The Effect of the Shear Flow on the Morphological Pattern of Particles in an Undercooled Melt

Abstract

The effect of shear flow on the morphological pattern of particles in an undercooled melt is studied by using the asymptotic method. The mathematical model of the particle includes the anisotropic interface kinetic undercooling. The asymptotic solution for the mathematical model of the particle shows that shear flow in an undercooled melt intensifies the deformation and distortion of the particle in the initial stage of crystal growth. Due to the shear flow, the growth rate of the interface increases in the shear direction of the flow and strengthens the inward decay of the part of the interface induced by the anisotropic interface kinetics in the initial stage of crystal growth. As the shear rate of the flow increases, the interface of the particle is seriously deformed and distorted until it breaks into smaller particles. The analytical result provides the prediction of the formation of interface microstructures during solidification through the change of processing parameters.

1. Introduction

The morphological pattern formation of particles is of fundamental importance in the solidified microstructures of materials. Theoretical and experimental investigations have made significant progress in understanding and controlling interface microstructures in the convective melt driven by the forced flow [1,2,3,4,5,6,7,8,9]. Li et al. [10] found that the crystal can directly nucleate and form spherical particles to a large scale from the convective melt induced by mechanical stirring and electro-magnetic stirring. Wang et al. [11,12,13,14,15,16] analyzed the formation mechanism of in situ iron nanoparticles during the solidification of molten melt and found that the refined particles induced by the shear effect of the forced flow greatly strengthen the matrix alloy, and the second-phase finer particles fabricated in the convective melt can be embedded between micron-sized grains to enhance the strength of metallic materials. Current experimental and numerical simulation studies of in-situ nano particle second phase strengthening materials have shown that the in-situ strengthened phases in the matrix alloy are obtained by the in-situ reaction method [17], molecular dynamics simulations [18], solute element addition [19], the lattice Boltzmann method [20,21], etc. A certain proportion of specific strengthening elements are added to directly form the second phase of in-situ nanoparticles during the solidification process of melts. These strengthened phases have strong interfacial adhesion with the matrix and high thermodynamic stability; however, it is hard to control the growth of additional particles during solidification, leading to the coarser and uneven interface microstructures of alloys and reduced physical properties of the materials. These studies suggest an essential role of the forced flow in the formation of interface microstructures during solidification.

The nanoparticle strengthening alloy problem is a fundamental theoretical subject, which has been preoccupying a number of researchers in experiments and numerical simulations. The problem under the discussion of the paper is not just a free boundary problem, where the interface shape is a part of the solution, but also intrinsically a singular boundary problem. It is hard to obtain the exact solution with numerical approaches accurately. Numerical methods have inherent defects in revealing the physical essence of the free boundary problem. Hence, it is not strange that the numerical simulations conducted so far cannot yield satisfactory, reliable information. To avoid these weaknesses, it may, after all, be accepted as an ideal and effective approach to seek an approximate analytical solution, with a concise and distinct physical essence.

In this paper, we focus on the direct calculation of the morphologies of microstructures for the dynamic model of particles in the melt with a forced shear flow. By using the asymptotic method [22,23,24,25], we obtain the analytical solution of particle growth and reveal the dependence of the formation of the interface microstructure of particles on the change of processing parameters during solidification.

2. The Mathematical Model

In the convective undercooled melt, as the temperature decreases below the liquidus, a great amount of crystalline nuclei with different sizes of nearly micro or nano scales forms. The nuclei are nucleating or growing simultaneously in the metastable molten melt. It is assumed that the melt is undercooled to a temperature $T_{\infty }$ ($T_{\infty }<T_M,$ $T_M$ is the melting equilibrium temperature of pure substance, the melt undercooling is defined as $\Delta T=T_M-T_{\infty }$), and the convection in the melt is driven by an external forced flow. Experiments [10,11] have demonstrated that under the influence of the forced flow, these nuclei deform or even split into smaller particles, in which the shear effect of the forced flow plays a key role in the formation of second phase nanoparticles. We focus on the morphological evolution of a particle with initial radius ${\overline{r}}_0$. For simplicity, the shear effect of the convection velocity $U$ is driven by the expression

$$U~S_{x}zi+S_{z}xk,$$

where $r=xi+yj+zk$, $i$, $j$ and $k$ are the unit vectors of the Cartesian coordinate frame in the Cartesian coordinate system $(x,y,z)$, respectively, the velocity gradient components $S_x$ and $S_z$ are constant velocity gradients. $S=(S_x+S_z)/2$ is the shear rate.

Let $\overline{\mathbf{U}}$ and $\overline{P}$ respectively denote the velocity and reduced pressure in the liquid phase, ${\overline{T}}_L$ and ${\overline{T}}_S$ respectively denote the temperatures in the liquid phase and solid phase. The interface of the particle is expressed as $\overline{R}=\overline{R}(\theta ,\phi ,\overline{t})$ in the spherical coordinate system, where $\overline{t}$ is the time. The continuity equation, Navier–Stokes equations, are:

$$\nabla \cdot \overline{U}=0 (\overline{r}>\overline{R}(\theta ,\phi ,\overline{t})),$$

(1)

$$(\overline{U}\cdot \nabla )\overline{U}=-\frac{1}{{\rho }_L}\nabla \overline{P}+\upsilon {\nabla }^2\overline{U} (\overline{r}>\overline{R}(\theta ,\phi ,\overline{t})),$$

(2)

where ${\nabla }^2$ is the Laplace operator, $\nabla$ is the gradient operator, $\overline{P}$ is the reduced pressure, $\upsilon$ is dynamic viscosity of melt, and ${\rho }_L$ is density of melt. The density of solid and liquid phases is assumed to be equal.

