Phase Field Equations with Memory: The Hyperbolic CaseAuthor(s): Horacio G. Rotstein, Simon Brandon, Amy Novick-Cohen, AlexanderNepomnyashchySource: SIAM Journal on Applied Mathematics, Vol. 62, No. 1 (May - Sep., 2001), pp. 264-282Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/3061906Accessed: 05/03/2010 12:15

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SIAM J. APPL. MATH. ? 2001 Society for Industrial and Applied Mathematics Vol. 62, No. 1, pp. 264-282

PHASE FIELD EQUATIONS WITH MEMORY: THE HYPERBOLIC CASE*

HORACIO G. ROTSTEINt, SIMON BRANDONt, AMY NOVICK-COHEN?, AND

ALEXANDER NEPOMNYASHCHY?

In memory of Chaim Charach (1949-1999) Abstract. We present a phenomenological theory for phase transition dynamics with memory

which yields a hyperbolic generalization of the classical phase field model when the relaxation kernels are assumed to be exponential. Thereafter, we focus on the implications of our theory in the hyper- bolic case, and we derive asymptotically an equation of motion in two dimensions for the interface between two different phases. This equation can be considered as a hyperbolic generalization of the classical flow by mean curvature equation, as well as a generalization of the Born-Infeld equation. We use a crystalline algorithm to study the motion of closed curves for our hyperbolic generalization of flow by mean curvature and present some numerical results which indicate that a certain type of two-dimensional damped oscillation may occur.

Key words. phase field equations, phase transitions, memory effects, mean curvature flow, crystalline algorithms

AMS subject classifications. 80A22, 74D10, 45K05, 35L70, 35R35, 65M60

PII. S0036139900369102

1. Introduction. In this paper we present a phenomenological theory for phase transitions with memory which arises by considering "memory" or delayed response of the system to thermal gradients and forces driving phase change. For simplicity and to gain intuition into the implications of the theory, we focus afterwards on the case in which the relaxation kernels are assumed to be exponential. With this assumption, our theory for phase transitions with memory reduces to a theory for hyperbolic phase transitions. Our theory allows us to describe the evolution of an interface between two phases in terms of its geometric and kinematic properties and to study its motion numerically. One feature of hyperbolic phase transitions is the possible appearance of interfacial oscillations, a phenomenon which has been previously modeled in 1991 by Gurtin and Podio-Guidugli, who in their efforts to explain experimentally observed oscillations [1, 2] developed a mechanical theory for the evolution of a two-dimensional interface possessing an effective inertia [3]. For isotropic materials in the absence of a bulk driving force the following equation was obtained:

(1) Vt + -V =K,

where v is the normal velocity of the interface, vt is the time derivative of v in the direction normal to the interface, and n represents the curvature of the interface.

*Received by the editors March 3, 2000; accepted for publication (in revised form) January 18, 2001; published electronically August 22, 2001. The third author wishes to acknowledge the support of the Technion V.P.R. Fund-Loewengart Research Fund and the Israel Science Foundation (grant 331/99).

http://www.siam.org/journals/siap/62-1 /36910.html tFaculty of Mathematics, Technion-IIT, Haifa 32000, Israel. Current address: Brandeis

University, Volen Center for Complex Systems, MS 015, Waltham, MA 02454-9110 (horacio@ cs.brandeis.edu).

tFaculty of Chemical Engineering, Technion-IIT, Haifa 32000, Israel (cersbsb@tx.technion.ac.il). ? Faculty of Mathematics, Technion-IIT, Haifa 32000, Israel (amync@tx.technion.ac.il,

nepom@tx.technion.ac.il).

264

PHASE FIELD EQUATIONS WITH MEMORY

Equation (1) is a hyperbolic generalization of the classical flow by mean curvature equation, v = n,, to which it reduces upon rescaling t' = t/y/ and taking the limit y -- oc, as can be seen by writing (1) in local coordinates.

Binder, Frisch, and Jackle [4] developed in 1986 a theory which describes phase transition in the presence of slowly relaxing internal variables. Among the systems and processes which they thought to be relevant in this context were (i) slow struc- tural relaxation in a viscous solvent such as a polymer melt with a miscibility gap in the proximity of the glass transition temperature; (ii) the strain fields associated with the lattice constant misfit of the two constituents in phase separating solid metallic alloys, which interacts with a dislocation network which relaxes to minimize energy; (iii) the vacancy concentration, which is not conserved because of vacancy/interstitial annihilation and creation events, and which affects the effective mobility and hence the phase separation process; and (iv) clusters of impurities already segregated out from the mixture, which also might have a tendency to relax after quench. In their work, rather than attempting to give an explicit description of the nature and dynamics of these slow variables, they are all treated by the theory on a common phenomenolog- ical level by considering dynamics based on a chemical potential which incorporates relaxational response. The approach which they develop can be seen as taking into account memory effects.

In the present paper, we adhere to the rationale of their approach, but adopt a more systematic form of relaxation, and propose the following phase field equations with memory:

(~2) | Ut + A (t = ai(t - r) Au(r)dT, l 3 5t=f {t-a2(t- ) [ - 32 + f +U ]}(f)dT

on a bounded region Q C R2 with a smooth boundary, with Dirichlet and Neumann boundary conditions for u and 0, respectively, where u represents a (dimensionless) temperature field and q$ an order parameter. In (2), A, v, ,3, and r1 are positive parameters, and f((0) is a real odd function with a positive maximum at 0*, a negative minimum at -?*, and precisely three roots in the closed interval [-a, a] located at 0 and ?a, for some positive constant a.1 The kernels a1 and a2 should be taken to be nonnegative and sufficiently well-behaved especially near 0 in order to guarantee existence and uniqueness in an appropriate framework [5]. Notice that when ai(t) = 6(t), system (2) reduces to the classical phase field model

() f Ut + A\t = Au,

l/320t = /2A, + f( + u.

