1
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
2
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’
3
Contents
1 Model2 Problem3 Main mathematical results4 Three ingredients - ODE analysis - Berg’s effect - Facet splitting criteria -5 Open problems
4
1 Model
Crystals grown from vapor
(snow crystal)
from solution (NaCl crystal)
<driving force : supersaturation>
(density of atoms outside crystal is small)
5
Stefan like Model
3 (crystal at ti( ) me )t t
: supersaturation e
e
C C
C
: concentration of atomsC
: saturated concentrationeC
6
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
7
unnormalized version :
c
Cv D V
n
: volume
of an atomcv
( )t
: diffusion
coefficient of
atoms in a solution
D
n
8
(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
"
9
Remark. If ( ) | | then
-div mean curvature
In 2 setting
-div ( )
: curvature
( ) (cos ,sin )
: stiffness coefficient
p p
D
K
K ( )t
10
We shall consider (1) - (5) for given
quasi-stationary
One phase Stefan problem with Gibbs-Thomson + kinetic effect
(0).
11
Solvability (smooth )
K. Deckelnik - C. Elliott ’99
( Hele Shaw type )
No … Friedman –Hu ’92
Liu – Yuan ’94
12
Others (No )
Kuroda-Irisawa-Ookawa ‘77
Stability of facets
Experiment e.g. Gonda-Gomi ’85
(No ) : Fingering :
Saffman-Taylor
R.Almgrem ’95
13
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
14
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
15
1Difficulty : is not
so the meaning
(5) = (n)
is not clear.
We shall interpret
(5) by using
subdifferentials
C
16
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
17
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
18
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
19
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
20
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
21
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
23
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
24
(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
25
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
26
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
27
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
28
•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
30
All my preprints
are in
Hokkaido University
Preprint Series
on Math.
http:coe.math.sci.hokudai.ac.jp