Modelling the Sintering Neck Growth Process of Metal Fibers under the Surface Diffusion Mechanism Using the Lattice Boltzmann Method

Abstract

In this paper, the sintering neck growth process of metal fibers under the surface diffusion mechanism is simulated by using the Lattice Boltzmann method (LBM). The surface diffusion model is developed considering the geometrical characteristic of metal fibers. Then, the LBM scheme is constructed for solving the developed surface diffusion model. The sintering neck growth process of two metal fibers with different fiber angles is simulated by LBM. The simulated morphologies of sintering metal fibers well agree with ones obtained by experiments. Moreover, the numerical simulation results show that the sintering neck radius of two metal fibers is increased with the increase of fiber angle, which implies that the initial geometrical characteristic plays an important role in the sintering neck formation and growth of metal fibers.

1. Introduction

Porous metal fiber materials have many excellent functional characteristics, such as filtration separation, energy absorption, sound absorption, efficient combustion and enhanced heat and mass transfer, and have been widely used in chemicals, textiles, medicine and electronics [1,2,3]. Generally, porous metal fiber materials are manufactured by a sintering process. The sintering process is very complex, and one or several sintering mechanisms, including evaporation–condensation (EC), surface diffusion (SD), grain boundary diffusion (GBD), volume diffusion (VD) and lattice diffusion (LD), may take place at different sintering stages [4,5]. The morphological evolution of sintering metal fibers is also very complex, as shown in Figure 1. This is because metal fibers have a unique geometrical structure and contain a large amount of strain energy. In addition, the bridging phenomenon of metal fibers in the loose state can greatly reduce their contact opportunity. Moreover, the high-temperature sintered metal fibers are formed to the shape of the bamboo node. So, the sintering neck, which is used to describe the junction connected by two metal fibers, is irregular and complex. The morphology and size of the sintering neck seriously affects the properties of porous metal fiber materials. Therefore, it is meaningful to study the formation and growth process of the sintering neck.

In recent years, many researchers have studied the sintering mechanism of metal fibers. Pranatis and Seigle [6] found that the main sintering mechanism of metal fibers is the combined action of surface diffusion and volume diffusion. Kostornov et al. [7,8] studied the sintering behavior of metal fibers based on viscous flow theory. Feng et al. [9] found that the dominating sintering mechanism of metal fibers at low temperature and relative densities is volume diffusion. Li et al. [10] found that surface diffusion is the dominant sintering mechanism of 316L stainless steel fibers at relatively high sintering temperatures for a short dwelling time, however, the dominant sintering mechanism at relatively low sintering temperatures for a long dwelling time is grain-boundary diffusion.

In addition, some numerical simulations of sintering metal fibers under a specific sintering mechanism have been investigated by using different numerical methods. Ma [11,12] modelled the morphological evolution of a long fiber via the surface diffusion mechanism. Chen et al. [13,14,15,16] established the oval–oval model for sintering metal fibers under the surface diffusion mechanism and simulated the growth process of the sintering neck by using the level set method. Also, under the surface diffusion mechanism, Song et al. [17] simulated the sintering process of two metal fibers by the finite difference method. So, the numerical simulation method is an effective way to investigate the sintering process of metal fibers. However, the finite difference method cannot deal well with the sharply topological changes. Although the level set method can naturally capture topology changes, it is very time-consuming and cannot maintain volume conservation.

The Lattice Boltzmann method (LBM) is proposed based on microscopic models and mesoscopic kinetic equations [18]. Due to its advantages, such as parallel computing and easy implementation, LBM has been widely used in multiphase flow [19], porous media flow [20], phase transformation [21] and grain growth [22]. In addition, LBM is successfully utilized to simulate the sintering process of metal particles [23]. Considering the advantages of LBM, it can also be utilized to simulate the sintering process of metal fibers.

In this paper, the surface diffusion model of metal fibers is developed based on its geometrical characteristic. Then, the LBM scheme is constructed for solving the developed surface diffusion model. The sintering neck growth process of two metal fibers with different fiber angles is simulated by LBM. Finally, the effect of fiber angle on sintering neck growth is discussed.

