数理解析研究所講究録
第 1924 巻 2014 年 11-20

11Evolution of crystal surface by a single screw
dislocation with multiple spiral steps
Takeshi Ohtsuka
Division of Pure and Applied Sciences,
Graduate School of Science and Technology,
Gunma University
4-2, Aramaki-machi, Maebashi, Gunma 371-8510, Japan
tohtsuka@gunma-u.ac.jp

1.

Introduction

This is a preliminary version to the mathematical results on the growth rateof the surface with rotating spiral steps evolving by an eikonal-curvatureequation.
The theory of crystal growth with aid of screw dislocations is proposedby [1]. According to the theory spiral steps on the growing crystal surface
are proposed by the screw dislocations across with the surface, and the steps
evolve with an eikonal-curvature flow
$V=v_{\infty}(1-\rho_{c}\kappa)$

(1.1)

,

denotes the normal velocity in the normal direction which is awhere
isdenoting spirals,
continuous unit normal vector field of the curveandare constants
the curvature of the spirals in the direction of $-n,$
denoting the mobility and the critical radius of the two dimensional kernel,
respectively. The steps evolve on the helical crystal surface going around thescrew dislocations, and then the the surface evolves in the vertical direction.
Burton, et al. [1] also give some speculations on the growth rate of theevolving crystal surface, in particular by co-rotating pair of spirals. They
pointed out if the distance of the pair is enoughly closer than , then thegrowth rate by the pair is twice of that with a single spiral step.
A level set formulation of evolving spirals is proposed by [10] or [11]. In
the formulation we regard the surface without spiral centers as the bounded. Then, the spirals are described as a zero point set of
domainwith an auxiliary function and the predetermined multiple-valued function
$V$

$n$

$\Gamma_{t}$$v_{\infty}$$\kappa$$\rho_{c}$$\rho_{c}$$\Gamma_{t}$$u-\theta$$W\subset \mathbb{R}^{2}$$u$$\theta(x)=\sum_{j=1}^{N}m_{j}\arg(x-a_{j})$(1.2)

which is introduced by [8] in the Allen-Cahn type equation for evolving
spirals, where
denotes a center of j-th screw dislocation for $j=1$ , . . . $N.$$a_{j}$

12
The right angle condition between$\Gamma_{t}$and$\partial W$, i.e.,(1.3)$\Gamma_{t}\perp\partial W$

is considered in [10] or [11]. Moreover, a simple method reconstructing thesurface from solutions to the level set equation is proposed by [11].
In this paper we investigate the vertical growth rate of the surface with$m(\geq 1)$ rotating spiral steps by $(1.1)-(1.3)$ .
The goal of this paper is todemonstrate that the growth rate of the surface evolving with rotating $m(\geq$1) spiral steps by the eikonal-curvature flow is $m$ times of that with a single
rotating spiral step under some suitable assumptions.
Ogiwara and Nakamura [9] obtained similar result of the above with the
Allen-Cahn type equation introduced by [8]. The crucial diﬀerence between
ours and the result in [9] is that the solution as in [9] forms $1/m$ times
rotational symmetric pattern. The solution in our level set method does notform such a pattern and keep the rotation angle of the curves in whole time.
This paper is organized as follows. We first introduce the level set method
for spirals and a way reconstructing the surface with spiral steps in \S 2. Then
we investigate the vertical growth rate of the surface with rotating spiralsteps in \S 3.

2.

Preliminaries

We first introduce briefly the level set formulation for spirals by [10] or [11].
Although we consider only a single spiral case in the next section, we here
consider the general case: with multiple screw dislocations and multiple spirals. In \S 2.2 we propose a way reconstructing the surface with spiral steps
from the solution to the level set equation, which is introduced by [11].

2.1.

