Dynamic correlation of fluctuations during spinodal decomposition

Phys#ca 103A (1980) 99-118 © North-Holland Pubhshmg Co

DYNAMIC C O R R E L A T I O N O F F L U C T U A T I O N S DURING SPINODAL DECOMPOSITION C BILLOTET* and K BINDER

lnstItut fur Festkorperforschung der Kernforschungsanlage Juhch, D-5170 Juhch, W-Germany Recoved 9 May 1980

The kinetics of the formation of order is studied for both a system with non-conserved one-component order parameter and a binary mixture w~th conserved order parameter ("spinodal decomposition") By a decouphng of two-point probabilities a closed kinetic equation for the structure factor S(k, h, t2) is obtained which describes the correlation of fluctuations at two times h, t2 after the start of the nonlinear relaxation process Both this dynamic structure factor and the cross section for inelastic scattering from such a decaying system far from equdibnum are calculated numerically for representative cases The validity of the approximations made are discussed as well as the experimental observabihty of the departures found from equlhbrium behavior

1. I n t r o d u c t i o n

The theory of nonlinear relaxation processes at phase transitions has received great attention recently (see e.g. refs, 1 and 2). In systems, where the order parameter of the transition is not conserved one can consider quenches of external system variables (like temperature T, magnetic field/4, etc.) where the order parameter relaxes from one value to another3). H o w e v e r , when a quench is performed from a disordered state right through the critical point to a temperature within the ordered state, the average order parameter stays zero: rather the system b e c o m e s inhomogeneous on a macroscopic scale, as domains of the various (degenerate) order parameter orientations form, which steadily grow with time4). The same process occurs in systems with conserved order parameter, like fluid or solid binary mixtures, where the resulting formation of order is k n o w n as "spinodal decomposition"2.4~s). In these cases both theoretical and experimental attention have been focused on the bahavior of the equal-time structure factor S(k, t). This spatial fourier transform of equal-time correlations of order parameter fluctuation depends on the time t elapsed after the quench, as expected for such nonlinear relaxation processes. In the present paper, however, we are rather concerned with the structure factor S(k, h, h) describing the correlation of fluctuations at two * Present address" Fachberelch Physik, Universltat Essen, 4300 Essen 1, Unlversitatsstr 2 99

100

C BILLOTET AND K BINDER

different times after the quench. This quantity has received httle attention explicitly so far, but its behavior is also quite interesting. This quantity is also experimentally accessible, as has recently been demonstrated by a hght scattering investigation of temporal fluctuations during the phase separation kinetics of a fluid binary mixture 7) In the present paper, we closely follow previous work LS) devoted to a study of S(k, t). In section 2, we present a generalization of the theory from which an equation of motion for the two-time structure factor S(k, tl, t2) is calculated. In order to obtain a closed equation decouplings 5) are necessary which are valid under quite restrictive conditions only, as discussed in I For the very initial stages of the process (which is not what is studied experimentally 7) this decoupling is expected to be accurate in our case, however~'5'6). In section 3, numerical results are presented and discussed for a variety of representative cases. Section 4 gives results for the associate inelastic scattering intensity, while section 5 contains our conclusions

2. Equation of motion for time-dependent correlation functions As in I, we consider a system which undergoes a second-order phase transmon at the critical temperature To, and focus attention on relaxation processes where the system is quenched at time t = 0 from a temperature To > T~ to T < Tc (fig. 1). Both in the case of a conserved order parameter and in the case of non-conserved order parameter the average order parameter will retain its original value (~b)= 0, and the system will separate into a mixture of two phases with order parameter -+ (~b) as indicated in fig. 1. We assume here that the ordered state is described by a one-component (i.e. scalar) order-parameter density ~b, at each point r With the notation ~b = {¢~,,, ~,~, } we can write down an equation of motion for the probability density p(~b, t) that a state ~b occurs at time t O-t P(@' t) =

d{@}[W(@'---> @)p(@', t) - W(@ ~ ~b')p(~b, t)].

