Anisotropic shrinkage induced by particle rearrangement in sintering

Anisotropic shrinkage induced by particle rearrangement in sinteringFumihiro Wakai *,Kentarou Chihara,Michiyuki YoshidaSecure Materials Center,Materials and Structures Laboratory,Tokyo Institute of Technology,R3-234259Nagatsuta,Midori,Yokohama 226-8503,JapanReceived 19January 2007;received in revised form 4April 2007;accepted 8April 2007Available online 4June 2007AbstractA microscopic model is presented which describes the anisotropic shrinkage induced by particle rearrangement during sintering.Two types of topological transitions in the rearrangement,that is the formation of a new contact,are illustrated in three-dimensional simu-lation of sintering of four spheres arranged in a rhombus.The shrinkage rate is shown to become anisotropic as the particle coordination number changes with rearrangement.A micromechanical principle is discussed with the use of sintering force and effective viscosity.The anisotropic shrinkage is caused by the additional sintering force acting on the new grain boundary formed at the rearrangement.Ó2007Acta Materialia Inc.Published by Elsevier Ltd.All rights reserved.Keywords:Sintering;Simulation;Micromechanical modeling1.IntroductionSintering is a thermal process that transforms a powder compact into a bulk material;it is used for mass produc-tion of complex-shaped components.Dimensional control of components is fundamental to the avoidance of geomet-rical distortion and to meet required tolerance specifica-tions.Non-uniform shrinkage occurs during sintering generally,because the process involves a number of com-plex phenomena.The origin of non-uniform shrinkage is classified into three types:heterogeneous structure in pow-der compacts,external fields and anisotropic microstruc-ture.Heterogeneous structures,such as inhomogeneous initial density distribution [1,2]and multilayered systems [3,4],cause distortion during sintering.The external stress [5,6],gravity [7]and temperature gradient also deform components.These macroscopic dimensional changes can be predicted by using a constitutive equation that involves the strain rate tensor,macroscopic viscosity tensor,stress tensor and isotropic sintering stress [1,8–13].On the other hand,anisotropic shrinkage can occur due to anisotropic microstructure,when elongated particles arrange them-selves in a preferred orientation in powder processing such as die-pressing [14],tape casting [15],injection molding [16]and extrusion.This anisotropic shrinkage can be described by using an anisotropic sintering stress tensor [10,17],which is affected by anisotropies in particle shape [18–20],pore shape [21,22],surface energy and arrangement of par-ticles.The viscosity tensor will be also affected by the aniso-tropic diffusion coefficient [23].While the sintering is simply described as densification macroscopically,it is really a microscopic process of com-plex evolution of many particles by atomic diffusion at ele-vated temperatures.The elementary processes in sintering can be described by analyzing microscopic evolution of arranged particles.The bond formation and neck growth in the initial stage of sintering are described by using the Frenkel [24]–Kuczynski [25]model of two spheres.Pore channel closure can be illustrated with the sintering of three spheres arranged in a triangle.The formation of a closed pore and its shrinkage are modeled as the sintering of four spheres arranged in a tetrahedron [26].The shrinkage in particle scale can be expressed as motion of the center of mass driven by the sintering force [27,28].Analysis of the forces which lie behind the microscopic evolution will pro-vide a unique insight into the anisotropic shrinkage,based on the knowledge of microstructure.1359-6454/$30.00Ó2007Acta Materialia Inc.Published by Elsevier Ltd.All rights reserved.doi:10.1016/j.actamat.2007.04.027*Corresponding author.Tel.:+81459245361;fax:+81459245390.E-mail address:wakai.f.aa@m.titech.ac.jp (F.Wakai)./locate/actamatActa Materialia 55(2007)4553–4566The aim of this paper is to investigate how anisotropic shrinkage at microscopic length scales is affected by inter-action with neighboring particles,specifically,by local rear-rangement of particles[29].We achieve this by three-dimensional computer simulation of idealized sintering. In the late1970s,Exner[30]experimentally studied viscous sintering of four glass spheres arranged in a rhombus (Fig.47in[30]),but there was no approach for an analyt-ical description of this complicated model at that time.In our simulation study we analyze the forces acting between particles and show that rapid anisotropic shrinkage is induced by particle rearrangement during sintering.The structure of this paper is as follows.In Section2we formulate evolution equations for surface and grain boundary in idealized sintering.Brakke’s Surface Evolver program[31]can be used as a tool to simulate the evolution of complex surface and grain boundary networks.Geomet-ric rules for topological transitions of grain boundary net-works are examined in Section3.Micromechanical principles of sintering will be presented in Section4.2.Simulation of ideal sintering2.1.Surface motion and grain boundary motionThe sintering mechanisms of crystalline particles are divided into densifying mechanisms(grain boundary diffu-sion,lattice diffusion when the grain boundaries act as source/sink of vacancies)and non-densifying mechanisms (surface diffusion,evaporation–condensation,lattice diffu-sion from/to the surface)[26].The rigid body motion of particles takes place in the densifying mechanisms[32]. On the other hand,the non-densifying mechanisms give the shift of mass center of particles,but not their relative motion.Recently,it has been shown that the micromechan-ical principle of sintering,i.e.,that the motion of mass cen-ter is driven by the sintering force,is common to various sintering mechanisms:viscous sintering[10,33],densifying mechanisms[9,11]and also non-densifying mechanisms [27,28].We aim to understand the dynamics at a micro-scopic length scale that is common to various sintering mechanisms.In real sintering,several diffusion mechanisms take place concurrently.The relative importance of these diffusion mechanisms to sintering varies with particle size, pore size and temperature.We consider a simple case of idealized sintering in the following sections to avoid these complications:(1)rigid body motion of particles is not con-sidered,by assuming that the grain boundaries do not act as sink/source of vacancies;(2)diffusion is assumed to be so fast that the process is interface controlled;and(3)sur-face energy,grain boundary energy,surface mobility and grain boundary mobility are assumed to be isotropic.Sin-tering by evaporation–condensation[27],interface-con-trolled surface diffusion[34]and interface-controlled bulk diffusion[35]are examples of ideal sintering.Mullins[36]proposed that the evolution of particle shape by diffusion mechanisms can be expressed as surface motion.From the above assumptions,the surface motion in ideal sintering is described by Cahn and Taylor’s equa-tion[34]t¼M s c sðjÀ jÞð1Þwhere t is the normal velocity,c s is the surface energy and M s is the surface mobility.The curvature j=1/r1+1/r2is defined by two principal curvatures.The average of curva-ture j is given byj¼1A sZA sj d Að2Þwhere A s is the total surface area.The surface motion by Eq.(1)conserves the total volume of particles.According to Burk and Turnbull’s assumption[37], grain boundary motion by curvature in grain growth is expressed byt¼M gb cgbjð3Þwhere M gb is the grain boundary mobility and c gb is the grain boundary energy.The microstructural evolution in ideal sintering is described only by the surface motion (Eq.(1))and the grain boundary motion(Eq.(3)).We study ideal sintering in the following sections,but the essential results obtained are general and can easily be ap-plied to various sintering mechanisms.2.2.Surface Evolver programThe microstructural evolution in ideal sintering can be simulated by using Brakke’s Surface Evolver program [31].The outline of the program is described here briefly. Both the surface and the grain boundary of particles are represented as a set of triangularfinite elements,or facets. Each facet consists of three edges and three vertices.The surface and the grain boundary have energies proportional to their area.The Surface Evolver program evolves the sur-face toward minimal energy by a gradient descent method under any constraint.The gradient of energy at a vertex is a force,which must be converted to a velocity vector for the motion.This conversion involves what may be called the mobility factor:how a vertex responds to the force on it. In order to approximate motion by curvature,the resis-tance to motion of a vertex is proportional to the area asso-ciated with the vertex(area normalization with effective area option).The actual motion is found by multiplying the velocity by a scale factor.The physical interpretation of the scale factor is the time step.The surface motion by the difference in curvature(Eq.