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Silver Nanoparticle Formation: Predictions and Verification of the Aggregative Growth Model Dirk L. Van Hyning,† Walter G. Klemperer,‡ and Charles F. Zukoski*,† Department of Chemical Engineering and Department of Chemistry, University of Illinois, Urbana, Illinois 61801 Received June 19, 2000. In Final Form: February 23, 2001 The mechanisms of growth of 5-30 nm silver particles produced from the reduction of silver perchlorate by sodium borohydride are explored. Evidence is provided that, within 5 s of mixing of silver perchlorate with borohydride, the concentration of ionic silver has dropped 2 orders of magnitude, initially resulting in the formation of ∼2.5 nm silver particles. We hypothesize that growth takes place over the next 20-50 min by an aggregative growth mechanism. This hypothesis is tested by comparing model predictions with experimental results of particle growth rates over a range of ionic strengths and reaction temperatures. The model assumes particles interact via electrostatic and van der Waals forces, and comparison of numerical solutions to population balance models for aggregation and coalescence demonstrates that particle growth rates and final sizes can be predicted with surface potentials within 20% of those previously reported under reaction conditions.
I. Introduction Descriptions of particle formation in precipitation reactions often focus on nucleation as the key step in controlling the final number density and thus final average particle size.1 However, numerous studies have demonstrated that aggregation can play a key role in determining the final particle number density.2-5 One consequence of this observation is that alterations in particle interactions due to changes in solution conditions during a reaction can result in large changes in the state of particle aggregation. In this work we take this observation one step further and show that particle interactions can be manipulated during the reaction to change the rates of particle growth and final particle sizes. We focus our attention on the experimental system involving nanoscale silver particles formed by the reduction of silver perchlorate with sodium borohydride. Particle formation in the silver-borohydride system follows three distinct stages.6 First, upon mixing, the reaction between borohydride and silver occurs extremely rapidly, resulting in the formation of small (∼2-3 nm) particles. These particles grow in the second stage to achieve sizes of 8-20 nm over time scales of 20 min to 1 h. In the final stage of the reaction, the borohydride is consumed by reaction with water. This results in the loss of the dominate stabilizing species BH4- and the suspension passing from a reducing to an oxidizing environment. The resulting changes in pair potential can drive the particle to aggregate in a catastrophic manner. In a previous study,7 the interaction potentials of growing silver particles were characterized as reagent † ‡
Department of Chemical Engineering. Department of Chemistry.
(1) LaMer, V.; Dinegar, R. J. Am. Chem. Soc. 1950, 72, 4847. (2) Wall, J. F.; Grieser, F.; Zukoski, C. F. J. Chem. Soc., Faraday Trans. 1997, 93, 4017-4020. Chow, M.; Zukoski, C. J. Colloid Interface Sci. 1994, 165, 97. (3) Bogush, G.; Zukoski, C. J. Colloid Interface Sci. 1991, 142, 19. (4) Lee, K.; Look, J. L.; Harris, M. T.; McCormick, A. V. J. Colloid Interface Sci. 1997, 194, 78-88. (5) Heard, S.; Grieser, F.; Barraclough, C.; Sanders, J. J. Colloid Interface Sci. 1983, 93, 545. (6) Van Hyning, D. L.; Zukoski, C. F. Langmuir 1998, 14, 7034. (7) Van Hyning, D. L.; Zukoski, C. F. Langmuir 2001, 17,3120. Van Hyning, D. L. Ph.D. Thesis, 1999.