(1)

Here W(@'--> ~ ) describes the transition probability density that a transition occurs from ~ ' to ~ per unit time. Similarly the equation of motion for the conditional probability pCO,d(@t~J,h; @a~, t2) that the system is in s t a t e I~(2) at time tz when it was in state @t~) at time tl is given

poood(,#.,, t,, .,2,. t2) = f dt,U'}[ W(*'--' ,t.'2').°°°d('t' ''', t,, *'. t2) -W(~ba) ~ ~')p¢°"d(Ik °), tl; tk~2~, h)].

(2)

DYNAMIC CORRELATION OF FLUCTUATIONS

T

.

.

.

O .

.

.

.

.

.

.

.

.

101

.

T I

-1

l

i

I

I

-(u/)

0

~

~

+

.1

Fig 1 Order parameter ~bplotted vs temperature T (schematic) A symmetric quench (for which (~b(t)) ~-0) from a temperature To > T~ to a temperature m the ordered region ,s indicated

In I the master equation, eq. (1), was reduced to a Fokker-Planck equation, v denoting the volume of our d-dimensional system, 0t p(~b, t) = -

ddr 8J,(~b, 8~/, t) '

(3)

[SF(~) 8p(@,&/,, t)I_, J,(~,t)=-c:[~p(~b,t)+kBT

(4)

v

where

F(~b) is the (coarse-grained)free energy functionalof the system, the symbol 8 denotes functional derivatives, and C (for the system with nonconserved order parameter) is a rate factor fixing the time scale. In the case of conserved order parameter, C has simply to be replaced by - M V ~ , where M is a mobility. Similarly eq. (2) leads to

at--~pc°nd(I//(l)'

tl;

i//(2)'

t2) = -

f

dar

8Jct°nd(~ (1), tl;

8~)

~bt2), t2) ,

(5)

V

with Jc°nd(~btl), t~; 6~2), t9 = - C [FSF(~ ~ t~)) pco.d(~,), t~; 10¢2),h)

+ ks T

8pc°"d(~°), t~; ¢,t2J, h) 1 &ha) _"

(6)

We use eq. (6) to obtain the equation of motion for the time-dependent correlation function

102

C BILLOTET AND K BINDER

S(r2 - rl, t2, t,) = (~br,(t,)~(t2)) - 6kr,(t,))(O,2(t,)) = S(r2-

(7)

rl, t2, t l ) - (~b,,(tt))(~kr2(t2)},

since the averages in eq. (7) can be expressed in terms of P(O, t) and pC°ha(O"), fl; 0 a), t2) as (~,2(t2)) --= f d{O} ~,~P(O, t2),

(8)

f d{O")f

• (9)

S(r2-

r l , t2, t l ) =

Note that S is translahonally mvariant with respect to spatial coordinates, and hence depends on r 2 - r l only rather than on r~, r 2 separately; S is not translationally invariant with respect to time, since we consider a nonequilibrium relaxation process. From eqs. (5), (9) we hence obtain 0t2

S ( r 2 - r,, t2, t , ) = - ~ d{O")}fJ d{0(2)}p(0(l) , t|) J f

8Jcr°nd(0¢l),tl; 0l/(2), t2) ds0),b(2)

(10)

1;

which, integrating by parts, becomes

0t2

f dfO")}/d{Oa)}p(O J ('), t,)fddrJ~,°nd(o0),

S ( r 2 - r t ) , t2, t , ) = J

t,;

I]/(2) , t2)

v X~ a

,h(l),t,(2 "vr, "t'r2) -_- i~ d { O °)}

f d{0(2)}p(0O, ' t,)

x J~,~(O "~, 6; 0 a), •"v~'r~ ~'"")+ ot2t, ~ ×

f

f

(11)

'') t,)J:~"a(O "), t,, 1~.(2),, "t.h(2) t2]tll

r2 •

From eqs. (3), (8) one similarly obtains [cf. I]

~t2(~,~(t,))= f d{O}J,~(O, t2),

(12)

and using eq. (4) this further yields

Ot2 (~,2(h)) = - C f d{0} ~aF(O) P(O, t2) Similarly we find from eqs. (6, 11) that

(13)

DYNAMIC CORRELATION OF FLUCTUATIONS

103

_0 $(r:c~t2

× ['SF(O ~2~) [ ~,t~ pc°"d(o"~, tl; 0 ~2~,t9 + kB T

~pc°nd(o(I), tl; 0 t2), t2) ] ~d,a~

v-t-r 2

3

-CSt,,2 f d{0 (I)} f d{0t2'}O(Otl/, tl)0~ , x[_[8F(O ~ ~2~)p ,o.d( 0 ~,~,t,; 0% t2)+kBT x ~SP'°~(Oa~' h; 0 (2~,t2)].

