A Mathematical Model for Crystal Growth by Aggregation of Precursor

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J. Phys. Chem. B 2005, 109, 23879-23887

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A Mathematical Model for Crystal Growth by Aggregation of Precursor Metastable Nanoparticles Timothy O. Drews,† Markos A. Katsoulakis,‡ and Michael Tsapatsis*,† Department of Chemical Engineering and Materials Science, UniVersity of Minnesota, 151 Amundson Hall, 421 Washington AVe. SE, Minneapolis, Minnesota 55455, and Department of Mathematics and Statistics, UniVersity of Massachusetts, 710 North Pleasant St., Amherst, Massachusetts 01003-9305 ReceiVed: July 7, 2005; In Final Form: September 22, 2005

A mathematical model is developed to describe aggregative crystal growth, including oriented aggregation, from evolving pre-existing primary nanoparticles with composition and structure that are different from that of the final crystalline aggregate. The basic assumptions of the model are based on the ideas introduced in an earlier published report [Buyanov and Krivoruchko, Kinet. Katal. 1976, 17, 666-675] to describe the growth of low-solubility metal hydroxides (e.g., iron oxides) by oriented aggregation. It is assumed that primary particles can be described as pseudo-species A, B, and C, which have the following properties: (1) fresh primary particles (colloidally stable inert nanoparticles, denoted as A), (2) mature primary particles (partially transformed nanoparticles at an optimum stage of development for attachment to a growing crystal, denoted as B), and (3) nucleated primary particles (denoted as C1). The evolution of primary particles, A f B f C1, is treated as two first-order consecutive reactions. Crystal growth via crystal-crystal aggregation K′

(Ci + Cj 98 Cij) is described using the Smoluchowski equation. The new element of this model is the inclusion of an additional crystal growth mechanism via the addition of primary particles (B) to crystals (Ci): K

(B + Ci 98 Ci+1). Two distinct, but constant, kernels (K * K′) are used. It is shown that, when K′ ) 0, a steplike crystal size distribution (CSD) is obtained. Within a range of K′/K values (e.g., K′/K ) 103), CSD with multiple peaks are obtained. Comparison with predictions of models that do not include the intermediate stage of primary particles (B) indicates pronounced differences. Despite its simplicity, the model is able to capture the qualitative features of CSD evolution that have been obtained from crystal growth experiments in hematite, which is a system that is believed to undergo oriented aggregation.

Introduction The synthesis of nanocrystals and their assemblies with controlled morphologies remains a subject of intense fundamental interest for applications ranging from catalysis to electronic and photonic materials.1-7 In certain cases of crystal growth from colloidal sols (mother liquors), aggregative mechanisms,8-10 including that referenced as epitaxial11 and oriented or imperfect (misoriented) aggregation,12 have been proposed to describe the formation and/or growth of crystals from coexisting primary particles.13-19 A common ingredient of these aggregative mechanisms is the postulate of direct participation of the primary particles to nucleation and/or growth by various coalescence processes. They are mainly motivated by electron microscopic examination of crystals indicating mosaic or highly branched morphologies with domains on the order of the primary particle. Although alternative scenarios based on diffusion-limited anisotropic growth mechanisms may also account for mosaic20 and branched morphologies,21 there are cases where strong additional evidence points to an aggregative mechanism of primary particles being dominant, at least at an early stage of crystal nucleation and growth.14,15,17,18 * To whom correspondence should be addressed. E-mail: tsapatsi@ cems.umn.edu. † University of Minnesota. ‡ University of Massachusetts.

The primary particles may be crystalline nanoparticles or microparticles with almost the same crystal structure and composition as the resulting crystals, or crystalline nanoparticles of a different structure than that of the final crystals, or nanoparticles with a narrow particle size distribution but unknown composition and structure.22-26 Moreover, primary particles may be pre-formed19 or self-assembled at time scales much smaller than that of crystal growth22 or continuously generated via a nucleation mechanism.8,9 Here, we introduce a mathematical model to describe aggregative crystal growth, including oriented aggregation, from evolving pre-existing primary nanoparticles with composition and structure that are different from that of the final crystalline aggregate. The basic assumptions of the model are based on the ideas introduced in refs 14 and 15 to describe the growth of low-solubility metal hydroxides (e.g., iron oxides) by oriented aggregation. We believe that these ideas may also be relevant for the early stages of nucleation and growth of the siliceous zeolite TPA-ZSM-5, [Si96O192]-MFI.27 Although the mathematical model is rather general, we introduce its main elements by associating it with the previously mentioned two systems. As noted in refs 14 and 15, precipitates consisting of primary particles with sizes 3-4 nm have been observed in mixtures created by pouring together solutions of FeCl3 and NaOH. Similar nanometer-sized particles are formed when tetraethyl orthosilicate (TEOS) is added and hydrolyzed in tetrapropyl-

