Simulated Dynamic Optical Response Strategy for Model

5588

Ind. Eng. Chem. Res. 2010, 49, 5588–5602

Simulated Dynamic Optical Response Strategy for Model Identification of Metal Colloid Synthesis Roberto Irizarry DuPont Electronic Technologies, 14 T. W. Alexander DriVe, Research Triangle Park, North Carolina 27709

A model for metal spherical particle formation is proposed, guided by optical kinetic data of monodisperse metal colloid synthesis. The dynamic optical response of these systems is characterized by broad bands, making their analysis difficult to interpret in terms of particles sizes. To overcome this problem, the data are analyzed in terms of a new strategy called simulated dynamic optical response. Kinetic data were generated using a silver-ethylenediamine complex reduced with isoascorbic acid as a model system. Using this strategy, it was found that autocatalytic formation of primary particles followed by a zone of very fast aggregation mechanism can describe the dominant dynamics during early stages. In later stages, the dominant mechanism switches to slower aggregation modulated by a stability factor. The presented model and identification strategy may be applied to other reaction precipitation systems to produce metal particles. 1. Introduction Chemical reaction precipitation is a very cost-effective and versatile way to manufacture monodisperse spherical metal particles. In particular, spherical silver powders in the range of 100 nm to 6 µm are used in very large quantities in electronic industries. As a result of their importance, a significant number of syntheses are continually being published and patented.1-4 The formation kinetics of these syntheses are usually very fast, making the resulting powder size distribution and morphology very sensitive to process conditions. Furthermore, the systematic development of new powders with desired properties is still mainly empirical in nature because of the lack of fundamental kinetic data. Given the fast dynamics, the system cannot be sampled in real time, making the kinetics of particle formation difficult to study. The author and co-workers have used a rapid mixing stopped-flow reactor with multidiode array to collect UV-vis absorption as a function of time during the precipitation process of different syntheses for gold and silver.5-7 The advantage of using UV-vis spectra to study the kinetics of metal particle formation is that it can be obtained very quickly and in a noninvasive fashion. The plasmon resonance characteristics of metal particles can be utilized to understand the formation of metal particles. More recently, optical response dynamics have been used to study the formation of nanoparticles (NP). Leon, Irizarry, and Castro-Rosario8 studied the formation of Ag2S nanoparticles in the size range of 2-10 nm using a stopped flow reactor. The absorbance near the band edge of the semiconductor NP is used to monitor the size of particles formed over time. Tiemann9 studied the growth of ZnS NP by precipitation from supersaturated aqueous solution using stopped-flow UV absorption spectroscopy. The absorption energy is correlated with the average particle radius derived from energy gap versus the bulk band gap energy. Mori10 studied the rate of perylene particle formation using a rapidflow reactor following the absorption peak around 434 nm, attributed to perylene monomer, and the absorption peak around 460 nm, the perylene nanoparticle band. In all these examples, the optical signal can be accurately correlated with * To whom correspondence should be addressed. E-mail: [email protected].

average size, making the evolution of particle size explicit, from which a mechanism can be proposed. Unlike synthesis of small NP (2-20 nm), the data obtained in our experiments for the formation of large colloidal metal particles are difficult to analyze because a typical experiment results in broad bands. The main difficulty is due to the system having a wider distribution of sizes. To overcome these limitations, in this work I describe an identification methodology that can effectively be used to extract macroscopic model mechanistic information from UV kinetic data. Using this approach, a new macroscopic model is proposed guided by the generated kinetic data for the synthesis of silver-ethylenediamine complex reduced with ascorbic acid.3 The paper is organized as follows: In section 2, the current state of modeling is briefly reviewed. In section 3, the equipment and procedure for the kinetic experiments are described. The simulated dynamic optical response (SDOR) strategy to analyze the kinetic data is introduced in section 4. In section 5, the experimental results of the test synthesis are discussed. In section 6, a new macroscopic population balance model of spherical particle formation is proposed. The capability of the model in reproducing the experimental data is considered in section 7. Section 8 presents the discussion and validation of results, and section 9 presents the main conclusions of this work. 2. Current State of Modeling of Spherical Particles Made by Reaction Precipitation Nucleation Mechanism. LaMer classical nucleation theory, originally developed to describe the formation of sulfur sols,11,12 has been applied to describe many other systems. In this mechanism, the concentration builds up slowly until a critical concentration is reached. Homogeneous nucleation occurs at a very high rate (burst nucleation). The large supersaturation immediately lowers the concentration of monomers in solution until the nucleation completely stops. At this point, growth occurs by diffusion of monomers to the particle. Although successfully applied to many systems, it has been recognized that many cases, in particular metallic nanoparticles, do not follow this mechanism. The rate of burst nucleation can be approximated using classical nucleation theory (CNT). In this theory, the formation of embryos of different sizes is controlled

