Growth	  rate	  of	  a	  surface	  by	  a	  co-­‐rota1ng	  pair	  of	  spiral	  steps	  evolving	  by	  an	  eikonal-­‐curvature	  equa1on	T.	  Ohtsuka(Gunma	  University)	  	  Mathema1cal	  aspects	  on	  boundary	  layer	  	  in	  reac1on-­‐diﬀusion	  phenomena,	  	  Meiji	  Univ.,	  Nov.	  28,	  2014	  	Joint	  work	  with	  Y.-­‐H.	  R.	  Tsai(Univ.	  Texas	  at	  Aus1n)	  and	  Y.	  Giga(Univ.	  Tokyo)	

Spiral	  crystal	  growth	  Burton-­‐Cabrera-­‐Frank	  (1951,	  BCF	  paper)		
Layer	  structure	  of	  a	  complete	  (without	  disloca1ons)	  crystal	

Screw	  disloca1ons	  provide	  helical	  surface	  structure	  and	  spiral	  steps	  on	  the	  crystal	  surface.	

・Holizontal	  evolu1on:	  Steps	  evolve	  with	
V = v1 (1 ⇢c )(steps	  =	  spiral	  curve t ⇢ R2 )	
V : Normal	  velocity,  : curvature	( v1 :	  velocity	  of	  straight	  steps,	  	  ⇢	  	  	  c	  	  :	  cri1cal	  radius	  (constants))	
・Ver1cal	  evolu1on:	  the	  step	  climbs	  the	  helical	  surface.	
Growth	  rate	  of	  the	  surface	  =	  angle	  velocity	  ×	  step	  height	Evolu1on	  of	  “crystal	  surface”	  Ques1on:	  How	  fast	  the	  “surface”	  	 evolve?	
Single	  spiral	  case	  (1	  screw	  disloca1on,	  1	  spiral	  step)	
Approxima1on	  by	  a	  rota1ng	  spiral:	t = {r(cos(✓(r) + !t), sin(✓(r) + !t)); r > 0}
⇒	  Growth	  rate:	
RS =!
! 1 v 1 h0
⇥ h0 =( h0 > 0 :	  step	  height)	
2⇡2⇡⇢cKnown	  results	  • BCF(1951):	  Approxima1on	  by	  Archimedean	  spiral	  ✓◆v1v11) r = 2⇢c ✓, ! =! = !1, !1 =2⇢c⇢c2p3p ⇡ 0.315.
• BCF(1951):	  Beder	  approxima1on	   !1 =2(1 + 3)
• Ohara-­‐Reid(1973):	  Shoo1ng	  method	  with	  ODE	  model:	  ) !1 = 0.330958061

Surface	  evolu1on	  by	  a	  co-­‐rota1ng	  pair	  THE	  GOAL	  of	  this	  talk	  is	  to	  know	  	the	  growth	  rate	  of	  the	  surface	  by	  a	  co-­‐rota1ng	  pair.	
Evolu1on	  equa1on	
V = v1 (1 ⇢c )⌦ ⇢ R2
a1a2⌦ ⇢ R2 :	  Crystal	  surface(view	  from	  above)	
a1 , a2 2 ⌦ :	  a	  pair	  of	  centers	  with	  co-­‐rota1ng	  spirals	) Rp = Rp (d) :	  Growth	  rate(ac1vity)	  depends	  on	  the	  distance	  	  	  	  d	  	  	  	  :=	  	  	  	  	  |a	  	  	  	  1	  	  	  	  	  	  	  a	  	  	  2	  	  |	  	  of	  a	  pair.	  	

Specula1ons	  by	  BCF:	  • [far	  apart]	  d > 2⇡⇢c ) Rp (d) ⇡ RS
• [limi1ng	  case]	  d ! 0 ) Rp (d) ! 2RS• [close,	  but…]	  	  	  	  	  d	  	  	  	  	  	  	  	  2⇡⇢	  	  	  	  	  	  	  c	  	  	  )	  	  	  	  	  	  No	  es1mates.	  	  Goal:	  Verify	  these	  specula1ons,	  and	  give	  an	  es1mates	  of	  Rp(d).	

