1116
Progre"' of Theoretical Physics, Vol. 59, No. 4, April 1978
A Simple Self-Consis tent Method in the Theory
of Spinodal Decomposit ion
Hiroyuki TOMITA
Department of Physics, School of Liberal Arts and Sciences
Kyoto University, Kyoto 606
(Received November 14, 1977)
A kinetic model of the spherical spin system is proposed and is applied to the problem
of the spinodal decomposition in the quenched mixture. The time-dependence of the struc·
ture function can be exactly calculated in a self-consistent manner, which serves as a late
stage approximation for the mode-couplings in the spin-conserved TDGL model. The wellknown sharp peak appears at temperatures below the transition point and it shifts to the
small wave-number region. The peak position obeys the weak power law in time. The
classical spinodal line is not found so far as the transient behaviour of the structure function
is concerned.
§ l.
Introduction
The spinoJal decomposition in binary mixtures 1s one of the interesting problems in the statistical mechanics of the states far from equilibrium. Many authors
ha\·e in,·estigated it, using the phenomenologi cal models and stochastic models.
One is the TDGL modeP 1 ~ 1 l and another is the kinetic Ising model.'l~n
In this paper a simple approximation based on the sum-rule of the correlation
function is proposed, which rcali;ces the follovving intuitive picture of the wellknown behcn-iour of the structure function. vVe have four points of ,-iew to understand the growth of the sharp peak of the structure function in the small wa ,-en umber regiOn. Let us consider the Ising spin system for simplicity:
(a) One has a sum-rule
sdqS(q, t) =(s'),=l,
where s denotes the Ising spin and
S(q, t)=(s(q)s(-q )),.
(b)
Because of the spm consen·ation one finds thai
S (0, t) =constant .
(c)
(d)
The short wavelength modes rapidly approach the equilibrium state,
i.e., the local equilibrium state.
In the true equilibrium state below the transition temperature one finds
Theory of Spinodal Decomposition
1117
a singularity Mo (q) at q = 0 corresponding to the existence of the
long range order M and it makes a part of the sum-rule (a), i.e.,
l S(q, =)dq=l-M
Jq~O
2 ,
or
l [S(q, t) -S(q, =)]dq=M
2 ,
Jq"cO
for any finite t.
These circumstances are shown m Fig. 1.
I
I
I
I
I
I
t =01
I
/
I
I
I
I
I
I
ll
q
Fig. 1. Schematic understanding of
the behaviour of the structure
function S(q, t) multiplied by
volume element 4n-q 2 • Corres·
ponding to the four points in the
text, one finds that (a) the area
under the curve is constant, (b)
the value at q=O is fixed to zero,
(c) the area in the large wave·
number region flows to the small
wave-number region and (d) there
exists a singular part M'tJ(q) m
the equilibrium state (t= oo).
To incorporate this intuitive picture into a theory, a solvable kinetic model
1s introduced in § 2, which is a dynamical extension of the spherical spin model
defined by Berlin and Kac. 8 l The results of the computation using this model
are summarized in § 3. In § 4 the approximate meaning of the present model is
discussed.
An application to the nucleation problem is to appear m a subsequent paper.*l
§ 2.
Kinetic models of Gaussian and mean spherical spin systems
In the problem of phase transition we have two solvable models of continuous
spin system, i.e., the Gaussian model and the spherical model. In this section it
is shown that these can be applied to the dynamical problems by the method of
master equation.
First let us introduce a dynamics of Gaussian spin system {s;} as follows:
*l
A brief report is in this issue, p. 1391.
1118
II. Tomita
Using an exchange type transition probability
lV<ijJ
(si-~si
+A,
sr--~sj-
A),
(2·1)
where (ij) represents the pair of lattice points, the master equation for the probability function P ( {si}, t)ILdsi is given by
(2·2)
where ~v is the number of spms. The assumption of detailed balance in the
equilibrium state restricts the form of the transition probability. The simplest
one may be taken as
(2·3)
where a'
1s
a positi\·e constant, Ju the exchange constant and ([J ( {si}) is gi\·en by
(2·4)
where J. is a positive constant and /3 = 1/kBT. An abbreviated notation ([J (si +A,
sj- A) =([J (s; + J, sj- A, {sk> k=fi, j}) is used in Eq. (2 · 3). One may restrict {Jij}
to the nearest-neig hbour interactions for simplicity, i.e.,
J.. {J0
=
for nearest-neig hbour pair (ij),
(>O)
L)
otherwise.
