Curvature-Driven 4th-Order Diffusion from Non-Invariant Grain Boundaries.

Philip Broadbridge

Acknowledgements to P. Tritscher, J. M. Goard, M. Lee, D. Gallage,

J. Hill, D. Arrigo, D. Triadis, W. Miller Jr., P. Vassiliou, P. Cesana

PDEs for evolution of curves and surfaces under isotropic and homogeneous processes should be invariant under Euclidean group.Simple example is evolution by mean curvature.Hypersurface of dimension n-1 embedded in Rn.

� ⇥� ⇤n

n = ‘inward’ unit normal vector.

n · @r(✓, t)@t

= B

This models surface of volatile metals e.g. Mg.Surfaces of stable metals e.g. Au, evolve by 4th ordersurface diffusion. In 2D,

@N

@t= �B

2

@2

@s2

HRTEM image of grain boundary. Z. Zhang et al, Science 302, 846-49 (2003)

−3 −2 −1 0 1 2 3−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

�b

�s

�

�s

m = tan(⇥) ; �b(T ) = 2�s(T )sin(⇥)

Tritscher & Broadbridge, 1995

!

Metal surface near grain boundary

� = particle density

� = mean volume per particle

v = mean drift velocity

v =�Ds

kT

���s

Nernst-Einstein Relation

� = chem pot’l per particle;

T = temp; k = Boltzmann const; Ds = surf diff const

J = ��vVolume flux on surface

� = ⇥ [�s(⇤) + ���s (⇤)]⇥.

Laplace-Herring Equation 1814 - 1950

�s = surface tension

� = arctan yx

Curvature

Sub this flux model in local cons of mass �N

�t+

�J

�s= 0

� =�yxx

(1 + y2x)3/2

Eq of continuity implies

⇥N

⇥t= B

⇥2�

⇥s2(B constant)

cos(⇥)⇤y

⇤t= B

⇤2�

⇤s2

�y�N

Now use � =����d2rds2

����

=

�⌥⌥⌃⇤

dx

ds

d

dx

1⇧1 + y2

x

⌅2

+

⇤dx

ds

d

dx

�1⇧

1 + y2x

yx

⇥⌅2

= f (�) �x

�[f �(�)]2 + [�f �(�) + f(�)]2

where � = yx , f(�) =1�

1 + �2= cos ⇥

Mullins Surface Diffusion Eq + bdry conds for grooving

yt = �B �x

⇤�1 + y2

x

⇥�1/2�x

yxx

(1 + y2x)3/2

⌅

�t = �B ⇥2x

⇤f(�) ⇥x

��xf(�)

⇧[f �(�)]2 + [�f �(�) + f(�)]2

⇥⌅

� � 0 , �x � 0, x�⇥.

⇤x

��xf(�)

⇤[f �(�)]2 + [�f �(�) + f(�)]2

⇥= 0 , x = 0

� = m ,x = 0, t > 0

� = 0 , t = 0 x � 0 ;

yx =

y� = ��x [D(yx)�x [E(yx)yxx]]

Anisotropic surface diffusion in terms of rescaled dimensionless variables,

� = 0; y = 0

x�⇥; y � 0 , yx � 0

x = 0; J = 0 �⇥ �x [E(yx)yxx] = 0

x = 0 ; yx = m(�).

m = tan(⇥) ; �b(T ) = 2�s(T )sin(⇥)

Surface tension

D(⇥) =�

� + ⇥, E(⇥) =

��

� + ⇥

⇥3

,

closest to isotropic model

E(�) =1

(1 + �2)3/2D(�) =1

(1 + �2)1/2,

when � = 2.026

Integrable model

�(�), � = tan�1✓,

E / [�00(�) + �]/[1 + y2x

]3/2 (Herring eq.)

0.5

1

1.5

30

210

60

240

90

270

120

300

150

330

180 0

Polar plot of surface tension vs angle for integrable model

D = D1(t)D2(yx); E = E1(t)E2(yx)

⌧ =

Zt

0D1(t)E1(t)dt

From explicit time dependence of mobility and surface tension, due to temperature change,

, new time coordinate

µ =�

� + ⇥z =

� x

0

� + ⇥

�dx

µ� = �µzzzz �1�

R(⇤)µz ,

where R(�) = �y� (0, �).

