finite element formulation of coupled grain-boundary and surface diffusion with grain-boundary...

Finite Element Formulation of Coupled Grain-Boundary and Surface Diffusion with Grain-Boundary MigrationAuthor(s): J. Pan, A. C. F. Cocks and S. KucherenkoSource: Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 453, No. 1965 (Oct.8, 1997), pp. 2161-2184Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/53044 .

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Finite element formulation of coupled grain-boundary and surface diffusion with

grain-boundary migration

BY J. PAN1, A. C. F. COCKS2 AND S. KUCHERENKO1

1Department of Mechanical Engineering, University of Surrey, Guildford, Surrey GU2 5XH, UK

2Department of Engineering, University of Leicester, Leicester LE1 7RH, UK

Finite element formulations are developed to model microstructure evolution by a combination of grain-boundary diffusion, grain-boundary migration and free surface diffusion. The formulations are based on a unified variational principle which allows fully coupled processes to be analysed. For example, a process can be analysed which involves grain-boundary diffusion along a curved and migrating grain-boundary network, coupled with surface diffusion along internal and/or external free surfaces which intersect with the grain-boundary network. The numerical solution provides the velocities of each individual grain and the velocities of grain-boundaries and migrating surfaces. The finite element formulations, when combined with a time integration algorithm, form a numerical technique which can be used to simulate microstructural evolution in polycrystalline materials. The technique can be applied to a wide range of physical problems including: sintering of powder compacts; grain- growth; diffusive void growth and crack propagation; superplastic deformation and the morphological evolution of electronic thin films. Various numerical examples are presented to demonstrate the effectiveness of the numerical technique.

1. Introduction

In many practical situations microstructure evolution in polycrystalline materials occurs by solid state diffusion and the migration of grain-boundaries. Creep deformation, creep damage accumulation, creep crack propagation, superplastic deformation, sintering of powder compacts, grain-growth, the instability of micro-electronic thin films can all be controlled by solid state diffusion, grain-boundary migration or a combination of these processes. Following the classic work of Herring (1951), who established the atomic chemical potentials in terms of grain-boundary stress and free-surface curvature, significant progress has been made in modeling material behaviour controlled by solid state diffusion. Similar progress has also been made in the study of grain-boundary migration, which has resulted in the development of macroscopic grain-growth models (Hillert 1965). It is not appropriate to give a comprehensive review of these areas here, but it can be said that the previous efforts have enabled us to relate many aspects of the macroscopic behaviour of engineering materials to their microstructures and the underlying physical processes. The situa- tion is, however, far from satisfactory, since real material systems are often complex

Proc. R. Soc. Lond. A (1997) 453, 2161-2184 ? 1997 The Royal Society Printed in Great Britain 2161 TIX Paper



J. Pan, A. C. F. Cocks and S. Kucherenko

and various, often unrealistic, assumptions have to be made in the material models in order to solve the mathematical equations. The recent development of computer simulation techniques has provided a powerful tool to gain a much deeper insight into the relationship between microstructure and macroscopic behaviour.

Based on the variational principle developed by Needleman & Rice (1980), Cocks suggested a finite element formulation for grain-boundary diffusion in a hexagonal grain-boundary network (Cocks 1989). This formulation has been used to investigate crack-tip fields (Pan & Cocks 1993a) and void growth (Cocks & Pan 1993) in creeping materials. The formulation was later extended to grain-boundary diffusion in arbitrary networks of grains with straight grain-boundaries (Pan & Cocks 1993b) and later combined with a finite difference scheme for coupled grain-boundary and surface diffusion (Pan & Cocks 1995). The numerical technique, when combined with a time integration algorithm, can be used to simulate morphological evolution of polycrystalline materials with internal and/or external free surfaces of any shape. It has been used to study microstructural evolution during superplastic deformation (Pan & Cocks 1993b) and the co-sintering of spherical particles of different sizes

(Pan et al. 1997). Important physical conclusions were drawn from these numerical studies. However, the numerical technique has certain limitations. For example, grain boundaries were assumed to remain straight and therefore grain-boundary migration driven by curvature effects was excluded; the surface diffusion part of the problem was solved by a finite difference scheme, which resulted in a complicated treatment of the coupling conditions between grain-boundary and surface diffusion.

Recently, Needleman & Rice's variational principle has been extended to problems involving grain-boundary and surface diffusion by Suo & Wang (1994) and Sun et al. (1996), and to the problem of grain-boundary migration by Cocks & Gill (1996). In this paper these variational principles are unified. A set of linear velocity finite elements are developed for surface diffusion and grain-boundary migration, respectively. Grain boundaries and free surfaces are discretized into a series of straight segments (elements), along which a linear variation of the migration velocity is assumed. The previously developed element for grain-boundary diffusion (Pan & Cocks 1993b) is used for each segment of a grain boundary to deal with diffusion along curved and migrating grain boundaries. The finite element formulations, combined with a time integration algorithm, form a numerical technique for computer simulation of morphological evolution at the level of grain size in porous polycrystalline materials.

2. The variational principle

As mentioned in the introduction, variational principles have been developed for grain-boundary diffusion by Needleman & Rice (1980), for surface diffusion by Suo & Wang (1994), for grain-boundary migration by Cocks & Gill (1996) and for coupled grain-boundary and surface diffusion by Sun et al. (1996). In this section, we simply unify these variational principles to allow fully coupled processes to be analysed. We consider a two-dimensional system consisting of a grain-boundary network Fgb intersected by internal and/or external free surfaces Fs. Along part of the external boundary of the system FF an external distributed force F is applied. The total potential energy E of the system is

E -= J YgbdF+ 7s dF- F UdF, (2.1) Pgb Rr rF

Proc. R. Soc. Lond. A (1997)

