Simulation of Batch Nanoparticle Growth by the Generalized Diffusion

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C: Physical Processes in Nanomaterials and Nanostructures

Simulation of Batch Nanoparticle Growth by the Generalized Diffusion Model Tianlong Wen, Xiaochen Zhang, Dainan Zhang, Chong Zhang, Qiye Wen, Huaiwu Zhang, and Zhiyong Zhong J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b07438 • Publication Date (Web): 31 Oct 2018 Downloaded from http://pubs.acs.org on November 5, 2018

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The Journal of Physical Chemistry

Simulation of Batch Nanoparticle Growth by the Generalized Diffusional Model Tianlong Wen*, Xiaochen Zhang, Dainan Zhang, Chong Zhang, Qiye Wen, Huaiwu Zhang, Zhiyong Zhong State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu, Sichuan, China 610054

KEYWORDS: Diffusional growth, monodisperse nanoparticles, batch nanoparticles, nanoparticle growth dynamics, Ostwald ripening, monomer concentration.

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ABSTRACT. Diffusional growth in solution is the most efficient method to grow monodisperse colloidal nanoparticles. Instead of calculating a single nanoparticle, here the growth of an ensemble of nanoparticles has been numerically simulated by the generalized growth rate of nanoparticles, which is deduced in our previous work from the very fundamental laws and equations. By doing that, the dynamic interaction among nanoparticles through monomer attachment and detachment can be clearly revealed. The constant and diminishing monomer concentration have been simulated separately and compared together. The simulations give the growth dynamics of batch nanoparticles across a broad range of conditions, where diffusivity, adsorption and concentration of monomers play a significant role. The relations between average radius and time, variance and time, and variance and average radius have been obtained for different growth conditions. The average radius-time curve can be well fitted to 𝐸[𝑅] = 𝐸[𝑅0] +𝛼𝑡𝑛 andE[R] = E[𝑅0] +

(𝑅𝑠 ― 𝐸[𝑅0])𝐿𝑎𝑛𝑔𝑒𝑣𝑖𝑛(𝛽𝑡) for constant and diminishing monomer concentration respectively. By the simulation, many hidden details about diffusional growth have been unveiled, such as the details of Ostwald ripening and the role of monomer concentration.


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1. Introduction Monodisperse nanoparticles are essential to obtain nanoparticle crystals of long range order 1-7 and to achieve uniform and tunable physical properties of nanoparticle ensembles,8-12 which will deeply affect their many applications in such as display technology,13 lithography,14 flash memory,15 catalysis,16-17, terahertz modulators,18 magnetic nanoparticle imaging and hyperthermia,19-20 and magnetic nanocomposites.20-22 Solution-based synthesis of functional nanoparticles23-26 has many advantages over physical deposition in vacuum27-28 or air29 since the former method has better control over the size, composition, uniformity, stability and selfassembly of the nanoparticles. To foster a mature nanoparticle in solution, it involves two steps: (1) the birth of a nucleus by the aggregation of monomers (atom-surfactant complex) in solution and (2) the growth of the nucleus by consuming the supersaturated monomers in solution.30 To prevent the abortion of the nucleus, their size should be larger than the critical size for nucleation, which is dynamically determined by the monomer concentration.31 LaMer and Dinegar proposed that the nucleation of all particles should occur in a very short period so that they have nearly the same growth history to achieve monodispersity.30Under diffusion-layer assumption, Sugimoto later deduced the growth rate of nanoparticles under two extreme cases of diffusion-controlled and adsorptioncontrolled growth.32 The analytical expression of a single nanoparticle indicates that the size distribution of nanoparticle ensembles can be either focused in diffusion-controlled growth or broadened in adsorption-controlled growth. However, the growth rate under the intermediate conditions is not obtained in Sugimoto’s deduction. Without the diffusion-layer assumption, we deduced a generalized growth rate of spherical nanoparticles from the fundamental Fick’s law of diffusion, conservation of mass and Gibbs-Thomson equation.31 The generalized growth rate can