(14)

From the definition of the probabilities p and pco.d it is obvious that / 9 ( 0 (2) ,

t2) = f d{O°)}p(O 0), tl)pe°na(O°', h; 0 <2),h),

(15)

and hence

8pc°"~(0"), t~; 0 ¢2~,t2) 8p(0% t2) ( &b~) = J d{O°)}p(O°), tO t~ )

(16)

With eqs. (15), (16) we rewrite the second integral on the right hand side of eq. (14) as (2) ~/.~,(2~ ( al.,.(2)u,.~2~ [ 8 F ( 0 '~') 8V(0 ~2,, t9 ] -Ct~tlt 2 J ~t'~w Jr,2 [ 8~b,,~'"(2) p,.. , t 2 ) - k a T ~ J

= ~,.,:f d{0 ~2~}J,,(O ~2~, t2)~,2,-<2~

(17)

where once more eq. (4) was used. Defining then equal-time correlation functions similar to eq. (7), S(r2 - r,, t) = (O,,(t)~bn(t)) - (~b,l(t))(~,~(t)) = ~'(r 2 -- r,, t) -- (~/,,,(t)) (~b~(t)),

(18)

we note that eq. (17) represents nothing but ~St~t~0S(r2 - r~, t2)/ Ot2. Integrating then the second term in the square bracket of the first integral of the right hand side of eq. (14) by parts yields a term CkaTSt~t28(r2- rO. Hence eq. (14) becomes __t9 ~(r2 _ rl, t2, tl) = - - C f d{O"'} f d{Ot2'}p(O t'', t0pC°"a(Ot', t, • 0 t2', t2)~trl,), ~F(0(2))

0t2

'

+ ~$,,t2[ 1 0 S ( r 2 - r , , t 2 ) + Ck~T~5(r2-r,)]. 0t2

80,~

(19)

104

C BILLOTET AND K BINDER

In order to make further progress, we assume that the free energy functional F(~b) contains a gradient energy term 1K(VAbr) 2 and a coarse-grained free energy density f(~sr), F ( ~ ) = ~ ddr [-~K(Vr~,)2+ f(#s,)].

(20)

u

Hence ~F(~)

-~.--

K V 2 0 r + af(~b~) = _ K V 2 ( 4 + o~/(~br) I

~0~

~,

~

+~

,,=0

~

t""n-I Pr-

~nf

I

.=2 (n - 1) ! ag~" *r=0

(21) and inserting this expression into eq. (19) we find, after some algebra, for the nonconserued case .to(2), t 2,V~,r, x.,.o) ~r22 8F(~b{2)) f d{l~'`I,}f d{~'(2)}p(l~`I), tl)pCond(~t(I,, 11;~p, )

af = K ~ i f d~kk2fdare'k('-'2'(O~,(t,)@r(t2))+(@)t,--~ I,=o

1 ,-1 o~f I + ,=2 (n - 1) v(VSr,(t,)[@~z(t2)] )-8-~ .~o"

(22)

From eq. (19), (21), (22) we note that the nonlinear terms in the expansion, eq. (21) will introduce higher order correlation functions (@(rl, t0[Kb(r2, t2)]'-I), and hence a whole hierarchy of equations of motion is generated, as expected. We here follow the usual strategy of truncating this hierarchy by a suitable decoupling of these higher-order correlations in terms of lower-order ones. As in I we use a decoupling similar to the scheme proposed by Langer, Baron and Miller (LBM)6). We consider the probability p(~s(,l~), tl; dr(2), ~'r2 t2) that at time tl we have the state wr~ -,-o) at rl and at time t2 the state @~)at r2. By definition we have p(Vs~), tl;

't'(2)h) = f d{Vs,,,,} ,1) f d{gt,,,,2}p(¢ ,2) (,,, t0p~°'d(~ °), tl; ,/,,2),h). -r-r 2 ,

The decoupling approximation then is [ A ~ ) -- @, p(~s~.), tl; a,(2) t2) = p(~s~~, t0p(~s~ ), t2) 1 + v. r2 ,

(23)

(@)t~]

S(r2 - ri, t2, tO AQsr, (I)A~s,~ (2)] ((A~/t)2)q ((Agt)2)t2 ].

