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Langmuir 2009, 25, 3786-3793
Modeling of Formation of Nanoparticles in Reverse Micellar Systems: Ostwald Ripening of Silver Halide Particles Diwakar Shukla,† Abhijeet A. Joshi, and Anurag Mehra* Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400076, India ReceiVed NoVember 6, 2008. ReVised Manuscript ReceiVed December 17, 2008 There are many possible size enhancement processes that affect the formation of nanoparticles in reverse micelles, such as coagulation and Ostwald ripening, and different physical systems are likely to follow one or more of these mechanisms depending upon the properties of the system. It has been suggested that silver halide particles, prepared from a reverse micellar system of AgNO3 and KCl in NP-6/cyclohexane solution, increase in size due to Ostwald ripening (Kimijima, K.; Sugimoto, T. J. Phys. Chem. B 2004, 108, 3735), which occurs due to the dependence of the solubility of the particles on the particle size so that the larger particles grow at the expense of smaller particles. This study provides a modeling framework to quantitatively analyze the ripening process of nanoparticles produced in reverse micellar systems.
Introduction Reverse micellar solutions are a much explored medium for the production of nanoparticles. An important issue in any particle production method is the control of the particle size distribution. Hence, many studies focus on assessing the factors that are likely to affect the size of nanoparticles during the formation process. A large number of experimental and modeling studies have been conducted using a reverse micellar system for the formation of nanoparticles. Copper nanoparticles,2 silver and silver halides,1-6 CdS, ZnS, PbS, and Ag2S,7-12 and metal oxides such as ZnO and TiO213,14 have been prepared by such means. Towey et al.15 established that the process of coalescence of micelles was responsible for the formation of nanoparticles. However, it was observed7 that the particles produced merely by collision of the micelles are of sizes much smaller than those observed experimentally. Particle coagulation was observed to have occurred, which resulted in rapid growth of the nanoparticles. For coagulating systems, Suzuki et al.12 modified the growth mechanism by proposing a reduction in the coagulation rate above a critical diameter. A simplified mechanism was proposed by Jain et al.,16 which incorporated a size restriction on the growth of nanoparticles based on the size of the micelle in which the * To whom correspondence should be addressed. E-mail: mehra@ iitb.ac.in. † D.S. is currently a graduate student in the Department of Chemical Engineering, Massachusetts Institute of Techology, Cambridge, MA 02139.
(1) Kimijima, K.; Sugimoto, T. J. Phys. Chem. B 2004, 108, 3735. (2) Pileni, M. P. Langmuir 1997, 13, 3266. (3) Bagwe, R. P.; Khilar, K. C. Langmuir 1997, 13, 6432. (4) Dirk, L.; Hyning, V.; Zukoski, C. F. Langmuir 1998, 14, 28. (5) Manna, A.; Imae, T.; Iida, M.; Hisamatsu, N. Langmuir 2001, 17(19), 6000. (6) Sugimoto, T.; Kimijima, K. J. Phys. Chem. B 2003, 107, 10753. (7) Hirai, T.; Sato, H.; Komasawa, I. Ind. Eng. Chem. Res. 1994, 33, 3262. (8) Khiew, P. S.; Radiman, S.; Huang, N. M.; Soot Ahmad, Md. J. Cryst. Growth 2003, 254, 235. (9) Motte, L. J. Mater. Sci. 1996, 31, 38. (10) Lianos, P.; Thomas, J. K. J. Colloid Interface Sci. 1987, 117(2), 505. (11) Petit, C.; Lixon, P.; Pileni, M. P. J. Phys. Chem. 1990, 94, 1598. (12) Suzuki, K.; Harada, M.; Shioi, A. J. Chem. Eng. Jpn. 1996, 29, 264. (13) Hingorani, S.; Pillai, V.; Kumar, P.; Multani, M. S.; Shah, D. O. Mater. Res. Bull. 1993, 28, 1303. (14) Wang, G. H.; Li, G. L. Nanostruct. Mater. 1999, 11(5), 663. (15) Towey, T. F.; Khan-Lodhi, A.; Robinson, B. H. J. Chem. Soc., Faraday Trans. 1990, 86(22), 3757. (16) Jain, R.; Shukla, D.; Mehra, A. Langmuir 2005, 21, 11528.