(1)),can be approximated by enforcing the constraint on conservation of the total volume of particles.This part consists of projecting the ver-tex forces to be orthogonal to the gradient of the volume [31].Where there is a constraint,the Surface Evolver pro-gram automatically calculates the Lagrange multiplier for the constraint.The Lagrange multiplier for afixed-volume constraint is the pressure or the average normal stress in the interior of particles immediately adjacent to the surface.4554 F.Wakai et al./Acta Materialia55(2007)4553–4566This pressure is identical to c s j,which is defined by Eq.(2). As vertices on the grain boundary are not constrained,the grain boundary motion by curvature(Eq.(3)),is approxi-mated.The Surface Evolver program had been applied to simulate sintering of two particles[27,38],pore channel clo-sure[39],pore shrinkage[35]and coarsening[40,41]and these results have beem summarized in a review[28].Here,we simulated ideal sintering for a case where the surface mobility M s and the grain boundary mobility M gb are the same.The particle volume wasfixed to avoid the complication of coarsening and grain growth in the simulation.The sintering of particles is described by intro-ducing dimensionless timetücsM s trð4Þwhere r0is the initial particle radius.This definition of dimensionless time is consistent with the scaling law of ideal sintering[42].3.Microscopic evolution in sintering3.1.Sintering of spheres arranged in a rhombusSintering of four spheres arranged in a rhombus is stud-ied to investigate the role of local arrangements in aniso-tropic shrinkage.The planar arrangements of four spheres are classified according to bond angle.Particle D in Fig.1a touches particles A and C and forms A–D and C–D bonds.The bond angle,/,is defined by the mass cen-ters of three particles A,D and C(black dots).Here,we study the case of an initial bond angle/0of70°in Fig.1.The ring of four spheres has a hole in the center. When spheres touch each other,circular grain boundaries are formed,as shown in the lower part of Fig.1a.The grain boundary intersects with the surface to form a circular sur-face triple junction(free surface–grain boundary–free sur-face).The radius of the surface triple junction is the neck radius.As the outward motion of the surface expands the neck radius,the hole changes its shape and shrinks,as shown in Fig.1b.When the A–D boundary meets the C–D boundary,a new semi-circular A–C boundary is created (Fig.1c).Then,the three grain boundaries meet to form a triple junction.The two semi-circular boundaries on both sides grow rapidly and coalesce(Fig.1d)and so the hole is closed.The area of the A–C boundary increases with time(Fig.1e).3.2.Tetragonal–rhombic transitionThe initial arrangement of spheres at/0=90°has tetragonal(D4h)symmetry,as shown in Fig.2a.ThesquareFig.1.Sintering of four spheres arranged at initial bond angle/0=70°:(a)dimensionless time t*=0.003;(b)t*=0.129;(c)t*=0.219;(d)t*=0.225;(e) t*=0.451(dihedral angle w=120°,c gb/c s=1).The upper part shows the process of hole closure.The lower part shows the formation and the coalescence of two new grainboundaries.Fig.2.Sintering of four spheres arranged at initial bond angle/0=90°:(a)dimensionless time t*=0.003;(b)t*=0.128;(c)t*=0.340;(d)t*=0.347;(e) t*=0.450(dihedral angle w=120°,c gb/c s=1).The upper part shows the closure of a square hole.The lower part shows the formation of a new rectangular grain boundary.F.Wakai et al./Acta Materialia55(2007)4553–45664555hole at the center shrinks and its shape changes from con-cave to convex (Fig.2b).The hole shrinks isotropically,keeping its tetragonal symmetry until it closes (Fig.2c).The symmetry of the particle cluster is broken after the hole closure (Fig.2d and e).A rectangular A–C boundary is created,but particles B and D remain separated.The shrinkage becomes anisotropic,so that the cluster has rhombic symmetry (D 2h ).This spontaneous symmetry breaking takes place because the formation of the new boundary decreases the total energy of the system.When the system reaches a critical state,i.e.,pore channel closure,the configuration with tetragonal symmetry becomes unstable and two lower energy states appear;one with an A–C boundary (Fig.2e)and the other with a B–D bound-ary.These two states are related to each other by a rotation and have the same energy.In other words,there is a degen-eracy.The actual breaking of the symmetry may easily occur by the effect of a small external asymmetric cause.The details of the topological transition of grain boundary and surface networks will be described in Section 3.4.3.3.Spheres arranged in two trianglesAt the minimum bond angle of 60°,the sphere centers are the vertices of two coplanar equilateral triangles shar-ing an edge,as shown in Fig.3a.Two holes locate in the centers of triangles.Particle A touches with particle C at the beginning and a circular A–C boundary is formed,as shown in the lower part of Fig.3a.As the two holes shrink,the A–C boundary has a convex lens shape (Fig.3b).Each hole closes quickly and forms a triple junction where three grain boundaries meet (Fig.3c).The area of the A–C boundary increases with time (Fig.3d and e).3.4.Topological transition of surface and grain boundary networksThe naturally evolving surface and grain boundary change their topology during sintering.Threeinterfaces,Fig.3.Sintering of four spheres arranged at initial bond angle /0=60°:(a)dimensionless time t *=0.003;(b)t *=0.045;(c)t *=0.052;(d)t *=0.129;(e)t *=0.450(dihedral angle w =120°,c gb /c s =1).The upper part shows the closure of two triangular holes.The lower part shows the formation of two triple junctions,where three grain boundaries meet.4556 F.Wakai et al./Acta Materialia 55(2007)4553–4566either surface or grain boundary,may meet along a line and four triple lines may meet at a point[43].The network structure of grain boundary and surface is described by lines,i.e.,triple junctions and surface triple junctions, and vertices.Wakai[28]illustrated topological transitions during sintering as shown in Fig.4a and b.In the forma-tion of a bond between two spheres,a circular loop of sur-face triple junction is created at the point of contact and it expands in the initial stage of sintering(Fig.4a).We term this the a process.Pore channel closure is modeled with the sintering of a ring of three spheres.When the hole closes, three grain boundaries meet at a vertex(Fig.4b),(2).SixA BABABF.Wakai et al./Acta Materialia55(2007)4553–45664557surface triple junctions meet at the vertex momentarily, then it splits into two vertices.A new triple junction is formed in Fig.4b,(3),so that four edges,either triple junc-tion or surface triple junction,meet at each vertex.We shall call this the b3process.The superscript indicates that three boundaries are involved in the process.Examining the sintering of four spheres arranged in a rhombus,the hole closure in Fig.1can be decomposed into two topological transitions.When two grain boundaries meet(Fig.1c),a new semi-circular boundary and a triple junction are created as schematically shown in Fig.4c. The new boundary is bounded by the triple junction and a surface triple junction.We shall call this the b2process because two boundaries are involved in the process.The formation of a new boundary by the b2process can also be seen in the sintering of three spheres[41].When the hole is closed(Fig.1d),two semi-circular boundaries on both sides coalesce with each other as schematically illustrated in Fig.4d.We shall call this the coalescence process.In a perfectly symmetrical arrangement of four spheres (Fig.2),four boundaries meet at a point as schematically shown in Fig.4e,(2).Eight surface triple junctions meet at the vertex momentarily,then it splits into four vertices.A new rectangular boundary is created at once as shown in Fig.4e,(3),then the symmetry is broken.But,this is an exceptional case.Hole closure in the sintering of a ring of four spheres can usually be decomposed into the b2pro-cess and the coalescence process.3.5.Singularity formation at topological transitionsThe surface may have a curvature singularity,where the local curvature diverges to infinity,during topological tran-sitions in sintering,i.e.,bond formation and pore channel closure.The average of curvature j,which is defined by Eq.