Level set formulation for spiralsLet
be a bounded domain. We here assume that there exist
spiral centers denoted by
for $j=1$ , . . . , $N$ . Letbea signed number of spirals associated with ; there exist
spirals associated withrotating with counter-clockwise (resp. clockwise) rotational
orientation if $m_{j}>0$ (resp. $m_{j}<0$ ) provided that $V>0$ . See [11] for detailsof signed number of spirals.
As we explained in the previous section, the spirals at time $t\geq 0$ , which
is denoted by , is described as the zero point set of $u-\theta$ with an auxiliary
continuous function . However, we have to remove an open neighborhoodoffrombecause of the singularity of at . Thus we now set$\Omega\subset \mathbb{R}^{2}$$N\in \mathbb{N}$$a_{j}\in\Omega$$m_{j}\in \mathbb{Z}\backslash \{O\}$$|m_{j}|$$a_{j}$$a_{j}$$\Gamma_{t}$$u$$U_{j}$$\Omega$$a_{j}$$\theta$$a_{j}$

13. Assume that

if $i\neq j,$$N$boundaries
for $j=1$ , . .. , orare smooth so that
Then, we now give a level set formulation for spirals as$W= \Omega\backslash (\bigcup_{j=1}^{N}\overline{U}_{j})$$U_{i}\cap U_{j}=\emptyset$$\partial\Omega$$\partial U_{j}$$\Gamma_{t}=\{x\in\overline{W};u(t, x)-\theta(x)\equiv 0 mod 2\pi \mathbb{Z}\},$, and theis smooth.

$\overline{U}_{j}\subset\Omega$$\partial W$$n=-\frac{\nabla(u-\theta)}{|\nabla(u-\theta)|}$

.

(2.1)

Although the above includes a multiple-valued function, (2.1) is locally similar as the usual level set method with locally smooth function $u-\theta$ . Thenand are represented as$V$$\kappa$$V= \underline{u_{t}} \kappa=-div\frac{\nabla(u-\theta)}{|\nabla(u-\theta)|}.$’$|\nabla(u-\theta)|$Thus, we obtain the level set equation for $(1.1)-(1.3)$ as follows.
$u_{t}-v_{\infty}| \nabla(u-\theta)|\{\rho_{c}div\frac{\nabla(u-\theta)}{|\nabla(u-\theta)|}+1\}=0$

in$(0, T)\cross W$$\nabla(u-\theta)\rangle=0$

on$(0, T)\cross\partial W$$\langle\vec{v},$

,

(2.2)
, (2.3)

whereis the outer unit normal vector field of. See [5] for detailsof the level set method.
When we define as in (2.1), we have to clarify “interior” and “exteriorof spirals. In [10] or [11] we introduce a covering space ofas$\vec{v}\in S^{1}$$\partial W$$n$$\overline{W}$$X=\{(x, \xi)\in\overline{W}\cross \mathbb{R}^{N};(\cos\xi_{j}, \sin\xi_{j})=\overline{x-a_{j}}$wherefunction$\xi=(\xi_{1}, \ldots, \xi_{N})$and$\overline{x}=x/|x|$for

for $j=1$ , . . . , $N\}$ ,$x\in \mathbb{R}^{2}\backslash \{0\}$(2.4). Then, the single-valued$\tilde{u}(t, x, \xi) :=u(t, x)-\sum_{j=1}^{N}m_{j}\xi_{j}$on$\tilde{O}_{t}$plays a.role ofon, and spirals
on is defined as$[0, \infty)\cross X$$u-\theta$$X$, and thus the (interior”$\tilde{I_{t}}$“exterior”’$X$$\tilde{\Gamma}_{t}$$\tilde{I_{t}} =\{(x, \xi)\in X;\tilde{u}(t, x, \xi)>0\},$$\tilde{O}_{t} =\{(x, \xi)\in X;\tilde{u}(t, x, \xi)<0\},$$\tilde{\Gamma}_{t} =\{(x, \xi)\in X;\tilde{u}(t, x, \xi)=0\}.$The definition of in (2.1) reflects the direction of the interior and exterior
as the above.
Much analysis of $(2.2)-(2.3)$ has been done. Note that the equation (2.2)
is degenerate parabolic and has singularities at where $\nabla(u-\theta)=0$ , so that$n$

14
the solutions to $(2.2)-(2.3)$ should be considered in viscosity sense; see [3],[2], [4] for details on theory of viscosity solutions. The comparison principle
which implies the uniqueness of the viscosity solutions, and the existence ofviscosity solutions to $(2.2)-(2.3)$ for a continuous initial data
globally-in-time are established by [10]. Then,is derived with followingway.
$u_{0}\equiv u(0, \cdot)$$\Gamma_{t}$(i) Construct$u_{0}\in C(\overline{W})$describing$\Gamma_{0}$in our level set formulation.