particle grew. Further modifications in the model were made by Shukla and Mehra,17 who also accounted for the effect of the size of particles on the rate of coagulation. The synthesis of nanoparticles using reverse micelles has been modeled using population balance methods and the Monte Carlo technique. Natarajan et al.18 proposed a stochastic, two-stage approach for modeling the reduction of a metal salt to form metal nanoparticles. Tojo et al.19 carried out a Monte Carlo simulation to study the influence of synthesis variables on the particle size distribution of the nanoparticles. Li and Park20 used Monte Carlo simulations to estimate the particle size distribution in the synthesis of nanoparticles using reverse micellar solutions. Singh et al.21 improved the simulation scheme proposed by Li and Park. Ramesh et al.22 proposed a stochastic population balance model for studying the effect of the nucleation rate, intermicellar exchange rate, critical nucleation number, and concentration of the reactants on the size of the particles obtained. Jain and Mehra23 studied the effect of different initial reactant distributions and intermicellar exchange protocols on the particle size distribution of the nanoparticles. Though coagulation explains most of the experimental observations on particle size enhancement, for some systems it is likely that another mechanism prevails. For instance, experimental studies by Sugimoto and Kimijima1,6 suggest that Ostwald ripening occurs in the formation of AgX nanoparticles. Guo et al.24 have also indicated the possibility of Ostwald ripening and aggregation in the formation of La2(CO)3 nanoparticles. Liu et al.25 have developed a model for Ostwald ripening for the formation of β-carotene nanoparticles. The present work is aimed at quantitatively studying the process of Ostwald ripening on nanoparticles produced in the reverse micellar system. Specif(17) Shukla, D.; Mehra, A. Nanotechnology 2006, 17, 261. (18) Natarajan, U.; Handique, K.; Mehra, A.; Bellare, J. R.; Khilar, K. C. Langmuir 1996, 12, 2670. (19) Tojo, C.; Blanco, M. C.; Lopez-Quintela, M. A. Langmuir 1997, 13, 4527. (20) Li, Y.; Park, C. W. Langmuir 1999, 15(4), 952. (21) Singh, R.; Durairaj, M. R.; Kumar, S. Langmuir 2003, 19, 6317. (22) Ramesh, A. K; Hota, G.; Mehra, A.; Khilar, K. C. AIChE J. 2006, 277, 1556. (23) Jain, R.; Mehra, A. Langmuir 2004, 20, 6507. (24) Guo, G.; Gu, F.; Wang, Z.; Guo, H. J. Cryst. Growth 2005, 277, 631. (25) Liu, Y.; Kathan, K.; Saad, W.; Prud’homme, R. K. Phys. ReV. Lett. 2007, 98, 036102.
10.1021/la803684y CCC: $40.75 2009 American Chemical Society Published on Web 02/10/2009
Ostwald Ripening of SilVer Halide Particles
Langmuir, Vol. 25, No. 6, 2009 3787
δn ) (xβ - xR)
4πR2 δR Vm
(2)
where Vm is the molar volume. This states that the flux at the interface is equal to the amount transferred from one phase to the other by the change in size of the particle. From the diffusion equation, the change in the number of moles is also given by
δn ) 4πR2D+
|
∂x δt ∂r r)R
(3)
where D+ is the diffusivity of the solute in the liquid phase. Equating the right-hand sides of eqs 2 and 3
(xβ - xR)
|
∂x 4πR2 δR ) 4πR2D+ δt Vm ∂r r)R
Because VmD+ ) D
D Figure 1. Schematic of Ostwald ripening.
ically, the mechanism of Ostwald ripening as it applies to the size enhancement of AgCl particles is considered in detail, and a model is proposed based on Wagner’s theory27 of Ostwald ripening.
