SIAM J. APPL. MATH.
Vol. 59, No. 6, pp. 2086–2107
c 1999 Society for Industrial and Applied Mathematics
SECOND-ORDER PHASE FIELD ASYMPTOTICS
FOR UNEQUAL CONDUCTIVITIES∗
ROBERT F. ALMGREN†
Abstract. We extend Karma and Rappel’s improved asymptotic analysis of the phase field
model to different diffusivities in solid and liquid. We consider both second-order “classical” asymptotics, in which the interface thickness is taken much smaller than the capillary length, and the new
“isothermal” asymptotics, in which the two lengths are considered comparable. In the first case,
if the phase field model is required to be gradient flow for an entropy functional, then for unequal
diffusivities it is impossible to construct a phase equation with finite kinetics which converges with
second-order accuracy to a Gibbs–Thomson equilibrium condition with infinitely fast kinetics. In the
second case, some error terms are pushed to higher orders, and it is easy to eliminate the remaining
errors with finite phase kinetics.
Key words. phase field asymptotics, diffusivity
AMS subject classifications. 80A22, 35K57, 35R35, 41A60
PII. S0036139997330027
1. Introduction. We consider the motion of an interface Γ(t), either a curve
in two dimensions, or a surface in three dimensions, which divides a fixed enclosing
domain Ω into two bulk regions Ω+ (t) and Ω− (t). For solidification, Ω+ is the liquid
region, Ω− the solid region, and Γ(t) the solid/liquid interface. The interface evolves
according to a modified Stefan problem, briefly presented below and extensively discussed in the literature (see [11, 12]).
Motion of the interface is controlled by diffusion of a dimensionless scalar field
u(x, t). For solidification of a pure material, u is the temperature relative to the
equilibrium freezing temperature, scaled by the latent heat of fusion and the heat
capacity. With the assumption of a constant miscibility gap, isothermal solidification
of a binary alloy can be mapped onto the same equations, in which u is the chemical
potential [20]. Indeed, the problem of alloy solidification motivates this work, since
solute diffusion coefficients typically differ by many orders of magnitude between solid
and liquid phases. For simplicity, we shall use only the language appropriate for a
pure material.
In the bulk phases, u solves two separate diffusion equations
(1)
ut = D± ∆u
in Ω± ,
in which the diffusion coefficients D± are in general different in solid and in liquid.
Equation (1) is to be supplemented by suitable initial and boundary conditions.
Across Γ, u is continuous. But advancing or retreating of the interface means
that material is changing phase and releasing or absorbing latent heat of fusion; thus
∗ Received by the editors November 14, 1997; accepted for publication (in revised form) August
25, 1998; published electronically August 31, 1999. This research was supported by the National
Science Foundation CAREER program under award DMS-9502059, by the NSF MRSEC program
under award DMR-9400379, and by an Alfred P. Sloan Foundation Research Fellowship.
http://www.siam.org/journals/siap/59-6/33002.html
† Department of Mathematics, The University of Chicago, Chicago, IL 60637 (almgren@math.
uchicago.edu).
2086
SECOND-ORDER PHASE FIELD ASYMPTOTICS
2087
∇u must satisfy the jump condition
(2)
∂u
V = − D
∂n
+
across Γ,
−
where V is the normal velocity of Γ, positive where solid is advancing into liquid. We
take the heat capacities to be the same in solid and in liquid, so that the difference in
diffusivities is entirely due to the different conductivities. If the heat capacities were
also different, then the latent heat of fusion would depend on temperature.
Finally, u must satisfy the Gibbs–Thomson condition on Γ:
(3)
u = −δ K − β V
on Γ,
where K is the mean curvature of Γ, positive where the solid is convex.
The material parameter δ is the capillary length and is proportional to the surface
energy per unit area. (The more usual notation d0 is confusing in the context of an
asymptotic expansion.) In general, this energy depends on the orientation of the
interface relative to an underlying crystal lattice in the solid, and thus K should
be replaced by an anisotropic smooth or crystalline “weighted mean curvature” [17].
Despite the physical importance of anisotropy, here we carry out our asymptotics only
for the isotropic case, deferring anisotropy to future work.
The kinetic parameter β ≥ 0 is an inverse mobility. If β > 0, then the motion law
may be written V = −M (u + δK) with M = 1/β: the speed is proportional to the
amount by which the interface is locally out of equilibrium. In this regime we may
loosely say that condition (3) is of “parabolic” type. This dynamics is appropriate
for large undercoolings.
In the small undercooling regimes of practical interest, it is more accurate to
describe the interface condition as an instantaneous equilibrium between temperature
and curvature, corresponding to setting β = 0. The system (1), (2), (3) remains
perfectly well-posed in this limit, and we would like any approximation method also
to remain well-behaved as β → 0.
The search for effective ways to solve this problem has a long history. One of the
most promising methods is the phase field model (4), (5) below. For accounts of the
evolution of phase field models and their application to realistic materials problems,
see [14, 10]. This model contains a small parameter, the interface thickness , and is
constructed so that the sharp-interface model (1), (2), (3) is recovered as → 0 for
suitable choices of the other parameters. This paper concerns the precise nature of
the convergence of the diffuse-interface model to the sharp-interface model.
Recently, Karma and Rappel [10] have shown how to obtain quantitatively accurate comparisons of phase field solution with sharp interface models, for equal diffusivities. In two dimensions, they have achieved agreement with boundary-integral
computations of dendrite tip speeds [9] at undercoolings near one-half, and in three
dimensions, they have obtained agreement with experiments [8].
Our primary aim in this paper is to extend Karma and Rappel’s analysis to unequal conductivities in solid and liquid. Because their formulation relied on odd/even
symmetries within the interface layer, this extension reveals more fully the underlying
structure of the approximation. We discover some surprising differences between the
cases of equal and unequal conductivities.
Karma and Rappel considered two different asymptotic expansions of the phase
field model. For equal diffusivities, these give the same conclusions, but for unequal
diffusivities they are different.
2088
ROBERT F. ALMGREN
In the first expansion, which we refer to as “classical” asymptotics, we take the
interface thickness → 0 while keeping finite values of the other parameters. In
particular, the interface thickness is assumed to be much smaller than the capillary
length. We present this model in section 2; in section 4 we carry out the asymptotics
to two orders in using the boundary-layer notation of section 3. In section 5, we show
how to construct interpolation functions that satisfy all the constraints required for
second-order accuracy; these interpolants necessarily have certain undesirable properties.
The same quadratic convergence has been shown rigorously for convergence of
solutions of the Allen–Cahn equation to motion by mean curvature [15]. Some of the
correction terms we derive were also obtained by [5]; the new feature of our work is
that we show how to eliminate the extraneous terms. Somewhat similarly, higherorder boundary conditions have been determined for electromagnetic scattering from
thin interfacial layers [1, 3].