concentration and reaction temperature were varied. Particle aggregation was interpreted in terms of pair potentials involving a summation of van der Waals attraction and electrostatic repulsions. The van der Waals attractions are assumed to be controlled by the Hamaker coefficient of bulk silver while electrostatic repulsions are controlled by the particle surface potential, ψs, and electrolyte concentration. As the bulk properties of the silver do not change, changes in rates of aggregation are interpreted in terms of alterations in ψs. Particles grow more rapidly while retaining their narrow size distribution when the temperature is increased and as the borohydride concentration is decreased. In addition, if an indifferent electrolyte is added, particles grow more rapidly and retain their narrow size distributions up to a critical ionic strength when catastrophic aggregation is observed. These observations support the notion that the silver sols grow by slow aggregation (i.e., repulsive barriers produced by the electrostatic forces are not sufficient to stop aggregation into the primary minimum on the time scale of the experiment.) Quantitative experiments aimed at determining the growing silver particle surface potential showed four general trends. First, AFM measurements between silver surfaces demonstrated that borohydride and the borate species generated by the oxidation of borohydride by water are potential-determining ions.6 Second, the surface potential produced by borohydride adsorption decreases substantially with temperature, from ∼120 mV at 2 °C to ∼80 mV at 60 °C when [NaBH4] ) 7.5 mM and [Ag] ) 0.250 mM. (For much of the previous work and that discussed here [Ag] is held constant at 0.25 mM and [NaBH4] is varied. We discuss effects of changing [NaBH4] in terms of the molar ratio, R, of [NaBH4] to [Ag]. Thus for R ) 30, ψs varies from 120 to 80 mV as the reaction temperature varies from 2 to 60 °C.) The reduced surface potentials account for the limited colloidal stability of the sols at elevated temperature.7 As BH4- is the chief potential-determining ion early in the reaction, it is not surprising that ψs moves toward zero with decreasing borohydride concentration. Third, during the final stage, when the reaction medium transforms from a reducing to an oxidizing environment, the particle surface potential
10.1021/la000856h CCC: $20.00 © 2001 American Chemical Society Published on Web 05/01/2001
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is reduced and shows no temperature dependence.7 One consequence of this change in surface properties is that, during the final stages of BH4- oxidation, at high temperatures the particles see a small change in surface potential and their stability is not greatly influenced by the loss of BH4-. At low temperatures, however, the sudden drop in surface potential which occurs at the end of the reaction often produces catastrophic aggregation. Armed with estimates of variations in ψs over the course of the reaction as BH4- is consumed and how ψs changes with reaction conditions, we here develop an aggregative growth model for the silver sols. First, we demonstrate that the vast majority of ionic silver has been reduced early in the reaction, suggesting that subsequent growth is dominated by aggregation and coalescence. A population balance model is developed to describe this process where the size dependent aggregation rate constants are calculated from a knowledge of the Hamaker coefficient of bulk silver and measurements of ψs discussed above. A comparison is then made between predicted rates of growth and absolute sizes with experimental results gathered on systems where step changes in temperature and ionic strength are made during the reaction. Due to assumptions made in reducing experimental data in the previously reported experiments, the values of ψs are subject to a number of approximations. As our calculations indicate that particle growth rates are extremely sensitive to ψs, the ability of the aggregative growth model to capture the full dynamics of particle growth is an extremely sensitive test of these measurements and the hypothesis that particles grow by aggregation. In section II, we develop the theoretical framework of the aggregative growth model. In section III, model predictions, as well as comparison with experimental data, are presented. In section IV, conclusions are drawn. II. Description of Aggregative Growth Model Aggregative growth models consist of a series of population balance equations, following the number density populations of particles containing i monomer units, ni. As given in the Smoluchowski theory,8,9 the time rate of the change in the number density of particles with i monomers is
dn1 dt dni dt
∞
i-1
)
β1ini + g(t) ∑ i)1
) -n1
(1)
∞
βi-j,jni-jnj - ni∑βijnj ∑ j)1 j)1
(2)
where ni is the number density of particles containing i monomers and βij is the binary aggregation rate constant for the aggregation of particles of size i and j. The first term in eq 2 accounts for the increase in number density of particles of size i due to aggregation of smaller particles of size i - j and size j. The second term accounts for a loss of particles of size i due to aggregation of those particles with particles of any other size. In eq 1 the rate of monomer nucleation, g(t), is included.3,4,10 Often, the kinetics of formation of “monomeric” species is slow compared to that of aggregation, as (8) Verwey, E. J.; Overbeek, J. Theory of the Stability of Lyophobic Colloids; Elsevier: New York, 1948. (9) Russel, W.; Saville, D.; Schowalter, W. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989. (10) Feeney, P. J.; Napper, D. H.; Gilbert, R. G. Macromolecules 1984, 17, 2520-2929.
