a phase-field paradigm for grain growth and recrystallization

A Phase-Field Paradigm for Grain Growth and RecrystallizationAuthor(s): Mark T. LuskSource: Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 455, No. 1982 (Feb.8, 1999), pp. 677-700Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/53397 .

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[i THE ROYAL iIU SOCIETY

A phase-field paradigm for grain growth and recrystallization

BY MARK T. LUSK

Division of Engineering, Colorado School of Mines, Golden, CO 80301, USA

Received 12 November 1997; accepted 14 May 1998

A new model is presented for the mesoscale modelling of grain-boundary motion. Grain boundaries are treated as regions of disorder across which orientation parameters change. The framework uses a set of lattice parameters to track grain orientation, and this is done with fewer equations than the more standard approach, which requires one equation for each grain orientation. An asymptotic analysis relates the theory to classical sharp-interface kinetics. A class of equilibrium states is analytically derived. The theory is numerically implemented in one dimension in order to illustrate the existence of stable grain structures. An ad hoc dislocation-substructure energy is also used to demonstrate how the associated driving force causes grain boundaries to move, an essential ingredient in simulating recrystallization.

Keywords: grain-boundary motion; recrystallization; phase-field; order-disorder; mesoscale; substructure

1. Introduction

The grain boundaries within a polycrystalline material can be induced to move provided such motion reduces the overall energy. In contrast to lattice deformation, such processes involve atomic restructuring of the material lattice itself (Humphreys & Hatherly 1995). Viewed from the continuum level, this additional degree of freedom calls for an associated balance principle, which is commonly referred to as a configurational or material force balance (Gurtin 1995; Maugin 1995; Aifantis 1986). Whether the driving force is associated with dislocation substructure, surface curvature, applied stress, or anisotropy, the underlying mechanism is that atoms dissociate from one lattice and attach themselves to a neighbouring structure. The kinetics and kinematics of individual atoms during such rearrangements are greatly affected by the orientation of each grain as well as by the relative orientation of the boundary between the grains. In sharp-interface models of grain-boundary motion, the kinematics of rearrangement are suppressed and only the end states are considered via jump conditions. The kinetics of motion are assumed to depend only on the state of the lattice on either side of the grain boundary. Large-scale numerical imple- mentations of such theories are difficult because of the singularities intrinsic to the model and because of the need to impose supplementary rules for handling multi- grain junctions, nucleation and annihilation of grains, and interaction with material boundaries. A second approach uses one or more order parameters to track the rearrangement of lattices across grain boundaries (Landau & Khalatnikov 1965; Allen & Cahn 1979). The most common strategy taken in these phase-field models is to use one order parameter for each grain orientation. Also referred to as diffuse-interface

Proc. R. Soc. Lond. A (1999) 455, 677-700 ? 1999 The Royal Society Printed in Great Britain 677 ThX Paper



M. T. Lusk

fields, these parameters simply identify the degree to which a given atomic arrangement conforms to one of the candidate lattices. An associated energy function makes it unfavourable for intermediate states to exist except in transition zones, i.e. across grain boundaries. With each of the M grain orientations is associated a well in the M-dimensional energy space. Such regularized models have been successfully used to study a number of different material processes (Fan & Chen 1997a-c; Jou & Lusk

1997). In a recent work, a recrystallization model of this type was placed within a framework that regularizes the notion of a configurational force balance,t and a scaling was provided which was shown to yield the classical sharp-interface kinetics as the interface zone is made increasingly sharp (Gurtin & Lusk 1999). The standard idea of using one order parameter per grain orientation was adopted in that work, and the focus was on a new field used to track dislocation substructure. While straightforward to implement, it has been shown that one must use 30 or more order parameters in order to obtain two-dimensional simulations that are insensitive to a more refined discretization of grain orientation (Fan & Chen 1997a). Inasmuch as each order parameter calls for the solution of a nonlinear partial differential equation, the numerical challenge is rather formidable to consider for three-dimensional simulations.

In the present work, a break is made from the one-to-one relationship between order parameters and grain orientations. This is accomplished by using a single parameter to distinguish the degree of order of the local atomic arrangement and a set of lattice parameters to characterize orientation. Only two wells exist in energy space: one for complete order, and one for complete disorder. Because the orientation of a grain does not affect the lattice energy, this approach might be viewed as more physically appealing than the standard theory, in which a set of orientations, all given the same energy level, are energetically preferred over all other orientations of the lattice. One must then view any intermediate orientation as non-physical within such a theory.T In the new paradigm, the energy function is sensitive to gradients in the lattice parameters so that rearrangements of the grain orientation are most easily accomplished in transition zones, i.e. grain boundaries, where the regularized lattice exhibits the lowest order.

A hybrid scheme has been considered wherein a single, vector-valued order parameter is used (Morin et al. 1995; Lusk 1999). The magnitude of this vector is equivalent to the order parameter presented here, while the direction of the vector represents lattice orientation. The associated energy surface, plotted in order-vector space, features a circular trough of unit radius. However, in order to allow exhibit grain boundaries, a symmetry-breaking undulation is superimposed on the trough, which discretizes lattice orientation. The result is N energy wells for N grain orientations but only a single, vector-valued order parameter. In the new model, lattice orientation does not enter into the energy function at all.

