Shaping Particles via Controlling the Diffusion of Building Blocks

Sep 21, 2015 - On the basis of the model of diffusion-limited aggregation, there exista a concentration gradient of BBs at the growth front, at the ...
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Shaping Particles via Controlling the Diffusion of Building Blocks Han Wang†,‡ and Yongsheng Han*,† †

State Key Laboratory of Multiphase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100190, China ‡ University of Chinese Academy of Sciences, Beijing 100049, China ABSTRACT: Using nanoparticles as building blocks to build up hierarchical structures is the next big thing in nanoscience, as well as a core issue of the emerging mesoscience. The building process involves the movement of building blocks, followed by interaction and binding between them. The diffusion of building blocks is an important kinetic factor influencing the shape development of materials, but this issue has not yet been well investigated by the material research community. In this paper, we take the preparation of silver particles as an example to investigate how the diffusion of building blocks influences the shape of silver products. The silver products are prepared by reducing silver ions in a mixture of ethylene glycol and water. Through a change in the volume ratio of ethylene glycol in water, the diffusion of building blocks is tuned, which results in the formation of silver products with various shapes from rectangular plates to spindle rods. This shape variation is attributed to the change of the diffusion rate of building blocks, and the mechanism underlying these phenomena is investigated by simulation using a newly developed diffusion−reaction model. Both the experimental and simulation results indicate that the diffusion of building blocks plays an important role in the shape development of particles, which is expected to arouse more attention to consider mass transfer in the controllable synthesis of materials.

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INTRODUCTION Controlling the shape of particles is a simple and effective means of tailoring the properties of materials in catalysis,1−3 electronics,4−7 or molecular sensing.8−10 Among the strategies in tuning the shape of particles, the kinetic approach is a green and promising technique to shape particles by regulating the rates of kinetic factors, such as the mass-transfer and reaction rates.11 Different approaches have been developed to regulate the kinetic factors, which include substantially slowing precursor decomposition or reduction,12,13 using a weak reducing agent,14−16 and coupling the reduction to an oxidation process.17−21 These kinetic strategies were extensively studied by Xia’s group, and metal particles with different shapes including sphere,22−24 cube,25,26 tetrahedron,21 plate,27,28 nanobar,29 and nanowire20,30 were successfully synthesized. It was found that, when the kinetic rate is slowed, the growth of particles experiences a transition from isotropic to anisotropic growth, resulting in the formation of diverse morphologies that deviate from the thermodynamically favored shape.31,32 Previous studies on the kinetic parameters mainly focused on the reaction rate. If we look at the formation of materials from an atomic-scale view, materials are formed by the transfer of building blocks (BBs; atoms, molecules, or clusters) to the growing surface, followed by binding or reaction. The transfer rate of BBs is another important kinetic factor involved in the formation of materials but was seldom investigated previously. In our recent studies, we have studied the effect of chemical diffusion on the morphology development of platinum particles, in which we found that the diffusion of chemicals determines the manner of crystallization.33 At diffusioncontrolled conditions, the crystals grow by the aggregation of clusters or particles, forming porous spheres with dendritic structures inside. Later, we verified the diffusion effect in the precipitation of calcium carbonates. The compromise between © 2015 American Chemical Society

diffusion and reaction leads to the formation of snow-shaped vaterite particles,34 which is a new morphology of calcium carbonate. These results promise that controlling chemical diffusion is a surfactant-free alternative to effectively regulating the shape of particles. Particle-based aggregation is a common crystallization scenario in nature and in laboratories. The diffusion of building particles should also play a role in building the hierarchical structures, which is not yet discovered. Here we take the synthesis of silver particles as an example to discover this issue. When the silver ion is reduced by sodium citrate at an enhanced temperature, the growth of silver products shows the character of the particle-based aggregation pathway. The synthesis of silver products is carried out in a mixture of water and ethylene glycol, in which the diffusion of BBs is tuned by changing the viscosity of the solvent through regulation of the volume ratio of ethylene glycol. Diverse morphologies of silver products are synthesized in different solvents. Through tracking of the shape development and analysis of the growth direction of crystals, the effect of the diffusion of BBs on the structure of products is discussed, and a plausible model is proposed to explain the diffusion effect. In the following, a statistical simulation is carried out to examine the proposed model.