in the Stober silica process3 or emulsion polymerization of polystyrene particles.10 In these processes, monomeric particles are continuously nucleating and must be accounted for in the growth mechanism. Of key importance to quantitatively determining the validity of an aggregative growth model in the silver-borohydride system is probing the rate at which reagents are consumed and nuclei are produced for inclusion into g(t). Final particle size distributions and growth rates are determined by the binary aggregation coefficients, βij. It is through these coefficients that particle interactions are incorporated into the model. The aggregation rate constants are determined from the rate two spheres diffuse together in the presence of a potential energy barrier:
βij ) 6πDijrij
(3)
where Dij is the relative diffusion coefficient of particles i and j. The radius of interaction, rij, is taken as the summation of the radii ri + rj. Incorporating the Stokes equation for drag on a sphere for the diffusion constants and using the Muller treatment8,9 for the diffusion of spheres in the presence of a barrier to aggregation, eq 3 becomes
βij )
2kT 1 (r + rj)(ri-1 + rj-1) 3µ i Wij
(4)
where µ is the solution viscosity, kT is the product of Boltzmann’s constant and absolute temperature, and Wij is the stability ratio, given by
∫r∞+r
Wij ) (ri + rj)
i
j
exp(Vij(R)/kT) R2G(R)
dR
(5)
where Vij(R) is the pair interaction potential between particles of size ri and rj with center to center separation R and G(R) is a hydrodynamic factor. In this work we set G(R) to 1.11 Calculation of the stability ratio requires knowledge of the pair interaction potential between particles i and j. For this study, the pair interaction potential will be modeled using the classical approximation of a summation of electrostatic repulsion and van der Waals attraction (termed the Derjaguin-Landau-Verwey-Overbeek or DLVO approximation).9 The van der Waals potential is well-described by the Hamaker approximation,12 which uses a pairwise summation of the molecular van der Waals interactions over the volume of the macroscopic body. This interaction, for two spheres of differing radii, is given by
VVDW(h) ) -
A12rj 12h
[
1
h 1+ 2(r1 + r2)
(h/rj)
+
h2 h + rj 4r1r2 h 1+ 2(r 2h h 1 + r2) ln jr r h2 h 1+ + rj 4r1r2
( )
+
{ ( )}] 1+
(6)
where A12 is the well-known Hamaker coefficient characterizing dispersion interactions,12 ri is the radius of sphere i, h is the center-to-center particle separation, and rj ) 2r1r2/(r1 + r2) is the geometric average of the particle (11) Kim, S.; Zukoski, C. J. Colloid Interface Sci. 1990, 139, 198. (12) Israelachvili, J. Intermolecular and Surface Forces; Academic Press: San Diego, 1985. Hamaker, H. C. Physica 1937, 4, 1058-1072.
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radii. For bulk silver, the value of the Hamaker coefficient is calculated to be approximately 30 × 10-20 to 50 × 10-20 J.12 In all of the calculations in this study, the lower value was selected. The electrostatic interaction can be described using the linear superposition approximation, given by9
( )(
VE(R) ) 4π0ψ02
2rirj ri + rj
2
)
exp(-κ(R - ri - rj)) R
(7)
where R is the center-to-center particle separation, ψ0 is the surface potential, ri is the radius of particle i, is the dielectric constant of the medium, 0 is the permittivity of free space, and κ is the Debye parameter, which is a measure of the range of electrostatic interactions. The Debye parameter, κ, is a function of the ionic strength in solution, given by
κ)
(
e2 0kT
∑
)
Fizi2
1/2
(8)
where Fi is the number concentration of ions of charge zi. In the system studied here, the value of κ under normal reaction conditions varies from 1 × 108 to 3 × 108 m-1. Values of rjκ in this study range from 0.4 to 1.0. For particles of this small size, the linear superposition approximation (LSA) is accurate to within 10% of the exact numerical solution to the nonlinear Poisson-Boltzman equation down to dimensionless separations (κh) of ∼1-2.13 For an ionic strength of 10-3 M this corresponds to a separation of 1-2 nm. With the surface potentials used in this study, we anticipate the exact value of VE would be underpredicted by 20-30%.13 For future reference we note here that, in determining Wij, the temperature dependencies of and µ must be taken into account. The unknowns required for calculation of the time evolution of particle size distributions with eqs 1-8 are the surface potential, salt concentration, initial particle size, and g(t). Using the values for surface potentials measured previously,3 the following computational method is used to evaluate eqs 1 and 2 and calculate predicted silver particle growth rates. (A) Description of Computational Methods. Equations 1 and 2 constitute an infinite series of linear differential equations. The number of equations required to model a process is determined by the size range of particles to be modeled. In processes where addition of soluble species to the particle surface is important, the operative sizes can range from angstroms to microns and the number of equations needed to describe the system can easily approach 109, making solution difficult.13-15 However, in our system, as discussed below, the smallest (“monomeric”) unit is approximately 2.5 nm in diameter. Therefore, incorporation of only 500 equations (corresponding to a maximum particle size containing 500 monomers) can follow a growth process up to particle diameters of 20 nm. Due to the limited size of the system of equations in this study, a direct numerical solution is used. The aggregative growth model is broken up into two independent parts. First, the matrix of aggregation rate constants, βij, is calculated using the appropriate surface (13) Carnie, S. L.; Chan, D. Y.; Stankovich, J. J. Colloid Interface Sci. 1994, 165, 116-128. (14) Gelbard, F.; Seinfeld, J. H. J. Colloid Interface Sci. 1979, 68, 363. Gelbard, F.; Tambour, Y.; Seinfeld, J. H. J. Colloid Interface Sci. 1980, 76, 541-556. (15) Kostoglou, M.; Karabelas, J. J. Colloid Interface Sci. 1994, 163, 420-431.