The second expansion (section 6) was newly proposed by Karma and Rappel;
it takes the capillary length and kinetic coefficient to scale proportionately to the
interface thickness. Following them, we call this “isothermal” asymptotics, since at
leading order the temperature is constant on the interface. We show how some of
the error terms identified in section 4 are pushed to higher orders in this limit. By
neglecting the higher-order errors, we must satisfy only a reduced set of integral
constraints and very effective models can easily be constructed.
2. Phase field model. Phase field models have their origin in order-parameter
models of critical dynamics [7]; their potential use in solidification modeling was
pointed out somewhat later [2, 6, 13].
These models introduce a phase variable φ(x, t), which takes the value 1 in liquid,
−1 in solid. The phase variable obeys a reaction-diffusion equation, in which a nonlinear source term pushes φ towards the bulk values ±1, while diffusion smears interface
layers to a thickness O(), where is an artificial small parameter. The temperature
u(x, t) obeys the diffusion equation, with a source term corresponding to release of
heat as φ changes.
In our model, the diffusion coefficient depends on the phase variable. Since the
diffusion coefficients D± have units of length2 /time, it is convenient to separate D±
into an overall diffusivity D and nondimensional values q+ and q− so that D± = Dq± .
Within the interface layer, we interpolate the thermal conductivity by some function
q(φ) with q(±1) = q± .
Then our system has the specific form
µ 2 1
β
0
0
(4)
+
u p (φ) + ∆φ,
φt = − 2 g (φ) −
δ
D δ
3 δ
1
ut + p̃(φ)t = D div q(φ)∇u .
(5)
2
The double-well potential g(φ) has wells of equal depth at φ = ±1. Both p(φ) and p̃(φ)
are forms of the internal energy function, with p(±1) = p̃(±1) = ±1. In order that the
position of the potential wells not change with temperature, we require p0 (±1) = 0,
while p̃0 (±1) is unconstrained.
Besides the approximation parameter , the only parameter in (4), (5) which does
not appear in the sharp-interface system (1), (2), (3) is the dimensionless number µ.
It must be carefully specified in order to obtain second-order accuracy (see (45) in
section 4.5.3) or to eliminate leading-order kinetic errors (see section 6).
SECOND-ORDER PHASE FIELD ASYMPTOTICS
2089
We identify the coefficient on the left of (4) as m/D, where the dimensionless
inverse mobility coefficient is
(6)
m =
Dβ
+µ .
δ
δ
The first term in m may be written P̄ −1 β V̄ /δ K̄, where the Péclet number P̄ = ρ̄V̄ /2D
measures the typical size of the solid object relative to its surrounding diffusion layer.
Here V̄ and K̄ = 2/ρ̄ are typical values of the normal velocity and mean curvature;
for example, with anisotropic surface energy, these might be values at the tip of a
steady dendrite.
Thus, if the kinetic contribution βV in (3) is small compared to the surface energy
term δK, then, for finite Péclet number, the first term in m is small relative to unity,
and the O() correction becomes relatively more important for finite values of /δ.
This effect becomes less significant as P decreases, typically corresponding to small
undercoolings.
The type of (4) depends on the value of m. If m < 0, it is ill-posed and has no
meaningful solution. Since β ≥ 0, this can happen for finite if µ < 0; conversely,
if µ < 0, then as β → 0 we require smaller and smaller values of /δ to maintain
well-posedness of (4).
If m > 0, then (4) is of parabolic type and can be solved by standard explicit or
implicit methods. If µ > 0, then this will be the case for all β ≥ 0 and finite > 0.
It is remarkable that it is sometimes possible for solutions of the system (4), (5) with
m > 0 to converge with second-order accuracy to (1), (2), (3) with β = 0.
If m = 0, then (4) is a nonlinear elliptic problem: at each moment of time, φ is
determined by the instantaneous temperature field. If µ = 0, then this is the relevant
case for β = 0; in other words, the type of (4) mirrors the “type” of (3). We shall see
that this is the typical behavior for phase field systems with a gradient structure.
If p̃ = p, then the system (4), (5) is a gradient flow for the negative entropy
functional
Z 1
2 2
1
2
|∇φ| + g(φ) +
u
dx,
F[φ, e] =
2
3δ
Ω
taking the order parameter φ to be nonconserved and the internal energy e = u +
(1/2)p(φ) to be conserved [16, 19]. As a consequence, F decreases monotonically in
time. Although this gradient structure is appealing theoretically [18], it seems to have
no concrete advantage in practice, and indeed has practical disadvantages [9].
A natural choice for g(φ) is the quartic
(7)
ḡ(φ) =
1
(1 − φ2 )2 .
2
The simplest choice for p(φ) is the odd cubic function
(8)
p̄(φ) =
1
φ 3 − φ2 ,
2
while for a nongradient model we may take p̃(φ) = φ. Conductivity interpolants q(φ)
are also easy to construct. For any reasonable choices of these functions, and for any
value of µ, solutions of (4), (5) converge to solutions of (1), (2), (3) with errors in
interface curvature and velocity of size O(). The purpose of this paper is to show
how to choose g, p, p̃, q, and µ so as to obtain the most accurate model possible.
2090
ROBERT F. ALMGREN
3. Boundary-layer asymptotics. For small , typical solutions to (4), (5),
possibly after an initial layer in time, consist of large “bulk” solid and liquid regions
in which φ is nearly equal to the stable stationary states φ = ±1 of the nonlinear
source term in (4). In the bulk regions, standard asymptotics [4] show that φ ∼ ±1
up to terms exponentially small in and that u satisfies the diffusion equations (1) at
all orders in .
For > 0, we define the interface Γ(t; ) to be the level set {φ = 0}. Let us
parameterize Γ(t; ) by s. For simplicity, we shall use notation appropriate only to
curves in two dimensions, but we shall point out the necessary extensions to three
dimensions. In a neighborhood of Γ(t; ), we define a signed distance function r(x, t; )
with the same sign as φ(x, t; ). We extend s to this neighborhood by giving s(x, t; )
its value at the foot of x in Γ(t; ). Then (r, s) form an orthogonal curvilinear coordinate system near Γ(t; ), satisfying |∇r| = 1, ∇r · ∇s = 0. If Γ is smooth, then this
change of coordinates is valid near Γ; in particular it is valid within a distance O().
With a slight abuse of notation, for each the dependent functions φ and u may be
viewed as functions either of (x, y, t) or of (r, s, t). Elementary computations (see [4])
give us the formulas for converting derivatives over (x, t) to derivatives over (r, s, t):
(9)
(10)
∆ = ∂rr + ∆r ∂r + ∆s ∂s + |∇s|2 ∂ss ,
∂t = ∂t − V ∂r + st ∂s .
On the left, ∆ denotes the Laplacian ∂xx + ∂yy , and ∂t denotes time derivative at
fixed x. On the right, ∂t denotes time derivative at fixed r, s. The heat conduction
term has the form
(11)
div q(φ)∇u = ∂r q(φ) ur + ∆r q(φ) ur + ∆s q(φ) us + |∇s|2 ∂s q(φ)us .