If one takes the limit A~O, the master equation
Planck type equation as usual:
f)
-1 -P(
{s;}, t)
a at
=
(2 · 2)
becomes a F okker-
-, f)
f)([J
f) }
2.:.;
-Aij { -+---P( {si}, t),
i, j
as,
as j
as j
(2·5)
where a= a' J,
( z (the number of nearest-neig hbours)
Jij = . -1
1
,0
for Z=J'
for the nearest-neig hbour pair (i, j),
otherwise ,
and 1n the wave-numbe r representatio n,
1 f)
{
f)([J
f)
}
--P({s(q
)},t)=:l. :-f) -A(q).
----+ --P({s(q)},t ),
a at
q as(q)
as(-q) as(-q)
where
(2·6)
1119
Theory of Spinodal Decomposition
_
s (q ) -
1
1~
vN
'\1
.:.....
e
iq·r
Sr
r
and
.:1 (q) = 2::;
eiq•r .:1; i+r •
r
The factor .:1 (q) is proportional to l for small q. This corresponds to the spin
conservation. For the time being the total spin IS taken to be zero:
which corresponds to the fifty-fifty mixture in the case of binary alloy.
The Fokker-Planck equation (2·6) is linear and can be exactly solved.
evolution equation for the variance, i.e., the structure function
The
S(q, t) =<s(q)s(-q)),
IS given by
1
-
a
f)
- S (q, t) = - 2.:1 (q) (A- (3J (q) )S (q, t) + 2.:1 (q).
at
(2·7)
Hereafter let us use the following notations:
r= (A-(3J(O) )/ K,
K=(3J,
and put
a=1/2K,
and then Eq. (2 · 7) is rewritten as
!_S(q, t) = -.:J(q) (r+.:l(q))S(q, t) +K- 1 .:1(q).
at
(2·8)
This linear evolution equation can be also solved exactly. The parameter r Is
the function of temperature and is negative below the transition temperature Tc
which is determined by
m the Gaussian model.
In this case Eq. (2·8) is unstable in the region of wavenumber .:J(q) <[r[, and one cannot find any steady state solution in this region,
which corresponds to the absence of equilibrium state below Tc in the Gaussian
model.
In the static problem of the phase transition this fault is avoided by introducing
a saturation effect, i.e., a restriction
(2·9)
1120
H. Tomita
as 1s proposed by Berlin and Kac, 81 and is called a spherical model.
One may apply it to the dynamical problem. This means that one should
soh·e the Fokker-Planck equation (2 · 6) on the N-dimensional sphere (2 · 9) or
(2-10)
The equation becomes a nonlinear one and very difficult to solve. Fortunately we
have a simple method in the static problem: Start with the Gaussian model and
determine the parameter r (or original },) so that the sum-rule in the averaged
form
(2·11)
instead of Eq. (2 -10) is satisfied. This condition of the mean spherical model 91 may
be applied to every instant of time. This simplified method can be expected to
be equivalent to the original problem so far as the condition
lim
N~oo
((_!_
.L; s/ -1) = 0,
N 1~
It
2
\
(2 -12)
or equivalently
(2 -12') *)
Is satisfied in the thermodynamic limit. The condition (2-12') can be examined
after the calculations of S (q, t).
The above method with the evolution equation (2 · 8) serves as a kinetic
spherical model. The same method has been applied to the critical dynamics by
Suzuki. 101 Here the parameter r should be determined in a self-consistent manner
by
r(t)
=
J
dq.d(q) {K- 1 -.d(q)S(q, t)}
j Jdq.d(q)S(q, t),
(2 -13)
so that
!!__ sdq S(q, t) =0.
at
where the integration is taken over the first Brillouin zone.
tion can be written in the following form:
The evolution equa-
!!__S(q, t) = -.d(q) {[r(t) +.d(q)JS(q, t)- [ro+.d(q)]Seq(q)},
at
-----""_"
*l
_ _ __
If the long-range order M exists, one finds that
2M'< (os(O)) '), +l_ ~<los(q) l')t'=finite.
Nq
(2-14)
Theory of Spinodal Decomposition
1121
by the use of the equilibrium solution (see the Appendix)
(2·15)
Seq(q) =1/K(ro+LI(q)),
where r 0 >0 at T> Tc and r 0 = 0 at T<Tc and the transition temperature Tc is
determined by
(2·16)
§ 3.