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RESULTS AND DISCUSSION Figure 1 shows the scanning electron microscopy (SEM) micrographs of silver particles synthesized at different volume ratios of alcohol (VRA), which demonstrates that the silver Received: Revised: Accepted: Published: 9742

June 9, 2015 September 19, 2015 September 21, 2015 September 21, 2015 DOI: 10.1021/acs.iecr.5b02084 Ind. Eng. Chem. Res. 2015, 54, 9742−9749

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Industrial & Engineering Chemistry Research

Figure 1. SEM micrographs of silver samples synthesized in the solvent of an alcohol/water mixture with different VRA: 0.2 (A1 and A2); 0.5 (B1 and B2); 0.8 (C1 and C2).

Figure 2. Shape evolution of silver particles synthesized at different solvents: (A1−A3) at a VRA of 0.2 and collected at reaction times of 1, 2, and 10 min, respectively; (B1−B3) at a VRA of 0.8 and collected at reaction times of 1, 2, and 10 min, respectively.

particles synthesized in different VRA conditions show distinctive shapes. When the reaction is carried out in low

VRA condition (VRA = 0.2), the particles obtained look like tetragonal plates with noticeable thickness (A1 and A2 in 9743

DOI: 10.1021/acs.iecr.5b02084 Ind. Eng. Chem. Res. 2015, 54, 9742−9749

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Figure 3. TEM characterization of the synthesized RP (A1−A3) and SR (B1−B3). A1 and B1 are TEM images; A2 and B2 are high-magnification TEM images; A3 and B3 are SAED patterns.

B2 in Figure 2. At a VRA of 0.2, these nanocrystals prefer to attach along two axes, leading to the formation of a RP (A2 in Figure 2). Later growth thickens the plate, as shown by A3 in Figure 2. At a VRA of 0.8, these initially formed nanocrystals seem to aggregate preferentially along one axis to form long structures, as shown by B2 in Figure 2. Later growth intensifies the anisotropic aggregation, forming spindle-like rods with a large aspect ratio, as shown by B3 in Figure 2. Therefore, we could conclude that both RPs and SRs follow the particle-based aggregation growth model, and their difference is the anisotropic aggregation preferentially along the long axis at VRA = 0.8. In order to get to know more details about the formation of RPs and SRs, we further characterize them by highmagnification transmission electron microscopy (TEM), as shown in Figure 3. Observation on TEM images of these two particles (A1 and B1 in Figure 3) tells us the growth of the SR takes a more anisotropic path than that of RP because the spindle has a higher aspect ratio, which agrees well with the SEM characterization. High-resolution TEM on the RP, as shown by A2 in Figure 3, shows that the BBs in the RP have almost identical crystallographic orientations, and they attach to

Figure 1). The basal planes are rectangular faces with a moderate aspect ratio. Such a particle is referred to as a rectangular plate (RP) in the following for the purpose of simplicity. When the VRA is increased to 0.5, the silver particles obtained show noticeable changes in shape, as shown by B1 and B2 in Figure 1. The four corners of the basal faces are truncated, leaving two narrowed ends for the plate. At the same time, the aspect ratio of the plate is increased. When VRA is increased to 0.8, the aspect ratio of the particles further increases remarkably, as shown by C1 and C2 in Figure 1. Two ends of particles are very sharp, demonstrating a quasi-onedimensional feature. Considering that the projecting profile of these particles looks more like an elongated spindle, we refer to these as spindle rods (SRs) in the following discussion. To elucidate the formation process of the particles, we have examined their shape evolution by collecting samples at different reaction times, as shown in Figure 2. In the initial reaction, nanoscale crystals are largely formed, as shown by A1 and B1 in Figure 2. In the following, these initially formed nanocrystals do not experience a traditional seed-mediated growth path to form large particles. Instead, they tend to aggregate to build up high-order structures, as shown by A2 and 9744