10.1021/jp0537299 CCC: $30.25 © 2005 American Chemical Society Published on Web 11/24/2005

23880 J. Phys. Chem. B, Vol. 109, No. 50, 2005 ammonium (TPA) hydroxide aqueous solution. The noncrystalline 3-5 nm hydrated TPA-silica nanoparticles form stable colloidal suspensions.16,18,22,28 In both of the previously mentioned cases, the nanometer-sized primary particles are not thermodynamically stable structures but do undergo spontaneous transformation with no pronounced change in particle size. Moreover, they have detectable composition changes that are possibly the result of internal rearrangements. For the case of iron oxides,15 these rearrangements have been described as depolymerization followed by formation of ordered regions. For the case of hydrated TPA-silica nanoparticles, the rearrangements may consist of the condensation of silicate species and incorporation of TPA in the interior of the nanoparticle.29,30 After a clearly identifiable and often extended (e.g., several months at room temperature for TPA-ZSM-5)30 induction period over which only primary particles are observed, detectable crystallinity (by X-ray or electron diffraction and imaging) and crystal growth are observed. In both systems, at early stages of growth, primary particles coexist with the growing crystals and the crystals have mosaic or branched morphologies.14,15,30 The aforementioned observations point to an aggregative growth mechanism. Moreover, the particle domains are almost aligned in a way19,30 that has been attributed to oriented aggregation.17 On the basis of the aforementioned general observations and the ideas put forward in refs 14 and 15, the following differentiation between primary particles can be made (see ref 15): (1) Fresh primary particles (A): these are colloidally stable nanoparticles that do not aggregate and do not contribute to crystal growth by attachment to crystals or nuclei. (2) Mature primary particles (B): these are partially transformed nanoparticles that are capable of attachment to a growing crystalline aggregate or nucleus but are not capable of aggregating with other mature primary particles. (3) Nucleated primary particles (hereafter referenced as “nuclei”) (C): these are nanoparticles that have the same crystal structure as the crystal aggregates that can attach to growing crystals and mature primary particles (B). Arguments supporting this differentiation of primary particles for the case of iron oxides are described in detail in refs 14 and 15 and, therefore, are not repeated here. For the case of TPAZSM-5, the stability of the fresh primary particles (A) has been attributed to steric stabilization by TPA present on their external surface22 and to a higher surface potential, compared to that of TPA-ZSM-5.31 The intermediate stability of mature primary particles (B) can be attributed to an intermediate surface charge (between that of fresh particles and zeolite), decreased surface stabilization that is due to TPA incorporation from the external surface to the interior of the nanoparticles, or decreased solubility (or rate of dissolution) that is due to a partially condensed and TPA-stabilized, zeolite-like internal structure. It is assumed that mature primary particles are at the optimum stage of development to continue aggregate growth by attachment, because they are destabilized, with respect to fresh primary particles, and they are not yet as fully developed as TPAZSM-5 nanocrystals to require precise matching for attachment during growth. Finally, the properties of nuclei (C) closely follow the known properties (surface charge)16 and ability for oriented attachment26 of TPA-ZSM-5 crystals. Although a statistically distributed population of primary particles, in terms of composition and possibly size, better reflects the evolutionary process of primary particles, the simple aggregation model that we formulate here using the A, B, and C pseudo-species captures the essential features of such a

Drews et al. TABLE 1: List of Parameters, Variables, and Reactions Associated with Mechanism I expression Reactions k1

A 98 B k2

B 98 C1 Ki

B + Ci 98 Ci+1 Ki ) constant ) K Parameters a ) k1/k2 b ) nA0K/k2 Variables xA ) nA/nA0 xB ) nB/nA0 y ) nj/nA0 f ) 1 - x A - xB τ ) k2t Equations xA ) e-aτ dy/dτ ) xB dxB/dτ ) ae-aτ - xB - bxBy i-1

eby - 1 byi )

