Atomic-scale investigation of coarsening kinetics by the phase-field crystal model

Coarsening, or Ostwald ripening, is a common phenomenon that occurs in a vast array of polydisperse two-phase mixture systems via a diffusive mass flow from small particles to large ones, such as self-assembling, catalysis, powder sintering, semi-solid casting, and heat treatment, etc. [1–6]. The coarsening process is driven by the decrease in interfacial energy per unit volume and often results in a substantial increase in the average particle size and a corresponding decrease in the particle number density [7]. On the one hand, coarsening is very important in improving grain morphology. On the other hand, coarsening often results in an increase of the average particle size which leads to the degradation of the mechanical strength of alloys [8] or the efficiency of the catalyst [4]. Therefore, a better understanding of grain coarsening kinetics is of great fundamental and practical interest.

The theory of coarsening was based on the classical work of Lifshitz, Slyozov, and Wagner which is known as LSW theory, and the coarsening model was built by assuming a steady-state diffusion field and an infinitesimally dilute second phase. Although a series of modified models had been proposed, we still cannot describe the coarsening kinetic process accurately for systems with high second phase volume fractions. Some recent studies suggest that the neglecting of the crystallographic features of the second phase might be the main reason for the failure of the prediction of the coarsening kinetics. The work of Tian [9] indicated that the coarsening kinetic coefficients are significantly different for amorphous nanoparticles and crystalline nanoparticles. Dake [10] reported the grain rotation phenomenon during the coarsening of semi-solid Al-Cu alloy and found the coarsening kinetic was influenced by the grain boundary misorientation. Wang [11] and Fife [12] found the coarsening rate constant is influenced by the grain boundary mobility. All these studies broadened our knowledge on the coarsening process, however, the coarsening kinetics and the underlying mechanisms of two-phase mixture systems with high second phase volume fractions remain largely unknown.

In this work, an atomic-scale numerical method called phase-field crystal (PFC) model [13] was used to study the coarsening process of nanoparticles in semi-solid regions. As a simplified classical density functional theory [14,15], PFC can describe the physical process that happened on atomic length scales and diffusional time scales [16–18], which has been successfully used in the study of grain growth, nucleation [19–21], and Ostwald ripening with low solid volume fractions [22]. Herein, the kinetics of coarsening with different solid volume fractions and initial size distributions were further studied by the PFC model.

The dimensionless free energy functional for PFC can be written as

$$\begin{equation} {\tilde F}=\int \left\{\frac{\psi }{2}\left[- \varepsilon +\left(1+\nabla^{2}\right)^{2}\right]\psi +\frac{\psi^{4}}{4}+\frac{\alpha }{2}\left| \nabla \overline{\psi }\right|^{2}\right\}\textrm{d}\overset{\rightarrow }{r}, \end{equation} \tag{ 1 }$$

where ψ is the dimensionless time-averaged atom number density, $\overline{\psi }=\int \textrm{d}\overset{\rightarrow }{r}\chi \big(\overset{\rightarrow }{r}- \overset{\rightarrow }{r}^{'}\big)\psi \big(\overset{\rightarrow }{r}\big)$ is a local spatial average of the density field ψ, ε is a parameter related to temperature and crystal anisotropy, α is a parameter that affects the property of the solid/liquid interface [13] and grain boundaries [23], herein, the alpha is set as a constant (${\alpha} = 20$). The equation of motion for ${\psi}$ is

$$\begin{equation} \begin{array}{c} \hspace{-12.7pc}\displaystyle\frac{\partial \psi }{\partial t}=\Gamma \nabla^{2}\displaystyle\frac{\delta \overset{˜}{F}}{\delta \psi }+\eta\\[10pt] =\Gamma \nabla^{2}\left[\left(- \varepsilon +\left(1+\nabla^{2}\right)^{2}\right)\psi +\psi^{3}+\mathrm{E }_{r}\right]+\eta, \end{array} \end{equation} \tag{ 2 }$$