DOI: 10.1021/acs.iecr.5b02084 Ind. Eng. Chem. Res. 2015, 54, 9742−9749

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On the basis of the model of diffusion-limited aggregation, there exista a concentration gradient of BBs at the growth front, at the diffusion-limited condition,37 which results in an anisotropic growth of particles, as schematically demonstrated in Figure 5. At high diffusion rate, the concentration gradient is

each other by both end-to-end and side-to-side attachment. The lattice distance along the long axis is 0.236 nm, which corresponds to the plane distance of {111} planes for silver crystals. To get the crystallographic orientation of the long and short axes, selected-area electron diffraction (SAED) patterns are taken on the edge of the plate, as marked by the red box in Figure 3 (A1). The SAED pattern shown in Figure 3 (A3) reveals that the long axis is along the ⟨111⟩ direction, while the short axis is along the ⟨220⟩ direction. Similar crystallographic information is obtained on the SRs. The lattice distance along the long axis of the SR is 0.238 nm, which is close to the plane distance of {111} planes, indicating that the long axis is along the ⟨111⟩ direction. The SAED pattern in Figure 3(B3) confirms the ⟨111⟩ direction of the long axis and the ⟨220⟩ direction of the short axis, which agrees with the growth directions of the plate discussed above. On the basis of the evidence indicated by high-resolution TEM and SAED patterns, we conclude that aggregation along both the ⟨111⟩ and ⟨220⟩ directions forms the RP shape, while dominant aggregation along the ⟨111⟩ direction forms the spindle shape of silver particles. The above results and discussion show that variation of the particle shape reflects anisotropic growth of particles at different VRA of the solvents. The change of the VRA from 0.2 to 0.8 leads to more than triple an increase of the viscosity of the solvents, from 0.525 to 1.745 mPa·s,35 as shown by the inset bar graph (upper right) in Figure 4. From the TEM image

Figure 5. Schematic demonstration on the anisotropic growth of crystals induced by the low diffusion rate of the BBs.

not noticeable along the growth front. In this case, the growth rates along different directions are identical. This growth scenario is assisted with the surface-smoothing process, which favors the layerwise step growth, leading to the formation of RP. At low diffusion rate of the BBs, the concentration gradient along the growth front becomes remarkable and dominates the aggregation of particles. Besides, the concentration gradient along the long axis of the growing particle is more striking than that along the short axis. As a result, the growth along the long axis is sped up by the gradient, leading to the formation of spindle-like particles. In order to verify the above-proposed model, we build up a two-dimensional diffusion−reaction model to discover the inherited mechanism on the anisotropic attachment of BBs in aggregation-based growth of particles. The simulation system is a two-dimensional round shape with reflecting boundary conditions. In the center of the system, we set an initial seed as the seeding site for aggregation. The initial seed is a tiny rod with an aspect ratio of 2 as an attempt to capture the asymmetric structure of the initial nucleus. In the model, aggregation of the particle can be characterized as four stochastic kinetic stages: (i) the birth of BBs around the inner border of the system by a zero-order-reaction manner with reaction rate constant k(0) (=1.0); (ii) the randomized diffusion of the BBs with diffusion coefficient D; (iii) the attachment of the BBs onto the surface of the growing front when the BBs encounter the growing aggregate by the bimolecular reaction between the BBs and the growing

Figure 4. Change of the solvent viscosity with the VRA.

of the BBs (upper-left inset image in Figure 4), the diameter of the BBs is approximately determined as 20 nm. With the viscosity of the solvent and the diameter of the BBs, the diffusivity of the BBs in the solvent can be calculated by the following Stokes−Einstein equation:36

D=

kBT 6πrη

(1)

where kB is the Boltzmann constant, T is the thermodynamic temperature, r is the diameter for the spherical particle, and η is the viscosity of the solvent. The diffusion coefficient of the BB decreases from 2.46 to 0.72 (10−11 m2/s) as the VRA is increased from 0.2 to 0.8, as shown by the bar graph in Figure 4. Therefore, the shape variation of silver products is accompanied by a change of the diffusion behavior of the BBs. 9745