Level	  set	  method	  for	  spiral	  steps	  (O(2003),	  O-­‐Tsai-­‐Giga(preprint))	  		 2nDomain: ⌦ ⇢ R bounded	  with	  smooth @⌦	  .	  
Centers:	   a1 , . . . , aN 22⌦ (N 1), 3N	  	[W =⌦\4B⇢j (aj )5
j=1

nt = {x 2 W ;⇢j > 0 (j = 1, . . . , N )r(u ✓)|r(u ✓)|u(t, x) ✓(x) ⌘ 0 mod 2⇡Z}, n =NX✓(x) =mj arg(x aj ) (mj 2 Z \ {0})j=1

ex. Archimedean spiral :2t = {x 2 R ;|x|arg x ⌘ 0mod 2⇡Z}Descrip1on	  of	  spirals	  Archimedean	  spiral:		
(Polar coordinate) : ✓ =(Level set) :t = {x;!t2	  Archimedean	  spirals:	

|x|

arg x ⌘ 0}u(t, x)(Polar coordinate) : ✓ =(Level set) :r + !tt = {x;r + !t, ✓ =2(!t|x|)

r + !t + ⇡2arg x ⌘ 0 mod 2⇡Z}

(Level set) :t = {x;u(t, x)✓(x) = arg(x✓(x) ⌘ 0 mod 2⇡Z}a1 ) + arg(xa2 )Claim:	  θ	  should	  be	  a	  mul1ple-­‐valued	  func1on.	

Level	  set	  equa1on	  	0 mod 2⇡Z}, n =t = {x 2 W ; u(t, x) ✓(x) ⌘It	  is	  roughly	  regard	  as	  the	  usual	  level	  set	  of	  	u ✓.
utr(u ✓))V =,  = div.
|r(u ✓)||r(u ✓)|r(u ✓)|r(u ✓)|Level	  set	  equa1on: V = v1 (1 ⇢c ) and	  right-­‐angle	  condi1on	  8⇢	  r(u ✓)><	   ut
v1 |r(u ✓)| ⇢c div+ 1 = 0 in (0, T ) ⇥ W,|r(u ✓)|⇒		  >	  :h~⌫ , r(u ✓)i = 0 on (0, T ) ⇥ @W.	  

(cf.	  Y.	  Giga,	  Surface	  evolu,on	  equa,on:	  a	  level	  set	  approach,	  Birkhäuser,	  2006)	

Remark	  • This	  equa1on	  works	  well	  because	  ∇θ	  or	  ∇2θ	  are	  single-­‐valued.	  
• The	  solu1on	  is	  in	  viscosity	  solu1on	  sense	  since	  the	  above	  is	  degenerate	  parabolic	  equa1on.	  	  (cf.	  Chen-­‐Giga-­‐Goto(1991),	  Evans-­‐Spruck(1991),	  Crandall-­‐Ishii-­‐Lions(1992).)	

Mathema1cal	  analysis	  ⇢r(u ✓)	v1 |r(u ✓)| ⇢c div+ 1 = 0 in (0, T ) ⇥ W,8>
< ut(LV)	
>: h~⌫ , r(u ✓)i = 0|r(u ✓)|on (0, T ) ⇥ @W.

Comparison.(’03 O) Assume that W is C 2 .
Let u U SC([0, T ) W ) and v LSC([0, T ) W ) bea viscosity sub- and super-solutions of (LV).
Then, if u(0, x) v(0, x) for x W , then u(t, x) v(t, x)for (t, x) (0, T ) W .

Existence	  and	  uniqueness(‘03	  O)	  For	  	  u	  	  	  0	  	  	  	  	  	  	  C(W	  	  	  	  	  	  	  	  	  	  )	  	  there	  exists	  a	  viscosity	  solu,on	  	  u	  	  	  	  	  	  	  	  C([0,	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  )	  	  	  	  	  	  	  W	  	  	  	  	  )	  	  globally-­‐in-­‐,me	  sa,sfying	u(0, ·) = u0 .
Global	  solu1on	  exists	  uniquely	  with	  respect	  to	  the	  con1nuous	  ini1al	  data.	
Problem:	

• Con1nuous	  ini1al	  data	  is	  NOT	  unique	  w.r.t.	  an	  ini1al	  curve.	  
• Construc1on	  of	  a	  con1nuous	  ini1al	  data	  is	  NOT	  trivial.	