To begin to clarify the implications of (2), we consider a particular case of (2), namely the following scaled hyperbolic phase field model:

(4) Utt + e2 Ott + 71 Ut + E2 y71 t = a Au, ( )l 62^c2 Ott +272 qt =62A +f()+ EU,

where -Y1, 7Y2, and a are nonnegative constants and 0 < e << 1. This system is obtainable from (2) by introducing the scaling A = 2, /3 = e1/2, and D/ = e, and

choosing the kernels as ai(t) = aa2e-7f 12t and a2(t) = voa2e-7202 t, then rescaling

1For the sake of simplicity and without loss of generality we take a = 1.

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ROTSTEIN, BRANDON, NOVICK-COHEN, AND NEPOMNYASHCHY

time according to t' = a/2 t, differentiating the system with respect to time, and dropping the ' from t. The scaling considered in (4) should be appropriate when the latent heat of fusion is small, when the time relaxation is more rapid for the order parameter than for the temperature field, and in the presence of nonnegligible hyperbolic effects in both fields.

Under the assumption that u = h = const, consideration of the second equation in (4) alone yields

(5) e2 tt + e2 '2 t = e2 A + f () + e h.

Analysis of (5) has been recently considered in the literature [6] in a bounded region Q C R2 with a smooth boundary, and with Neumann boundary conditions. Here h acts as an externally imposed driving force. Equation (5) is a hyperbolic generalization of the Allen-Cahn equation

(6) c2 qt = e2LA + f(q) +f h,

for which it has been shown by Allen and Cahn [7] and Rubinstein, Sternberg, and Keller [8] that coarsened curved fronts with moderate curvature move in the plane with normal velocity proportional to their curvature,

(7) v = K.

The same law governs the motion of interfaces for (3) if u is of order e initially as well as order e along the boundary uniformly in time [9, 10].

Initially circular interfaces which evolve according to (7) remain circular, the radius R(t) obeys Rt = -1/R(t), and a circle shrinks to a point in finite time. For more generally shaped interfaces, there exist analytical results which show that some of the behavior is qualitatively similar. Gage and Hamilton [11] proved that convex curves embedded in the plane shrink to points in finite time when evolving according to

(7) and that such curves remain convex and become circular as they shrink. Grayson [12] extended their results by showing that embedded plane curves become convex without developing singularities, shrinking smoothly to points with round limiting shapes.

For (5), interfaces were shown to move according to the (damped) Born-Infeld equation [6, 13],

(8) vt + 72v(1-v2)- (1-v2) + h(1 -v2)- = 0,

where h is linearly proportional to h. Initially circular interfaces governed by (8) remain circular if the initial velocity is uniformly prescribed, and the radius evolves according to

(9) PRtt =-( + 72Rt (1-R 2)-h (1-Rt2).

Equation (9) was analyzed in [6] when h = 0, and it was shown that if Rt = 0 initially, then -1 < Rt < 0 for all t < tc, where tc is the extinction time, i.e., the earliest time tc > 0 such that R(tc) = 0. In addition it was shown numerically that circles shrink to points in finite time, and approximations to tc were obtained which showed that tc - 72/2 as 72 -+ oo.

266

PHASE FIELD EQUATIONS WITH MEMORY

This manuscript is organized as follows. In section 2 we outline the rationale behind the formulation of the memory phase field equations and present the scaling leading to the hyperbolic phase field equations. In section 3 we obtain by means of a formal and self-consistent asymptotic analysis that (8) with h = 0, i.e.,

(10) Vt + 72V(1 - v2) - (1 - v2) = 0,

describes the dynamics of interfaces in two dimensions for (4) when the temperature field is assumed to be zero (or at most of order e) initially as well as on the boundary. In section 4 we define precisely a concept of motion by hyperbolic crystalline curva- ture and present numerical results which were obtained by implementing this kind of algorithm for (10) and which demonstrate some of the effects of inertia.

2. Derivation of the equations.

2.1. Classical phase field models. Let us consider a material which can be in either of two phases, solid or liquid, and which occupies a fixed region in space, Q C R2, having a smooth boundary. Let us assume, moreover, that the two phases are separated by a transition zone of finite thickness that can be approximated by a sharp interface in a manner to be made more precise in the next section. We assume the temperature to be described by the function e3: Q x [0, To] -* R for some To > 0. It is convenient to work with a reduced temperature defined by T(x, t) := E(x, t) - TM, where TM denotes the planar melting temperature, so that for planar interfaces T(x, t) > 0 indicates that the point x C Q is in the liquid phase, whereas T(x, t) < 0 implies that x is in the solid phase.