In order to focus attention on the key features of this new theory, only the grain motion itself will be considered. The approach is easily extended to track other fields, such as temperature, mechanical stress, dislocation back stress, and a material- hardening parameter. The theory is presented in an isothermal, two-dimensional

t Configurational force balances are discussed by Fried & Gurtin (1993) and Gurtin (1996). $ See Lusk (1999) for a discussion of regularized lattice objectivity, where such energy functions

are viewed as regularized versions of a function that is, essentially, infinitely dense in preferred well orientations.

Proc. R. Soc. Lond. A (1999)

678




setting. This geometry allows grain orientations to be characterized by a single angle that is, in turn, described by the set of lattice parameters. It is important to point out that the approach can be used for three-dimensional settings as well, and this will be discussed.

A set of evolution equations is derived from basic thermodynamic postulates and is subsequently non-dimensionalized and scaled. The resulting system is subjected to an analysis, where it is shown that the model is asymptotically equivalent to the standard sharp-interface theory of grain-boundary motion. A one-dimensional numerical implementation is then presented in order to illustrate two key features of the model:

(i) there is a stable, characteristic grain-boundary width that the system will evolve to;

(ii) under the influence of an artificial driving force for dislocation substructure, grain boundaries will move so as to reduce the system energy.

Within the one-dimensional setting, an exact analytical equilibrium solution is derived. An example is provided to show that the evolving fields tend towards such equilibria, and this suggests that these equilibrium states exhibit some form of stability.

2. A phase-field model

The new phase-field paradigm idealizes the formation and motion of grain boundaries as being associated with a dynamic balance between opposing concepts of energy minimization:

(i) an ordered lattice of atoms has lower energy than a disordered arrangement;

(ii) gradients in lattice orientation are energetically unfavourable.

A parameter, W, tracks the order of a lattice with =- 1 implying perfect order and p = 0 implying complete disorder. Perfect lattices of differing orientation are distinguished by a set of lattice parameters s = {(s, s2,..., sN}. The evolution of these N + 1 parameters is used to simulate grain-boundary motion. Associated with each of these fields is a system of microforces:

r(") ... microforce associated with p,

(') ... microstress associated with 'p,

ir(st) ... microforce associated with si, i = 1,..., N,

(Si)... microstress associated with si, i = 1,..., N.

Denote by v the outer normal to the boundary, 0P, of an arbitrary material region, P. A tilde ( ) indicates that the associated field has not been non-dimensionalized. The superscript '.' implies the derivative of a quantity with respect to time. The divergence of a field is represented by div while gradient and Laplacian operators are denoted by V and A, respectively. A subscripted variable indicates partial differen- tiation with respect to that quantity, and this is equivalent to the prefix symbol, 0, subscripted with the same variable.


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The N + 1 microforce balances (Fried & Gurtin 1993) below constitute a phase- field regularization of the balance of a single set of configurational forces. They are followed by a dissipation inequality, which is simply an isothermal statement of the Second Law of Thermodynamics. Appearing in this inequality is the stored energy, W, per unit volume of material. The governing equations are

J (O) . vda+ (rP) dv = 0,

j(si) -vda+ ir(si) dv = 0 (2.1)

N

{V / dv Z X si).vs' da+ v u)' da. P i1 p--

The localized versions of these equations can be written as

div(+('P)) + r() = 0,

div(+(Si)) + (si) = 0, dvN N (2.2)

-W - , -

i(si)S + () VP + (Si) Vs ? 0. i=l i=l

Allowing all fields to be functions of (p, g?, VX7, si, Vsi, and by using a Coleman- Noll argument (Coleman & Noll 1963), the following constitutive restrictions can be made:

7r(w) =-W - p(fp)(

r((si) = - p(SiS (2.3)

(Si = Wvs . 3)

Here the same symbol has been used for both field and constitutive function. Note that the kinetic resistance coefficients, /3() and P(si), represent a particular choice of a more general, non-negative definite kinetic resistance tensor. These results allow the microforce balances of equation (2.2) to be written as

(v).=_ -WV + div(Wvv), (

s = M_(S) [-W ci + div(WvTv,)],

where M(si) = 1//(si) is the mobility of the associated lattice parameter. A basic physical assumption is next made that the energy of the system should depend, at most, on the gradient of the lattice parameters, but not on the values of these parameters themselves. This is reasonable, because two perfect grains should have the same energy if they are only rotated versions of one another, and the lattice parameters are intended to describe the lattice orientation. A specific form of internal energy is chosen:

A N W = fof(g) + g(G) + IYI|V2 + h() Vs 2 (2.5) 2 1,7 S,

12r (2-51


680 M. T. Lusk




where fo, q, and W have the same dimensions and where h is dimensionless. The

exchange energy, f, makes it energetically favourable to be in either a state of perfect order (o = 1) or a state of complete disorder (p = 0). The disorder energy, g, represents the difference in energy between perfect crystals and a completely disordered arrangement of atoms. The mixture function, h, characterizes the degree to which some amount of disorder can allow the restructuring of a lattice to occur more easily. These functions must obey the following restrictions:

g(0) = 0, g'(0) = g(1) = 0, 26

h(1) = 0, h"(1) 0, h'((p) < 0 VW.

