Thermal gradient crystallization of an amorphous CdGeAs 2 alloy
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anv(R,t) Ot +~
a
(nv (R,t) 9 G(R,t)) = 0
[1]
where
[7] Substituting Eq. [3] into the right side of Eq. [7] gives
0
t=
[81
7 :JR"
L
I n s e r t Eq. [8] into Eq. [6]:
nv(R,t) = the s i z e d i s t r i b u t i o n function such that nv(R,t) dR g i v e s the n u m b e r of p a r t i c l e s per unit v o l u m e whose size is b e t w e e n R and (R + dR) at t i m e t, and G(R,t) = the local growth r a t e (shrinkage r a t e ) of p a r t i c l e s of size R at t i m e t, defined as the r a t e of change of R, the d i m e n s i o n used to s p e c i f y the p a r t i c l e s i z e . The growth path for a p a r t i c l e in a s t r u c t u r e is the t i m e sequence of its g e o m e t r i c s t a t e s ; in s i m p l e s y s t e m s , it is the c u r v e r e p r e s e n t i n g the v a r i a t i o n of the s i z e of a p a r t i c l e as it e v o l v e s f r o m its n u c l e a t i o n t i m e . If growth paths f r o m d i f f e r e n t n u c l e a t i o n t i m e s do not c r o s s , then the growth r a t e of a p a r t i c l e of size R at t i m e t, i . e . , G(R,t), has a c l e a r m e a n i n g . If the paths c r o s s , say, p a i r w i s e , for e x a m p l e , then p a r t i c l e s of the s a m e size at a given t i m e will have two d i f f e r e n t growth r a t e s , depending upon to which g r o w t h path they belong, i . e . , when they n u c l e a t e d . It is the p u r p o s e of this note to d e r i v e a g e n e r a l i z e d c o n t i n u i t y equation which is v a l i d for the case of both c r o s s i n g and n o n c r o s s i n g growth p a t h s . Define
Rrn N V >(R,t) =
fR nv ( R ' , t ) d R '
[2]
where R m is the size of the l a r g e s t p a r t i c l e at t i m e t and R ' is a d u m m y v a r i a b l e of i n t e g r a t i o n . N V >(R,t) g i v e s the n u m b e r of p a r t i c l e s p e r unit v o l u m e whose s i z e s a r e l a r g e r than R at t i m e t. It follows that
>(R,t4
[
a-ff
[3]
]t- -n.(n,t).
The n u m b e r d e n s i t y function, nv(R,t), may u s u a l l y be modelled a s a continuous function of R and t if g e o m e t r i c a l i m p i n g e m e n t can be n e g l e c t e d . In such a c a s e , it follows f r o m Eq. [2] that NV> (R,t) is a c o n t i n uous single valued function of R and t. T h u s one can w r i t e a continuity equation r e l a t i n g t h e s e t h r e e v a r i ables:
~
/a = - \
~
7t \'-~]Nv
>
aNv
>'R,t).'~
-o7
= nv,R,t )
JR
a(~t)
at
O
OR
[9]
Eq. [9] is the g e n e r a l i z e d continuity equation applicable to both c r o s s i n g and n o n c r o s s i n g growth paths, since no a s s u m p t i o n r e g a r d i n g the n a t u r e of the growth paths was r e q u i r e d in its d e r i v a t i o n . If the growth paths do not c r o s s , then [10] >
Substituting this r e s u l t into Eq. [9] y i e l d s Eq. [1]. In the t h e o r e t i c a l d e v e l o p m e n t of growth models in which the b e h a v i o r of i n d i v i d u a l p a r t i c l e s v a r i e s with both the p a r t i c l e size and t i m e of o
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