated using the expression D
fc)
i
f cij ∑ ∑ i)1 j)1
in a bin depending upon its particle size. In the current investigation, the bin size is taken to be 0.1 nm.
(15)
where D is the diameter of the largest particle in the system at any time in angstroms. The frequencies of coagulation of all of the bins involved in a particular coagulation event are updated at the end of the event. The resulting larger nanoparticle is placed
The parameters used for simulation are calculated using the data reported by Hota et al.1 The parameters for the simulation are reported in Table 1. These parameters are similar to the values used in the literature.8-10,13 The water present in the aqueous pools is of two types: free water, which is not associated with the polar headgroups of surfactants, and the water associated with the surfactant heads. Therefore, the bulk values of physical constants may be applied when the amount of free water is significant as compared to the amount of water associated with the surfactant headgroups. Therefore, for parameters such as the solubility product, bulk values are used because sufficient free water is present. The value of the collision efficiency is influenced by various factors such as the nature of the surfactant, the size of the micelles, and the concentration of the dissolved species. Because the mean occupancy of ions is very high in the reported experiments,1 a high value of β is chosen for simulation. After the addition of micellized AgNO3, shell formation starts. Initially, about 50% of the shell is deposited by the iondisplacement mechanism. The number of ion-displacement steps would depend on the availability of the free core surface for displacement, the mean occupancy of Ag+ ions, and the [Ag+]/ [S2-] ratio. The availability of free core surface depends on the radius of core nanoparticles used for shell-layer deposition. Therefore, the core particle size and distribution affect the final core-shell particle distribution in that large core particle diameters would lead to an increase in the rate of ion displacement. Thus, larger mean core nanoparticle sizes will lead to a larger fraction of the shell being deposited by the ion-displacement mechanism. With time, the silver ions are consumed by the sulfide ions to form liquid Ag2S molecules; therefore, the fraction of shell deposited as a result of the ion-displacement mechanism decreases with time. The variation of the fraction of shell deposited due to the ion-displacement mechanism with the conversion of the
Figure 4. Fraction of Ag2S deposited as a result of ion displacement as a function of conversion for various values of the water-to-surfactant molar ratio (R).
Modeling Shell Formation in ReVerse Micelle Systems
Langmuir, Vol. 22, No. 23, 2006 9505
Figure 5. Fraction of Ag2S deposited as a result of ion-displacement as a function of conversion for various values of the ion ratio (X) [Ag+]/[S2-]).
Figure 6. Fraction of Ag2S deposited as a result of ion displacement for various values of the ion ratio (X) [Ag+]/[S2-]) when 100% conversion is reached.
limiting reactant, for different values of the water-to-surfactant molar ratio, is shown in Figure 4. The decrease in the R value corresponds to an increase in the total number of micelles and therefore a low mean occupancy of reactants. The probability of a micelle with a particle and another micelle with silver ions is higher for smaller R-value systems. For high R-value systems with a high mean occupancy, the ionic reaction takes preference over displacement because there is a high probability of contact between reactant molecules per (intermicellar) collision. The period in which ion displacement remains active and the fraction of shell deposited by ion displacement depends on the [Ag+]/ [S2-] ratio used in the experiments; a lower values of this ratio would create a preference for heterogeneous nucleation compared
to ion displacement. The fraction of Ag2S deposited as a result of ion displacement as a function of conversion for various ion ratio values (X ) [Ag+]/[S2-]) is illustrated in Figure 5. Figure 6 shows the fraction of Ag2S deposited because of ion displacement as a function of the ion ratio. The linear increase in the fraction of shell molecules deposited by ion displacement with respect to the increase in X is expected. An increase in X would imply a high mean occupancy of Ag+ as compared to S2ions, which leads to a proportionate increase in the collisions between micelles containing Ag+ ions and those containing core particles. A very high value of X implies that only the iondisplacement mechanism operates during shell deposition. This would lead to the formation of a monolayer shell.
