A MAGNETIC WETTING TRANSITION

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A MAGNETIC WETTING TRANSITION

C. Walden, B. Györffy

To cite this version:

JOURNAL DE PHYSIQUE

Colloque C8, Suppl6ment au no 12, Tome 49, dgcembre 1988

A

MAGNETIC WETTING TRANSITION

C. J. Walden and B. L. Gyorffy

H. H. Wills Physics Laboratory, University of Bristol, Royal Fort, Tyndall Avenue, Bristol BS8 1 TL, G.B.

Abstract. - We discuss, within a continuum Landau theory, the possibility of an anisotropy-reIated wetting transition in

a semi-infinite cubic ferromagnet. This is then investigated for a specific value of anisotropy parameter, where it is shown to divide into two Ising model problems.

1. Introduction

To focus attention on the possible surface phase transitions in magnetism we recall the simple Ising case. We then study the object of our principal con- cern: the wetting transition in a more realistic vector model. The mean field theory of a semi-infinite Ising model in the presence of a localized surface field HS is well known [I, 21. Such a model may serve to describe a ferromagnet in contact with a magnetically harder substrate. For the case where HS competes with the order parameter in the bulk, the possibility of a wet- ting transition [3] is predicted, wherein (for sufficiently strong HS) a domain of opposite order is induced a t ("wets" ) the ferromagnet-substrate interface. In the limit where the bulk magnetic field H -* 0- (HS

>

0) the thickness of this wetting domain diverges to infin- ity ("complete wetting" ). As a function of tempera- ture the transition to this state may be first order or second order ("critical wetting" ) and occurs at a tem- perature Tw, the wetting temperature, below which the interface is said to be "partially wet"

.

Where a first order transition is exhibited it extends into the region H f: 0 for temperatures T

>

T,

and repre- sents a transition between two interfacial structures, both now of finite thickness. This transition between a thin and a thick wetting layer has been termed the "prewetting" transition [2], its locus in the

T-H

plane (the "prewetting line" ) terminating at a prewetting critical point

(T,,

,,..,

H,, ),,,

.

A continuum Landau theory of such a model is readily solved since a first in- tegral of the Euler-Lagrange equation can be explicitly found.

2. A vector model

In this paper, we generalize t o a model with classi- cal vector spins. Here, for a cubic crystal, a wetting domain develops if, in rotating from the surface to the bulk direction, the magnetization passes through an intermediate alignment along a crystal axis, where it tends t o be pinned. The vector model exhibits this wetting mechanism in addition to the usual Ising sce- nario, which is seen when H and H' are antiparallel.

Complete wetting in this latter case occurs for surface magnetizations M S

>

-MI,, where Mb is the magne- tization in the bulk.

An appropriate Landau free energy functional (di- vided by the product of temperature and surface area) is

where

t E (T

-

Tc)

/

Tc and the prime denotes differentia- tion with respect to z , the perpendicular distance from the interface. M, and M y are the a; and ycomponents of the magnetization averaged over directions parallel to the interface, their values at z = 0 being denoted by M: and M:. M, is not included since [4], by the Maxwell equation V.B = 0, we do not expect it to be energetically favourable for the plane averaged value to be *dependent. X is a measure of anisotropy and equals unity for an isotropic system. The bulk mag- netic field at zero temperature is related to the reduced quantity which appears in (1) by

Here

04%

is the excess free energy per unit surface area of a 90' domain wall in an infinite ferromagnet for the special case X = 3. The significance of this case is clear in the context of an infinite system, where it repre- sents a bifurcation point from a regime involving two 90' walls to one involving a single Ising-like ("180°" ) wall. In short, the energy of the two 90' walls is equal to that of one 180' wall. is the zero temperature correlation length for this case. An expression similar t o (4) holds for the interfacial field

H

'

.

All distances

C8 - 1636 JOURNAL DE PHYSIQUE

z in (1) are in units of

t o .

The parameter c may be re- garded as a measure of "surface temperature"

,

i.e. as a measure of how much faster/slower than in the bulk, the surface magnetization vanishes with temperature [5]. Minimization of (1) yields a pair of coupled Euler- Lagrange equations, which, however, [6] for X = 3 can be decoupled when expressed in terms of the variables P

=

M, + M y and Q r Mx - M y . We then obtain

2P" = t P

+

p3

-

( H ~

+

H y )

₍₅₎

Fig. 2. - Coexisting thin (dashed) andthick (solid) vetting profiles on the P-prewetting line, for

\HI

= 0.0001 directed

29" = tQ

+

Q3

-

(Hz

-

H;) _{(6) along -0%. (Other parameters as for Fig. 1).} with the boundary conditions

Thus, for X = 3 we can divide the problem into two Ising model. problems, the behaviour of each of which is readily investigated by means of a graphical con-

@

, . 2 1 0 , 6

\*---

____

struction [3]. In particular we have studied the case

-___---

0.2 where H is directed along -Ox and H' is at an angle 0.0

8' = z

/

6 t o 402. Figure 1 shows the two prewetting 0 8

,

16 24

lines exhibited for

IH'I

/

to

= 0.1, c /

to

= 0.001. Fig-

ures 2 and 3 show coexisting magnitude and angular Fig. 3.

-

Coexisting profiles on the Qprewetting line. (Pa- profiles for IHI = 0.0001. rameters as for Fig. 2).

[I] For a review, see Binder, K., Phase Transitions and Critical Phenomena, Vol. 8, Eds. C. Domb and J. L. Lebowitz (Academic Press) 1983. [2] Pandit, R. and Wortis, M. Phys. Rev. B 25

Fig. 1. - Prewetting lines for P Mx

+

My and Q G (1982) 3226.

I -SI

[3] See e.g. Sullivan, D. E. and Telo da Gama, M. M., Mx

-

My for C / t o = 0.001, and surface field H

/

t o = _{Fluid Interfacial Phenomena, Ed. C . A. Croxton}

8 8

0.1 at angle ~ ~ = a / 6 to the aiaxis. (wiley) 1986.