Uniqueness	  of	  level	  sets	  Comparison of interior.(’08 Goto-Nakagawa-O)	
Assume that ⇥W is C . Let u ⌅ U SC([0, T ) W ) and2v ⌅ LSC([0, T ) W ) be a viscosity sub- and super-solutionsof (LV). Then, if{(x, ) ⌅ X; ũ(0, x, ) > 0} ⇥ {(x, ) ⌅ X; ṽ(0, x, ) > 0},then {(x, ) ⌅ X; ũ(t, x, ) > 0} ⇥ {(x, ) ⌅ X; ṽ(t, x, ) > 0}

for t ⌅ (0, T ). Also, if{(x, ) ⌅ X; ũ(0, x, ) < 0} ⇤ {(x, ) ⌅ X; ṽ(0, x, ) < 0},then {(x, ) ⌅ X; ũ(t, x, ) < 0} ⇤ {(x, ) ⌅ X; ṽ(t, x, ) < 0}for t ⌅ (0, T ).

X = {(x, ⇠) 2 W ⇥ RN ; ⇠j ‘ = ’ arg(xu(t, x, ⇠) = u(t, x)NX
j=1aj ) (j = 1, . . . , N, ⇠ = (⇠1 , . . . , ⇠N ))}=helical	  (crystal)	  lapce	  of	  atoms	

mj ⇠j ⇡ u(t, x)✓(x)→	  The	  level	  set	  is	  unique	  with	  respect	  to	  an	  ini1al	  curve.	

Surface	  height	  Assump1on:	  There	  is	  no	  horizontal	  	 and	  enough	  small	  ver1cal	  displacement	  of	  atom	  by	  screw	  disloca1on.	  	  The	  surface	  height	  func1on h(t, x) sa1sﬁes	  h = h0 div t n in W.
8< h = 0 a.e. in W,
h0 > 0 jump discontinuity only on t,:
in the direction ofn.
If t = {x 2 W ; u(t, x) ✓(x) ⌘ 0 mod 2⇡Z},	  	h0h(t, x) =✓ t (x),2⇡✓ t (x) = ⇥(x) + 2⇡k(t, x) + ⇡#(u(t, x) [⇥(x) + 2⇡k(t, x)])	  ( ✓ t (x)	  	  is	  a	  branch	  of	  θ	  whose	  discon1nuity	  is	  only	  on t 	  )	# : Heavyside function, ⇥(x) : a branch of ✓ (fix),k(t, x) : a branch number of the location of the step, i.e.,k(t, x) 2 Z s.t.
⇡  u(t, x) (⇥(x) + 2⇡k(t, x)) < ⇡.

Growth	  rate	  Z
1Mean	  growth	  height:	 H(t; t0 )	
=[h(t, x)h(t0 , x)]dx|W | W) Growth rate ⇡ H 0 (t; t0 ).

From	  numerical	  data	  we	  derive	  two	  kinds	  of	  the	  growth	  rate.	
(1)[Quan1ty]	  Linear	  approxima1on	

Sample:	  surface	  evolu1on	  by	  a	  single	  spiral	  step	
Evolu1on	  equa1on:	
V = 6(1) RS = 10.5347220.03)Approxima1on:	   H(t; 0) ⇡ R` t + bwith	  data	  in	  [0.3,1].	  	
) R` = 10.606435

Graph	  of	  the	  growth	  rate	  Surface	  evolu1on	  by	  a	  single	  spiral	  	  	  	  	  	  	  	  	  	  	  	  0.03)	  	  	  	  	  	  	  	  	  	  	  	  .		 step	  with	  	  V	  	  	  	  	  =	  	  	  	  	  6(1Diﬀerence	Diﬀerence	  of	  the	  numerical	  data	0
R(t)=(H⇤Calculatewµ ) with	
⇢Cµ cos4 (t/µ) (|t/µ| < ⇡/2,µ (t) =0(otherwise).
Z(Cµ is the constant s.t.
µ = 1.)) Rw ⇡ H 0