2162



Coupled grain-boundary and surface diffusion

where rgb and rs are specific energies for grain boundaries and free surfaces, respectively, and U is the displacement of FF with respect to a reference configuration. The system evolves to reduce E. The grains are assumed to be rigid and only three processes that dissipate energy during the evolution are considered: diffusion along grain boundaries; diffusion along free surfaces; and grain-boundary migration. The diffusive flux, defined as volume of matter flowing across unit area per unit time, is referred to as jgb for grain-boundary diffusion and to js for surface diffusion. The velocity of a migrating grain boundary is referred to as vm. The rate of energy dissipation by the three processes can be calculated as

f FgbjgbdF + Fsjs dr + j Fmm d, rgb bs Jrgb

in which Fgb, Fs and Fm are the driving forces for grain-boundary diffusion, surface diffusion and grain-boundary migration, respectively. For any virtual variation of jgb, js and Vm, the principle of virtual work requires that

Fgb6jgb F d r d J+ Fsjs dF (6E) 0. (2.2) b s fgb

dt

Assuming linear kinetic laws for solid diffusion and grain-boundary migration,

jgb = MgbFgb, js = MFs, vm = MmFm, (2.3)

where Mgb, Ms and Mm are the mobilities associated with grain-boundary diffusion, surface diffusion and grain-boundary migration, respectively, we obtain

I_ 1 . f (dE / g? gb' drs + J vjs~,v dP + 6 mm) +-

.b gb Jgb dgs Igsb Mm ? dt =0

which can be rewritten as

I _j2

/ .2

/ 1 2 dE1

6[ d2 &bjdP? + v

dm d + - 0.

[ 2MgJ2bd+b J 2Mp sd8+ 2Mm m

dt

This leads to the following variational principle.

Principle 2.1. Among all the virtual velocities of migrating grain boundaries and virtual diffusive fluxes that satisfy matter conservation, the actual velocity and flux fields minimize the functional

.2= /1 .2 1 2 d-V2 /d 2Mg +Jgb2 d + 2 dd+ * (2.4) Jrgb 2Mgb g- d r 2M- - s b2 Mm m dt

It can further be shown that there is only one stationary solution to 61H = 0 and that this solution satisfies the following four conditions.

(1) It provides the well-known expressions for the driving forces of the three energy dissipation processes,

Fgb -- . Fs = -' Fm = 'YgbKgb, (2.5)

where a is the stress normal to a grain boundary, ';gb and ns are the curvatures of the grain boundaries and free surfaces, respectively, and s is the curvilinear coordinate along the grain boundary.


2163




, free-surface

free-surface

Figure 1. Discretization of grain-boundaries and free surfaces.

(2) The equilibrium conditions between interfacial tensions at junctions of grain boundaries and at junctions between grain boundaries and free surfaces are satisfied.

(3) The grain-boundary stresses and external forces are in equilibrium. (4) The continuity of the chemical potential where a grain-boundary meets a free

surface and at the junctions of grain-boundaries. This means that any assumed solution does not have to satisfy the above four con-

ditions when using the variational principle. The assumed solution, however, must satisfy matter conservation. This requirement is often inconvenient when construct- ing approximate solutions. For example, it is difficult to construct an approximate velocity field which automatically satisfies matter conservation at grain-boundary junctions. The problem can be solved by introducing Lagrange multipliers to the functional at locations where matter conservation is not automatically satisfied (Cocks 1989):

I/_j2 1 I2 J 2 dE (~ ) *=J 2M gbd+ 2Msd gb 2Mm dt

(2.6) The outer summation is over all the locations where matter conservation is violated and the inner summation is over all the fluxes flowing into such a location. The last term ensures that the diffusive fluxes into any location vanish through variation with respect to the Lagrange multiplier if the assumed fluxes cannot guarantee matter conservation. It can be demonstrated that A takes the value of the atomic chemical potential where it is defined. Equation (2.6) forms the basis for the finite element formulations presented in the next section.

It is interesting to point out that only the general linear forms of the kinetic laws (i.e. equations (2.3)), not the actual kinetic laws (which are the combinations of (2.3) and (2.5)), are required in the implementation of the variational principle. As noted in the first condition above, the actual driving forces are provided by the variational principle.

3. Discretization

As shown in figure 1, grain boundaries and free surfaces are represented by a series of straight elements. Along a free surface, the migration velocity ,/s is defined at each


2164




node and the diffusive flux js is defined across the mid-point of each element. Also, a Lagrange multiplier As is introduced at each node. Similarly, along a grain boundary, the migration velocity Vm is defined at each node; the diffusive flux jgb is defined across the mid-point of each element; and a Lagrange multiplier is introduced at each node. Furthermore, a separation velocity, Vgb, is defined at each node which is directly related to the velocities of the two grains either side of the grain boundary. The velocity of each grain is defined at the 'centre' of the grain which can be defined arbitrarily.

In the following sections, the functional H* is discretized in terms of the degrees of freedom described above.

4. Finite element formulation for surface diffusion

As matter diffuses along a free surface, atoms are deposited on, or removed from, a particular part of the surface, resulting in a migration of the surface at a velocity vs. It is important to realize that Vs is only imaginary since it is not the velocity of any matter point. The migration velocity only makes sense if a one to one correspondence between points on the 'old' surface and those on the 'new' surface is defined. For small time intervals, this one to one correspondence can be defined by the unit vector n normal to the 'old' surface. This implies that Vs is always in the direction of n. Matter conservation requires that

+ vs =0, (4.1)

where js is the surface diffusive flux and s is the curvilinear coordinate along the free surface.

As shown in figure 1, a free surface can be approximated by a series of straight segments. We refer to these segments as elements and the junctions between any two elements as nodes. The free surface is then defined by the coordinates of the nodes. For the discretized surface, we first define the velocities of the nodes and then assume a linear variation of the migration velocity along each element. To define the nodal velocity, we have to define the vector normal to the surface at a node. As shown in figure 2, a circle is drawn through the node under consideration and its two neighbouring nodes. The unit vector pointing away from the material and along the line which links the centre of the circle and the node is taken as the normal vector. The velocity of the node is then always in the direction of the normal vector.