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be automatically reduced to the Sugimoto’s expression under the extreme conditions.31 The generalized growth rate allows us to simulate the growth of batch nanoparticles under general growth conditions, which will provide us many clues for the fabrication of monodisperse nanoparticles. Equation 1 shows the generalized growth rate of nanoparticles, where R is the radius of nanoparticles at time t, Ω is the volume occupied by an atom, D is the diffusivity of the monomers, k is the adsorption constant, T is the absolute temperature, Cb is the monomer concentration in the bulk solution, Ceq is the equilibrium monomer concentration, and C∞ is the equilibrium monomer concentration at the flat surface. 31 𝑑𝑅 𝑑𝑡

where A =

=

𝐷𝐴Ω

(1)

𝑅2

𝑘𝑅2(𝐶𝑏 ― 𝐶𝑒𝑞)

(1a)

𝐷 + 𝑘𝑅

( )

and 𝐶𝑒𝑞 = 𝐶∞𝑒𝑥𝑝

2𝛾Ω 𝑅𝑘𝐵𝑇

(1b)

where equation (1a) is valid under quasi-static assumption, or if 4DΩkR(𝐶𝑏 ― 𝐶𝑒𝑞)/(𝐷 + 𝑘𝑅)2 ≪ 1, which is true if monomer concentration in solution is much lower than that in the solids.31 If solving the differential equation (1), the radius of a nanoparticle at any time t, R, can be expressed as 𝑅 = 𝑅(𝑡, 𝑅0,𝐷, 𝑘, 𝐶𝑏,𝑇, Ω, 𝛾, 𝐶∞)

(2)

where R0 is the initial radius of the nanoparticle right after the nucleation event. Namely, the growth of a given nanoparticle is determined by a group of parameters of t, R0, D, k, Cb, T, Ω, γ,


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and C∞. Among them, R0 is the nucleation parameter, D and k are the growth parameters, Cb is the precursor parameter, and T is the reaction condition parameters, and Ω, γ and C∞ are the material parameters. Consequently, the final product of nanoparticles is sensitive to the growth time, initial nucleation, solvent, type and concentration of precursors, reaction conditions and chosen materials. Variation of these parameters could lead to discrepancy of the final product. Apparently, it is of challenge to obtain the analytical expression of R in equation (2). In this work, instead of considering only a single nanoparticle, the growth of an ensemble nuclei is numerically simulated using the generalized growth model to unveil the many characteristics of the batch nanoparticle growth. Through the simulation, we can obtain the details of the growth dynamics of the batch nanoparticles across various growth conditions. By the simulation, we expect the growth of batch nanoparticles could be revealed clearly so that it can be finely tuned for better control over their size and uniformity. 2. Simulation Method In the simulation, we assume the synthesis follow the LaMer’s mechanism,30 namely, all nanoparticles nucleate simultaneously at the beginning, and the monomers are quickly consumed by nucleation to reach below the critical supersaturation to prevent secondary nucleation.30 As a result, the size of nanoparticles follows a unimodal distribution. Nanoparticles are subjected to a combination of finite diffusion and a reversible growth reaction with finite kinetics. The growth rate is given by equation (1), and in an ad hoc manner, the dissolution rate is assumed to follow equation (1) as well when Cb < Ceq. Further, during the growth, nanoparticles do not coalesce, aggregate, split, compete locally for monomers, or otherwise interact other than through their effect on the bulk monomer concentration.


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After the one-time nucleation event, the radius of the nucleus ensemble is assumed to obey the unimodal log-normal distribution as follow,33

𝑓(𝑅0) =

(―

1 2𝜋𝜎𝑅0𝑒𝑥𝑝

(ln (𝑅0) ― 𝜇)2

)

2𝜎2

(

(3)

𝑉𝑎𝑟[𝑅0]

)

μ = 𝑙𝑛 (E[𝑅0]) ―0.5𝑙𝑛 1 + (𝐸[𝑅 ])2

(

𝑉𝑎𝑟[𝑅0]

0

)