2. Surface Diffusion Model for Sintering Metal Fibers

The sintering mechanism plays an important role in the modelling and simulation of the sintering process. Based on the experimental research for sintering particles or fibers, their morphology evolution can be divided into two stages [24,25,26]. In the first stage, material is transported from the surface toward the neck, and the densification does not occur. In the second stage, material is moved from the interior towards the neck, and the densification occurs. Correspondingly, the sintering mechanism of the first stage includes EC and SD. The sintering mechanism of the second stage includes GBD, VD and LD. In particular, a large number of experiments have proofed that SD is the dominant sintering mechanism of particles or fibers at the initial stage [10,27,28,29]. So, in order to investigate the sintering neck growth process of metal fibers at the initial stage, the SD is assumed to be the only sintering mechanism in this study. The surface flux is generated by a chemical potential gradient which is proportional to the surface curvature. Based on Fick’s law, the mathematical model of surface diffusion is deduced by Mullins [30]. The dimensionless surface diffusion model can be expressed as:

$$\frac{\partial y}{\partial t}+\frac{\partial J_{\mathrm{s}}}{\partial x}=0,$$

(1)

where y represents the normal vector of the surface. J_s is surface flux, and it can be expressed as:

$$J_{\mathrm{s}}=-\frac{1}{{(1+y^{\prime })}^{1/2}}\frac{\partial K}{\partial x},$$

(2)

where K represents the surface curvature, and it can be expressed as:

$$K=-\frac{y^{″}}{{(1+{y^{\prime }}^2)}^{3/2}}.$$

(3)

Compared to sintering metal powders, the sintering neck of metal fibers is not completely symmetrical. The metal fibers are represented by cylinders to study the formation and growth process of the sintering neck. Figure 2 shows two metal fibers in polar coordinates. α and β represent the fiber angle and polar angle, respectively. Figure 3 shows the schematic section of two metal fibers in rectangular coordinates. Y is the length of the sintering neck. O₁ and O₂ represent two ovals, and their formulas can be expressed as:

$$O_1:y^2{\cos }^2(\frac{\mathsf{\alpha }}{2}-\mathsf{\beta })+{(x+R_1)}^2=R_1^2,$$

(4)

$$O_2:y^2{\cos }^2(\frac{\mathsf{\alpha }}{2}+\mathsf{\beta })+{(x-R_2)}^2=R_2^2,$$

(5)

where R₁ and R₂ are the radii of O₁ and O₂, respectively.

The sintering neck growth process of two metal fibers with different fiber angles or sections can be characterized by varying α or β. So, the surface diffusion model for the sintering of metal fibers can be established by combining the traditional surface diffusion model and oval equations.

3. Lattice Boltzmann Method for Surface Diffusion Model

In order to realize the numerical simulation of the sintering neck growth process, the LBM is utilized to solve the surface diffusion model for sintering metal fibers. In LBM, the distribution functions are defined on the lattice with several discrete velocity directions and collided at the lattice nodes. During the collision step, the information is transported along these lattice directions.

In this research, the so-called one-dimensional three-velocity (D1Q3) model, as shown in Figure 4, is used to solve the surface diffusion model. The particle distribution function f_i(x, t) and equilibrium distribution function $f_i^{\mathrm{eq}}(x,t)$ in the ith direction of each lattice point are introduced to describe the probability of a particle with a velocity of c_i appearing at time t, and the speeds in three directions are c₁ = 0, c₂ = c, c₃ = −c, respectively.

The discretized Lattice Boltzmann equation can be expressed as:

$$f_i(x+\mathsf{\epsilon }{c}_i,t+\mathsf{\epsilon })-f_i(x,t)=-\frac{1}{\mathsf{\epsilon }}\left(f_i(x,t)-f_i^{\mathrm{eq}}(x,t)\right).$$

(6)

The Taylor expansion of the left-hand side of Equation (6) is:

$$f_i(x+\mathsf{\epsilon }{c}_i,t+\mathsf{\epsilon })-f_i(x,t)={\displaystyle \sum _{n=1}^{\infty }\frac{{\mathsf{\epsilon }}^n}{n!}{\left(\frac{\partial }{\partial t}+c_i\frac{\partial }{\partial x}\right)}^n}{f}_i(x,t).$$

(7)

Then, the Chapman–Enskog expansions of f_i(x, t) and time t for Knudsen number ε can be expressed as:

$$f_i=f_i^{(0)}+\mathsf{\epsilon }{f}_i^{(1)}+{\mathsf{\epsilon }}^2{f}_i^{(2)}+O({\mathsf{\epsilon }}^3),$$

(8)

