Physica 124B (1984) 251-254 North-Holland, Amsterdam

CUSP POINTS IN SURFACE FREE ENERGY: FACETING AND FIRST-ORDER PHASE TRANSITIONS

Nicolrls GARCIA, Juan Jos6 S.,~ENZ and Nicol~is CABRERA Departamento de Fisica Fundamental, Universidad Aut6noma de Madrid, Cantoblanco, Madrid-34, Spain

Received 5 May 1983

An analytical expression for the geometry of a crystallite in equilibrium between two phases for a general surface tension is presented. The interaction between surface steps of hcp 4He is discussed and some experiments proposed. It is shown that cusp points stabilizing plane surfaces play a similar role in first-order phase transitions.

A recent work [1] has studied the roughening transition [2] in connection with the faceting observed in the crystallization of superfluid to hcp 4He [3]. This was discussed in terms of the role played by the cusp points (see the pioneer- ing work by Wulff [4], Herring [5] and Cabrera [6]) in the surface free energy fl(p, q) between two phases, where p =-OZ/ax and q = - a Z / a y are the slopes or step density of the equilibrium surface Z(x, y) referred to the plane stable sur- face. A variational analysis of the ther- modynamical potential shows [1,4,7] that the equilibrium surface reads:

(1)

with

2 0/3 2 0/3 (2) x = Apr ap ' Y = Apt aq '

where [1] ApT = Apz - Apg(Z- Z0) = const. The difference in pressure between the two phases is APz at the value Z and Apg(Z-Zo) takes ac- count for the gravitation effects [1], Ap being the difference in density between phases and g the gravity.

In this paper we obtain the solution to Z(x, y) for a general form of /3(p, q) and apply it to

estimate the free energy of hcp 4He from the shape and dimension of the observed crystallites [3]. Furthermore, we show that the existence or non-existence of flat regions in Z(x, y) (which is the criteria for the roughening transition) is generalizable to first order phase transitions as existence or non-existence of cusp points in the Gibbs energy.

In order to clarify the different contributions to the free energy we write for convenience, but without loss of generality for small (p, q), the following expansion in moduli (see Cabrera [6]):

/3(p, q, T) = ~ ~np(T)lPl" +/~,~(T)lqln), n=0

(3)

where T is the temperature of the system. The cusp points correspond to the terms n = 1. If /31p(T)>0 and /31q(T)>0 a plane stable facet exists at p = q = 0. The terms/31p and/31q are the surface energies for the formation of steps; /32p and fl2a correspond to the energy of interaction between steps, and so on. The roughening tran- sition occurs at T = TR such that ~lp(T~ TR)~O or ~lq(T~ TR)~O, i.e. the temperature TR at which the cusp point disappears. The critical behaviour of this transition was given by Van Beijeren [8] for the BCSOS model.

It is easy to show from eqs. (1)-(3) in com- bination with the criteria for surface stability [6]

0378-4363/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

252 N. Oarcfa et al. / Cusp points in surface free energy

that the general solution Z(x, y) for/3(p, q) is

Z x + y (-1 + n){/3.plpl" +/3~lq["}

n=O

+ (-1 + 20(q))fl~{ql"-~}] -' (4)

for Ipl > Ip01, Iql >lq01; and Z = Z0 facet ( 0 < x < XF; 0 < y < YF) for p = q = 0, where 0(&) is the Heaviside function. The values of P0 and q0 are given by

+ qo p=~ - - p=po q=qo q=qo

(5) and by a straightforward calculation it can be seen that if fl2p > 0 and/32q > 0 we obtain Ip01 = 0 and Iq01--0. Otherwise, Ip01 and [q0] >0 . This means physically that there is a discontinuity in the derivative of the surface Z(x, y) at the con- tours of plane surface defined by the coordinates (XF, YV). That is to say that certain crystallo- graphic planes or otherwise step densities (p, q) are unstable. We also have