The temperature field governing equations of liquid phase and solid phase are expressed as:

$$\frac{\partial {\overline{T}}_L}{\partial \overline{t}}+(\overline{U}\cdot \nabla ){\overline{T}}_L={\kappa }_L{\nabla }^2{\overline{T}}_L (\overline{r}>\overline{R}(\theta ,\phi ,\overline{t})),$$

(3)

$$\frac{\partial {\overline{T}}_S}{\partial \overline{t}}={\kappa }_S{\nabla }^2{\overline{T}}_S (\overline{r}<\overline{R}(\theta ,\phi ,\overline{t})),$$

(4)

where ${\kappa }_T$ and ${\kappa }_S$ are the thermal diffusion coefficients of liquid phase and solid phase, respectively.

The corresponding interface conditions are the total mass conservation and tangential non slip of the interface

$$\overline{U}\cdot n=0, \overline{U}\cdot \mathsf{\tau }=0 (\overline{r}=\overline{R}(\theta ,\phi ,\overline{t})),$$

(5)

where $n$ and $\mathsf{\tau }$ are the unit normal vector and unit tangent vector of the interface, respectively.

The interface temperature is controlled by the Gibbs–Thomson condition and energy conservation condition at the interface, i.e.,

$${\overline{T}}_L={\overline{T}}_S,$$

(6)

$${\overline{T}}_S={\overline{T}}_M(1+\frac{2\gamma }{\Delta H}\overline{K})-\frac{1}{\mu }{\overline{U}}_I\left[1+\beta (n_1^4+n_2^4+n_3^4)\right],$$

(7)

$$(\Delta H+2\gamma \overline{K}){\overline{U}}_I=k_S\frac{\partial {\overline{T}}_S}{\partial n}-k_L\frac{\partial {\overline{T}}_L}{\partial n},$$

(8)

where ${\overline{U}}_I$ is the moving velocity of the interface, $\overline{K}$ is the average curvature of the interface, $\Delta H$ is the latent heat per unit volume, $\gamma$ is the isotropic surface free energy, $\mu$ is the interface kinetics coefficient, $\beta$ is the anisotropy parameter of interface kinetics, $k_L$ and $k_S$ are the heat conduction coefficients in the liquid phase and solid phase, respectively.

The above solidification system in Equations (1)–(8) constitutes a free boundary problem for the shape of the crystal–melt interface, and hence for the evolution morphology of the particle.

For the asymptotic analysis, the dimensionless variables are defined by scaling velocity with the characteristic velocity of the crystal–melt interface $V_p=k_L\Delta T/({\overline{r}}_0\Delta H)$, here $\Delta H$ is the latent heat per unit volume, the length with the initial radius ${\overline{r}}_0$, the time with ${\overline{r}}_0/V_p$, the reduced pressure with ${\rho }_L{V}_p{\kappa }_L/{\overline{r}}_0$, the temperature with $\Delta T$. With the nondimensionalization transformation:

$$U=\frac{\overline{U}}{V_p},P=\frac{\overline{P}}{{\rho }_L{V}_p{\kappa }_L/{\overline{r}}_0}, T_L=\frac{{\overline{T}}_L-T_M}{\Delta T},T_S=\frac{{\overline{T}}_S-T_M}{\Delta T},r=\frac{\overline{r}}{{\overline{r}}_0}, t=\frac{\overline{t}}{{\overline{r}}_0/V_p}.$$

(9)

Equations (1)–(4) are transferred into the dimensionless governing equations:

$$\nabla \cdot U=0,$$

(10)

$$\epsilon (U\cdot \nabla )U=\mathrm{Pr}{\nabla }^{2}U-\nabla P,$$

(11)

$$\epsilon \frac{\partial T_L}{\partial t}+\epsilon (U\cdot \nabla )T_L={\nabla }^2{T}_L,$$

(12)

$$\epsilon {\lambda }_S\frac{\partial T_S}{\partial t}={\nabla }^2{T}_S,$$

(13)

where $\epsilon$ is the relative undercooling parameter,

$$\epsilon =\frac{\Delta T}{\Delta H/(c_{pL}{\rho }_L)},$$

${\lambda }_S={\kappa }_L/{\kappa }_S$, $c_{pL}$ is the specific heat in the liquid phase, ${\rho }_L$ is the density in the liquid phase, ${\kappa }_L$ and ${\kappa }_S$ are the thermal diffusivities in the liquid and solid phases, respectively.