10.1021/ie100644h  2010 American Chemical Society Published on Web 05/17/2010

Ind. Eng. Chem. Res., Vol. 49, No. 12, 2010

by the free energy function, with two terms to account for the phase change and the creation of a new surface. The critical size is reached when the number of atoms in an embryo have maximized the free energy change, and larger clusters grow irreversibly.13-15 Turkevich and co-workers studied the formation of colloidal gold by heating the solution in the presence of sodium citrate.16 In this case, the sodium citrate is used as both the reducing agent and the stabilizer of formed particles. Given the slow kinetics of this system, they measured the nucleation rate using electron microscopy images of samples. They found that particle formation consist of an induction period, followed by a fast increase in the number of particles, followed by a rapid decrease in rate. As this system has been well-studied and all kinetic parameters and reaction mechanisms are available, detailed modeling of this system has been reported recently.17 Similar findings on nucleation rates were described in the work of Finke and co-workers on the reduction of transition-metal salts under H2.18 They proposed a mechanism of cluster formation consisting of slow nucleation followed by fast autocatalytic surface growth.18,19 Their mechanism consists of the formation of nuclei from precursors followed by fast autocatalytic growth: A f B then A + B f 2B, where the first step represents nucleation and the second step represents growth. Model for the Formation of Large Colloidal Spherical Particles. Dirksen, Benjelloun, and Ring20 studied and modeled the precipitation of copper oxalate in a batch reactor. From the morphology of final particles, it was suggested that they were formed by aggregation mechanism. They developed a mathematical model assuming atomistic growth and particle aggregation. They postulated a population balance model in which only primary particles aggregate with larger particles. Growth by aggregation mechanism has been observed in other colloidal systems as well.21 In the case of monosized spherical metal particles, there is a lack of experimental data on the formation dynamics. Most modeling work so far has been based on inferences from the final particle examination. The work of Privman, Goia, Matijevic, and co-workers introduced a model that focuses on the master equations for the secondary particle concentration. The primary particles are produced by burst nucleation at a rate given by classical nucleation theory. The primary particles are then consumed by secondary particles by a diffusion-dominated mechanism. They proposed this mechanism as the main route to monodispersed colloid synthesis for spherical particles.22-28 There is a large amount of experimental evidence of final particle morphology suggesting that these particles are formed by the assembly of smaller units (primary particles). To better understand the mechanism governing this process, kinetic data are needed. 3. Kinetic Experiments: UV Absorption Spectroscopic Kinetic Experiments Synthesis of Metal Particles. In this study, the reduction of silver-ethylenediamine (EDA) complex with ascorbic acid is used as a model system.3 Ascorbic acid is a versatile reducing agent used in the development of many systems to make spherical and crystalline silver particles.2,3,7,29 Ethylenediamine is an example of many possible amines that can effectively complex the silver to reduce the redox potential to make larger colloidal particles. Depending on the experimental conditions, the size of these particles ranges from 0.1 to 2.0 µm. The overall reaction is

+

5589

2[Ag(C2H8N2)] + C6H8O6 f 2Ag + C6H6O6 + 2C2H8N2 + 2H+ 0

The final morphology of the particles consists of nanounits.3 Similar spherical particles have been developed using EDA and hydroquinone as reducing agent7 (see Figure 1). Kinetic Experiments. To eliminate the mixing effects, I generated the kinetic data using a stopped-flow reactor (SFR), model SX.18MV (Applied Photophysics Ltd., London, UK), designed for fast kinetics. In a typical experiment, the silver nitrate/EDA solution and the ascorbic acid solution are injected in equal volumes into a T-mixer that fills an optical cell. The equipment micromixing time is 10 ms with a diode array that allows full spectrum recording from 250 to 800 nm at a rate of 1000 spectra per second. The photodiode array (PDA) uses a linear 256-element diode array with a diode separation of about 2.17 nm. The optical cell is illuminated by a xenon arc lamp (ozone-free). A light path length of 10 mm in the observation cell is used. The particle images are analyzed using FESEM (Jeol 6700). 4. Model Identification Using the Simulated Dynamic Optical Response Strategy (SDOR) Nanosize silver particles effectively interact with light as a nonlinear function of their size relative to the mean free path of conducting electrons, and the frequency of the complex dielectric function in the metal. When an electromagnetic field interacts with the particle, a coherent oscillation of electrons in the conduction band is induced (surface plasmon resonance). The surface plasmon resonance has been extensively used to characterize NPs.30-33 For monosized colloidal systems, a welldefined plasmon peak allows the qualitative correlation of UV absorption and nanoparticle size and shape. As shown in the next section, the kinetics of the formation of larger particles is characterized by broad bands. This is a result of particles of very different sizes (nm-µm) present at the same time during the course of the kinetics of particle formation. Given the fact that the optical response is highly nonlinear with size, one cannot deconvolute the size distribution of the particles. One example of experimental data demonstrating the uniqueness problem when deconvoluting UV data is given in Henglein and Giersig,34 in which monosized and aggregated silver (by TEM) showed a very similar symmetric plasmon band, whereas for other conditions, aggregate silver particles (by TEM) showed a broad band. The use of UV absorption spectroscopic kinetic experiments for model identification is limited by the fact that the data cannot be deconvoluted into a size distribution or average size. In this work, a new strategy is proposed to overcome this problem. Model Identification Using Simulated Dynamic Optical Response (SDOR). In this strategy, given a “trial” or “test” model that predicts the evolution of a particle population, simulated absorption and/or scattering spectra are compared with experimental spectra. The identification is achieved by testing different “trial” models and comparing the simulated extinction/ scattering with the experimental values. This identification strategy is shown schematically in Figure 2. The procedure is summarized in the following steps (for the case of absorption data): (1) Generate a set of absorption spectroscopic kinetic data. For each experiment, save the absorption data in the matrix ABSexp(ti, λj) where the indexes i and j run over the time steps

5590


Figure 1. Scanning electron micrographs (SEM) of monodisperse silver particles obtained with the EDA method using a different reducing agent.7

If the trial model cannot reproduce the experimental data, go to step 2 and propose a new trial model. Notice that in this procedure, the experimental data and the corresponding SDOR may cover a set of different experiments, which are indexed as ic ) 1, Nic (omitted for brevity). I propose two strategies for the calculation of extinction coefficient Cext,p(V, λj) used in step 4. SDOR-1. In the first version, called SDOR-1, Cext,p(V, λj) is calculated by solving the Maxwell equations for a spherical particle in the far field (Mie theory). For a qualitative analysis, there is no need to find the value of K. For a quantitative analysis. the spectrophotometer is calibrated with a wellcharacterized NP to find the value of K (only one monosized colloidal solution is needed for this purpose). SDOR-2. In the second version, called SDOR-2, a set of wellcharacterized monosized colloids is utilized (each colloid with a different size). The absorption spectrum of each colloid is measured using the same equipment used for the kinetic experiment. Each spectrum is then normalized by the number of particles in the colloidal solution, and saved in a table, ABSext,p(Vk, λj) . With this information, eq 1 is approximated as ABSsim(ti, λj) )