Computations
Let us introduce the following continuum approximation to simplify the computations:
L1 (q)
(3 ·1)
= q'
and
(3· 2)
where the cutoff wave-number qc is defined by
(3. 3)
Further the wave-number and the other quantities are scaled as
lj=q/qc,
l=tq/,
K=Kq/
and hereafter they are used without bar.
dimension D becomes
and
r=r/q/,
The equation (3 · 2)
for the arbitrary
(3·4)
and the transition point given by Eq. (2 ·16) is obtained as
Kc=D/ (D-2)
111
for
D>2,
The evolution equation for the structure function is rewritten
this new frame.
as
f)
-S(q, t)
fJt
=
-l(r(t) +q')S(q, t) +q'/K,
(3. 5)
with the sum-rule
(3· 6)
or equivalently
(3·7)
1122
H. Tomita
The initial state 1s chosen at T = oo, i.e.,
S(q,O)=l,
(3·8)
and 1s suddenly quenched to the lower temperature T.
Then one finds
(3·9)
where
The initial \·alue r(O)
1s
negative vvhen
D(D+2)
--T -<- -- - -Tc
(D-2) (D+4)
8
=1+- - ---(D-2) (D+4)
As is shown in Fig. 2, r (t) turns to positive value (or can be always positive
from the beginning) when T> Tc and approaches the equilibrium ,·alue r 0 • On
the contrary it is always negative vvhen T~Tc and approaches r 0 = 0.
r(t)
t
1000
Fig. 2.
The behaviour of r (t) calculated from Eq. (3 · 7) for D
=
3.
Defining the time-dependent growth rate in Eq. (3 · 5) by
r(q,t)=q'([r (t)[-q 2 )
r(t)<O,
when
we can thus find an asymptotic stability, i.e., the unstable region
q 2 <[r(t)
I
becomes narro,ver and narrovver with decreasing r (t) [.
The results of the numerical computations for D = 3 are shown in Fig. 3.
Below the transition point Tc it is shown that a sharp peak grows limitlessly
and shifts to the small wave-number region. The position of the peak qm (t) is
f
Theory of Spinodal Decomposition
25
1123
I
I
I
I
:I
\CD
loo
I
I
I
\
10
(b)T=Tc
20
(alT=0.5Tc
5
15
t=O
1
0
10
10
0.5
q
'
''I
\en
\
(c)T=1.25Tc
5
5
1~~~~==~~~~
o~----~--~----o~.5~~~q~--~~
Fig.3.
The behaviour of S(q,t) for D=3 at (a) T=0.5T, (b) T=T, and (c) T=l.25T,, respectively.
found to obey a power law
(3 ·10)
m the very wide range of time 1,000<t<5,000 (the upper limit of the present
computations). The exponent a' is calculated as
a'~{0.21
at T<Tc,
0.24
at T=Tc,
and it may be compared with
a'= { 1/ (D + 2)
1/(4-r;)
at T<Tc
for D=3,
at T=Tc
and Yj=O,
which is suggested by Binder.!l) However this does not seem to hold for other
dimensions as is shown in Table I.
The parameter r (t) is also found to obey the power law
lr (t) I=rb',
m the same range of time.
results suggest that
(3 ·11)
The exponent b' 1s tabulated m Table II and the
b';;:;2a'.
1124
H. Tomita
The exponent a'.
Table I.
D
T/T,~=-~
The errors are found within ±0.002 in all cases.
3
4
5
2.5
0. 213
0. 217
0.239
o. 210
0.210
0.214
_____________
0.1
0.5
1. 0
0. 206
0.240
Table II. The exponent b'. The errors are found within ±0.002 in all cases.
--y~r-------'
D
3
4
5
2.5
0.441
0.433
0.497
0.439
0.422
0.576
0.438
0.444
'
----
0.1
0.5
1.0
Above the transition point a broad peak appears, but in this case it remams
finite. A simple understanding can be also obtained by the four points of view
mentioned in § 1 with ]}1 = 0 in (d). Further it can be shown that the initial
broad peak appears at
which is independent of the temperature including T <Tc. Similar results have
been shown by using the kinetic Ising Model. 12)
Corresponding to the intuitive picture of Fig. 1 the behaviour of the structure
function multiplied by the volume element 3q 2 is shown in Fig. 4 for T = Tc and
T = 0.5Tc with D = 3.
So far we have restricted the problem to s = 0.
function is defined by
In the case s=f=O the structure
S(q, t)=<s(q)s(-q))t-s 2 o(q).
