Physica A 199 (1993) 243-253

North-Holland

SDZ: 0378-4371(93)E0179-I

A 2D Ising drop in a corner

J. De Conincka, J. Frutterob and A. Ziermanna “Vniversitt de Mom-Hainaut, Place du Part 20, B-7000 Mom, Belgium bCentre de Physique Thtorique, CNRS Luminy, Case 907, 13288 Marseille Cedex 9, France

Received 13 April 1993

By using numerical simulations based on the Kawasaki algorithm, we study the shape of a

drop of + spins in a corner formed by two walls with a different surface field. Different

shapes for the drop are observed as a function of the opening angle S of the corner. This is in

complete agreement with thermodynamic arguments based on the Winterbottom or on the

Summertop construction. Moreover, it is shown that for the same temperature and wall free

energies a complete wetting regime may appear by varying the opening angle S. This confirms

the general validity of previous results derived for SOS models.

1. Introduction

The study of wetting phenomena has been a very active field of research in statistical mechanics. Different methods have been proposed: the grand canonical approach [l] related to a gas of droplets on top of a planar substrate, the canonical approach [2] for a sessile drop and the renormalization group approach [3] related to the thickness of a given film. A great part of this research has been devoted to the study of the wetting transition, i.e. when the contact angle of a sessile drop becomes zero [2] or when the thickness of the film becomes infinite [3] as a function of the nature of the wall-fluid interactions.

An open question in this field of research is the influence of the geometry of the substrate on these wetting conditions. A few studies have been developed within a renormalization group approach for particular geometries (spherical, cylindrical, . . .) [4]. Th e case of a drop in a corner (fig. 1) belongs to that class of problems. Let us point out here that this problem is particularly interesting in the canonical approach since we can then study the variations of the shape of the drop.

It has been shown in [S], using thermodynamic arguments, that there should appear, for a given opening angle 6, a transition between convex and concave

0378-4371/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved

244 J. De Coninck et al. I A 20 lsing drop in a corner

Fig. 1. A drop in a corner between two substrates with different inclinations. The second substrate is inclined by an angle 6 with respect to the horizontal axis.

shapes. This property has been verified within a microscopic approach of the solid on solid (SOS) type. In particular the phase diagram has been computed as a function of the opening angle 6 and of the difference of wall free energies A,a = maw - usw for a given Aia = dew - mBsw (here we use ~rs to denote the free enkrgy pei unit area of a substrate S that is covered by a phase P). The interesting property which has been pointed out in this paper is that there appears in this diagram a small region where the second wall is completely wet for certain values of 6.

This means that, for a given wall-fluid interaction and temperature, the geometry of the corner, i.e. the value of the opening angle 6, could affect the wetting conditions.

To establish the validity of this property for more realistic models, we study within a 2D Ising ferromagnet the equilibrium shape of a drop with fixed volume, preferentially attracted by two planar substrates that form an angle 6.

We thus perform a Monte Carlo analysis in a ferromagnetic Ising model with nearest neighbour interactions. The effect of the walls is modeled by a surface field. From a thermodynamic point of view the equilibrium shape is the shape that, with a fixed volume, minimizes the surface free energy. This corresponds to an algorithm that preserves the number of particles. The results presented here are therefore based on Kawasaki’s method [6]. The thermodynamical variational problem has already been investigated in [7,8]. In particular, one can find in [7] a complete discussion of equilibrium shapes for general values of the wall free energy densities. For weak wall attractions one has a convex equilibrium shape, which is a piece of the Wulff shape obtained by the Winterbottom construction (see below). If the wall attractions are large enough, but still in a partial wetting regime, one obtains a concave equilibrium shape, which is described by a construction that is, in a certain sense, inverse to the Winterbottom construction (following [7] we shall call it the Summertop construction). Some rigorous statistical proofs of the Wulff construction have been performed within an Ising ferromagnet using low temperature cluster expansion [9] and large deviation arguments [lo] and within SOS-type random

J. De Coninck et al. I A 20 Ising drop in a corner 245

walk models [ll]. However, for more complicated models, for instance to take into account the presence of a substrate, and at temperatures not accessible by those methods, Monte Carlo simulation may be a useful means to study equilibrium shapes, as has been recently shown in [12].