Change of variables

Transforms governing eq to linear PDE

=⇥ z = 0, µzzz =�R(⇤)

� + m(⇤).

Z = z +1�

y(0, ⇤) , µ� = �µZZZZ

which has scaling symmetry

Z = e�Z, ⇥ = e4�⇥, µ = µ.

In terms of canonical coords,

Y = Y, S = S + ⇥, µ = µ.

Then separation of variables is possible

µ = F (S)G(Y ).

Y = Z��1/4, S = log(�1/4),

µ = F (S)G(Y ).

+K2jY 1F3

�⇤14� j

4

⌅,

⇤12,34,54

⌅,

Y 4

256

⇥

+K3jY21F3

�⇤12� j

4

⌅,

⇤34,54,32

⌅,

Y 4

256

⇥

+K4jY31F3

�⇤34� j

4

⌅,

⇤54,32,74

⌅,

Y 4

256

⇥.]

µ0(Y ) +�⇧

j=0

⇥ j/4 K1j 1F3

�⇤�j

4

⌅,

⇤14,12,34

⌅,

Y 4

256

⇥

[x 1

y(0, ⇥) = �⇥1/4��

i=0

bi⇥i/4

1F3([a], [b1, b2, b3], z) =��

k=0

(a)k

(b1)k(b2)k(b3)k

zk

k!

(a)k = a(a + 1)(a + 2) . . . (a + k � 1)

Generalized hypergeometric function

For m const, we have similarity solution of form

y��1/4 = H(x��1/4)

which has scaling symmetry

For m varying with t, assume

These expansions for y(0,t) and µ(Z,t) can solve the free boundary problem with boundary conditions specified at unknown location

x = 0 () Z = y(0, ⌧)/�.

In order to implement boundary conditions at infinity,match Laplace transform of power series solution with

µ = exp

��p

14 Z⇥2

⇥⇤C3(p) cos

�p

14 Z⇥2

⇥+ C4(p) sin

�p

14 Z⇥2

⇥⌅+

1p

µ = exp

��p

14 Z⇥2

⇥⇤C3(p) cos

�p

14 Z⇥2

⇥+ C4(p) sin

�p

14 Z⇥2

⇥⌅+

1p

By matching with , C3(p) and C4(p) can be deduced. With j fixed, K1j, K2j, K3j and K4j are linearly dependent since they are functions of the two independent power series coefficients in

µ(0, ⌧), & µZ(0, ⌧)

L�1C3(p) and L�1p1/4C4(p)

_________ t=0.0002 ……………t=0.1 ---------------t=1

m(�) = 1/2 + 1/2 �1/4

Mullins 1957 theory of evaporation-condensation.Lateral mixing of vapour keeps pressure close to that above flat surface. Linear model of non-equilibrium evaporation rate gives

�⌅N

⌅t= �J =

��(peq � p)⇥2⇥mkT

,

Laplace showed that energy per unit volume of surface material is

2�s⇥

Omega being spec vol per particle.

so energy platform per particle is . E = 2��s⇥

Gibbs-Thompson formula

implies evaporation rate proportional to mean curvature.

peq � p = p(eE/kT � 1) ⇡ 2p⌦�/kT

�y�N

cos(�)⇥y

⇥t=

1(1 + y2

x)1/2B

yxx

1 + y2x

�y

�t= B

yxx

1 + y2x

Around linear groove, z-direction is irrelevant. Evaporation equation reduces to curve-shortening equation.