In the classical phase field model equations [14], the two phases are assumed to be characterized by different values of a dimensionless physical quantity called an order parameter, ?, which is referred to as a nonconserved order parameter (NCOP) if no global constraints are imposed on it and as a conserved order parameter (COP) if global constraints (such as mass conservation) are imposed. For solid/liquid phase transitions, it is appropriate to consider the nonconserved classical phase field model which assumes an NCOP to vary smoothly though steeply across the interface, as- suming approximately the value -1 in the solid (T < 0) and +1 in the liquid (T > 0). The resultant description of the process is given by a system of two coupled partial differential equations, for the temperature and the order parameter [14, 15].

A temperature or energy balance equation is derived by assuming the internal energy e(x, t) to be given by

(11) e(x, t) :=pCp (T +

where p is density, Cp is the specific heat, and I is the latent heat of fusion (normalized by Cp). Following the classical theory of heat conduction it is assumed that heat transfer for homogeneous and isotropic bodies is governed by Fourier's law [14, 16]

(12) q(x, t) =-k VT,

where q = q(x, t) is the thermal flux and the constant k > 0 is the thermal conduc- tivity. Invoking the first law of thermodynamics, when there is no heat supply to the body by an external source (and when certain other assumptions hold2) the internal

2For example, pcp is time independent

267

ROTSTEIN, BRANDON, NOVICK-COHEN, AND NEPOMNYASHCHY

energy e = e(x, t) and the flux are related by

(13) e = -div q.

Combining (11), (12), and (13) now yields that

(14) (14) Tt + l 0t = KAT,

where K = k/pCp is the thermal diffusivity. Equation (14) is coupled to a phenomenological equation (for q) based on en-

ergetic (free energy) penalization which drives the evolution of the system toward equilibrium states. More specifically, the free energy is assumed to be given by an expression such as FT:

FT(q) = (Vq)2 + -Sc T dx,

where ~ is a correlation length [14, 17], the term involving the gradient reflects long range interactions [18], the function F(q) is a double well potential3 (the free energy density at zero temperature of a spatially homogeneous system [18]), 7T is constant, Sc

is an entropy coefficient [19], and the term in which it appears, -ScT?b, is the entropic contribution to the free energy due to the difference in the entropy densities of the liquid and solid phases. The functional derivative of FT with respect to X is given by

rFT F'(q$)2ST (15) + Sc T.

This expression may be considered as a generalized force which appears as a conse- quence of the tendency of the free energy to decay toward equilibrium. In classical phase field theory it is assumed that +(x) responds by decreasing by a rate propor- tional to this generalized force; i.e.,

(16) Qt I 6.FT

where T is a relaxation time. Employing (15) in (16) and calling f*(Q) = -F'(0), one obtains

(17) rT)t ^2A^? + Sc T.

Note that f*(4) is a function of X which vanishes with negative slope at I = +1 (for example, in Ginzburg-Landau theory f* (0)) = -(03 - ) [20]).

The classical phase field model is not strictly relaxational in the sense that the free energy functional .FT need not always decrease on solution paths unless the tem- perature is constant. This fact has led to thermodynamically consistent formulations which generalize the classical phase field equations and are based on monotone in- crease of the system entropy (see [22] and references therein). For simplicity, we constrain our attention in the present paper to the classical model and its hyperbolic and memory generalizations.

3In classical Ginzburg-Landau theory, F(4) = -(02 - 1)2 [20, 21].

268

PHASE FIELD EQUATIONS WITH MEMORY

2.2. Heat conduction in materials with memory. Classical theory of heat conduction assumes for homogeneous and isotropic bodies that heat transfer is based on Fourier's law (12). If the internal energy is taken to be given by e = pCpT, the classical linear heat equation

(18) T = KAT

follows from Fourier's law (12) and the first law of thermodynamics (13). Note that

(18) coincides with (14) when 1 = 0, i.e., when there is no latent heat release. This theory suffers two main defects. First of all, it does not take into account memory effects which are present in some materials [23]. Second, it predicts that all thermal disturbances at a given point in the body are felt immediately throughout the whole body; i.e., all disturbance propagates with infinite speed [16, 23, 24, 25].

Conduction of heat is the result of the diffusive motion of molecules (in gases), lattice vibrations (in solids), and the motion of electrons (in conducting materials); the maximal speed of heat propagation is therefore of the same order as the velocity of the sound waves. The physical fact that heat flow occurs at finite velocities indicates that the mathematical description of heat transport might be more appropriately hyperbolic rather than parabolic. Possible hyperbolic effects were first considered in 1948 by Cattaneo [24], who looked at the problem from the point of view of the kinetic theory of gases and obtained the following (telegraph) equation for heat conduction:

(19) a Ttt+ Tt = K T-,

where a is a characteristic relaxation time. Equation (19) was also proposed inde- pendently by Band and Meyer [26] and Osborne [27] to account for the appearance of a second sound in helium II as seen in the experiments of Peshkov [29] within the context of dissipative wave formulation. (Second sound had been predicted by Tisza [28] and Landau [30] but within a less realistic nondissipative formulation.) Offhand, second sound effects should be most pronounced under conditions which maximize the thermal conductivity, as well as at very high temperatures on very short time scales. For a discussion of second sound effects, see, e.g., [31, 32]. More recently Gurtin and Pipkin [16] developed a theory of heat conduction with finite wave speeds for isotropic materials, which in a sense generalizes the idea of Cattaneo. The constitutive assump- tion hypotheses of their nonlinear memory theory depend on the summed history of the temperature and temperature gradient fields as well as on the present values of the temperature field, but not on the present value of the temperature gradient field.

