Article pubs.acs.org/Langmuir
On the Two-Step Mechanism for Synthesis of Transition-Metal Nanoparticles Siva Rama Krishna Perala and Sanjeev Kumar* Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India S Supporting Information *
ABSTRACT: The two-step particle synthesis mechanism, also known as the Finke−Watzky (1997) mechanism, has emerged as a significant development in the field of nanoparticle synthesis. It explains a characteristic feature of the synthesis of transition metal nanoparticles, an induction period in precursor concentration followed by its rapid sigmoidal decrease. The classical LaMer theory (1950) of particle formation fails to capture this behavior. The two-step mechanism considers slow continuous nucleation and autocatalytic growth of particles directly from precursor as its two kinetic steps. In the present work, we test the two-step mechanism rigorously using population balance models. We find that it explains precursor consumption very well, but fails to explain particle synthesis. The effect of continued nucleation on particle synthesis is not suppressed sufficiently by the rapid autocatalytic growth of particles. The nucleation continues to increase breadth of size distributions to unexpectedly large values as compared to those observed experimentally. A number of variations of the original mechanism with additional reaction steps are investigated next. The simulations show that continued nucleation from the beginning of the synthesis leads to formation of highly polydisperse particles in all of the tested cases. A short nucleation window, realized with delayed onset of nucleation and its suppression soon after in one of the variations, appears as one way to explain all of the known experimental observations. The present investigations clearly establish the need to revisit the two-step particle synthesis mechanism.
■
INTRODUCTION
The classical LaMer model1 proposed in 1950 provides a framework for understanding particle synthesis for a variety of systems in different fields of investigations.2,3 The two-step mechanism of Watzky and Finke,4 proposed for the synthesis of transition metal nanoparticles that do not follow the classical route, is a significant breakthrough in the field. The new mechanism has been applied to the synthesis of transition metal nanoparticles in general, including that of Ir, Au, Ag, Pt, Rh, Pd, and Ni.4−9 A typical experimental data set, pointing to applicability of the two-step mechanism for the synthesis of iridium nanoparticles, is shown in Figure 1. The precursor concentration remains nearly unchanged for a long time and then undergoes rapid sigmoidal decrease. The LaMer model, reduction of precursor to form reduced atoms, which form nuclei as per the classical nucleation theory and then deposit on them to grow them into particles, explains neither the long induction time nor the rapid sigmoidal decrease in precursor concentration thereafter. A number of other reports have also questioned the applicability of the classical nucleation theory.10−12 The two-step mechanism with slow continuous first-order nucleation and rapid autocatalytic particle growth successfully explains both the long induction time and the rapid sigmoidal decrease in precursor concentration, as shown by the solid line © 2014 American Chemical Society
Figure 1. Iridium precursor concentration versus time data from Figure 4 of Widegren et al.,5 and predictions of F−W model (eq 7) with k1 = 0.017 h−1 and k2 = 3.30 × 102 h−1 M−1.
in Figure 1. The elegance of the two-step mechanism and its widespread applicability to a number of systems are indeed its most attractive features. The mechanism also explains aggregation of a class of proteins.13 The significance of the Received: August 11, 2014 Revised: September 25, 2014 Published: October 2, 2014 12703
dx.doi.org/10.1021/la503199m | Langmuir 2014, 30, 12703−12711
Langmuir
Article k1
two-step mechanism and its simple mathematical description are highlighted in a number of (review) articles in various fields9,14−17 in the intervening years. The mechanism is widely believed4,7,14,18 to lead to synthesis of relatively monodispersed particles. This critical aspect of the two-step mechanism has not been tested however due to the limitations of the original mathematical description. In the present work, we examine the two-step mechanism rigorously, using the framework of population balances.19 The need to examine the mechanism carefully is as follows. The Brust−Schiffrin method, widely used for the synthesis of highly stable thiol protected metal nanoparticles,20 is recently found to follow continuous nucleation growth-capping mechanism.21,22 The continuous nucleation in this mechanism leads to synthesis of monodisperse particles only because the growing particles simultaneously get capped with thiol molecules, and a fully capped particle does not grow further. A short burst of nucleation, a hallmark of the classical LaMer model, is otherwise essential for the synthesis of relatively monosized particles through separation of nucleation and growth processes. Continued and slow nucleation for nonautocatalytic and nondiffusion controlled growth of particles leads to high polydispersity.23 The nonclassical organizer mechanism for citrate synthesis of gold nanoparticles24 also realizes a burst of nucleation through rapid decay of the organizer.25 It is unclear therefore if continuous nucleation with autocatalytic growth of particles can produce monodisperse particles. We begin with a brief discussion of the two-step mechanism, followed by the development of population balance models to progressively do away with the limitations of the earlier description. As the predictions of the rigorous models do not explain the experimental findings, a number of alternatives are also explored using models at the end.