𝜎2 = 𝑙𝑛 1 + (𝐸[𝑅 ])2 0

(3a)

(3b)

where E[𝑅0] and Var[𝑅0] are the expectation value and variance of the radius, which determine the parameters µ and σ in log-normal distribution as in equation (3a) and (3b). Among the ensemble, the nuclei of radius R0 𝑅𝐶). Let Nt and N be the total number of nuclei per unit volume after the one-time nucleation and the total number of nuclei per unit volume for growth respectively. Thus N = 𝑁𝑡(1 ― 𝐹(𝑅𝐶)), where F(R) is the cumulative distribution function of the corresponding log-normal distribution function. In this work, the software Matlab was used for numerical simulation of batch nanoparticles, and the numbers of the nucleus having radius R0i for growth in unit volume is digitized as dN(𝑅0𝑖) = 𝑁𝑡𝑓(𝑅0𝑖|𝑅0𝑖 > 𝑅𝐶)𝛿𝑅0

(4)

𝑅0(𝑖 + 1) = 𝑅0𝑖 +𝛿𝑅0

(4a)


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where i is positive integer, 𝑅01 = 𝑅𝐶 +𝑂(𝑅)≅𝑅𝐶, 𝛿𝑅0 = 0.02 𝑛𝑚. For a given material (Ω, γ and C∞) and reaction condition (T), Ceq is specified for each radius R0i according to equation 1(b). The size distribution of a specific nanoparticle ensemble at a given time t and constant temperature T is then determined by the growth parameters D, k and the precursor concentration Cb. In this work, two typical situations in the synthesis are considered, namely, (I) monomer is continuously supplied during the nanoparticle growth to keep the monomer concentration constant, and (II) monomer is gradually consumed by the nanoparticle growth without extra addition.

Case I: Monomer is continuously supplied during the growth, and Cb=constant For this case, equation (1b) is substituted into equation (1a), which is subsequently substituted into equation (1) to obtain the growth rate of nanoparticles at specific radius. The time is set at t = 0 second right after the nucleation event. The radius of nanoparticles at time t with initial radius of 𝑅0𝑖 is calculated accumulatively from 𝑅0𝑖 in a small time interval δt by

𝑅0𝑖(𝑡) = 𝑅0𝑖(𝑛𝛿𝑡) = 𝑅0𝑖((𝑛 ― 1)𝛿𝑡) +𝛿𝑡 ×

|

𝑑𝑅 𝑑𝑡 𝑅 = 𝑅 ((𝑛 ― 1)𝛿𝑡) 0𝑖

(5)

where 𝑅0𝑖(0) = 𝑅0𝑖, n is positive integers.

Case II: Monomer is gradually consumed, and Cb is gradually diminishing.


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For this case, assume the total volume of the solution is Vb, and the total number of monomer in solution is equal to Cb(t)Vb at time t. The radius of a nanoparticle at time t is also calculated by equation (5). However, when calculate dR/dt at time t in equation (5), Cb is time dependent. And

[𝐶𝑏((𝑛 ― 1)𝛿𝑡) ― 𝐶𝑏(𝑛𝛿𝑡)]𝑉𝑏 1 = Ω

∞

∑4𝜋3 [𝑅

3 0𝑖(𝑛𝛿𝑡)

― 𝑅30𝑖((𝑛 ― 1)𝛿𝑡)] ×

dN(𝑅0𝑖) × 𝑉𝑏 =

𝑖=1

1 Ω

∞

∑4𝜋3 [𝑅


― 𝑅30𝑖((𝑛 ― 1)𝛿𝑡)]𝑁𝑡𝑓(𝑅0𝑖|𝑅0𝑖 > 𝑅𝐶)𝛿𝑅0𝑉𝑏

𝑖=1

=

𝑁𝑡𝑉𝑏𝛿𝑅04𝜋 Ω

3

∞

∑ [𝑅


― 𝑅30𝑖((𝑛 ― 1)𝛿𝑡)]𝑓(𝑅0𝑖|𝑅0𝑖 > 𝑅𝐶)