(24)

As previously, the motwation for this decoupling is to approximate the two-point probability as a product of one-point probabilities, and correct for correlation effects self-consistently by the use of the two-point correlation function. Since the latter, in thermal equilibrium, is defined via ,t,a) • x.~.,.o)A.,.(2) S~q (r: - r,, t2 - t,) -- f d V/,l,) .If a,t,a),.,~,l,(i)_.,.,~,.,~.,,,t,; "t" r 2 , t2Iz-IIPrl ~.a~r 2 , d

(25)

DYNAMIC CORRELATION OF FLUCTUATIONS

105

we immediately realize that then the use of eq. (24) in eq. (25) leads to an identity. The idea then is to use the knowledge on fluctuations m equilibrium states, and use eq. (24) for the equation of motion of S(r2-rj, t2, h), which describes th dynamics of fluctuations far from equilibrium. Combining hence eqs. (24), (22) we find

{~,,(tl)[~b~.(t2)]"-I) =

(lll)t,([lll]n-l)t 2 q"

S(r2r~, t2, tl) (~,_lA~)t2, ((Ai//)2)t~

(26)

and further 1

=

t

t

(~b)t' a~-~f~L=0 + ,~_-2(n - 1)., (~,,( ~)[4'~( 9

Of

].-t

a"f

)~

I.=o

S(r2- r" t2' t') (A~b fl~rf' a~u/

= (1~),(~-~)t2-I" '

(27)

((A 1/1)2)t2

Defining then spatial fourier transforms S(k, t2, h) = (l/v)

f ddr e'~'S(r, t2, t0,

(28)

Eq. (19) is rewritten with the help of eqs. (22), (27) as

~---S(r2-r,,t2, t , ) = - C { K (2~)d f ddkk2e-"('2-r')S(k,h,t,)

clt 2

+ (~0),,

,:

((a,/,)bt2

[l aS(r2- r,, + 8tit2 ~ at2

,2

t2) ÷ Cka T,(r,- r,)]

(29)

With the definitions

we then find

c3 S(k, t2, t,) = - C[ Kk2S(k, t2, t,) + Seq (k" t2- t') lA~b-~ I 0t2 ((A~)2)t2 t2

+ (21r)dvcS(k)(0)t'(afl-~f~}t2 }+ ~,2,,[~ aS(k,at2/2)1-.Ck~T].

(31)

Using then the relation

S(k, t2, tO = S(k, t2, t , ) - (#J),2(~)t, ~ we finally get

8(k)

(32)

106

C BILLOTETAND K BINDER

--8t2

tL

\

8"~af/d

t0} (33)

Ot2

Defining A(t2) ------((A@)2)721 A@ ~-~ t2'

(34)

we find that for t2 > t~ the solution of eq. (33) is given by t2

ydt Kk2÷A(t, 1

(35,

tl

In the case of conserved order p a r a m e t e r the calculation is completely analogous to the derivation of eqs. (22)-(35), the only difference being that C is replaced by -MV~, and hence taking spatial fourier transforms this yields a factor Mk2.The final results, eqs. (33)-(35), are valid for the conserved case as well if we just replace C by M k 2. It Is interesting to note that eq. (35) has a form completely analogous to that which one obtains in thermal equilibrium from generalized Landau theory for this case, t2

tl

= Seq (k) exp[ - C(t2 - t O K ( k 2 + ~-2)1,

(36)

where ~ is the correlation length of order parameter fluctuations. Hence V~(h)/K plays the r61e of an effective time-dependent correlation length of fluctuations, which determines the decay rate of fluctuations during the nonlinear relaxation process [note, however, that A(t2) may become negative for some times due to the initial instability of the system after the quench and then fluctuations increase over some range of time rather than decreasel. Also the amplitude of the fluctuations, which is measured by the equal-time structure factor, S(k, h), is not a constant but depends on the time h after the quench. The time evolution of the equal-time structure factor is found from eq. (33) putting tl = t2, ~---S(k, t) = - 2 C { [ K k 2 + A ( t ) ] S ( k , t) - kBT[v}, at