Given different time scales t₀ = t, t₁ = εt, t₂ = ε²t, it becomes:

$$\frac{\partial }{\partial t}=\frac{\partial }{\partial t_0}+\mathsf{\epsilon }\frac{\partial }{\partial t_1}+{\mathsf{\epsilon }}^2\frac{\partial }{\partial t_2}+O({\mathsf{\epsilon }}^3).$$

(9)

Substituting Equations (7)–(9) into Equation (6) can deduce the equations on a time scale as:

$$O({\mathsf{\epsilon }}^0):f_i^{\mathrm{eq}}=f_i^{(0)},$$

(10)

$$O({\mathsf{\epsilon }}^1):D{f}_i^{(0)}=-\frac{1}{\mathsf{\tau }}{f}_i^{(1)},$$

(11)

$$O({\mathsf{\epsilon }}^2):\frac{\partial f_i^{(0)}}{\partial t_1}+\left(\frac{1}{2}-\mathsf{\tau }\right)D^2{f}_i^{(0)}=-\frac{1}{\mathsf{\tau }}{f}_i^{(2)},$$

(12)

where $D=\left(\frac{\partial }{\partial t_0}+c_i\frac{\partial }{\partial x}\right)$. Suppose:

$$\{\begin{array}{c}{\displaystyle \sum _i{f}_i^{(k)}}=0,k=1,2,3,\hfill \\ {\displaystyle \sum _i{f}_i^{(0)}}={\displaystyle \sum _i{f}_i^{(\mathrm{eq})}}=y,\hfill \\ {\displaystyle \sum _i{c}_i{f}_i^{(\mathrm{eq})}}=J_{\mathrm{s}},\hfill \\ {\displaystyle \sum _i{c}_i{c}_i{f}_i^{(\mathrm{eq})}}=m.\hfill \end{array}$$

(13)

Based on Equations (11)–(13), the conservation laws in different time scales can be expressed as:

$$\frac{\partial y}{\partial t_0}=-\frac{\partial J_{\mathrm{s}}}{\partial x},$$

(14)

$$\frac{\partial y}{\partial t_1}=-\left(\frac{1}{2}-\mathsf{\tau }\right)\left(\frac{{\partial }^{2}y}{\partial t_0^2}+2\frac{{\partial }^2{J}_{\mathrm{s}}}{\partial t_0\partial x}+\frac{{\partial }^{2}m}{\partial x^2}\right),$$

(15)

Then:

$$\frac{\partial y}{\partial t_0}+\mathsf{\epsilon }\frac{\partial y}{\partial t_1}=-\frac{\partial J_{\mathrm{s}}}{\partial x}-\mathsf{\epsilon }\left(\frac{1}{2}-\mathsf{\tau }\right)\left(\frac{{\partial }^{2}y}{\partial t_0^2}+2\frac{{\partial }^2{J}_{\mathrm{s}}}{\partial t_0\partial x}+\frac{{\partial }^{2}m}{\partial x^2}\right),$$

(16)

Based on Equations (9) and (14), Equation (16) can be rewrote as:

$$\frac{\partial y}{\partial t}+\frac{\partial J_{\mathrm{s}}}{\partial x}=-\mathsf{\epsilon }\left(\frac{1}{2}-\mathsf{\tau }\right)\frac{\partial }{\partial x}\left(\frac{\partial J_{\mathrm{s}}}{\partial t_0}+\frac{\partial m}{\partial x}\right).$$

(17)

If $\frac{\partial J_{\mathrm{s}}}{\partial t}+\frac{\partial m}{\partial x}=0$ approaches Equation (16) approaches Equation (1) with $O({\mathsf{\epsilon }}^2)$ precision. So, the LBM model for Equation (1) can be expressed as:

$$\{\begin{array}{c}{f}_i^{(0)}=\frac{1}{2c^2}(J_{\mathrm{s}}c+m)\hfill \\ f_2^{(0)}=\frac{1}{2c^2}(-J_{\mathrm{s}}c+m)\hfill \\ f_0^{(0)}=y-f_1^{(0)}-f_2^{(0)}\hfill \end{array}$$

(18)

The LBM model is solved by the finite element method, and the expressions for J_s, K and m can be expressed as:

$$J_{\mathrm{s}}(j,n)=-\frac{K(j+1,n)-K(j-1,n)}{2h{\left(1+\frac{{(y(j+1,n)-y(j-1,n))}^2}{4h^2}\right)}^{\frac{1}{2}}},$$