Convolu1on	  with	  the	  deriva1ve	  of	  a	  molliﬁer	

Remark: Rw around	  ini1al	  and	  terminal	  1me	  should	  be	  avoided.	  	Technical	  remarks	  on	  discre1za1on	  For	  numerical	  simula1ons	  we	  d	iscre1ze	  the	  level	  set	  equa1on	  with	  an	  usual	  explicit	  ﬁnite	  diﬀerence	  scheme.	  
Technical	  remarks	  • Equa1on:	  	   V = 6(1 ⇢c ) (v1 = 6, ⇢c > 0(parameter))• Regulariza1on	  of	  the	  curvature	  t"erm:	  #r(u ✓)r(u ✓)ki,j := div⇡ div p|r(u ✓)| i,j"2 + |r(u ✓)|2 i,j• Lapce	  points	  ⌦ ⇡ [ 1, 1]2 ⇡ Ds = {(i x, j x); 100s  i, j  100s},
x = 1/(100s), s = 1, 2, 4.
• Centers	  are	  located	  only	  on Ds	  ,	  and	  only	  centers	  are	  removed.	  	  (⇢j = x)• Single	  case: a1 = (0, 0),	  	  • Co-­‐rota1ng	  pair:	  a1 = ( k x, 0), a2 = (k x, 0), 2  k  50s.
• Step	  heights: h0 = 1. (Growth	  rate	  =	  Rota1ng	  number)	  	  	Co-­‐rota1ng	  pair	  Evolu1on	  eq.	 V = 5(1 0.02	 ) (v = 5, ⇥c = 0.02).

t=0	  t=0.05	  t=0.1	  t=0.5	  Surface	  evolu1on	  by	  a	  co-­‐rota1ng	  pair	  	⇢ = 0.03 (2⇡⇢ ⇡ 0.1884)cc

H(t; 0)d = |a1 a2 |d = 0.04
d = 0.08
d = 0.14

Rw (t)

tt
d = 0.20
d = 0.30
d = 1.00

2 ⇥c

The	  evolu1on	  is	  faster	  than	  the	  single	  one	  when	  d>2πρc	  but	  d≒2πρc.	  ⇢c = 0.05 (2⇡⇢c ⇡ 0.314)
H(t; 0)d = 0.04
d = 0.12
d = 0.22
d = 0.32
d = 0.56
d = 1.00

Rw (t)t⇢c = 0.07 (2⇡⇢c ⇡ 0.4396)
H(t; 0)td = 0.04
d = 0.16
d = 0.30
d = 0.44
d = 0.76
d = 1.00

Rw (t)

tt

Ac1vity	  formula	  by	  a	  group	  Assume	  that	  a	  group	  N –single	  screw	  disloca1ons	  are	  on	  line	  with	  length	  L.		
“single”	  “group”	
a2

a1

a4

aN

a3

Then,	  the	  ac1vi1es	  of	  a	  group	  and	  a	  single	  are	  as	  follows:	
N h0
Ac1vity	  of	   R = ! h = !1 v1 h0
Ac1vity	  of	  RL =S02⇡2⇡⇢cTa	  single	
a	  group	  L
TL is	  a	  dura1on	  a	  single	  spiral	  go	  around	  the	  line	  with	  length	  L.	
! = !1 v1 /⇢c is	  the	  angle	  velocity	  of	  a	  single	  spiral.	

• T1 : time for half turn at the terminal of L. 	
• T2 : both way on L.(Claim: the velocity is v1 .)	
2⇡
2⇡⇢c2LT1 ==, T2 =!
!1 v 1v12⇡⇢c2L2⇡⇢c
TL =+=(1 + L!1 /(⇡⇢c ))!1 v 1v1!1 v 1
N = 2,L = d	⇒Activity by aN! 1 v 1 h0N) RL =·=RS
co-rotating pair	
1 + L !1 /(⇡⇢c )2⇡⇢c
1 + L !1 /(⇡⇢c )T L = T1 + T 2 :Rela1ve	  errors 	  2⇥c	Ac1vity	  formula	  by	   Rp (d) =RS (d <)1 + d ⇤1 /( ⇥c )⇤1
1 = 0.330958061

a	  co-­‐rota1ng	  pair	Rela1ve	  errors	  between	  	  R	  	  	  	  	  (t	  	  	  	  	  	  	  	  	  [0.3,	  	  	  	  	  	  	  	  1])	  	  	  	  	  	  and	  R	  	  	  	  p	  	  (d)	  	  	  	  	  	  or	  	  R	  	  	  S	  	  	  .	
|R (d) Rp (d)||R (d) RS |ep (d) =,eS (d) =Rp (d)
RSc = 0.04
c = 0.03
errors	errors	

1100ddc = 0.05

c = 0.06

errors	

1100dc = 0.08

c = 0.07

11
Where	  is	  the	  cross	  point	  of	  	  e	  	  p	  	  	  (d),	  	  	  	  	  	  	  	  e	  	  S	  	  	  (d)	  	  	  	  	  	  	  ?	