The rather severe restrictions on h( ) simplify the ensuing asymptotic analysis and cause the grain-boundary structure to be somewhat more compact than a non- singular coefficient function. The governing equations are then:

'(P) = -fof'(p) - g' (p) + AP + h2(7) | " w z=l~ > (2.7)

s = (si) div h(i) ' }

(a) Non-dimensional form of the governing equations

The evolution equations (2.7) are now non-dimensionalized and, in the process, scaled. Let I and v be characteristic length and speed, respectively. The following non-dimensional quantities are also used:

e2 -eq, (2.8) lfo ' fo

___ 2 __ E- 2

=2fo 2fo = e, M(i) = lfo (2.9) 12 f 12 f If 'fo

The evolution equations can now be written in non-dimensional form with differential operators henceforth taken with respect to the non-dimensional time and space variables:

e9? = -6-lf(p) - qg'(p) + e-yAp + eAh() E ( )2

i=1 (2.10)

es = AM(si) div ( Vsi \

As will be shown, this scaling implies that these equations are asymptotically equivalent to a sharp-interface equation of motion by mean curvature. It can also be shown that a different scaling is asymptotically equivalent to the motion of a boundary region of finite thickness, and this will be the subject of a future work.

(b) A specific class of materials

In order to consider numerical examples, a specific class of materials will be chosen. For such applications, take the exchange energy, f, to be the standard double-well


681



M. T. Lusk

potential,

f(pO) = 2 - (p)2, (2.11) and suppose that the disorder energy and mixture function are given by

g(P) = W2(2 _- 1), h(() = 2(1 -o 2). (2.12) Then the governing equations are

N

e* = -e-lf'(p) + 2qs(1 - s) + ey/A - (Ap1 V | 2) - V ( i=- 2) (2.13)

S AM(si) div ( 2

(i) The mobility functions

The mobilities, M(si), are positive functions of the constitutive set {(p, V(, Vsi}, but they are taken to be constants in the initial implementation of this theory. In multi-grain boundary simulations, though, these mobilities play a key role in insulating one grain boundary from another. They are taken to be functions of the lattice order, s, so that the lattice mobility goes to zero as lattice order increases. For values of sp greater than some critical value, the mobilities should be taken to be zero. This makes physical sense, since lattice re-orientation requires that some degree of disorder be present, as is true at a grain boundary but not within the bulk of a perfect grain. This is illustrated in the last numerical application.

3. Correlation to sharp-interface equation

In order to relate the phase-field model to experimentally measurable parameters, a matched asymptotic analysis is used to show how the governing equations collapse to a standard sharp-interface kinetic equation in the limit as the grain-boundary thickness shrinks to zero. Within this simple setting, the sharp-interface equation simply predicts grain-boundary motion driven by surface energy, written,

V = -n, (3.1) where V is the grain-boundary normal velocity and n is the mean curvature of the grain boundary (Gurtin & Lusk 1999). In the present work, the phase-field model does not attempt to capture the dependence of grain-boundary energy on lattice mismatch and boundary orientation (Humphreys & Hatherly 1995), and so this surface energy is found to be a constant. The required asymptotic analysist has been performed on equations (2.13) with a sketch of this work given in Appendix A. The sharp-interface equation (3.1) is recovered in the limit, as e goes to zero. It is important to note that the phase-field model can be easily extended to include the energy associated with dislocation substructures (Gurtin & Lusk 1999). In that case, the asymptotic result gives motion driven by both mean curvature and the jump in dislocation energy across grain boundaries. Such a substructure energy is introduced, in an ad hoc way, in order to illustrate grain-boundary motion in one dimension. This is done in the following section.

t A very elegant, alternative approach for accomplishing such asymptotic analyses is presented in Fried & Gurtin (1996) and Fried (1997), and was applied to a previous work in recrystallization by Gurtin & Lusk (1999).


682




9p

left grain

-1.0

right grain

-0.5 0 0.5

left grain

1.0 x

right grain

motion

motion

x -1.0 -0.5 0 0.5 1.0

Figure 1. The stabilization of a single grain boundary that is initially, from an energetic perspective, too wide. The curves show the evolution of both the order parameter and the lattice parameter to the final (narrowest) state.

4. Numerical implementation in one dimension

The phase-field model of equation (2.10) has been implemented in a one-dimensional numerical algorithm to illustrate three essential features:

(i) grain boundaries have a stable, characteristic width;

(ii) grain boundaries evolve so as to reduce the size of grains with higher substructure energy;

(iii) multiple-lattice parameters can be tracked.

For the sake of convenience, this initial implementation was performed by using Mathematica 3.0 software.