[∫

∑

k (Vk+1+Vk)/2

(Vk-1+Vk)/2

]

n(V, ti)dV ABSext,p(Vk, λj) (SDOR) (2)

Inverse Problem. A more rigorous identification scheme is the solution of the following combinatorial optimization problem Nic

min

Figure 2. Model Identification Using SDOR Strategy.

and a set of wavelengths {λ1, λ2, ..., λm} are recorded in each kinetic experiment. (1.1) The system should be sufficiently diluted that the independent scatterers assumption is valid. (1.2) Other species present during the reaction should not interfere with surface plasmon spectrum, or a correction for this interaction should be made before saving the data in the ABSexp(ti, λj) matrix. (2) Propose a mechanism and parameters to build a “trial” model. (3) Solve the corresponding “trial” model to generate a theoretical particle size distribution as a function of time, n(V,ti)for a set of time steps, ti. (4) Generate the SDOR. For each time, ti, and a set of wavelength range {λ1, λ2, ..., λm} calculate the simulated dynamic optical response predicted by the trial model ABSsim(ti, λj) )

∫

∞

-∞

n(V, ti)Cext,p(V, λj)dV/K (SDOR) (1)

where ABSsimis the simulated absorption spectra, Cext,p(V, λj) is the extinction coefficient of a single particle of volume V, and K is a constant that converts the extinction coefficient into the same arbitrary units as the experimental detector. Notice that the assumption of independent scatterers is implicit in this equation. (5) Compare the simulated absorption with the experimental absorption. The analysis can be qualitative or quantitative. Some of the important features to consider are the rate of change of the absorption peak, the peak shift, and the shape of the spectra.

all models, M

F)

∑ { ∑ ∑ (ABS

ic sim(ti, λj)

ic)1

i

-

j

}

ABSicexp(ti, λj))2

SDOR(M, ic), for ic)1, Nic

ST (3a)

(3b)

where F is the objective function to be minimized, and SDOR(M, ic) is the SDOR for a model, M, and experiment, ic. This is a combinatorial optimization problem where the search space covers a set of possible models as a well as their corresponding parameter values. The combinatorial nature of this problem and the set of constraints (eq 3b), makes the solution of this optimization problem a challenging one. The solution of this problem is considered in a concurrent work,35 using stochastic optimization algorithms previously developed to solve this type of problem.36,37 Because SDOR-1 involves theoretical calculations of absorption and scattering, the solution of this problem is described next, including possible extensions to more complex particles. 4.1. Modeling the Optical Response of Spherical Nanoparticles. The Mie theory consists of the analytical solution of the Maxwell equations for the absorption and scattering of light by a spherical particle.38-40 The particles have to be large enough so that classical electromagnetic theory applies. For a spherical particle of radius a, the extinction and scattering coefficients Cext and Csca are related to the Mie scattering coefficients an, bn through ∞

Cext )

∑

2π (2n + 1)Re(an + bn) k2 n)1

(4)


5591

Figure 3. Kinetic data: (a) 24, (b) 16, and (c) 9.6 mM. ∞

Csca )

∑

( | |)

2π (2n + 1) |an k2 n)1

2

(5)

where k is the wavenumber given by k2 ) ω2µ, is the permittivity, µ is the permeability, and ω is the frequency. The permittivity can be expressed in terms of the permittivity of free space, 0, and the dielectric function, ε, by ) 0ε. The operator Re is the real part operator. The Mie scattering coefficients an and bn are functions of the particle radius and the wavelength in terms of the Ricatti-Bessel functions: m jn(x)d[xjn(x)]/dx - jn(x)d[mxjn(mx)]/dx

εplas(ω) ) 1 -

(1) m jn(mx)d[xh(1) n (x)]/dx - hn (x)d[mxjn(mx)]/dx

bn )

2

jn(x)d[xjn(x)]/dx - jn(x)d[mxjn(mx)]/dx (1) jn(mx)d[xh(1) n (x)]/dx - hn (x)d[mxjn(mx)]/dx

(6)

Γ(r) ) γ +

VF a

(9)

where VF is the electron velocity at the Fermi surface, and a is the particle radius. The modified collision frequency is incorporated into the measured dielectric functions to determine the size-dependent dielectric function

(7)

where x ) nma, m ) np/nm is the ratio of indices of refraction of the particle and medium, and jn(x) and hn(1)(x) are Bessel functions. In these equations, it is assumed that the magnetic permittivities of the medium and the particle are the same. For nonspherical particles, numerical methods for solving the Maxwell equations are needed: discrete dipole approximation,41,42 finite difference time domain methods,43 or T-matrix methods.40,44 For very small particles (nanoparticles), size-dependent dielectric functions are needed to correct for the confinement effect. When the electrons oscillate in response to the applied electromagnetic field, their motion is damped via inelastic collisions that occur with the frequency γ, given by γ ) τ-1 b where τb is the relaxation time of the bulk metal. The plasmon contribution to the dielectric constant is approximated using the Drude model

(8)

ω2 + iωγ

where ωp is the frequency of the plasma oscillation of the bulk metal. For particles comparable to the mean free path of the electrons, an additional damping mechanism results from the collision of conduction electrons with the particle surface. This collision frequency is given by

2

an )

ω2p

2

+ |bn

ε(a, ω) ) εexp(a, ω) +

ω2p ω2 + iωγ

-

ω2p ω2 + iωΓ

(10)