potential, salt concentration, reaction temperature, and physical properties (viscosity, dielectric constant, etc). The stability ratios, Wij (from eq 5), used in calculation of the βij matrix are numerically integrated using the method of Riemann sums:19
exp(Vt/kT) ∆h (h)2
(ri + rj)
(9)
using a separation step (∆h) of 0.1 Å. The second part of the model uses the βij matrix and an initial particle size distribution or nucleation rate to calculate ni(t). Each particle population density is calculated at each time step using Euler’s method, where
ni(t+∆t) ) ni(t) +
∂ni ∆t ∂t
(10)
The simulation program was optimized to select the maximum time step, ∆t, without compromising computational accuracy. Further details on the computational procedure can be found elsewhere.16 III. Results and Discussion The aggregative growth model predicts size distributions and growth rates based on inputs characterizing particle interactions, of which we allow only the surface potential and Debye length to vary, the rate of silver reduction and initial particle size, which set r1 and g(t), and solution temperature, which sets solution viscosity and dielectric constant and influences the surface potential. In previous work,3 surface potentials were characterized for a number of different solution conditions. Thus, we have first estimates of this essential parameter. Of substantial significance to the model are r1 and g(t). Below we discuss experiments aimed at determining these parameters. (A) Rate of Reduction of Ionic Silver. In this study we are interested in distinguishing between the rate of particle growth by adsorption and reduction of Ag+ species on the growing particle surfaces and the rate of growth produced by aggregation of clusters of Ag0. Our hypothesis is that the reaction of Ag+ with BH4- is diffusion-limited (and catalyzed by the presence of Ag0 particles). Thus, we anticipate that the vast majority of the Ag+ in solution will be reduced to Ag0 in the first few seconds after mixing silver perchlorate and sodium borohydride solutions. To support this hypothesis, we measure the concentration of ionic silver early in the reaction by two methods. In the first, newly reacted silver sols were injected with a NaCl solution and evidence of the production of AgCl was sought. The final concentration of NaCl was chosen to be 0.25 mM so as not to induce substantial aggregation of the growing silver particles. Silver chloride has a solubility product of 1.77 × 10-10 at 25 °C.26 With a chloride concentration of 0.25 mM, we anticipate seeing an AgCl precipitate for Ag+ concentrations above ∼7 × 10-7 M. (16) Van Hyning, D. L. Ph.D. Thesis, 1999. (17) Ruppin, R.; Yatom, H. Phys. Stat. Sol. 1976, 74, 647. (18) Skillman, D.; Berry, C. J. Chem. Phys. 1967, 48, 3297. (19) Doyle, W. T.; Agarwal, A. J. Opt. Soc. Am. 1965, 55, 305-309. (20) Kleeman, W. Z. Phys. 1968, 215, 113-120. (21) Fornasiero, D.; Grieser, F. J. Chem. Phys. 1987, 87, 3213. (22) Mie, G. Ann. Phys. 1908, 25, 4. (23) Kerker, M. The Scattering of Light and Other Electromagnetic Radiation; Academic Press: New York, 1969. (24) van de Hulst, H. C. Light Scattering by Small Particles; Wiley: New York, 1957. (25) Henglein, A.; Ershov, B.; Janata, E. J. Phys. Chem. 1993, 97, 339.
Aggregative Growth of Silver Sols
If NaCl is added to a silver perchlorate/sodium borohydride solution 5 s after the reduction is initiated, no visible changes in solution color or turbidity were observed. In addition, absorbance spectra recorded before and after addition of the salt show no change. This test suggests that the highest possible concentration of free silver in solution (based on the solubility of AgCl at 25 °C) is ∼7 × 10-7 M. At this point in the reaction, the particles are ∼2-3 nm in size, and over the course of the next 20-60 min, they grow to 10-20 nm in size. The amount of Ag+ left in solution after 5 s is clearly too small to account for any appreciable particle growth by adsorption and reduction of Ag+ onto the particle surfaces. To confirm that no free or bound ionic silver is present in solution, two further experiments were performed. Two solutions were reacted normally: one under conditions of [Ag] ) 0.250 mM, R ) 6, and T ) 2 °C and the second under conditions of [Ag] ) 0.250 mM, R ) 12, and T ) 25 °C. Within 10 s of initiating the reactions, sodium perchlorate was added at a concentration in excess of that required to rapidly aggregate the particles in solution. The solutions were rapidly filtered and the filtrates submitted for elemental analysis. The filtrates showed no sign of silver sol formation after filtration, even after being left overnight (i.e., the surface plasmon band giving the solutions a yellow color did not develop). As control studies indicate that excess sodium perchlorate does not interfere with Ag+ reduction by borohydride, the lack of a surface plasmon band shows that prior to filtration the concentration of Ag+ has been reduced by 2 orders of magnitude. This conclusion is further supported by the elemental analysis of the filtrates where the silver concentration was that of the blanks. In combination with the lack of evidence for AgCl precipitation, we conclude that the ionic silver concentration has been reduced by at least 2 orders of magnitude in