A linear version of the Gurtin-Pipkin theory yields a linearized constitutive equa- tion for the heat flux

/0oo (20) q(t) = -k a(t')VT(t - t')dt',

where a(t) is assumed [16] to be a differentiable scalar-valued function on (0, oo) with the dimensions of 1/time such that a(oo) = 0. In their theory the function a(s) is called the heat flux relaxation function and a(0) the instantaneous conductivity. Redefining

/00 k -> k a(s)ds,

the equilibrium conductivity, (20) reduces to (12) when VT is independent of time. By combining (20) with (11) and (13) in the absence of an external source of heat

269

ROTSTEIN, BRANDON, NOVICK-COHEN, AND NEPOMNYASHCHY

release and by assuming any "instantaneous" contributions to the specific heat to be included in the function a(t), the integro-differential equation

(21) Tt = K j a(t- s)AT(s)ds -00

is obtained for the heat conduction. Note that when a(t) = 6(t), (21) reduces to the classical heat equation. For simplicity and to gain some initial intuition, it is possible to consider the prototype kernel

(22) a(t) = a e-7t,

where a and -y are nonnegative constants. Substituting (22) into (21) yields the following expression for the evolution of the temperature:

Ttt + Tt = a AT,

which gives (19) by rescaling time and space as r = t/? and z = (K/a)1/2x and setting a = 1/-y2.

2.3. Hyperbolic phase field equations. Memory effects have been included in the temperature evolution equation in phase field models for both NCOPs and COPs [25, 33, 34, 35]. In those models, (13) applied to the internal energy given by (11) and the heat flux given by the functional (20) with a heat flux relaxation function A1 yields

23) (23) Tt + 2ct = K A1(t- s) AT(s)ds.

To the best of our knowledge, memory effects have not been included in the order parameter evolution equation. We proceed by assuming that the response of b to the tendency of the free energy to decay toward a minimum is not instantaneous but delayed, thereby obtaining

ft .FT -rt-t^ A2(t--s) -- (s) ds,

or more specifically,

( 4 - f*() - (24) rot - A2(t - s) 2A4 + f ) +ScT (s)ds,

00 _ ? -

where A2 may be considered as a generalized force relaxation function with charac- teristics similar to A1. Equations (23) and (24) will be henceforth called the phase field equations with memory. Of course when A1(t) = A2(t) = 6(t), they reduce to the classical phase field equations (14) and (17).

Philosophically, it may seem problematic to consider the current state of the system as dependent on the entire history of the system. This issue can be resolved or at least bypassed by writing (23)-(24) as

T + t= K f t A1 (t - s)AT(s) ds + g(t),

'Tt = fo A2(t - s) [e2 + + ST] (s) ds + g2(t),

270

PHASE FIELD EQUATIONS WITH MEMORY

where g (t) = K f 0 A(t - s) AT(s) ds and g2(t) = f_? A2(t - s) [2Aq + f( +

SCT](s) ds. In this manner, the state of the system at times prior to t = 0 is seen to act via (prescribable) forcing terms on the system at times t > 0.

As noted in the introduction, the phase field equations with memory can perhaps be considered to have their roots in the works of Binder, Frisch, and Jackle [4] (see also the earlier works of Jickle and Frisch [36, 37]), who considered systems in which phase change took place in the presence of additional slow structural relaxation variables within a common phenomenological framework. In [4], the authors considered that under isothermal conditions the chemical potential ,u would contain a relaxational part which could be expressed as a function of space and time as

14lrel = 92fOa2 (X, t) x A(t- s) 0

(x, s)ds, - \

0--2 equil- ca

where A is a relaxation function which is normalized so that A(0) = 1 and lim_o o(T) = 0. Here q0 indicates a phase field variable (defined in [4] relative to a constant

equilibrium value), and (4 )equii indicates the value of (2f4) at quasi-static equilib- rium, i.e., when X has become locally equilibrated relative to the other variables of the system.

Note that formally the above expression may be viewed as an approximation to

I1 ret J A(ts) ( - ( ) (x, s)ds -0 \9 3/I / equil-

Integrating by parts with respect to time and identifying A2(t) = -tA(t), we see that

=Of_ (Of'\ -1re ] A2(t)- 1 yX(x, s)ds.

/- (\ )/ equil-

If one generalizes the above expression by considering first variations of the free energy rather than partial derivatives of the free energy with respect the phase variable (in the sense that this was undertaken by Cahn and Hilliard [38] in extending the Gibbsian concept of free energy), then

t 6jFT A\rel= X A2(t - S) T(X, s) ds.

Here FT has been normalized so that 6- FT 0 at quasi-static equilibrium. If this quantity is taken to be the driving force in the phase relaxation, i.e.,

T(ft = -Plrel,

and the possible presence of an instantaneous response is added into the memory kernel, one now obtains the second equation in the phase field equations with memory, (23)-(24), discussed above.

We remark that Binder, Frisch, and Jackle were in fact considering an isothermal conserved phase variable influenced by nonconserved relaxational variables, and they combined /ltre with an instantaneous contribution to the chemical potential (rather

271

ROTSTEIN, BRANDON, NOVICK-COHEN, AND NEPOMNYASHCHY

than adding instantaneous contributions directly into the relaxation kernels as we have chosen to do), then used mass conservation to obtain an evolution equation. It is noteworthy that under appropriate simplifying assumptions, the Binder, Frisch, Jackle construction reduces to a linear differential equation [37] which reoccurs as the differential operator appearing in the linearization of the viscous Cahn-Hilliard equation, derived in [39].

