简介概要

Modeling sintering behavior of metal fibers with different fiber angles

来源期刊：Rare Metals2018年第10期

论文作者：Dong-Dong Chen Zhou-Shun Zheng Jian-Zhong Wang Hui-Ping Tang

文章页码：886 - 893

摘要：The formation of sintering necks between two metal fibers was investigated using the oval-oval model with respect to the fiber angle range of 0°-90°. Surface diffusion was assumed to be the predominant mechanism in every section of the junction of two metal fibers in this model, which was addressed numerically using the levelset method. The growth rates of the sintering necks in the direction of the bisector of obtuse angle, the bisector of acute angle and the fiber axis were discussed in detail. It is found that the growth rate of the sintering necks decreases with fiber angle increasing in the direction of the fiber axis and the bisector of acute angle. However, an opposite variation in growth rate of sintering necks can be found in the direction of the bisector of obtuse angle. The numerical simulation results show that the growth rate of the sintering necks is significantly affected by the initial local geometrical structure which is determined by the fiber angle.

稀有金属(英文版) 2018,37(10),886-893

Modeling sintering behavior of metal fibers with different fiber angles

Dong-Dong Chen Zhou-Shun Zheng Jian-Zhong Wang Hui-Ping Tang

School of Mathematics and Statistics, Central South University

State Key Laboratory of Porous Metal Materials, Northwest Institute for Nonferrous Metal Research

作者简介：*Zhou-Shun Zheng e-mail: zszheng@csu.edu.cn;

收稿日期：10 April 2015

基金：financially supported by the National Natural Science Foundation of China (Nos. 51174236 and 51134003);the National Basic Research Program of China (No. 2011CB606306);the Opening Project of State Key Laboratory of Porous Metal Materials (No. PMM-SKL-4-2012);

Modeling sintering behavior of metal fibers with different fiber angles

Dong-Dong Chen Zhou-Shun Zheng Jian-Zhong Wang Hui-Ping Tang

School of Mathematics and Statistics, Central South University

State Key Laboratory of Porous Metal Materials, Northwest Institute for Nonferrous Metal Research

Abstract：

The formation of sintering necks between two metal fibers was investigated using the oval-oval model with respect to the fiber angle range of 0°-90°. Surface diffusion was assumed to be the predominant mechanism in every section of the junction of two metal fibers in this model, which was addressed numerically using the levelset method. The growth rates of the sintering necks in the direction of the bisector of obtuse angle, the bisector of acute angle and the fiber axis were discussed in detail. It is found that the growth rate of the sintering necks decreases with fiber angle increasing in the direction of the fiber axis and the bisector of acute angle. However, an opposite variation in growth rate of sintering necks can be found in the direction of the bisector of obtuse angle. The numerical simulation results show that the growth rate of the sintering necks is significantly affected by the initial local geometrical structure which is determined by the fiber angle.

Keyword：

Metal fiber; Surface diffusion; Fiber angle; Initial local geometrical structure; Initial evolution speed;

Received： 10 April 2015

1 Introduction

Owing to its filtration separation,energy absorption,sound absorption,efficient combustion,enhanced heat and mass transfer,porous metal fiber materials have been widely used in the fields of electronics,chemicals,textiles,machinery,food and medicine [ 1, 2, 3] .

Previously,researchers have carried out some works on the preparation and application of the porous metal fiber materials [ 4, 5] .Tang et al. [ 6] studied the sound absorbing properties of stainless steel fiber porous materials and found that the sound absorption coefficient increases with the increase in porosity and thickness of fibrous materials.Wang et al. [ 7] studied the fractal dimension for porous metal materials of FeCrAl fiber and found that the fractal dimension decreases with the increase in the magnification and increases continuously with the porosity enhancing.Zhu et al.established both three-and four-point bending setups to characterize the bending properties of porous metal fiber sintered sheet and found that both three-and four-point bending strengths decreased with porosity increasing from 70%to 90%and higher sintering temperature produced higher bending strength for the porous metal fiber sheet sintered at the temperature of 700-1000℃ [ 8] .Xu et al. [ 9] studied the consolidation process of SiC_f/Ti-6Al-4V composites by matrix-coated fiber method via hot pressing using finite-element modeling and found that the higher fiber content will lower the consolidation rate.Kostornov et al. [ 10, 11, 12] studied the sintering process of metal fiber systematically with different materials and wire diameters,and the sintering mechanism of metal fiber was revealed according to the theory of viscous flow of sintering metal powder.