The autocatalytic growth of particles B directly from precursor A is the second step of the two-step mechanism. k2
The diffusion of A to particle surface is not expected to be kinetically limiting for the slow syntheses considered here. The rate of change of precursor and particle mass for the approximation that particle growth is proportional to their volume (mass) instead of surface area (which facilitates representation of growth of all of the particles collectively) is captured by Watzky and Finke4 as −
= k1[A] + k 2[A]([A]0 − [A])
−
d[C] = k C[B][C] dt
(8)
For [A]0 = [C]0/1400, [C]0 − [C] = 1400[B], and 1400 times faster consumption of hydrogen for the reporter reaction, eq 8, is recast4 as k2 d[C] = k1[C] + [C]([C]0 − [C]) dt 1400
(9)
Equation 9 permits experimentally more accurate time variation of [C] to be used to fit rate constants k1 and k2. The fitted rate constants also lead to a prediction of [A] and [B] versus t through eq 7. The ability of the two-step mechanism to fit time variation of precursor and cyclohexene concentration is amply demonstrated in the literature.4,5,7 The autocatalytic particle growth consumes precursor at 3 orders of magnitude larger rates than that required for the birth of nuclei, for the typical values of the fitted rate constants. The kinetically dominant role of particle growth over nucleation, expressed through large values of ratio k2/k1, is assumed to separate nucleation and growth of particles in time4,6,7,26,27 and explain the synthesis of relatively monodispersed particles, observed through measurement of size distributions at the end of the synthesis in a number of cases.4 The large values of ratio k2/k1, expected to lead to the synthesis of monodispersed particles, are in the range of 105 M−1 for an initial precursor concentration [A]0 of 1.2 mM.26 The separation of nucleation and growth phases and negligible role for aggregation and ripening of particles serve as the starting point for quantitative prediction7 of time variation of mean particle size. The number of nuclei born in the induction/nucleation phase is taken to be equal to the number of particles present at the end of the synthesis. The final population of particles is determined empirically from the amount of precursor added and the measured value of mean particle size at the end of the synthesis. The time variation of particle size is thus estimated to be (for t > tind):7
slow
(1)
fast
(2)
nucleation
(3) autocatalytic growth
A + Pn + 2.5H 2 ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ Pn + 1 + H+
(7)
In physical terms, this approximation changes a surface reaction to one that permits atoms inside a particle also to participate in growth reactions. The balance equation for cyclohexene to cyclohexane conversion on particles’ surface, a reporter reaction, is consistently written as
TWO-STEP MECHANISM AND FINKE−WATZKY MODEL Watzky and Finke4 reduced [Bu4N]5Na3 [(1,5-COD) Ir(P2W15Nb3O62)] (named precursor A) with excess hydrogen to synthesize iridium nanoparticles. The reduction of cyclohexene on the surface of particles, which consumes hydrogen at 1400 times faster rate than that required for particle synthesis, was used as a reporter reaction. Please refer to the Supporting Information for more information on the experimental setup and measurements. The reaction steps involved in the synthesis of iridium nanoparticles from precursor A, as explained by Watzky and Finke,4 are
nP ⎯⎯⎯⎯⎯⎯⎯⎯⎯→ Pn
d[A] d[B] = dt dt = k1[A] + k 2[A][B]
■
A(acetone)2 + 2.5H 2 ⎯→ ⎯ cyclooctane + P + H+
(6)
A + B → 2B
−
A + 2acetone ⎯⎯⎯→ A(acetone)2
(5)
A→B
(4)
In the above reactions, Pn represents an iridium nanoparticle with n iridium atoms in it. The two-step mechanism was proposed4 in the context of experimentally observed long induction time followed by sigmoidal decrease in iridium precursor concentration. In the two-step mechanism, the first three reactions are combined into one pseudo first-order step leading to formation of nuclei directly from precursor A. 12704
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⎛ [A] − [A]t ⎞1/3 d p(t ) = d p,f ⎜ 0 ⎟ [A]0 ⎝ ⎠
dMj /3 dt
(10)
(14)