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Figure 6. Simulation results from a newly developed diffusion−reaction model. Parts A1−A3 represent the shape of the obtained aggregate. Parts B1−B3 are the contours of the concentration distribution of BBs. Parts C1−C3 are plots of the concentration distribution along the long and short axes of the obtained aggregate. The diffusion constants for the left, middle, and right columns of this figure are 2.46, 1.31, and 0.72 (10−11 m2/s), respectively.

ability of the computing system and the simple model adopted. We are on the way to discovering more details on these dynamic processes in the computer. Both the experimental and simulation results indicate that the changes of the diffusion constant of the BBs influence the shape variation of the obtained particles. Also decreasing diffusion rate of the BBs favors the anisotropic growth along the long axis, which leads to an increase in the aspect ratio of the particle, thus leading to the formation of longer rods. The concentration distribution around the growing aggregate is also computed by our model. To make the results of the concentration distribution statistically meaningful, the concentration is calculated by counting the number of BBs per area during 1 × 105 steps. The results are shown by the contour map and plot lines in Figure 6. When the diffusion constant is high, the BBs around the formed aggregate with square shape are almost homogeneously distributed, and no remarkable concentration gradient exists on the boundary of the aggregate, as shown in parts B1 and C1 of Figure 6. When the diffusion constant is reduced, striking concentration gradients along the long and short axes of the aggregate are formed, as shown in parts B2, B3, C2, and C3. Besides, the concentration gradient along the long axis is higher than that along the short axis. The above phenomenon can be explained by the following facts. At higher diffusion rate, the transport of BBs from their birth regime to the boundary of the aggregate is very fast, so the BB

aggregate; (iv) the fusion of the BBs into the lattice vacant site of the growing aggregate. In order to investigate the influence of the diffusion rate of the BBs on the shape development of the aggregate, we should exclude the influence of the reaction rate of the bimolecular reaction. Therefore, the constant of the reaction rate for the bimolecular reaction is set at a high level to make sure that bindings between the BBs and the aggregate take place once the BBs diffuse onto the surface of the aggregate. Also, the diffusion rate of the BBs is set at 0.72, 1.31, and 2.46 (10−11 m2/s), which corresponds well with the diffusivity displayed in Figure 4. As shown in Figure 6A, after 1.6 × 106 dynamic steps, the initial seeds evolve to particles with rectangular shapes, by epitaxial growth along the short and long axes. When the diffusion constant is increased to 2.46 × 10−11 m2/s, the shape of the obtained aggregate is almost square with a moderate aspect ratio. When the diffusion constant is decreased to 1.31 × 10−11 m2/s, the aspect ratio of the aggregate increases to 2; when the diffusion constant is decreased to 0.72 × 10−11 m2/s, the aspect ratio of the aggregate increases to almost 3. The above results from simulation partly confirm the experimental results. We have to admit that the simulation partly succeeded in reproducing a tendency of anisotropic growth of crystal nuclei. However, other features such as the truncated rectangular shape and the spindle-like shapes found in a thicker solution are not reproduced by the simulation, which is caused by the limiting 9746

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TEM using a JEM-2100 (UHR) high-resolution transmission microscope at an accelerating voltage of 200 kV. Simulation on the Growth of the Aggregate. The objects represented in the newly developed diffusion−reaction model are the BBs moving in two-dimensional space. The attributes of the BBs, such as size, type, mobility, and position, may change over time. The diffusion refers to the motion of the objects, and the reaction refers to the birth of the BBs and the binding between the BBs and the growing aggregate. The theoretical foundation, governing the diffusion−reaction dynamics of the BBs is as follows. The diffusion of the BBs obeys the isotropic Brownian dynamics governed by the modified Langevim equation:38

is homogeneously distributed around the growing aggregate. At lower diffusion rate, the transport of BBs from their birth regime to the boundary of the aggregate takes longer time. Once the BBs diffuse to the boundary of the aggregate, they are bounded on the surface of the aggregate quickly, leading to lower concentration near the boundary of the aggregate compared to that far from the aggregate. By comparing the concentration distribution along the long and short axes in part C3 of Figure 6, we find that the concentration gradient along the long axis has a sharper slope than that along the short axis. This difference explains the preferential growth along the long axis at the low diffusion rate of BBs. The high concentration gradient along the long axis means that more BBs are delivered to the surface of the long axis than to the area near the short axis. As a result, more BBs attach to the top surfaces of the long axis, making the growth along the long axis much faster than that along the short axis. Therefore, the preferential growth along the long axis at a low diffusion rate of BBs is attributed to the higher concentration gradient of BBs along the long axis.