(a) Grain-boundary stabilization

The parameter set,

e = 0.0707, q = 0.1414, y = 0.2, A = 0.6,


(4.1)

683



M. T. Lusk

left grain H, 0.8 right grain

0.6

0.4

motion _ motion

x -1.0 -0.5 0 0.5 1.0

s

1.0-

left grain

motion

motion

0.2

-1.0 -0.5 0 0.5 1.0

Figure 2. The stabilization of a single grain boundary that is initially, from an energetic perspective, too narrow. The curves show the evolution of both the order parameter and the lattice parameter to the final (widest) state.

was used to solve equations (2.13) under the two sets of initial conditions. The first set of conditions represents a single grain boundary that is, from an energetic perspective, too wide. The evolution and stabilization of the order parameter and the single lattice parameter are shown in figure 1. The same calculation was then re-done with initial conditions that would cause the grain boundary to widen with time. The results, shown in figure 2, show that the grain boundary does indeed widen to an equilibrium width. Both calculations were run until the total energy of the system stabilized. The maximum (dimensionless) time in the results shown is 25 s, but calculations were also performed out to 50 000 s without any observable changes in the final state noted. Simulations were also performed with increasingly fine discretization to ensure that the solutions were converging as discretization density was increased. Sig- nificantly, the final states for both the wide and narrow initial-state calculations are, within numerical accuracy, identical. These states are shown in figure 3, and this


684




9p

1.0 -

0.8 -

0.6

0.4

x -1.0 -0.5 0 0.5 1.0

Figure 3. The two initial and two final states associated with the stabilization of grain boundaries that are too wide and too narrow. Note that the final states, shown in bold, cannot be distinguished from each other.

suggests the existence of a stable grain-boundary width. The existence of a small but finite gradient in the lattice fields away from grain boundaries is addressed and corrected at the end of ? 6.

(b) Motion of a grain boundary driven by substructure energy

To be a reasonable model for studying recrystallization, grain boundaries should, in the absence of surface effects, move so as to eliminate grains of higher substructure energy. In an earlier work (Gurtin & Lusk 1999), a scalar field was used to explicitly account for dislocation substructure, but this equation has been suppressed in the present work for the sake of clarity. However, it has been shown that the effect of substructure can be approximately accounted for by simply adjusting the height of the wells in the exchange energy (H.-J. Jou, personal communication). Such an approach is taken here in order to demonstrate that the desired grain-boundary motion is captured. The parameter set,

e = 0.05, q = 2.0, 7 = 0.2, A = 0.1334, (4.2)

was used to solve equations (2.13) under the two sets of initial conditions. The simulation was carried out to 20 (non-dimensional) seconds, the results are shown in figure 4. Note that the edges of the grain boundary move at the same speed once the boundary has stabilized. This is consistent with the results of the asymptotic analysis given in Appendix A.

(c) Stabilization and motion of a system of lattice parameters

The theory presented allows for several lattice parameters to be tracked by using only two energy wells. For the sake of clarity, only one lattice parameter has been considered. In order to demonstrate the utility of this model, though, a simple numerical


685



M. T. Lusk

9,

left grain

-1.0 -0.5 0

-0.5

-1.0

Figure 4. A single grain boundary moves in response to higher substructure energy in the grain on the right. The substructure energy is modelled by letting the well heights in the exchange energy be functions of position.

implementation has been accomplished by using two lattice parameters along with the single-order parameter.

The results are shown in figures 5 and 6. To generate figure 5, parameter values of

= 0.0707, q = 0.1414, y = 0.2, A = 0.1571, (4.3)

were used to solve equations (2.13). The simulation was carried out for 50 (non- dimensional) seconds. To generate figure 6, the parameters

e 0.05, q 2.0, y = 0.2, A 0.1334, (4.4)

were used to solve equations (2.13). Again, the simulation was carried out for 50 (non-dimensional) seconds. These two figures are only intended to demonstrate that any number of lattice parameters can be tracked by using the same methodology as was developed for one lattice parameter. As has already been mentioned, the reason


right grain

x

s

right grain

x

686




9

motion

-1.0 -0.5

motion

0 0.5

1.0

0.8

0.6 :- motion

motion

-1.0 -0.5

Figure 5. A single grain boundary narrows because it is energetically favourable to do so. Here two lattice parameters are used but any number of parameters are possible using the same double-well exchange energy.


right grain

x

1

right grain

left grain

left grain

left grain

x 1.0

right grain

0.5 x

1.0

687



M. T. Lusk

left grain

left grain

left grain

right grain

motion

x

Si

right grain

motion

x

S2

1.0-

right grain

x

-1.0 -0.5 0 0.5 1.0

Figure 6. A single grain boundary moves in response to higher substructure energy in the grain on the right. The substructure energy is modelled by letting the well heights in the exchange energy be functions of position. Here two lattice parameters are used but any number of parameters are possible using the same double-well exchange energy.


688




for the small gradient in the lattice fields away from grain boundaries is addressed and corrected at the end of ? 6.