Since |∇r| = 1, ∆r is the mean curvature of the level surfaces of r, normal
translations of Γ. Standard results of differential geometry then give
(12)
(∆r)(r, s, t) =
K + 2Πr
∼ K0 + K1 − ρ K02 − 2Π0 + · · ·
2
1 + Kr + Πr
as → 0.
We construct a formal asymptotic analysis based on the assumption that as → 0,
Γ(t; ) smoothly approaches its limit Γ(t; 0). Thus its normal velocity V , its mean
curvature K, and its Gauss curvature Π have regular expansions in :
(13)
(14)
V (s, t; ) = V0 (s, t) + V1 (s, t) + · · · ,
K(s, t; ) = K0 (s, t) + K1 (s, t) + · · · ,
(15)
Π(s, t; ) = Π0 (s, t) + Π1 (s, t) + · · · .
(Π is the product of the two principal curvatures and K is their sum.)
In the inner region, |r| ∼ O(), we look for a solution in terms of the stretched
variable ρ = r/:
(16)
(17)
r
, s, t + Φ1 , s, t + · · · ,
r
r
u(x, t; ) = U0 , s, t + U1 , s, t + · · · .
φ(x, t; ) = Φ0
r
SECOND-ORDER PHASE FIELD ASYMPTOTICS
2091
In the intermediate zone, where 1 |ρ| −1 and |r| 1, the inner
and outer constructions must describe the same solution. This gives the matching
conditions
lim U0 (ρ) = u±
0 (0)
(18)
ρ→±∞
and the far-field conditions
±
U1 (ρ) ∼ ∂r u±
0 (0) ρ + u1 (0) + o(1),
(19)
ρ → ±∞.
For φ, we have
(20)
lim Φ0 (ρ) = ±1,
ρ→±∞
lim Φj (ρ) = 0,
ρ→±∞
j ≥ 1.
By our definition that φ = 0 on Γ, we also have the condition at each order that
Φ0 (0, s, t) = Φ1 (0, s, t) = · · · = 0.
Within the inner layer, the derivatives ∆s, ∇s, and st are within O() of their
values on Γ, so we shall simply use these symbols to refer to their values on Γ.
4. Classical asymptotics. In the inner region, where |ρ| ∼ O(1), we first use
(9)–(11) to rewrite (4), (5) in curvilinear coordinates (r, s, t) and then rescale r to
ρ = r/, obtaining
µ 1
β
+
φt − V φρ + st φs
(21)
δ
D δ
1 0
1 2 1
1
1
= − 2 g (φ) +
u p0(φ) + 2 φρρ + ∆r φρ + ∆s φs + |∇s|2 φss ,
3 δ
(22)
ut −
1
1
1
p̃(φ)t − V p̃(φ)ρ + st p̃(φ)s
V uρ + st us +
2
1
1
= D 2 ∂ρ q(φ) uρ + ∆r q(φ) uρ + ∆s q(φ) us + |∇s|2 ∂s q(φ) us .
We now substitute the inner expansions (16), (17) and the interface expansions (13)–
(15) with the far-field matching conditions (18)–(20). We collect terms with matching
powers of and solve the “φ equation” (21) and the “u equation” (22) to two orders,
beginning at O(−2 ) and working down to O(1). At O(−2 ) we determine the interface
structure. At O(−1 ) we determine the leading-order laws that govern the motion
of the interface. At O(1) we determine the first-order corrections to the motion
laws, which we shall be able to eliminate by choosing the interpolation functions
appropriately.
4.1. φ at O(−2 ).
(23)
Φ0ρρ − g 0 (Φ0 ) = 0.
For double-well g(φ), (23) has a unique solution Φ0 (ρ, s, t) = ψ(ρ) satisfying ψ(±∞) =
2
±1, with ψ 0 = 2 g(ψ). The translational degree of freedom is removed by the condition that ψ(0) = 0. We shall assume that g(φ) is such that this solution is monotone
increasing and approaches its limiting values exponentially as ρ → ±∞.
For g = ḡ of (7), we have the odd solution
(24)
ψ̄(ρ) = tanh ρ.
2092
ROBERT F. ALMGREN
4.2. u at O(−2 ).
∂ρ q(Φ0 ) U0ρ = 0,
which integrates to
U0ρ =
C(s, t)
.
q(ψ)
Since U0ρ → 0 as ρ → ±∞, and q has finite limits there, the constant C must be zero,
and U0ρ is zero for all ρ. Thus we have the inner/outer matching condition
U0 (ρ, s, t) = U0 (s, t) = u±
0 (r = 0),
(25)
and the leading-order outer solution u0 is continuous across the interface.
4.3. φ at O(−1 ).
(26)
β
2 1
∂ρρ − g (ψ) Φ1 = − K0 + V0 ψ 0 (ρ) −
U0 p0 ψ(ρ) .
δ
3 δ
00
We obtain the solvability condition by multiplying by ψ 0 (ρ) and integrating over ρ
from −∞ to ∞; the left side vanishes and so must the right. Thus K0 and V0 must
satisfy
(27)
−K0 −
1
β
V0 −
U0 = 0,
δ
Iδ
where we shall assume from now on that
Z
Z
3 1p
3 ∞ 0 2
(28)
2 g(φ) dφ = 1
ψ (ρ) dρ =
I =
4 −∞
4 −1
(this is true for g = ḡ and may always be enforced by scaling g(φ)), and we have
evaluated
Z ∞
Z ∞
0
d
0
p ψ(ρ) ψ (ρ) dρ =
(29)
p ψ(ρ) dρ = 2.
dρ
−∞
−∞
Then with (25), (27) becomes the Gibbs–Thomson condition
(30)
u0 Γ = − δ K0 − β V0 .
This is the leading-order version of (3), connecting the leading-order part of the outer
temperature field at the interface with the leading-order interface curvature and velocity. It holds for any choice of the constitutive functions p(φ), q(φ) satisfying the
endpoint conditions at φ = ±1.
The self-adjoint even operator L = ∂ρρ − g 00 (ψ) has a well-defined inverse into
the space of bounded functions which vanish at the origin. Thus, with (30), Φ1 is
uniquely determined by p(φ) as
2 0
U0 −1 0
(31)
L
ψ − p (ψ) .
Φ1 =
δ
3
Let us observe in passing that if p(φ) is the simple odd cubic function (8) and g = ḡ,
then the right side of (31) vanishes. Then Φ1 ≡ 0 and there is no O() correction
to φ.
2093
SECOND-ORDER PHASE FIELD ASYMPTOTICS
4.4. u at O(−1 ). Substituting Φ0 and U0 , we obtain
∂ρ q(ψ) U1ρ
which we integrate once to find
(32)
= −
V0
p̃(ψ)ρ ,
2D
V0 (s, t)
A(s, t)
−
p̃(ψ)
D
2D
q(ψ) U1ρ =
and a second time to obtain
A(s, t)
U1 (ρ, s, t) = ū(s, t) +
D
(33)
Z
ρ
0
dσ
V (s, t)
− 0
2D
q ψ(σ)
Z
ρ
0
p̃ ψ(σ)
dσ.
q ψ(σ)
The “constants” of integration ū and A are to be determined by matching to the outer
solution.