For each element, a local coordinate ( is defined, as shown in figure 3. The origin of ( is located at the mid-point of the element and the positive direction of ( always makes an angle of 90? anticlockwise with the normal to the element, ne (which always points away from the material). ( is normalized by the half length of the element so that ( = -1 at the first node of the element and ( = 1 at the second node. The migration velocity between the two nodes can be approximated by

s(C)= [N1(C) N2()] [ ' ], (4.2) [Vs,2 J

in which vs, and vs,2 are the two nodal velocities, and N1 (() and N2(() are the shape functions,

NV(C) = (1- ), N(O = (1 + ). (4.3)


2165




ni

Figure 2. Definition of the surface normal at a node.

is,1 nl

,, _ IXn Aes, v~~~~~~~~~~~~~~~~~~~~~~~' s ,1

Figure 3. A surface diffusion element.

From (4.1) we have

is(() =-s vs(C)d +js,o (4.4) /o

where Is is the half length of the element and js,o is the diffusive flux across the origin of the local coordinate. Substituting (4.2) into (4.4), we obtain

Vs,2

js()= [-2lsC-/sC2 -2s(C+ IsC2 1] vs, . (4.5)

s,0 _

The contribution of the current element to the functional 1* is then

Je 2?s = 2Md ji (?)d = [vs,2 v, jis,][K] (4.6)

s,o _

where [Ks] is a 3 x 3 viscosity matrix for surface diffusion. At each node, matter conservation is not guaranteed by expression (4.5). A La-

grange multiplier is therefore introduced at each node. The contribution of each


2166

.\




Vs,2

Figure 4. A special surface diffusion element joining the grain-boundary.

element to the last term of the functional H* is

[A2 A1] s (-1) =[A2 Al][Cs] Vs , (4.7)

- js,O _

where [cs ] is a 2 x 3 complementary matrix. The two contributions of the current element to the functional, (4.6) and (4.7),

can be combined into the following form:

Vs,2

1 vs,1 [V,2 Cs,l Js,O As,2 As,l] ,0 isO = [Us]T [As][Us], (4.8)

As,2

in which [Us] is the vector of elementary unknowns and [As] is the generalized viscosity matrix of the element under consideration. The detailed expression of [As ] is given in Appendix A.

The contribution from the current element to the term dE/dt of the functional 17* is simply

s, 1 Vs,

here, nl and n2 are the normals to the free surface at the first and the second nodes of the element defined earlier in this section, t is a unit vector pointing from the first node to the second node of the element and ys is the specific free surface energy. We refer to [ Fs ] as the elementary force matrix for surface diffusion. It simply represents the surface line tension in the plane of the element.

Expressions (4.8) and (4.9) are valid for all the surface elements, except for those which meet grain boundaries. Figure 4 shows such an element where the second node of the element is at the junction between the free surface and the grain boundary.


2167




At the junction, the equilibrium condition between the surface and grain-boundary tensions leads to a discontinuity in the normal to the surface. The direction of the surface velocity at the junction is therefore unknown. The origin and physical im- plications of this problem have been discussed in our previous paper (Pan & Cocks 1995). A special surface element is constructed as shown by figure 4. The second node of the surface element is given two degrees of freedom, referred to as Vsx,2 and Vsy,2, which are the components of the junction velocity in the global coordinate system x, y. The migration velocity of the element can then be written as

Vsx,2

vs(()= [neXN2(() ne,yN2() Nl(0)] v y,2 (4.10) - vs,1

in which ne,x and ne,y are the x and y components of the elemental normal ne, and

N1(C) and N2(() are given by (4.3). From (4.4) we have

Vsx,2

js(() - [ne,x(-- ls- lsC2) ne,y(- -1S( ls2) -/s+ 1]

Vy2 .(4.11) J

- 2^ 4 2s 4/ , 2 s +4 s ,

1] . 411)

rvs,1

Js,o

Substituting (4.11) into the functional H* produces a 4 x 4 viscosity matrix [ K ] and a 2 x 4 complementary matrix [ C ], so that the contribution of the special element to H* takes the form

Vsx,2

Vsy,2

Ks* Cs*T Vs,2

I[Vsx,2 Vsy,2 Vs, is,O As,2 As,i] [S S S [U [A][U], (4.12)

is,0

As,2

where [ Us ] and [ A* ] are the vector of unknowns and the generalized viscosity matrix, respectively, for the special element. The detailed expression of [A]* is provided in

Appendix A. The contribution of the special element to dE/dt of the functional 17* is

Vsx,2 Vsx,2

[stx ysty -n.l t%Y] Vsy,2 =[Fs*] vsy,2 , (4.13)

Vs,1 _ _ Vs,1

here tx and ty are the x and y components of the unit vector t, and [ F ] is the force matrix of the special element.

Since the velocity of the second node of the special surface element is also associated with the rate of grain-boundary extension (or contraction), there are additional contributions from Vsx,2 and vsy,2 to dE/dt related to the specific grain-boundary


2168




Figure 5. A grain-boundary migration element.

energy 7gb. This will be discussed in the next section when we develop a special element for grain-boundary migration.

The total contribution to the functional 7* from all the free surfaces is the summation of (4.8) and (4.9) over all the standard surface elements and the summation of (4.12) and (4.13) over all the special elements that meet grain boundaries.