Theory of Ostwald Ripening
(5)
Now the diffusive, mass-transport equation in the liquid surrounding a spherical particle in spherical coordinates is written as
(
∂x ∂2x 2 ∂x )D 2 + ∂t r ∂r ∂r
)
(6)
and the boundary conditions for t g 0 are
The process of Ostwald ripening occurs because of the variation of the solubility of particles with the particle size. In this process, the smaller particles dissolve in the bulk fluid phase due to higher solubility and this dissolved material gets deposited on the larger particles which are relatively less soluble in the bulk phase. This difference in solubility is due to the Gibbs-Thomson effect. Thus, Ostwald ripening occurs by mass transfer through the bulk phase (Figure 1). The first theoretical model for Ostwald ripening was proposed independently by Lifshitz and Slyzohov26 and Wagner.27 They derived an expression for the growth rate of the average particle radius in the system. The size distribution of particles in the system evolves according to the following population balance equation:
∂n(R) ∂ + (n(R) G(R)) ) 0 ∂t ∂R
|
∂x dR ) (xβ - xR) ∂r r)R dt
(4)
(1)
where n(R) represents the number density of the particles having a radius between R and R + dR and G is the growth rate of a single particle of size R. The growth rate expression proposed by Wagner27 is considered here to explain the process. This particular approach is also known as the critical radius model.28 According to this model, there exists a critical radius Rcrit below which all particles in the system lose mass and above which all particles gain mass. The model is based on the process of Ostwald ripening in a two-phase system, a particle phase and an liquid (oil) phase. A detailed treatise on ripening has been given by Ratke and Voorhees.27 Consider a spherical particle of radius R in the liquid phase. The concentration of AgCl at any radius r and time t is denoted by x in the liquid phase. Let the equilibrium concentration at the interface (liquid side) be xR and the equilibrium concentration inside the particle be xβ. Let n be the number of moles present in a particle. The mass balance for the growing particle at the interface is given by (26) Lifshitz, I. M.; Slyozhov, J. J. Phys. Chem. Solids 1961, 19, 35. (27) Ratke, L.; Voorhees, P. W. Growth and Coarsening: Ripening in Material Processing; Springer: New York, 2002. (28) Finsy, R. Langmuir 2004, 20, 2975.
x(r)R, t) ) xR
(7)
x(rf∞, t) ) x
(8)
0
Solving eq 6 with the conditions in eqs 7 and 8 and assuming a quasi steady state, for the motion of the solid-liquid boundary, the solution becomes
x(r) ) x0 + (xR - x0)
R r
(9)
Evaluating the derivative of the mole fraction at r ) R and substitution into eq 5 yields
dR D x0 - xR ) dt R xβ - xR
(10)
From the Gibbs-Thomson effect, the solubility of the particle of radius R is given by
xR ) xR,e +
lR R
(11)
where lR is the capillary length. Similarly, for the bulk concentration of the dissolved material
x0 ) xR,e +
lR Rcrit
(12)
where Rcrit is the critical radius for the system. Thus, substituting eqs 11 and 12 into eq 10 results in
(
dR 1 1 DlR ) β dt (x - xR)R Rcrit R
)
(13)
Multiplying by 4πR2 to express the growth rate in terms of the rate of change of the particle volume gives
(
)
dVp 4πDlR R -1 ) β dt (x - xR) Rcrit
(14)
or equivalently in terms of the number of molecules in a particle
3788 Langmuir, Vol. 25, No. 6, 2009
Shukla et al.