However,the formation of the sintering necks of metal fibers is unlike metal powder,which can be characterized by ball-ball model or ball-plate model in two-dimensional space [ 13] ,and the metal fiber is represented by a cylinder to investigate the formation of the sintering necks,as shown in Fig.1.

In this study,an oval-oval model is developed to simulate the sintering behavior of two metal fibers intersected at different fiber angles.Different mechanisms may occur during the sintering of two metal fibers,leading to the formation of sintering necks.To simplify the simulation process,it is assumed that surface diffusion is the predominant mechanism for the sintering process.In addition,it is considered only the sintering of long metal fibers (i.e.,having a substantial ratio of length-to-diameter) so that the end effect of each metal fiber can be neglected.The morphological evolution of a long cylinder or a long fiber via the surface diffusion mechanism was studied previously by modeling in the broad context of the heat treatment of metals and ceramics [ 14, 15] .However,these fundamental studies dealt with only the morphological evolution of one long cylinder or fiber.To the best of our knowledge,the formation of sintering necks between two fibers or two long cylinders has rarely been dealt with.In the present study,the oval-oval model was employed to assess and analyze the sintering behavior of two metal fibers in three directions with four fiber angles.On this basis,the relationship between the fiber angle and the growth rate of the sintering necks was discussed.Also,the effects of the initial local geometric al structure and the initial evolution speed for the growth rate of the sintering necks were investigated.

2 Oval-oval model established by level-set method

2.1 Level-set method

Level-set method was proposed by Osher and Fedkiw [ 16] and developed further during the past several years [ 17, 18] .Level-set method is a kind of front capturing methods in which the interface is implicitly described within a fully Eulerian approach [ 19] .An additional phase function is required and the motion of the interface is studied by solving the convection problem under a given velocity field.The main advantage of this approach is that all the topological changes are taken into account naturally by the numerical technique.The other advantage of the method is related to the ease of computation of geometrical quantities such as the normal vector and the curvature [ 19, 20] .However,its main drawback is related to the volume conservation which cannot be ensured just by transporting the level-set function.In this paper,the mathematical model is solved by level-set method to achieve the numerical simulations of metal fibers.

2.2 Oval-oval model

Figure 2 is the polar coordinate system for two metal fibers.Assume that the fiber angle isβand the bisector of obtuse angle represents polar axis.The polar coordinate system is established by rotating the polar axis counterclockwise,andαrepresents the angle in polar coordinates.The sections of the sintering crunode are taken from every direction,and they may be circular,oval and rectangular.Figure 3 is the section of two metal fibers.The Cartesian coordinates system is established in the cross section.O₁ represents the oval in the above,O₂represents the oval under O₁,and O₃ represents the circle which is tangent to O₁ and O₂ simultaneously.ρis the radius of O₃,a is the radius of metal fiber and r is the length of the sintering necks.

According to the geometric relationship shown in Figs.2 and 3,the functions of O₁ and O₂ can be expressed as:

Fig.1 Scanning electron microscopy (SEM) images of sintering metal fibers:aΦ12μm,1200℃,2 h;bΦ8μm,1200℃,3 h

Fig.2 Polar coordinates system for two metal fibers

Fig.3 Section of two metal fibers

where the change inα(0≤α≤π) describes the cross section in every direction.

Assuming that surface diffusion is responsible for surface movement,the mathematical model introduced by Mullins [ 21] can be expressed as:

where r_n is the normal vector of the surface,t represents time,K is the surface curvature,S is the arc length and B is a coefficient defined by:

where D_s is surface diffusion coefficient,γis surface free energy per unit area,Ωrepresents atomic volume,δ_s is the surface diffusive width,k is Boltzmann's constant and T is absolute temperature.The dimensionless model can be expressed as:

The dimensionless variables can be obtained by aKhrhilau ndK^*=a,weea is te inita rdisa aof metal fiber.

Then,the velocity of the surface normal to itself is proportional to the pergence of the surface flux,which is the Laplacian of the curvature [ 22, 23] .The implicit functionΦ(x,y,t) represents the interface,and its changes make the interface evolve.In order to define the evolution of the functionΦ(x,y,t),the evolution equation is expressed as:

In the case of surface diffusion,

where F is evolution speed and the Laplacian of the curvature is expressed as:

where K is the curvature which can be expressed as

,and K_x,K_y,

K_xx,K_xy,and K_yy are the first-order and second-order spatial derivatives of curvature K,respectively.