5. Experimental Results Early Stages. A series of kinetics experiments was performed in which the silver-amine complex and the ascorbic acid solution were mixed in the stopped-flow reactor at ambient temperature. To understand the early nucleation stages, I performed a series of short-time experiments using different silver concentrations while keeping the molar ratios of all other chemicals constant. The amine/silver molar ratio was 3 and the ascorbic acid/silver ratio was 1:1. The silver-EDA solution was aged for 30 min at room temperature before the experiment. Figure 3a-c shows absorption spectra for silver concentrations 24, 16, and 9.6 mM, respectively. For all dilutions, at wavelengths of 390-600 nm, a broad band of the surface plasma resonances was observed. The position of the peak around 420 nm remains constant at

5592


Figure 4. Absorbance at 420 nm as a function of time.

these short time intervals; only the intensity changes. This is a good indication that the number of particles of different sizes is increasing. Figure 4 shows the absorption at 420 nm as a function of time. The system starts slowly (induction time), then there is a very fast increase in rate. To collect kinetic data, I diluted the system to more than the optimized conditions. For the conditions of these experiments, the particles are analyzed using a field-emission scanning electron microscope (FESEM Jeol 6700). As shown in Figure 5a, even for these low concentrations it can be observed that the particles are formed by aggregation of primary particles. Under the conditions of the kinetic experiments, the morphology of the primary particles is spherical (Figure 5b); however, the particles exhibit increased agglomeration, as compared to particles processed under optimal conditions. Late Stages. To understand the behavior over longer time periods, an experiment was performed using the same molar ratio, with kinetics followed for 500 s. To see the transition from dominated production of primary particles to dominated growth by particle aggregation, I selected a more diluted system using 6 mM Ag solution. As shown in Figure 6, at short times a similar behavior was observed (a broad band increasing in intensity without a major peak shift). At intermediate times, the position of the peak shifts toward longer wavelengths. At longer times, the position of the peak continues shifting toward higher wavelengths, and there is a reversal in intensity of the peak, whereby it starts decreasing steadily. This later stage can be explained as an aggregation stage. These experiments have been repeated using different methods for preparing the silver-amine complex. It was found that the kinetics was affected by the mode of preparation, such as temperature and age of the solution. Nevertheless, in all cases, the general features of the kinetic data presented in this section remained the same. Furthermore, similar data have been obtained for other syntheses to form spherical silver and gold particles,5-7 also showing similar trends. 6. Macroscopic Population Balance Model of Spherical Particle Formation The population balance model introduced in this section was developed using the SDOR identification strategy with the kinetic data presented in the last section. The model consists of a pseudomechanism for the formation rate of NPs (primary particles) and a mechanism for the aggregation of these primary particles to form the final particles (secondary particles). The model for primary particle formation is described in section 6.1 and the aggregation mechanism for secondary particles enlargement is described in section 6.2. The model is shown schematically in Figure 7. 6.1. Minimal Pseudomechanistic Model for the Generation of Primary Particles. A model based on a minimal pseudomechanism is proposed for the formation of primary particles. This

Figure 5. Field-emission electron micrographs of silver particles obtained with silver-EDA complex reduced with ascorbic acid at the concentration of the kinetic experiments (24 mM): (a) ×130 000 magnification and (b) ×10 000 magnification.

model attempts to capture the combined effect of reaction, nucleation, and cluster-cluster aggregation to form primary particles with the following compact set of elementary reaction steps Ag+ + e- f Ag0

(11)

n(Ag0) f C

(12)

C + C 98 P + I

(13)

other species

n(Ag0) + I f C + I

(14)

There are two pseudospecies, C and I, introduced in this pseudomechanism. The pseudospecies, C, represents the lumping of a population of silver clusters of different types and sizes into a single variable. The other pseudospecie, I, is another lumped variable representing small clusters of silvers with ligands. It is assumed that all primary particles have the same volume, Vo (containing Z atoms).


5593

Figure 6. Kinetic data, 6 mM for 500 s.

d[C] 2 2 ) k1[Ag0] - 2k2[C]2 + k3[Ag0][I] dt Z Z

(16)

d[I] ) k2[C]2 dt

(17)

The rate of production of primary particles into the system is given by Jp(t, Vo) )

Figure 7. Illustration of the formation mechanism based on a fast coarsening of fine particles and a much slower aggregation of larger particles stabilized using DLVO theory. A pseudomechanism predicts the rate of primary particle formation.

In this pseudomechanism, the silver ions are reduced by a chemical reaction, as shown in eq 11. The silver atoms form silver clusters, C, as shown in eq 12. As described in eq 13, the clusters interact with each other (cluster-cluster aggregation and cluster restructuring) to form a final primary particle P. During the interactions between silver clusters C, the presence of other species (ligands, counterions, surfactants, etc.) will result in the formation of small clusters, I, including silver and the other species. The pseudospecies, I, accelerates the generation of more clusters in eq 14. The kinetic model is derived directly from the elementary steps proposed in the pseudomechanism. In the current version of the kinetic model, steps in eqs 11 and 12 of the pseudomechanism are combined into a single step (reaction and nucleation are represented by a single rate equation). The formation of C is proportional to the silver concentration: k1[Ago]. The clusters aggregate at a rate k2[C][C] to form primary particles, P, and intermediate clusters, I (if ligands are present). The presence of the intermediate clusters, I, can further accelerate the formation of clusters, C, at a rate k3[Ago][I]. The constant k1 captures the combined effect of chemical reaction and nucleation. The constants k2 and k3 relate to the aggregation mechanism. Arbitrarily, I have assumed that the size of C is half of the size of the primary particle volume (n ) Z/2). With these assumptions, the following kinetic model is proposed for the minimal pseudomechanism d[Ag0] ) -k1[Ag0] - k3[Ag0][I] dt

(15)

d[P] ) k2[C(t)]2 dt

(18)

where Jp(Vo, t) is the rate of formation of primary nanoparticles of size Vo and time t. 6.2. Population Balance-Based Aggregation Model for Final Particle Formation. The primary particles of size Vo supplied at a given rate (eq 18) grow to form larger particles mainly by aggregation. The population balance model describing this process is given by45

