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Scaling functions for 3d critical wetting
E. Brézin, T. Halpin-Healy
To cite this version:
E. Brézin, T. Halpin-Healy. Scaling functions for 3d critical wetting. Journal de Physique, 1987, 48 (5), pp.757-761. �10.1051/jphys:01987004805075700�. �jpa-00210495�
Scaling functions for 3d critical wetting
E. Brézin and T. Halpin-Healy (*)
Département de Physique, Ecole Normale Supérieure, 24, rue Lhomond, F-75231 Paris, Cedex 05, France (Reçu le 25 novembre 1986, accepig le 22 décembre 1986)
Résumé. 2014 Les formes d’échelle de l’aimantation de surface et de la susceptibilité sont déterminées pour le
mouillage critique tri-dimensionnel. Les lois d’échelle phénoménologiques sont bien confirmées pour les
systèmes de type I (basse tension interfaciale), mais il y a des corrections logarithmiques pour les systèmes de type II et III. Nous discutons brièvement aussi les questions soulevées par des simulations numériques récentes
de ce système.
Abstract. 2014 The scaling forms of the surface magnetization and susceptibility are determined for critical
wetting in three bulk dimensions. The naive phenomenological scaling predictions are confirmed for model I systems, but there are logarithmic corrections associated with systems exhibiting model II, III behaviour
Questions raised by a recent numerical simulation are also briefly addressed.
Classification
Physics Abstracts
68.45G - 64.60C
1. Introduction and results.
Critical wetting with short range forces leads to very
interesting theoretical developments [1-3, 4-6] but is, unfortunately, presently outside the realm of tradi- tional experimental physics. Nonetheless, recent numerical simulations exhibiting critical wetting in
the 3-D Ising model have been performed by Binder
et al. [7], who observe strictly mean-field behaviour
(interface correlation length exponent v = 1), de- spite a contention that they’d reached the critical
regime. In fact, the theory of what they have
measured has not been worked out explicitly and they called upon phenomenological scaling laws in
order to compare their data to the theory. Indeed,
we find that the scaling laws utilised by them are not exactly correct in these systems ; there are small logarithmic deviations which arise from the capillary
wave fluctuations that play a crucial role in distorting
the interface in 3d, the upper critical dimension of the critical wetting problem. In this note we report the calculation of these logarithmic corrections and conclude with some comments regarding the more
fundamental discrepancies suggested by the numeri- cal experiments.
In the magnetic language appropriate to the Ising
model let us recall briefly that a field-driven wetting transition occurs when, at a fixed temperature below Tc, a surface magnetic field hl, opposite to the bulk spontaneous magnetization, increases in magnitude
up to a value hl c’ at which point the thickness of the surface layer of flipped spins diverges ; the transition is continuous for a finite range of temperatures T, T Tc. In the numerical simulation, a finite bulk field h was applied and the enhancement in surface magnetization Am, = m, (h) - m, (0), as
well as the surface susceptibility amll ah, were
measured as a function of h, at the critical value
h, c of the surface field. In this note we report the results for these quantities ; they depend, as all the previous results [5, 6] in this system, on the surface tension parameter w ,
in which a is the interfacial tension per unit area and
6b is the bulk correlation length. In the regime in
which T is not far from Tc, w is independent of
T and universal with a value [8] close to 0.8.
Scaling arguments [2] would lead to
in which v is the exponent of the capillary length ) of the interface. However, since the behaviour of ) is not simply characterized by an exponent, the
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004805075700
758
scaling law (2) will be slightly modified by logarith-
mic corrections. The results are
whereas
which agrees with the scaling law (2) ;
with
with
The surface susceptibility aml/ah is easily deter-
mined since it is given by the derivative of Ami (h )
with respect to h.
2. Outline of the method.
We use, as in the previous work on the subject [5, 6],
a simple Hamiltonian for an interface z(x) in which
x is a two-dimensional vector parallel to the wall. It
consists of a surface tension energy, of the wall- interface interaction and of the energy due to the
coupling of the spins to the external field ; in units in which the interfacial tension per unit area is taken
equal to one, we have
A simple analysis [3] leads to the model I potential
that is the sum of two exponentials
in which a is proportional to (hl, - hl ) and b is a positive parameter ; a is inversely proportional to
the bulk correlation length and, in the given system of units, is simply related to the parameter
w defined above by to = a 2/(4 7r). The wall is located in the plane z = 0 and thus z (x ) is limited to the half-space z > 0. Nevertheless, since the poten-
tial is strongly repulsive for negative z, we expect the important fluctuations of the interface to be those
near the minimum of the potential, which, for small
a and h, is obtained for a large positive value of z.
Hence, we will omit the restriction z > 0 and in the renormalization process will allow for arbitrary fluc-
tuations. It has been shown by Brdzin, Halperin and
Leibler [5] (BHL) that this is legitimate in the range 0 -- w 1/2.
At the scale of the characteristic correlation length
of the interfaces the potential V, is renormalized
by the fluctuations into
In the presence of the field the interface is located at the minimum z * of V 1, e + hz, and 6 is determined as
-2 = V[J j (z * ). These equations determine the sur-
face free energy per unit area
One then obtains .åml (h) as 9f, (a, h)l aa I a = 0 .
