On the liquid phase instability criterion

Physica 109A (1981) 357-363 North-Holland Publishing Co.

ON THE LIQUID PHASE INSTABILITY CRITERION

V.N. RYZHOV and E.E. TAREYEVA Institute of High Pressure Physics, Academy of Sciences, Moscow, USSR

Received 17 February 1981

The criterion of instability of liquid towards solid formation (differentfrom the Kirkwood's one) is derived on the base of investigation of the exact closed integral equation for the singlet distribution function. This criterion is also obtained directly from the first equation of the BBGKYhierarchy. The kinetic equation approach which gives the same criterion is discussed briefly.

The problem of the formation of the ordered crystalline distribution from the uniform and isotropic fluid state is often discussed from two closely related points of view. Since Kirkwood's study 1) attempts were made to obtain a response of the fluid to a weak external field with the crystalline symmetry2~). In this approach the instability of the fluid towards crystalline phase formation is associated with the susceptibility divergence. An alternative approach is based on the investigation of nonlinear integral equations for the singlet distribution function p(r)5-H'J4). These equations always have a constant solution p(r) = pe appropriated to the liquid phase. However, because of the nonlinearity, the constant solution is not unique and as the parameters vary some new solutions appear which can have the crystalline symmetry. Usually these solutions are associated with crystalline phases. It is obvious that Bogolubov's concept of quasi-averages tz) plays an important role in the approach to the crystallization phenomena as the broken space symmetry problem. A number of authors (see e.g. 13-16) emphasized the close relation between both of these approaches. In ~4-16)the discrepancies in the results obtained with the use of different approximations for the kernels of integral equations have been analysed. In 15) the bifurcation point has been shown to coincide with the point of mechanical instability of liquid against a weak external field. In this paper we derive the instability criterion for the liquid phase using the precise closed nonlinear integral equation for p(r) 17'18) (see alsolg). We also demonstrate how this criterion can be obtained directly from the first equation of the B B G K Y hierachy2°). Finally, we discuss briefly the kinetic equation approach and (under certain assumptions) derive the same instability criterion. 0378-4371/81/000~l$02.75

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1981 North-Holland

358

V.N. RYZHOV AND E.E. TAREYEVA

Let us consider first the nonlinear integral equation for the singlet distribution function 0(r)17'Js): In

__P_~) = ~=1 ~ ~.1 fJ Sk+t(r, . . . . . rk+,)p(r2).., p(rk+, dr2.., drk+1 --= ~:(r,, {p(r)}).

(1)

Here Z is the activity, Sk+~(rl. . . . . rk+0 the irreducible cluster sum of Mayer functions connecting (at least doubly) k + 1 particles. The quantity Z can be defined with the use of the normalization condition

1 f p(r) dr = p,

(2)

V3

where p is the mean number density in a volume V. It is worthwhile to notice that no approximation has been made to derive eq. (1) and no unknown functions except p(r) appear in it. The quantity ~(rl, {p(r)}) is the generating functional for the set of direct correlation functions c,(rl . . . . . r,) introduced by Percus2~): .-1 ~(rl, {p(r)})

c,(rl . . . . . r,) = ~o(r-~- ~- ~ "

(3)

Let o(r)= Oe be the known constant solution of eq. (1). At the bifurcation point another solution o(r)= pc(1 + h(r)) bifurcates continuously from the initial one, h(r) having the symmetry of the proposed crystalline lattice. The bifurcation points of the eq. (1) are defined by the eigenvalues of the linearized eq. (1) for small h(r). Using (2) and (3) and the fact that

+ f h(r)dr=O, we obtain the linearized equation in the form {p(r)}) o(r)=0,h(r2) h(rl) = p~ f dr2 t$~:(rl, 6p(r2)

= p~ f c(r12)h(r2) dr2,

(4)

with the direct correlation function c(r12) for liquid in the kernel. After the Fourier transformation the bifurcation condition takes the form

1 -- pee(k) = O, where

~(k) = ~. / c(r) e-ikr dr.