(37)

which has been studied in I. From these previous results on S(k, t) and A ( t ) the time-displaced structure factor S(k, t 2, tO is hence readdy obtained, using

DYNAMIC CORRELATION

OF FLUCTUATIONS

107

eq. (35) and performing the quadrature numerically These results will be discussed in the next section. 3. Numerical results for the time-dependent structure factor in quenching experiments

For the explicit calculation we have used a free-energy density f ( ~ ) of G m z b u r g - L a n d a u form. It is convenient to cast the results in a scaled universal form, i.e. measure time in such units ts and momentum In units ks, -r = t / h ,

q = k/ks,

S ( q , Ti, 72) = S ( k , tl, t2)/[2Seq (k = 0)],

(38)

for which eq. (36) takes the following dimensionless form [cf. I] Seq (k, t) = Seq (k) exp[ - 7(q 2 + 2)].

(39)

Figs. 2-6 show the resulting structure factor S ( q , 7b 72) for quenches from infinite temperature to a temperature T slightly below Tc (in the chosen scaling renormalizatlon the dependence on T is absorbed in the scales, cf I) in the case of nonconserved order parameter. It is seen that for small 71 the structure factor is markedly different from its thermal equilibrium form, eq. (36), where the time-dependence is a simple exponential decay. For larger 71 and not too small q the structure factor fairly quickly reaches its asymptotic form, which is (AT - 72 - 71)

S (q,~l ~2 }

15

10

0 0

5

10

I 15

I 20

A

Fig 2 N o r m a h z e d structure function S(q, "rt, T2) at a scaled time ~-j = 0 l after a quench f r o m infinite temperature plotted vs A~ ~ ~'2- ~J f o r various q (order parameter not conserved)

108

C BILLOTET AND K BINDER

S(q,z

30

20 =0238 't1=2 I /q=0 476

5

10

15

20

&'t

Ftg 3 N o r m a h z e d structure function S(q,r~, re) at a scaled ttme r~ = 2 after a q u e n c h from mfimte t e m p e r a t u r e plotted vs Ar ------r2 - r~ for various q

S (q.'q ,~2 )

150

7

100

50

-5

0

5

10

15

20 A't,

Fig. 4 N o r m a h z e d s t r u c t u r e functton S(q, rh r,) at a scaled time ~1 = 5 after a que nc h from infimte t e m p e r a t u r e plotted vs d r ~ r2 x rt for various q

DYNAMIC CORRELATION

OF FLUCTUATIONS

Slq,'~T"~2)

109

S(q.'tl "t2)

T.15

1150

q238q=0 =0 476

-20

-10

10

20

30 A~

Fig 5 N o r m a h z e d s t r u c t u r e f u n c t i o n S(q, r~, ~'2) at a s c a l e d t i m e 7, = 20 a f t e r a q u e n c h f r o m m f i m t e t e m p e r a t u r e p l o t t e d v s A~" ~ ~'2- ~~ for v a r i o u s q

S{q,'q "~2~

-100

-50

0

50

100 a't

F i g 6 N o r m a h z e d s t r u c t u r e f u n c t i o n S(q, ¢1, r2) at a s c a l e d tmme ~-~= 100 a f t e r a q u e n c h f r o m m f i m t e t e m p e r a t u r e p l o t t e d v s A~" -= ~'2 - ~'~ for v a r i o u s q

110

C BILLOTETANDK

BINDER

hm S(q, rl, ~'9 = expl - q21Arl]/q 2.