Ix~IAPT= 2 ~'~ n~..Ip01"-', n = l

lyFIAPT = 2 ~ n/J.~lq01 "-1 . (6) n = l

This completes the necessary ingredients for obtaining the surface Z(x, y) that minimizes the thermodynamical potential for a given surface tension f l (p ,q) [1,4,7]. Now we apply the theoretical results (1)-(6) to obtain the surface tension of hcp 4He by fitting the measured meniscus of the 4He crystallite [3]. From holo- graphic interferometry Avron et al. [3] observed clearly the coexistence of faceting and rough curved surfaces. Preliminary calculations for a restricted expansion (3) were presented in ref. [1[, but from a more general expansion new interesting questions can be raised for experi- mentalists. In figs. la and lb we present the

Y~n~

2 2fin/6

-.0/.s I 0

I z(m"46 1'2°11 /..23 I 2 -3./,0 3 I r~) 1.05 /.

~' b)' "~'. -'~" J

0 - . 2 '

-.lO ~: 33.27

-106.80! 146.85 -95.13

23.75

Fig. 1. (a) Circles represent experimental results. The line is the "bes t" theoretical surface for/32 > 0. In this case all the surface slopes are stable. We assume that the dark circles are spurious. The relative values 2/3,,/Ap are in mm. (b) The same as in (a) but with/32 < 0 by considering that the dark circles are as precise as the others. This surface has unstable slopes near the flat region (see text) and in the region indicated by the insert circles.

experimental points [3] by circles for a section to Z(x, y) with x - - 0 , because of this only infor- mation for fl,~ can be obtained. The points show clearly a stable facet p = q = 0 and then a rough surface appears after ]YF]--~ 1.2 mm. This result is clear, but in the region (dark circles) 1.2 mm [y[ ~ 2 . 2 mm there are a set of scattered points on which we want to concentrate because of their theoretical importance. We have written a program [9] that optimizes the best fitting Z(x, y) from the experimental data and by using for- mulas (1)-(6) we can obtain the best fl(p, q) directly. This is a method for obtaining directly, from experimental data, the surface tension. Continuous lines in figs. la and lb are the best- fitted surfaces to the experimental points - f ig , la

N. Oarcfa et al. / Cusp points in surface free energy 253

by considering that the points 1 . 2 m m < y < 2.2 mm are spurious and may be due to experi- mental errors and fig. lb by assuming that these points are as precise as the other ones. If this is the case we notice a discontinuity in the slope q (step density in the y direction) at yv ~ 1.2 mm. This is important because it tells us if the step interactions are attractive, /32q < 0 (fig. lb), or repulsive,/32q > 0 (fig. la). We believe that this is a fundamental theoretical point in a basic system as 4He that can be easily answered by precise measurements, i.e. by giving or checking the points in the critical region near the plane facet.

We proceed further by showing the interesting theoretical result that the cusp points are a general property of any first-order phase tran- sition, i.e. ferromagnetic, gas--liquid or solid, etc. Therefore, the analysis presented above is ap- plicable to a general first-order phase transition. To prove this property of cusp points, take the expression (3) for the free energy with /3,p = 0 (case of the calculation for 4He). We claim that the following mathematical equivalences exist for the ferromagnetic transition taken as an example:

(i) /3(q, T) is equivalent to the Gibbs free energy G(H, T).

(ii) q is equivalent to the applied magnetic field

OF - - n ~ - - - -

OM"

(iii) (APT/2)Z(Y, T) is equivalent to the mag- netic Helmholtz free energy F(M, T);

y = APT Oq

is equivalent to the magnetization

OG OH"

For example, notice that formula (1) with the equivalences given implies

F(M, T) = G(H, T) - OG(H, T) H, OH

which is a well-known relation. Therefore the transition ferro~paramagnetic appears when the cusp point energy goes to zero. In particular we can calculate, as an example, the Gibbs energy for Landau's free energy (fig. 2b)