∫

∂n(V, t) 1 x ) Jp(t, Vo)δ(V - Vo) + q(V - V′, V′)n(V ∂t 2 0 ∞ V′, t)n(V′, t)dV′ - n(V, t) 0 q(V, V′)n(V′, t)dx′ (19)

∫

where the number density, n, is the number of particles per unit volume, Jp(t, Vo) is the primary particle addition rate, q is the aggregation rate, V and V′ are particle volumes, and t is the time. The first term on the right-hand side is the nucleation term. The second and third terms are the birth and death of particles of volume V due to aggregation. It is assumed that the dominant coagulation mechanism is Brownian motion with kernel given by qB(V, V′) )

2kT 1/3 (V + V′1/3)(V-1/3 + V′-1/3) 3µ

(20)

where qB is the Brownian collision kernel, k is Boltzmann’s constant, µ is the viscosity, and T is the reactor temperature. The Brownian collision kernel is corrected with two functions to account for two phenomena observed in our experimental data: (a) fast aggregation of small particles at early stages and (b) stabilized aggregation at larger stages. The proposed modified kernel is given by q(V, V′) )

W1(V, V′)qB(V, V′) WDLVO(V, V′)

(21)

where WDLVO is the stability factor given by the DLVO theory, and W1 is a new phenomenological function introduced in this work to account for the fast aggregation of small particles with any other particle.

5594


The function W1 introduced in this work is defined as W1(V, V′) ) 1 + Ae-Bmin[V,V′]

(22)

where the parameters A and B are model adjustable parameters. This function accelerates the aggregation of smaller particles to any particles by a factor W1; this effect becomes negligible for larger particles. The so-called stability factor, using standard DLVO theory to account for the attractive and repulsive forces between particles46-48 is given by W(V, V′) ) 2

∫

∞

0

exp(VT(x, V, V′)/kT) (x + 2)2

(23)

dx

In this equation, x is the shortest distance between particle surfaces scaled by the average radius between the two particles. VT is the potential between two colloidal particles. The interaction energy VT is the sum of all forces between particles. In particular, the effects of van der Waals attractive forces and electrostatic repulsive forces are considered (VT ) Vvdw + Velec). The van der Waals potential between two particles is given by Vvdw ) -

[

AH 2a1a2 2a1a2 + 2 + 6 R2 - (a - a )2 R - (a1 + a2)2 1 2 R2 - (a1 + a2)2 (24) ln 2 R - (a1 - a2)2

]

where AH is the Hamaker constant, a is the particle radius, and R is the distance between two particle centers. There are several formulations for the electrostatic interaction. To simplify the problem, the following expression is used: Velec ) 4πεψ20

a1a2 ln[1 + exp(-kx)] a1 + a 2

(25)

where ψo is the surface potential, k is the Debye-Huckel parameter, and ε is the dielectric constant of the medium. For a reacting system, the actual experimental values of the DLVO parameters are difficult to obtain. Furthermore, in many systems there are dramatic differences between the measured and calculated values of the stability constants using DLVO theory. One example is the heterocoagulation of spherical PVC latex and chromium hydroxide particles.26 The DLVO model assumes a uniform charge distribution on the surface of interacting particles, which is not the case in metal formation with various ionized groups on their surfaces. To overcome these limitations, I used the DLVO equations in this model, relaxing the parameters to become adjustable parameters determined from kinetic data and final particle size distribution (AH(a), κ(a), ψ(a)), where the superscript (a) indicates adjustable parameters. To minimize the number of parameters in this work, I adjusted only κ(a), and the other parameters were kept constant using representative experimental values. 7. SDOR Results The model presented in section 6 was found using the SDOR identification strategy shown in Figure 2. The dominant mechanisms of the model were identified by a qualitative comparison of the features of the kinetic data and the SDOR-1. The features used during the identification process were the rate of change of the absorbance peak (around 420 nm), the peak shift, and the spread of the spectra (band broadness). The model param-

eters were subsequently adjusted by simple trial and error, this time using SDOR-2. Alternatively, these parameters could be found by formal optimization, but this route was not pursued here. Final particle sizes are also generated and compared with experimental data reported in the literature.3 To illustrate the capability of the found model, I also presented SDOR for a limited set of “test” models, in particular, models using classical nucleation theory, Brownian aggregation, and diffusional aggregation as “trial” mechanisms. As shown in this section, the behavior of all these alternative mechanisms does not reproduce the experimental data as well as the proposed mechanism. 7.1. Simulation Details. SDOR-1 Parameters. The theoretical absorbance of the spherical particles was calculated using Mie theory with the following optical parameters. The dielectric constant of the reacting system was approximated using the refractive index of water.49 The experimental values of the dielectric constant for bulk metals are given by Irani, Hurn, and Wooten50 and Johnson and Christy.51 These constants were modified to correct for the finite size of the NPs as described in section 4.1 using the following parameters: pωp ) 6.1 eV, pγbulk ) 0.074 eV, and VF ) 0.914 eV nm.52 SDOR-2 Colloids. Four colloids were purchased (20, 40, 60, and 80 nm) from British BioCell International (Ted Pella, Inc.). The reported size and particle concentration were used as received. The measured absorption spectra in the stopped flow reactor were normalized by the reported particle concentration to generate the matrix ABSext,p(dk, λj) for d1 ) 20 nm, d2 ) 40 nm, d3 ) 60 nm, d4 ) 80 nm. This matrix was used in eq 2 to generate the SDOR. The SDOR-2 calculation is constrained to model PSD predictions in the same range covered by the monosized colloids (in this work, 0). All other possible collisions between secondary particles are set to zero (qB(V1, V2) ) 0, V1,V2 * V0). The nucleation rate in eq 9 is JCNT. SDOR using this model in Figure 15 results in a sharp plasmon peak, with the intensity of the peak increasing with time until a final intensity is reached. This is in contrast to the experimental data, in which the intensity increases and then decreases at longer times.