(40)

rl, 72~

As noted already m I, our decoupling a p p r o x l m a h o n yields unphyslcal results at late times, and hence eq (40) is e x p e c t e d to be reasonable for q ~ 1 only, which Is not the regime of interest here. What we rather expect to happen is that S(k, t2, tl) ts basically g w e n by S~q (k, t 2 - t,) provided the conditions q2rl >> 1,

tA~'t~ rl

(41)

hold. For small enough q one hence can see transient effects in the timedependent structure factor as long as r, ~< 1/q 2, i.e for tames m a r k e d l y longer than the relaxation time in thermal equilibrium. The case of c o n s e r v e d order p a r a m e t e r is quahtatively similar, figs 7-9. Again we find that for large q the dynamic structure factor reaches fairly quickly its a s y m p t o t i c behavior, which in our a p p r o x l m a h o n is lira S(q, z,, r2) = exp[ - q'lArl]/q 2,

(42)

~'1, ~'2~

while the correct behavior would be again g w e n by eq (36), with C = Mk 2 and an additional ~-functlon smgularity at q = 0, t~ = tl. N o w the condition that one ms close to the a s y m p t o t i c regime where S(q, r~, ~2) depends on IAr[ only and no longer on ~'r, r2 separately, reads >> 1,

S{q.x~,x21 _

la l

(43)

r,. 500

1000

-

10

i

1500

2000

I

q~= ~

....

2500 Ax

~

]

f,........~

S(q.'t

10

20

A~: 30

Fig 7 N~rma~izedstru~turefunctl~n~(q~r2~'r2)atasca~dtlme~t=~afteraquen~hfr~mmfimte t e m p e r a t u r e plotted vs A~" --- r2 - r~ for various q (order p a r a m e t e r c o n s e r v e d )

DYNAMIC CORRELATION OF FLUCTUATIONS

Ill

S {q,'t 1 "t2)

86I

I q=O 476

4

//k~

`~'~o ,

~._

~

,

0

10

J

20

..

<° ~8 ,

I

~

30

40

~,z

Fig 8 N o r m a h z e d structure function S(q, ~'), z2) at a scaled t~me r~ = 5 after a q u e n c h from mfimte t e m p e r a t u r e plotted vs Az ------r 2 - rj for various q

S (q,x ,'t 2 ) 40

30

20

.t~:lO0 I

q =0~76"

10

j q =0 71l /~q=0952 -100

0

j 100

200

300

/,'~

Fig 9 N o r m a h z e d structure function S(q, ~-~,¢2) at a scaled time ~'t = 100 after a q u e n c h from mfimte t e m p e r a t u r e plotted vs z1~-~ ~'2- ~t for v a n o u s q

112

C BILLOTET AND K BINDER

For small enough q one hence can see transient effects In the time-dependent structure factor as long as 71 ~< 1/q 4. From fig. 9 it is obvious that for small enough q the deviations from equilibrium then are drastic even for AT of order unity. At late stages one expects, however, that the characteristic w a v e v e c t o r obeys a L l f s h i t z - S l y o z o v 8) law qmaxOC t -113 rather qmax~ t -~t4 inherent in the present treatment9); hence for a given q the present approximations somewhat over-estimate the time needed for the dynamic structure factor to reach its equilibrium behavior, eq. (36). It is also Instructive to examine the behavior of the fourier transform S(q, o2, rl) defined by +~

(44)

$(q, to, zl) = / d(Ar) e '°'~" S(q, TI, Z2)

In thermal equilibrium $(q, to, rl) is independent of rl, due to translational invariance of S(q, T1, rE) with respect to time. In addition, S(q, T~, T2) is an even function of AT, and hence the imaginary part of S(q, to, zl) then vanishes identically. In the nonequilibrium case, where S(q, zi, T2) depends on both AT and TI, it is clear that S(q, T~, T~) will be no longer an even function of AT (cf. figs. 2-9), and hence in general the imaginary part of S(q, to, rt) is nonzero Also the real part of S(q, to, T0, which in our case for thermal equilibrium [eq. (36)] is a simple lorentzian, 2toe(k)

Seq (k, o)) : Seq ( k ) 0) 2 _1._[toc(k)]2,

toe(k) = CK(k 2 + E-s),

(45)

may b e c o m e quite complicated (see e.g. fig. 10). From eq. (35) it is easy to derive the following general expression

i

tl+t

0

tl tl

4 S(k, O) e -~'~'+crk2)''

i~70-~)]

[ exp/C f dt'a(t')] 0

0

tl

-t I

tl+t

where A(0) is related to the correlation length in the critical state above T~. From eq. (46) we conclude that oscillatory terms e x p [ - ltotl], although damped, will in general not cancel out when A(t) and S(k, t) are time-dependent {for A(t) =--~-2 and S(k, t) =-- Seq (k) we recover the equilibrium result, eq. (45),

DYNAMIC CORRELATION OF FLUCTUATIONS

113

15

10

real part ag~nary pa, t OS

L7

15

.

