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Faceted Crystals Grown from Solution

A Stefan Type Problem with a Singular Interfacial Energy

Yoshikazu GigaUniversity of Tokyo and

Hokkaido University COE

Joint work with Piotr Rybka December , 2005 Lyon

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A basic problem from pattern formation in the theory of crystal growth.

In what situation a flat portion (a FACET) of crystal surface breaks or not ?

Goal : We shall prove :

‘All facets are stable near equilibrium for a cylindrical crystal by analysizing a Stefan type problem’

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Contents

1 Model2 Problem3 Main mathematical results4 Three ingredients － ODE analysis － Berg’s effect － Facet splitting criteria －5 Open problems

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1 Model

Crystals grown from vapor

(snow crystal)

from solution (NaCl crystal)

<driving force : supersaturation>

(density of atoms outside crystal is small)

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Stefan like Model

3 (crystal at ti( ) me )t t

: supersaturation e

e

C C

C

: concentration of atomsC

: saturated concentrationeC

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3

(1)

quasi-steady approximation of

diffus slowion eq. (process is )

=0 in \ ( )t

| |

(2)

lim ( (given, ) )x

x t

(3) Stefan condition

(conservation of mass)

on ( )

V tn

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unnormalized version :

c

Cv D V

n

: volume

of an atomcv

( )t

: diffusion

coefficient of

atoms in a solution

D

n

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(4) Stiffness and

mobility on surface

div on ( )

: Cahn-Hoffman

V t

3

(5) ( ) ( ( ))

: a given interfacial energy

( ) ( ) , , 0

: convex

= ( ) 0 given

kinetic coefficient

1/ : mobility

x n x

p p p

n

"

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Remark. If ( ) | | then

-div mean curvature

In 2 setting

-div ( )

: curvature

( ) (cos ,sin )

: stiffness coefficient

p p

D

K

K ( )t

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We shall consider (1) － (5) for given

quasi-stationary

One phase Stefan problem with Gibbs-Thomson + kinetic effect

(0).

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Solvability (smooth )

K. Deckelnik - C. Elliott ’99

( Hele Shaw type )

No 　　… Friedman –Hu ’92

　　 Liu – Yuan ’94

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Others (No )

Kuroda-Irisawa-Ookawa ‘77

Stability of facets

Experiment e.g. Gonda-Gomi ’85

(No ) : Fingering :

Saffman-Taylor

R.Almgrem ’95

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2.Problem (specific to ours)

1

1 2 3

2 2 23 1 2

3

: may not be . We assume

(6) ( , , )

| | ,

so that the Frank diagram

{ | ( ) 1}

positive

consists of

const

TB

C

x x x

r x r x x

F p p

two straight cones

with common basis

1/ TB

1/1not C

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3

| | 1

2 2 21 2 3 1 2

3

We take such so that

equilibrium shape

Wulff shape

= { | ( )}

{( , , ) | ,

| | }

i cylinde rs a

m

TB

W

x x m m

x x x x x

x

TB

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1Difficulty : is not

so the meaning

(5) = (n)

is not clear.

We shall interpret

(5) by using

subdifferentials

C

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3

| |

(0) givenInitial condition

(5 ') ( )

Our prob

( (

(1) =0 in ( )

(2) lim 0

)) on (

(3) on ( )

(4) div on ( )

(6) Frank

(7)Symmetry assumption

)

lem

x

x n

t

V tn

V

x

t

t

0, 0, ( )T B i in

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Postulate that

velocity of

each facet

, ,

is so that

th

Red

e cylinder stays

as a cylind

consta

uced

nt

Pro

e

b em

r.

l

T BS S S

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Then

integrating

(4) on

and taking

the average

yields

iS

21(8)

| |

, ,i

i i ii S

d VS

i T B

H

2( , )

crystalline curvature

TBT

R L R

S BS

( )R t

( )L tTS

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3. Main Math Results

1

0

1

Th (Rybka-G '02)

Actually,

the problem

is reduced to

ODEs fo

local-in-time

solution

{ ( )} for

(1)-(4),(6)

r

( ( ), ( )).

equilibrium

2 2

,( (8)

( ,

7)

)

, .

TB

R t L t

t

z

In general

this not

fulfill (5)

may

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Th (Rybka-G ‘04)

If

is close to the Equilibriumthen the solution solves the original problem(1),(2),(3),(4),(5),(6),(7)Near equilibriumFacet does not break.

( (0), (0))R L

0,z

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Reduction to ODE

0 in

on ,

( ( ) 0)

, 0

( )

ci

iij j

i

i ii I

f

fS n

f

Vn

V f

n

n

{ , , }I T

22

(( , )) | | |

By definition of

for all

Integrating ( ) over yields

Here (( , ))

Note

| | |

ODE for ( , )

ci

c

i j i j j j

i

i

S

j j j

T

I

j

i

V f f S S V S

R

f

f

S

f h f h

dLV

L

,dR

Vdt dt

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e

0 0

Unique local solvability

( , ) (( , )) Lipschitz.

( )

R ( ) {0}

(0) 0

| ( ) ( ) | ( ) | |

if | |,| |

where (0) 0 a

Phase

det A

Potrait near

<0

i jR L f f

Rz z

L

dzAz F z

dtA

F

F x F y x y

y

z

x

nd [0, )C

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(J. Hale 88 Book Appendix)

1 : unstable mfd (C curve)uW

1: stable mfd ( curve)

This structure

(det 0) is very

helpful to bound

sW C

A

1

2

and

( )on { | | 1, 2}

( )

T

T

T

V V

V V

Aee e

Ae

L

R

0z

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2 21 2 3

c

33

Th (R

( ( ) 0)

on

Berg's effect

= ( , )

=0 in

(a) 0 0 for 0

(b)

ybka-G '03

0 0 o

(

)

)

n

ii

T

B

T

B

TS

x x x

Vn

V xx

V S S

V V

r

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r small

large

r

3x

2 2 21 2

Applications

average of

over { }

r

TS x x r

If 0,

then 0 for [0, ]R r

V

r R

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22

(5 ) on if 0

1(0 ) ( ) ( )(1 )

2We set

so that LHS / 2

Bound for impli

Berg's

es

lower

( )

and upper bound

ef

for

1 11 2( 1)

2

'

fect

T

R r

T

T

T T

S V

rr n

R

Ra aV dr

V

V

d

R Ra V d V

r r da

n

Near equilibrium : close to zero / bounded away from zero

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•Existence of solution of the Original problem is widly open if is not near equilibrium (Even if is given M.-H. Giga – Y. Giga ’98 graphs) ( : constant M.-H. Giga – Y. Giga ‘01 level set approach : unique existence of generalized sol (2-D))•Uniqueness of the solution of the original problem (Sol is unique for Reduced problems)

5. Open problems

(0)

29

i

2 2

S

2 2

If the evolution is

‘Sufficiently regular’

then must be a

of

Min{ | div |

n(x) . .

( ), div ( )}.

What is a natural class of

s

m

olutions so that solution exist

ini

s

f r

mizer

o

i

i i

d

a e x S

L S L S

H

Belletinni e

the origina

t al : 3-D.

l eq

d

?

ivV

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All my preprints

are in

Hokkaido University

Preprint Series

on Math.

http:coe.math.sci.hokudai.ac.jp