(ii) Solve $(2.2)-(2.3)$ with $u|_{t=0}=u_{0}.$(iii) Extract$\Gamma_{t}$given by (2.1).

It remains two problems: how to establish (i), and uniqueness ofwithrespect to choice of . On (i), the existence ofdescribing
is proved in [6], and a practical way of the construction
is proposed by[11]. The uniqueness of with respect tois also established by [6]. The
crucial result is the comparison of interior in , which is as follows.
$\Gamma_{t}$$u_{0}\in C(\overline{W})$$u_{0}$$\Gamma_{0}$$u_{0}$$\Gamma_{0}$$\Gamma_{t}$$X$

Lemma 2.1 ([6]) Let and be a
sub-and supersolution to $(2.2)-$(2.3) onthat $u(O,$ or $v(O,$ arefor some $T>$ O.
continuous, and are upper and lower semicontinuous,
pectively. If$u$$[0, T)\cross\overline{W}$

$u$

$v$

$v$

$visco\mathcal{S}ity$$As\mathcal{S}ume$

$)$

$)$

$re\mathcal{S}$$\{(x, \xi)\in X;\tilde{u}(0, x, \xi)>0\}\subset\{(x, \xi)\in X;\tilde{v}(0, x, \xi)>0\}$$(resp. \{(x, \xi)\in X;\tilde{u}(0, x, \xi)<0\}\supset\{(x, \xi)\in X;\tilde{v}(0, x, \xi)<0$

then$\{(x, \xi)\in X;\tilde{u}(t, x, \xi)>0\}\subset\{(x, \xi)\in X;\tilde{v}(t, x, \xi)>0\}$$($

resp. $\{(x, \xi)\in X;\tilde{u}(t, x, \xi)<0\}\supset\{(x, \xi)\in X;\tilde{v}(t, x, \xi)<0$

for $t\in(0, T)$ .
Note that only the comparison of interior sets is proved in [6], but the comparison of exterior sets is also derived with parallel argument in [6].

2.2.

Height function

A simple way to construct the height function $h(t, x)$ of evolving crystal
surface for solution to $(2.2)-(2.3)$ is introduced by [11].
$u$15
From the theory of linear elasticity (see e.g. [7]), the crystal surface canbe described by the graph of a function $h(t, x)$ which satisfies$\{\begin{array}{l}\triangle h=0 except on \Gamma_{t},h has jump discontinuities with height a>0 only on \Gamma_{t}.\end{array}$

The equation for$h$is represented as$\triangle h=-adiv(\delta_{\Gamma_{t}}n)$

.

(2.5)

In [11] the function$h(t, x)= \frac{a}{2\pi}\theta_{\Gamma_{t}}(x)$is introduced instead of solving (2.5) with a suitable boundary condition,
where
is a branch of whose discontinuities are only on . In fact, theabove is a solution to (2.5) by straightforward calculation. A symple way
to construct
is proposed by [11]. Let $k=k(t, x)\in \mathbb{Z}$ be such that$\theta$$\Gamma_{t}$$\theta_{\Gamma_{t}}$$h$$\theta_{\Gamma_{t}}$$-\pi\leq u(t, x)-(\Theta(x)+2\pi k(t, x))<\pi,$where$\Theta=\sum_{j=1}^{N}m_{j}\Theta_{j}(x)$$\arg(x-a_{j})$and$\Theta_{j}:\overline{W}arrow[-\pi, \pi$

) is the principal value of. Then,$\theta_{\Gamma_{t}}(x)=\Theta(x)+2\pi k(t, x)+\pi\theta(u(t, x)-(\Theta(x)+2\pik(t, x$

is our desired function, where(2.6)

is the Heaviside function, i.e.,; heredenotes the indicator function for. The continuity ofonis established from the definition.
To investigate the evolution of the surface by spiral steps we define themean growth height $H(t)$ of the surface as$\theta(\sigma)=1_{[0,\infty)}(\sigma)-1_{(-\infty,0)}(\sigma)$$J\subset \mathbb{R}$$\theta_{\Gamma_{t}}$$\theta:\mathbb{R}arrow \mathbb{R}$$1_{J}$$\overline{W}\backslash \Gamma_{t}$$H(t)= \frac{1}{|W|}\int_{W}(h(t, x)-h(0, x))dx$

3.