𝑖=1

As a result

𝐶𝑏(𝑛𝛿𝑡) = 𝐶𝑏((𝑛 ― 1)𝛿𝑡) ―

4𝜋𝑁𝑡𝛿𝑅0 3Ω

∞

∑𝑖 = 1[𝑅30𝑖(𝑛𝛿𝑡) ― 𝑅30𝑖((𝑛 ― 1)𝛿𝑡)]𝑓(𝑅0𝑖|𝑅

0𝑖

> 𝑅𝐶)

(6) And 𝐶𝑏(0) = 𝐶𝑏0 is the initial monomer concentration in bulk solution. As a result, the monomer concentration in this case is a function of nanoparticle concentration. As growth proceed, the monomer will be gradually consumed. For a given size, when 𝐶𝑏(𝑡) = 𝐶𝑒𝑞, the growth is ceased; when 𝐶𝑏(𝑡) < 𝐶𝑒𝑞, the nanoparticle will shrink according to equation 1. And the dissolved part of the shrinking nanoparticles will become monomers in the bulk solution again, where equation (6) is still valid. For shrinking nanoparticles, when R0i(t) is below the critical size for nucleation at the corresponding 𝐶𝑏(𝑡), the dN(R0i) is set to be zero, resulting in reduction of nanoparticle concentration.


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For both cases, the expectation value and variance of the nanoparticle ensemble at time t is given by equation (7a) and (7b) respectively. ∞

E[R(t)] = E[R(nδt)] = ∑𝑖 = 1𝑅0𝑖(𝑡) ×

{

∞

Var[R(t)] = ∑𝑖 = 1𝑅20𝑖(𝑡) ×

𝑑𝑁(𝑅0𝑖) 𝑁

𝑑𝑁(𝑅0𝑖)

(7a)

𝑁

} ― {𝐸[𝑅(𝑡)}

2

(7b)

N and dN(R0i) are dynamically adjusted in the Matlab program if complete dissolving of nanoparticles occurs. To obtain the size distribution function, let F(R, t) be the cumulative function of the number of nanoparticles at size R and time t, where F(R, t) = ∑𝑅

0𝑖(𝑡)

≤𝑅

𝑑𝑁0𝑖

As a result, the size distribution function of the nanoparticle ensemble is given by

𝑓(R,t) =

∂𝐹(𝑅,𝑡) ∂𝑅

(8)

Integration of equation (8) from R=0 to R=∞ will give N, the total number of nanoparticles per unit volume at time t. 3. Results and Discussion 3.1 Simulation parameters The initial size distribution ((𝑓(𝑅0|𝑅0 > 𝑅𝐶)) and nucleus concentration (N) are determined by the operation of experimentalist as well as nucleation method, for example, injection method34 or rapid formation of monomers upon heating/reduction.35-36 In this work, we only concentrate on the growth rather than the nucleation. And for all simulations in this work, the expectation


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value and variance of the nucleus ensemble are set at 2 nm and 1 nm respectively right after the one-time nucleation. And the nuclei of radius R Eaη ) monotonously with reaction temperature. As a result, solvent with viscosity of mild temperature dependence is preferred to obtain small D/k ratio and considerable growth rate according to equation (10)&(11) at elevated reaction temperature. Higher monomer concentration will effectively reduce the size distribution for large D/k ratio, whose effect is minute for small D/k ratio. To prevent broadening effect due to Ostwald ripening,