(5)

ON THE LIQUID PHASE INSTABILITY CRITERION

359

The condition (5) is the same as derived previously by Lovett ~4''5) and occurs to be the condition of mechanical instability of the system against the action of an infinitesimal external field. L e t us derive now the condition (5) from the first equation of the B B G K Y hierarchy2°): V1 In p(rO + [3 f

VlCP(r12)g(rl,r2)p(r2) dr2 + [3VI U(rl) = O.

(6)

Here U(r) is the external field potential and t/'(r~2) is the intermolecular potential and g(rb r2) is connected with the pair distribution function p(2)(rl, r2): p(2)(rl, r2) =

g(rl, r2)p(rOp(r2).

The pair correlation function g(rt, r2) is a functional of p(r). As is shown in 17 this functional has the form

(1

1),

f ~gSk+l(rl,

0jl;

x p(r3).., p(rk+Od r 3 . . ,

, rk+O

drk+l,

(7)

with fl2 = e -~°12- 1. If

p(r) =

We(1+

h(r))

(8)

then up to the first order in

h(r)

we can write

r2)l ..... g(rl, r2)=ge(rl:)+Pe f Sg(rl, ~--;-, op(r3)-c !Ip(r)=pentr3)ur3.

(9)

Here ge(r~2) is the radial distribution function for liquid. Note that Kirkwood') assumed the variations of the functions p(r) and g(r~, r2) to be independent from one another. Let us insert (8) and (9) into eq. (6) and linearize the resulting equation with respect to h(r). We have now

V~h(r,) + [3pef dr2h(rE){V~(r~E)g(r~2) +pef dr3V,dp(r,3)Sg~(~r.br3)[

~+flV,U(rO=O.

(10)

Up (r2) Ip(¢)=p,J

Note that the direct correlation function can be written in the form ~8)

c2(rl, r2)= ~,=

(k -1)! l

fSk+l(r,,

.

. rk+l)p(r3)...p(rk+,)dr3 . . . . .

drk+l

(11)

V.N. RYZHOV AND E.E. TAREYEVA

360 and

1

Vlc2(rl, r2) = k=l (k - 1)!

f [_flVl~12e_Oa,12]

OSk+l(rl. . . . . rk+l) Ofl2

X p(r3) • • • p(rk+t) d r 3 . . , drk+l + k=2

(k -1 2)! f [-/3V1~13 e -00'3]

OSk+j(r~afl3 . . . . . rk+l)

× p(r3) • • • p(rk+O d r a . . , drk+t,

(12)

while from (7) we have

~g(rl, r3) _ e_~,bl3f ~ 8p(r2)

k=2 (k

l__ f O&+l(rl. . . . . rk+l) 2)!

0~3

× p(r4) • • • p(rk+l) d r 4 . . , drk+l.

(13)

Using (7), (13) and (12) we obtain

6g(rl, /'3) Vlc2(rl, r2) = -- ~Vlqbl2g(rl, r2) -- ~ f V 113~ ~--~2) p('r 3J"d r 3,

(14)

or in the case of liquid with p(r) = pc,

6g(rl, r3) ptr)=pe dr3.

Vic(rl2) = - flVlt~12ge(rl2) - ~Pe f Vl(~)13 ~p(r2)

(15)

The substitution of (15) into (10) gives Vlh(rl)

Pe f

VlC(rlE)h(r2) dr2 +/3Vl U(rl) = 0,

(16)

or after Fourier transformation, -/30(k)

h(k) = 1 - pef.(k) and the instability criterion has again the form (5). L e t us consider now the instability of the liquid phase from the kinetic point of view. The first equation of the nonequilibrium hierarchy is2°)"

Op(xl,att)+ glmVlO(X,, ,) = f ~71(~)12+ p(2)(Xl, X2, t) dx2.