(
)
dnp 4πDlRNA R -1 ) dt V (xβ - xR) Rcrit m
(15)
Stokes-Einstein equation is used to calculate this diffusion coefficient Dm:
where NA is the Avogadro number and Vm is the molar volume. The capillary length is further given by27 R
l )
2VmσxR(1 - xR)
(16)
(xβ - xR)RgT
where σ is the specific surface energy of solid AgCl, Rg is the gas constant, and T is the absolute temperature. Substituting this into eq 15 yields
(
)
dnp 8πσDNAxR(1 - xR) R -1 ) dt Rcrit (xβ - xR)2R T g
(17)
Substituting xR ) VmC∞, where C∞ is the normal, size-independent solubility of the solute in the liquid
(
)
dnp 4πDNAVm(1 - xR)C∞ R -1 ) dt Rcrit (xβ - xR)2R T g
(18)
At infinite dilution, xR , 1 and for a pure substance xβ f 1; therefore, (1 - xR)/(xβ - xR)2 f 1. Thus
(
)
dnp R - 1 C∞ ) 4πDNAR0 dt Rcrit
(19)
where R0 ) 2σVm/RgT. The equilibrium solubility C∞ is not expected to change with a change in the particle size distribution. Applying material balance using the equation derived above gives the value of Rcrit as the arithmetic mean of the particle radii at any time:
Rcrit
∫0∞ n(R) dR ) ∫0∞ n(R) R dR ∑
kBT 6πηR
(22)
Using the above equation for the diffusivity of the micelle-particle complex in the material balance equation
∫0∞ n(R)
dnp dR ) 0 dt
(23)
which states that the net loss or gain of solute by the particles is zero. Substituting the expression for dnp/dt from eq 19 in the above equation yields for the reverse micellar system
(
)
∫0∞ 4πR0DNA RRcrit - 1 C∞n(R) dR ) 0 ∫0∞
(
)
(24)
4πR0NAkBT R - 1 C∞n(R) dR ) 0 6πηR Rcrit
(25)
dR ∫0∞ n(R) dR ) ∫0∞ n(R) R
(26)
1 Rcrit
N
∑
1 1 1 1 ) ) Rcrit N p)1 R Rh
(27)
Second, C∞ should be replaced by some hypothetical equilibrium surface solubility as seen by the particles. This value for the solubility, to be used for the reverse micellar case, is calculated by equating the net flux of solute to the particle surface, due to collisions with other micelles, to the flux obtained by using the classical Ostwald ripening theory. The total number of collisions that a particle of radius R undergoes per unit volume per second is given by
(20) KR ) βqNmicn(R) dR
N
1 Rcrit ) R ) Rmean N p)1
Dm )
(21)
The first-principles model reported above is applicable to the system of particles suspended in a liquid where solute transfer takes place via diffusion through this liquid. In this case, the smaller particles shrink in size due to their enhanced solubility arising from their higher curvature, as compared to larger particles. In the reverse micellar system, when the particle size is comparable to the micellar size, the micelle-particle complex can be treated as a single entity which diffuses through the oil/liquid and transfers the solute from one particle to another through intermicellar collisions. When the micelle-particle complex encounters another particle-containing micelle, the dissolved AgCl molecules get transferred from one particle to another depending upon the difference in their sizes. Meanwhile, the particle which loses molecules from its surrounding water pool undergoes dissolution to compensate for the loss and replenishes the equilibrium concentration of AgCl in its water pool. No barrier to dissolution or incorporation of AgCl is considered in this model, and both the dissolution and incorporation events are assumed to be instantaneous. For this mechanism of solute transfer between particles to be consistent with the fundamental Ostwald ripening theory, the following aspects need to be considered. First, the molecular diffusivity of the solute should be replaced by the diffusivity of the micelle-particle complex. The