FL(M, T) = Fo(T)- aM 2 + bM 4 , (7)

the obtained results is

GL(H, T)= a 2 1 Fo(T) - -~ Mo - M01HI- 4da IHI2

+ 1 1 4 32a:M0 IS] 3 - 64a3M~ [HI

+ (8)

where M0 is the permanent magnetization. O b v i o u s l y , G L ( H , T) presents a cusp for T < Tc where M0g0 . The complete expansion for GL(H, T) can be obtained by the procedure des- cribed before for the shape of a crystallite and

a) A5

H

AF;

b) /'~ / /

/ / ! , /

\ \ /,,/

XX~x /L ' , ~ ' , /

M

Fig. 2. (a) Gibbs energies Op(P) GL(L) (L is for Landau's theory) given in the text. The discontinuous lines are the metastable and unstable regions. (b) The corresponding Helmholtz free energies. Note the cusp in AF for the curve P.

254 N. Garcfa et al. / Cusp points in surface free energy

the result is presented in fig. 2a. Of course, the value for GL(H, T) is a particular case of the general expansion for G(H, T):

G(H, T)= ~ G,,(T)IHI", (10) n=O

Acknowledgement

We thank Prof. F.C. Frank and J.M. Soler for discussions. This work was supported by the CAICYT through contract number 1426-82.

GL(H, T) is the one that produces an analytical expression for FL(M, T) at M = 0. This in general is not true; i.e. the Helmholtz free energy is not analytic and may present also a cusp point at M = 0 . In fig. la (thick line) we present Gp(H, T) = Go(T)- M0tHI- (X/2)H 2, x being the magnetic susceptibility. The resulting Up(M, T) = Fo~F) + (1/2X)(IMI - Mo) 2, IMI > M0, which obviously presents a cusp in the analytical prolongation (metastable region) at M = 0. The above expression for Fp(M, T) suggests that the expansion in IMI should be around M0. The expansion for G(H, T) and F(M, T) is im- portant because it determines the nucleation processes. This has been analysed by Cahn and Hiiliard [10, 11] by introducing Van der Waals term (VM) 2 for FL(M, T). We also have worked out the nucleation (metastable region) and the spinoidal (unstable region) for different expres- sions in F(M, T) and it is clear that nucleation energies, magnetization in the clusters and spinoidal points depend very much on the form of G(H, T) and F(M, T). Also from recent work by Cahn [12] it seems that the theory outlined here can be applied to phase grain boundaries and equilibrium.

References

[1] N. Cabrera and N. Garcfa, Phys. Rev. B25 (1982) 6057. [2] W.K. Burton and N. Cabrera, Disc. Faraday Soc. 5

(1949) 33; W.K. Burton, N. Cabrera and F.C. Frank, Philos. Trans. Roy. Soc. London Sr. A243 (1951) 299.

[3] J. Landau, S.G. Lipson, L.M. Mhhtthnen, L.S. Balfour and D.O. Edwards, Phys. Rev. Len. 45 (1980) 31; J.E. Avron, L.S. Balfour, C.G. Kuper, J. Landau, S.G. Lipson and L.S. Schulman, Phys. Rev. Lett. 45 (1980) 814.

[4] G. Wulff, Z. Kristallogr. 34 (1901) 449. [5] C. Herring, Phys. Rev. 82 (1951) 87. [6] N. Cabrera, Am. Soc. Test Mater. Proc. 340 (1962) 24;

Surf. Sci. 2 (1964) 320. [7] L.D. Landau and E.M. Lifshitz, Statistical Physics (Per-

gamon, New York, 1980). [8] H. van Beijeren, Phys. Rev. Lett. 38 (1977) 993. [9] Juan Jos(~ Sfienz, N. Garcfa and N. Cabrera, to be

published. [10] J.W. Cahn and J.E. HiUiard, J. Chem. Phys. 28 (1958)

258; 31 (1959) 688. [11] J.W. Cahn and J.E. HiUiard, Acta Metall. 19 (1971) 151. [12] J.W. Cahn, Transitions and Phase Equilibria among

Grain Boundary Structures (preprint).