We have the far-field expansions
)
(
A
V0 F̃±
V0
AG±
U1 (ρ) ∼
(34)
+ o(1), ρ → ±∞,
+
∓
ρ + ū +
D±
2D±
D
2D
with
G+ =
(35)
G− =
Z
∞
0
Z
0
−∞
1
1
−
q+
q ψ(ρ)
!
1
1
−
q−
q ψ(ρ)
dρ,
!
dρ,
∞
F̃+ =
Z
F̃− =
Z
0
0
−∞
!
p̃ ψ(ρ)
1
dρ,
−
q+
q ψ(ρ)
!
p̃ ψ(ρ)
1
dρ.
+
q−
q ψ(ρ)
For monotone q(φ), G+ and G− both have the same sign as D+ − D− .
The inner/outer matching conditions (19) then give
A
V0
∓
= ∂r u±
0 (0),
D±
2D±
(36)
ū +
V0 F̃±
AG±
+
= u±
1 (0).
D
2D
The first of these gives the heat conservation condition
−
∂u0
V0 = D
(37)
,
∂r +
which is the leading-order version of the jump condition (2).
We may also determine
A =
and
u1
+
−
1
−
D+ ∂r u+
0 + D− ∂r u0
2
V0
A
= (G+ − G− ) + (F̃+ − F̃− )
D
2D
1
−
=
(G+ − G− − F̃+ + F̃− )q+ ∂r u+
0 + (G+ − G− + F̃+ − F̃− )q− ∂r u0 .
2
2094
ROBERT F. ALMGREN
Thus, though the leading-order outer temperature u0 is always continuous by (25),
for general functions p̃(φ) and q(φ) the O() correction u1 has a discontinuity whose
magnitude varies with the local gradients of u0 .
In order to obtain second-order accuracy in the Gibbs–Thomson condition, we
need u1 to be continuous across the interface. This will always be true if and only if
(38)
G+ = G− = G
and
(39)
F̃+ = F̃− = F̃ ,
which are two integral conditions on p̃(φ) and q(φ). If q(φ) is constant, then (38) is
immediate and (39) is easily satisfied by taking p̃(φ) odd as in [9].
If (38), (39) are satisfied, then (36) relates the temperature of the inner solution
at the center of the interface ρ = 0 to the common inner limit of the outer solution
as r → 0.
4.5. φ at O(1). Using solutions obtained at previous orders, we have
1
β
∂ρρ − g 00 (ψ) Φ2 = ρ(K02 − 2Π0 ) ψ 0 − K0 + V0 Φ1ρ + g 000 (ψ) Φ12
(40)
δ
2
2 1
2 1
µ
β
−
U0 p00 (ψ) Φ1 −
U1 (ρ) p0 (ψ) − K1 +
V0 + V1 ψ 0 .
3 δ
3 δ
Dδ
δ
We determine the solvability condition for Φ2 as for Φ1 : by multiplying both sides
by ψ 0 and integrating over ρ. We break the resulting algebraic condition into three
classes of terms.
4.5.1. Anomalous curvature terms. The first class is the single term (K02 −
R∞
2
2Π0 ) −∞ ρ ψ 0 dρ. We shall assume that g(φ) has been chosen to give a ψ(ρ) with
Z ∞
2
(41)
ρ ψ 0 dρ = 0.
−∞
For example, any even g(φ) gives an odd ψ(ρ) satisfying (41). If (41) is not satisfied,
then terms containing K2 and Π appear in the Gibbs–Thomson condition at O().
From now on, we shall assume that ψ(ρ) is odd.
4.5.2. Anomalous temperature terms. The second class of terms is those
which contain Φ1 or Φ1ρ . One way to eliminate these, as we have noted following
(31), is to choose p(φ) and g(φ) by (7), (8) so that Φ1 ≡ 0. More generally, integrating
the last two such terms by parts, using the defining equation (26) and the solvability
condition (27), we determine
Z ∞ 2 1
1 000
β
2
00
U0 p (ψ) Φ1 ψ 0 (ρ) dρ
− K0 + V0 Φ1ρ + g (ψ) Φ1 −
δ
2
3 δ
−∞
Z
2U0 ∞ 00
ψ (ρ) Φ1 (ρ) dρ.
= −
δ −∞
To interpret this, we recall that Φ1 is determined from ψ and p by (31). We split
p(φ) into symmetric and antisymmetric parts
ps (φ) =
1
p(φ) + p(−φ) ,
2
pa (φ) =
1
p(φ) − p(−φ) .
2
SECOND-ORDER PHASE FIELD ASYMPTOTICS
2095
With ψ(ρ) odd, pa contributes nothing, and we have
Z
4 U02 ∞ 00
ψ (ρ) L−1 p0s ψ(ρ) dρ.
2
3 δ
−∞
Integrating by parts, multiplying and dividing by ψ 0 (ρ), and again integrating by
parts, we see that this is equivalent to
Z
4 U02 ∞ d
1
−1 00
−
L
ψ
(ρ)
ps ψ(ρ) dρ = 0.
2
0
3 δ
ψ (ρ)
−∞ dρ
Now,
0
L ρψ 0 = 2ψ 00 + ρ ψ 00 − g 0 (ψ) = 2ψ 00
using the defining relation (23), and so
1
L−1 ψ 00 /ψ 0 = ρ.
2
Thus, the net contribution from the terms containing Φ1 is −(4/3)(U02 /δ 2 )L, where
the new solvability integral is
Z R
Z
1
1 ∞
ps ψ(ρ) dρ =
(42)
p ψ(ρ) dρ,
lim
L =
2 −∞
2 R→∞ −R
under the assumption that ψ(ρ) is odd. If p(φ) is odd, then L = 0.
4.5.3. Anomalous kinetic terms. The third class is the two remaining terms.
Of these, the complicated one is
Z
2 1 ∞
−
U1 (ρ) p0 ψ(ρ) ψ 0 (ρ) dρ
3 δ −∞
(
)
Z
Z
Z
dσ
A ρ
2 1 ∞
V0 ρ p̃ ψ(σ)
d
−
dσ
ū +
p ψ(ρ) dρ
=−
3 δ −∞
D 0 q ψ(σ)
2D 0 q ψ(σ)
dρ
= 1 + 2 + 3 .
We evaluate the three terms in sequence. As in (29), the first term is simply
4 ū
.
1 = −
3 δ
For 2 and 3 , interchanging the order of integration establishes that for any
function f (σ),
Z ∞Z ρ
d
f (σ) dσ p ψ(ρ) dρ
dρ
−∞ 0
Z 0
Z ∞
1 + p ψ(σ) f (σ) dσ.