where dp,f is the final, experimentally measured mean particle size.4 Given the experimentally determined value of induction time tind, the number of atoms in a nucleus can be estimated as the number of atoms in a particle of diameter dp(tind). The foregoing mathematical description of the two-step synthesis mechanism7 is named F−W model here. Watzky et al.7 reported one set of detailed experimental data covering variation of precursor concentration, cyclohexene concentration, mean particle size, standard deviation, and size distribution with time. They established the ability of the F−W model to capture particle synthesis by showing that the model predictions for this data set agree with the experimentally measured variation of mean particle size and concentrations of cyclohexene and precursor with time. The particles synthesized were quite monodisperse (less than 15% polydispersity) during and at the end of the synthesis. As the F−W model does not predict the number of nuclei born and size distribution or its breadth, the ability of the two-step mechanism to explain experimentally observed particle synthesis fully has not yet been tested. We have used the same data set in the present work to test the two-step mechanism comprehensively, for its ability to explain precursor consumption and particle synthesis, using population balance models.
M0 and M1 correspond to the total number and the total volume of particles per unit reactor volume. M1/3 and M2/3 relate to the sum of particle diameters and total surface area of particles. The rate of nucleation, consistent with the two-step mechanism, has first-order dependence4 given by Ṅ =
dM1 = k1[A]A v vIr + k 2[A]M1 dt
−
−
v jn(v , t ) dv
⎡⎛ 6 ⎞ M ⎤1/3 d v = ⎢⎜ ⎟ 1 ⎥ ⎣⎝ π ⎠ M 0 ⎦
1/2 ⎛⎡ 1/3 ⎤2 ⎛ 6 ⎞2/3 M ⎞ M ⎛ ⎞ 6 1/3 2/3 ⎟ ⎥ −⎜ ⎟ σ = ⎜⎜⎢⎜ ⎟ ⎝π ⎠ ⎢⎝ ⎠ M 0 ⎥⎦ M 0 ⎟⎠ ⎝⎣ π
(11)
(17)
(18)
(19)
(20)
Here, dm and dv are diameter and volume averaged mean size of particles. The model equations are nondimensionalized (please refer to the Supporting Information) and solved to obtain predictions for variation of concentration of cyclohexene and cyclooctane, number of nuclei born, mean particle size, and breadth of size distribution as a function of time. As discussed earlier, the two-step mechanism is tested here using the same data that is used by Watzky et al.7 to establish the two-step mechanism. The model equations are solved for the following initial conditions: [A]0 = 1.2 mM, [B]0 = 0 mM, [C]0 = 1650 mM, and Mi/3 = 0 for i = 0, 1, 2, and 3. The cyclohexene concentration versus time data from the plot (Figure 3b of Watzky et al.7) is read and used to fit parameters k1 and k2 using nlinfit of Matlab 7.11.0.584 (R2010b), a nonlinear regression subroutine that uses the Levenberg− Marquardt algorithm. The best fit values of parameters are k1 = 0.0259 h−1 and k2 = 3.443 × 103 h−1 M−1 for the degree of fit shown in Figure 2A. Given the small errors that are unavoidable in reading experimental data from a plot, the fitted values of rate constants agree well with the reported values7 for the F−W model (k1 = 0.0282 h−1 and k2 = 3.379 × 103 h−1 M−1). Figure 2A shows that the predictions of the F−W and PBM−v models for the conversion of cyclohexene are identical, as expected, and
(12)
where Hv = k2[A]. The time variation of jth moment of the size distribution, defined as ∞
k2 d[C] = k1[C] + [C]([C]0 − [C]) dt 1400
⎡⎛ 6 ⎞1/3 M ⎤ 1/3 ⎥, d m = ⎢⎜ ⎟ ⎢⎣⎝ π ⎠ M 0 ⎥⎦
PBM−v: GROWTH RATE PROPORTIONAL TO PARTICLE VOLUME In the first model, we relax the assumption that nucleation has no role after the induction period,4,7 and let the model control the birth of nuclei during the synthesis. The particle growth is still assumed to be proportional to particle volume. This model is named PBM−v. Thus:
∫0
d[A] d[B] = = k1[A] + k 2[A]([A]o − [A]) dt dt
Equations 17 and 18 are identical to the equations for [A], [B], and [C] in the F−W model. The PBM−v model is thus identical to the F−W model with respect to [A], [B], and [C]. The present formulation additionally provides for prediction of mean particle size and breadth of size distribution (σ) as
■
M j (t ) =
(16)
and correspondingly, as explained earlier:
where Ṅ (t) is rate of nucleation, vc is volume of the critical nucleus, and G(v, t) is growth rate of particles of volume v at time t.