(28)
where β is the collision efficiency, q ) 8kBT/3η is the Brownian collision frequency, and Nmic is the number density of micelles. The number of collisions between micelles containing particles of radius R with those containing particles of radius R′ is given by
KRR ) βqNmicn(R) dR
(∫
n(R′) dR′
∞
0
n(R′′) dR′′
)
(29)
In a collision between a micelle containing a particle of radius R with another micelle containing a particle of radius R′, the number of molecules transferred from one particle to another is calculated by setting the final dissolved solute concentration in their respective water pools as the average concentration value. For example, if a micelle containing n1 molecules of dissolved solute collides with another micelle containing n2 dissolved solute molecules, then each daughter micelle after collision contains (n1 + n2)/2 molecules. The expression for the number of moles of solute transferred from one micelle to the other in a single collision event is given by
( R′1 - R1 )
nRR ) NAVmicCw∞R0
(30)
where, Vmic is the volume of water in a single micelle and Cw∞ is the bulk solubility in water. The solubility in water pools of micelles is calculated using the Kelvin equation:
Ostwald Ripening of SilVer Halide Particles
Langmuir, Vol. 25, No. 6, 2009 3789
C(R) ) Cw∞ exp(R0/R) ≈ Cw∞(1 + R0/R)
(31)
Table 1. Properties of the AgCl System property
Therefore, the total number of moles of solute transferred per second to the particles of radius R is given by
KRRnRR ) 4 βqNmicn(R) dR ∞ NAVmicCw∞R0 1 1 - n(R′) dR′ ∞ 0 4 R′ R n(R′′) dR′′
NR )
∫0∞
[∫
∫0
(
)
]
(32)
This must also be equal to the rate of change of the number of molecules for all the particles of radius R, calculated using eq 19, so that
NR ) n(R) dR
(
NR ) 4πR0DmNA
dn dt
(33)
)
R - 1 C∞n(R) dR Rcrit
(34)
Using the Stokes-Einstein equation for the particle diffusivity from eq 22
NR )
( (
) )
4πR0kBTNA R - 1 C∞n(R) dR 6πηR Rcrit
(35)
2R0kBTNA R - 1 C∞n(R) dR 3ηR Rcrit
(36)
NR )
Equating the right-hand sides of eqs 32 and 36 results in
(
)
2R0kBTNA R - 1 C∞ ) 3ηR Rcrit βqNmic
∫0∞ n(R′′) dR′′
[∫
∞
0
NAVmicCw∞R0 1 1 - n(R′) dR′ 4 R′ R
(
)
]
(37)
Simplifying the above equation by substituting the value of q and evaluating the integrals
(
)
(
)
R R - 1 C∞ ) βNmicVmicCw∞ -1 Rcrit Rh
(38)
From material balance, Rcrit ) Rh (eq 27) and C∞ is a constant. Thus, the expression for C∞ is given by
C∞ ) βNmicVmicCw∞
(39)
The above equation for C∞ can also be written as
C∞ ) β
VwCw∞ Vw + Vo
(40)
where Vw and Vo are the total volumes of water and oil in the reverse micellar solution. When Vo goes to zero, C∞ is equal to Cw∞. On subsituting the expression for C∞ from eq 39, the final equation for the rate of change of the number of molecules is given by
(
)
dnp R -1 ) 4πβR0NADmNmicVmicCw∞ dt Rcrit
(41)
The rate of change of the number of molecules in a particle is related to the radius of the particle (assuming a spherical particle).