~t s

3

-05

Fig 10 Normahzed fourier transform S(q, to, ~'a) plotted versus o~t2 at rl = 5, q = 0 357 (order parameter conserved)

from eq. (46), as expected}. The numerical results (e.g. fig. 10) present clear evidence for the oscillatory terms. We think that this behavior is not an artefact of our decouphng approximation but a real effect, since the decoupling should be accurate for small r~, T2, and hence S ( q , to, rn) is expected to be reliable for small ~1 and large tots.

4. Inelastic scattering from decaying states For thermal equilibrium states, the intensity of inelastic scattering of light, neutrons, etc., which involves a m o m e n t u m transfer h k and energy transfer hto, is simply proportional to S,q (k, to). For nonequilibrium situations, the intensity at time tv is not simply related to S(k, to, h). This had to be expected, since S(k, to, tl) is no longer a real (and positive) quantity, as S,q (k, oJ) has to be in order to relate to scattering intensities. Furukawa n°) has already considered this problem and showed that the double-differential scattering cross section is expressed as tl

d2cr(tl.__~) d ~ dto

/.

Re I dt e '~u'-° S(k, tl, t).

(47)

114

c BILLOTET AND K BINDER

It 1s straightforward to show that in equilibrium one obtains from eq. (47) the standard result H) d2°req cc ~q (k, to), dO dto

(48)

while m the nonequihbrium situation we obtain from eq. (35), putting the proportionahty constant in eq. (47) equal to unity, d2tr(t0 S(k, O) e -crk2t, dO dto = 602 + C2[ K K 2 + A(0)] 2 {to sm totl - C[ Kk 2 + A(0)] cos toh} tl

0 tI P

t1

+ / d t cos[to(t1- t)]S(k, t) e x p [ - C K k 2 ( t , - t ) - C I dt'A(t~)] 0

t

(49) From this result it is obvious that oscillatory terms ( ~ sin totb cos tot0 occur also in the cross section. Of course, these terms can be seen only for frequencies of order 1/t~, which are extremely small for physical quenching times. However, since for small k S(k, t) needs a large time to reach its equihbrium value Seq (k), it is clear from eq. (49) that the cross section will deviate from its equilibrium structure [eq. (45)] for to ~ toe(k) if S(k, tO still differs appreoably from Seq (k). Figs. 11-14 show some numerical examples for the case of both nonconserved and conserved order parameter. Pronounced deviations from lorentzian line-shape as well as the effect of the oscillatory terms can be clearly recognized. However, the present approximations seem still to be rather poor as they do not comply with the condition that the cross section must be strictly non-negative 5. Conclusions

In this paper time-dependent correlation functions have been studied for systems which are quenched from a d~sordered state to a state below the critical temperature of an order-disorder or unmixing transition, respectively. A decoupling approximation for the two-point probability distribution is used to derive a closed equation of motion for the structure factor S(k, t2, tl), eq. (33). In the limiting case of infinitesimal small quenches (thermal equilibrium) the resulting structure factor S(k, t 2 - t l ) is that of ordinary (generalized)

DYNAMIC CORRELATION

OF FLUCTUATIONS

115

Ginzburg-Landau theory, eq. (36). Extensive numerical calculations are presented for the nonequilibrium case It is shown that for very small k the structure factor S(k, tz, fi) differs from the equihbrium result S(k, h - t l ) significantly for fairly late ttmes tl after the quench, while for large k correlation of fluctuations rapidly approaches equilibrium behavior These results are m accord wtth the expectation that the dynamics of fluctuattons ts different if their wavelength ts comparable to the lengths on which the system is still inhomogeneous These effects can be seen experimentally m principle by studying the inelastic scattering cross section. The scattering contains also terms oscillating with cos(cob) factors where t~ is the time after the quench. This latter effect ts hardly observable experimentally Since the decouphng approximation used here is insufficient to describe the behavior at late times rehably, our results should be considered as quahtattve only. Therefore a comparison wtth recent light scattering experiments, where the dynamics of fluctuattons during the spinodal decomposition of a fluid binary mtxture was studiedV), cannot yet be attempted on a quantitative basis