The crystal–melt interface conditions (5)–(8) are transferred into the dimensionless interface conditions:

$$U\cdot n=0,U\cdot \mathsf{\tau }=0,$$

(14)

$$T_L=T_S,$$

(15)

$$T_S=2\Gamma K-E^{-1}M{U}_I\left[1+\beta (n_1^4+n_2^4+n_3^4)\right]$$

(16)

$$(1+2\Gamma EK)U_I=(k\nabla T_S-\nabla T_L)\cdot n,$$

(17)

where $\Gamma$ the surface energy parameter, $\Gamma =\gamma T_M/({\overline{r}}_0\Delta H\Delta T)$, $E=\Delta T/T_M$, $M=V_p/(\mu T_M),$ and $n_i$ is the value of surface normal in the i-direction ($i=1,2,3$), $\beta$ is the anisotropy strength parameter.

The far-field conditions are: $r\to \infty$

$$U~s_{x}zi+s_{z}xk,$$

(18)

$$T_L\to -1,$$

(19)

where $s_x$ and $s_z$ are dimensionless velocity gradients, $s_x=S_x{\overline{r}}_0/V_p$, $s_z=S_z{\overline{r}}_0/V_p$. $s=(s_x+s_z)/2$ is the dimensionless shear rate.

Finally, the initial condition for the interface is that, at time $t=0$,

$$R(\theta ,\phi ,0)=1.$$

(20)

For the sake of simplicity, it is assumed that the densities of the liquid and solid phases are equal, and the buoyancy effects are neglected. The separation or split into two particles or the interaction between particles does not involve them.

3. Asymptotic Solution and Analysis

The quantity $\Delta H/(c_p{\rho }_L)$ is typically several hundred degrees K. For example, for Fe, $\Delta H=2.404\times {10}^9$ J m⁻³, $c_p=477$ J kg⁻¹ K⁻¹, ${\rho }_L=7874$ kg m⁻³, $\Delta H/(c_p{\rho }_L)=640.06$ K; for Cu, $\Delta H=1.830\times {10}^9$ J m⁻³, $c_p=390$ J kg⁻¹ K⁻¹, ${\rho }_L=8930$ kg m⁻³, $\Delta H/(c_p{\rho }_L)=525.45$ K. In a practical solidification system, the relative undercooling parameter $\epsilon$ is small, $\epsilon <<1$. We take $\epsilon$ as a small parameter and seek the asymptotic solution for the free boundary problem for the shape of the crystal–melt interface in Equations (10)–(20). The anisotropic interface kinetics is assumed to be of the same order of magnitude to the small parameter, $\beta =\Theta \epsilon$, $\Theta =O(1)$. We seek the following asymptotic solution for the growth system of the particle

$$\left\{\begin{array}{c}\begin{array}{cc}U~U_{L0}+\epsilon U_{L1}+\cdots & P~P_{L0}+\epsilon P_{L0}+\cdots \end{array}\\ \begin{array}{cc}{T}_L~T_{L0}+\epsilon T_{L1}+\cdots & T_S~T_{S0}+\epsilon T_{S1}+\cdots \end{array}\\ R~R_0+\epsilon R_1+\cdots \end{array}\right.,$$

(21)

where each order approximation is further expanded into a series of spherical harmonics. The interface curvature is expanded into:

$$K~-\frac{1}{R_0^{}}+\frac{\epsilon }{2R_0^2}\left[\frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial }{\partial \theta }\right)+\frac{1}{{\sin }^2\theta }\frac{{\partial }^2}{\partial {\phi }^2}+2\right]R_1+\cdots .$$

(22)

Substituting (21) together with (22) into Equations (10)–(20), we derive the equations and boundary conditions for each order approximation. The leading order approximations for the flow field and the temperature fields satisfy the equations:

$$\mathrm{Pr}{\nabla }^2{U}_{L0}=\nabla P_{L0}, \nabla \cdot U_{L0}=0,$$

(23)

$${\nabla }^2{T}_{L0}=0, {\nabla }^2{T}_{S0}=0,$$

(24)

which are subject to the interface conditions: at the interface,

$$U_0\cdot n=0, U_0\cdot \mathsf{\tau }=0,$$

(25)

$$T_{L0}=T_{S0},$$

(26)

$$T_{S0}=-\frac{2\Gamma }{R_0}-E^{-1}M\frac{d{R}_0}{dt},$$

(27)

$$(1-\frac{2\Gamma E}{R_0})\frac{d{R}_0}{dt}=k\frac{\partial T_{S0}}{\partial r}-\frac{\partial T_{L0}}{\partial r}.$$

(28)

The far-field condition is, as $r\to \infty$,

$$U_{L0}\to s_{x}zi+s_{z}xk,$$

(29)

$$T_{L0}\to -1.$$

(30)

The initial condition for the interface is, at time $t=0$,

$$R_0(0)=1.$$

(31)