dv = k 2[A]v = H vv dt
(15)
Because M1/AvvIr = [B], eq 16 can be recast as
POPULATION BALANCE MODELS The framework of population balances, already in use in the literature to model synthesis of nanoparticles,25,28−31 recognizes particles of different sizes explicitly, and allows prediction of variables related to particle size distribution. Let n(v, t) dv represent the number of particles in size range v to v + dv per unit physical volume at time t. The governing population balance equation for nucleation and growth of particles is given by
G (v ) =
dM 0 A v = v Ir ·k1[A] dt vc
where Av is Avogadro’s number and vIr is the volume of an iridium atom. Equation 14 for j = 3 yields
■
∂n(v) ∂ [G(v , t )n(v)] = Ṅ (t )δ(v − vc) + ∂t ∂v
= Nv̇ c j /3 + (j /3)H vMj /3 for j = 0, 1, 2, and 3
(13)
is obtained by multiplying eq 11 by vj and integrating it with respect to v from 0 to ∞. Thus: 12705
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nearly constant, at about 0.4 nm, for all of the reported measurements, except for the measurement at 1.71 h, which appears to be erroneous. For a mean final particle size of about 2.7 nm, these measurements suggest synthesis of near monodispersed6 particles (less than
On the Two-Step Mechanism for Synthesis of Transition-Metal Nanoparticles Siva Rama Krishna Perala and Sanjeev Kumar* Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India S Supporting Information *
ABSTRACT: The two-step particle synthesis mechanism, also known as the Finke−Watzky (1997) mechanism, has emerged as a significant development in the field of nanoparticle synthesis. It explains a characteristic feature of the synthesis of transition metal nanoparticles, an induction period in precursor concentration followed by its rapid sigmoidal decrease. The classical LaMer theory (1950) of particle formation fails to capture this behavior. The two-step mechanism considers slow continuous nucleation and autocatalytic growth of particles directly from precursor as its two kinetic steps. In the present work, we test the two-step mechanism rigorously using population balance models. We find that it explains precursor consumption very well, but fails to explain particle synthesis. The effect of continued nucleation on particle synthesis is not suppressed sufficiently by the rapid autocatalytic growth of particles. The nucleation continues to increase breadth of size distributions to unexpectedly large values as compared to those observed experimentally. A number of variations of the original mechanism with additional reaction steps are investigated next. The simulations show that continued nucleation from the beginning of the synthesis leads to formation of highly polydisperse particles in all of the tested cases. A short nucleation window, realized with delayed onset of nucleation and its suppression soon after in one of the variations, appears as one way to explain all of the known experimental observations. The present investigations clearly establish the need to revisit the two-step particle synthesis mechanism.