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three strategies can be used: (1) collect nanoparticles well before the depletion of monomers for diminishing monomer concentration, (2) supply new monomers during synthesis by syringe pumping or producing monomers by slow chemical reaction in the solution, and (3) reduce the numbers of nuclei. More nuclei are often obtained for quick injection or large ramp rate of temperate for ‘hot-injection’ and ‘heating-up’ method respectively, and vice versa. Secondary nucleation event could occur during synthesis to yield a bimodal distribution of nanoparticle size. Under this circumstance, longer reaction time should be used to deplete the monomer concentration to the level such that the larger nanoparticle in the bimodal distribution still can grow while the smaller nanoparticles will dissolve. By doing this, the monodispersity could be achieved via Ostwald ripening.48 However, the process is usually slow compared to the focusing effect achieved by diffusional growth, and it can be expedited for large value of D and k (higher temperature). In this work, the temperature is maintained constant (T=573 K) during simulation. By substituting equation (12)-(14) into equation (1), the effect of reaction temperature for ‘hotinjection’ method and the temperature ramp rate for ‘heating-up’ method could be clearly revealed. The above bimodal distribution problem could also be simulated by the generalized growth rate of nanoparticles. Finally, if the generalized growth rate was extended to the diffusion and adsorption of two types of monomers, the growth of compound nanoparticle ensembles could also be simulated, which will have broader applicability. These works are in progress. 4. Summary The growth of batch nanoparticles, following unimodal log-normal size distribution, has been numerically simulated using the generalized diffusional model. The cases of constant and


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gradually reduced monomer concentration have been simulated. Our systematical simulations have unveiled the general rules of batch nanoparticle growth by diffusional growth. The E[R]-t, Var[R]-t, and Var[R]-E[R] relationships have been shown for a broad range of growth conditions. This work reveals many hidden details of batch nanoparticle growth, which cannot be easily revealed by experimental method, such as the two stages of Ostwald ripening and the role of monomer concentration at small D/k ratio. As a result, we expect the growth of batch nanoparticles by diffusional growth be well guided by our simulation. Future works include the simulation of temperature dependence, bimodal distribution and compound nanoparticles.


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FIGURES.

Figure 1. The size distribution of the nanoparticle ensemble at different time under: (a) adsorption controlled growth (D=1X10-10 m2/s, k=5X10-7 m/s with D/k=2X105 nm), (b) diffusion controlled


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growth (D=1X10-15 m2/s, k=5X10-1 m/s with D/k=2X10-6 nm), and (c) the intermediate growth conditions (D=1.5X10-14 m2/s, k=1X10-6 m/s with D/k=15 nm). For the simulation, N=1X1015 /m3, C∞=3X1025 /m3, Cb=4.5X1025 /m3, γ=1 J/m2, Ω=1X10-30 m3, T=573 K and δt=0.2 s.

Figure 2. The variance and average radius of the nanoparticle ensembles as a function of time at different monomer concentration: (a)&(b) adsorption controlled growth (D=1X10-10 m2/s, k=5X10-7 m/s with D/k=2X105 nm), (c)&(d) diffusion controlled growth (D=1X10-15 m2/s, k=5X10-1 m/s with D/k=2X10-6 nm), and (e)&(f) the intermediate case (D=1.5X10-14 m2/s, k=1X10-6 m/s with D/k=15 nm). For this simulation, N=1X1015 /m3, C∞=3X1025 /m3, γ=1 J/m2, Ω =1X10-30 m3, T=573 K and δt=0.2 s. The following monomer concentration is simulated (otherwise specified): Cb=4.5X1025 /m3, 6X1025 /m3, 7.5X1025 /m3, 9X1025 /m3, 12X1025 /m3, and the orange arrows indicate the ascending order of the monomer concentration.


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Figure 3 (a), (b) and (c) The variance of the nanoparticle ensembles as a function of radius at different D/k ratio and monomer concentration, where D and k were varied. The variance as a function of (d) growth time and (e) radius at D/k=15 nm and different D & k, the color dots in (e) indicate the size at 3600 seconds for corresponding D&k. (f) the Var[R]/E[R] ratio as a function of radius at D/k=15 nm. The variance as a function of radius at different monomer concentration for (g) D/k=0.01 nm (h) D/k=10 nm and (i) D/k=1000 nm respectively. For this simulation, C∞=3X1025 /m3, Cb=4.5X1025 /m3 (otherwise specified), γ=1 J/m2, Ω=1X10-30 m3, T=573 K and δt=0.2 s.