00Cri1cal	  disntance	  of	  a	  co-­‐rota1ng	  pair	  ⇥c	dc =, ⇤1 = 0.330958061
Cri1cal	  distance	  of	  a	  co-­‐rota1ng	  pair	  ⇤1Rp (dc ) = RS

Numerical	  cri1cal	  distance	  	  d	  ˜	  c	  	  	  is	  such	  that	  ep (d) < eS (d) if d < d˜c ,ep (d) > eS (d) if d > d˜c(Roughly, ep (d˜c ) = eS (d˜c ))
⇥c
0.030
0.040
0.050
0.060
0.070
0.080

2 ⇥c
0.188496
0.251327
0.314159
0.376991
0.439823
0.502655⇥c /⇤10.284773
0.379697
0.474621
0.569545
0.664469
0.759394d˜c0.284813
0.379700
0.474650
0.569574
0.664486
0.759396←Numerical	  cri1cal	  distance	  is	  almost	  agree	  with	  ours.	

Limi1ng	  case	  Consider	   N = 1, a1 = 0 2 ⌦, and	 W = {x 2 R2 ; ⇢ < |x| < R}.	  
Level	  set	  equa1on:	  	(LV)mut

v1 |r(uhr(u⇢r(um✓0 )| 1 + ⇢c div|r(um✓0 )m✓0 )|m✓0 ), ~⌫ i = 0 on (0, T ) ⇥ @W.

= 0 in (0, T ) ⇥ W,(✓0 (x) = arg x)

Spirals:	
t = {x 2 W ;=m[1j=0( =:u(t, x)m✓0 (x) ⌘ 0{x 2 W ; u(t, x)m[1mod 2⇡Z}m✓0 (x) ⌘ 2⇡j0,t

mod 2⇡mZ}j,t ).

j=0Claim.	  	 i,t \j,t = ; for t > 0 if i 6= jprovided that u(0, ·) 2 C(W ).

m 1,t1,tGrowth	  rate	  by	  N-­‐spirals	  with	  a	  single	  center	  		  Theorem.	

Let H and Hj respectively be the mean growth heightby m-spiral steps
Pm t , and only a single spiral step j,t .
Then, H(t) = j=0 Hj (t).
In particular H(t) = mH0 (t) if there exist ↵j > 0 for
j = 1, 2, . . . , m 1 such that R↵j 0,0 = j,0 , where R↵
is the rotation matrix with angle ↵ 2 R.

Claim:	  H	  and	  Hj	  is	  uniquely	  determined	  w.r.t.	  choice	  of	  solu1on	  u.	
Key point:j,t = {x 2 W ;vj (t, x)✓0 (x) ⌘ 0

mod 2⇡Z}u(t, x) 2⇡jwith vj (t, x) =m=) vj is a solution to (LV)1 .

Claim.	  This	  result	  means	  that	  there	  is	  no	  interac1on	  between	  spirals.	  
(This	  is	  diﬀerent	  from	  the	  behavior	  by	  Allen-­‐Cahn	  type	  equa1on	  by	  Kobayashi	  (Ogiwara-­‐Nakamura(‘02).)	  Summary	  • Rough	  es1mate	  rate	  by	  a	  co-­‐rota1ng	  pair.	  	  	8 of	  the	  growth	  2R̃p (d) =<RS(d < dc = ⇡⇢c /!1 ),1 + d !1 /(⇡⇢c ): R(d dc )S!1 = 0.330958061(Ohara-Reid’s constant)• The	  rela1ve	  error	  between	  the	  above	  and	  the	  numerical	  results	  are	  less	  than	  7%	  for	   ⇢c 2 [0.03, 0.08].
• Introduce	  a	  new	  deﬁni1on	  and	  quan1ty	  of	  the	  cri1cal	  distance	  of	  a	  co-­‐rota1ng	  pair.	  
2• Deﬁni1on:	   d = dc s.t Rp (dc ) :=RS = R S
1 + dc !1 /(⇡⇢c )• Quan1ty:	   dc = ⇡⇢c /!1 > 2⇡⇢c .
• Further	  problem	  • Mathema1cal	  analysis	  on	  the	  growth	  rate	  and	  the	  cri1cal	  distance(ﬁnding	  exact	  limit	  and	  give	  a	  proof.)