5. Finite element formulation for grain-boundary migration

At any particular point of a grain boundary, the velocity of grain-boundary migration can only be in the direction normal to the grain boundary. The reason for this is exactly the same as that discussed at the beginning of the last section for free-surface migration. In the finite element formulation, grain-boundary migration is treated in a similar way to surface migration. Each of the grain boundaries is divided into a series of straight elements. At each node, the normal of the grain boundary is defined by drawing a circle through the node and its two neighbouring nodes. For each grain boundary, the normals are selected to point to a chosen side of the boundary. Fig- ure 5 shows a grain-boundary element which is allowed to migrate. The element only has two degrees of freedom, i.e. the migration velocities of the two nodes, referred to as Vm,1 and Vm,2, respectively. A linear velocity profile is assumed along the element,

vm() = [N2(() ( N1()] Vm,l , (5.1)

here ( is the local coordinate along the element which is the same as that used for surface diffusion, and Nl(() and N2(() are given by (4.3). The contribution of the current element to the functional 1* is

Ib2M vm Idmf=i2M nd'Vm d( Vm,2 Vm,l ][Km] ]

Jr^, 2M m -1 Vm,r 1

2 [ Um ] [ Km][U], (5.2)


2169




Figure 6. A special grain-boundary migration element joining the free surface.

in which lgb is the half length of the grain-boundary element, [Um] = [Vm,2 Vm,l] is the vector of unknowns of the element and [Km ] is a 2 x 2 viscosity matrix for grain-boundary migration. The detailed expression of [ K ] is given in Appendix A.

The contribution of the current element to dE/dt is

[(n2 t)'gb -(nl . t)gb] IVm22 [Fm][Um], (5.3) Vm,l

in which nl and n2 are the normals to the grain boundary at the first and the second nodes of the element, t is the unit vector pointing from the first node to the second node and [ Fm] is the elemental force matrix for grain-boundary migration.

Again, (5.2) and (5.3) are not valid for a grain-boundary element which meets a free surface, since the normal to the grain boundary does not exist at the junction. As discussed in the last section, the junction has two degrees of freedom which are shared by the two joining elements for surface diffusion and the element for grain- boundary migration. Again, a special element is needed, which is shown in figure 6. The element has three degrees of freedom, [UM ]T = [Vm,2 Vmx,i1 Vmy, ], in which

vmx,l and Vmy,l are the x and y components of the migration velocity of the junction which are the same as vsx,2 and Vsy,2 of the special surface element that meets the

grain boundary at its second node. The migration velocity of the element can be written as

vm()=[ N2(() ne,xNi () ne,yN(C) ][ U ]. (5.4) The contribution of the special element to the functional H* is

dr = 2 d = [Um [ UT[ Um ], (5.5) JFgb2 2m j m 2(

in which [ Km ] is a 3 x 3 viscosity matrix of the special element for grain-boundary migration. Its detailed form is provided in Appendix A.

The contribution of the special element to the term dE/dt is

Vm,2

[(n2' t)7Ygb -t7xgb -tyVgb] Vmx,l [ Fm][ Um ] (5.6)

- my,1

in which tx and ty are the x and y components of the unit vector t pointing from


2170




the first node to the second node of the element, and [ Fm] is the force matrix of the special element for grain-boundary migration.

The total contribution of grain-boundary migration to the functional H* is the summation of (5.5) and (5.6) over all the grain-boundary elements that meet a free surface and the summation of (5.2) and (5.3) over the rest of the grain-boundary elements.

6. The dihedral angle In the current formulation, the equilibrium conditions between surface tensions

and grain-boundary tensions at junctions between grain boundaries and free surfaces are satisfied through the variational principle. Therefore, the assumed velocity fields do not have to satisfy the dihedral angle. The variational principle ensures that the dihedral angle is recovered by surface diffusion. Locally at a junction, the dihedral angle should be recovered almost instantaneously. However, for each timestep the characteristic length scale of the recovery process is often very small and the numerical solution can only recover the dihedral angle over the length scale of the element size used. If the element size is much larger than the characteristic length scale associated with instantaneous recovery, recovery can only be observed in the numerical solution over many timesteps, or it may never be observed if other kinetic processes dominate.

Almost all previous numerical schemes enforce the dihedral angle 'precisely' in one way or another at each timestep. The mesh size is usually much larger than the characteristic length scale over which the effect of the dihedral angle can be observed within a single timestep. The 'precise' enforcement of the dihedral angle is therefore misleading. The current finite element method allows the recovery process to compete with other kinetic processes. This is important for problems in which surface diffusion is slow and a 'non-equilibrium' profile of the surface results.

7. Finite element formulation for grain-boundary diffusion

A finite element formulation has been developed by the authors for diffusion along networks of straight grain boundaries (Pan & Cocks 1993b, 1995). When dealing with grain-boundary migration in the last section, a curved grain boundary is approximated by a series of straight segments (elements). In fact, the finite element formulation for grain-boundary diffusion that has previously been developed can be directly applied to each of the straight elements.

During grain-boundary diffusion, matter is either removed from, or deposited onto, a particular location of a grain boundary. This matter redistribution results in a 'separating' or an 'approaching' velocity, referred to as Vgb, of the grains either side of a grain boundary in the direction normal to the boundary. Matter conservation requires that Vgb and the diffusive flux jgb along the grain boundary satisfy the relationship

Ojgb Vgb + - = 0, (7.1)

in which s is the curvilinear coordinate along the grain boundary. Figure 7 shows three segments of a grain boundary where the relative motion between the two grains has been exaggerated. The 'centres' of the two grains, C1 and C2, have been arbitrarily chosen where the translational velocities, ui and vi, and the rotational velocity


2171




wi (i = 1, 2) of the two grains are defined. For a grain-boundary diffusion element, its separation velocity can be related to the degrees of freedom of the two associated grains in a linear form (Pan & Cocks 1993b),

U1

Vl

01

U2

V2

(M2

(7.2)

in which, as shown in figure 7, nx and ny are the components of the normal of the grain-boundary element in the global coordinate system, lgb is the half length of the element, ( is the local coordinate along the grain-boundary element normalized by lgb and ei (i = 1, 2) is the offset distance of the grain centre from a line drawn through the origin of the local coordinate along the direction of the elemental normal.

From equation (7.1), we have

jgb(() = - gb Vgb(() dC + jb,0.

Substituting (7.2) into the above expression gives

jgb(C) = [ nxlgb nylgbS -(0.512b(2 -+gbel C)

-nxtgb -nylgbC (0.51bC2 + lgbe2C) 1 ]

x

U1

Vl

01

U2

V2

W2

. gb,O .