8.1. Found Model. The found model describes the kinetic data and the final size distribution more accurately than other tested models. The particle size coarsening predicted by the found model (see Figure 10 and 11) has been observed experimentally in related systems with slow dynamics where TEM images have been taken.34,39,56 For the case of silver reduced by the citrate method, large-sized (50-100 nm) silver crystallites display a broad band with a peak around 420 nm.34,56 Other reduction methods that produce smaller particles (5-20 nm) display a sharp surface plasmon absorption in the range of 380-400 nm.34,56 The pseudomechanism proposed in this work is a crude approximation of the actual governing mechanism. Nevertheless, it reproduces the dominant dynamics of the experimental data. This pseudomechanism can be further improved with the addition of the experimental or theoretical knowledge of species formed8 and chemical kinetics rate expressions.57 The pseudomechanism steps can be related to real mechanistic steps and real species in the system. In the synthesis studied in this work, it may be assumed that pseudocluster I could be a measurement of the presence of silver-amine organometallic clusters. To test this hypothesis, experiments were made with a deficient EDA/ silver molar ratio (1.5 molar ratio vs 3 molar ratio in previous experiments). As shown in Figure 16, autocatalytic behavior is not observed under these conditions. This example shows how changes in the chemistry can completely change the nucleation mechanism. Identification schemes like SDOR are useful tools for screening mechanistic changes for different reactionprecipitation syntheses.


5599

Figure 14. Simulation results using CNT to describe the nucleation rate: (a) SDOR-2 and (b) absorbance at 420 nm as a function of time.

Figure 15. Simulation results for the approximation to the diffusional model: (a) SDOR-1 and (b) absorbance at 420 nm as a function of time.

8.2. SDOR Identification Method. The SDOR model identification methodology introduced in this work has proven to be an effective tool to interpret optical data in terms of

possible governing mechanisms. This identification strategy depends on the quality of SDOR-1 and SDOR-2 predictions. The SDOR-1 is limited by the Mie approximation and param-

5600


Figure 16. Absorbance at 420 nm as a function of time.

eters used (medium dielectric constant, etc.). For qualitative analysis, as shown in this work, this is a good first-order approximation. For quantitative analysis, this approach underpredicts the broadness of the absorption spectra. After calibration with a well-characterized colloid, the absorption peak and the peak shift can be used as metrics for a more quantitative identification process. The SDOR-2 is preferred for quantitative analysis because it eliminates the modeling approximation by using model colloids. In this work, both strategies have proven to be useful for model identification. To validate the SDOR assumption of independent scatterers for the concentration range used in this work, 20-nm nanoparticles were prepared using the polyol process.1,58 In this synthesis, the first solution is prepared by dissolving 1.4 g of silver nitrate in 80 g of ethylene glycol at room temperature and raising the solution temperature to 80 °C. The second solution was prepared by dissolving 1.4 g of Daxad 19 in 440 g of ethylene glycol at room temperature. The solution temperature was then increased to 170 °C. The two solutions are mixed under agitation to form the nanoparticles. The nanoparticles are estimated to be about 20 nm in diameter as measured by LM-10 equipment (Nanosight,U.K.), which monitors the Brownian motion of particles. As a second comparison, a perfect match was observed between the prepared colloid and the purchased 20 m colloid. The colloidal solution was diluted several times to generate a calibration curve. Figure 17a shows the absorption spectra as a function of dilutions. A linear relationship between the number of particles and the measured absorbance was observed, as shown in panels b and c in Figure 17. The observation of this linear relationship between the number of particles and the measured absorbance validates the assumption of the absorbance being the accumulated contribution of independent scatterers. Furthermore, theoretically it is known that the plasmonic near-field coupling decays very rapidly as 1/r3, where in this case r is the distance away from the particle. 9. Conclusions New kinetic optical data was generated and analyzed using the SDOR strategy to identify the dominant mechanism during particle formation. In this strategy, instead of trying to deconvolute the broad band response in average sizes, a trial population balance model was used to generate a simulated dynamic optical response. Comparing the SDOR with experimental kinetic data, it was found that the nucleation is better described using a reduced-order mechanistic model than with standard nucleation theory. It was also found that, initially, small

Figure 17. Calibration using 20 nm particles made by the polyol process: (a) surface plasmon absorbance at different dilutions, (b) calibration curve at 340 nm, and (c) calibration curve at 410 nm.

particles (including primary particles) aggregate very quickly, transitioning to a slower aggregation rate modulated by DLVO stability at later stages. The model reproduced the main features of the experimental kinetic optical data and the final particle size distribution and morphology. The simulated optical response combined with high resolution optical kinetic experiments has proved to be a powerful tool for mechanism identification. This technique and the model