∑(b y /k!) k k

k)1

eby

distribution. It will be shown that this model is able to capture some of the qualitative experimental trends, indicating that extension to a model that incorporates elements of a moredetailed treatment of oriented aggregation,17,32 and a description of the primary particle evolution that is coupled with solutionphase chemistry, is a worthy undertaking. Although the aging and coalescence mechanism simulated here is similar to previously studied mechanisms in some ways, there are fundamental differences in it. A novel element in the model is the differentiation of the state of primary particles, with intermediate, partially transformed primary particles being the optimal building materials for aggregates. Model Formulation The primary particle aging mechanism is a two-step mechanism. Coalescence is permitted when a particle is either a mature primary particle, a nucleus, or a crystal. The coalescence of a mature primary particle and any size crystal can be governed by one type of kernel, while the nuclei-crystal and crystalcrystal coalescence can be governed by another. The mechanism simulated here is more general than that proposed in refs 14 and 15, in that coalescence of crystals is permitted (in the references, only mature primary particle addition to nuclei and crystals is proposed). Each particle in the population can be a fresh primary particle (A), a mature primary particle (B), or a crystal (Ci, where i denotes the number of particles that comprise the crystal with i ) 1 for nuclei). The population balance model starts at t ) 0 with all particles as fresh primary particles. As particles coalesce, they form crystals in a manner that conserves mass in the system (i.e., when the particles and crystals coalesce, the mass of the particles is assumed to be additive). Mechanism I. In mechanism I, which is summarized in Table 1, the fresh primary particles (A) are consumed only via the k1

aging process A 98 B:

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dnA ) -k1nA dt

The mature primary particles (B) are consumed via the aging k2

Ki

and coalescence processes B 98 C1 and B + Ci 98 Ci+1 respectively:

dnB dt

dy1 ) xB - bxBy1 dτ

(1)

dyn-1 ) bxB(yn-2 - yn-1) dτ

ni ∑ i)1

nA0

d(

ni) ∑ i)1 dt

) k2nB

(3)

Equation 3 has a summation of cluster numbers to nA0. It is possible to rewrite eq 3 as nA0-1

d(

ni) ∑ i)1 dt

) k2nB

(4)

dyn-1 ) b(yn-2 - yn-1) dy

(for n g 3)

(9b)

The system of equations shown in eq 9 can be solved to give a general equation for yi: i-1

eby - 1 -

∑

k)1

byi )

( ) bkyk k!

(10)

eby

where yi ) ni/nA0. One approach to obtain the CSD is to solve eqs 6 and 7 numerically and then use eq 10 to obtain the yi values. We followed this approach here. Alternatively, one can use the quasi-steady-state (QSS) approximation for xB and find y(τ) and yi(τ) analytically. For some of the parameters studied here, the QSS approximation was not valid and, therefore, was not used. Mechanism II. Mechanism II, which is summarized in Table 2, contains all of the components of mechanism I, but it also allows for the coalescence of crystals of any size, TABLE 2: List of Parameters, Variables, and Reactions Associated with Mechanism II Reactions k1

A 98 B Ki

B + Ci 98 Ci+1 Ki

(5)

B + Ci 98 Ci+1 Kij

nA0-1 where nj ) ∑i)1 ni and nA0 is the initial number of fresh primary particles. Equations 4 and 5 can be nondimensionalized:

dxB ) ae-aτ - xB - bxBy dτ

(6)

dy ) xB dτ

(7)

where xB ) nB/nA0, y ) nj/nA0, a ) k1/k2, b ) nA0K/k2, and τ ) k2t. Equations 6 and 7 and the solution of eq 1 (xA ) e-aτ, where xA ) nA/nA0), can be solved to give the total number of fresh primary, mature primary, and crystalline particles as a function of time, but give no information about the crystal size distribution (CSD). To obtain the CSD, a kinetic equation that describes the rate of change of the number of particles of each size i, as a function of time, is written. We determined that these kinetic equations have an analytical solution that provides yi, where yi ) ni/nA0, if y is known. A kinetic equation can be written for each particle size