5. Lattice parameters and grain orientation

The two-well paradigm allows any number of lattice parameters to be tracked, and these fields may take on any value away from grain boundaries. This last point is

especially significant since previous phase-field models necessarily tie values found away from grain boundaries to energy minima. For instance, a single lattice parameter may be taken to be the lattice orientation angle relative to some fixed reference frame. This angle may take on any value, so that lattice orientations are not limited to a set of discrete values and there are only two equations that need be solved in such simulations. However, the penalty on lattice parameter gradients implies that grain-boundary width is approximately proportional to the mismatch in orientation angle. This implies that the ratio of maximum to minimum mismatch angles is also, approximately, the ratio of maximum to minimum grain-boundary widths. The number of equations has been drastically reduced, but the new system would result in a

dynamic range problem that would have to be addressed in determining the appropriate domain size and discretization. This is also a concern in the model proposed by Morin et al. (1995) and Lusk (1999).

From a physical perspective, the orientation of a lattice is a feature associated with a number of changes in the lattice structure. Tracking lattice parameters is therefore a more reasonable proposition than tracking the lattice orientation explicitly, and the dynamic range issue is only significant in this latter case. One simplistic relation that uses N lattice parameters to describe 2N twist boundary orientations, 0, is

N

0 - 2max s 2i_-1 (5.1) i=1

Here the lattice parameters, si, are allowed to range between 0 and 1, so this for- mula amounts to an N-bit binary description of the lattice orientation. For example, only six evolution equations are required for tracking 32 grain-boundary orientations, and this is, approximately, of the order of the number of orientations previously identified as appropriate for two-dimensional simulations (Fan & Chen 1997b). The difference in strategies is made clear by noting that, in the old approach, each grain orientation is described by a set of order parameters that are all equal to zero, except for one parameter. Such an M-bit binary description therefore requires M order parameters because every binary word contains only one non-zero digit. In the new model, every digit in the binary descriptor can be used to characterize an angle, so N parameters can be used to differentiate 2N orientations. Although the relationship used here is non-physical, it serves to illustrate the idea that a finite set of lattice parameters can be used to deliver the orientation of grains.

With the N-bit binary description of orientation angle, the ratio of maximum to minimum grain-boundary widths is only N1/2. For example, for the 32 orientations suggested for two-dimensional applications, the ratio of widths is, approximately, a factor of 2.5. This does not seem to represent a significant computational issue.


689



M. T. Lusk

feff ()

increasing

I ?

a b c 1

Figure 7. A graphical interpretation of the effective exchange energy and the parameters used in obtaining an elliptic integral solution, equation (6.6), to the equilibrium equations (6.1). Note that the key features of the energy function have been grossly exaggerated in order to better illustrate the relevant ideas. As a grain boundary narrows, the parameter, C, increases, and this raises the second well in the effective exchange energy.

6. Equilibrium states

The system of equations presented in this work allows for non-trivial equilibria with ordered lattices on either side of a grain boundary. In contrast, the evolution of two inwardly moving fronts associated with the generalized Ginzburg-Landau equation,

? = -f'(p) + 2qo(1 - ) + 7AW,

will always result in the pair annihilation of those fronts and a resulting trivial state of p = 1 everywhere (for q > 0). In the present work, equilibria for the system of equations (2.13) are identified for the case of one spatial dimension by finding non-trivial solutions to

0= -e-'f'() + 2qp(1 - o) + e-Y/xxP - _ 12)20s' O9s,

~(1- ^~2)2Ox j (6.1)

0Oax 2

where f(o) is given by equation (2.11). Only one lattice parameter is considered, since the result can be immediately extended to an arbitrary number of such variables. The problem is addressed on the finite domain, -L < x < L, with boundary conditions


690




of

c9(-L) = 9 lb -1, p(+L) = b 1, (6.2) s(-L) = Sbl, s(+L) Sb2, Sbl, Sb2 E {0, 1}.

Substitution of equation (6.1)2 into equation (6.1)i then gives 0 = -e-lf'() - qg'(p) + AC2h'(p) + eaOxxqp

= -6-f'() + 2q((1 - 9W) - 4AC2? + ey/,xx9, (6.3) where the constant, C, must satisfy

L Ce-1/2 h(Wp(x)) dx = (Sb2 - Sbl) (6.4)

-L

An effective exchange energy is next introduced,

feff((p) = f(p) + eqg(p) - eAC2h(o), = 12P[(1 - p)2 + eq(4p - 2) + e4AC2] - e2AC2, (6.5)

so that multiplication of equation (6.3) by 9Ox and double integration from x = 0 to an arbitrary x < L then gives

r '(X) do x - =??E (6.6) V2 (o0) V/feff() - feff(f(0)) The term in the radical is fourth order in q, and such integrals can always be reduced to the standard Legendre form by using Cayley's method (Bowman 1961). That is the approach, for instance, used by Novick-Cohen & Segel to study equilibria associated with the Cahn-Hilliard equation (Novick-Cohen & Segel 1984). In the current problem, several equilibria can be found by this method depending on the types of roots assumed for the fourth-order term, and these solutions are typically in terms of elliptic functions. The solution matching that of the numerical simulations, however, is found to have a very simple representation. Assume that equation (6.6) is taken to be equivalent to

x- Vf~(x) ) do 2

1(0) v/feff () - feff(q(0))

[e x f(() do5 (6.7)

=?''J (q{f - c)x/(0 - a)( - b)' The roots of the expression in the denominator are such that a < b < c, and they have the graphical interpretation shown in figure 7. The integral can then be evaluated, and the resulting equality can be solved for Sp as a function of position. The solution is

(x) - c(a - b) + 2[2ab - c(a + b)] exp(x/eV/y) + c(a - b) exp(2x/e/-y) a - b + 2(a +b - 2c) exp(x/ev/y) + (a - b) exp(2x//7) '

x 0,

(x) c(a - b) + 2[2ab - c(a + b)] exp(-x/ev-y) + c(a - b) exp(-2x/ez/y) a - b + 2(a + b - 2c)exp(-x/ev/y) + (a - b) exp(-2x/e/ty)

x < 0.