For each cross section,the level-set function [ 18, 19] is defined as:

where R² is the two-dimensional space,Γ(t) is the curve(interface) determined by Eqs.(1) and (2),and Q(t) represents the inner region surrounded by interface.Figure 4describes the internal area and interface of the cross section.

The oval-oval model is defined by Eqs.(6),(7) and (9),which is solved by level-set method to obtain the evolution of the interfaces in different directions,and then,the threedimensional geometry structure of the sintering necks is established by reconstituting these interfaces.

The process of three-dimensional reconstitution is expressed as follows.

Step 1:extract the position coordinates of interfaceΓ(t).According to the two-dimensional numerical simulation results in every section,the position coordinates of interface,which are expressed as (x',y'),can be extracted from these sections

Fig.4 Internal area and interface of cross section

Step 2:coordinate transformation.According to the coordinates obtained from Step 1,these coordinates can be transformed to three-dimensional coordinates (x,y,z).The coordinate transformation formulation can be expressed as follows:

Step 3:plot three-dimensional figure.According to the three-dimensional coordinates obtained from Step 2,the sections can be plotted in a three-dimensional figure.The three-dimensional geometry structure can be reconstituted when all sections are plotted in the threedimensional figure

2.3 Algorithm program based on level-set method

As an alternative approach,the velocity extension methodology [ 18] is used.This method assumes that the velocity is known everywhere on the interface,and it is extended outward in the direction normal to the interface.The corresponding algorithm for the oval-oval model is similar with the algorithm in Ref. [ 20] .

3 Numerical simulation results and discussions

The growth processes of the sintering necks of two metal fibers with the fiber angle range of 0°-90°were simulated by oval-oval model,respectively.The speed function of curve evolution can be described by Eq.(7) with B=1.The sintering simulation was carried out through MATLAB.The meshing is 500×500,and Ax=Δy=1.The radius of metal fiber is 100.Three sections in the direction of the bisector of obtuse angle (α=0°),the bisector of acute angle (α=90°) and the fiber axis were taken to be simulated.The step number of each figure is1500,and the interface is marked for every 300 steps.Then,the relation between the fiber angle and the growth rate of the sintering necks was discussed.

3.1 Numerical simulation results

For two metal fibers with the fiber angles of 30°,45°and60°,the sections of two metal fibers are shown in Figs.5,6and 7,respectively.The neck radius in the direction of the bisector of acute angle is the largest,while that in the direction of the bisector of obtuse angle is the smallest.For two metal fibers with the fiber angle of 90°,the section of two metal fibers is shown in Fig.8.The difference is small between the neck radius in the direction of the bisector of right angle and that in the direction of the fiber axis.

3.2 Discussion

For two metal fibers with the fiber angles of 30°,45°,60°and 90°,the sintering necks radius in three directions (the bisector of obtuse angle,the bisector of acute angle,and the fiber axis) are plotted as a function of time in Fig.9.For the studied fiber angles,the growth trend of neck radius in three directions is same.In the initial stage of sintering,the growth rate of sintering necks is fast and the sintering necks radius almost grows linearly with the time increasing.In the direction of the bisector of obtuse angle,when the fiber angle is 90°,the growth rate of sintering necks is the fastest and the neck radius is the largest.And the growth rate of sintering necks is the slowest and the neck radius is the smallest for the fiber angle of 30°.However,an opposite variation of growth rate of sintering necks can be found in the directions of the bisector of acute angle and the fiber axis.So,it can be concluded that the growth rate of the sintering necks decreases with fiber angle increasing in the direction of the fiber axis and the bisector of acute angle.

Fig.5 Sections of two metal fibers with fiber angle of 30°:a in direction of bisector of obtuse angle forα=0°,b in direction of bisector of acute angle forα=90°,and c in direction of fiber axis forα=75°

Fig.6 Sections of two metal fibers with fiber angle of 45°:a in direction of bisector of obtuse angle forα=0°,b in direction of bisector of acute angle forα=90°,and c in direction of fiber axis forα=64.5°

Fig.7 Sections of two metal fibers with fiber angle of 60°:a in direction of bisector of obtuse angle forα=0°,b in direction of bisector of acute angle forα=90°,and c in direction of fiber axis forα=65°

Fig.8 Sections of two metal fibers with fiber angle of 90°:a in direction of bisector of right angle forα=0°(α=90°) and b in direction of fiber axis forα=45°

Fig.9 Sintering neck radius in direction of a bisector of obtuse angle,b bisector of acute angle and c fiber axis versus time with fiber angles of30°,45°,60°and 90°