(8b)

(9a)

expression

because dnnA0/dt ) 0 for all times, except possibly at the end of the simulation (or the end of the physical process). The solution of eq 1 can be substituted into eq 2,

dnB ) k1nA0e-k1t - k2nB - KnBnj dt

dy1 ) 1 - by1 dy

(2)

where Ki is the kernel for coalescence of a mature primary particle with any crystal and ni is the particle density of crystal i. Here, we consider the simple case where all Ki are the same for the coalescence of a mature primary particle and any size crystal, so Ki ) K. The summation on eq 2 is only to (nA0 - 1), because, after there is a particle in bin nnA0, all of the initial particles will be a component of that crystal. Next, an equation for all crystals is written. It is convenient to add the numbers of all crystals and then determine how the total number of crystals varies over time. The rate of change of the total number of crystals is

(for n g 3)

Equation 7 can be substituted into eqs 8 to give

nA0-1

) k1nA - k2nB - KinB

(8a)

Ci + Cj 98 Ci+j Ki ) constant ) K Kij ) constant ) K′ a ) k1/k2 b ) nA0K/k2 c ) nA0K′/k2 xA ) nA/nA0 xB ) nB/nA0 y ) nj/nA0 f ) 1 - x A - xB τ ) k2t

Parameters

Variables

Equations xA ) e-aτ -aτ dxB/dτ ) ae - xB - bxBy dy 1 ) xB - cy2 dτ 2

()

dyk dτ

(∑

nA0

i)k-1

) by1yk-1 - by1yk + c

1

2

i)1 j)k-i

yiyj - yk

∑y i)1

)

i

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Figure 1. Normalized number of fresh primary particles (xA ) nA/nA0), mature primary particles (xB ) nB/nA0), the sum of all crystals (y ) nj/nA0), and crystallinity (f ) 1 - xA - xB), as a function of τ, for (a) a ) 0.01 and b ) 10, (b) a ) 0.01 and b ) 1000, (c) a ) 1 and b ) 10, (d) a ) 1 and b ) 1000, (e) a ) 100 and b ) 10, and (f) a ) 100 and b ) 1000. In panel a, xB is multiplied by a factor of 50, in panel b, xB is multiplied by a factor of 100 and y is multiplied by a factor of 10, in panel d, xB and y each are multiplied by a factor of 5, and in panel f, y is multiplied by a factor of 5. Kij

Ci + Cj 98 Ci+j (which is called higher-order coalescence). When higher-order coalescence events are permitted, the kernel for the coalescence of any two crystals is also nonzero and equations analogous to eq 6 and 7 can be written to track the total number of mature primary particles (B) and the total nA0-1 number of crystals (nj ) ∑i)1 ni), as a function of time. Smoluchowski’s coagulation equation can be written as33

dnk dt

) Kij

(

nA0

1i)k-1

2

)

ninj - nk∑ni ∑ i)1 i)1

j)k-i

(11)

nA0

d(

∑nk)

k)1

dt

)-

() 1

2

K′(

nA0

nk)2 ∑ k)1

(12)

Equation 12 can be nondimensionalized and combined with eqs 6 and 7 to give an equation set to describe xB and y as a function of time when K′ > 0:

(13)

1 dy ) xB - cy2 dτ 2

(14)

()

where c ) nA0K′/k2. The CSD can also be computed numerically for the case where K and K′ are nonzero by solving eq 13, eq 14, and

dyk dτ

Here, we consider the simple case where Kij ) K′. When eq 11 is summed over all sizes of nucleated particles (i.e., over all k), eq 11 reduces to

dxB ) ae-aτ - xB - bxBy dτ

) by1yk-1 - by1yk + c

(

1i)k-1

2

nA0

)

yiyj - yk∑yi ∑ i)1 i)1

j)k-i

(15)

The population balance simulations were performed by solving the discrete form of the Smoluchowski equation, and these simulation results were verified with an analytical solution for the Smoluchowski equation for special cases. The ordinary differential equation (ODE) models presented in the Model Formulation section were solved numerically. The numerical simulations were performed with Matlab software, using an ordinary differential equation solver (ODE45). Results and Discussion Mechanism I. Simulations were first performed with mechanism I, which is summarized in Table 1, to determine the