(17)

here p(xl, t) and pt2)(Xl, x2, t) are singlet and pair nonequilibrium distribution functions; x = (p, r); dx = dp dr. Let us consider the kinetic stage of the evolution of the system with 2°)

p~Z)(x~,x2, t) = p~Z)(xj,x2, {p(x, t)}).

ON THE LIQUID PHASE INSTABILITY CRITERION

361

Let po(X) be the equilibrium singlet distribution function

po(x) = tp(p)p(r),

q~(p) = (27rmO) -3/2 e -p2/2mO

and let

o~2~(x,, x2) = #2~(Xl, x2, {p0(x)}) be the equilibrium pair distribution function. Suppose that

p(x, t) = po(x) + Ap(x, t) = p0(x)(1 + h(x, t))

(18)

and consider a small deviation from the equilibrium. In the linear approximation pt2)(xl, x2, t) can be written in the form p(2)(Xl, X2, t) = p~2)(Xl, X2) + Ap(2)(XI, X2, t),

Ap~2)(x~, x2, t) = f '~P~2~(xl' x2, I J ~p(X3,t) t)l~m,o=oZaP(x3' t) dx3. Now we assume that

= ~pg:)(xl,x9

~p~2~(xl, x2, t) I

~---P'~3,-~ lap(x,t)=o

8p0(X3)

(19)

In the equilibrium state we have

8p~2)(xl, x2) = 8(xl - x3)po(x2)g(rl, r2) + 8(xz - x3)po(xOg(rl, r2) 8po(X3) 8g(rl, r2)

+ po(Xl)Po(X2) ~pO(X3) and (19) gives

Ap~E)(Xl, X2, t) = po(Xz)g(rt, r2)Ap(xl, t) + po(xOg(rl, r2)Ap(x2, t) +

po(Xl)Po(X2) f .ax3 8g(rl, -o0(x;3r2) Ap(x3, t)

(20)

dx3.

Using the normalization condition fq~(p)dp = 1 it is easy to obtain from (11) the expansion

c2(rl, r2) = ~--l (k -1 1)! f Sk+l(rl, . . . rk+l)Po(X3) . . . • po(Xk+I) . dx3. The similar expansion for g(rl, rE) c a n be obtained from (7). Now Vic2(rl, re) = - [3Vltlbl2g(rl, r2)--[3 f Vl*13 -- 8g(rl, r3) p0~3jdx3 r_ 8p0(x2)

• dX/+l.

362

V.N. RYZHOV AND E.E. TAREYEVA

and

fApt2)(Xl, X2, t)V14)~: dx2 -OAp(xl, t)Vl In p(rl) -Opo(xO f Ap(x2, t)Vlc2(rl, r2) dx2, so that finally the linearized kinetic equation for O

(21)

Ap(x, t) is

{Ap(Xl, t)V~ In p(rO

+ p0(Xl)Vl f C2(1"1,lr2)Ap(x2, t) dx2} = 0.

(22)

Let us consider the relaxation towards the uniform distribution p0(x) = Pe¢(P)In this case eq. (22) takes the form

0A

pl ViAp(rl, pl, t)

P' f

- Oe~(Pt) m VI

c(r12)Ao(r2,p2, t) dr2 dp2 = 0.

(23)

This equation can be solved with the use of Fourier and Laplace transformations in standard way. For the quantity

= f dp p,ok(p),

~ with

P°'k(P)=f ei~'pk(p't)'

Ap(r'P' t) = I~'~V k e'k" pk(p, t),

0 we obtain ~k=J

f pk(p, 0)~(p)

i(pk/m-~o)

dp/(1-

pe~(k) f ~o(p)pk/m pk/m-to dp).