where Γ is a mobility parameter, $\mathrm{E }_{k}=\alpha k^{2}\exp (- k^{2}/\lambda )\hat{\psi }$ is the Fourier form of $\mathrm{E }_{r}$, and η is a conserved stochastic noise, $\langle \eta (\overset{\rightarrow }{r},t)\eta (\overset{\rightarrow }{r}^{'},t')\rangle =- \xi \nabla^{2}\delta (\overset{\rightarrow }{r}- \overset{\rightarrow }{r}^{'})\delta (t- t')$. In this work, eq. (2) was solved using the semi-implicit Fourier spectral method with the grid space $\textrm{d}x={\pi}$/4 and time step $\textrm{d}t=1.5$. All our simulations are performed in a square domain of $L_{x}\times L_{y} = 2048\textrm{d}x\times 2048\textrm{d}x$. The simulation parameters $(\psi_{\textit{solid}},\psi_{\textit{liquid}},\varepsilon )$ in this work are chosen based on the phase diagram in ref. [24], $\varepsilon =- 0.3$, $\psi_{\textit{solid}}=0.31188$, $\psi_{\textit{liquid}}=0.37595$. The solid volume fraction was $\textit{f} ={}$0.1–0.7, the initial number of solid nanoparticles was 100, the initial size distribution of the particles was constructed based on the Gaussian distribution,

$$\begin{equation} f\left(\rho \right)=\frac{1}{\sigma \sqrt{2\pi }}\exp \left[\frac{- \left(\rho - \mu \right)^{2}}{2\sigma^{2}}\right], \end{equation} \tag{ 3 }$$

where $f(\rho )$ is the frequency, $\rho =R/\langle R\rangle$is the normalized particle radius, μ and σ are parameters describing the mean and standard deviation of the distribution. It is about 20–30 particles at the end, and 3 parallel runs are performed for each volume fraction.

Figure 1 shows the snapshots of atomic configurations of the nanoparticle coarsening process of different solid volume fractions with $\sigma =0.2,$ where ${f} =0.1$ (fig. 1(a)), 0.3 (fig. 1(b)), 0.5 (fig. 1(c)), and 0.7 (fig. 1(d)). The color coding indicates the averaged local lattice orientation for each atom, the initial lattice orientation and position of the particles were given by a random number generator. These figures demonstrate that, on average, the particles coarsen with time. For lower solid volume fractions (f < 0.1), the particles are rounded and isolated crystalline domains, the shrinking and growing of the particles are through the surface diffusion driven by the local curvature. With the increase of f ($0.1<\textit{f}<0.3$), part of these particles may coalesce, and their shape will distort from circular. The work of Moats et al. [22] and Fife et al. [25] showed that the temporal evolution of the characteristic length of the coarsening systems with non-circular particles still complies with the LSW theory. As for f larger than 0.5, almost all the particles connect and form a skeleton-like structure, and the particles are connected by grain boundaries, the atomic-scale structure and morphology of grain boundaries might play a role in the coarsening process.

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The normalized scaled particle size distribution (PSD) with solid volume fraction ${f}=0.3$ and $f=0.5$ are shown in fig. 2(a) and fig. 2(b), respectively. The symbols represent the data of PSD at different time while the solid black lines are the smooth fits to these data. From fig. 2(a), different from the LSW theory [26] and the phase-field result of Li [5], we found the size distribution is fairly symmetric and can be fitted with a Gaussian function. The standard deviation of the Gaussian fit of PSD is 0.226 for $f=0.3$, our result is in agreement with the previous simulation by Moats [22] and experiments by Seul [27]. For ${f}=0.5$, as shown in fig. 2(b), the size distribution is similar to the curve in fig. 2(a) which can also be fitted with a Gaussian function for t < 30000, and the standard deviation is 0.189. When t = 40000, the size distribution starts to become wider and deviates from the initial Gaussian. Further comparison with the Takajo [28] distribution, in which two particles are considered to form one larger particle by coalescence, shows that the distribution for t = 40000 is narrower than that of this pure coalescence model. We suggest this is due to the Ostwald-type mechanism and coalescence mechanism occur simultaneously for ${f}=0.5$ (fig. 1), and it is Ostwald-type predominant for short holding time, it is coalescence predominant for long holding time. This is in agreement with the results by Poirier [29] and Bender [30]. As a result, the size distribution for t = 40000 lies between the initial Gaussian and Takajo [28]. Another aspect of the coarsening theory is the average size $\langle R(t)\rangle$ of a coarsening system that follows the rule of $\langle R(t)\rangle =[\langle R(0)\rangle^{3}+Kt]^{1/3}$ for the steady state, where $\langle R(0)\rangle$ is the average radius of the initial state, K is the coarsening rate. Figure 3(a) shows the evolution of $\langle R(t)\rangle^{3}- \langle R(0)\rangle^{3}$ as a function of time with different solid volume fractions (f). For f < 0.5, the curves appear in a linear dependence of the third power of average radius changes over time, which means the coarsening kinetics of crystalline nanoparticles also follow the rules of the LSW theory. While, for f > 0.5, the validity of this relationship is decreased, as the local slope of the curves changes in time. Figure 3(b) is the plot of the coarsening rate as a function of solid volume fractions f, the solid line is our simulation results, the dotted line is the theory results of Marsh and Glicksman model [31], all the data has been normalized to the theoretical curve of LSW. We can see that, for low solid volume fractions (f < 0.5), the coarsening rate of particles increases with f, and it is in agreement with the theoretical analysis of Marsh and Glicksman. While, for f larger than 0.5, our simulations showed that the coarsening rates decreases with f which runs counter to the theoretical model. Recent experiments [32,33] reported similar results with our simulations and explained this phenomenon by a phenomenological model [34].