In order to put problem (23)-(24) in dimensionless form, we choose T*, the supre- mum of the difference between the initial temperature and the melting temperature, as the unit of temperature, and L*, the maximum diameter of the domain of interest or the minimum radius of curvature of the initial interface if it is not planar (the bigger of the two), as a unit of length. Thus we define the dimensionless variables

(25) x' := t:= L* ' (L*) T*

We also introduce the dimensionless parameters and functions

f* rTK I (26) f '-ScT L*VScT* v:= 2 2T*'

A ~(t)K A2(t)K (27) a (t') (L*)2 a2(t') A2(t)K

After a little algebra and dropping the prime (') from the variables in (25)-(27), we obtain

( Ut+ A qt f a( ( t- r) Au(r)dT, (28) 2

t = f_ a2(t -

r) [32A + + ] (r)dr.

Suppose we now focus on exponentially decaying kernels of the form al (t) = a1 e-7 t

and a2 (t) = a2(0)e-2t, with 2 = v-1 a2(0), and consider the scaling A = 2, 3 = 2

and -7 = e in (28). By substituting these kernels and scalings into (28), differentiating with respect to t and rearranging terms, then rescaling by means of the transformation

- 1/2 -1/2 -1/2 t = 2 t, 1 = a2 71, 72 = a2 / 72, setting a = a,1/a2, and dropping the bar (-) from t, 72, and 71, system (4) is obtained.

3. Asymptotic analysis. In this section we follow Caginalp and Fife [19] in carrying out a formal asymptotic analysis for (4), taking e to be a small positive parameter and assuming 71, 72, and a to be 0(1). We demonstrate that under these assumptions, the motion of interfaces for (4) is described by (10). Note that for <vl << 1, (10) reduces to (1) which might be considered as a natural hyperbolic extension of the flow by mean curvature equation (7). We note that for the hyperbolic phase field equations

f 3/2Utt + 65/20tt + Ut + e(t = a AU,

\9/2ott + 3ot = 63LA + f () + 262u,

which arise by considering A = (9(e), v = 0(1), /3 = O(e1/2), and r = 0(e2), and relaxation kernels of the form ai = biyi exp(-yit), where bi = 0(1) and 7yi = O(e-3/2), the limiting motion of the interface is described by [40]

e3/2Vt + V = K,

which constitutes a form of hyperbolically perturbed motion by mean curvature.

272

PHASE FIELD EQUATIONS WITH MEMORY

3.1. Assumptions and definitions. As in [19], we treat interfaces as moving internal layers of thickness 0(e), and we focus on the dynamics of a fully developed layer and not on the process by which it was generated. Let us assume that for some eo > 0 and To > 0 for all e E (0, 60) there exists a solution {u(x,t), (x,t)} of (4) defined on (x, t) c Q x [0, To] which contains an internal layer. More precisely, we assume that the domain Q can be divided into two open regions Q+ (t; e) and Q_ (t; e) separated by a smooth curve F(t; e) defined by F(t; e) := {x e Q | X(x, t; e) = 0}, which does not intersect 9Q. The curvature and velocity of this curve are assumed to be bounded independently of 6, and the function q is assumed to vary continuously though steeply across the interface, tending far from the interface to 1 in Q+(t; e), and to -1 in Q_(t;e).

Let us set x = (x1, x2), and let d(x) = dist(x, F) be the distance from x to F. We define a local orthogonal coordinate system (r, s) in a neighborhood of F by

r (X E}- d(x) if ?(x) > 0, r(x,t; ) = { -d(x) if ?(x) < 0,

and s(x, t; e) is a smooth function of t, which measures the arclength along F(t; e) from a point which is assumed to move normally to F as t varies. The assumed initial smoothness of F(t; e) implies, at least initially, that r is a smooth function in a sufficiently small neighborhood of F. We expect that the geometric form of the solution which we obtain shortly is in fact independent of the choice of parameterization; see, e.g., [41].

We assume outer expansions for u and 0 of the form

u = u(x, t;) = u?0(x, t) + e u(, t) + 0(e2),

q = +(X, t; e) = 0?(x, t) +- 61 (X, t) + 0(62).

In order to define the inner expansions we introduce the inner variable

z(x, t; e) : (x, t; )

and assume the corresponding inner expansions to be given by

u = U(z,s,t; ) = U?(z,s,t) + Ul(z,s,t) + 0(62),

? = 1(z, S, t; e) = 4?(z, s, t) + 6e '(z, s, t) + 0(62).

The definition of F implies that (I(0, s, t; e) = 0. In what follows we employ the notation

g r?= limr g(r,s, t;e)

gr Ir+?= limr gr(r, s,t; e)

to refer to the evaluation of any variable g as F is approached from either side (r > 0 or r < 0). The following matching conditions [9] are assumed to hold as z - ?o:

(29) GO (z, s, t) = g?(0?, s, t),

(30) G1(z, s, t) - g (0?+ s, t) + Z go (0+, s, t), where Gi represents the ith term in the inner expansion and gi represents the ith term in the outer expansion.

273

ROTSTEIN, BRANDON, NOVICK-COHEN, AND NEPOMNYASHCHY

3.2. Hyperbolic phase field equations. Let us now see the implications of these assumptions and expansions for the phase field equations (4), if u is assumed to vanish initially and along the boundary for all t E [0, To].