For the flow field, when it is superimposed addictively, the linear shear flow, the flow field throughout the melt, is modified by the additional fluid velocities. After carrying out the differentiation algebra, we have the leading order approximations $U_{L0}=\left(u,v,w\right)$ and $P_{L0}$ in the rectangular coordinates

$$\left\{\begin{array}{c}u=(s_x+s_z)\left(r-\frac{5}{2}\frac{R_0^3}{r^2}+\frac{3}{2}\frac{R_0^5}{r^4}\right)\sin \theta \cos \theta \cos \phi \\ v=\left(s_{x}r-(s_x-s_z)\frac{R_0^3}{2r^2}-(s_x+s_z)\frac{R_0^5}{2r^4}\right){\cos }^2\theta \cos \phi -\left(s_{z}r+(s_x-s_z)\frac{R_0^3}{2r^2}-(s_x+s_z)\frac{R_0^5}{2r^4}\right){\sin }^2\theta \cos \phi \\ w=-\left(s_{x}r-(s_x-s_z)\frac{R_0^3}{2r^2}-(s_x+s_z)\frac{R_0^5}{2r^4}\right)\cos \theta \sin \phi \\ P_{L0}=-5(s_x+s_z)\mathrm{Pr}\frac{R_0^3}{r^3}\sin \theta \cos \theta \cos \phi +const\end{array}\right..$$

(32)

The solution of Equation (24) which obeys the conditions (26)–(31) is

$$\left\{\begin{array}{c}{T}_{L0}=-1+R_0(R_0-2\Gamma E)\frac{d{R}_0}{dt}\frac{1}{r}\\ T_{S0}=-\frac{2\Gamma }{R_0}-E^{-1}M\frac{d{R}_0}{dt}\\ t=\frac{1}{2}(R_0^2-1)+(2\Gamma -2\Gamma E+E^{-1}M)\left[(R_0-1)+2\Gamma \ln \frac{R_0-2\Gamma }{1-2\Gamma }\right]\end{array}\right.,$$

(33)

where $R_0$ is the implicit solution of the ordinary differential equation that satisfies the initial condition (31)

$$\frac{d{R}_0}{dt}=\frac{R_0-2\Gamma }{R_0(R_0-2\Gamma E+E^{-1}M)}.$$

(34)

From Equation (34), as $R_0<2\Gamma$, $\frac{d{R}_0}{dt}<0$, the particle decays, whereas as $R_0>2\Gamma$, $\frac{d{R}_0}{dt}>0$, the particle grows. Setting $\frac{d{R}_0}{dt}=0$ yields a critical value of the particle $R_0=2\Gamma$. Returning to the dimensional quantity of the critical value, $R_0=2\Gamma$, we obtain the critical nucleation radius of the particle $R_{*}$ as:

$$R_{*}=\frac{2\gamma T_M}{\Delta H\Delta T}.$$

The first order approximations for the temperature fields satisfy the equations:

$$\frac{\partial T_{L0}}{\partial t}+(U_{L0}\cdot \nabla )T_{L0}={\nabla }^2{T}_{L1}, {\lambda }_S\frac{\partial T_{S0}}{\partial t}={\nabla }^2{T}_{S1},$$

(35)

which are subject to the interface conditions at the interface,

$$T_{L1}=T_{S1}+\frac{R_0-2\Gamma E}{R_0}\frac{d{R}_0}{dt}{R}_1$$

(36)

$$T_{S1}=\frac{\Gamma }{R_0^2}(\Lambda +2)R_1-E^{-1}M\frac{\partial R_1}{\partial t}-E^{-1}M\Theta \frac{d{R}_0}{dt}(n_1^4+n_2^4+n_3^4),$$

(37)

$$(1-\frac{2\Gamma E}{R_0})\frac{\partial R_1}{\partial t}+\frac{(\Lambda +2)\Gamma E}{R_0{}^2}{R}_1\frac{d{R}_0}{dt}=k\frac{\partial T_{S1}}{\partial r}-\frac{\partial T_{L1}}{\partial r}-\frac{2(R_0-2\Gamma E)}{R_0^2}\frac{d{R}_0}{dt}{R}_1,$$

(38)

where

$$\Lambda =\frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial }{\partial \theta }\right)+\frac{1}{{\sin }^2\theta }\frac{{\partial }^2}{\partial {\phi }^2},$$

and $P_n^m(\cos \theta )$ is the associated Legendre polynomial of degree $n$ and order $m$.

The initial condition for the interface is, at time $t=0$,

$$R_1(\theta ,\phi ,0)=0.$$

(39)

The solution of Equation (35) which obeys the conditions (36)–(39) is:

$$T_{L1}=\frac{A_{0,0}}{r}+\frac{r}{2}\left[(2R_0-2\Gamma E)\frac{d{R}_0}{dt}+R_0(R_0-2\Gamma E)\frac{d^2{R}_0}{d{t}^2}\right]$$

$$+\frac{A_{2,1}}{r^3}{P}_2^1(\cos \theta )\cos \phi +(s_x+s_z)R_0(R_0-2\Gamma E)\frac{d{R}_0}{dt}(-\frac{5R_0^3}{24r^2}-\frac{R_0^5}{12r^4}+\frac{r}{12})P_2^1(\cos \theta )\cos \phi$$

$$+\frac{A_{4,0}}{r^5}{P}_4(\cos \theta )+\frac{A_{4,4}}{r^5}{P}_4^4(\cos \theta )\cos 4\phi ,$$

$$T_{S1}=B_{0,0}+\frac{{\lambda }_S{r}^2}{6}\left[{\left(\frac{d{R}_0}{dt}\right)}^2+(R_0-2\Gamma E)\frac{d^2{R}_0}{d{t}^2}\right]+B_{2,1}{r}^2{P}_2^1(\cos \theta )\cos \phi$$

$$+B_{4,0}{r}^4{P}_4(\cos \theta )+B_{4,4}{r}^4{P}_4^4(\cos \theta )\cos 4\phi ,$$

$$R_1=g_{0,0}^{}+g_{2,1}{P}_2^1(\cos \theta )\cos \phi +g_{4,0}{P}_4(\cos \theta )+g_{4,4}{P}_4^4(\cos \theta )\cos 4\phi ,$$