1 − p ψ(σ) f (σ) dσ −
=
−∞
0
Thus, we find
!
Z 0
1 − p ψ(ρ)
1 + p ψ(ρ)
dρ −
dρ
q ψ(ρ)
q ψ(ρ)
0
−∞
4 1 A
1
=−
G + F+ − F− ,
3 δ D
2
2 1 A
2 =−
3 δ D
Z
∞
2096
ROBERT F. ALMGREN
using the definitions (35) along with the analogous
!
Z 0
Z ∞
p ψ(ρ)
1
dρ, F− =
−
F+ =
q+
q ψ(ρ)
−∞
0
and the compatibility condition (38). In addition,
4 1 KV0
3 =
,
3 δ 2D
with
K+ =
(43)
!
p ψ(ρ)
1
dρ,
+
q−
q ψ(ρ)
1
K = (K+ + K− ),
2
Z
K− = −
∞
0
Z
0
−∞
1 − p ψ(ρ) p̃ ψ(ρ)
dρ,
q ψ(ρ)
1 + p ψ(ρ) p̃ ψ(ρ)
dρ.
q ψ(ρ)
If p, p̃, and ψ are odd, then q constant implies K+ = K− = K. For p(φ) monotonic
and ψ(ρ) odd, K+ , K− , and K are positive.
Then, using (28), the total second-order solvability condition is
LU 2
β
1
AG A(F+ − F− ) KV0
µ
− 20 −
+
−
V0 + V1 = 0,
ū +
− K1 +
δ
δ
D
2D
2D
Dδ
δ
which with (36) becomes
(44)
u1 Γ = −δ K1 − β V1 −
V0
1
A
Lu20
1
− (F+ − F− )
−
.
µ − (F̃ + K)
2
D
2
D
δ
For general µ, the leading-order velocity V0 appears as a kinetic term in the boundary
condition for u1 , giving an -dependent effective kinetic coefficient. The observation
of Karma and Rappel is that we may eliminate this spurious term by choosing the
special value
(45)
µ=
1
(F̃ + K).
2
In addition, to eliminate the term A/D (recall that A depends on the local gradients ∂r u±
0 ), we need to require
(46)
F+ = F− ,
and to eliminate the nonlinear temperature term u02 we need
(47)
L = 0.
If (45)–(47) are satisfied, then with (30) we have the total condition
(48)
(u0 + u1 )Γ = −δ (K0 + K1 ) − β (V0 + V1 ).
Provided that the constitutive functions are properly chosen, the Gibbs–Thomson
condition (3) is satisfied to two orders in .
SECOND-ORDER PHASE FIELD ASYMPTOTICS
2097
4.6. u at O(1). Finally, we show that heat can also be conserved to second
order. We have
∂ρ q(ψ) U2ρ
= −∂ρ q 0 (ψ) Φ1 U1ρ − K0 q(ψ) U1ρ
− ∆s q(ψ) U0s − |∇s|2 ∂s q(ψ) U0s
V0
V1
V0
1
U1ρ −
p̃(ψ)ρ −
∂ρ p̃0 (ψ) Φ1 .
U0t + st U0s −
+
D
D
2D
2D
Recall that ∆s and |∇s| represent their values on the interface; the dependence on ρ
would appear at higher order in . We substitute (32) and integrate once to find
B
K0 A
K0 V0
−
ρ+
P̃ (ρ) − Q(ρ) ∆S u0
D
D
2D
V0
V1
V0 0
1
u0t + st u0s ρ −
U1 −
p̃(ψ) −
p̃ (ψ) Φ1
+
D
D
2D
2D
q(ψ) U2ρ = −q 0 (ψ) Φ1 U1ρ +
with
P̃ (ρ) =
Z
ρ
0
p̃ ψ(σ) dσ,
Z
Q(ρ) =
ρ
q ψ(σ) dσ,
0
and in which B(s, t) is another constant of integration. Here
∆S u0 = |∇s|2 u0ss + ∆s u0s
is the surface Laplacian of the leading-order temperature.
We determine the boundary condition for ∂r u±
1 (0) by expanding U2ρ as ρ → ±∞.
First we observe that
P̃ (ρ) ∼ ±ρ − H̃± + o(1),
as ρ → ±∞, where
Z ∞
1 − p̃ (ψ(ρ)) dρ,
H̃+ =
0
Z 0 1 + p̃ (ψ(ρ)) dρ,
H̃− =
Q(ρ) ∼ q± ρ − J± + o(1)
J+ =
J− =
−∞
Z
0
Z
∞
0
−∞
q+ − q ψ(ρ) dρ,
q ψ(ρ) − q− dρ.
As in (35), for monotonic q(φ), both J+ and J− have the same sign as D+ − D− and
are zero if q(φ) is constant.
Then, using (34), (38), and (39) and recalling that Φ1 (ρ) → 0 as ρ → ±∞, we
determine the far-field expansion
D± U2ρ ∼
+
(
(
)
V0
A
∓
ρ
D±
D±
)
!
AG V0 F̃
1
+
ū +
∓ V1 + o(1)
D
2D
2
1
−AK0 ± K0 V0 − D± ∆S u0 + u0t + st u0s − V0
2
1
B − K0 V0 H̃± + J± D ∆S u0 − V0
2
as ρ → ±∞. Then the O(1) term in the matching condition
2098
ROBERT F. ALMGREN
Table 1
The effect of violating individual integral conditions.
Condition
(38/57)
(39/56)
(41)
(45/59)
(46/55)
(47/53)
(50/54)
(51/58)
Error at O() if violated
Uniform temperature jump [u]
[u] proportional to V
K2 and Π in Gibbs–Thomson
Anomalous kinetic term in Gibbs–Thomson
Constant term in Gibbs–Thomson
u2 in Gibbs–Thomson
Temperature source term KV
Surface diffusion ∆S u
±
U2ρ (ρ) ∼ ∂rr u±
0 (0)ρ + ∂r u1 (0) + o(1),
ρ → ±∞,
gives the heat conservation condition at next order:
(49)
D
∂u1
∂r
−
= V1 +
+
1
(H̃+ − H̃− ) K0 V0 − (J+ − J− ) D ∆S u0 .
2
The term K0 V0 is the “interface stretching” term of [5]. To eliminate the spurious
terms, we need both
H̃+ = H̃−
(50)
and
(51)
J+ = J− .
We then recover (2) to two orders:
(52)
D
∂(u0 + u1 )
∂r
−
= V0 + V1 .
+
Condition (50) is the analogue of (47) and says that neither p nor p̃ should give the
interface a net heat content.
In (51), J+ − J− is the “deficit” between the conductivity integrated across the
boundary layer and the value that would hold if the bulk values were extended up to
the interface as constants. If this deficit is nonzero, then heat diffuses anomalously
within the interface layer of thickness O().
Table 1 summarizes all the integral conditions we have introduced and the nature
of the error introduced at O() if the condition is violated.