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Figure 2. Crystal size distribution (CSD) at various values of τ for (a) a ) 0.01 and b ) 10, (b) a ) 0.01 and b ) 1000, (c) a ) 1 and b ) 10, (d) a ) 1 and b ) 1000, (e) a ) 100 and b ) 10, and (f) a ) 100 and b ) 1000.

behavior of the simulation model with the aging model and mature primary particle addition only. Of the mechanisms presented here, mechanism I is the closest to that described by Park et al.,34 in that they present a monomer addition model with a source; however, direct comparison with their results is complex, because their aging model and coalescence kernel are fundamentally different than ours and, therefore, it will not be attempted here. In mechanism I, crystal growth ceases when all fresh and mature primary particles have been consumed. In the simulations, the parameters a and b were selected to show the behavior of the model over a wide range of parameter space. In Figure 1, xA, xB, y, and the degree of crystallinity (f, which is defined as 1 - xA - xB) are shown as a function of τ (fresh primary particles (A) and mature primary particles (B) are considered to be noncrystalline, or with a different crystal structure than particles Ci). In Figure 2, the CSD is shown for several values of τ for each parameter set. For presentation purposes, the CSD is plotted as a function of i1/3, where i is the number of particles in a crystal and represents a lower limit of crystal radius (i.e., assuming no void space present in the crystalline aggregate). In reality, considerable internal porosity is present in the particles, and, therefore, the actual radii may be significantly increased. Moreover, for fractal crystalline aggregates, the crystal size may scale as i(1/n) with n < 3, e.g., n ) 2.5.35 In all of the simulations shown in Figure 1, there is an exponential decrease in xA as a function of τ. All of the simulations show a peak in xB and the peak height in xB increases as a increases. Because a is the ratio of k1 to k2, increasing a is analogous to increasing the rate of conversion of fresh primary particles to mature primary particles, relative to the rate of conversion of mature primary particles to C1 crystal nuclei, which explains why the peak in xB is higher when a is increased,

i.e., more mature primary particles accumulate at early stages of nucleation and growth. The degree of crystallinity of the dispersed solid particles, which can be experimentally deduced by collecting all particles and analyzing them by X-ray or electron diffraction, can be estimated in the simulations from the quantity f ) 1 - xA - xB. In all of the simulations shown in Figure 1, f starts at zero when τ ) 0 and monotonically increases toward f ) 1 (i.e., up to complete conversion of all the metastable primary particles to crystals). The shape of f is sigmoidal and it resembles typical experimental observations.36 The highest rate of increase for f is established near the time when xB reaches its maximum value. This high crystallization rate is retained for some time and is then reduced due to primary particle depletion. The total number of crystals (y) increases rapidly when xB is large and approaches a plateau value as xB approaches zero. When all mature primary particles (B) have been consumed, then the simulation is complete. In typical experiments, other phenomena, including Ostwald ripening and aggregation, will sustain the evolution of the CSD. Here, we do not account for Ostwald ripening, but aggregation between crystals will be considered in mechanism II. As parameter a is increased, the dimensionless time required to reach the end of the aggregation process is shortened. Moreover, for a given value of a, as b is increased, larger particles are formed, as indicated by the smaller plateau value of y in these cases. This is shown more clearly in Figure 2. Figure 2 shows that there are two different situations that arise in the growth of the crystals. When b is small, which is the case in Figure 2a, 2c, and 2e, most of the crystals that have formed are small and the number of crystals of each size decreases as the crystal size increases. At the point when all of

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Figure 3. Normalized number of fresh primary particles (xA ) nA/nA0), mature primary particles (xB ) nB/nA0), the sum of all crystals (y ) nj/nA0), and crystallinity (f ) 1 - xA - xB), as a function of τ, for (a) a ) 0.01, b ) 10, c ) 0.01; (b) a ) 0.01, b ) 1000, c ) 1; (c) a ) 1, b ) 10, c ) 0.01; (d) a ) 1, b ) 1000, c ) 1; (e) a ) 100, b ) 10, c ) 0.01; and (f) a ) 100, b ) 1000, and c ) 1. In panel a, xB is multiplied by a factor of 50, in panel b, xB is multiplied by a factor of 100 and y is multiplied by a factor of 10, in panel d, xB and y are each multiplied by a factor of 5, and in panel f, y is multiplied by a factor of 5.