found will be used in future work to study different metal particle reaction-precipitation syntheses in more detail, identifying mechanism and kinetics parameters. To further expand this analysis, future work will consider the possibility of nonspherical particles shapes and multidimensional PBE modeling in the SDOR-1 and SDOR-2 calculations. Acknowledgment The stopped-flow runs were performed in DuPont’s powder development lab by Lourian Burwell and John Cole under the supervision of the author. The FESEM analysis of the particles was performed in a DuPont analytical lab by Ann Scheuer. Literature Cited (1) Sugimoto, T. Fine Particles, Synthesis, Characterization, Mechanism of Growth; Marcel Dekker: New York, 2000. (2) Irizarry R.; Rivera V.; Glicksman H. Process for making highly dispersable, spherical silver powder particles that are very high solids and highly ordered. U.S. Patent 7,648,557, January 19, 2010. (3) Halaciuga, I.; Goia, D. V. Preparation of silver spheres by aggregation of nanosize subunits. J. Mater. Res. 2008, 23, 1776–1784. (4) Suber, L.; Sondi, I.; Matijevic´, E.; Goia, D. V. Preparation and the mechanisms of formation of silver particles of different morphologies in homogeneous solutions. J. Colloid Interface Sci. 2005, 288, 489–495. (5) Murillo L. E., Viera O., Vicun˜a E., Briano J. G., Castro M. E., Ishikawa Y., Irizarry R., Sola´ L. Growth Kinetics of Gold Nanoparticles. In Technical Proceedings of the 2002 International Conference on Computational Nanoscience and Nanotechnology; San Juan, Puerto Rico, April 21-25, 2002; Nano Science and Technology Institute: Danville, CA, 2002; Vol. 2, Chapter 16, pp 435-438. (6) Mercado, L., Castro, W., Vicun˜a, E., Ishikawa, E., Briano, J. G., Irizarry, R., Sola´, L., Castro, M. E. Real Time Kinetic Measurements of Silver Nanocluster Growth. In Technical Proceedings of the International Conference on Computational Nanoscience and Nanotechnology; San Juan, Puerto Rico, April 21-25, 2002; Nano Science and Technology Institute: Danville, CA, 2002; Vol. 2, Chapter 16, pp 439-442. (7) Irizarry R. Rivera V. A Method to Make High Purity Spherical SilVer for Fodel Application; DuPont report 2005-MCM-01; DuPont: Wilmington, DE, 2005; parts of this report are available from the author under appropriate nondisclosure agreement. (8) Leon, M.; Irizarry, R.; Castro-Rosario, M. E. Nucleation and growth of silver sulfide nanoparticles. J. Phys. Chem.B. 2009, 114 (13), 5839– 5849. (9) Tiemann, M.; Marlow, F.; Brieler, F.; Linden, M. Early Stages of ZnS Growth Studied by Stopped-Flow UV Absorption Spectroscopy: Effects of Educt Concentrations on the Nanoparticle Formation. J. Phys. Chem. B 2006, 110, 23142–23147. (10) Mori, J.; Miyashita, Y.; Oliveira, D.; Kasai, H.; Oikawa, H.; Nakanishi, H. Stopped-flow analysis on the mechanism of perylene nanoparticle formation by the precipitation method. J. Cryst. Growth 2009, 311, 553–555. (11) LaMer, V. K.; Dinegar. Theory, Production and Mechanism of Formation of Monodispersed Hydrosols, R. J. Am. Chem. Soc. 1950, 72, 4847. (12) LaMer, V. K. Nucleation in Phase Transitions. Ind. Eng. Chem. 1952, 44, p. 1270. (13) Kashchiev, D.; van Rosmalen, G. M. Review: Nucleation in solutions revisited. Cryst. Res. Technol. 2003, 38, 555–574. (14) Roelands, C. P. M.; ter Horst, J. H.; Kramer, H. J. M.; Jansens, P. J. Analysis of nucleation rate measurements in precipitation processes. Cryst. Growth Des. 2006, 6, 1380–1392. (15) Ring, T. A. Nano-sized cluster nucleation. AdV. Colloid Interface Sci. 2001, 91, 473–499. (16) Turkevich, J.; Stevenson, P. C.; Hillier, J. Farraday. Discuss. Chem. Soc. 1951, 11, 55. (17) Kumar, S.; Gandhi, K. S.; Kumar, R. Modeling of formation of gold nanoparticles by citrate method. Ind. Eng. Chem. Res. 2007, 46, 3128– 3136. (18) Watzky, M. A.; Finke, R. G. Nanocluster Size-Control and “Magic Number” Investigations. Experimental Tests of the “Living-Metal Polymer” Concept and of Mechanism-Based Size-Control Predictions Leading to the Syntheses of Iridium(0) Nanoclusters Centering about Four Sequential Magic Numbers. Chem. Mater. 1997, 9, 3083–3095.