.

(2.7)

Surface evolution by multiple spirals associated with a single centerIn this section we investigate the surface evolution by a single screw dislocation with multiple spiral steps.
Letbe a bounded domain with smooth boundary. Assume that$0\in\Omega$
and letbe an open neighborhood of $0\in\Omega$ with smoothboundary. Set
and thenis also smooth.
$\Omega\subset \mathbb{R}^{2}$$U\Subset\Omega$$W=\Omega\backslash \overline{U}$$\partial W$

16
Assume that $m(\geq 1)$ spirals which are denoted by. Let
ciated with a single center$\Gamma_{0}=\bigcup_{j=0^{\Gamma_{j)}}}^{m-1}0$

asso-$0\in\Omega$$\Gamma_{0,0}=\{P(\sigma)\in\overline{W};\sigma\in[0, \sigma_{0}]\}$

be a continuous curve which has no self-intersections. Assume that there) for $j=0$ , 1, . . . , $m-1$ such thatexists$\alpha_{j}\in[0, 2\pi$(A1) $0=\alpha_{0}<\alpha_{1}<\alpha_{2}<\cdots<\alpha_{m-1}<2\pi,$(A2)$R_{\pm\alpha_{j}}\overline{W}=\overline{W}$(A3)$\Gamma_{j,0}=R_{\alpha_{j}}\Gamma_{0,0}$(A4)$\Gamma_{i,0}\cap\Gamma_{j,0}=\emptyset$

where$R_{\beta}$

is thefor

for $j=0$ , 1, 2, . . . , $m-1,$
for $j=0$ , 1, 2, . . . , $m-1,$
if $i\neq j,$$2\cross 2$$A\subset \mathbb{R}^{2}$$\mathbb{R}^{2};x\in A\}$

$R_{\beta}A=\{R_{\beta}x\in$rotation matrix with angle. In this section we consider two kind of the surface$\beta\in \mathbb{R},$evolution:(E1) by $m$-spiral steps on

condition$\Gamma_{t}$which evolves with (1.1), (1.3) and the initial$\Gamma_{t}|_{t=0}=\Gamma_{0},$(E2) by only a single spiral step onwhich evolves with the same equationsto (E1) and the initial condition
for $j=0$ , 1, 2, . . . , $m-1.$$\Gamma_{j,t}$$\Gamma_{j,t}|_{t=0}=\Gamma_{j,0}$,
We now denote the mean growth height of (E1) or (E2) as $H(t)$ andrespectively. The goal of this section is to prove $H(t)=mH_{0}(t)$ .
The situation (E1) is described by our level set formulation as follows.
$H_{j}(t)$Step 1. We first construct$u_{0}\in C(\overline{W})$satisfying$\Gamma_{0}=\{x\in\overline{W};u_{0}(x)-m\theta_{0}(x)\equiv 0 mod 2\pi \mathbb{Z}\}$

where $\theta_{0}(x)=\arg(x)$ . In particular we construct$u_{0}\in C(\overline{W})$

,

(3.1)

satisfying(3.2)$\Gamma_{j,0}=\{x\in\overline{W};u_{0}(x)-m\theta_{0}(x)\equiv 2\pi j mod 2\pi m\mathbb{Z}\}$by the method as in [11].