5. The integral constraints. We now analyze the interpolation functions permitted by these integral conditions. We assume
that ψ(ρ) is monotone increasing.
Then, setting φ = ψ(ρ) so dρ = dφ/ψ 0 ψ −1 (φ) , the integral conditions of Table 1 are
equivalent to
SECOND-ORDER PHASE FIELD ASYMPTOTICS
(53)
(54)
(55)
(56)
(57)
(58)
0
dφ
1 + p(φ)
ψ0
−1
Z 0
dφ
1 + p̃(φ)
ψ0
−1
Z 0
p(φ) dφ
1
+
q−
q(φ) ψ 0
−1
Z 0
p̃(φ) dφ
1
+
q−
q(φ) ψ 0
−1
Z 0
1
1
dφ
−
q−
q(φ) ψ 0
−1
Z 0
dφ
q(φ) − q−
ψ0
−1
Z
=
=
=
=
=
=
2099
Z 1
dφ
1 − p(φ)
,
ψ0
0
Z 1
dφ
1 − p̃(φ)
,
ψ0
0
Z 1
1
p(φ) dφ
,
−
q+
q(φ) ψ 0
0
Z 1
1
p̃(φ) dφ
,
−
q+
q(φ) ψ 0
0
Z 1
1
1 dφ
−
,
q(φ) q+ ψ 0
0
Z 1
dφ
q+ − q(φ) 0 .
ψ
0
Each of these conditions expresses a symmetry between φ < 0 and φ > 0, and in
the case of equal diffusivities when q(φ) is constant, they are all easily satisfied by
choosing ψ(ρ), p(φ), and p̃(φ) to be odd. We expect them to be substantially more
difficult to satisfy in the case of unequal diffusivities.
The value of µ required by (45), with the definitions (35) and (43) and with
condition (57), is given by the functional
1
µ[p, p̃, q] =
4
(59)
Z
1
−1
1 − p(φ) p̃(φ) dφ
.
q(φ)
ψ0
Now we would like to find p, p̃, and q satisfying (53)–(58) so that µ is as large as
possible; in particular we would like to achieve µ > 0.
For a given positive q(φ) with q(±1) = q± , let us define
1
q(φ) − (q+ + q− )
2
p0 (φ) =
,
1
(q+ − q− )
2
(60)
and note that p0 satisfies (53) and (54) if q satisfies (57), and p0 satisfies (55) and
(56) if q satisfies (58). We readily calculate
1
dφ
q+ q−
− q(φ)
q+ + q− −
q(φ)
ψ0
−1
(Z 0
1
1
1
dφ
=
q
−
q(φ)
+
q
q
−
−
+ −
(q+ − q− )2
q
q(φ)
ψ0
−
−1
)
Z 1
1
dφ
1
−
q+ − q(φ) + q+ q−
+
q+
q(φ)
ψ0
0
1
µ[p0 , p0 , q] =
(q+ − q− )2
Z
= 0
if q satisfies (57) and (58).
Let us write
p(φ) = p0 (φ) + π(φ),
p̃(φ) = p0 (φ) + π̃(φ).
2100
ROBERT F. ALMGREN
We require π(±1) = π̃(±1) = 0,
Z
(61)
1
1
dφ
=
ψ0
Z
π(φ) dφ
=
q(φ) ψ 0
Z
π(φ)
−1
π̃(φ)
−1
dφ
= 0
ψ0
to satisfy (53), (54), and
(62)
Z
1
−1
1
−1
π̃(φ) dφ
= 0
q(φ) ψ 0
to satisfy (55), (56). Then (59) gives
(63)
µ[p0 + π, p0 + π̃, q] = −
1
4
Z
1
−1
π(φ) π̃(φ) dφ
.
q(φ)
ψ0
The mobility correction is a negative inner product of the two internal energy perturbation functions.
5.1. The gradient case. If the system (4)–(5) is gradient flow, then we require
p̃ = p. In this case, it is not possible to have µ > 0.
Theorem 1. For q+ 6= q− , the largest value of µ attainable by any set p̃ = p and
q that satisfy (53)–(58) is µ = 0.
Proof. With p̃ = p, we require π̃ = π, so that (63) becomes
1
µ[p0 + π, p0 + π, q] = −
4
Z
1
−1
π(φ)2 dφ
.
q(φ) ψ 0
This takes its maximum value µ = 0 when π ≡ 0, that is, p = p̃ = p0 .
Clearly, if p(φ) is monotone, then p(φ)2 ≤ 1 in (59), and hence, µ > 0. Thus
it is impossible to find monotone p̃(φ) = p(φ) satisfying all the integral constraints.
Nonmonotone p(φ) means that when the interface is moving, latent heat is being both
released and absorbed within the interface layer.
Our task now is to find reasonable interpolation functions q(φ) which satisfy (57),
(58). If these functions also have q 0 (±1) = 0, then p = p0 as given by (60) is the best
choice for p. We restrict our attention to g = ḡ, so ψ = ψ̄ = tanh ρ and ψ 0 = 1 − φ2 .
Let us consider a matching pair of the form
1
1
(q+ + q− ) + (q+ − q− ) p0 (φ),
2
2
1
2
p0 (φ) = φ(3 − φ ) + a φ(1 − φ2 )2 ,
2
q(φ) =
(64)
which satisfies all the desired boundary conditions including p0 (±1) = 0. We take
π = π̃ = 0, so if q satisfies (57), (58), then p, p̃ automatically satisfy (53)–(56) and
give µ = 0. We thus apply no correction to the kinetic coefficient; the only result of
the second-order asymptotics is the correct choice of a, which will be different from
the naive choice a = 0.
Since p0 is odd, q automatically satisfies (58). Condition (57) must be integrated
numerically, yielding one nonlinear equation for a which is readily solved. There is one
positive and one negative solution for a; the positive one gives the smoother function
p(φ). Figure 1 gives the numerically determined roots for q− < q+ ; the value of a
is unchanged under the exchange q− ↔ q+ . As q− /q+ → 0, a tends to the limiting
2101
SECOND-ORDER PHASE FIELD ASYMPTOTICS
1.5
a = 1.450
a = 1.510
a = 1.245
a = 1.066
1
a
a = 0.907
0.5
a = 0.375
0
0
0.2
0.4
0.6
0.8
1
q−
Fig. 1. Numerically determined values of the coefficient a as a function of q− with q+ = 1,
with values for q− = 0, 0.05, 0.1, 0.2, 0.5, and 1.
1
p(φ)
0.5
0
−0.5
−1
−1
−0.5
0
φ
0.5
1
Fig. 2. The specific energy interpolants with q+ = 1, q− = 0.1. The solid line is the interpolant
p(φ) for the gradient case; the diffusivity interpolant q(φ) has the same shape. The two dashed lines
are example interpolants p(φ) and p̃(φ) for the nongradient case.
value 3/8 for which p(φ) and q(φ) become monotone. The limiting value as q− → q+
is determined by an asymptotic analysis which we omit here. The resulting q(φ) for
q+ = 1, q− = 0.1 is shown in Figure 2. Numerical computations in one dimension
confirm the second-order convergence of this scheme.