3.2.1. Outer problems. Substituting the outer expansions into (4) and equat- ing coefficients of the corresponding powers of e, we obtain

31 1 ut +^41 +u? = l Au?, (31) )

and

(32) f9 ut + u7 = a, Au', (0 f()

1 +U0 = 0

as the outer problems at 0(1) and 0(e), respectively. The unique solution of the 0(1) outer problem for the assumed initial and boundary conditions is u? = 0, q? = ?1. For the 0(e) outer problem, the unique solution is ul = 0, $1 = 0.

3.2.2. Inner problems. Employing the (z, s, t) coordinate system, (4) may be written as

((a - r2)Uzz -- + [a ArUz - 2rtUzt - rttUz - 2rtstUzs - lrtUz] = 0(c2),

(33) (1 - r2)zz + f() 3+ e [-2rtzt - rttz - 2rtst<zs - Y2rtTz + ArD, + U] = (e2).

Substituting the inner expansions into (33) and equating coefficients of the corre- sponding powers of e, the following inner problems are obtained at 0(1) and 0(c), respectively:

{ (a - r2) UO? - 0,

and

(a - r2) Uzlz = (rtt + 71 rt - aZr)Uz + 2rtUzt + 2rtstUz,, (35) (1 - r2)z + f'(0o)>JD = (rtt + y2rt - r)>0 + 2rtbto + 2rtstDos + U?.

Note that U? _ 0 constitutes a bounded solution of the first equation in (34) and also satisfies the matching conditions (29). For simplicity, we now adopt U? = 0 as our solution to the first equation in (34). To solve the second equation in (34), we assume that r2 < 1 and define a new variable

(1-r: )

In terms of ((, s, t), the second equation in (34) may be written as

(36) 4? + f((?) = 0.

There exists a unique solution to (36) which satisfies the matching conditions ? (Ioo) = ?1 and the normalization condition 4?(0) = 0. We denote this unique solution by I(() and obtain that

D o ( Z )

\-r2 2

274

PHASE FIELD EQUATIONS WITH MEMORY

Substituting j?0 into (35) yields

37 ( (-r2) Uzlz = 0, ( 37) (1 - r2) Lz + f'(0) Il = (rtt + 72 rt - Ar) I0 + 2 rt (zt.

Noting that U1 = 0 satisfies the first equation in (37) as well as the matching con- ditions (30), we set U1 0. The second equation in (37), expressed in the ((, s, t) coordinate system, reads as

2 rt2 rtt (tt - {?

rtj- Ar

(38) f (1) + + + (1 - Ar

It is easy to check that ''(() satisfies the homogeneous equation <IQ1 + f/((?) C(1 = 0

and that the operator A : 2 + f'(? (()) in L2(-oc, oo) has a simple eigenvalue at the origin with 4' as the corresponding eigenfunction (see [9]). Hence the solvability condition for (38) gives

(39) 2 r2 rtt (D'O ' rtt + 7Y2 rt - Ar 00 2,,

(39) o t 2 ) 32

(V + @, V d( + 2) ' I')2d( = 0.

Notigthatf(1 -_o (lr)2 oo

Noting that df0_ 4'"dd = -2 f -o(j')2d?, it follows from (39) that

(40) r2 + 72 rt - Ar. 1 rt

Along the interface Ar = n and rt = -v, where n is the curvature of the interface and v is its normal velocity [22]; hence (40) implies (10).

We remark that the same limiting motion can also be derived for the system

f Utt + 63 (tt + Ut + 62 ot = Au, l 62 Ott 2 + 2 72 t = 62 AO + f(X) + e U,

which can be obtained from (4) by taking 71 = a = 1/e. We comment that the assumption that r2 < 1 is at least reasonable since, as

noted in the introduction, it holds in the radial case if the initial velocity is zero. Note, furthermore, that if the initial velocity is zero, then the definition of C implies that at least initially the inner profile is widening.

4. Motion of simple closed polygonal curves by hyperbolic crystalline curvature. In this section we present a crystalline algorithm designed to approxi- mate motion by hyperbolic mean curvature (10), which is essentially as simple as the crystalline algorithm which has been used to approximate motion by mean curvature. We then implement this algorithm numerically and obtain some preliminary indica- tions as to how (10) differs from classical motion by mean curvature and what the effects of inertia are.

Crystalline algorithms based on the concept of crystalline curvature were intro- duced as a means for studying the evolution of polygonal curves, or crystals [43, 44, 45]. Crystalline curvature measures the rate of change of surface area with volume under certain admissible deformations as the height of these deformations tends to zero [46]. In this sense crystalline curvature is analogous to ordinary curvature and motion by crystalline curvature is analogous to motion by ordinary curvature [46]. Moreover, motion (of polygonal curves) by crystalline curvature can be used to approximate

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ROTSTEIN, BRANDON, NOVICK-COHEN, AND NEPOMNYASHCHY

motion by curvature of smooth curves. To do so, a faceted (polygonal) approxima- tion of the curve is constructed, and the resultant evolution law leads to a system of nonlinear ordinary differential equations for the lengths of the facets. Convergence of a crystalline algorithm for the motion of simple closed curves and graphs to motion by weighted mean curvature, which is an anisotropic generalization of motion by mean curvature, has been proven in [47, 48] under suitable assumptions. In [49] a hyperbolic analogue of motion by crystalline curvature was introduced to approximate motion of closed convex polygonal curves according to (1). In this section we generalize the approach introduced in [49] to treat (10).

