Spinodal decomposition in adiabatically closed systems: theory

Physics LettersA 158 (1991) 307—312 North-Holland

PHYSICS LETTERS A

Spinodal decomposition in adiabatically closed systems: theory J. Schmelzer Fachbereich Physik der Universitdt Rostock, Universitatsplatz 3, 0-2500 Rostock 1, Germany

and A. Milchev Institute ofPhysical Chemistry, Bulgarian Academyof Sciences, Sofia 1040, Bulgaria Received 27 March 1991; accepted for publication 8 July 1991 Communicatedby A.A. Maradudin

The initial stages of spinodal decomposition are investigated for the case that the phase-separating system is adiabatically isolated. It is shown that the latent heat ofthe transition and the resulting variation in temperature lead to significant changes of this process, as compared with the isothermal Cahn—Hilliard theory. This may give a new clue to the understanding of experimental results.

1. Introduction Nucleation and spinodal decomposition are two major conceptsfor the theoretical description of firstorder phase transitions [1,2]. Though the theoretical foundations were laid about a hundred years ago by Gibbs [3] and van der Waals [4], a kinetic theory of spinodal decomposition including surface effects was formulated only at the end of the fifties by Hillert [5], Cahn and Hilliard [6], notwithstanding the fact that experimental evidence of such a type of transition was already accumulated 20—30 years earher (see ref. [7]). The original Cahn—Hilliard theory ofspinodal decomposition [6] and also the majority of subsequent approaches presume that the decomposition process proceeds under isothermal conditions [8]. This assumption is appropriate when diffusion processes proceed slowly in comparison with thermal equilibration and exchange processes. Later, this condition was modified to allow for a prescribed temperature variation in the course of the transition [9]. However, if diffusion and thermal conduction processes proceed with comparable rate constants,

the decomposition process itself results in temperature variations due to the latent heat of the transition. Such effects are expected to be of particular importance for adiabatically isolated systems since in corresponding investigations of nucleation and growth processes in adiabatically closed systems both quantitative and qualitative modifications of the kinetics due to temperature variations in the systems could be observed [10,11]. It is the aim of the present and subsequent papers to develop a theoretical description of spinodal decomposition for adiabatically closed systems and to compare the results with the predictions of the Cahn— Hilliard and other theories developed for isothermal conditions. The conclusions will be illustrated by Monte Carlo simulations [12]. As a model system a binary regular solution [131 with an upper critical solution point was chosen. The theory can be applied, of course, in a slightly modified form to other systems. 2. The Cahn—Hilliard theory: basic equations and solutions According to Caha [141 the spatio-temporal evo-

0375-9601/91/S 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

307

Volume 158, number 6,7

PHYSICS LETTERS A

9 September 1991

lution of the concentration c of one of the components in a binary solution can be described in a hinearized approximation by

we may write, e.g.,

Oc/Ot=M(02g/8c2) I ~~C—

It can be seen that for isothermal systems the amplification rate depends only on the actual value of the wave vector and not on the degree of evolution

2M,cV4c.

(1)

c is the local concentration of, say, component A in relative units (c ~ 1), c 0 the initial concentration of the unstable homogeneous solution, T the temperature, g the free enthalpy per particle, K an expansion coefficient ofof g with respectMa to mobility Vc reflecting the inhomogeneity the system, coefficient. A special solution of (1) can be written as

A(k’,t’)=A(k’,0)exp[k’2(k~2—k’2)t’]

.

(10)

of the considered or other modes. The amplification factor, 2 (k’~2 k’2) (11) R ) = k’in fig. 1. is (k’ shown The temporal evolution of the spectral function —

,

c=c 0+A(k, t) cos(k-r)

(2)

.

Substitution of (2) into (1) shows that the evolution of the amplitude A is determined by the differential equation 2A[(02g/0c2)IT~+2,ck2] (3) 0A(k,t)/0t=—Mk The solution of (3) reads A(k, t)=A(k, 0) exp[R(k)t] R(k) = —Mk2[(O2g/0c2)I,~~+2Kk2]. ,

(4) (5)

It follows from (5) that a critical wave number k~ exists defined by =

—

(1/2K) (0 2g/0c2 ) I

~

(6)

The amplitude of the harmonic fluctuation grows with time for a value of the wave vector k2 k~.For k2=k~the amplitude remains constant. The relations (3)— (6) are also valid for the general solution of (1),

A(k’, t’) for isothermal conditions is illustrated in fig. 2. Assuming A (k’, 0) = const in agreement with (11), an amplification is found for k’
R(kl106

a3 c(r, t)=c 0+ (2)3/2 JA(k, I) exp(ik-r) dk, (7) evolving from the initial concentration fluctuation, 3 c(r, 0) =c a 0 + (2)3/2 A(k, t=0) exp(ik-r) dk.