CSE in differentiated form

Fujita 1952-54 presented exact similarity solutions for u(x,t) with

Bluman and Reid 1988: these come from “potential” symmetries, or from point symmetries of the system

yt

= D(yx

)yxx

✓t

= @x

[D(✓)✓x

] = @2x

[arctan(✓)]; ✓ = yx

; D(✓) = B/(1 + ✓2)

yx

= ✓ ; yt

= D(✓)✓x

D(✓) =1

a✓ + b, D(✓) =

1

(a✓ + b)2, D(✓) =

1

(a✓ + b)2 + c

P. Tritscher & PB,I.J. Heat Mass Transf 1994

Broadbridge 1989: Exact CSE parametric similarity solution for grain boundary groove � ⇥� (�, ⇥) = (x/2

⇤t, y/2

⇤t)

⇥ = ��1/2⇤sin[F (⇤; �)] + (1� ⇤2 � �ln⇤)1/2cos[F ] ; 0 ⇥ � ⇥ �⇥

⇥ = ��1/2⇤sin[G(⇤; �)] + (1� ⇤2 � �ln⇤)1/2cos[G] ; �⇥ ⇥ � ⇥ 1

⌅ = m[⇥��H(⇤; �)]

F (⇥; �) =� �

0(1� q2 � �lnq)�1/2dq

G(⇥; �) = tan�1m� F (⇥; �) + F (⇥m; �)

H(⇥; �) = ��1/2m�1⇥ sec[F (⇥; �)] ; 0 � � � �⇥

H(⇥; �) = ��1/2m�1⇥ sec[G(⇥; �)] ; �⇥ � � � 1

�⇥ = m�1tan[F (1; �)] ; � =⇥2

m � 1�m2

⇥2mln ⇥m

2F (1; �)� F (⇥m; �) = tan�1m

� = m�1tan[F (⇥; �)] ; 0 � � � �⇥

� = m�1tan[G(⇥; �)] ; �⇥ � � � 1

Arrigo et al 1997: for classical diffusion, groove depth F(0) proportional to m but for CSE with m sufficiently large,

�ln(

m

2⌅

�) +

14

⇥1/2

� 12⇥ |y(0, t)|⇤

(Bt)⇥

�2ln(

m⌅�

)⇥1/2

Groove depth due to surface diffusion.

zt = B[zrr

(1 + z2r )1/2

+1

rzr]

Evaporation at axi-symmetric surface by mean curvature:

✓ = zr; ⇢ = rt�1/2; ✓ = f(⇢)

zr = m, r = at1/2 ; z = 0, t = 0 ; z ! 0, r ! 1

r=0 z=0 r=at1/2

phase boundary

Gallage, PB, Triadis and Cesana use inverse methodpreviously applied to 1D nonlinear diffusion J. Philip (1960).

D(✓) =�0.5B�1 d⇢

d✓

R ✓0 ⇢d✓

1 + ✓(1 + ✓2)d ln(⇢)/d✓

Isotropic model

Well-known solutions (mostly trivial) for curve-shortening eq.

Straight lines are steady states.

Shrinking circles radius R(t)=[2B(tc-t)]1/2 .Tubular pipe of outer,inner radii R,r becomes simply connectedat time r2/2B and disappears at time R2/2B

Calabi “grim reaper” travelling wave

y + ct =1clog(cos(cx))

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−9

−8

−7

−6

−5

−4

−3

−2

−1

0

.

−5 0 5−6

−5

−4

−3

−2

−1

0

Obtuse open-angle solution = grain boundary solution.

Other similarity reductions give rotating spirals, or Abresch-Langer 1986 “flowers”with n intersecting petals.

All of these solutions can be recovered by symmetry reductions from potential symmetries of CSL, i.e. symmetries of system

yx = u ; yt = D(u)ux

=� yt = D(yx)yxx

and ut = �x[D(u)ux]

Doyle & Vassiliou 1998 classified nonlinear diffusion equations

ut = �x[D(u)ux]

according to compatibility with functional separation of variables

�(u) = f(x)g(t)

For D(u)=1/(1+u2), this leads to

(1) u=yx=tan(x) -> y=log(cos x) +t +const (vertical grim reaper),

(2) u=x/[2(t0-t) -x2]1/2 -> shrinking circle

(3) u=[e2(x-t) -1]-1/2 -> rotated (horiz) grim reaper

u2 + 1u2

= (1 + e2t) cosec2(x),(4)

->

(K,x0,t0 arbitrary)

y =1K

log

�⇤exp(2BK2[t� t0]) + cos2(K[x� x0]) + cos(K[x� x0])

exp(BK2[t� t0])

⇥.