(2),is plotted as a function of dimensionless time in Fig.5a–c.The sign is chosen so that a sphere with radius r0has a negative curvatureÀ2/r0.When spheres are bonded,the surface near the neck has a positive curvature. j increases with time in the initial stage of sintering, because the surface area with positive curvature increases as the neck radius increases.The curvature inside the hole diverges to infinity when the hole closes.While curvature at the hole is inversely proportional to the hole radius,the surface area near the hole is proportional to the hole radius.Then, j has a peak at the hole closure,but remains finite. j decreases after the hole closure because the surface area with large positive curvature disappears.The average of curvature on the surface of i th particle is defined asj i¼1S iZS ij d að5Þwhere S i is the surface area of i th particle.The average of curvature for particle A, j A,and that for particle B, j B, may differ due to asymmetric arrangement of particles. For the case of initial bond angle/0of70°(Fig.5a), j A and j B are identical in the initial stage of sintering. j A in-creases and j B decreases suddenly at the b2process(dotted arrow),because the hole with positive curvature is sur-rounded by particles A and C only,and particles B and D are separated from the hole as shown in Fig.1c. j A has a peak at the hole closure(solid arrow)by boundary coalescence and decreases afterward.Particles become par-tial spheres with constant curvature at any point in the equilibrium.But, j A and j B are different at the equilibrium under constraint offixed volume.Generally,the average curvature of a particle increases with increasing particle coordination number[28].Since the coordination number of particle A is three and that of particle B is two after the b2process, j A is higher than j B.At the initial bond angle/0of90°in Fig.5b, j A and j B are identical from symmetry.The symmetry is broken after the hole closure so that j A becomes higher than j B due to the difference in the coordination number.Since particles A and C are already bonded at the initial bond angle/0of 60°(Fig.5(c)), j A is higher than j B from the beginning.The product of the surface energy and the difference in curvature between two particles c sð j AÀ j BÞis the driving force for coarsening[40]and the A–B boundary will be curved.However,coarsening and grain growth do not occur because this simulation is performed under the con-straint offixed volume.Physically,this corresponds to the sintering of a composite in which particles B and D are the second phase.10φφ0= 60°φ0= 70°4558 F.Wakai et al./Acta Materialia55(2007)4553–45664.Micromechanics of sintering4.1.Total energyThe total energy of a particle cluster is the sum of ener-gies associated with the surface area A s and the grain boundary area A gbE¼cs A sþc gb A gbð6ÞThe sintering process depends on the dihedral angle w,which is determined bycgb¼2c s cos w=2ðÞð7ÞWhen w>0°(c gb/c s<2),the total energy E of a cluster ofN particles is lower than that of separated spheres NE0,where E0is the energy of a single sphere.The normalizedreduction in energy is defined asCDBA0.01CDBACDBACDBAF.Wakai et al./Acta Materialia55(2007)4553–45664559D EüðEÀNE0Þ=NE0ð8ÞThe energy reduction in the sintering of rings of four spheres is plotted as a function of dimensionless time in Fig.6.The energy decreases with time and the particle clus-ters reach the equilibrium state under the constraint of fixed particle volume.The equilibrium energy is indepen-dent of the initial bond angle/0,but it is dependent on dihedral angle.In the following sections we show only the results at w=120°(c gb/c s=1)for simplicity.The dot-ted arrow in Fig.6indicates boundary formation by the b2 process and solid arrows indicate hole closure.The energy decreases more rapidly after these topological transitions.4.2.Anisotropic shrinkageShrinkage during sintering can be defined by the posi-tion of the mass center of a particle.As shown in Fig.1, the mass center of a particle shifts toward the center of the ring because of mass transport from the spherical part to the neck region.The position of the mass center of a par-ticle is calculated through a surface integral using the diver-gence theorem of Gauss[38].The black dot inside a particle in Figs.1–3indicates the mass center.The relative measure of the linear shrinkage,or strain,is defined ase i¼ðR iÀR0i Þ=R0ið9Þwhere R i is the distance of i th particle from the center.Theinitial distance R0i is expressed by the sphere radius r0andthe initial bond angle/0R0A¼2r0sinð/0=2Þð10ÞR0B¼2r0cosð/0=2Þð11ÞThe shrinkageÀe i is plotted as a function of dimensionless time in Fig.7a–c.The hole closure is indicated by a solid arrow in thefigure.Although particles are arranged in a rhombus in Fig.7a,the shrinkage of particles A and B is the same in the initial stage of sintering;the shrinkage is isotropic.The significant anisotropic shrinkage starts after the b2process(dotted arrow).While the bond angle/is kept constant in isotropic shrinkage,it changes with time due to anisotropic shrink-age,as shown in Fig.8.The bond angle in thefinal equilib-rium state is/eq=62.0°and is independent of the initial bond angle/0.The distances of particles A and B in thefinal equilibrium are R eqA ¼0:82r0and R eqB¼1:20r0,respec-tively.Although thefinal shape is independent of/0,the final shrinkage strain,which is calculated by Eqs.