(19)

$$K(j,n)=-\frac{y(j+1,n)-2y(j,n)+y(j-1,n)}{h^2{\left(1+\frac{{(y(j+1,n)-y(j-1,n))}^2}{4h^2}\right)}^{\frac{3}{2}}},$$

(20)

$$m(j,n)=m(j-1,n)-\Delta x\frac{J_{\mathrm{s}}(j,n)-J_{\mathrm{s}}(j,n-1)}{\Delta t}.$$

(21)

In addition, the particle distribution functions in different directions can be calculated by:

$$f_i(x_1,t)=f_i^{(0)}(x_1,t)+\left[f_i(x_1+\Delta x,t)-f_i^{(0)}(x_1+\Delta x,t)\right],i=0,1,2$$

(22)

$$f_i(x_2,t)=f_i^{(0)}(x_2,t)+\left[f_i(x_2+\Delta x,t)-f_i^{(0)}(x_2+\Delta x,t)\right],i=0,1,2$$

(23)

4. Simulation Results and Discussion

The numerical simulations of the sintering neck growth process for two metal fibers with different fiber angles are conducted using Matlab. The simulation results are compared with the experimental results, and the effect of fiber angle on the sintering neck radius is discussed.

4.1. Simulation Results and Verifications

In these numerical simulations, the radii of two metal fibers are 1, and the step size of the space is 0.04. The sintering neck interfaces at 0, 0.00005, 0.0005, 0.005 and 0.05 simulation time are marked to describe its growth process. Figure 5 shows the sintering neck growth process of two metal fibers with different fiber angles (α = 0~π/2, β = 0). As shown in the figure, the sintering neck is gradually formed and grown up. Obviously, the atoms located in the area near the sintering neck move to the contact zone along the surface of metal fiber and accumulate to form the sintering neck. This is because the concentration of atoms at the contact zone is lower than that around the contact zone, so the atoms diffuse from the surrounding zone to the contact zone. From the perspective of curvature-driven diffusions, the curvature of the contact zone is gradually decreased with the increase of sintering neck growth to reduce surface energy and maintain stability.

In addition, the simulation results are compared with the experimental results to verify the reasonability and effectivity of the developed surface diffusion model and LBM. Figure 6 shows the section morphologies of sintering metal fibers with different fiber angles (0~π/2) at 1200 °C for 2 h. From the experimental results, it can be found that the sintering neck is formed between two metal fibers, and the geometry of the sintering neck is smooth. Also, the sintering neck is increased with the increase of fiber angle. Although the geometry of the metal fiber is not a standard cylinder, the morphology of the experimental results agrees well with that of the simulation results, which implies that the numerical simulation can accurately describe the growth process of the sintering neck.

4.2. Discussion

Figure 7 shows the sintering neck radii of two metal fibers with different fiber angles. Clearly, the sintering neck radius rapidly increases at the initial stage. Then, its growth rate gradually slows down and finally approaches zero. This is because the curvature difference at the contact zone is very large at the initial stage, which provides a large driving force for atoms. With the growth of the sintering neck, the curvature difference is gradually reduced, so the driving force is also reduced.

In addition, the effect of fiber angle on the growth process of the sintering neck is discussed. Obviously, the sintering neck radius is increased with the increase of fiber angle. This is because the distance between two fibers near the contact zone is decreased with the increase of fiber angle. The small distance of two fibers near the contact zone is beneficial to decrease the migration length of atoms, which can accelerate the formation and growth process of the sintering neck. So, the initial geometrical characteristic obviously influences the sintering neck formation and growth process of metal fibers.

5. Conclusions

The numerical simulation research about the sintering process of metal fibers under the surface diffusion mechanism is conducted by LBM, and the numerical simulation results are verified by experimental ones. The main conclusions can be summarized as follows:

(1) The surface diffusion model for metal fibers is developed based on the geometrical characteristic. The sintering neck growth process of two metal fibers with different fiber angles or sections can be characterized by the developed model.

(2) The LBM scheme is constructed to solve the surface diffusion model for sintering metal fibers. The numerical simulations of the sintering neck growth process for two metal fibers with different fiber angles are conducted by LBM.

(3) The morphology of experimental results agrees well with that of the simulation results, and the sintering neck radius of two metal fibers is increased with the increase of fiber angle.