(5) u=cosh(x)/[e-2t-sinh2 x]1/2 -> rotation of (4) above

- A periodic solution that can satisfy boundary conds y=0 at x=0,L -J. R. King,

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

x/l

y/l

Each peak or valley is asymptotic to a grim reaper as Initial conds may resemble diffraction grating.

t⇥ �⇤

---- approx bounds y = ±K [t0 � t] + log(2)/K

B(t-t0)=-0.07

(6) u= ± sin x [cos2 x-e2t ]-1/2 ->

cosh(y)� et0�tcos(x) = 0

x2 + y2 = 2(t0 � t) + O([t0 � t]2)

x0 1

y 0

1

2

Figure 1: Evolution by heat shrinking flow of cosh y-5cos x=0

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

asymptotically approaches two coalescing grim reapers as .t⇥ �⇤

t-t0=-10

grimgrim

Long before extinction, ymax ⇥ t0 � t + log(2),and near extinction, ymax ⇥

�2(t0 � t).

Anisotropy

Physical requirement:

After rotation of axes by !/2, 0 < B0 = D(0) <�

0 < B0 = D(0) <�

D(yx) =1

yx2D

��1yx

⇥⇥ B0y

�2x

After rotating back,

yt = B(yx)yxx

1 + y2x

= D(yx)yxx

Doyle-Vassiliou classification of diffusion eqs compatible with functional separation

log �(u) = v(x) + w(t)

gives one more physically realistic model:

D(u) = D0cos(z(Au))

Au =� z

0(cos s)�3/2ds ; � �/2 < z < �/2

D(u) =1

1 + u2+ O(u4) u small

= (u/�

2)�2 + O(u�4) u large

With A= 20.5,

−6 −4 −2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u

D

−3 −2 −1 0 1 2−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

x

y

t=−2.0

−15 −10 −5 0 5 10 15

−10

−5

0

5

10

x

y

t=−8.0

Functionally separated solution approaches grim reaper at large negative times, like others that we saw above !!

Grooving by evaporation-condensation solved exactly: PB 1989

Grooving by surface diffusion, with slope yx(0,t) constant, on nearly isotropic surface, solved by Tritscher & PB 1995.

Solved exactly when m depends on t, due to temperature dependence of surface tension-PB & Goard 2010.

For what anisotropic materials do Angenent ovals, grim reapersand diffraction grating solutions exist ?—PB & P. Vassiliou 2011

Set up reliable semi-discrete approximationfor boundary conditions that can’t be treated analytically- Zhang & Schnabel 1993, Lee 1997, PB & Goard 2012.

Integrable nonlinear discretised model with self-adapting non-uniform grid- PB, Kajiwara, Maruno & Triadis, in progress.

P. B. & J. M. Goard, “Temperature-dependent surface diffusion near a grain boundary”, J. Eng. Math., 66(1-3), 87-102 (2010).

D.J. Arrigo, P. B., P. Tritscher & Y. Karciga, "The Depth of a Steep Evaporating Grain Boundary Groove: Application of Comparison Theorems", Math. Comp. Modelling, 25 (10), 1-8 (1997).

P. Tritscher & P. B., "Grain Boundary Grooving by Surface Diffusion : an Analytic Nonlinear Model for a Symmetric Groove", Proc. Roy. Soc. A, 450, 569-587 (1995).

P. B. & P. Tritscher, " An integrable fourth order nonlinear evolution equation applied to thermal grooving of metal surfaces", I.M.A. J. App Math., 53,249-265(1994).

P. B., "Exact Solvability of the Mullins Nonlinear Diffusion Model of Groove Development", J. Math. Phys. 30, 1648-51 (1989).

P. B. & J. M. Goard, “When central finite differencing gives complex values for a real solution !”, Complex Variables and Elliptic Equations, 57 (2-4), 455-467 (2012).

J. M. Goard & P. B., "Exact solution of a degenerate fully nonlinear diffusion equation", Journal of Appl. Math. and Physics (Z.A.M.P.) 55(3), 534-538 (2004) .

P. B. & P. J. Vassiliou, “The Role of Symmetry and Separation in Surface Evolution and Curve Shortening”,Symmetry, Integrability & Geom: Methods & Applics (SIGMA), 7, 052, 19 pages (2011).