To first check the validity of our simulation, we compared our results with exact calculations performed in [13], where the difference of the wall free energy densities of a horizontal or vertical wall is computed as a function of the surface field. Using thermodynamic arguments, we may accordingly construct the equilibrium shapes and compare them with the results we obtain by simulation. This part is developed in section 2.

Choosing the same surface field at both walls and keeping it constant while increasing the temperature, we observe a transition from convex shapes given by the Winterbottom construction to concave shapes given by the Summertop construction. At the temperature where the contact angles are expected to be equal to 1~14 and the surface curvature changes sign, we indeed observe a straight interface. It is thus interesting to use this method to find equilibrium shapes for more complicated models, e.g. inclined walls, Van der Waals interaction or, as has been done in [12], in the presence of gravity. In section 3 we concentrate on the study of the effect of the geometry of the walls on their wetting condition. We perform several simulations with a fixed interaction between the wall and the particles, changing only the angle between the walls.

2. The model

Consider a square lattice A, = [-n, n] x [-n, n] E Z* and two walls of inclination 6, E [0,7r/4] and 6, E [a,, 6, + ~1. We investigate a spin system SE{-l,l}” in the region A={i~(i,,i,)EA,Ii,sins,~i,coss,, i,sin&~ i, cos S,} with the Hamiltonian

H(s) = - c sisj - x hiSi ) (1) (i.i)CA (i,j):iEA

iEZZ\A

where we use (i, j) to denote a couple of nearest neighbour sites. The field hj is defined as

if jEE*\A,,

if jEA, and j2a0,

if jEA, and j2<0, (2)

where a, and a2 are positive numbers. The second sum in (1) provides an

246 .I. De Coninck et al. I A 20 Ising drop in a corner

attractive interaction of the particles with the two considered walls and a repulsion of particles at the margins of the box A,,.

At low enough temperature (inverse temperature p > 4 ln(1 + fi)) this model exhibits two stable thermodynamic phases with spontaneous magnetiza- tion *m(p), where

m(P) = ~~~~~~~,’ (sinh2(2/3) - 1)) 1’8

Fixing the number of positive spins on A, one may expect the formation of two regions A+ and A_ such that the mean value of a single spin si equals m(p) if i E A+ and equals -m(p) if i E A_. Since the bulk free energies of the two phases are the same, the shape of those regions is given by minimizing that contribution to the free energy which is due to the walls and to the interface between A+ and A_. This can be obtained by Monte Carlo methods that preserve the number of + spins since the associated algorithm (Kawasaki) satisfies the detailed balance condition and is ergodic [6]. To check the consistence of this approach and to study the amplitude of the fluctuations, we have first studied the case of one horizontal wall and one vertical wall with the same field a. With the wall free energy densities that have been computed in [13], the Young relation for the contact angles takes the explicit form

tan 0(a, p) = #(a> P) - 144% P>l

v{[cosh(2p)]‘lsinh(2P) - +[$(a, 0) + +(a, P))‘]}” - 1 ’

(4)

where

+(a, P) = e2’ cosh(2P) - cosh(2up)

sinh(2P) (5)

One can easily see that this function is increasing with the inverse temperature /3. Since the Winterbottom and Summertop constructions may lead only to convex and concave shapes, respectively, the contact angle indicates concavity or convexity. At a transition temperature T,,(u), defined by tan 0(u, &,(a)) = 1, we obtain a flat (straight) interface. Above To(u) (tan 8 < 1) the droplet is concave and below To(u) (tan 8 > 1) it is convex. This gives rise to a simple phase diagram with temperature driven transition between convexity and concavity.

The field at which, for a given temperature, the shape changes from concavity to convexity is given by

J. De Coninck et al. I A 20 Ising drop in a corner

a cc = T& arcosh cosh(2p) - $ee2’ cosh’(2P)

e -4p cosh4 (2p) - 4 -e -4p sinh2(2p)) .

247

(6)

For completeness, we also give the formula for the minimal a, field which ensures complete wetting of a horizontal wall,

a, = & arcosh[cosh(2@) - ezPsinh(2P)] .