the mature particles have been consumed and crystal growth is complete, the CSD is narrow. When b is increased, which is analogous to increasing K and is shown in Figure 2b, 2d, and 2f, the CSD eventually evolves as a moving front toward larger crystals, leaving behind a constant number of crystals of smaller size. When crystal growth is complete, a wider CSD is established that may be described as steplike. Note that the mechanism proposed in refs 14 and 15 is closely replicated by mechanism I (i.e., no higher-order coalescence is permitted). It is clear that this mechanism alone cannot account for the multiple peaks in the CSD reported in refs 14 and 15 and elsewhere. Mechanism II. In mechanism II, which is summarized in Table 2, all of the features of mechanism I are retained and the additional feature of coalescence between any size crystal is considered. In the simulations, the parameters a and b were selected to show the behavior of the model over a wide range of parameter space. Parameter c was initially selected as 0.001 × b, to represent a perturbation of mechanism I. In Figure 3, xA, xB, y, and the degree of crystallinity f are shown as a function of τ. In Figure 4, the CSD is shown for several values of τ for each parameter set. At first glance, it may appear that the trends shown in Figure 3 are identical to those shown in Figure 1. There is an exponential decrease in xA as τ increases. The behavior of xB is similar in all cases; the peak in xB is located at almost the same τ and the peak height is almost the same. The trends in f are almost the same in Figures 1 and 3. The main difference in the trend in y is that y has a maximum in the simulations in Figure

3. The maximum is not obvious in the plots shown in Figure 3, because only small values of τ are plotted. Although the trends in xA, xB, f, and y (for small τ) were similar for mechanisms I and II for the same values of a and b, the trends in the CSD between the two mechanisms for the same values of a and b are different. In Figure 4a, the CSD for τ ) 10 and τ ) 100 seem to be similar to those for the same values of τ in Figure 2a. However, in Figure 4a, at τ ) 1000, the CSD shows that several large crystals have been formed by the consumption of small crystals. In Figure 4b, the CSD for values of τ ) 100 and τ ) 1000 appear very different from the corresponding values in Figure 2b. Closer inspection indicates that substantial differences exist, even at τ ) 10. In Figure 4b, at τ ) 10, the footprint of the moving front observed in Figure 2b can be seen, but the shape of the curve has been distorted. At τ ) 100, the CSD in Figure 4b has a peak at i1/3 ≈ 3, whereas there is no such peak in the CSD of Figure 2b. When τ ) 1000, there are more small particles in the system than there are larger particles, because only a few large particles have been formed. Figure 4c and 4d use the same values of a and b that Figure 2c and 2d use, but the simulations in Figure 4c and 4d were performed for larger values of τ, to observe the effect of including higher-order aggregation. In Figure 4c at τ ) 10, the CSD appears similar to the corresponding CSD in Figure 2c. As τ increases in Figure 4c, the small crystals aggregate to form larger crystals. In Figure 4d, at τ ) 10, the CSD curve is already distorted, in comparison to the corresponding curve shown in

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Figure 4. CSD at various times for (a) a ) 0.01, b ) 10, c ) 0.01; (b) a ) 0.01, b ) 1000, c ) 1; (c) a ) 1, b ) 10, c ) 0.01; (d) a ) 1, b ) 1000, c ) 1; (e) a ) 100, b ) 10, c ) 0.01; and (f) a ) 100, b ) 1000, c ) 1.

Figure 2d. The distortion consists of removing particles from the plateau regime of the CSD in Figure 2d and adding them to the region where the CSD is almost zero. This distortion results in the formation of one local maximum at i1/3 ≈ 3. At later times, additional local maxima appear in Figure 4d. In the CSD for τ ) 50 and τ ) 100, there seem to be two peaks in the distribution. This is an important finding, because it agrees with the experimental findings of two peaks in the CSD reported in refs 14 and 15 (see full discussion later). It is important to note that the simulations shown in Figures 3 and 4 could be run for much longer values of τ. Eventually, one large crystal aggregate would be formed that would be composed of nA0 particles. The simulations presented here were not performed to that point, because the simulations here emphasize the early stages of growth. As τ increases, other processes, such as Ostwald ripening, precipitation, and sizedependent aggregation (where K and K′ are not constant) cannot be neglected. Such processes are known to create maxima in CSD and may further complicate the CSD appearance. The question we address next is how the peaks in the CSD are created. In Figure 5, the processes of aging and coalescence solely by the addition of primary particles and higher-order coalescence have been separated to study the effect on the CSD. A simulation was performed with the parameters used in Figure 4d, where, first, mechanism I was simulated up to τ ) 10 (when mechanism I was almost complete) and then the resulting CSD was used as an initial condition for a simulation where only higher-order coalescence was considered. The resulting CSD are identical in shape to those shown in Figure 4d, which indicates that, in Figure 4d, the aging and coalescence by the addition of primary particles occurs first and is then followed by the slower process of higher-order coalescence, meaning that these processes are, to a good approximation, occurring sequentially, because of a