15 10

$

(8) The length parameter a is introduced in (7) and (8) to allow a transformation to dimensionless quantities. It will be specified later. Defining the reduced variables 4, (9) k’ =ak, r’ =r/a, t’ =2t(Mic)/a 308

05

6~

Fig. 1. The amplification rate R(k’) as a function of k has for isothermal conditions a time-independent form as shown in this figure (compare (11)). The critical wave number k~in reduced variables was chosen as k~ =0.09.

Volume 158, number 6,7

PHYSICS LETTERS A

9 September 1991

interaction of the particles is decreased in the course of the transition. Moreover, for regular solutions the parameter ic has the form [6]

A(R’t)

(16) where a is typically of the order of the intermolecular distance in the solution. Eq. (16) completes the definition of the length parameter, introduced in (7)—(9). Substitution of (14) and (16) into (6) yields 0.5 0.1 k Fig. 2. Evolution in time of the spectral function A (k, t’) for isothermal conditions. The different curves refer to different mo-

k~= 2

~T

I \ 2co c0 (1 c0 )) For regular solutions the spinodal curve is deter—

—

ments of time (I) t’ =l0~(2) t’ =3x 10~(3) t’ =5x lOg; (4) = 7 x l0~.The typical features of spinodal decomposition under isothermal conditions are clearly to be seen: all curves coincide at a common point of intersection, the maxima of thedifferent curves are found at the same position for all moments oftime.

the different modes, characterized by different wave vectors k, grow or decay independently from each other, at least, so far as (1) is a sufficiently accurate approximation. Hereby the relation 2(k~—k’2)t’ (13) ln[A(k’, t’)/A(k’, 0)]=k’ is fulfilled.

3. Spinodal decomposition and temperature

mined by kT________

=

2co c(l—c) Physically reasonable values for k~are obtained only for values of the initial concentration c0 inside the spinodal curve. Moreover, the inequality k~<2 (19) is fulfilled always. The concentration dependence ofthe potential energy per particle in a regular solution is given further by [6] u=wc(1—c)+ic(Vc)2.

variations in regular solutions For homogeneous regular solutions [13] the free enthalpy per particle g can be written as g=wc(l—c)+kBT[clnc+(l-—c)ln(1—c)],

(18)

(20)

As a consequence of (20) the change ofthe potential energy per particle in the final stage ofthe transition can be expressed, neglecting interfacial contributions,as

(14) where kB is the Boltzmann constant. The parameter o can be interpreted as a measure of the difference between the potential energy of the interaction of particles of different kinds. It is defined as (15)

In this equation z is the number of nearest neighbours, VAA, VBB and VAB are the interaction potentials between like and unlike particles, respectively. In addition, V~= VBB is assumed. A decomposition in regular solutions is possible only for w>-0. It means that the potential energy of

~UW(CaCo)(CpCo)<0,

(21)

where Ca and cp are the concentrations of the evolving macroscopic phases. Again, it follows from the necessary condition Ca
(22)

The specific heat per particle C~is taken equal for both types of particles and is assumed to be tem309

Volume 158, number 6,7

PHYSICS LETTERS A

perature-independent. Decomposition processes in a regular solution are thus accompanied for adiabatic constraints by an increase of temperature. More generally the total decrease of the potential energy i~U~ connected with the evolution of a concentration profile C(r, I) can be expressed as

~iJ

L~U 1

C)+K(VC)

0(l

C0)]

dv, (23)

which after some transformations yields dV{_w(C_Co)2+K[V(C_C

—

Assuming an internal thermal equilibrium and a temperature-independent specific heat of the solution similar to (22), we obtain (29) E~T=(A2co/2C~)(l—~cx2k2) and 2k2) (30)

2WC

[WC(l

9 September 1991

2}.

i.~T= VCp JIdkA(k, t)A(—k,t)(! —~a ~

for both considered cases (26 )and (28), respectively. Consequently, also in the general case the decomposition is accompanied by an increase in temperature.