(9)–(11),is dependent on/0.Thefinal shrinkage strain of par-ticle A is larger than that of particle B at/0>/eq(e.g., /0=70°in Fig.7a and90°in Fig.7b)and it is smaller at/0</eq(e.g.,/0=60°in Fig.7c).The strain rate of i th particle is given by_e i¼d e i=d tÃð12ÞThe shrinkage rateÀ_e i is plotted as a function of dimension-less time in Fig.9a–c.For the case of/0=70°in Fig.9a,the shrinkage ratesÀ_e A of particle A andÀ_e B of particle B are the same at t*<0.05.The value ofÀ_e B is slightly higher than À_e A from t*=0.05until the b2process at t*=0.22.This stage corresponds to the particle configuration in Fig.1b. The surface curvature at the hole on particle B is different from that on particle A.Since the surface velocity is depen-dent on its curvature,the mass center of particle B shifts fas-ter than that of particle A as particle B is elongated to the hole center in Fig.1b.When semi-circular A–C boundaries are formed by the b2process in Fig.1c,À_e A increases very rapidly as shown in Fig.9a.The shrinkage rateÀ_e A has a maximum at the hole closure,then decreases gradually.On the other hand,À_e B decreases after the b2process.At/0=90°in Fig.7b,shrinkagesÀe A andÀe B are identical from symmetry until hole closure at t*=0.34. After particles A and C are bonded together,Àe A becomes higher thanÀe B.Particles B and D move apart as the A–C boundary grows,thenÀe B decreases after t*=1.17.This symmetry breaking at the hole closure can be also seen in the shrinkage rate in Fig.9b.The shrinkage rateÀ_e B changes its sign at t*=1.17.At/0=60°in Fig.7c,the shrinkageÀe A is larger than Àe B before the hole closure since particle A touches particle C from the beginning.The shrinkage rateÀ_e A is higher thanÀ_e B as shown in Fig.9c and they have peaks at the hole closure.The shrinkage rateÀ_e A decreases rapidly after the hole closure and it changes its sign at t*=1.18.4.3.Sintering forceThe motion of mass centers can be described as a response to the force acting between particles.The sinter-0.0010.0110= 70°4560 F.Wakai et al./Acta Materialia55(2007)4553–4566ing force was originally defined by Gregg and Rhines [44]for equilibrium states in which the external force just stops the sintering contraction along one axis of the sinter body.The sintering force in equilibrium is given by the change in energy of the system with respect to the displacement of particles,i.e.,the slope of a potential energy curve [38].Alternatively,Beere [45]expressed sintering force as the sum of the pressure acting through the grain boundary area and the surface tension acting between two particles.Beere’s idea of sintering force carries over to a non-equilib-rium process of sintering [27].Here,we consider the sinter-ing force vector acting on a particle in the sintering of four particles in Fig.10.Particles C and D are omitted to show the internal structure.The A–B boundary may be slightlycurved when jB is different from j A ,then,we cut it with a plane along the surface triple junction.When the plane cuts particle B,the sintering force vector acting on the cross-section is given by [28]F ¼c s j B A 0gb e þZC c s ðn Ât Þd s þ12Z Lc gb ðn Ât Þd s ð13ÞCDBACDBACDBA0.01DIMENSIONLESS TIME, C DBAF.Wakai et al./Acta Materialia 55(2007)4553–45664561where A 0gb is the cross-sectional area of the A–B boundary,e is the unit normal vector to the plane,n is the unit normal vector to the surface/grain boundary and t is the unit tan-gent vector along the surface triple junction/triple junction.The first term on the right-hand side of Eq.