The contribution of the grain-boundary diffusion element to the functional HT* is

1 . 1 /j2d gb

~ j2b (() d( 2Mgb g =

2Mgb J1 g

2 [ul1 Vl W1 U2 V2 W2 jgb,O ] [gb]

(7.3)

(7.4)


U1

Vl

W1

U2

V2

W2

j gb,0 O

(7.5)

2172

Vgb(() = [-nx - ny (IgbC + el) nx ny - (lgb + e2) ]




in which [ Kgb ] is a 7 x 7 viscosity matrix for grain-boundary diffusion. Its detailed form is provided in Appendix A.

Similar to surface diffusion, a Lagrange multiplier is introduced at each node to enforce matter conservation, since this is not guaranteed by equations (7.4). The contribution of each grain-boundary element to the last term of 7* is

Ul

Vl

031 F ,W

[Ag,2 gbb,2 , gb) Agb,2 Agb, ][Cgb] 2 (7.6) L g

V2

V2

_ jgb,0 _

where [ Cgb] is a 2 x 7 complimentary matrix, which is given in Appendix A. From (7.5) and (7.6), the two contributions of the current element to the functional H* can be combined together into the following form:

U1

Vl

Wl C31

T U2

U[l Vl wi U2 V2 w2 Jgb,o Agb,2 Agb,l] [g 0 V2 I Cgb 0

Jgb,o

Agb,2

Agb,1

[ U ]Agb][ Ugb], (7.7)

in which [Ugb] is the vector of elementary unknowns and [Agb] is the generalized viscosity matrix for the grain-boundary diffusion element.

Where a grain boundary meets a free surface, a Lagrange multiplier, Atip, is introduced to enforce matter conservation, i.e. a term

Atip(jgb + j + Js)tip (7.8)

is introduced into 7*, where (jgb)tip, (j+)tip and (js)tip are the diffusive fluxes into the junction from the grain boundary and the two free surfaces, respectively. The contributions of surface diffusion and grain-boundary diffusion to H* related to Atip are given by (4.7) and (7.6). Atip is, in fact, the chemical potential at the junction, which is directly related to the so-called 'capillarity stress' and the 'pore tip curvature'. A detailed discussion of Atip can be found in our previous paper (Pan & Cocks 1995).

As shown in figure 8, at the grain boundary/free surface junction, a new surface is created and the grain boundary shortens as the grain boundary 'opens up' (i.e. as atoms are inserted into the grain boundary). This produces a contribution to the term dE/dt in the functional H*. The process provides an extra degree of freedom


2173



2174 J. Pan, A. C. F. Cocks and S. Kucherenko

I V2

2

.gb

e\ C v\,

Figure 7. A grain-boundary diffusion element.

::x::i_:~~i:a:_:iii? i ~(a) (b) i,ii

i newly created surface

..................., '

,....... /... .

'new' grain-boundary

Figure 8. Change of free energy at a grain-boundary/surface junction associated with the 'opening up' of the grain-boundary.

for the dihedral angle to be recovered locally, as demonstrated by figure 8a. However, this recovery process is not taken into account by the approximate velocity fields for the surface and grain-boundary migration and is therefore ignored. The intersection between the extensions of the two free surfaces is taken as the new position of the grain boundary/free surface junction, as shown in figure 8b. The associated value of dE/dt can be calculated as

dE { sin Wi, + sin T2 ]

_ [ sin ii + sin 2 1 b Vb-

Lcos - - sin f2 + cos fT2 sin ;n j V J open,

? ?,ii',','~?=',i'0'ii i'?/t t t \ #|:iiiiii iii: i i0i- i 4iii: i iiiiiaiiiiiii

in which fi and @2 are defined in figure 8b and Vopen is the opening velocity of the grain boundary at the junction. Vopen is related to the velocities of the two grains





either side of the boundary through equation (7.2). When the equilibrium dihedral angle is reached, i.e. for 1i = @2 = P = arcsin(-ygb/2ys), the above equation reduces to the more familiar expression

dE d = (Ys COS ) Vopen,

which is simply the work rate done by the surface tension as the grain boundary opens up.

8. Global equations and time integration

The functional H7* has been discretized step by step in the above sections, which can be combined to give

- = [U] [A][U] + [F][U], (8.1)

in which: [U] is the global vector of unknowns, consisting of the velocities of individual grains (two translational and one rotational for each grain), nodal migration velocities of grain boundaries and free surfaces, diffusive fluxes across the mid-points of surface and grain-boundary diffusion elements, and Lagrange multipliers at every nodes of the surface and grain-boundary diffusion elements; [A] is the generalized global viscosity matrix which is assembled from [As] of equation (4.8), [A*] of equation (4.12), [Kin] of equation (5.2), [Km] of equation (5.5), and [Agb] of equation (7.7); and [F] is the generalized global force vector which is assembled from

[Fs] of equation (4.9), [Fs*] of equation (4.13), [Fm] of equation (5.3), [F*] of equation (5.6), the contribution from equation (7.9), and the contribution from external forces.

Taking the variation 67* - 0 then gives

[A][U] = -[F]. (8.2)

Equation (8.2) can be solved using standard numerical matrix methods. A time integration scheme is then required to update the position of the grain boundaries and the free surfaces. In the examples described in the following section it proves appropriate to use a direct Euler method for its simplicity.

9. Examples

(a) Non-dimensionalization

It is convenient to present the numerical results in a non-dimensionalized form. First we define a characteristic 'strain rate', es, as

s= d (9.1)

where d is a characteristic length scale of the problem. We can then adopt the following non-dimensionalizations:

__ S __ - i _ *Js jgb S=- t=t ?s, , = -d, js- sd2', gb -d2,

Vs Vgb Vm - Ygb rd

d 9gb gVm d I7gb - , r -, (9.2)

5sd Vgb d = E= -5=sd' g %s M - Mgb - Mmd2

Mgb-Proc. - So . A


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Finally, the kinetic laws take the non-dimensionalized form A0 - - -

Js = , gb = Mgb a Vm = Mm'ngbIgb. (9.3)

(b) Elliptical cavity: surface diffusion alone

The finite element formulation for surface diffusion can be validated by checking the numerical results against the analytical expressions for the initial velocity of surface migration, flux and chemical potential of an elliptical cavity. An elliptical cavity can be represented in the following parametric form:

x = asin0, y = bcos0.