The evolution equation is just the same as Eq.
be changed into
(3 ·12)
(3 · 5), but the sum-rule should
(3 ·13)
It should be noted here that, defining a scaled function by
S' (q, t)=S(q, t)/(1-s 2) ,
(3·14)
a kind of correspondence rule can be found as follows:
The evolution equation
becomes
i_s' (q, t)
at
=
-l(r (t)
+ l)S' (q, t) + q 2/
K (1-s 2) ,
(3 ·15)
Theory of Spinodal Decomposition
1000
1125
(a)T=0.5Tc
3
2
(b)T=Tc
3
2
OIL-~~;_---~T---~~L_-=~--_j
0.8
0.2
0.4
0.6
Fig. 4. The structure functions multiplied by 3q 2 for D=3 at (a) T=0.5T, and (b) T= T,.
The difference of the situations between (a) M=Ij,fl and (b) M=O is clearly shown.
with the sum-rule
and the initial condition
S' (q, 0) =S(q, 0)/ (1-.s') =1.
The parameter r (t) is determined by Eq. (3 · 7) which Is rewritten as
r(t)
=
il
qD+l[K-1(1-s2)-1-q2S'(q, t)Jdql
r
qD+lS'(q,t)dq.
1126
H. Tomita
Thus the problem of given s=f=O and T is exactly transformed into that of s = 0
and
T' = T I (1- s 2 )
•
(3 ·16)
This correspondence rule is shown in Fig. 5. In this sense we cannot find the
classical spinodal line, i.e., we have no discontinuity on the horizontal line T =canst.
Approaching the spontaneous magnetization curve 1'11(T) is equivalent to approaching the critical temperature Tc.
Fig. 5. Correspondence rule for s~O. All
problems with (s, T) on the broken
line T= T' (1-s') can be reduced to
the problem with (0, T').
§ 4.
Discussion
Langer et al. 4 > obtained just the same equation as Eq. (3 · 5) starting from
a spin-conserved TDGL model and approximating the non-linear mode-coupling
terms by
Thus the time-dependent parameter r (t), or A (t) in their notations is related to
( s 2)t. ( s4 ) 1 etc. and can be determined by the one-point distribution function, the
equation of which is reduced from the full master equation. In their theory <s 2) 1 is
time-dependent and the sum-rule Eq. (3 · 6) is not satisfied because the coarsegrained density of the order parameter is used in the TDGL model. However <s 2)t
becomes saturated on account of the nonlinear effect in the sufficiently later stage.
In this stage the sum-rule can be expected to be satisfied approximately, and the
present method based on it serves as a simple approximation for the TDGL model
in later stage. The exponent a' in § 3 is in good agreement with Langer's result
a' =0.212±0.005
for T<Tc.
The temperature dependence of the saturation and fluctuation of the local
Theory of Spinodal Decomposition
1127
order parameter becomes important near the critical point, and further the saturation value should be taken to be zero above the critical temperature. In these
regions it is difficult to find a relation between these two methods.
The values of the exponent a' ( ~0.21) seem to be independent of the dimension D within the present numerical computations. To find the reason of this
fact one has to investigate the non-linear equation (3 · 5) analytically. This problem is left to the future.
Appendix
--Equilibrium Properties of the Spherical Model-The partition function with an external field h = {3H defined by
Z=
SdNs exp {- ~ ~ (r+ J (q))s(q)s( -q) + v' Nhs(O)},
(A·1)
can be easily calculated as
1
1
N- 1 ln Z=-ln 27!"-2
2N
~ ln
K(r+ J(q))
q
h
+--.
2
2Kr
(A·2)
Then one finds that
M=
(~ ~ s..) =h/Kr,
(A·3)
1
( - h ) 2tJ(q)
<s(q)s(-q))=- --------+
K(r+J(q))
Kr
(A·4)
and
(A·5)
Define
(A·6)
and the sum-rule 1s rewritten as
M2=l-~o(r)_
K
(1)
(A·7)
.
The transition temperature is determined with the conditions r = 0 and h
l.e.,
Kc=Ko (0).
(2)
Above Tc the temperature is given by
K=K0 (r)
(r>O)
= 0,
(A·S)
H. Tomita
1128
(3)
as a function of r in the case h = 0.
Below Tc one finds
lim r=O
n~o
by the condition
lim M (h, T) =f=O .
h~O
Then the spontaneous magnetization is given by
( 4)
In the case of finite field h=f=O, one finds
M (K, r)
=
j~------K~r}
and h = Krl'vl(K, r).
Then r can be determined as a function of K (or T) and h.
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1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
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