symbol
1,6
surface energy viscosity1,6 mean molar volume1,6 number density of micelles1,6 volume of the micellar core1,6 temperature1,6 bulk solubility in water1,6 collision efficiency17
value 2
σ η Vm Nmic Vmic T Cw∞ β
0.1 J/m 0.99 mPa s 25.78 cm3/mol 2.382 × 1024 m-3 2.266 × 10-27 m3 298 K 8.22 × 10-7mol/dm3 0.01
Therefore, the rate of change of the radius of a particle with np molecules is given by
( )
(
)
kBT dR R 1 NmicVmicCw∞ -1 2 ) βR0Vm dt 6πηR Rcrit R
(42)
Methodology Sugimoto and Kimijima1,6 have synthesized AgCl nanoparticles using reverse micelles to elucidate the particle formation mechanism. They conclude that Ostwald ripening is the primary size enhancement process for this system. The system under consideration is formation of AgCl particles from AgNO3 and KCl in NP-6/cyclohexane solution. Instead of starting from the nanoparticle formation process, a system of nanoparticles with different size distributions is used as an initial condition for simulation. The model proposed here applies to the period after the nucleation and growth (due to precipitation of liquid AgCl on existing particles) of the particles is completed and Ostwald ripening is the only active growth mechanism. The particle size distribution (PSD) or standard deviation in the mean diameter is not reported by Sugimoto and Kimijima.1,6 The standard deviation in the mean particle diameter (before ripening) has been taken to be equal to the values observed experimentally in reverse micellar systems.3,16,17,23,29,32 The rate of ripening depends on the breadth of the particle size distribution. For example, a distribution which resembles a δ function will not show any growth due to ripening as all particles have the same solubility. On the basis of experimental information, two Gaussian distributions, one with a small mean and variance and another with a large mean and variance, are combined to define the initial distribution. The differential equation for each particle was solved along with the overall mass balance equation using the explicit Runge-Kutta method. The time step was chosen to be 0.01 s. The number of particles was increased from 100 to 1000 to obtain the critical number of particles above which the results are independent of the number of particles. This value was found to be around 800. Table 1 shows the material and system properties used in simulation. The procedure used to simulate the growth of the nanoparticles due to Ostwald ripening is as follows: (1) The initial population of particles is distributed according to the assumed initial distribution into bins of different radii. (2) The rate of the growth of each particle is calculated on the basis of its radius and the critical radius of the system at that time. (3) The number of molecules in a particle is updated on the basis of its growth rate. (4) Particles above the critical radius gain mass, particles below the critical radius lose mass, and the radii of all particle are updated. (5) Steps 2-4 are repeated at each time step.
Results and Discussion: Time Evolution The computed results obtained from the model, derived from first principles, are indeed able to explain Ostwald ripening phenomena. The experimental results closely match the simulation estimates in terms of mean diameter variation with respect to time as shown in Figure 2. The time scale for reaction between (29) Husein, M.; Rodil, E.; Vera, J. Langmuir 2003, 19, 8476. (30) Shukla, D.; Mehra, A. Langmuir 2006, 22, 9500. (31) Jain, R.; Shukla, D.; Mehra, A. Ind. Eng. Chem. Res. 2006, 45, 2249. (32) Sugih, A. K.; Shukla, D.; Heeres, H. J.; Mehra, A. Nanotechnology 2007, 18, 035607.
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Figure 4. Moments of the particle size distribution as a function of time. Figure 2. Variation of the mean diameter with time for the AgCl system. The predicted time evolution profile is compared with the experimental results of Sugimoto and Kimijima.1,6 The time evolution of the corresponding particle size distributions is shown in Figure 3.
silver and chloride ions on mixing the two reverse micellar solutions of AgNO3 and KCl is expected to be around 100 ms.16,17,23 The subsequent growth of particles can take place due
Figure 3. Evolution of the particle size distribution of AgCl nanoparticles with time.
Ostwald Ripening of SilVer Halide Particles
Langmuir, Vol. 25, No. 6, 2009 3791
The variation of the particle size distribution with respect to time is shown in Figure 3. The evolution shows that the spread of the distribution increases with time. The process of Ostwald ripening will cause smaller particles to dissolve and larger particles to grow. All the dissolving particles do not dissolve completely, giving rise to particles of smaller sizes which did not exist earlier. It can also be seen that the peak in the PSD for the small size particle decreases rapidly due to the high solubility of small particles as compared to the larger ones. Similarly, the large particles also grow to larger radius particles though at a much slower rate than small particles due to their large surface area. The particle size distribution evolves rapidly initially, and then the growth rate diminishes, which results in slow evolution of the particle size distribution.