In the following discussion, we identify b as the characteristic length scale of the problem, then

a = - sin 0, y = cos 0. (9.4)

At t = 0, the curvature of the ellipse can be calculated as

_ xy-yx = + (9.5) (s- 2 + y2)3/2 '

where a dot represents d/dO. The surface diffusive flux and migration velocity can be calculated as

_J O~s O~- o 3 ai =O = as' a (9.6)

and

- _s a a_ 0 g ao a a0s 90 9as _Vs = (9_ 7) s 9s = 90

0, = 90 90 09

' '

in which

0- -[(b) cos20 + sin20]. (9.8)

For the free surface, the Lagrange multiplier is simply For the free surface, the Lagrange multiplier is simply

As --S. (9.9)

The actual expressions of As, js and vs are not presented here. Figures 9a-c show the comparisons between equations (9.6), (9.7) and (9.9) and results obtained from the finite element analysis when 16 elements are used along a quarter of the ellipse. Figures 10a-c show similar comparisons when 30 elements are used. It can be seen that the finite element formulation can provide very good accuracy even with a coarse mesh.

Comparison is also made between the finite element method and the finite difference method described by Pan & Cocks (1995). As surface diffusion proceeds, the elliptical cavity evolves towards a circle. It is found that the two methods give identical cavity profiles at any prescribed time.

For the problem of surface diffusion alone, the finite element method provides no obvious advantage over the finite difference method. It is for the coupled problems that the finite element method offers its distinct advantages.


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Figure 9

6

4

js

2

Coupled grain-boundary and surface diffusion Figure 10

A (a) 6 5

5-

4-

J. 3-

2-

1-

A (b)

. ? ? * ? , I

I , , I, Y,

2177

A (b)

. 11? ? I I . -11 e

60-

40-

20-

0

60-

40 - s -

20 -

0

As A As \ -2.0 - \-2.0 -

0 -3.0> -3.0 , , , , . 0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2

0 0

Figure 9. Comparisons between the finite-element and analytical solutions for: (a) diffusive flux, (b) migration velocity, and (c) Lagrange multiplier along the surface of an elliptical cavity. Sixteen elements are used along a quarter of the cavity in the finite-element analysis: * * , FEM analysis for (a)-(c). --, equation (9.6) for (a); (9.7) for (b); and (9.9) for (c). Figure 10. Comparisons between the finite-element and analytical solutions for: (a) diffusive flux, (b) migration velocity, and (c) Lagrange multiplier along the surface of an elliptical cavity. Thirty elements are used along a quarter of the cavity in the finite-element analysis: * *, FEM analysis for (a)-(c). --, equation (9.6) for (a); (9.7) for (b); and (9.9) for (c).

(c) Grain-boundary grooving: coupling between surface diffusion, grain boundary and grain-boundary diffusion

To demonstrate the finite element formulation for the coupling between surface diffusion and a grain boundary, we consider the problem of thermal grooving of a free surface at a grain boundary controlled by surface diffusion (i.e. no diffusion is allowed along the grain boundary). Assuming that the slope of the thermal groove is small, the problem was solved by Mullins (1957). Here the same problem is solved using the finite element method (FEM) without the small slope assumption. The comparison between Mullins solution and the FEM solution is given in figure 11 at


vs




?* , * * t ?** - Finite element solution showing nodal positions

Mullins' solution

4.''. * 'I * " ' ' ."*

* -4 ' *. *-" ' . . '. *i.. '

. *? * - *- *,

b) 41 it III III *t t1 II1 e4 1 .l I 11 11 41 41 i I . 1 III 11 . II 1 , t : It I11 4 ' 41 1 Il

(a)

(b) : .j , : *....'I :, : I 'I 4, .-1I . :41.

. )* '- , " " ' ,' , - ' - - - ' |- ' 4' ' - " - '- f' -

(c)

(d)

Figure 11. Comparison between the finite-element solution and Mullins's solution for grain- boundary grooving controlled by surface diffusion. The dihedral angle used in this example is 60?. Grain-boundary diffusion is 'switched off' by setting Mgb = 0.00001 in the numerical analysis: (a) t= 0; (b) t= 5.643 x 10; (c) t= 2.183 x 103; (d) t= 3.241 x 104.

iZi

h >

V

Figure 12. The geometry analysed by Thouless. A two-dimensional array of equal-sized grains lie on the surface of a substrate. No diffusion can occur along the interface between them. The initial configuration is the one of equilibrium, so the surfaces have a uniform curvature.

several non-dimensionalized times. It can be seen that the two methods give almost identical results at the early stage of grooving and differ slightly at the later stage.

To further demonstrate the FEM for the coupling between surface diffusion and

grain-boundary diffusion, we consider a 'film' problem studied by Thouless (1993). As shown in figure 12, a 'film' consists of two-dimensional grains of uniform length 1, with grain boundaries of height h perpendicular to a substrate. Both grain-boundary diffusion and surface diffusion were considered and a remote stress cr, or strain-rate ?oo, was applied to the system perpendicular to the grain boundaries. No matter transportation was allowed along the film-substrate interface. Unlike Mullins solution




Coupled grain-boundary and surface diffusion 2179

_. ..... ***'.. *.N.-? Finite element solution showing nodal positions

Thouless's solution

(a)

- __

h (a) ) -= 1.951 x102; *(c) = 5.401 X 102; (d) = 1.718 x* '.