4.1. Definitions. Basically, we rely on the framework developed in [43] and [46] in order to make precise our definition of motion of polygonal curves by hyperbolic crystalline curvature. Let ~b be defined as the interfacial energy. For any n EA K, the set of unit vectors in R2, let ((n) := 4(cos-l(n ex)), where ez is the unit normal in the x-direction. Thus, we define its Wulff shape, W, as

W = {xI x . n < (n) for all n E A/}.

Since W is the intersection of half spaces, it is convex. The Wulff shape for general positive, real-valued, and convex 0 can be shown to be the unique solution (up to translation) to the problem of surrounding a given area (volume) with a curve with least possible surface energy using only "admissible directions," i.e., directions cor- responding to the exterior normals of W [50]. In the present context, the interfacial energy is isotropic and we may set /(0) = 1 for any 0 < 0 < 27r. For _ - 1, the Wulff shape is a unit circle centered at the origin. Let us now take bn to be an approximation to $ such that its Wulff shape

Wn = {xx. * n < n(n) for all n C JV}

is an n-sided polygon inscribed in the unit circle, and we proceed to work with the approximations l,n and Wn. For our present purposes, we shall call the set of exterior normals to Wn the set of admissible directions, and we define A(n) to be the length of the edge of Wn having normal n if there is such an edge and 0 otherwise.

As our initial conditions, let us consider a closed polygonal curve P(t), t > 0, with side segments L1,... , Ln numbered consecutively in a counterclockwise fashion, labeling Lo = LN and LN+1 = L1. P(t) can be seen as representing a crystalline approximation of a grain of solid material, which evolves under the action of the phase field equations with memory. Let us define ni(t) to be the unit exterior normal to Li. We denote by li(t) the length of Li, and we define Vi(t) = vi(t) ii(t) to be the normal velocity of line segment Li, where vi(t) denotes the speed in the normal direction. The corner ci, which is located where Li and Li+l meet, will be said to be of regular type if the interior angle between Li and Li+, is acute, and of inverse type otherwise. We define Xi to be 1 if the corners ci-1 and ci are both of regular type, -1 if the corners ci-1 and ci are both of inverse type, and 0 if ci-1 and ci are of different types. We set AOi = i - Oi-1, where 0i is the angle that the normal ni makes with the x-axis, and define Ai = A(ni).

In order to provide a framework in which crystalline curvature will be sensibly (nontrivially) defined for all facets of the polygonal curve as it evolves, we define admissible polygons as follows [46, 50].

DEFINITION 4.1. P(t) is said to be admissible at time t if li(t) > 0 for i = 1,... ,N, li(t) > 0, and ni(t), ni+1(t) correspond to consecutive normals (either in the clockwise or counterclockwise sense) of W,.

276

PHASE FIELD EQUATIONS WITH MEMORY

Note that by allowing consecutive normals to be taken either clockwise or counter- clockwise, the set of admissible polygons includes both convex and nonconvex polyhe- dral curves. In order to guarantee that P(t) is admissible for all t E [0, To], we assume that P(O) is initially admissible and that the orientation of the normals is invariant under the flow; i.e., the ni (t), i = 1,..., N, are time independent. It remains now to determine the velocities (and lengths) of the various sides of the evolving curve. This is done as follows.

DEFINITION 4.2. An admissible polygon P(t) is said to evolve on the interval

[0, To] according to hyperbolic crystalline curvature (the geometric crystalline curva- ture version of the Born-Infeld equation) if

^ r vz0i + 72i(1-2) -

vi ) _ .- v2) -= o, (41) i = [cot AOi + cot Oi+l]vi - csc AOzi vi- - csc AOi+lvi+

for all t E [0, To]. It is easy to show (see [42] for details) that, given an initial admissible polygonal

curve with a prescribed initial velocity, there exists a unique family of polygonal curves (P(t)) which satisfy the initial conditions and move according to (41) until the first time at which the length of one of the sides shrinks to zero. Afterwards it is possible to rerun the algorithm with a smaller number of sides if the resultant polyhedral curve still constitutes an admissible polygon.

For the classical crystalline algorithm, studies have been undertaken to see when stability criteria corresponding to motion by mean curvature or by weighted mean curvature would indicate that a facet should "break" and hence that the number of facets should instantaneously change at some time during the evolution. In particular, it has been proven that the framework of the crystalline algorithm is adequate to uniquely follow motion by weighted mean curvature for polyhedral curves evolving in R2 which are initially compatible with the Wulff diagram; in R3 this framework has been shown to be insufficient in general [51]. Thus, before the robustness of the implications of the hyperbolic crystalline algorithm can be ascertained, similar facet breaking stability analysis must be undertaken.

4.2. Numerical results. In order to study the evolution of polygonal curves by hyperbolic crystalline curvature, we solved the system of ODEs (41) numerically using the Runge-Kutta method of order four until the first instance at which one or more sides of the polygon vanishes. We have not yet resolved the question of convergence of the algorithm for motion of curves by the hyperbolic crystalline curvature to motion by hyperbolic curvature as prescribed by (10), although we conjecture that there is convergence. In order to support this conjecture we analyzed the extinction times, tc, for regular polygons with a large number of sides (N) and compared them with the extinction times for circles which were calculated in [6]. The results are presented in Table 1, where it is possible to see that as the number of sides increases, the extinction times approach that of the circle.