■
INTRODUCTION
The classical LaMer model1 proposed in 1950 provides a framework for understanding particle synthesis for a variety of systems in different fields of investigations.2,3 The two-step mechanism of Watzky and Finke,4 proposed for the synthesis of transition metal nanoparticles that do not follow the classical route, is a significant breakthrough in the field. The new mechanism has been applied to the synthesis of transition metal nanoparticles in general, including that of Ir, Au, Ag, Pt, Rh, Pd, and Ni.4−9 A typical experimental data set, pointing to applicability of the two-step mechanism for the synthesis of iridium nanoparticles, is shown in Figure 1. The precursor concentration remains nearly unchanged for a long time and then undergoes rapid sigmoidal decrease. The LaMer model, reduction of precursor to form reduced atoms, which form nuclei as per the classical nucleation theory and then deposit on them to grow them into particles, explains neither the long induction time nor the rapid sigmoidal decrease in precursor concentration thereafter. A number of other reports have also questioned the applicability of the classical nucleation theory.10−12 The two-step mechanism with slow continuous first-order nucleation and rapid autocatalytic particle growth successfully explains both the long induction time and the rapid sigmoidal decrease in precursor concentration, as shown by the solid line © 2014 American Chemical Society
Figure 1. Iridium precursor concentration versus time data from Figure 4 of Widegren et al.,5 and predictions of F−W model (eq 7) with k1 = 0.017 h−1 and k2 = 3.30 × 102 h−1 M−1.
in Figure 1. The elegance of the two-step mechanism and its widespread applicability to a number of systems are indeed its most attractive features. The mechanism also explains aggregation of a class of proteins.13 The significance of the Received: August 11, 2014 Revised: September 25, 2014 Published: October 2, 2014 12703
dx.doi.org/10.1021/la503199m | Langmuir 2014, 30, 12703−12711
Langmuir
Article k1
two-step mechanism and its simple mathematical description are highlighted in a number of (review) articles in various fields9,14−17 in the intervening years. The mechanism is widely believed4,7,14,18 to lead to synthesis of relatively monodispersed particles. This critical aspect of the two-step mechanism has not been tested however due to the limitations of the original mathematical description. In the present work, we examine the two-step mechanism rigorously, using the framework of population balances.19 The need to examine the mechanism carefully is as follows. The Brust−Schiffrin method, widely used for the synthesis of highly stable thiol protected metal nanoparticles,20 is recently found to follow continuous nucleation growth-capping mechanism.21,22 The continuous nucleation in this mechanism leads to synthesis of monodisperse particles only because the growing particles simultaneously get capped with thiol molecules, and a fully capped particle does not grow further. A short burst of nucleation, a hallmark of the classical LaMer model, is otherwise essential for the synthesis of relatively monosized particles through separation of nucleation and growth processes. Continued and slow nucleation for nonautocatalytic and nondiffusion controlled growth of particles leads to high polydispersity.23 The nonclassical organizer mechanism for citrate synthesis of gold nanoparticles24 also realizes a burst of nucleation through rapid decay of the organizer.25 It is unclear therefore if continuous nucleation with autocatalytic growth of particles can produce monodisperse particles. We begin with a brief discussion of the two-step mechanism, followed by the development of population balance models to progressively do away with the limitations of the earlier description. As the predictions of the rigorous models do not explain the experimental findings, a number of alternatives are also explored using models at the end.
The autocatalytic growth of particles B directly from precursor A is the second step of the two-step mechanism. k2
The diffusion of A to particle surface is not expected to be kinetically limiting for the slow syntheses considered here. The rate of change of precursor and particle mass for the approximation that particle growth is proportional to their volume (mass) instead of surface area (which facilitates representation of growth of all of the particles collectively) is captured by Watzky and Finke4 as −
= k1[A] + k 2[A]([A]0 − [A])
−
d[C] = k C[B][C] dt
(8)
For [A]0 = [C]0/1400, [C]0 − [C] = 1400[B], and 1400 times faster consumption of hydrogen for the reporter reaction, eq 8, is recast4 as k2 d[C] = k1[C] + [C]([C]0 − [C]) dt 1400
(9)