(6.8)


691



M. T. Lusk

I.U

-1.0 -0.75 -0.5 -0.25 0

s

1.0

0.8

0.4

0.2

-1.0 -0.75 -0.5 -0.25 0 0.25 0.5 0.75

Figure 8. Comparison of a long time state obtained numerically with the equilibrium solution of equations (6.8) and (6.9). The numerical solutions are those shown in the previous figures for grain-boundary stabilization.

The lattice parameter solution is then given by x

s(x) = 2CE -1/2 [1 _- i2(y)] dy + Sbl -L

(6.9)

Note that the roots, a, b, and c, are in terms of the parameter, C, which must be determined from (6.4) and is approximated, in Appendix A, by equation (A 15). The relative sizes of the parameters in equation (6.8) imply that the grain-boundary width is of order +/c, and this is also proved in the asymptotic analysis.

In order to compare the equilibrium solution with the end states of the numerical simulations, the following parameters were used:

= 0.0707, q = 0.1414, y = 0.2, A = 0.6, (6.10)

p(0) = 1.96932 x 10-4 L = 1, 9b =0.9933, Sbl = 1.0, Sb2 = 0.0. (6.11)

These parameters are the same ones used for the numerical implementation to illustrate grain-boundary stabilizations and gave a numerical value of e-1/2C = -0.74784. This compares well with the value of e-1/2C = -0.745 predicted by


0.25 0.5 0.75 1.0

1.( , .. , . I . , . . -1 . . . . . ? m I 7

692

1 t1

0.8 -

0.6

0.4

0.2 -

v .0

3O x

. / ~~~~~~~~~~~~~




x -1.0 -0.5 0 0.5 1.0

Figure 9. The mixture function, h, is modified so that it has a zero at <0 = c, where c is the largest root in the denominator expression developed as part of the equilibrium solution. This modification removes the small but finite slope in the lattice parameter profile away from grain boundaries. The solution shown should be compared with the equilibrium states of figures 1, 2 and 8.

equation (A 15) of the asymptotic analysis. All parameters were inserted into equation (6.8), with the equilibrium grain-boundary shape plotted in figure 8. Also re- plotted in that figure are the final states shown in figure 3. Clearly, the analytical solution and the two numerical solutions are, for all practical purposes, equivalent. This result supports the conjecture that the solution given by equation (6.8) is a locally, linearly stable equilibrium state associated with the grain boundary. A detailed con- sideration of the local, linearized stability of both types of equilibria will be the subject of a future work with E. van Vleck.

One of the reasons that the equilibrium issue is so important is that the parameters in the evolution equations can be chosen so that grain boundaries narrow until they collapse. The well-defined grain-boundary region is then replaced by a gradual change in lattice orientation distributed throughout the bulk. Inasmuch as this is physically unreasonable, parameter values should be chosen so that such collapses do not occur. The existence of an analytical expression for stable grain-boundary profiles allows this to be accomplished in a straightforward manner.

Both the numerical and analytical solutions exhibit a small but finite slope associated with the lattice parameter at the ends of the domain, and this sort of behaviour is also observable in the transient solutions previously shown. This slope results because the order parameter does not actually approach a unit value but, rather, the value of the double root, p = c. The root is a well-defined function of the system parameters, so that the mixture function, h, can be modified to have a zero at W = c rather than at Wp = 1. This results in both the order and lattice parameters having a slope approaching zero at the domain boundaries. The change in mixture function was used to obtain the numerical equilibrium state shown in figure 9 using the same parameters as those used to produce figures 1, 2 and 8. Note that the finite slope is not present in the modified model. With such an improvement, the equilibrium solution on an infinite domain is just a special case of the finite-domain solution already presented, and the small gradients in lattice parameter(s) observed in previous results will no longer be present.


693



M. T. Lusk

1.0

0.8

C 0.6

8 0.4

0.2

0 0.2 0.4 0.6 0.8 lattice, (p

Figure 10. The lattice-mobility function, M(), plotted above, is used with Neumann boundary conditions to show that grain structure is preserved even when Dirichlet boundary conditions are not imposed.

9' s _ ̂ Q final final .o

0.8. initial

0.4 -0.4

F -gue1.~~~in0 -0.5 0 0.5 1.0 -1.0 -0.5 0 0.5 1.0ia -i.0 -0.5 0 '0.5 ....11.0 -0.5 0 0.5 1.0

Figure 11. These plots of the order parameter and lattice function were generated by using Neumann boundary conditions along with the mobility function of the previous figure. The single grain boundary equilibrates to a non-trivial structure. Distant grain boundaries are, in this manner, insulated from one another through the lattice mobility.