(40)

where

$$A_{0,0}=R_0{B}_{0,0}+\frac{{\lambda }_S{R}_0^3}{6}\left[{\left(\frac{d{R}_0}{dt}\right)}^2+(R_0-2\Gamma E)\frac{d^2{R}_0}{d{t}^2}\right]+(R_0-2\Gamma E)\frac{d{R}_0}{dt}{g}_{0,0}$$

$$-\frac{1}{2}{R}_0^2(2R_0-2\Gamma E){\left(\frac{d{R}_0}{dt}\right)}^2-\frac{1}{2}{R}_0^3(R_0-2\Gamma E)\frac{d^2{R}_0}{d{t}^2},$$

$$A_{2,1}=R_0^5{B}_{2,1}+R_0^2(R_0-2\Gamma E)\frac{d{R}_0}{dt}{g}_{2,1}+\frac{5}{24}(s_x+s_z)R_0^5(R_0-2\Gamma E)\frac{d{R}_0}{dt},$$

$$A_{4,0}=R_0^9{B}_{4,0}+R_0^4(R_0-2\Gamma E)\frac{d{R}_0}{dt}{g}_{4,0},$$

$$A_{4,4}=R_0^9{B}_{4,4}+R_0^4(R_0-2\Gamma E)\frac{d{R}_0}{dt}{g}_{4,4},$$

$$B_{0,0}=\frac{2\Gamma }{R_0^2}{g}_{0,0}-E^{-1}M\frac{d{g}_{0,0}}{dt}-{\alpha }_{0,0}{E}^{-1}M\Theta \frac{d{R}_0}{dt}-\frac{{\lambda }_S{R}_0^2}{6}{\left(\frac{d{R}_0}{dt}\right)}^2-\frac{{\lambda }_S{R}_0^2}{6}(R_0-2\Gamma E)\frac{d^2{R}_0}{d{t}^2},$$

$$B_{2,1}=-\frac{4\Gamma }{R_0^4}{g}_{2,1}-\frac{E^{-1}M}{R_0^2}\frac{d{g}_{2,1}}{dt},$$

$$B_{4,0}=-\frac{18\Gamma }{R_0^6}{g}_{4,0}-\frac{E^{-1}M}{R_0^4}\frac{d{g}_{4,0}}{dt}-{\alpha }_{4,0}\frac{E^{-1}M\Theta }{R_0^4}\frac{d{R}_0}{dt},$$

$$B_{4,4}=-\frac{18\Gamma }{R_0^6}{g}_{4,4}-\frac{E^{-1}M}{R_0^4}\frac{d{g}_{4,4}}{dt}-{\alpha }_{4,4}\frac{E^{-1}M\Theta }{R_0^4}\frac{d{R}_0}{dt}.$$

From the interface condition (38), it follows that $g_{nm}=g_{nm}(t)$ satisfy the following ordinary differential equations, respectively.

For the mode $n=0$,

$$\frac{d{g}_{0,0}}{dt}=-\frac{1}{R_0^2}\Re (R_0,0)g_{0,0}+\frac{k{\lambda }_S{R}_0^2}{3D(R_0,0)}{\left(\frac{d{R}_0}{dt}\right)}^2+\frac{k{\lambda }_S{R}_0^2}{3D(R_0,0)}(R_0-2\Gamma E)\frac{d^2{R}_0}{d{t}^2}$$

$$-\frac{R_0(2R_0-2\Gamma E)}{D(R_0,0)}{\left(\frac{d{R}_0}{dt}\right)}^2-\frac{R_0^2(R_0-2\Gamma E)}{D(R_0,0)}\frac{d^2{R}_0}{d{t}^2}-{\alpha }_{0,0}\frac{E^{-1}M\Theta }{D(R_0,0)}\frac{d{R}_0}{dt},$$

where

$$\Re (R_0,n)=\frac{1}{D(R_0,n)}\left[R_0(R_0-2\Gamma E)\frac{d{R}_0}{dt}-(n+2)(nk+n+1)\Gamma +(n+2)\Gamma E{R}_0\frac{d{R}_0}{dt}\right],$$

$$D(R_0,n)=R_0-2\Gamma E+(nk+n+1)E^{-1}M.$$

For the mode $n=2$,

$$\frac{d{g}_{2,1}}{dt}=\frac{1}{R_0^2}\Re (R_0,2)g_{2,1}-\frac{5(s_x+s_z)}{24}\frac{R_0^2(R_0-2\Gamma E)}{D(R_0,2)}\frac{d{R}_0}{dt}.$$