Here we present the results of three runs with a surface field a, = a2 =

0.65324 and with p = 0.63, 1, 1.5. For this field the theoretical values for the contact angle are tan 0(p = 0.63) = 0.24, tan e(p = 1) = 1 and tan 0( p = 1.5) = 1.85. So we expect convexity for p = 1.5, a straight interface for p = 1 and a concave one for /3 = 0.63. In fig. 2 we present the shapes that we obtained as an average over the final configurations of 11 statistically independent simula- tions of 2 x lo6 Monte Carlo steps, where we start with a triangular bloc of approximately 500 particles placed into the corner between the two walls.

40 Fig. 2. Equilibrium shapes of a drop in the corner between a vertical

30 and a horizontal wall with the same surface field a = 0.65324. We

20 observe a temperature driven transition of this shape. For /3 = 1.5 the

10 shape of the droplet is convex (a), for p = 1 the surface is a straight line

(b) and for p = 0.63 the shape of the droplet is concave (c). The shapes

0 obtained by simulation are in good agreement with the shapes predicted

0 10 20 30 40 50 by thermodynamics, which are marked by a full line.

248 _I. De Coninck et al. I A 20 Ising drop in a corner

Since the difference of those pictures from the theoretical shapes is only of the order of a few molecules, we may conclude that our method works correctly and we use it to study the equilibrium shape for nonvertical walls.

3. Inclined walls and corners

The shape of a drop in a corner can be deduced from the Wulff shape by drawing two straight lines with slopes tan 6, and tan 6, at a distance aAwl - aBwl and c,w - oBBw2, respectively#‘. From the Wulff point 0 we obtain, up to a dilatation, the shape of the desired drop as the part of the Wulff shape that lays in front of both walls, i.e. in the region {(x, y)]y cos 6, -x sin 6, 2 Arc, y cos 6, - x sin S, 3 AZ’+} (here and in the following we use the abbreviation b,~=~~v,, -rBsw,, i=l, 2).

If this region does not intersect the Wulff shape, which happens at large wall attractions A,a, one has to draw two lines --x sin 6, + y cos 8, = -Ai(r, i = 1, 2. The equilibrium shape is then given as the region between those lines and the Wulff shape (for a detailed description of this construction see ref. [7]).

Consider the Wulff shape that is defined by the surface tension (+ of a 2D Ising ferromagnet, which has been computed in ref. [14]. It is easily verified that it is given by the graphs of the functions [13]

1 y(x) = t p arcosh

cosh2(2P) sinh(2P)

Let us suppose that the droplet is attached to a horizontal wall IV, with a free energy density A,o and a wall W, of inclination angle 6 and free energy density A2~ G Aio. The constructions described above imply the existence of an angle 6,, below which the wall W, is dry and above which it starts to be partially wet. There exists another angle S,, > 6,, below which the equilibrium droplet takes

a convex Winterbottom shape and above which it takes a concave Summertop shape. From (7) we get

cos 6,, = A,a A,u - ydy2 + (A~(T)~ - (A2’+)2

(AP)~ + y2

and

(8)

*I In order to describe also the case rAW, - CT,,; < 0, i = 1, 2, it would be better to say: draw a

straight line given by the equation --x sin 6, + y cos 6, = gAW, - a,,,.

J. De Coninck et al. I A 20 Ising drop in a corner 249

cos 6 = 4~ A,a + Y&* + GW2 - @24*

cc (A~u)* + y2 ’ (9)

where y = y-‘(Aia) (here y-l is the reciprocal function of y). If the wall attraction A1cr satisfies ~(0) < A,cr < (+(IT/~), the wall IV, is completely wet at 6, = 0 and 6, = 1r/2 and partially wet at S, = 7r/4. Fixing the wall W, at 6, = 1r/4 and taking o(O) < A,a < A,(+ < a(7r/4), we observe a transition angle S,, > S,, such that the wall IV, is partially wet if S,, < 6, <S,,,, and completely wet if

82 > qw It is easy to see that SPpw satisfies

a@,,) = A,(+ . (10)

The angles S,, and S,, are defined analogously as in the case 6, = 0. We just insert y = [x,, + y(x,,)]/*, where x0 is the solution of y(x) =x + e A2w, into (8) and (9). In fig. 3 we show the phase diagrams of the different regimes of partial and complete wetting for the cases S, = 0 and 6, = IT/~.