5601

(19) Lint, Y.; Finke, R. G. Novel Polyoxoanion- and BudN+Stabilized, Isolable, and Redissolvable, 20-30-A Ir-300-900 Nanoclusters: The Kinetically Controlled Synthesis, Characterization, and Mechanism of Formation of Organic Solvent-Soluble, Reproducible Size, and Reproducible Catalytic Activity Metal Nanoclusters. J. Am. Chem. Soc. 1994, 116, 8335–8353. (20) Dirksen, J. A.; Benjelloun, S.; Ring, T. A. Modelling the precipitation of copper oxalate aggregates. Colloid Polym. Sci. 1990, 268, 864– 876. (21) Bogush, G. H.; Zukoski (IV), C. F. Uniform Silica Particle Precipitation: An Aggregative Growth Model. J. Colloid Interface Sci. 1991, 142, 19–34. (22) Privman, V.; Goia, D. V.; Park, J.; Matijevic, E. Mechanism of Formation of Monodispersed Colloids by Aggregation of Nanosize Precursors. J. Colloid Interface Sci. 1999, 213, 36. (23) Mozyrsky, D.; Privman, V. Diffusional Growth of Colloids. J. Chem. Phys. 1999, 110, 9254. (24) Libert, S.; Gorshkov, V.; Privman, V.; Goia, D.; Matijevic´, E. Formation of Monodispersed Cadmium Sulfide Particles by Aggregation of Nanosize Precursors. AdV. Colloid Interface Sci. 2003, 100-102, 169. (25) Gorshkov, V.; Goia, D.; Matijevic, E.; Privman, V.; Libert, S. Model of Controlled Synthesis of Uniform Colloid Particles: Cadmium Sulfide. Langmuir 2003, 19, 10679. (26) Matijevic, E. Nanosize Precursors as Building Blocks for Monodispersed Colloids. Colloid J. 2007, 69, 29–38. (27) Robb, D. T.; Halaciuga, I.; Privman, V.; Goia, D. V. Computational Model for the Formation of Uniform Silver Spheres by Aggregation of Nanosize Precursors. J. Chem. Phys. 2008, 129, 184705. (28) Robb, D. T.; Privman, V. Model of Nanocrystal Formation in Solution by Burst Nucleation and Diffusional Growth. Langmuir 2008, 24, 26–35. (29) Velikov, K. P.; Zegers, G. E.; van Blaaderen, A. Synthesis and Characterization of Large Colloidal Silver Particles. Langmuir 2003, 19, 1384–1389. (30) Evanoff, D., Jr.; Chumanov, G. Size-controlled Synthesis of Nanoparticles. 1. “Silver-only” Aqueous Suspensions via Hydrogen Reduction. J. Phys. Chem. B 2004a, 108, 13948–13956. (31) Evanoff, D., Jr.; Chumanov, G. Size-controlled Synthesis of Nanoparticles. 2. Measurement of Extinction, Scattering, and Absorption Cross Sections. J Phys. Chem. B 2004b, 108, 13957–13962. (32) Sudeep, P. K.; Kamat, P. V. Photosensitized growth of silver nanoparticles under visible light irradiation: a mechanistic investigation. Chem. Mater. 2005, 17, 5404–5410. (33) Merga, G.; Wilson, R.; Lynn, G.; Milosavljevic, B. H.; Meisel, D. Redox Catalysis on “Naked” Silver Nanoparticles. J. Phys. Chem. C 2007, 111, 12220–12226. (34) Henglein, A.; Giersig, M. Formation of Colloidal Silver Nanoparticles: Capping Action of Citrate. J. Chem. Phys. B. 1999, 103, 9533–9539. (35) Irizarry R. SDOR strategy to solve Inverse problems in population balance identification problems. 2010, in preparation. (36) Irizarry, R. A generalized framework for solving dynamic optimization problems using the artificial chemical process paradigm: Applications to particulate processes, discrete dynamic systems. Chem. Eng. Sci. 2005, 60, 5663–5681. (37) Irizarry, R. Hybrid dynamic optimization using artificial chemical process: Extended LARES-PR. Ind. Eng. Chem. Res. 2006, 45, 8400–8412. (38) Bohren, C. F. Huffman, D. R. Absorption and Scattering of Light by Small Particles; John Wiley: New York, 1983. (39) Feldheim, D. L. Foss, C. A. Metal Nanoparticles: Synthesis, Characterization and Applications; Marcel Dekker:New York, 2002. (40) Mishchenko, M., Travis, L. D., Lacis, A. A. Scattering, Absorption, and Emission of Light by Small Particles; Cambridge University Press: Camrbidge, U.K., 2002. (41) Purcell, E. M.; Pennypacker, C. R. Scattering and absorption of light by nonspherical dielectric grains. Astrophys. J. 1973, 186, 705–714. (42) Draine, B. T.; Flatau, P. J. Discrete-dipole approximation for scattering calculations. J. Opt. Soc. Am., A 1994, 11, 1491–1499. (43) Taflove, A. Hagness, S. C. Computational Electrodynamics: The Finite-Difference Time-Domain Method, third ed.; Artech House: Norwood, MA, 2005; 02062. (44) Barber, P. W.; Chang, R. K.; Massoudi, H. title. Phys. ReV. B. 1983, 27, 7251–7261. (45) Ramkrishna, D. (2000) Population Balances,: Academic Press, San Diego. (46) Russel, W. B., Saville, D. A., Schowalter, W. R. (1989), Colloidal dispersions, Cambridge University Press, Cambridge, U.K. (47) Israelachvili, J. Intermolecular and Surface Forces; Academic Press: New York, 1991.

5602


(48) Goodwin, J. Colloids and Interfaces with Surfactant and Polymers; John Wiley: New York, 2004. (49) Daimon, M.; Masumura, A. Measurement of the refractive index of distilled water from the near-infrared region to the ultraviolet region. Appl. Opt. 2007, 46, 3811–3820. (50) Irani, G. B.; Hurn, T.; Wooten, F. Optical constants of silver and gold in the visible and vacuum ultraviolet. J. Opt. Soc. Am. 1971, 61, 128– 129. (51) Johnson, P. B.; Christy, R. W. Optical constants of noble metals. Phys. ReV. B 1972, 6, 4370–4379. (52) Taneja, P.; Ayyub, P. Size dependence of the optical spectrum in nanocrystalline silver. Phys. ReV. B 2002, 65, 245412–125416. (53) Kumar, S.; Ramkrishna, D. On the solution of population balance equations by discretization: I. A fixed point technique. Chem. Eng. Sci. 1996, 51, 1311. (54) Irizarry, R. Fast Monte Carlo methodology for multivariate particulate systems - I: Point ensemble Monte Carlo. Chem. Eng. Sci. 2008, 63, 95.

(55) Irizarry, R. Fast Monte Carlo methodology for multivariate particulate systems - II: τ-PEMC. Chem. Eng. Sci. 2008, 63, 111. (56) Shin, H. S.; Yang, H. J.; Kim, S. B.; Lee, M. S. Mechanism of growth of colloidal silver nanoparticles stabilized by polyvinyl pyrrolidone in γ-irradiated silver nitrate solution. J. Colloid Interface Sci. 2004, 274, 89–94. (57) Ortega, E. V.; Berk, D. Precipitation of Silver Powders in the Presence of Ethylenediamine Tetraacetic Acid. Ind. Eng. Chem. Res. 2006, 45, 1863–1868. (58) Jitianu, A.; Kim, M.; Andreescu, D.; Goia, D. A simple preparative route to highly stable dispersions of uniform silver nanoparticles. J. Nanosci. Nanotechnol. 2008, 8, 1–6.

ReceiVed for reView March 17, 2010 ReVised manuscript receiVed May 3, 2010 Accepted May 5, 2010 IE100644H

Simulated Dynamic Optical Response Strategy for Model

Recommend Documents