d x(t ) = dt

2D

d η(t ) ∇V [x(t )] −D dt kBT

(2)

where x(t) is a two-dimensional vector indicating the instantaneous position of a particle at time t, kB is the Bolzmann constant, and T is the thermodynamic temperature. The first term on the right-hand side of eq 2 is the stochastic force depending the diffusion constant (D) of the object and the Wiener process [η(t)] distribution, which depicts the independent random process of the object. The second term on the right-hand side of eq 2 is the deterministic force, which depends on the interaction between objects, where ∇V represents the gradient (spatial derivative) of the potential induced by the interaction force (repulsive or attractive) between objects. Equation 2 can be solved numerically by employing a Euler discretization with constant time step Δt, obtaining a discrete sequence of object configuration x(t) in time t, related by

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CONCLUSION In conclusion, the effects of the diffusion of BBs on the structure development of silver products were investigated in this paper. The diffusion rate was regulated by changing the volume ratio of ethylene glycol in water. At low VRA, RPs with a low aspect ratio were largely formed, while spindle-like rods with a high aspect ratio were obtained at high VRA. Highresolution TEM and SAED patterns indicate that the shape variation was a result of the preferential aggregation of BBs, which was caused by the existence of a microscopic concentration gradient around the growth fronts. Simulation results confirmed the role of diffusion in creating a concentration gradient around the growing aggregate. With a decrease of diffusion, the concentration gradient played a more and more important role, which led to an anisotropic aggregation of particles, forming SRs. This study showed that the diffusion of BBs plays a significant role in the shape development of particles, which promises a versatile and alternative pathway for shape control on materials.

x t +Δt = x t +

2DΔt ηt − ΔtD

∇V [x(t )] kBT

(3)

where ηt is represented by the probability density function f (x ) =

1 −x 2 /2 e 2π

(4)

and the gradient of potential ∇V is calculated from the Lennard-Jones (L-J) pseudopotential between two objects when they come to their interaction range. The L-J potential governs the interaction between two adjacent objects in a way that they repel at close range and then attract and is eventually cut off at some limiting separation rc. In our work, the distance for the minimum potential energy is set as 2 times the object diameter, and the distance for the cutoff potential (rc) is set as 4 times the object diameter. Assuming that l is the distance vector that object A deviates from its balance position with object B, then its gradient of potential ∇V is obtained by 0.5PA−Bl2, where PA−B is the interaction strength between A and B. Reactions in our simulation here encompass two events: the birth of BBs and the binding between the BBs and the growing aggregate. The birth of BBs is understood as a zero-oder reaction, which is generally described by the differential equation

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EXPERIMENTAL SECTION Synthesis of Silver Particles in Different Solvents. Reagents (analytical grade) used in the experiment were purchased from Sigma (Beijing), and purified water with a resistivity higher than 18 MΩ was used. A solution of AgNO3 (0.2 M) was prepared by dissolving 0.0340 g of analytically pure AgNO3 reagent in 1 mL of purified water. A solution of sodium citrate (7 mM) was prepared in a flask by dissolving 0.0925 g of sodium citrate in 50 mL of a mixture of ethylene glycol and water with different VRA (0.2, 0.5, and 0.8). The flask was heated to 80 °C for 30 min under vigorous stirring. Then a AgNO3 solution was injected into the flask to initiate the reaction. The color of solution changes generally from colorless to yellow and finally to turbid. After 50 min of reaction, the products were sampled. Characterization of the Shape of Samples. The shape of samples was examined by a JSM-6700F scanning electron microscope fitted with a field-emission source at an accelerating voltage of 20 kV. Samples were spread on double-sided carbon wafer tape and coated with gold prior to microscopic characterization. The products were further investigated by

dC b(t ) (0) = k macro dt

(5)