Figure 5. CSD for simulations where mechanism I is performed with a ) 1 and b ) 1000, up to τ ) 10. At τ ) 10, the CSD is used as the input to the mechanism II model and performed for various values of τ with a ) 0, b ) 0, and c ) 1 to show the effect of only higher-order coalescence on the CSD.

considerable difference in time scales (note that b/c ) 103). In other words, the peaked CSD can be created using the outcome of mechanism I as an initial condition for the solution of the Smoluchowski equation. Comparison with Mechanisms that Do Not Include Mature Primary Particles. Other mechanisms were briefly investigated for the purpose of comparison using the parameters from Figure 4d as a basis. In the first mechanism, shown in Figure 6, all primary particles were nuclei at the beginning of the simulation. There was no evolution of fresh or mature primary particles considered. This is the well-known case of aggregation with a constant kernel and the initial condition that all of the particles are monomers, for which the Smoluchowski equation has an analytical solution.37 As it is well-known, large particles are formed over time, but no peaks form in the CSD. In Figure 7, a variation on this mechanism was simulated, where all primary particles were initially fresh and they evolved to nuclei in a one-step process and were then permitted to undergo higher-order coalescence. This model does not include the role of intermediate (mature) primary particles. In the simulation

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Figure 6. CSD for simulation of mechanism II with the initial condition C1 ) 1 at t ) 0, and using a ) 0, b ) 0, c ) 1.

Figure 8. Comparison of (a) experimental and (b) simulated CSD at various times. In panel a, the experimental data are from Figure 1c in ref 14 and an rpp value of 4 nm was used. In panel b, the simulation is with panel a (a ) 10, b ) 1 × 105, c ) 100).

k

K′

Figure 7. CSD for various values of τ for the mechanism A 98 C1 98 Cn with the initial condition xA ) 1 at τ ) 0: (a) a ) 0, b ) 0, c ) 100; (b) a ) 0, b ) 0, c ) 1; and (c) a ) 0, b ) 0, c ) 0.01. Here, τ ) kt and c ) nA0K′/k.

results shown in Figure 7, the CSD resembled those in Figure 6 and did not show any peaks. Comparison of Model Predictions with Experimental Results. Experimental data from Figure 1c in ref 14 for hematite growth at 80 °C for 24 and 48 h are plotted in Figure 8a. A primary particle size of 4 nm was used to nondimensionalize the crystal size in Figure 8a. We performed simulations of mechanism II to determine if our model can qualitatively capture the essential features of a CSD evolution. The parameters in the simulations were chosen as follows: (1) b/c was fixed at 103 for the reasons presented previously in this report; (2) a was set to 10, to reflect the fact that the nucleation rate constant (k2) should be somewhat lower than the rate constant associated with the partial transformation of fresh primary particles to mature primary particles (k1); and (3) b was used as a parameter to match the local minimum in the experimental data, which occurs at rcryst/rpp ≈ 9.2 and is indicated by an arrow in Figure 8a. This procedure produced the CSD curves shown in Figure 8b for τ ) 10, corresponding to 24 h, and τ ) 20, corresponding to 48 h (note that rcryst/rpp ) 1.2i1/3 was used, according to ref 8). Given the simplicity of the model, its ability to predict the