For the mode $n=4$,

$$\frac{d{g}_{4,0}}{dt}=\frac{3}{R_0^2}\Re (R_0,4)g_{4,0}-{\alpha }_{4,0}\hspace{1em}\frac{(5+4k)E^{-1}M\Theta }{D(R_0,4)}\frac{d{R}_0}{dt},$$

$$\frac{d{g}_{4,4}}{dt}=\frac{3}{R_0^2}\Re (R_0,4)g_{4,4}-{\alpha }_{4,4}\frac{(5+4k)E^{-1}M\Theta }{D(R_0,4)}\frac{d{R}_0}{dt}.$$

From the initial condition (39), it follows that $g_{nm}=0$. Then, the solutions of Equation (40) are solved, respectively, as follows,

$$\begin{gathered} g_{0,0}=\frac{k{\lambda }_S}{3H(R_0,0)}{\displaystyle {\int }_1^{R_0}\frac{{\tau }^{2}H(\tau ,0)}{D(\tau ,0)}\frac{d}{d\tau }\left(\frac{(\tau -2\Gamma E)(\tau -2\Gamma )}{\tau (\tau -2\Gamma E+E^{-1}M)}\right)d\tau }-\frac{{\alpha }_{0,0}{E}^{-1}M\Theta }{H(R_0,0)}{\displaystyle {\int }_1^{R_0}\frac{H(\tau ,0)}{D(\tau ,0)}d\tau } \\ -\frac{1}{H(R_0,0)}{\displaystyle {\int }_1^{R_0}\frac{\tau H(\tau ,0)}{D(\tau ,0)}\frac{d}{d\tau }\left(\frac{(\tau -2\Gamma E)(\tau -2\Gamma )}{\tau -2\Gamma E+E^{-1}M}\right)d\tau }, \end{gathered}$$

(41)

$$g_{2,1}=-\frac{5(s_x+s_z)}{24H(R_0,2)}{\displaystyle {\int }_1^{R_0}\frac{{\tau }^2(\tau -2\Gamma E)H(\tau ,2)}{D(\tau ,2)}d\tau },$$

(42)

$$g_{4,0}=-{\alpha }_{4,0}(5+4k)E^{-1}M\Theta \frac{1}{H(R_0,4)}{\displaystyle {\int }_1^{R_0}\frac{H(\tau ,4)}{D(\tau ,4)}d\tau },$$

(43)

$$g_{4,4}=-{\alpha }_{4,4}(5+4k)E^{-1}M\Theta \frac{1}{H(R_0,4)}{\displaystyle {\int }_1^{R_0}\frac{H(\tau ,4)}{D(\tau ,4)}d\tau },$$

(44)

in which,

$$H(R_0,n)=\frac{{(R_0-2\Gamma )}^{c_n}}{R_0^{d_n}{[D(R_0,n)]}^{b_n}},$$

$$c_n=\frac{1}{2}\frac{(n-1)(n+2)(nk+n+1)(2\Gamma -2\Gamma E+E^{-1}M)}{2\Gamma +(nk+n+1)E^{-1}M-2\Gamma E},$$

$$d_n=\frac{(n-1)(n+2)}{2}-\frac{n(n-1)[(n+2)k+n+1]\Gamma E}{(nk+n+1)E^{-1}M-2\Gamma E},$$

$$b_n=(n-1)\frac{[2+n(n+2)(k+1)](\Gamma -\Gamma E)+(nk+n+1)E^{-1}M}{2\Gamma -2\Gamma E+(nk+n+1)E^{-1}M}+\frac{n(n-1)[(n+2)k+n+1]\Gamma E}{(nk+n+1)E^{-1}M-2\Gamma E},$$

$${\alpha }_{0,0}=\frac{3}{5}, {\alpha }_{4,0}=\frac{2}{5}, {\alpha }_{4,4}=\frac{1}{420}.$$

Consequently, we have obtained the asymptotic solution of the particle growth,

$$\left\{\begin{array}{c}\begin{array}{cc}U=U_{L0}+O(\epsilon )& P=P_{L0}+O(\epsilon )\end{array}\\ \begin{array}{cc}{T}_L=T_{L0}+\epsilon T_{L1}+O({\epsilon }^2)& T_S=T_{S0}+\epsilon T_{S1}+O({\epsilon }^2)\end{array}\\ R=R_0+\epsilon R_1+O({\epsilon }^2)\end{array}\right..$$

(45)

It can be easily testified that the above solution in (45) does not satisfy the initial conditions for the temperature fields. The cause is that the assumption that $\partial /\partial t=O(1)$. In order to solve the early-time behavior, we introduce the time fast variable, $\widehat{t}=t/\epsilon$, and seek the inner solution in the sense of time. By matching the inner solution with the above solution, we obtain the solutions which obey the initial conditions for the temperature fields.

$$\left\{\begin{array}{c}{T}_L^{(inner)}=-1+\epsilon (1-2\Gamma )\frac{1}{r}erfc(\frac{r-1}{2\sqrt{\widehat{t}}})+o(\epsilon )\\ T_S^{(inner)}=-2\Gamma +{\displaystyle \sum _{n=1}^{\infty }{C}_{Sn}{e}^{-\frac{1}{{\lambda }_t}{n}^2{\pi }^2\tau }\frac{\sin n\pi r}{nr}}+O(\epsilon )\end{array}\right.,$$