. - . * : .-.' - * . . .

profile, as shown by figure 12** . Similar to Mullins solution, a small slope assumption Figure 13. Comparison between the finite element solution and Thouless's solution for the problem described by figure 12. The dihedral angle is 90? in order to satisfy the small-slope assumption made by Thouless. Other parameters used in the examples are MS =- Mgb = 1, h/l = 5, oo/s = 1.0 x 10-4 and the characteristic length scale d is identified as O.lh: (a) t= 0; (b) t= 1.951 x 102; (c) t = 5.401 x 102; (d) t= 1.718 x 103.

for grain-boundary grooving, Thouless's solution was started from the equilibrium profle, as shown by figure 12. Similar to Mullins solution, a small slope assumption was made by Thouless. H ere the same problem is solved using the finite element method. In order to satisfy Thouless's small slope assumption in the s imulation presented here, we assume 'gb = 0, SO that the equilibrium dihedral angle is 90? and the initial top surface is flat. The following geometric and material data are used:

h Mb_ 'Yg O and o- 1.0 x 10-4

I M s 7s The grooving profiles obtained from the FEM analysis are compared with Thou-

less's solution in figure 13 at several non-dimensionalized times. It can be seen that the two methods give almost identical results at the early stage of grooving but differ significantly from each other at the later stage. From figure 13, it can be seen that Thouless's solution predicts a faster grooving than that predicted by the FEM. This is expected, since the small slope assumption made in Thouless's solution leads to an overestimation of the value of curvature. The driving force and therefore the rate for grooving is overestimated by Thouless's solution.

(d) Thermal grooving at a migrating grain boundary: coupling between grain-boundary migration and surface diffusion

The finite element formulation for grain-boundary migration can be verified by comparing the numerically obtained migration velocity with that predicted by equation

Vm = Mm7Ygb^gb





-2.0 -- 8.0 Y

(a) X-'0gb 10 . . : j rsb

(b)

groove depth

(c) ? . . . . ? . v . ? *... .

.

(c) Vm

--.- Finite element solution. Steady state solution by Mullins & Suo.

Figure 14. Thermal grooving at a migrating grain boundary: (a) the initial geometry of the film; (b) the finite-element solution at a particular time; (c) the comparison between the finite-element solution and the steady-state solution by Mullins and Suo. The dihedral angle used in this example is 60?. Other parameters used in this example are given in the text: (a) initial profile and dimensions of the film; (b) FEM solution at t = 4.706 x 10-3; (c) the steady-state profile is reached by the numerical solution at t = 0.524.

for a section of perfectly circular grain boundary for which /gb is simply 1/R, where R is the radius of the arc. It was found that the FEM solution is almost identical to that predicted by the above equation within the machine accuracy even when a coarse FE mesh is used.

To demonstrate the FEM for the coupling between grain-boundary migration and surface diffusion, we consider the problem of thermal grooving at a migrating grain- boundary in a thin film, as shown in figure 14a. The interfaces between the film and the substrate are assumed to have different specific energies on either side of the grain boundary, which are referred to as ysbi and Ysb2, respectively. This difference in interfacial energy drives the grain boundary to migrate toward the part of interface possessing higher energy in order to reduce the total free energy of the system. At the same time, grain-boundary grooves at the free surface. Assuming steady state and small slope for the grooving profile, the problem was solved by Mullins (1958) and Suo (1996), who provided the groove depth, the migration velocity of the grain boundary, the profile of the free surface (by Mullins) and the profile of the grain boundary (by Suo).

Here the same problem is solved using the FEM. We start from the film geometry shown in figure 14a, which is far from the steady state solution. A very small value of Mgb is used in the numerical analysis to 'switch off' grain-boundary diffusion. Two set of material data are used, which are listed in table 1. For the first set of data, Ygb is chosen to be much smaller than ys and the difference between 7sbl and Ysb2 is chosen to be small. A particular value of Mm is chosen such that the steady state solution predicts a groove depth of O.lh.


2180




Table 1. Material data used in the FEM analysis of thermal grooving at a migrating grain boundary and comparisons between the numerical and analytical solutions at the steady state

set 1 set 2

'gb 0.2 1.0 Ysbl 1.1 1.5 sb2 1.0 1.0

Mgb 0.00001 0.00001 Mm 2.7196 10.376 steady state groove depth (analytical) 0.lha 0.2h steady state groove depth (numerical) 0.106h 0.1657h steady state Vm (analytical) 0.296 4.6296 steady state Vm (numerical) 0.285 4.48

a h: the film thickness

This set of data is designed to satisfy the small slope assumption made in the analytical solution so that a good agreement is expected between the FEM solution and the analytical solution at the steady state. It was found that the agreement is very good indeed and the grooving profiles obtained from the two solutions at the steady state are almost identical. As shown in table 1, the FEM solution predicts a value of 0.285 for the steady state migration velocity of the grain boundary, which can be compared with the value of 0.296 predicted by the analytical solution. The numerical solution predicts a steady state groove depth, as defined by figure 14c, of 0.106h which can be compared with 0.lh predicted by the analytical solution; here h is the thickness of the film.

The second set of data listed in table 1 is chosen to violate the small slope assumption and to see the difference between the analytical solution and the numerical solution. A particular value of Mm is chosen such that the analytical solution predicts a groove depth of 0.2h. Figures 14b and 14c show the thermal grooving process obtained from the numerical analysis. The numerical simulation starts from an initial profile with a dihedral angle of 90?. On the other hand, the equilibrium between surface and grain-boundary tensions requires the dihedral angle to be 60?. It is observed that the numerical solution can quickly recover the dihedral angle and then maintain it throughout the process. Initially the grain-boundary grooves at the free surface and migrates at the film-substrate interface without interfering with each other, as shown by figure 14b. As the process proceeds, interaction between grain- boundary grooving and migration occurs and finally the system reaches a steady state, as shown by the doted line in figure 14c. The analytical solution at the steady state is also shown in figure 14c by the solid line. It can be seen that there exists a certain difference between the two solutions and that the analytical solution overes- timates the grooving process for the same reason discussed in the previous section. As shown in table 1, for the second set of data the analytical solution predicts a value of 4.63 for the migration velocity of the grain boundary. This can be compared with the value of 4.48 predicted by the numerical solution. The numerical solution predicts a value of 0.1675h of the groove depth which can be compared with 0.2h predicted by the analytical solution.