When q+ = q− , formula (63) does not apply and the above theorem is not true.
In that case, as observed by Karma and Rappel, it suffices to take q(φ) ≡ 1 and any
odd function p(φ). For example, with the cubic corresponding to a = 0 above, we
have µ = 19/60. Since for finite the behavior of the system for q− close to q+ must
be close to its behavior for q− = q+ , the limits → 0 and q− → q+ do not commute.
2102
ROBERT F. ALMGREN
5.2. The nongradient case. By relaxing the constraint that p̃ = p, we can
easily achieve µ > 0 in (63) by choosing π̃ = −π. We again restrict our attention to
g = ḡ, with ψ 0 = 1 − φ2 , and choose q and p0 by (64). The values of a so that q(φ)
satisfies (57), (58) are as above.
We consider odd perturbation functions of the form
π(φ) = b2 φ(1 − φ2 )2 + b3 φ(1 − φ2 )3 ,
which give p0 (±1) = 0 for all choices of b2 and b3 . Condition (61) is automatically
satisfied by symmetry, and (62) reduces to a linear condition between the bk . Passing
over the details, Figure 2 shows one example of the resulting p and p̃. These functions
have µ > 0, but are otherwise rather impractical since their oscillations give the
interface a very fine internal structure.
If q− = q+ , then an effective model can be constructed by taking the odd functions
p(φ) = (1/2)φ(3 − φ2 ), p̃(φ) = φ, which give µ = 5/12.
6. Isothermal asymptotics. Karma and Rappel have proposed an asymptotic
analysis slightly different than the second-order one considered above. In the case of
equal diffusivities the result is substantially the same, but with unequal diffusivities
the conclusions are different.
Let us define λ = /δ and β1 = β/. We rewrite (4) as
2
1
µλ
0
0
(65)
φt = − 2 g (φ) − λ u p (φ) + ∆φ
β1 λ +
D
3
and keep (5) as before. We carry out an asymptotic expansion as in section 4, now
holding λ, β1 ∼ O(1) fixed as → 0, so that the capillary length and kinetic coefficient
go to zero. In this expansion, we assume that the interface curvatures and velocity
remain finite in the limit; this will be true for some solutions of the limiting system
though not for all.
In the u equation at O(−2 ), we find, as in section 4.2, that U0 is constant in the
interface region. In the φ equation at O(−2 ), we find the solvability condition (23)
plus an additional term
(2/3)λU0 p0 (Φ0 ); this equation has a solution Φ0 = ψ(ρ) if
and only if U0 = u0 Γ = 0.
In the u equation at O(−1 ), the analysis proceeds exactly as in section 4.4. The
inner temperature profile U1 (ρ) is again given by (33); heat conservation requires the
same jump condition (37) on [∂u0 /∂n] and we obtain the same condition for [u1 ]. For
continuity of u1 , we require again the two compatibility conditions G+ = G− and
F̃+ = F̃− (38), (39).
The leading-order problem for u0 is the classical Stefan problem with zero surface
tension and zero kinetics, which has ill-posed solutions. The only regularization is the
boundary conditions on u1 ; these come from the φ equation at O(−1 ), which retains
much of the flavor of the previous expansion at O(1) in section 4.5. In place of (26),
we obtain
2
µ
λ V0 ψ 0 (ρ) − λ U1 (ρ) p0 ψ(ρ) .
∂ρρ − g 00 (ψ) Φ1 = − K0 + β1 +
D
3
Substituting U1 (ρ), multiplying by ψ 0 , and integrating lead now to the solvability
condition
A
V0
1
1
1
F+ − F−
(66)
u1 (0) = − K0 − β1 V0 − µ − (F̃ + K)
−
,
λ
2
D
2
D
SECOND-ORDER PHASE FIELD ASYMPTOTICS
2103
which is similar to (44) without the final term Lu02 /δ; we do not need to assume that
ψ(ρ) is odd or to impose the symmetry condition (41) on ψ(ρ). To eliminate the
constant, we again require F+ = F− (46); we eliminate the “spurious” kinetic term
by again choosing µ by (45). We never need to impose (47) for symmetry of p(φ) to
eliminate L. The Gibbs–Thomson condition now appears as u1 (0) = −(δ/)K0 −β1 V0 ,
or
(u0 + u1 )Γ = −δ K0 − β V0 .
We have now determined each of the
two sharp-interface conditions to the leading
+
nonzero order in : ∂u0 /∂n − and u1 Γ . Although we may stop here if we choose, it
is nonetheless interesting to continue the expansion and derive as much information
as possible about the nature of the corrections to the boundary conditions we have
obtained.
The expansion of the u equation at O(1) proceeds as in section 4.6, except that
the terms involving u0 are absent. The final boundary condition (52) now becomes
(67)
D
∂(u0 + u1 )
∂r
−
+
1
= V0 + V1 + (H̃+ − H̃− ) K0 V0 .
2
The surface diffusion term ∆S is absent whether or not (51) is satisfied; the only
source of error is the interface stretching term K0 V0 . If we want to eliminate O()
errors in the heat conservation condition, we must again require H̃+ = H̃− (50).
At this point we must stop: expansion of the φ equation at O(1) would require
knowledge of U2 (ρ) and is impractical. We have no information about the nature of
the errors in the Gibbs–Thomson condition.
6.1. Interpolation functions. We now construct interpolation functions satisfying the “base” conditions (38/57), (39/56), and (46/55) (see Table 1). Our asymptotic analysis gives us no reason to try to satisfy (41), (47/53), or (51/58). We choose
not to try to satisfy (50/54), accepting the O() interface stretching term. Under this
reduced set of constraints, it is easy to construct interpolants giving µ > 0. We take
g(φ) = ḡ(φ) from (7) so ψ(ρ) is odd (thereby satisfying (41)).
Gradient case. We write interpolation functions
1 1
1
1 1 1
1
=
φ + bφ(1 − φ2 ) ,
+
−
+
q(φ)
2 q+
q−
2 q+
q−
1
p̃(φ) = p(φ) = φ(3 − φ2 ) + a(1 − φ2 )2 .
2
Since 1/q is odd about its midpoint, it satisfies (57) for any b. Then (55), (56) give
the algebraic condition
3
5
q+ − q−
b −
a =
.
α,
α=
5
4
q+ + q−
For example, b = 12 is the largest value for which q(φ) is monotone; in this case, we
have a = −(19/20)α and, from (59),
1
3,173 (q+ − q− )2
19
−
.