(46)

where $erfc(x)$ is the complementary error function, $erfc(x)=\frac{2}{\sqrt{\pi }}{\displaystyle \underset{x}{\overset{+\infty }{\int }}{e}^{-s^2}ds}$, and

$$C_{Sn}=2n{\displaystyle \underset{0}{\overset{1}{\int }}(T_S^{*}+2\Gamma )r\sin n\pi rdr},$$

in which $T_S^{*}$ is the initial temperature in the solid phase at time $t=0$. Moreover, it is seen that the above asymptotic solution in (45) does not satisfy the vanishing condition in the far field region $T_{L1}\to 0$ as $r\to \infty$ and then is only valid near the particle. In the far field region, we introduce the slow variable, $(\overline{r},\theta ,\phi )$, where $\overline{r}=\sqrt{\epsilon }r$. By using the multiple scales expansion method we find the asymptotic solution that obeys the diminishing condition in the far field. Finally, we have obtained the uniformly valid asymptotic solution in the whole melt region in (21). Because the phase transformation occurs mainly near the interface, we use the asymptotic solution in the whole melt region in (45), in which the interface shape is expressed as:

$$\begin{gathered} R=R_0+\epsilon \frac{k{\lambda }_S}{3H(R_0,0)}{\displaystyle {\int }_1^{R_0}\frac{{\tau }^{2}H(\tau ,0)}{D(\tau ,0)}\frac{d}{d\tau }\left(\frac{(\tau -2\Gamma E)(\tau -2\Gamma )}{\tau (\tau -2\Gamma E+E^{-1}M)}\right)d\tau } \\ -\epsilon {\alpha }_{0,0}\frac{E^{-1}M\Theta }{H(R_0,0)}{\displaystyle {\int }_1^{R_0}\frac{H(\tau ,0)}{D(\tau ,0)}d\tau }-\epsilon \frac{1}{H(R_0,0)}{\displaystyle {\int }_1^{R_0}\frac{\tau H(\tau ,0)}{D(\tau ,0)}\frac{d}{d\tau }\left(\frac{(\tau -2\Gamma E)(\tau -2\Gamma )}{\tau -2\Gamma E+E^{-1}M}\right)d\tau } \\ -\epsilon \frac{5(s_x+s_z)}{24H(R_0,2)}{\displaystyle {\int }_1^{R_0}\frac{{\tau }^2(\tau -2\Gamma E)H(\tau ,2)}{D(\tau ,2)}d\tau }{P}_2^1(\cos \theta )\cos \phi \\ -\epsilon {\alpha }_{4,0}(5+4k)E^{-1}M\Theta \frac{1}{H(R_0,4)}{\displaystyle {\int }_1^{R_0}\frac{H(\tau ,4)}{D(\tau ,4)}d\tau }{P}_4(\cos \theta ) \\ -\epsilon {\alpha }_{4,4}(5+4k)E^{-1}M\Theta \frac{1}{H(R_0,4)}{\displaystyle {\int }_1^{R_0}\frac{H(\tau ,4)}{D(\tau ,4)}d\tau }{P}_4^4(\cos \theta )\cos 4\phi +O({\epsilon }^2). \end{gathered}$$

(47)

With the analytical solution (45), we show the interface morphology of the particle under the influence of the shear flow. We used the following physical parameters of the Cu–Fe alloy with face-centred cubic (f.c.c) structures and analyze the interface morphology of the particle under the influence of anisotropic interface kinetics and the shear effect of the flow.

The physical parameters of the Cu–Fe alloy: $T_M=1812$ K (Fe), $k_L=386$ J s⁻¹m⁻¹ K⁻¹ (Cu), $k_S=80.4$ J s⁻¹m⁻¹ K⁻¹ (Fe), $c_p=390$ J kg⁻¹ K⁻¹ (Cu), $c_{pS}=477.3$ J kg⁻¹ K⁻¹ (Fe), $\gamma =0.1010$ J m⁻¹ (Fe), $\Delta H=2.409$ J m⁻³ (Fe), ${\rho }_L=8930$ kg m⁻³ (Cu), ${\rho }_S=7874$ kg m⁻³ (Fe).

Figure 1, Figure 2 and Figure 3 show the cross-sectional curves of the interface morphology of a particle growing in an undercooled melt under the influence of anisotropic interface kinetics. Figure 1 shows the cross section curves of the interface morphologies of a particle under the influence of anisotropic interface kinetics at different times during initial crystal growth after nucleation. It is seen that under the influence of anisotropic interface kinetics, during the initial growth times from $t=0$ to $t\approx 4.5$, some part of the interface of the particle in the <110> growth directions decays inward. However, when the inward decay proceeds up to a certain distance at a time $t\approx 4.5$, the part of the interface of the particle in the <110> growth directions ceases to grow. During the initial growth process, the inward decay of the part of the interface induced by the anisotropic interface kinetics causes the smaller inner radius, which is less than the critical nucleation radius $R_{*}$. The local inward growth and outward growth of the particle deforms the interface of the spherical crystal and forms the remarkable ear-like interface shape with some concave parts and other convex parts. The particle with a smaller inner radius than the critical nucleation radius $R_{*}$ tends to be locally re-melted or broken.