In Figure 1 the evolution (shrinkage) is portrayed of a regular polygon of 120 sides which initially approximated a circle of radius 1. The behavior is, as expected, similar to the behavior of a circular interface.

Figure 2 shows the evolution (shrinkage) of a polygon of 30 sides for various values of 72- In Figures 2(a) and 2(b) (72 = 0 and 72 = 1) we see that the polygon starts shrinking, initially becoming more regular, but later, due to inertia, losing regularity. It is interesting to note that the roles of the major and minor axes are interchanged as the polyhedral curve evolves. This can be seen as a type of two-dimensional relaxation

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ROTSTEIN, BRANDON, NOVICK-COHEN, AND NEPOMNYASHCHY

TABLE 1 Extinction time tc for motion by hyperbolic (Born-Infeld) crystalline curvature for regular

polygons with N sides which are initially circumscribed by a circle of radius 1, for various values of Y2.

'72 N = 96 N = 120 N = 150 j Circle I 0.0 1.5700 1.5703 1.5705 1.5708 1.0 1.7353 1.7357 1.7360 1.7363 2.0 1.9770 1.9776 1.9780 1.9785 5.0 3.0804 3.0814 3.0822 3.0833 10.0 5.3558 5.3578 5.3592 5.3614

FIG. 1. Motion by hyperbolic (Born-Infeld) crystalline curvature. Shrinkage of a regular poly- gon of 120 sides for Y2 == 0 and t = 0, 0.8, 1.2, 1.45.

oscillation. In Figure 2(c) (7Y2 - 5) we see that once the polygon becomes round, it appears to maintain this regularity afterwards. Moreover, as 72 increases, the motion seems to become more similar to classical motion by crystalline curvature in which convex curves become round and shrink to points in finite time and irregular polygons shrink while gaining regularity. Thus Y2, which appeared originally as the exponential decay rate in the relaxation kernel in the equation for q, acts here as a damping coefficient.

Technically, it also would be easy to implement our algorithm for nonconvex polyhedral curves, such as for dumbbell shaped polyhedral curves, as was done in [45] in the context of the classical crystalline algorithm. In Figure 5a of [45], the middle "bar" section of the dumbbell can be seen to thicken until it coalesces with the "top" of one of the "bells" and the inner sides of that bell disappear. In this manner, the evolving polyhedral curve becomes more convex and remains so thereafter. For the hyperbolic crystalline curvature algorithm, if the initial velocity is zero, the middle bar section can be expected to thicken until it coalesces with the "top" of one or more bells; however, inertia can be expected to render the coalescence (and increased convexity) instantaneous only, as there is no reason in the hyperbolic case to expect the velocities of the coalescing segments to be identical at the time of coalescence. Thus, facet breaking would be an immediate consequence of coalescence. We postpone further

278

PHASE FIELD EQUATIONS WITH MEMORY

0.5 -0-.0

-0.5 - -0.5

-1 - - .1 -

-1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5

(a) (b)

-1.5 -1 -0.5 0 0.5 1 1.5

(C)

FIG. 2. Motion by hyperbolic (Born-Infeld) crystalline curvature. Shrinkage of a polygon of 30 sides for (a) 72 = 0 and t = 0, 0.8, 1.1, 1.3; (b) y2 = 1 and t = 0, 0.9, 1.3, 1.6; (c) Y2 = 5 and t = 0, 1.3, 2.2, 2.8, 3.1.

numerical experimentation until questions of convergence, stability, and uniqueness are clarified.

5. Conclusions. We have presented a memory-type extension of the classical phase field equations (28) which systematically takes into account delayed response of the system due to slowing relaxing internal variables and can predict finite speed of propagation for thermal fluctuations. For the case of exponential kernels (22), for which the integro-differential system reduces to a hyperbolic one (4), we have analyzed the motion of interfaces. For (4), by means of formal asymptotic analysis we obtained

(8) as a description of the dynamics of an interface between two different phases in two dimensions. A crystalline algorithm was used in order to study the motion of smooth curves evolving according to this law. This was achieved by breaking the smooth curves into facets and studying the motion of the resultant set of polygonal grains which approximate the smooth ones. The crystalline approach has the advantage that it requires fewer calculations than classical methods of solving PDEs. Numerical results were presented demonstrating that inertial effects can induce a type of two- dimensional damped oscillation.

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ROTSTEIN, BRANDON, NOVICK-COHEN, AND NEPOMNYASHCHY

Further work is ongoing in order to improve our understanding of the scope of applicability of the model and the analytical implications both of the phase field equations with memory (2) and of hyperbolic motion by mean curvature (10); see, e.g., [5]. In terms of the numerics it would be interesting to study the convergence of the algorithm as well as to formalize an analogous algorithm for graphs. It would also be interesting to extend the results obtained in this paper for a more general class of kernels. First steps in this direction have been undertaken in [40], [42], and [52].

Appendix. From Cartesian to (r, s, t) and (z, s, t) coordinates.. To con- vert from Cartesian to (z, s, t) coordinates, we transform derivatives as follows (re- calling that |Vrl 1):

?>t = (t + -ezDrt + 1sst,

Ott := tt + 2- 4ztrt + 2)s5tst + -2(zzr + 4)ss2 + -4zrtt + 4sStt + 2-q)Zrtst,

and

E2 1

where the operators V and A refer only to the spatial variable x.

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