To study the microscopic details of these thermodynamic considerations, we have to investigate a drop of + spins in a corner with an opening angle S and appropriate surface fields. However, we do not know the surface attractions as a function of the surface field and the inclination angle (except for S = +km, k E Z). Since we expect the wall free energy to be a monotonous function of the surface field a, we expect the validity of the same kind of phase diagram. By symmetry and monotonicity we get sgn[A,u(a,, S)] = sgn(a,). From this, the appearance of the transition angles S,, and S,, is obvious for a, > 0 and a2 = 0. Since A2~ is increasing with a2, the angles S,, and S,, are decreasing with a*. Let us concentrate on the more interesting case of 6, = n/4. To answer the question, if there exists a closed region of complete wetting similar to region 4 in fig. 3, we need to find a surface field for which we obtain complete wetting at 6 = 0, but partial wetting at 6 = 1r/4. We have therefore studied the behaviour of a droplet within a strip of horizontal width of 300 sites between two parallel walls of inclination 1r/4 with an attractive surface field a2 = 0.93 at the right margin of the box, which models a substrate, and a repulsive field a, = -1 at the left margin of the box. To realize an infinite strip we put periodic boundary conditions, i.e. we identify the points (i, 0) and (i + 120, 121), with the points (i + 120, 120) and (i, 1) respectively. We start with a triangular droplet of 1100 particles at the attractive wall and let it spread for the time of 4 X lo7 MC steps. To study the wetting condition, we have measured for the first four molecular layers above the attractive wall the number of particles that are in each of those layers as a function of time and we observe, as indicated in fig. 4, that it gives a clear signature of partial wetting. Indeed, those numbers stabilize

250 J. De Coninck et al. I A 20 Ising drop in a corner

&I- 4

(al -

SE 4

1.’

--_.

(b)

simulated __f(

% III

$j

:! .e g f

I -.._

f 1, ’ I 0 1 1.5

Fig. 3. (a) The phase diagram of wetting for a fixed horizontal wall IV, with wall attraction A,o = 1.5. The transition angles a,,, and SC, are drawn as a function of the wall attraction A,a of

the inclined wall W, on the interval [0, 1.51. We obtain three different regions. For small inclination

angles, the second wall W, remains dry (region I). For inclination angles 6, between a,, and 6,,

(region II) we observe that the wall W, is also partially wet in spite of being less attractive than W, and the equilibrium shape is convex. For 6, larger than SC, (region III), the equilibrium droplet is

concave.

(b) The phase diagram of wetting for a fixed wall W, of inclination 6, = a/4 with wall attraction

A,a = 1.77. The transition angles addpr a,,, S,, and C& are drawn as a function of the wall attraction

Azu of the inclined wall W,. We obtain four different regions. For small inclination angles, the

second wall W, remains dry (region I). For inclination angles 6, between Sddp and SC, (region II) we

observe that the wall W, is also partially wet in spite of being less attractive than W, and the equilibrium shape is convex. For 8, larger than S,, the equilibrium droplet is concave. If the wall

attraction A,u is large enough but still less than A,o, we observe a region of complete wetting of

the wall W, for 6 E [S,,.,, &] (region IV). A second region of complete wetting that appears for

6 E IS,, + n/2, &, + -a/2] is not contained in the diagram.

.I. De Coninck et al. I A 20 Ising drop in a corner 251

at a level that is clearly different from 120, which is the maximal possible value and which should be reached in the case of complete wetting. Here we took the average over 5 statistically independent MC experiments.

In fig. 5 we present the average picture of the final configurations of 9 statistically independent experiments of 6 x 10’ MC steps, where we started with a triangular droplet of 700 particles in the corner between two walls I%‘, with S, = ~14 and a, = 0.93 and W, with S, = 37~14 and a2 = 0.928; the length of both walls is 60 sites. We observe a concave droplet with nonvanishing contact angles in the corner and microscopic layers over the walls, which is to be related to a partial wetting regime.