Step 2. We next solve the level set equation $(2.2)-(2.3)$ with the initial data$u(0, \cdot)=u_{0}$ and;$\theta(x)=m\theta_{0}$$u_{t}-| \nabla(u-m\theta_{0})|\{div\frac{\nabla(u-m\theta_{0})}{|\nabla(u-m\theta_{0})|}+C\}=0$

in$(0, T)\cross W$$\nabla(u-m\theta_{0})\rangle=0$

on$(0, T)\cross\partial W.(3.4)$$\langle\vec{\nu},$, (3.3)

17
Then,$\Gamma_{t}$is extracted as$\Gamma_{t}=\{x\in\overline{W};u(t, x)-m\theta_{0}(x)\equiv 0 mod 2\pi \mathbb{Z}\}.$

Step 3. Construct the surface height function
height $H$ from as (2.7).

$h$to define the mean growth$u$When we consider the situation (E2) in our formulation, rigorously wehave to execute the procedure listed above provided that $m=1$ andinstead of . Hence we consider the equation (3.3), (3.4) with $m=1$ , i.e.,$\Gamma_{j,0}$$\Gamma_{0}$

(3.5)

in

$(0, T)\cross W$$\nabla(u-\theta_{0})\rangle=0$

on$(0, T)\cross\partial W$$\langle\vec{\nu},$

Then

,

$u_{t}-| \nabla(u-\theta_{0})|\{div\frac{\nabla(u-\theta_{0})}{|\nabla(u-\theta_{0})|}+C\}=0$$\Gamma_{j,t}$. (3.6)is described as$\Gamma_{j,t}=\{x\in\overline{W};v_{j}(t, x)-\theta_{0}(x)\equiv 0 mod 2\pi \mathbb{Z}\}$

with a solution$v_{j}$to (3.5), (3.6) with $v_{j}(0, \cdot)=v_{j,0}(x)$ satisfying(3.7)$\Gamma_{j,0}=\{x\in\overline{W};v_{j,0}(x)-\theta_{0}(x)\equiv 0 mod 2\pi \mathbb{Z}\}$and the same orientation of interior and exterior on$X$

as (3.2), i.e.,
(3.8)
(3.9)

,
$\{(x, \xi)\in X;\tilde{u}_{0}(x, \xi)=2\pi j\}=\{(x, \xi)\in X;\tilde{v}_{j,0}(x, \xi)=0\}$ ,$\{(x, \xi)\in X;\tilde{u}_{0}(x, \xi)>2\pi j\}=\{(x, \xi)\in X;\tilde{v}_{j_{)}0}(x, \xi)>0\}$where $\tilde{u}_{0}(x, \xi)=u_{0}(x)-m\xi$ , and $\tilde{v}_{j,0}(x, \xi)=v_{j,0}(x)-\xi$ . However, ourlevel set formulation enables us to obtain the above situation easily. In fact,$:=(u(t, x)-2\pi j)/m$ is also a viscosity solution to (3.5), (3.6) if is$:=(u_{0}(x)-2\pi j)/m$a viscosity solution to (3.3) and (3.4). Moreoversatisfies (3.7), (3.8) and (3.9). Hence we obtain$v_{j}(t, x)$$u$$v_{j,0}(x)$$\Gamma_{j_{\}}t}=\{x\in\overline{W};v_{j}(t, x)-\theta_{0}(x)\equiv 0 mod 2\pi \mathbb{Z}\}$by the uniqueness result in [6]. Thus the height of the surface $h_{j}(t, x)$ andthe mean growth heightare defined similarly as and $H$
only by
by $v_{j}=(u-2\pi j)/m.$
We are now in the position to state our main result.
$H_{j}(t)$Theorem 3.1$As\mathcal{S}ume$$\Gamma_{j,t}$$h$that $(Al)-(A4)$ . Then $H(t)=mH_{0}(t)$ for $t>0.$

18
We here state only the idea of the proof of Theorem 3.1. We first prove thefollowing fundamental result.
Proposition 3.2 The formula $H(t)= \sum_{j=0}^{m-1}H_{j}(t)$ holds for $t\geq 0.$