(13)arises fromthe average normal stress,or pressure c s jB .The A–B boundary in Fig.10is bounded by the surface triple junc-tion (dotted line)and the triple junction (solid line).The second term is an integration of surface tension along the surface triple junctionC and the third term is an integra-tion of grain boundary tension along the triple junction L .The grain boundary tension term is multiplied by 1/2,because this force is shared by two particles,B and C.The sintering force on a particle is a vector sum of forces acting through grain boundaries with neighbor particles.In the initial stage of sintering of four spheres,the forces F b from particle B and F d from particleD act on particle A as shown in Fig.11a.The net sintering force vector on par-ticle A,F A =F b +F d ,points to the hole center.Before the b 2process,grain boundaries are bounded by surface triple junction only,then the sintering forces F b and F d can be calculated by using only the first and the second terms in Eq.(13).For the case of /0=70°,the sintering forces on particle A,F A ,and particle B,F B ,are plotted as functions of dimensionless time in Fig.11a.F A and F B are different from the beginning,because the sintering force is depen-dent on bond angle from geometry in the initial stage of sinteringF A ¼2F 0sin ð/0=2Þð14ÞF B ¼2F 0cos /0=2ðÞð15Þwhere F 0is the absolute value of the sintering force acting between particles A and B.After the b 2process,particle A interacts with three neighbor particles –B,C,and D –which exert the sintering forces F b ,F c ,and F d ,respectively.Each sintering force can be calculated by Eq.(13)including the grain boundary tension term.The net sintering force vector F A =F b +F c +F d points to the hole center and it increases rapidly because the sintering force F c from parti-cle C is added at the b 2process.F A has a maximum at thehole closure,then decreases gradually.On the other hand,F B decreases after the b 2process.Both F A and F B become zero at the equilibrium.At /0=90°in Fig.11b,F A and F B are identical from symmetry until the hole closure.The symmetry is lost after particles A and C are bonded to-gether.Both F A and F B decrease with time,but F A is higher than F B due to F c from the A–C bond.F B changes its sign and reaches zero at equilibrium.At /0=60°in Fig.11c,F A is larger than F B before the hole closure since F c con-tributes to F A from the beginning.Both F A and F B decrease after the hole closure.4.4.Constitutive equationThe relation between the sintering force and the shrink-age rate is plotted in Fig.12a–c.The details of sintering force–shrinkage rate curves are complicated,but the shrinkage rate increases with the sintering force approxi-mately linearly in the initial stage of sintering.In viscous sintering [10,33]and in sintering by the ‘‘densification mechanism’’[9,11],the strain rate of i th particle is expressed as _ei ¼ÀF iið16Þwhere k i is the effective viscosity.Fig.12shows that the motion of the mass center can be approximately described by Eq.(16),even in sintering by the ‘‘non-densifying mech-anism’’.The slopes of curves give 1/k i and they are different for particles A and B;for example,at /0=70°in Fig.12a.While sintering forces F A and F B are dependent on bond angle according to Eqs.(14)and (15),the strain rates inthe initial stage of sintering are the same _eA ¼_eB as already shown in Fig.9a.From Eq.(16),the effective viscosity is dependent on bond angle in the initial stage of sintering k A ¼k 0sin ð/0=2Þð17Þk B ¼k 0cos ð/0=2Þð18Þwhere k 0is a coefficient.The corrected shrinkage rates_eA sin ð/0=2Þand _eB cos ð/0=2Þat various initial bond angles are plotted as a function of sintering force in Fig.13.All curves have similar slopes,i.e.,the coefficient k 0in Eqs.(17)and (18)is approximately the same,independent of /0.The sintering force–shrinkage rate curve is in the form of a loop,then shows hysteresis.We approximate the sin-tering force–shrinkage rate curves by using Eq.(16)in all stages of sintering.The effective viscosity is not a constant,but varies slightly with time.The curve may show an anomaly at topological transitions,for example,the curve becomes non-linear just before the b 2process for particle A in Fig.12a.At /0=90°in Fig.12b,the curves for par-ticles A and B are identical from symmetry.The curves split into two after the symmetry breaking at the hole clo-sure.The sintering force and also the shrinkage rate of par-ticle B become negative at the final stage of sintering and become zero at equilibrium.The shapes of curvesatFig.10.Geometry of a cluster of four particles after the hole closure.Only two particles are illustrated to show the internal structure.4562 F.Wakai et al./Acta Materialia 55(2007)4553–4566。