For the same fields one obtains complete wetting of a horizontal wall (cf. (4)). From the results presented in fig. 4 it is clear that for a, = 0.93, a2 = 0.928 and 6, = S, = ~/4 we obtain partial wetting of W,. From the theoretical results reviewed in section 2 we know that we obtain complete wetting of W, for the same fields, the same 6, but S, = rr/2, i.e. there occurs a region of complete wetting similar to region IV in fig. 3. By symmetry we should expect that W, is partially wet also for S = 3~14, which is indeed confirmed by fig. 5. Hence the region of complete wetting is closed from below and from above. The concave shape presented in fig. 5 shows also that there exists a transition angle S,, such that for S, < S,, the shape of the equilibrium droplet is convex and for 6, > S,, it is concave.

1 e+07 3e+07 number of MC-steps

Fig. 4. The number of particles in each of the first four molecular layers on a substrate of

inclination angle of 7~14 and of length 120 as a function of time. We observe that those numbers

stabilize at a level that is considerably less than the length of the wall. Once we consider 1100

particles in a complete wetting regime, the drop should spread until it entirely covers the wall by a film of a thickness around 9.

252 J. De Coninck et al. I A 20 Ising drop in a corner

0 0 60

Fig. 5. The average configuration of 9 independent experiments of 6 x 10’ MC steps with a droplet

in a corner between two walls IV, with 6, = v/2 and a, = 0.93 and IV, with S, = 37~/2 and a2 = 0.928.

We observe a concave droplet in the corner and microscopic layers that cover the walls. After this

number of steps the number of molecules for the first four layers on top of the substrates remained

indeed constant within the time.

This clearly demonstrates that the value of the opening angle S will influence the wetting condition.

As a concluding remark we would like to stress that this Kawasaki technique may be very helpful to get a better understanding of the influence on wetting of more complicated geometries for the substrate.

Acknowledgements

The authors thank the NATO, the FNRS, the ERASMUS project and the Communaute Francaise de Belgique for their financial support which made this collaboration possible.

This text presents research results of the Belgian program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming. The scientific responsibility is assumed by the authors.

References

[l] D.B. Abraham and J. De Coninck, J. Phys. A 16 (1983) L333

[2] J. De Coninck and F. Dunlop, J. Stat. Phys. 47 (1987) 827.

J. De Coninck et al. I A 2D Ising drop in a corner 253

[3] S. Dietrich, Wetting phenomena, in: Phase Transitions and Critical Phenomena, vol. 12, C.

Domb and J.L. Lebowitz, eds. (Academic Press, New York, 1988).

[4] P.J. Upton, J.O. Indekeu and J.M. Yeomans, Phys. Rev. B 40 (1989) 666.

[5] J. De Coninck, J. Fruttero and A. Ziermann, Physica A 196 (1993) 320.

[6] K. Binder and D.W. Heermann, Monte Carlo simulation in statistical physics, in: Solid State

Sciences, vol. 80, M. Cardona, P. Fulde, K. von Klitzing and H.-J. Queisser, eds. (Springer,

Berlin, Heidelberg, 1988).

[7] R.P. Zia, J.E. Avron and J.E. Taylor, J. Stat. Phys. 50 (1988) 727.

[8] J.K. Lee and M.I. Aaronson, Surf. Sci. 47 (1975) 692.

[9] R. Dobrushin, R. Kotecky and S. Shlosman, The Wulff construction: a global shape from

local interactions, in: AMS Translations of Mathematical Monographs, vol. 104 (Am. Math.

Sot., Providence, RI, 1992). [lo] C.E. Pfister, Helv. Phys. Acta 64 (1991) 953.

[ll] J. De Coninck, F. Dunlop and V. Rivasseau, Commun. Math. Phys. 121 (1989) 401.

[12] S.S. Manna, H.J. Herrmann and D.P. Landau, J. Stat. Phys. 66 (1992) 1155.

[13] D.B. Abraham, J. De Coninck and F. Dunlop, Phys. Rev. B 39 (1989) 4708.

[14] D.B. Abraham and P. Reed, J. Phys. A 10 (1977) L121.