Also the binding between the BBs and the growing aggregate is understood as the catalyzed transformation of the BBs on the surface of the aggregate from moving objects to stationary ones 9747

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the State Key Laboratory of Multiphase Complex Systems (MPCS-2014-D-05). Financial support from the National Natural Science Foundation of China (U1462130) is appreciated.

that subsequently become part of the aggregate, as described by the reaction equation B b − b + A agg = A agg + A agg

(6)

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where Bb‑b refers to the moving BB and Aagg refers to the constitute block of the aggregate that has no mobility. Therefore, the binding can be regarded as a second-order reaction from reactants of one Bb‑b and Aagg to two Aagg, which can be described by the differential equation dC A agg(t ) dt

(2) = k macro C B b−bC A agg

k(0) macro

(1) Enache, D. I.; Edwards, J. K.; Landon, P.; Solsona-Espriu, B.; Carley, A. F.; Herzing, A. A.; Watanabe, M.; Kiely, C. J.; Knight, D. W.; Hutchings, G. J. Solvent-free oxidation of primary alcohols to aldehydes using Au-Pd/TiO2 catalysts. Science 2006, 311, 362. (2) Nam, Y. S.; Magyar, A. P.; Lee, D.; Kim, J.-W.; Yun, D. S.; Park, H.; Pollom, T. S.; Weitz, D. A.; Belcher, A. M. Biologically templated photocatalytic nanostructures for sustained light-driven water oxidation. Nat. Nanotechnol. 2010, 5, 340. (3) Khalavka, Y.; Becker, J.; Sonnichsen, C. Synthesis of rod-shaped gold nanorattles with improved plasmon sensitivity and catalytic activity. J. Am. Chem. Soc. 2009, 131, 1871. (4) Plissard, S. R.; van Weperen, I.; Car, D.; Verheijen, M. A.; Immink, G. W. G.; Kammhuber, J.; Cornelissen, L. J.; Szombati, D. B.; Geresdi, A.; Frolov, S. M.; Kouwenhoven, L. P.; Bakkers, E. P. A. M. Formation and electronic properties of InSb nanocrosses. Nat. Nanotechnol. 2013, 8, 859. (5) Shankar, S. S.; Rai, A.; Ankamwar, B.; Singh, A.; Ahmad, A.; Sastry, M. Biological synthesis of triangular gold nanoprisms. Nat. Mater. 2004, 3, 482. (6) Kongkanand, A.; Tvrdy, K.; Takechi, K.; Kuno, M.; Kamat, P. V. Quantum dot solar cells. Tuning photoresponse through size and shape control of CdSe-TiO2 architecture. J. Am. Chem. Soc. 2008, 130, 4007. (7) Ithurria, S.; Tessier, M.; Mahler, B.; Lobo, R.; Dubertret, B.; Efros, A. L. Colloidal nanoplatelets with two-dimensional electronic structure. Nat. Mater. 2011, 10, 936. (8) Giugni, A.; Torre, B.; Toma, A.; Francardi, M.; Malerba, M.; Alabastri, A.; Zaccaria, R. P.; Stockman, M.; Di Fabrizio, E. Hotelectron nanoscopy using adiabatic compression of surface plasmons. Nat. Nanotechnol. 2013, 8, 845. (9) Soleymani, L.; Fang, Z.; Sargent, E. H.; Kelley, S. O. Programming the detection limits of biosensors through controlled nanostructuring. Nat. Nanotechnol. 2009, 4, 844. (10) Anker, J. N.; Hall, W. P.; Lyandres, O.; Shah, N. C.; Zhao, J.; Van Duyne, R. P. Biosensing with plasmonic nanosensors. Nat. Mater. 2008, 7, 442. (11) Yu, T.; Kim, D. Y.; Zhang, H.; Xia, Y. Platinum Concave Nanocubes with High-Index Facets and Their Enhanced Activity for Oxygen Reduction Reaction. Angew. Chem., Int. Ed. 2011, 50, 2773. (12) Xiong, Y.; Siekkinen, A. R.; Wang, J.; Yin, Y.; Kim, M. J.; Xia, Y. Synthesis of silver nanoplates at high yields by slowing down the polyol reduction of silver nitrate with polyacrylamide. J. Mater. Chem. 2007, 17, 2600. (13) Langille, M. R.; Personick, M. L.; Zhang, J.; Mirkin, C. A. Defining rules for the shape evolution of gold nanoparticles. J. Am. Chem. Soc. 2012, 134, 14542. (14) Washio, I.; Xiong, Y.; Yin, Y.; Xia, Y. Reduction by the end groups of poly (vinyl pyrrolidone): a new and versatile route to the kinetically controlled synthesis of Ag triangular nanoplates. Adv. Mater. 2006, 18, 1745. (15) Xiong, Y.; Washio, I.; Chen, J.; Cai, H.; Li, Z. Y.; Xia, Y. Poly (vinyl pyrrolidone): a dual functional reductant and stabilizer for the facile synthesis of noble metal nanoplates in aqueous solutions. Langmuir 2006, 22, 8563. (16) Lim, B.; Camargo, P. H.; Xia, Y. Mechanistic study of the synthesis of Au nanotadpoles, nanokites, and microplates by reducing aqueous HAuCl4 with poly (vinyl pyrrolidone). Langmuir 2008, 24, 10437. (17) Xiong, Y.; McLellan, J. M.; Chen, J.; Yin, Y.; Li, Z. Y.; Xia, Y. Kinetically controlled synthesis of triangular and hexagonal nanoplates of palladium and their SPR/SERS properties. J. Am. Chem. Soc. 2005, 127, 17118.