O025 d2o d~dto O02

J,xl=O 1j 0015

q=1190 001

0 0 0 ,~

0

0

I

I

15

3

tots

Fig 11 N o r m a h z e d scattering c r o s s section d2d/d/~ dto plotted v e r s u s tot, at ~'t = 0 1 and v a r i o u s q (order p a r a m e t e r n o t c o n s e r v e d ) [ordinate m u m t s of d2o-eq/dO dto (k = 0, to = 0)l

116

C BILLOTET AND K BINDER

10 d2o 8

I "~1:5 I 6

4

O0

~1

5

3

wto

Ftg 12 Normahzed scattermg cross section d26-/d~2 dto plotted versus tots at rj = 5 and various q (order parameter not conserved)

002 d2o dQd~ 0 015

001

0005

3

tots

Fig 13 Normahzed scattering cross section d2d'/df/d~o plotted versus ~ots at ¢~ = 0 I and various q (order parameter conserved)

DYNAMIC CORRELATION OF FLUCTUATIONS

117

d2o

1 ,:51

d~dw

2

N~q

=0357

~t

o

15

t

3

wt~

Fig 14 Normahzed scattering cross section d2#/d~ dto plotted versus (order parameter conserved)

tot2at ~'~= 5 and various q

C l e a r l y , a m o r e p o w e r f u l t h e o r y of s u c h n o n h n e a r r e l a x a t i o n p r o c e s s e s IS called for, w h i c h does n o t yet exist at p r e s e n t . H o w e v e r , the p r e s e n t p a p e r w a s i n t e n d e d to d r a w a t t e n t i o n to the m f o r m a t i o n w h i c h c a n be g a m e d f r o m c o n s i d e r i n g the t w o - t i m e s t r u c t u r e f a c t o r S(k, tl, rE) a n d the a s s o c i a t e inelastic s c a t t e r i n g c r o s s s e c t i o n d u r i n g such r e l a x a t i o n p r o c e s s e s far f r o m e q u i h b n u m . B y the p r e s e n t m e t h o d s , o n e e x p e c t s to o b t a i n S(k, t~, h) m o r e r e h a b l y if q u e n c h e s w e r e p e r f o r m e d s t a y i n g e n t i r e l y w i t h i n the o n e - p h a s e region A v a i l a b l e e x p e r i m e n t s ~2) c o n s i d e r the e q u a l time c o r r e l a t i o n S(k, t~) o n l y w h i c h is in good a g r e e m e n t with t h e o r e t i c a l expectationsg).

References I) C Bdlotet and K Binder, Z Physlk B32, (1978) 195 This paper will hereafter be denoted as | 2) K Binder, Sohd State Comm 34 (1980) 191 3) See e g K Binder, Phys Rev B8 (1973) 3423 4) See e g K Kawasakl, M C Yalablk and J D Gunton, Phys Rev AI7 (1978) 455, E LevJch and V Yakhot, Phys Rev B15 (1977) 243 5) See e g J W Cahn, Trans Metall Soc AIME 242, (1968) 166 for a general review, recent experimental work ~s found m V Gerold and J Kostorz, J. Apply Cryst 11 (1978) 376, W 1 Goidburg, C -H Shaw, J S Huang and M S. Pliant, J Chem Phys. 68 (1978) 484; N C Wong and C.M Knobler, J. Chem Phys 69 (1978) 725 6) K Binder, C Bfllotet and P Mirold, Z Physlk B30 (1978) 183, J S Langer, M Baron and H D Miller, Phys Rev. A11 (1975) 1417 7) M W KIm, A J Schwartz and W 1 Goldburg, Phys Rev Lett 41 (1978) 657

118 8) 9) 10) 11) 12)

C BILLOTET AND K BINDER I M LIfshttz and V V Slyozov, J Phys Chem Sohds 19 (1961) 35 K Binder, Phys Rev BI5 (1977) 4425 H Furukawa, Progr Theor Phys 58(1977)!127 L van Hove, Phys Rev 95 (1954) 249 N C Wong and C M Knobler, Phys Rev Lett 43 (1979) 1733