0)]

(24)

N is the total number of particles and V the volume of the system. For harmonic spatial fluctuations of the form given by (2) we obtain for the average decrease of the potential energy per particle 2(—w+,ck2). (25) i.~u=~A For regular solutions this expression is equivalent to L~u=~A2w( —1 + ~k’2) (26) .

For all harmonic fluctuations of the form (2) the decomposition process is accompanied thus by a decrease of the potential energy or an increase of temperature condition if the reduced wave number obeys the 2> 1 or k’ 4z!. Consequently, relation (27) is fulfilled for all modes having a physical relevance. For the general solution, given by (7) and (8), the change of the potential energy is a functional of the actual concentration field. Indeed, substitution of the general solution (7) into (24) results in .

—

6 ~wa

J

dk A(k, t)A( —k, t)(!

—

~a2k2) (28)

310

It can be checked easily that with the particular choice of the spectral function A(k

312/2a3] 1, t)—~A(k1,t)[(2~) x [ô(k—k 1 ) +o(k+k1 ) I (3!) the general equations (7), (28) and (30) are reduced to the special expressions (2), (25) and (29), respectively. In the calculations the integration has to be performed with respect to k 1. If a discrete set of modes evolves in the system, then instead of (3!) and (29) one obtains (2x 1, t) La) 3/2 A(k1, t)—. ~ ~A(k~’ x [ö(kW~k 1 )+ö(k~’~+k1 )] and ,~

L~T=

~—

~

t)(l—~a2k”2).

(32) (33)

4. Kinetics of spinodal decomposition for the case of an internal thermal equilibrium We consider now the temporal evolution of the concentration field C( r, t) taking into account ternperature variations of the system due to the latent heat of the transition. The temperature variations are described by (29) or (30). This implies the condition of an internalthermal equilibrium to be fulfilled. To demonstrate qualitatively the influence of temperature variations on the kinetics of the decom-

Volume 158, number 6,7

PHYSICS LETTERS A

position process first the evolution ofone single mode of the form given by (2) is analyzed. The evolution of the amplitude A can again be expressed by (34), (34) OA(k’, t’ )/Ot’ =Ak’2(k~—k’2) .

9 September 1991

Ln~

However, in contrast to isothermal conditions, only for small values of A the solution behaves as described by (13) since the temperature T increases and the critical wave number decreases with time according to k~=2{l—(kBT/2w)[l/Co(!—CoJ},

(35)

T=To+(A2w/2C~)(l—~k’2)

(36)

.

__________________________________

0

500l~

Denoting by k~( T 0) the critical wave vector for an isothermal decomposition process at a temperature T0, (34) can be transformed into 0 In A (k’, t’

2) \

=k~2(k~2(To)_k12_A2(k~,t~) 2C

k~(l—~k’ 0(l—C0)C~)

Fig. 3. Dependence of In [A(k’, C) IA(k’, 0)J versus time for (a) isothermal and (b) adiabatic conditions. While according to (13) for isothermal conditions a linear dependence is found, for adiabatic conditions the increase of temperature results in a decrease of the growth rate and in a deviation from a straight line (compare (37)). For both cases the initial amplitude was taken equal to A(k’, 0)=0.05, k’ is equal to k’=k~/.,J~ with k~= 0.09.

(37) A(k,t’)

The of and eq. (37) the values interesting 2(Tsolution2>0 small for initial A(k’,case 0) k’ 0)—k’ reads [A(k’, I’) (a2—b2A2(k’, 0) lnLA(k~o)~a2_b2A2(k’,t~))

]

02

2

=a2t’,

(38)

with 0.1k

1

a2=k’2[k~2(To)—k’2I b2=k’2k

2)/2C 3(! —ik’ 0( 1 —Co)C~, 20 (k’
(39) 11/2

(40)

~05

0.1

0.15 k

0)).