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For a deeper understanding of the reason for the decrease of coarsening rate at high solid volume fractions, we further tracked the growth process of all the particles, and three different kinds of coarsening mechanisms were found. For illustration purposes, fig. 4 shows the snapshots of the competitive growth process of the crystalline particles with different surroundings, the color coding indicates the averaged local lattice orientation for particles. We found the growth and shrinking of the connected crystalline particles with large initial lattice orientation differences are controlled by surface diffusion. As shown in fig. 4(a) and the Supplementary Movie Movie1.mov , the lattice orientation difference between particle A1 and A2 is 25.8 degrees, its coarsening process is through surface diffusion. We suggest that the decrease of coarsening rate now is due to the formation of grain boundary which will reduce the free solid/liquid interface, and this coarsening process can be described qualitatively by the liquid film migration model [34]. Figure 4(b) and Supplementary Movie Movie2.mov show another kind of particle growth mode, grain rotation, which often appears between particles with small initial lattice orientation differences. For the competitive growth process, the smaller particle B1 rotates around its centroid and results in a decrease of the lattice orientation difference between B1 and B2, and finally, the two particles merge into one larger particle when their lattice orientation becomes the same. Besides, we found the size of the B1 decreases with time during the grain rotation process, this indicates that the surface diffusion and grain rotation could occur simultaneously. Figure 4(c) and Supplementary Movie Movie3.mov show the competitive growth process for two particles with similar size, both C1 and C2 rotate around their respective centroids and merge into a large particle, this process is grain rotation mechanism dominated. Figure 4(d) and fig. 4(e) show the third growth mode, volume diffusion or grain boundary diffusion. As shown in fig 4(d) and Supplementary Movie Movie4.mov , the competitive growth of particle D1 and D2 was done by two steps: first, particles rotate around their respective centroids, when the neck radius (the black arrow region) between the two particles reaches its maximum value, the rotation stops; then the particle D1 continues to grow through volume diffusion. Figure 4(e) and Supplementary Movie Movie5.mov show the competitive growth process of three connected particles, the lattice orientation for particles E1, E2, and E3 are 31.6°, 37.0°, and 42.5°, respectively. Although the lattice orientation difference between these particles is very small, the grain rotation is suppressed by the triple line, and the growth of E3 is purely volume diffusion dominated. As the activation energy of volume diffusion is much higher than surface diffusion [9], which could also result in a decrease in the coarsening rate.

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In conclusion, the coarsening kinetic processes were investigated by the phase-field crystal model. We found that, for low solid volume fraction conditions (f), the coarsening process of crystalline nanoparticles is in agreement with the LSW theory and the detail of the atomic-scale nature of particles does not affect the kinetics of coarsening. While, for f > 0.5, the coarsening kinetics of crystalline particles will not agree with the coarsening models based on mean-field theories. The present simulation results are in qualitative agreement with the modified liquid film migration model by Manson-Whitton [34] when $f_{\mathrm{S0}} = 0.5$, however, since this model does not take the influence of solid-solid interface properties on coarsening into consideration, it is hard to describe the coarsening kinetic process quantitatively by this model. By checking the competitive growth process of all the particles, we found the coarsening process for high solid volume fractions is far more complicated than the prediction of classical coarsening models or its extensions, and three main mechanisms are found: surface diffusion, volume diffusion, and grain rotation. The definite growth mode of the particles is affected by the lattice structure, lattice orientation, size distribution, and topological domain structure of the particles. And the influence of crystalline characteristics or atomic-scale nature on coarsening kinetics should also be considered for further development of modern coarsening theory for high solid volume fractions, and more numerical or experimental investigations about coarsening kinetic processes are needed.

This work was supported by the National Natural Science Foundation of China (Grant No. 51801154), the fund of the State Key Laboratory of Solidification Processing in NWPU (Grant No. SKLSP201813), and the Projects of major innovation platforms for scientific and technological, and local transformation of scientific and technological achievements of Xi'an (20GXSF0003). Conflict of interest: The authors declare that they have no conflict of interest.

Data availability statement: All data that support the findings of this study are included within the article (and any supplementary files).