9506 Langmuir, Vol. 22, No. 23, 2006
Shukla and Mehra
based on the assumption that the shell grows over the core nanoparticle by ion displacement and a heterogeneous nucleation mechanism. A Monte Carlo scheme combining both of these mechanisms responsible for shell growth in core-shell nanocrystals has been proposed. The layer-by-layer deposition of shell on the core nanoparticle surface is necessary to correctly estimate the number of surface ions that can be displaced. The predicted core and core-shell distributions are close to those observed experimentally. Nomenclature
Figure 7. Model prediction and experimental PSD of core-shell nanocrystals.
The model prediction for the PSD of core-shell nanoparticles and the experimental PSD has been illustrated in Figure 7. The predicted particle size distribution for core-shell nanoparticle compares well with the reported distribution. Our simulations exclude the very few but large core-shell nanoparticles formed in the experiments of Hota et al.1 The formation of these particles is likely due to the coagulation of the particles during washing and separation of the nanoparticles from the continuous phase. The formation of few Ag2S nanoparticles also takes place because of homogeneous nucleation. It must be pointed out that wherever the size of the particles is comparable (or greater than) the native size of a (empty) micelle it could be argued that the micelle structure is destroyed and it ceases to be a micelle. The micelle, or rather a surfactant “cage”, then merely acts as a confined environment for the precipitation and growth of particles. If we consider the partial microemulsion route for shell deposition, then it can be seen that this route can be treated as a special case of the proposed model. The direct addition of Ag+ solution to the core particle-containing solution would lead to the transfer of Ag+ ions to the micelles before any coalescenceredispersion can take place because the time scale of mass transfer to the micelle is much shorter with respect to the coalescence time scale. If a micelle contains S2- ions, then this would lead to the formation of Ag2S liquid product molecules, and if the micelle contains a core particle, then ion displacement occurs. Therefore, for this case only the initial micelle contents will be different from that of the post-core route.
Conclusions A model has been developed on the basis of a purely stochastic Monte Carlo scheme to model the formation of core-shell nanocrystals, using the post-core method, via the inverted microemulsion route. The model is predictive in nature and is
A ) preexponential factor of the nucleation rate in micelles, s-1 B ) exponential term in the nucleation rate expression D ) diameter of the largest particle in the system at any time, angstroms dm ) diameter of a micelle, m dip ) diameter of a nanoparticle corresponding to the ith bin, nm f c ) total coagulation frequency, s-1 fcoal ) total fission-fusion or coalescence frequency of all micelles, s-1 f cij ) frequency of coalescence between micelles containing nanoparticles of MAN values i and j, s-1 fnuc ) total nucleation frequency of all micelles, s-1 Ksp ) solubility product of the precipitate, mol2 m-6 kB ) Boltzmann’s constant kn ) nucleation rate constant, s-1 ki ) rate of nucleation of the ith micelle NA ) Avogadro’s number ) 6.023 × 1023 mol-1 Nmic ) number density of micelles, m-3 N ) population of micelles included for simulation n* ) critical nucleation number nc ) number of molecules in the core nanoparticle ns ) number of molecules in the shell pc ) probability of a coagulation event in a micelle qm ) Brownian collision frequency between micelles, m3 s-1 rc ) radius of the core molecules, m rs ) radius of the shell molecules, m Rc ) radius of the core nanoparticles, m Sc ) surface area of the core, m2 Ss ) projected surface area of the shell molecules over the core surface, m2 R ) water-to-surfactant molar ratio T ) temperature on absolute scale, K Vm ) volume of a precipitate molecule, m3 Vmic ) volume of the micellar core, m3 X ) ion ratio [Ag+]/[S2-] Greek Letters β ) coalescence efficiency βd ) rate of decrease of the logarithm of the frequency of coagulation with respect to the harmonic mean of the diameter of coagulating particles, nm-1 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 the nucleus and the micellar core liquid, Pa m-1 LA061499Z