It is worthwhile to point out that all the above numerical results are insensitive to


2181




the various controlling parameters; for example, the timestep length, the mesh size and the criteria for re-meshing, etc.

10. Concluding remarks

The numerical examples presented here are for the purpose of verifying the method and the computer program. The finite element method is being used to investigate a wide range of physical problems including the pore/grain-boundary separation process during later stage sintering of ceramic powder compacts, the sintering kinetics of large pores, the mechanism of superplastic deformation and the cavity growth process during creep deformation of engineering alloys. These results will be published in forthcoming papers.

The variational principle and the finite element formulation can be readily extended to include more mechanisms for mass transport, such as evaporation and condensation and electromigration, and can be coupled with other deformation mechanisms such as power law creep, thereby allowing processes involving all these mechanisms to be simulated by computer.

This research is supported by EPSRC Grant GR/K78102 to the University of Surrey which is gratefully acknowledged.

Appendix A. List of generalized viscocity matrices

This appendix contains details of the various matrices presented in the text. All the matrices are symmetric except for [Cgb ].

(a) Surface diffusion For ordinary element,

1 [As]- -

- 23 12 17 13 120 s 120 s

23 13 120 s

1 12 6 s

2ls

For elements joining grain boundary, -23 2 13 23 3 17 3

120 e,xls 120 e,y e, s 120 e,x s 23 2 i3 1n 3

120 e,ys 120 e,ys ~~~~~1 -~23 3

[A] =-- 120 ls

-Ms

Ms -Ms 0 0

0

-tn 12 6 e,x s

6ne,yls

2ls

112

21S

-3 Msne,xls -3Msne,yls

Ms 0

(A1)

- Msnexls -

-4Msne,yls

-M 0

0

(A 2)

(b) Grain-boundary migration For ordinary element,

(A3)


2182

/gb 2 1

[Km] -- - 3~




For element joining free surface,

2 1 1 -

?r>'* /__gb 2 2 2 3 e , 3 e,x3 e,yA [Km] ^ = gb 2e,z2 (A 4) mm 22

_ e,y

(c) Grain-boundary diffusion

[Kgb] = Mgb

2 1gb* ene,g -ne,zell*g -nee,l* ne,x * 0 e,x gb n-e,xele,yb e,e, gb -xne,ygb xe2b 0 0

n2 1 1* l l n2 1* n l* 0 e,y gb gb--nee,l gb -e,yxgb e,y b gb

.llgb + el gb ne,xel b ne,yellb (O b + ee2b) -0-gb

n2 l* exe* n l* 0 e,x gb ne,gb -ne,xe2gb

ne,gb -ne,ye2l*g 0 e,y gb --ne2 gb 0.115 +e21* -0.5/g 0gb

+ e2 gb gb

21gb _

(A5) in which gb = 3gb.

[Cg - _-ne,xlgb -ne,ylgb 0.512b + ellgb ne,xlgb ne,ylgb -(0.5 ellgb) -1

gb ] -ne,xlgb -ne,ylgb -0.51gb + ellgb ne,xlgb ne,ylgb 0.5gb + e2lgb 1 j

(A 6)

References

Cocks, A. C. F. 1989 A finite element description of grain-boundary diffusion processes in ceramic materials. In Applied solid mechanics (ed. I. M. Allison & C. Ruiz), vol. 3. New York: Elsevier.

Cocks, A. C. F. & Gill, S. P. A. 1996 A variational approach to two dimensional grain-growth. I. Theory. Act. Metall. 44, 4765-4775.

Cocks, A. C. F & Pan, J. 1993 Void growth ahead of a dominant crack in a material which deforms by Coble creep. Int. J. Fracture 60, 249-265.

Herring, C. 1951 Surface tension as a motivation for sintering. In The physics of powder metal- lurgy (ed. W. E. Kingston). New York: McGraw-Hill.

Hillert, M. 1965 Acta Metall. 13, 227.

Mullins, W. W. 1957 Theory of thermal grooving. J. Appl. Phys. 28, 333-339.

Mullins, W. W. 1958 The effect of thermal grooving on grain-boundary migration. Act. Metall. 6, 414-427.

Needleman, A. & Rice, J. R. 1980 Plastic creep flow effects in the diffusive cavitation of grain- boundaries. Acta Metall. 28, 1315-1332.

Pan, J. & Cocks, A. C. F 1993a The effect of grain-size on the stress and velocity fields ahead of a crack in a material which deforms by Coble creep. Int. J. Fracture 60, 121-134.

Pan, J. & Cocks, A. C. F. 1993b Numerical simulation of superplastic deformation. Comput. Mater. Sci. 1, 95-109.

Pan, J. & Cocks, A. C. F. 1995 A numerical technique for the analysis of coupled surface and grain-boundary diffusion. Acta Metall. 43, 1395-1406.

Pan, J., Le, H., Kucherenko, S. & Yeomans, J. A. 1997 Effects of powder size distribution on stage one sintering of ceramic compact by solid state diffusion. I. Modelling. Acta. Metall. (Submitted.)

Sun, B., Suo, Z. & Cocks, A. C. F. 1996 A global view of structural evolution, case study: a row of grains. J. Mech. Phys. Solids 44, 559-581.


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Suo, Z. 1996 Motions of microscopic surfaces in materials. Adv. Appl. Mech., 33.

Suo, Z. & Wang, Z. 1994 Diffusive void bifurcation in stressed solid. J. Appl. Phys. 76, 3410- 3421.

Thouless, M. D. 1993 Effects of surface diffusion on the creep of thin films and sintered arrays of particles. Acta Metall. 41, 1057-1064.

Received 17 December 1996; accepted 9 May 1997




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