The evolution of the amplitude for one special value of k’ < k~( T0) is shown in fig. 3. For comparison, the results are given also for the isothermal case, when the evolution of the amplitude is described by (13). Furthermore, three conclusions can be drawn from (34)—(37): (i) the increase of temperature results in the decrease of the amplification rate for a given wave

Fig. 4. Time evolution of the spectral function A (k’, C) for 15 modes ofthe form c(r, t)=co+A(k’, t’)cos(k-r) in an adiabatically closed system. The initial valueofthe critical wave number is set equal to k~( T0) = 0.1, the values of the wave numbers of the other modes are k~ =0.1k(T0), k1=0.2k~(T0) 1 .5k~(T0). For all modes the initial amplitude was the same 4 (3) t’=5xl04 (4) (A (0) = 0.05). The different curves t’= 10’; (1) (5) t’=4X l0~(6) t’ correspond =13x iO~.Ttoothe givefollowing a better time13x steps: t’=O; (2) t’=3xl0 impression of the behaviour, the discrete set of points for each time is connected by a continuous curve.

31!

Volume 158, number 6,7

PHYSICS LETTERS A

number of the fluctuations compared with isothermal conditions; (ii) the critical and spinodal wave numbers k~and k’rn are shifted to lower values in the course of the transition; (iii) the spectral functions A (k, I) corresponding to different values of time do not intersect at a common point, as happens for isothermal conditions (compare fig. 2). These conclusions are illustrated by fig. 4 for the general situation that the evolution of the concentration profile is described by (7) and (8). For fluctuations ofthis form the amplification rate of a single mode is determined by (34) again. However, the temperature T depends at each moment of time on the degree of evolution of all modes and, according to (30), we have to write 2a2). T=T0+ C~V dkA(k, t)A( —k, t)(l —3k (41)

J

~

Eq. (37) is replaced then by O lnA

k’2 (k~(To)k~2

References [1] J.D. Gunton, M. San Miguel and P.S. Sahni, in: Phase

[21K. BinderandD. Stauffer, Adv. Phys. 25(1976). [3] J.W. Gibbs, The collected works, Vol. 1. Thermodynamics

[41J.D. van der Waals, J. Stat. Phys. 20

I

vJ dkA(k)A( —k) (1— ~k2a2)). (42)

As an example, in fig. 4 the evolution of a discrete set of harmonic modes with the wave numbers 0.lk~(T0),0.2k~(T0), l.5k~(T0)and the same value A (0) = 0.05 of the initial amplitude is demonstrated. For this case, (42) yields 2 (k~2( T 0 In A (k, ~ = k 0) Ot’ ...,

—

____________

kB

312

The results of the outlined analysis show that the latent heat released in the course of spinodal decomposition may significantly influence the characteristics of this process. In a forthcoming paper [121 the principal results are illustrated by Monte Carlo calculations. Possible further steps to be carried out in the future are the extension ofthe analysis to include non-linear terms in the basic equation (1) [8,15], the analysis of other classes of solutions, relinguishing the assumption of an internal thermal equilibrium.

(1979) 197.

C~Co(1—C 0)

—

taken equal to 0.1.

5. Discussion

(1928).

k~a6

was

transitions phenomena, Vol. 8, eds. C. Domb, M.S. Green and and critical J.L. Lebowitz (1983).

Ot’

—

Here k~ ( T0)

9 September 1991

2C~Co(l—C0)

v ~

2 )A2(k (I

—

~k~a

1, t’)).

)

[5] M. Hillert, Ph.D. MITJ. (1956). [6] J.W. Cahu and J.E.thesis, Hilliard, Chem. Phys. 28 (1958) 258. [7] J.W. Cahn, Trans. Metall. Soc. AIME 242 (1968) 166. [81 S. Komura, Phase Transitions 12 (1988) 3. [91E.L. Huston, J.W. Cahn and J.E. Hilliard, Acta Metall. 14 (1966) 1053. [lO]J.(1989) Schmelzer 104. and H. Ulbricht, J. Colloid Interf. Sci. 128 [11]J. Schmelzer and F. Schweitzer, Z. Phys. Chemie (Leipzig) 270 (1989) 5. [12]A. Milchev and J. Schmelzer, Spinodal decompositon in adiabatically closed systems: Monte Carlo simulations, to be published. [l3]R.Becker,Ann.Phys.32 (1938) 128. [14]J.W. Cahn, ActaMetall. 9 (1961) 795. [15]J.S. Langer, M. Bar-on and H.D. Miller, Phys. Rev. A 11 (1975) 1417.