Figure 5. Rate of growth plotted for different particle sizes for three different critical radii.
to either coagulation or Ostwald ripening. Jain et al.16 have shown that the evolution of the mean diameter of CdS nanoparticles reached a constant value at around 10 s. Figure 2 also shows that the time scale for particle growth is in hours, which implies that particles grow via a slow growth process such as Ostwald ripening. The particle size of 5.4 nm is chosen as the zero point of the aging time on the basis of the experimental results of Sugimoto and Kimijima.1,6 The mean particle size at 6 h is approximately 8 nm, which is much larger than the micelle size of 1.51 nm. Therefore, the mean particle size for AgCl nanoparticles is apparently not limited by the native size of the micelle.
The evolution of the moments of the particle size distribution is shown in Figure 4. The zeroth moment indicates that the total number of particles in the system decreases with time. The first moment, which signifies the mean diameter of the particles, increases with time, signifying particle growth. The second moment (not shown) indicates that the area decreases with time due to the decrease in the number of small particles and subsequent increase in the mean diameter. The third moment, which denotes the total volume of particles, remains constant. The mass balance and hence the volume balance are satisfied as seen from the constant value of unity for the third moment. The growth of the particles is due to the difference in solubilities of the particles on account of their curvature, so the process of Ostwald ripening does not bring about an overall increase in the average particle mass, but only increases the mean size of the particles by redistributing the mass among the particles; thus, the particle concentration has no role to play in the growth process. The
Figure 6. Variation of the mean diameter with time for the AgCl system as reported experimentally and as obtained by simulation for two cases of initial particle size distribution. Case 1 represents the initial PSD with a large variance, and case 2 represents the initial PSD with a small variance.
3792 Langmuir, Vol. 25, No. 6, 2009
Figure 7. Time evolution of the particle size distribution of AgCl nanoparticles for case 1, which has an initial PSD with a large variance.
growth of the particles is taken to occur via mass transfer due to collisions between micelles. It is observed that the initial distribution plays a crucial role in determining the course of evolution of the particle size via Ostwald ripening. Typical characteristics of the initial distribution that may lead to considerable ripening are analyzed. From eq 42, it can be seen that the growth rate of a particle is a function of the critical radius Rcrit, which is a harmonic mean of the radii of all the particles present in that system (eq 27). Further, the growth rate has an inverse R3 dependence. Figure 5 illustrates typical relative growth rates of particles (dR/dt) as a function of the radius. Particles of very small size are expected to have a high growth rate and, therefore, shrink much faster (Figure 5) as compared to larger particles. Smaller particles dissolving at a fast rate give rise to an effective rise in the mean diameter. When the particle size decreases below a certain threshold diameter, it is considered to be dissolved. The threshold diameter is chosen to be on the order of 5 × 10-10 m (size of a AgCl molecule). The dependence of the rate of change of the radius on the critical radius of the system is linear as expected from eq 42. Growth of the mean diameter of the system is mainly accounted for by the dissolution of smaller particles rather than size growth of large particles. Figures 3 and 5 corroborate this observation. To test the dependence of the growth rate on the initial particle size distribution, two extreme cases of particle size distribution are tested. The initial distribution for the AgCl system was constructed by combining two Gaussian distributions, one with a small variance and the other with a large variance. If the number of particles comprising one of the Gaussian curves is changed significantly, the time scale of ripening varies considerably. Simulations are performed for the system with high variance (case 1) and another system with low variance in the mean particle size. Figure 6 shows the time evolution of the mean particle diameter for these systems. For case 1, which comprises particles with a large variance, shows a time evolution profile very similar to the experimental data. For case 2, which comprises small variance particles, the growth rate is very small due to the small difference in solubility of the particles, which leads to a time evolution profile very different from the experimental data. The particle size distributions for the two cases are shown in Figures 7 and 8, respectively. These results clearly show that the results depend on the width of the
Shukla et al.