shape of the CSD curves and its agreement with CSD evolution over time is remarkable, suggesting that the model captures the essential processes of oriented aggregation. A variety of kernels has been studied in different models.9,34,37,38 For the hydrothermal systems that motivated this work, the Brownian37 and Derjaguin-Landau-Verwey-Overbeek (DLVO)9 kernels are most relevant. In the Brownian kernel, one takes into account the changes in diffusivity and capture area with particle size, whereas in the DLVO kernel, electrostatic and van der Waals interactions, as a function of size, also are incorporated. Preliminary calculations have shown that, for particle sizes of interest (i.e., particles with a radius of 2.5-50 nm), the Brownian kernel can differ by a factor of 10. In contrast, a DLVO kernel can be a much stronger decreasing function of the particle size. Note that the comparison of simulation results using the kernel that incorporates DLVO and Brownian physics with experiments is not straightforward, because the model assumes that the interacting particles are smooth, whereas for aggregative growth, the particles have considerable roughness. These issues will be addressed in a future publication. Conclusions A simple model was formulated to describe the kinetic processes that have been proposed to occur during nucleation and crystal growth in the presence of precursor primary particles (i.e., particles with structure and/or composition different from that of the growing crystal). The model introduces the concept of an evolving primary particle population toward nuclei assuming that nucleation occurs gradually within a primary particle. Following nuclei formation, the model describes aggregative crystal growth by two coalescence processes: (1) attachment of nuclei to nuclei, nuclei to crystals, and crystals to crystals, and (2) addition of partially transformed primary particles to nuclei and crystals. The first coalescence process is a well-known element of many aggregative crystal growth models. The second coalescence process is modeled for the first time here. The model treats the primary particles as pseudospecies, their evolution to nuclei as two first-order consecutive reactions (A f B f C1), and the coalescence processes are assumed to have two different, but constant (i.e., particle-sizeindependent), kernels.

Crystal Growth of Precursor Metastable Nanoparticles We have shown that the presence of an intermediate stage of primary particles, along with relatively slow higher-order coalescence (between crystals), are necessary ingredients for the formation of multiple peaks in a crystal size distribution (CSD). Simulation results are shown to be in good agreement with reported CSD evolution in an experimental system that is believed to undergo oriented aggregation. The model developed in this study lays the groundwork for further studies where experimental data from small-angle X-ray scattering (SAXS) and transmission electron microscopy (TEM) can be compared in a quantitative manner with simulations that may incorporate additional details regarding the morphology of the crystals, particle-particle interactions, and detailed kinetics coupled with solution chemistry. Acknowledgment. The authors acknowledge funding from NSF Grant No. NSF/NIRT (CTS-0103010). M.T. acknowledges helpful discussions with Dr. R. Bedard, UOP-LLC, regarding zeolite nucleation and growth and for providing refs 14 and 15. We acknowledge the authors of refs 19 and 31 for providing early versions of their manuscripts. The research of M.K. is partially supported by NSF-DMS-0413864 and NSF-ITR0219211. M.K. would like to thank the Institute for Mathematics and its Applications at the University of Minnesota, where part of this work was conducted during the program “Mathematics of Materials and Macromolecules: Multiple Scales, Disorder, and Singularities”. List of Symbols A ) concentration of fresh primary particles, (number of particles)/volume B ) concentration of mature primary particles, (number of particles)/volume Ci ) concentration of crystals of size i, (number of particles)/ volume CSD ) crystal size distribution K ) collision frequency factor (kernel) for coalescence of mature primary particle and Ci, volume/time K′ ) collision frequency factor (kernel) for coalescence of Ci and Cj, volume/time Ki ) collision frequency factor (kernel) for coalescence of mature primary particle and Ci, volume/time Kij ) collision frequency factor (kernel) for coalescence of Ci and Cj, volume/time V ) volume (m3) a ) k1/k2 b ) nA0K/k2 c ) nA0K′/k2 f ) degree of crystallinity; f ) 1 - xA - xB k1 ) rate constant for the aging of a fresh primary particle to a mature particle (s-1) k2 ) rate constant for the aging of a mature primary particle to a crystal (s-1) nA0 ) initial concentration of fresh primary particles, (number of particles)/volume ni ) concentration of particles of size i, (number of particles)/ volume nj ) concentration of all crystals, (number of particles)/volume rcryst ) effective radius of a crystal (m) rpp ) radius of a primary particle (m) t ) time (s) xA ) nA/nA0 xB ) nB/nA0 y ) nj/nA0

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