µ =
30
63,000
q+ q−
q+ + q−
2104
ROBERT F. ALMGREN
1
1
p(φ)
0.8
q(φ)
0.5
0.6
0
0.4
−0.5
0.2
−1
−1
−0.5
0
φ
0.5
1
0
−1
−0.5
0
φ
0.5
1
Fig. 3. Interpolants for “isothermal” asymptotics with diffusivity ratio 10 : 1. The dashed line
is p̃(φ) for the nongradient case.
p(φ) is monotone for q− /q+ > 0.434; µ is positive for q− /q+ > 0.069. These functions
are shown in Figure 3 for q+ = 1, q− = 0.1, for which µ = 0.205.
As q− → q+ , these interpolants reduce smoothly to the standard ones for equal
diffusivities.
We can eliminate the interface stretching term by choosing b so as to satisfy (54),
giving b = 25/12. But then q(φ) is nonmonotone: it exceeds its bulk values and in
fact 1/q(φ) touches zero for rather mild values of q− /q+ .
Nongradient case. We keep q(φ) and p(φ) as above but choose the slightly simpler
form
p̃(φ) = φ + ã(1 − φ2 ).
Condition (55) determines a as above. From (56) we find
1
b − 1 α.
ã =
3
For b = 21 , we have ã = −(5/6)α and
1
493 (q+ − q− )2
5
−
.
µ =
6
12,600
q+ q−
q+ + q−
For q+ = 1, q− = 0.1, we have µ = 0.469; since µ is larger, this system will be easier
to solve than the gradient case. The corresponding p̃(φ) is shown in Figure 3. In this
case, satisfying (54) requires b = 3, even worse than above.
6.2. Relation to standard asymptotics. The conclusions of the above asymptotic analysis can also be obtained by letting δ, β → 0 in the results of section 4. Under
our assumption that interface curvature and velocity remain bounded in this limit,
we then also have u0 Γ → 0. The conditions that we imposed in sections 4 and 5 that
we did not impose in this section are (47), (50), and (51).
The errors introduced by violation of (47) and (51) are, respectively, a nonlinear
term (/δ)u02 in the Gibbs–Thomson condition (48) and a surface diffusion term ∆S u0
in the heat conservation condition (52). As u0 → 0, both of these terms vanish: the
SECOND-ORDER PHASE FIELD ASYMPTOTICS
2105
first is of size λ2 and is smaller than the leading term u1 of size O(); the second is
of size 2 and is two orders smaller than the leading term [∂u0 /∂r] of size O(1).
If (50) is violated, then an interface stretching term K0 V0 of size appears in (52).
Though small as long as the interface thickness is small compared to the radius of
curvature, this term is formally the largest error remaining. We may choose whether
or not to attempt to eliminate it.
7. Conclusions. We have carried out the asymptotic analysis of the phase field
system (4), (5) in the sharp-interface limit → 0 with finite capillary length δ and
kinetic coefficient β and identified all the error terms that appear at O(). In section
4, we have shown how, if the interpolation functions satisfy a list of integral constraints, all these errors can be eliminated; the diffuse-interface model then yields a
second-order accurate approximation to the sharp-interface system (1), (2), (3). In
section 5, we have considered the possible interpolation functions that satisfy all these
constraints.
In Karma and Rappel’s case of equal diffusivities, a “miracle” occurred: in the
realistic physical regime when the kinetic coefficient β = 0 in the Gibbs–Thomson
condition (3), second-order accuracy could easily be obtained with a positive kinetic
coefficient m = µ/δ in the phase equation (4). Thus a nonequilibrium parabolic
equation could be used to accurately approximate a sharp-interface equilibrium condition.
For unequal diffusivities, this miracle does not occur. If the overall dynamics is
constrained to be gradient flow for an overall scalar functional F, then second-order
accuracy cannot be obtained with µ > 0. If the constraint of gradient structure
is relaxed, then it is possible to take µ > 0, but the necessary oscillations in the
interpolation functions make them not very useful. The limit of diffusivities becoming
equal differs from the behavior with exactly equal diffusivities.
In section 6, we have also analyzed the phase field model under Karma and
Rappel’s model that the capillary length δ and kinetic coefficient β scale with as
→ 0. In this limit, some of the error terms discovered at second order in section 4
appear at what becomes the leading order, while others remain at higher orders. If
we choose to eliminate only the leading-order errors, then the interpolation functions
must satisfy only a subset of the previous integral constraints; in section 6.1 we
have shown how to construct interpolation functions satisfying this reduced set giving
µ > 0. In section 6.2 we have shown how the additional error terms of section 4 are
pushed to higher orders as δ, β → 0.
To further illustrate the relationship between these two asymptotic analyses, let
us reverse the nature of our analysis. Suppose that we plan to carry out computations using a given system of phase field type, embodying a particular choice of all
interpolation functions and coefficients. We want, first, to decide what sharp-interface
model best describes the motion of the zero level set of φ for this model and, second,
to assess the nature of the discrepancies to be expected between the sharp and the
diffuse solutions.
From the structure of the given phase field model, we can identify the interpolation
functions p(φ), p̃(φ), and q(φ), and we can compute the integral quantities F± , G± ,
etc. We shall suppose that G+ = G− , F̃+ = F̃− , and F+ = F− (from (38), (39), (46));
if not, then error terms appear of a nature we have not considered. Then F , G, K,
L, etc. are determined. From the values of the constants in our phase field model, we
unambiguously identify , λ, δ, m, and D (scaling g(φ) to satisfy (28)).
We now determine the effective sharp-interface Gibbs–Thomson condition u =
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ROBERT F. ALMGREN
−δ∗ K − β∗ V which best describes the resulting computation. The capillary length
δ∗ = δ is unambiguously determined, regardless of which asymptotics we believe we
are using.
For the kinetic coefficient, we cannot unambiguously identify either β or µ individually, but only the linear combination Dβ + µ = mδ, using either (6) or (65).
However, both the asymptotics of section 4 and that of section 6 yield the same
effective kinetic coefficient
δ
λ
F̃ + K
m−
β∗ =
,
D
2
regardless of the choices made for β and µ separately. Thus, the best sharp-interface
model of Gibbs–Thomson form is unambiguously determined, regardless of which
asymptotics we use to interpret the phase field model. As noted in [10], this kinetic
coefficient may be zero or negative with m positive.
For each integral condition (53)–(58) which is not satisfied by the constitutive
functions used in the model, a “corrected” sharp-interface model should contribute
additional boundary terms of size O(). If δ and β happen to be small along with ,
then some of these error terms will be much smaller than others; those are the ones
which do not appear in section 6.
Finally, we point out that our asymptotic analysis has not included anisotropy of
surface energy. In the second-order asymptotics, inclusion of anisotropy will generate
complicated new error terms, since the anisotropy must be expanded to two orders.
In the small-δ, β asymptotics, no additional difficulties arise: the surface energy is
simply expanded to leading order as in [10].
An additional extension would be to include different heat capacities in solid and
liquid, which might be important in certain physical problems.
Acknowledgment. We thank Alain Karma for many helpful conversations, and
especially for the frogs and rats and elephants.
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