Figure 8. Time evolution of the particle size distribution of AgCl nanoparticles for case 2, which has an initial PSD with a small variance.
distribution of particles. In this paper, we have shown that for distributions with experimentally observed values of variance, the model is able to predict the experimental results accurately. The results shown for the AgCl system above support the claim that the particle growth mechanism proposed in this paper is responsible for AgCl particle growth. The reduction in the particle size in the reverse micellar system is mainly due to the small growth rate of nanoparticles, which was caused by a very low value of bulk solubility felt by the particles in the reverse micellar solution. The current model is different from the classical Ostwald ripening theory. In this model, the exchange of material between particles takes place via collision between particlecontaining micelles and not via diffusion through the bulk phase. In our previous work16,17,30,31 on the growth mechanism of CdS nanoparticles, it was shown that the adsorption of surfactant on the particle surface limits the growth of particles. The mechanism responsible for growth of CdS nanoparticles is coagulation. For AgCl particles, it has been experimentally observed that, despite high adsorption of surfactant molecules on the particle surface, it does not inhibit the growth of particles. This shows a difference in the particle formation mechanism between CdS and AgCl. For CdS particles the adsorption of surfactant molecules on the surface hinders the coagulation of two colliding particle-containing micelles, whereas for AgCl particles which grow via a diffusional process such steric hindrance is not important. Therefore, the dominant particle growth mechanism for a particular system depends on the material properties.
Conclusion This study presents a framework to simulate Ostwald ripening of nanoparticles formed in reverse micellar or microemulsion systems. The simulation results demonstrate that it is possible to predict the Ostwald ripening of nanoparticles in reverse micellar systems from first principles by incorporating an intermicellar solute transfer mechanism from one particle to another in the general theory of Ostwald ripening. The experimental data of Sugimoto and Kimijima1,6 are found to be in good agreement with the computed results. Acknowledgment. This study was partially supported by the research grant titled “Engineering Aspects of Ultrafine Particle
Ostwald Ripening of SilVer Halide Particles
Technology” and made available under the IRPHA scheme of the Department of Science & Technology, Government of India, New Delhi.
Notation C∞ Cw∞ D Dm d0 G(R) Kn kB KR KRR′ Ksp lR MAgCl NA N Nm Nmic nB np n(R)
solubility of AgCl in the RM system, mol/dm3 solubility of AgCl in water, mol/dm3 diffusion coefficient of AgCl in the oil phase, m2/s diffusion coefficient of a micelle in the oil phase, m2/s diameter of a molecule of AgCl, nm growth rate of a particle of radius R, m/s nucleation frequency for a micelle with a MAN value of i, s-1 Boltzmann constant, 1.3806503 × 10-23 (m2 kg)/(s2 K) total number of collisions of a micelle with a particle of radius R per unit volume, m-3 number of collisions of a micelle with a particle of radius R with micelles with particles of radius R′ per unit volume, m-3 solubility product of the precipitate, mol2 m-6 capillary length, m molar weight of AgCl Avogadro’s number, 6.023 × 1023 mol-1 number of particles included for simulation number of micelles included for simulation number density of micelles, m-3 number of moles of AgCl in a particle number of molecules of AgCl in a particle number density of particles having a radius between R and R + dR
Langmuir, Vol. 25, No. 6, 2009 3793
R Rg Rcrit Rmean Rh t T V V0 Vm Vmic Vp x x0 xR xβ
radius of a particle of AgCl, nm universal gas constant critical radius for Ostwald ripening, nm mean radius of the particles, nm harmonic mean radius of the particles, nm time, s temperature on the absolute scale, K mean volume of the system at any time t, m3 mean volume of the system at time t ) 0, m3 molar volume of AgCl, m3 volume of the micellar core, m3 volume of a particle of AgCl, m3 concentration of the particle at any radius r and time t, mol/dm3 concentration of the particle at any radius r and time t ) 0, mol/dm3 equilibrium concentration of AgCl in the particle phase, mol/dm3 equilibrium concentration of AgCl in the oil phase, mol/ dm3 Greek Letters
β η F σ
collision efficiency viscosity of the continuous phase, Pa s density of the particle, kg m-3 interfacial tension between the particle and bulk phase, Pa m-1
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