We here mention the idea of the proof of Proposition 3.2 in a few words. The
and $t>0,$
crucial property is the following description: for$k,$ $k_{j}\in \mathbb{Z}$$\theta_{\Gamma_{t}}(x)=m\xi-\pi(2k-1)$

if $(x, \xi)\in\{(y, \eta)\in X;-2\pi k\leq\tilde{u}(t, y, \eta)<-2\pi(k-1$$\theta_{\Gamma_{j,t}}(x)=\xi-\pi(2k_{j}-1)$

if $(x, \xi)\in\{(y, \eta)\in X;-2\pi k_{j}\leq\tilde{v}_{j}(t, y, \eta)<-2\pi(k_{j}-1$
where $\tilde{u}(t, x, \xi)=u(t, x)-m\xi$ and $\tilde{v}_{j}(t, x, \xi)=v_{j}(t, x)-\xi$ . The above implies$\theta_{\Gamma_{t}}=\sum_{j=0}^{m-1}\theta_{\Gamma_{j,t}}+\pi(m-1)$

and then we obtain $H(t)= \sum_{j=0}^{m-1}H_{j}(t)$ .
Thus, it suﬃces to prove $H_{j}(t)=H_{0}(t)$ for $j=0$ , 1, . . . , $m-1$ and $t>0.$
For this purpose we prove the rotational relations between the domain where
and that on , i.e.,
is not step on$\Gamma_{0}$$\Gamma_{j}$$\{(x, \xi)\in X;-2\pi k<\tilde{v}_{j,0}(x, \xi)<-2\pi(k-1)\}$$=\tilde{R}_{\alpha_{j}}\{(x, \xi)\in X;-2\pi k<\tilde{v}_{0,0}(x, \xi)<-2\pi(k-1)\}$$:=\{(R_{\alpha_{j}}x,\xi+\alpha_{j})\in X;-2\pi k<\tilde{v}_{0,0}(x, \xi)<-2\pi(k-1)\}$$=\{(x, \xi)\in X;-2\pi k<\tilde{v}_{0,0}(R_{-\alpha_{j}}x, \xi-\alpha_{j})<-2\pi(k-1$

Note that$\tilde{v}_{0,0}(R_{-\alpha_{j}}x, \xi-\alpha_{j})=\frac{u_{0}(R_{-\alpha_{j}}x)}{m}+\alpha_{j}-\xi,$which is well-defined by (A2). Then, the function$v_{j}(t, R_{-\alpha_{j}}x)+ \alpha_{j}=\frac{u(t,R_{-\alpha_{j}}x)}{m}+\alpha_{j}$is also well-defined by (A2), and a viscosity solution to $(3.5)-(3.6)$ with initial
data $v_{j}(0, R_{-\alpha_{j}}x)+\alpha_{j}=u_{0}(R_{\alpha_{j}}x)/m+\alpha_{j}$ by the rotation invariance on$(3.5)-(3.6)$ . Then, by Lemma 2.1 we obtain$\{(x, \xi)\in X;-2\pi k<\tilde{v}_{j}(t, x, \xi)<-2\pi(k-1)\}$$=\{(x, \xi)\in X;-2\pi k<\tilde{v}_{0}(t, R_{-\alpha}jx, \xi-\alpha_{j})<2\pi(k-1$(3.10)

19
Thus we obtain $h_{j}(t, x)=h_{0}(t, R_{-\alpha_{j}}x)+a(P+\alpha_{j}/2\pi)$ with a constantwhich implies$\ell\in \mathbb{Z},$$H_{j}(t)= \frac{1}{|W|}\int_{W}(h_{j}(t, x)-h_{j}(t, x))dx$$= \frac{1}{|W|}\int_{W}(h_{0}(t, R_{-\alpha_{j}}x)-h_{0}(t, R_{-\alpha_{j}}x))dx$$= \frac{1}{|W|}\int_{W}(h_{0}(t, x)-h_{0}(t, x))dx=H_{0}(t)$for $j=0$ , 1, . . . , $m-1$ and $t>0$ . Hence we obtain Theorem 3.1.

Remark 3.3 It is known that $(2.2)-(2.3)$ is derived from asymptotic expansion of the Allen-Cahn type equation due to [8]. It is obtained in [9]that solutions to the Allen-Cahn type equation converges to the solution describing a stationary rotating spirals with $1/m$-rotational symmetric pattern,which is diﬀerent behavior of that the rotational relation (3.10) expresses.

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