Fig.10 Initial local geometrical structure in direction of a bisector of obtuse angle,b bisector of acute angle and c fiber axis with fiber angles of30°,45°,60°and 90°

4 Calculated results of factors influencing formation of sintering necks

The numerical investigations about two sintering metal powders have been proposed,and the simulations are always simplified to circle-circle model in two-dimensional space because of its symmetrical property [ 22, 23] .However,the influence of initial local geometrical structure on the growth rate of the sintering necks is rarely discussed.The micro structural evolution of sintered body may be affected by the initial local geometrical structure and the initial evolution speed.In this section,because the sections of sintering crunodes of two metal fibers with the fiber angles of 0°-90°are different with oval-oval geometrical structures in different directions,the influences of initial local geometrical structure and the initial evolution speed on the growth rate of the sintering necks were discussed.

4.1 Influence of initial local geometrical structure

The initial local geometrical structure of the upper fiber is considered.According to Eq.(1),the expression of the initial local geometrical structure is:

The initial local geometrical structures in three directions(the bisector of obtuse angle,the bisector of acute angle,and the fiber axis) with the fiber angles of 30°,45°,60°and90°are shown in Fig.10.The larger the fiber angle is,the closer to each other the two fibers are.Then,the smaller the distance of two fibers near the contact point is,the faster the growth rate of the sintering necks is.In the direction of the bisector of obtuse angle,the larger the fiber angle is,the faster the growth rate of the sintering necks is.However,in the direction of the fiber axis and the bisector of acute angle,the smaller the fiber angle is,the faster the growth rate of the sintering necks is.These results are in accordance with the conclusions presented in Sect.3.2.

4.2 Influence of initial evolution speed

In Eq.(6),the evolution speed is The other

form of the evolution speed is:

Fig.11 Initial evolution speed (F) in direction of a bisector of obtuse angle,b bisector of acute angle and c fiber axis versus x with fiber angles of 30°,45°,60°and 90°

where the curvature .Then,the initialevolution speed can be expressed as:

where

The parameter setting is same with that in Sect.3.It was set B=1.The radius of metal fiber is 100.

The initial evolution speeds (F) in three directions (the bisector of obtuse angle,the bisector of acute angle,and the fiber axis) are plotted as a function of x with the fiber angles of 30°,45°,60°and 90°in Fig.11.The initial evolution speed is negative,which means that the curve evolves to the negative x direction.In physics,it means that the atoms move to the neck along the surface.As shown in Fig.11a,the difference of the initial evolution speed near the contact point for different fiber angles is small.The initial evolution speed near the contact point is the slowest for the fiber angle of 30°;however,when the fiber angle is60°,it is the fastest.Based on the diffusion kinetic rules,the growth rate of the sintering necks is the fastest for the fiber angle of 60°rather than for the fiber angle of 90°,and it is the slowest for the fiber angle of 30°.This is contradicted to the conclusions presented in Sect.3.2.From Fig.11b,c,the results are also contradicted to the conclusions presented in Sect.3.2.

4.3 Discussion

Based on above analysis,the growth rate of the sintering necks is greatly affected by the initial local geometrical structure of two metal fibers.However,the influence of initial evolution speed is not in conformity with the numerical simulation results.

In mathematics,the surface diffusion model can be characterized as a partial differential equation with an initial value problem.In the simulation of sintering metal powders by surface diffusion,the initial value condition is circle-circle geometrical structure in two-dimensional space,while it becomes oval-oval geometrical structure for the simulation of sintering metal fibers in two-dimensional space.It is obvious that the initial value condition affects the solution of partial differential equation,which means that the growth rate of the sintering necks is greatly affected by the initial local geometrical structure

5 Conclusion

Based on the oval-oval model proposed in this study,the sintering behavior of metal fibers was investigated with respect to different angles and sections,including the effects of the initial local geometrical structure and the initial evolution speed.The growth rate of the sintering necks decreases with the increase in fiber angle in the direction of the fiber axis and the bisector of acute angle.However,an opposite variation in growth rate of the sintering necks can be found in the direction of the bisector of obtuse angle.The initial local geometrical structure of the two metal fibers plays an important role in the growth rate of the sintering necks.

Acknowledgments This work was financially supported by the National Natural Science Foundation of China (Nos.51174236 and51134003),the National Basic Research Program of China (No.2011CB606306) and the Opening Project of State Key Laboratory of Porous Metal Materials (No.PMM-SKL-4-2012).