7. The role of lattice mobility

As has already been discussed, the lattice-mobility functions, M(S'), play a crucial role in insulating distant grain boundaries from one another. In the application examples considered thus far, Dirichlet (fixed-value) boundary conditions were imposed, only single grains were considered, and the lattice mobility was assumed to be constant. In general, though, the mobilities should be zero where the lattice has perfect order (qp = 1), and this is illustrated in a simple numerical example. The equilibration of a single grain boundary is considered, but Neumann boundary conditions (zero gradient) are imposed for both 9p and a single lattice parameter, s. The mobility function used is shown in figure 10 and the simulation results are given in figure 11. As predicted, the boundary values of both qp and s equilibrate to non-trivial values because of the lattice-mobility function. The formulation can be immediately applied to large-scale simulations of polycrystalline systems.


694




internal lattice , transition, grain transition. lattice

I I I

- s I

expansion is I Ireg I I I

I:' % I

I I I l \ I l / I

Ix I

Figure 12. Key regions associated with the asymptotic analysis are shown. An outer expansion is performed on each of the lattice regions as well as in the internal grain region An inner ex pansion is performed in both of the transition regions.

8. Discussion

A new phase-field paradigm has been presented that can be used to model recrystal- liza tion and grain growth. The number of grain orientations that can be considered

is not related to the number of energy minima; in fact, only two energy wells are employed. A matched asymptotic analysis shows that, under the right scaling, the phase-field model reduces to motion by mean curvature. A different parameter scaling asymptotically correlates to a kinetic equation for the motion of a boundary of finite width. An analytical solution has been found for equilibrium states, and numerical simulations suggest that these states are locally stable. Dislocation substructure energy is introduced in an ad hoc way in order to demonstrate that mismatches in such bulk energy functions induce grain-boundary motion. This idea will be for- malized in an ongoing effort to simulate recrystallization processes. Grain-boundary annihilation is captured inherently in this paradigm, since grain boundaries that come sufficiently close together will find it energetically favourable to form a single boundary. This also applies to a single grain shrinking to reduce surface energy within a larger grain. Its closed grain boundary will eventually self-annihilate, leaving only the larger grain.

The mobility of lattice parameters is used to insulate distant grain boundaries from one another by making it impossible for the grain orientation to change unless there is some disorder in the lattice.

The lattice parameters are used to deliver the lattice orientation, and a simple binary scheme is provided to illustrate this. The scheme allows 2N orientations to be described with N + 1 evolution equations. This is an improvement over the more standard modelling paradigm, which uses M equations to track M orientations. The model can be extended to three dimensions by using the same binary characteri- zation of lattice orientation along with any coordinate description that allows the orientation to be discretized. More physical relationships between lattice parameters and orientation are needed and will be investigated. A goal of such future work will be to completely decouple the number of lattice parameters used from the number of allowed grain orientations.


695



M. T. Lusk

A clear deficiency of this new model is that grain-boundary energies and mobilities are not functions of misorientation, and this will be generalized in a future work. Also, the number of equations is not an appropriate metric for assessing the efficiency of a numerical scheme, and direct numerical comparisons with the standard phase-field approach obviously need to be performed. A two-dimensional implementation of the new model has been undertaken for this purpose. That numerical algorithm will be subsequently extended to account for dislocation substructure and, eventually, inelastic deformation.

I am indebted to several individuals who have, through discussions, significantly aided the development of this work. I acknowledge, first, many fruitful interactions with G. Stiehl, G. Grach, and E. van Vleck throughout this work. D. Fan, C. Carter, L. Holm, D. Hughes, J. Cahn and R. Kobayashi have also provided a positive influence for which I am very grateful. This work was supported, in part, by the US Department of Energy under contract no. DE-AC04- 94AL85000 and by National Science Foundation Career Award no. CMS-9502409.

Appendix A. Asymptotic analysis of the phase-field model

A matched asymptotic expansion is performed on the phase-field equations (2.10) to show that, as the non-dimensional parameter, e, tends to zero, the sharp-interface equation (3.1) is recovered. The analysis is carried out for a single lattice parameter, s, and the results are then extended to N lattice parameters. The relevant equations are

e = -e-lf1'((o) - qg'(P) + efyA + EAh (o) 2, IV 1 * h(7p)2 (Al)

S = e- 1AM(s) div h(s).

As shown in figure 12, it is assumed that the order and lattice fields can be developed in parts:

(i) lattice (outer) solution at distances far from grain boundaries;

(ii) internal (outer) grain solution within grain boundaries;

(iii) transition (inner) solution associated with the edges of grain boundaries.

(a) Outer expansion

An outer expansion is used to consider both the lattice and internal grain regions. Let

w = WO + E(P+ E 2W23+ 6 3 3??

p=Oo+eO?e(p2+C23+/2 , (A2) s = so + e1/2s + 2 + 6 3 + }

(A 2)

Multiplying both sides of equation (A 1)1 by h2(o) gives the leading order (e-1) perturbation equation of

0 = h2((o)f'(o).