The elementary excitation energies are defined by the equation 1 p_e~k~ ~ ) j [p -¢(p)pk/m ~dp=0. _

The point of instability is given by the condition to = 0. The instability criterion again takes the form (5): 1 - p~(/~) --- 0. As was pointed out by the authors of ~3-15)eq. (5) has no solutions in the case of a hard sphere potential for one and three dimensions. This fact originated

ON THE LIQUID PHASE INSTABILITY CRITERION

363

t h e p o i n t o f v i e w t h a t the b i f u r c a t i o n a p p r o a c h c a n n o t b e u s e d in the p r o b l e m o f c r y s t a l l i z a t i o n . In f a c t this i m p o s s i b i l i t y a r i s e s if t h e c o n t i n u o u s v a r i a t i o n o f t h e s i n g l e t d i s t r i b u t i o n f u n c t i o n at t h e t r a n s i t i o n p o i n t is s u p p o s e d . H o w e v e r , in 16'19) it is s h o w n t h a t it is p o s s i b l e to o b t a i n t h e i n s t a b i l i t y in t h e h a r d s p h e r e s y s t e m in t h r e e d i m e n s i o n s if a s m a l l b u t finite d e n s i t y c h a n g e is c o n s i d e r e d . In t h a t v e r s i o n 16'19) o f t h e b i f u r c a t i o n a p p r o a c h to t h e c r y s t a l l i z a t i o n t h e o r y eq. (5) p l a y s a c r u c i a l role.

Acknowledgements T h e a u t h o r s w o u l d like to t h a n k N . N . B o g o l u b o v f o r h e l p f u l d i s c u s s i o n s and E.N. Yakovlev for valuable comments.

References 1) J.G. Kirkwood, in: Phase Transformations in Solids, R. Smoluchowski, J.E. Mayer and W.A. Weyl, eds. (Wiley, New York, 1951) p. 67. 2) R. Brout, Phase Transitions (Benjamin, New York, 1965) chap. 4. 3) W. Kunkin and H.L. Frish, J. Chem. Phys. 50 (1969) 181. 4) T. Naitoh and K. Nagai, J. Stat. Phys. 1 (1971) 391. 5) J.G. Kirkwood and E. Monroe, J. Chem. Phys. 9 (1941) 514. 6) S.V. Tyablikov, JETP 17 (1947) 386. 7. A.A. Vlasov, Many-Particle Theory and its Application to Plasma (New York, 1961). 8) H.J. Ravech~ and C.A. Stuart, J. Chem. Phys. 63 (1975) 1099, 65 (1976) 2305. 9) T. Yoshida and H. Kudo, Progr. Theor. Phys. 59 (1978) 393. 10) J.D. Weeks, S.A. Rice and J.J. Kozak, J. Chem. Phys. 52 (1970) 2416. I 1) J.J. Kozak, Adv. Chem. Phys. 40 (1979) 229. 12) N.N. Bogolubov, Phys. Abh. S.U. 6 (1962) 1,113, 229. 13) H. Miiller-Krumbhaar and J.W. Haus, J. Chem. Phys. 69 (1978) 5219. 14) R. Lovett and F.P. Buff, J. Chem. Phys. 72 (1980) 2425. 15) R. Lovett, J. Chem. Phys. 66 (1977) 1225. 16) V.N. Ryzhov and E.E. Tareyeva, Theor. Math. Phys. (Moscow) 49 (1981) N3. 17) E.A. Arinshtain, Dokl. Akad. Nauk (USSR) 112 (1957) 615. 18) F.H. Stillinger and F.P. Buff, J. Chem. Phys. 37 (1962) 1. 19. V.N. Ryzhov and E.E. Tareyeva, Phys. Lett. 75A (1979) 88. 20) N.N. Bogolubov, Problems of the Dynamical Theory in Statistical Physics (MoscowLeningrad, 1946). 21) J.K. Percus, The Equilibrium Theory of Classical Fluids, M. L. Frish and J.L. Lebowitz, eds. (Benjamin, New York, 1964).