(7)

k(2) macro

For eqs 5 and 7, and represent the reaction rate constants for zero- and second-order reactions, respectively, which represent the inverse mean time needed for the birth of Bb−b and the BBs (Bb−b) to decay into product (Aagg). In the particle-based simulation, it is impractical to compute the concentration of particles for eqs 5 and 7, and time is discretized into segments of Δt. The reaction rate therefore has to be converted into the probability for specific reaction events that take place during a given time interval. In our simulation, the reaction probability for a reaction event with macro reaction rate constant kmicro during time interval Δt is then related by the Poisson probability equation

p(Δt ) = 1 − e−kmacroΔt

(8)

Having the above theory describing the diffusion and reaction in our model, our simulation runs are based on the following algorithm. 1. Start at time t = 0, an initial particle configuration xt=0 is set. The diffusion and reaction events take place on a twodimensional space with sphere shape. Also, we exert a reflecting boundary condition at the border of the system. In the center of the system, a tiny seed is set as the seeding site for the growing aggregate. 2. Repeat for N steps (total simulation time NΔt) by running the following diffusion and reaction algorithm: (i) Advancing the Brownian dynamics of the BBs by one step of time length Δt, based on their diffusion constants D and the interaction potential V(x) between BBs. We should note that there is no movable BB in the system at time t. (ii) Creating a list of the two reaction events that occur in the current time interval Δt. For each reaction event, its probability to occur depends on its rate constant, which is executed with probability equation (8). For the zero-order reaction, the newly born BBs are randomly distributed near the border of the system. For the second-order reaction, the conversion from Bb−b to Aagg only occurs when a moving Bb−b contacts with a stationary Aagg. (iii) The newly formed Aagg objects join the vacant lattice site in the aggregate. The lattice inside the aggregate is a rectangular coordinate system that corresponds to the cubic crystallographic system for silver particles.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This study was supported by the Hundreds Talent Program from the Chinese Academy of Sciences and the project from 9748

DOI: 10.1021/acs.iecr.5b02084 Ind. Eng. Chem. Res. 2015, 54, 9742−9749

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DOI: 10.1021/acs.iecr.5b02084 Ind. Eng. Chem. Res. 2015, 54, 9742−9749