Equation 9 permits experimentally more accurate time variation of [C] to be used to fit rate constants k1 and k2. The fitted rate constants also lead to a prediction of [A] and [B] versus t through eq 7. The ability of the two-step mechanism to fit time variation of precursor and cyclohexene concentration is amply demonstrated in the literature.4,5,7 The autocatalytic particle growth consumes precursor at 3 orders of magnitude larger rates than that required for the birth of nuclei, for the typical values of the fitted rate constants. The kinetically dominant role of particle growth over nucleation, expressed through large values of ratio k2/k1, is assumed to separate nucleation and growth of particles in time4,6,7,26,27 and explain the synthesis of relatively monodispersed particles, observed through measurement of size distributions at the end of the synthesis in a number of cases.4 The large values of ratio k2/k1, expected to lead to the synthesis of monodispersed particles, are in the range of 105 M−1 for an initial precursor concentration [A]0 of 1.2 mM.26 The separation of nucleation and growth phases and negligible role for aggregation and ripening of particles serve as the starting point for quantitative prediction7 of time variation of mean particle size. The number of nuclei born in the induction/nucleation phase is taken to be equal to the number of particles present at the end of the synthesis. The final population of particles is determined empirically from the amount of precursor added and the measured value of mean particle size at the end of the synthesis. The time variation of particle size is thus estimated to be (for t > tind):7
slow
(1)
fast
(2)
nucleation
(3) autocatalytic growth
A + Pn + 2.5H 2 ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ Pn + 1 + H+
(7)
In physical terms, this approximation changes a surface reaction to one that permits atoms inside a particle also to participate in growth reactions. The balance equation for cyclohexene to cyclohexane conversion on particles’ surface, a reporter reaction, is consistently written as
TWO-STEP MECHANISM AND FINKE−WATZKY MODEL Watzky and Finke4 reduced [Bu4N]5Na3 [(1,5-COD) Ir(P2W15Nb3O62)] (named precursor A) with excess hydrogen to synthesize iridium nanoparticles. The reduction of cyclohexene on the surface of particles, which consumes hydrogen at 1400 times faster rate than that required for particle synthesis, was used as a reporter reaction. Please refer to the Supporting Information for more information on the experimental setup and measurements. The reaction steps involved in the synthesis of iridium nanoparticles from precursor A, as explained by Watzky and Finke,4 are
nP ⎯⎯⎯⎯⎯⎯⎯⎯⎯→ Pn
d[A] d[B] = dt dt = k1[A] + k 2[A][B]
■
A(acetone)2 + 2.5H 2 ⎯→ ⎯ cyclooctane + P + H+
(6)
A + B → 2B
−
A + 2acetone ⎯⎯⎯→ A(acetone)2
(5)
A→B
(4)
In the above reactions, Pn represents an iridium nanoparticle with n iridium atoms in it. The two-step mechanism was proposed4 in the context of experimentally observed long induction time followed by sigmoidal decrease in iridium precursor concentration. In the two-step mechanism, the first three reactions are combined into one pseudo first-order step leading to formation of nuclei directly from precursor A. 12704
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⎛ [A] − [A]t ⎞1/3 d p(t ) = d p,f ⎜ 0 ⎟ [A]0 ⎝ ⎠
dMj /3 dt
(10)
(14)
where dp,f is the final, experimentally measured mean particle size.4 Given the experimentally determined value of induction time tind, the number of atoms in a nucleus can be estimated as the number of atoms in a particle of diameter dp(tind). The foregoing mathematical description of the two-step synthesis mechanism7 is named F−W model here. Watzky et al.7 reported one set of detailed experimental data covering variation of precursor concentration, cyclohexene concentration, mean particle size, standard deviation, and size distribution with time. They established the ability of the F−W model to capture particle synthesis by showing that the model predictions for this data set agree with the experimentally measured variation of mean particle size and concentrations of cyclohexene and precursor with time. The particles synthesized were quite monodisperse (less than 15% polydispersity) during and at the end of the synthesis. As the F−W model does not predict the number of nuclei born and size distribution or its breadth, the ability of the two-step mechanism to explain experimentally observed particle synthesis fully has not yet been tested. We have used the same data set in the present work to test the two-step mechanism comprehensively, for its ability to explain precursor consumption and particle synthesis, using population balance models.