This has solutions of po = 0, 1. Suppose that To = 1 is assigned to lattice regions (away from the grain boundary), implying that the lattice order is perfect, to first


696




order, in such areas. Then the next-order (e?) perturbation equation from equation (Al) is

IVs0o2 = 0. (A 3)

Here the characteristics of h(o) in equation (2.6) have been used. Away from grain boundaries, therefore, the lattice parameter must be constant to first order. The third-order (e1) equation is trivially satisfied, and the fourth-order (e2) equation gives a relationship between the first-order terms for order and lattice parameters:

- [IVS112 1/3 (A4)

The same outer expansion is next used to consider the internal grain-boundary regions. Suppose that So0 = 0 within the grain-boundary zone. The order (e?) equation for the lattice parameter is then, simply,

0 =Aso. (A 5)

In a one-dimensional setting, the lattice parameter may therefore vary linearly across grain boundaries.

(b) Inner expansion

An inner expansion is carried out for the transition zones on either edge of a grain boundary. Let

C = (pO + 6^1 + e2 2 + eaC3 + ,

S = So+ 8 + 1/2l + E2 + e3/23 + .

Further, a local coordinate system is adopted with coordinates r and s associated with unit vectors r and s, which are normal and tangential to the interface zone, respectively, i.e. r and VWo are in the same direction and r s = 0. Finally, introducing the stretched variable, z = re-1, the gradient and Laplacian operators can be written as (Jou et al. 1998)

Vr7 =l-riA +( l + ) st3Srl

1 ^ / 1

/', - o,,0 + -9+ a e e

e

1 + eZZ ( + eZ)3( + 1

Ze)2

The time derivative of the order parameter can also be written in a local, expanded form:

-V (? = --09Zo + O(1), (A 6)

where V = -&tr is, to first order, the interface normal velocity already seen in equation (3.1). Then the leading-order (e?) equation obtained from equation (2.10)2 is

aZ h(7o)^ )=0 (A 7)


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M. T. Lusk

so that

_^ = 4C, (A8) h(^o)

where C is a constant. This constant is assumed to be of order one in the inner analysis with the following justification. If the problem geometry is one-dimensional, then equation (A 1)2 implies that

a -s e-/2C (A9)

with C not a function of position. By equations (A 1)2, (A 2), (A 4), this term must be of order one. Equation (A 8) then follows from the inner solution rescaling. For higher-dimensional problems, it is reasonable to assume that the magnitude of the term

lVsl

h(W)

is still of the same order, since the most significant changes in the associated fields should occur in the direction normal to the (local) grain boundary.

The leading-order equations (e-1, e?) from (2.10)1 are

0 - -f'(Po) + yaZZ?o,

-VOz io = - f" (o)Si + Y i + O9zzl - qg'(bo) + "Y&OzO + Ah' (2A 10)

To develop a kinetic relation from equations (A 10), introduce the linear operator,

L(0o)[01] '- -f"(%o)0i, + 7y0zz =I -Vz9o + qg'(0o) - 7TrZ9zo - Ah'(o)C2. (All)

The extreme right-hand side of this expression can be viewed as an eigenvalue of the linear operator, L, with an associated eigenfunction of @1. Likewise,

L( o)[Oz o]= 0

by equation (A 10)1, so that the eigenvalue associated with an eigenfunction of 0z,o is 0. The standard orthogonality condition for test functions of a linear operator (Courant & Hilbert 1953) then implies that

r+00

/ (O,z o)[-V z o + qg'(o) - y0o - Ah'( o)C2] dz = 0. (A 12) -00

As is common practice in such asymptotic analyses, define the conversion modulus as

:=J (z, o)2 dz= -

v/f() do. (A 13) -oo V Jo

Carrying out the integration of equation (A 12), then matching the inner and outer solutions implies that, to 0(e),

rVL = -qg(l) - Ah(0)C2 - a, left,

rVR = qg(l) + Ah(0)C2 - ar, right.


698




Here the surface energy, a, is given by

a = Fy,

and VL and VR are the velocities of the left and right edges of the grain boundary, respectively.

If VL > VR, then equations (A 14) and (A 9) imply that drslI < V/-qg(1)h(O)/eA in the middle region but, by construction, this slope must be increasing. Conversely, if VL < VR, then 0rsl > V/-qg(l)h(O)/eA in the middle region but, by construction, this slope must be decreasing. The evolving grain boundary therefore seeks a morphology for which the edges move at the same rate. In this case,

C - = (1) (A 15)

and the single kinetic equation for grain-boundary motion is, to leading order,

rV = -UK.

In the limit, the, grain boundaries simply move in accordance with a mean curvature equation. The relationship between the parameter, C, and the slope of the lattice parameter implies that the stable grain-boundary width is of order V/e. For the scaling chosen, the sharp-interface kinetic equation is not dependent on the lattice parameter, s. The jump in the lattice parameter across grain boundaries is provided by equation (6.4):

+00

[s] -1-/2C h('o(z)) dz + 0(e). (A 16) -00

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