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Figure 4. The size distribution of the nanoparticle ensembles at 0, 1200, 2400, and 3600 s at different nanoparticle concentrations under different conditions: (a)(b)&(c) adsorption controlled growth (D=1X10-10 m2/s, k=5X10-7 m/s with D/k=2X105 nm), (d)(e)&(f) diffusion controlled growth (D=7X10-15 m2/s, k=1X10-1 m/s with D/k=7X10-5 nm), and (g)(h)&(i) the intermediate case (D=1.5X10-14 m2/s, k=1X10-6 m/s with D/k=15 nm). For the simulation, C∞=3X1025 /m3, Cb0=4.5X1025 /m3, γ=1 J/m2, Ω=1X10-30 m3, T=573 K and δt=0.2 s.


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Figure 5. The average size and variance of the nanoparticle ensembles as a function of time at different nanoparticle concentration from 1015 to 1020 /m3 under: (a)(d) adsorption controlled growth (D=1X10-10 m2/s, k=5X10-7 m/s with D/k=2X105 nm), (b)(e) diffusion controlled growth (D=7X10-15 m2/s, k=1X10-1 m/s with D/k=7X10-5 nm), and (c)(f) the intermediate case (D=1.5X10-14 m2/s, k=1X10-6 m/s with D/k=15 nm). The orange arrows indicate the ascending order of nanoparticle concentration. The effect of monomer concentration on the E[R]-Var[R] curves is shown for (g) D/k=0.01 nm, (h) D/k=10 nm and (i) D/k=1000 nm at N=1018 /m3. For this simulation, C∞=3X1025 /m3, Cb0=4.5X1025 /m3 (otherwise specified), γ=1 J/m2, Ω=1X10-30 m3, T=573 K and δt=0.2 s.


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Figure 6. (a) The three-stage Var[R]-t curve obtained for intermediate D/k ratio (D=5X10-14 m2/s and k=5X10-6 m/s with D/k= 10 nm) and large nanoparticle concentration (N=1X1018 /m3). (b) shows details of the three stages. For the simulation, C∞=3X1025 /m3, Cb=4.5X1025 /m3, γ=1 J/m2, Ω=1X10-30 m3, T=573 K and δt=0.2 s.


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Figure 7. The comparison of Var[R]-E[R] curves between the case I and II growth at (a) D/k=0.01, (b) D/k=10 nm and (c) D/k=1000 nm. (d) shows the two stages of Ostwald ripening in the Var[R]E[R] curve. For the simulation, N=1X1018 /m3, C∞=3X1025 /m3, Cb=4.5X1025 /m3 (otherwise specified), γ=1 J/m2, Ω=1X10-30 m3, T=573 K and δt=0.2 s.


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Figure 8. The 2D D-k map of variance at different size. (a), (b) an (c) show the variance of case I growth at 5 nm, 10 nm and 15 nm respectively, and (d), (e) an (f) show the variance of case II growth at 5 nm, 10 nm and 15 nm respectively. The lines of 10 seconds, 1 hour and 10 hours were indicated, on which the corresponding time should be taken to obtain the radius. For this simulation, N=1X1018 /m3, C∞=3X1025 /m3, Cb=4.5X1025 /m3, γ=1 J/m2, Ω=1X10-30 m3, T=573 K and δt is reduced for large D and k.


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ASSOCIATED CONTENT Supporting Information. The average radius and variance obtained at different time step, the saturation value of variance as a function of monomer concentration for adsorption controlled growth in case I growth, the E[R]-t curve fitting, the Cb(t)-E[R] curve, and 2D D-k map of Var[R]/E[R] ratio. AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] (Prof. T. Wen) Notes The authors declare no competing financial interest ACKNOWLEDGMENT This work was financially supported by National Natural Science Foundation of China (Nos. 51772045, 51401046, 51572042, 61734002), National Key Research Development Program (No. 2016YFA0300801), Science Challenge Project (No. TZ2018003), and Fundamental Research Funds for the Central Universities (ZYGX2016J045).


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Table of Contents Graphic


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Simulation of Batch Nanoparticle Growth by the Generalized Diffusion

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