M0 and M1 correspond to the total number and the total volume of particles per unit reactor volume. M1/3 and M2/3 relate to the sum of particle diameters and total surface area of particles. The rate of nucleation, consistent with the two-step mechanism, has first-order dependence4 given by Ṅ =
dM1 = k1[A]A v vIr + k 2[A]M1 dt
−
−
v jn(v , t ) dv
⎡⎛ 6 ⎞ M ⎤1/3 d v = ⎢⎜ ⎟ 1 ⎥ ⎣⎝ π ⎠ M 0 ⎦
1/2 ⎛⎡ 1/3 ⎤2 ⎛ 6 ⎞2/3 M ⎞ M ⎛ ⎞ 6 1/3 2/3 ⎟ ⎥ −⎜ ⎟ σ = ⎜⎜⎢⎜ ⎟ ⎝π ⎠ ⎢⎝ ⎠ M 0 ⎥⎦ M 0 ⎟⎠ ⎝⎣ π
(11)
(17)
(18)
(19)
(20)
Here, dm and dv are diameter and volume averaged mean size of particles. The model equations are nondimensionalized (please refer to the Supporting Information) and solved to obtain predictions for variation of concentration of cyclohexene and cyclooctane, number of nuclei born, mean particle size, and breadth of size distribution as a function of time. As discussed earlier, the two-step mechanism is tested here using the same data that is used by Watzky et al.7 to establish the two-step mechanism. The model equations are solved for the following initial conditions: [A]0 = 1.2 mM, [B]0 = 0 mM, [C]0 = 1650 mM, and Mi/3 = 0 for i = 0, 1, 2, and 3. The cyclohexene concentration versus time data from the plot (Figure 3b of Watzky et al.7) is read and used to fit parameters k1 and k2 using nlinfit of Matlab 7.11.0.584 (R2010b), a nonlinear regression subroutine that uses the Levenberg− Marquardt algorithm. The best fit values of parameters are k1 = 0.0259 h−1 and k2 = 3.443 × 103 h−1 M−1 for the degree of fit shown in Figure 2A. Given the small errors that are unavoidable in reading experimental data from a plot, the fitted values of rate constants agree well with the reported values7 for the F−W model (k1 = 0.0282 h−1 and k2 = 3.379 × 103 h−1 M−1). Figure 2A shows that the predictions of the F−W and PBM−v models for the conversion of cyclohexene are identical, as expected, and
(12)
where Hv = k2[A]. The time variation of jth moment of the size distribution, defined as ∞
k2 d[C] = k1[C] + [C]([C]0 − [C]) dt 1400
⎡⎛ 6 ⎞1/3 M ⎤ 1/3 ⎥, d m = ⎢⎜ ⎟ ⎢⎣⎝ π ⎠ M 0 ⎥⎦
PBM−v: GROWTH RATE PROPORTIONAL TO PARTICLE VOLUME In the first model, we relax the assumption that nucleation has no role after the induction period,4,7 and let the model control the birth of nuclei during the synthesis. The particle growth is still assumed to be proportional to particle volume. This model is named PBM−v. Thus:
∫0
d[A] d[B] = = k1[A] + k 2[A]([A]o − [A]) dt dt
Equations 17 and 18 are identical to the equations for [A], [B], and [C] in the F−W model. The PBM−v model is thus identical to the F−W model with respect to [A], [B], and [C]. The present formulation additionally provides for prediction of mean particle size and breadth of size distribution (σ) as
■
M j (t ) =
(16)
and correspondingly, as explained earlier:
where Ṅ (t) is rate of nucleation, vc is volume of the critical nucleus, and G(v, t) is growth rate of particles of volume v at time t.
dv = k 2[A]v = H vv dt
(15)
Because M1/AvvIr = [B], eq 16 can be recast as
POPULATION BALANCE MODELS The framework of population balances, already in use in the literature to model synthesis of nanoparticles,25,28−31 recognizes particles of different sizes explicitly, and allows prediction of variables related to particle size distribution. Let n(v, t) dv represent the number of particles in size range v to v + dv per unit physical volume at time t. The governing population balance equation for nucleation and growth of particles is given by
G (v ) =
dM 0 A v = v Ir ·k1[A] dt vc
where Av is Avogadro’s number and vIr is the volume of an iridium atom. Equation 14 for j = 3 yields
■
∂n(v) ∂ [G(v , t )n(v)] = Ṅ (t )δ(v − vc) + ∂t ∂v
= Nv̇ c j /3 + (j /3)H vMj /3 for j = 0, 1, 2, and 3
(13)
is obtained by multiplying eq 11 by vj and integrating it with respect to v from 0 to ∞. Thus: 12705
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Langmuir
Article
nearly constant, at about 0.4 nm, for all of the reported measurements, except for the measurement at 1.71 h, which appears to be erroneous. For a mean final particle size of about 2.7 nm, these measurements suggest synthesis of near monodispersed6 particles (less than