Figure 2 shows the cross-sectional curves of the interface morphology of the particle under the influence of anisotropic interface kinetics at different times after the initial inward growth of some parts of the interface. When the inward decay proceeds up to a certain distance from the time $t\approx 4.5$, the part of the interface of the particle in the <110> growth directions begins to grow outward. During the initial growth process, the inward decay of the part of the interface induced by the anisotropic interface kinetics causes the smaller inner radius, which is less than the critical nucleation radius $R_{*}$. The local inward and outward growth of the particle deforms the interface of the particle and forms the remarkable ear-like interface. According to the classical nucleation theory, the particle with a smaller inner radius than the critical nucleation radius $R_{*}$ tends to be locally re-melted or broken. The inward distance induced by the anisotropic interface kinetics is defined as ‘the melting depth’.

Figure 3 shows that under the influence of the shear flow, the interface of the particle with an ear-like shape is deformed. As the shear rate of the flow increases, the interface of the particle with an ear-like shape is distorted. It implies that the shear effect of the flow aggravates the local melting tendency, which is caused by the smaller inner radius than the critical nucleation radius $R_{*}$, leading to the splitting of the distorted particle during the initial crystal growth to form more fine particles.

Figure 4 shows the variations of the temperature near the interface along different growth directions under the shear effect of the forced flow at different times. It is seen that when the shear flow is exerted on the Oxz plane, the shear flow around the particle increases the temperature gradient along the direction viewed from the polar angle and accelerates the growth of some parts of the interface near $\theta =\pi /4$ but depresses the growth of other parts of the interface near $\theta =-\pi /4$. With the growth of the particle (the interface where the arrows point is moving right), the interface of the particle is distorted. Because the inward growth of the interface in the initial stage of crystal growth induced by the anisotropic interface kinetics leads to a smaller inner radius than the critical nucleation radius of the particle, the interface of the particle splits into several smaller crystals in the initial stage of crystal growth.

With the resulting theoretical result, we calculate the microstructure formation of the second phase nanoparticles in a centrifugal casting experiment of as-cast Cu–Fe–Co alloys [11,16,23]. The relevant physical parameters are that: the solidification equilibrium temperature of Fe $T_M=1728$K, the latent heat per unit volume of Fe $\Delta H=2.404\times {10}^9$ J m⁻³, the specific heat of Fe $c_p=477.3$ J kg⁻¹ K⁻¹, the density of Fe ${\rho }_L=7874$ kg m⁻³, the surface tension of Fe is ${\gamma }_0=0.1010$ J m⁻², the mole mass fraction of Fe $M_0=56$ kg mol⁻¹, the sound velocity $V_0=4970$ m s⁻¹, the gas constant $R_g=8.314$ J kg⁻¹ K⁻¹, the kinetic coefficient of Fe: $\mu =3.09$ m s⁻¹ K⁻¹ (calculated from $\mu =\Delta H{M}_0{V}_0/({\rho }_L{R}_g{T}_M^2)$), the anisotropy parameter of interface kinetics $\beta =0.25$; the solidification equilibrium temperature of Cu is taken as $T_{\infty }=1356$K. From the heat preservation to the end of solidification, the undercooling near the surface of Fe particles in the matrix of the Cu alloy is in the range of 110~375 K, the relatively undercooling parameter $\epsilon$ is in the range of 0.20~0.58, the critical radius for nucleation $R^{*}$ is in the range of 0.33~0.85 nm. According to our theoretical result, the nanoparticles formed in the Cu–Fe–Co alloy melt are of the same order of magnitude as the critical radius for nucleation. Because the anisotropy of interface kinetics induces the melting depth of the nuclei formed after nucleation, the nuclei have radii less than the critical radius of nucleation $R_{*}$ and then tend to split. Under the shear effect of the forced flow, the nuclei are deformed and distorted. As a result, the interface of the particle splits or is broken into several more fine particles in the initial stage of growth after nucleation. The more fine particles repeat the same process to form the particles with nano scales. We should understand that even if one nucleus has split into several smaller particles in the melt, it is hard to judge whether these smaller particles have just split from this particle. By contrast, it is seen that if the shear flow is not exerted in the melt, the dispersed iron nano particles second phase and nano grains is difficult to obtain in the Cu–Fe–Co alloy matrix [11,23].

4. Conclusions

We have investigated the morphological pattern of particles in the convective melt driven by the shear flow. The mathematical model of the particle includes the anisotropic interface kinetic undercooling. By using the asymptotic method, we obtain the asymptotic solution for the mathematical model of the particle. It is concluded that the shear component of the forced flow in the convective melt intensifies the deformation and distortion of the particle. Due to the shear effect of the flow, the growth rate of the interface increases in the shear direction of the flow and strengthens the inward decay of the part of the interface induced by the anisotropic interface kinetics in the initial stage of crystal growth. As the shear rate of the flow increases, the interface of the particle is seriously deformed and distorted until it breaks into smaller particles. The analytical results reveal the dependence of the formation of interface microstructures on the change of processing parameters during solidification and provide the prediction of the formation of interface microstructures during solidification through the change of processing parameters.