In the range > 1/2, the entropic repulsion of the
interface by the wall is important and one cannot use
the simple potential (7). For 1/2 -- w ,- 2, the renor-
malized potential is the sum of an exponential and a gaussian ; for w > 2, it is the sum of two gaussians,
but the calculations are similar.
3. Calculation of the scaling functions.
Model I. - Unlike the thermally-driven wetting
transition (a - TW - T) considered previously by
BHL, in which the system deep in the bulk lies on
the coexistence curve (h = 0), the case at hand
suffers from somewhat less tractable mathematics.
In particular, the set of coupled equations that fix, in principle, the position of the interface as well as the correlation length cannot, unfortunately, be solved
in closed form for these quantities. An expansion in
powers of a is in fact necessary. With this end in mind, we set u = e- "Z in our renormalized model I
potential and write
Note that we need only expand to this order because it is the term in Is proportional to a that gives rise to
the enhanced surface magnetization Aml(h). The coupled equations we obtain are
the first allows us to write an expression for the
surface free energy density
with a piece explicitly linear in a. To zeroeth order in a, the above equations yield
These expressions will be needed to complete our
calculation of fl
To determine ul, we expand (lla, b) to linear order in a, obtaining
These results, taken with those at the previous order, then imply
which agrees precisely (no Log correction) with the
phenomenological scaling law (åm1 = h 1 - 1/2 JI) emp-
loyed by Binder et al., provided the model I expo- nent v = (1 2013 Cù )-1 is used [9]. Remembering that
the upper critical dimension of the critical wetting problem is three [5, 6], one should be a bit surprised by the absence of a logarithm. After all, for bulk
critical phenomena at the upper critical dimension, the scaling functions are modified by multiplicative logarithmic factors induced by fluctuations. (The surprise should not be too great, though, since 3d
critical wetting exponents are not mean-field ex-
ponents, suggesting that any analogy between sur-
face and bulk critical phenomena is quite limited.) It
turns out that the missing logarithm is a feature unique to model I - the interfaces of models II and III being insufficiently stiff to depress the fluctua-
tions that give rise to logarithmic corrections.
Model II. - As shown first by BHL, a self-consis- tent renormalization of the model I potential (lead- ing to Eq. (9)) is only valid for w : 0.5. For
W ::> 0.5, the exponential tail of the repulsive piece
of the potential is not relevant and the relatively rare
excursions which bring a piece of this more supple
interface into the vicinity of the wall give rise to a
renormalized repulsion that is Gaussian [5, 6]. For
w : 2, the functional integrity of the attractive
exponential is nonetheless maintained and an entire-
ly consistent renormalization is possible, resulting in
the following modelII renormalized potential at length scale ) :
where 5 2 =(1 /2 v )In C. Following the same pro-
cedure as we did for model I, we extremize
VII, c + hz and set 6 - 2 = V,, (ZO), obtaining the coupled equations,
The first allows us to write
which implies
where we have taken the liberty to write
8 = 5 0 (1 + 81). It is easy to calculate the first term
explicitly since it only involves evaluation of equa- tions (12a, b) to zeroeth order in a, for which we have,
Dropping small terms (of order In In), we find o = h- l2, zo = (2 ’Tr)- 1/2 In 1/ h. Substituting these
results back into the equations to retrieve In In llh
corrections, we discover )o unaltered, but
Using the fact that (s’ð/zo) approaches a constant in
the limit of vanishing h, the equality a = (4 7T (JJ )1/2,
and our recent findings, we have
We made no further mention of the remaining terms
in fl because it can be shown, with a bit of tedious
algebra, that
so the terms hzl, 2 hsl o/zo, and hzl f,6/zo are all
down by a factor l/(ln 1 /h ) relative to the leading
In. Returning to our expression for Am,, we stress
that there is, in fact, a logarithmic correction. Aside from this, however, the phenomenological scaling
formula is correct since
which is precisely the correlation length exponent of
model II, as first obtained by BHL.
760
Model III. - For co :::. 2, the renormalization of model II is no longer self-consistent because the attractive exponential is not renormalized to a
simple exponential, but rather to a Gaussian. In this
way, BHL were led to introduce a double gaussian (model III) potential which, when renormalized by capillary wave fluctuations up to a length scale g, takes the form
with 5 2 =(1 /2 7r )In . Proceeding as before, we obtain the coupled equations
which imply
Since the critical value in this case is nonzero,
f * = 9 e - , J3 2 7T , the appropriate expansion par-
ameter is f - f * and we have :
+ field terms + Gaussian piece.
Expanding (13a, b) to zeroeth order in this quantity yields
which result in precisely the same asymptotic be-
haviour for zo and )o as in the previous model.
Nonetheless, this results in quite different, but not at all unexpected, behaviour for the enhanced surface
magnetization
Hence, again, aside from the logarithm, the phenomenological scaling form is corroborated, pro- vided we set v = oo , which is consistent with the critical behaviour of the modelIII correlation
length. In contrast to the previous case, though, one
can show that the magnetic field terms in 1st scale in
the above manner as well,
remembering, of course, that the ratio -so2/zo is not
critical for h = 0. (The same holds true for the
Gaussian piece.) Thus, they too contribute to our
expression for Ami, simply changing the coefficient of proportionality, since it is apparent that they
don’t sum to zero.
In the interest of firmly establishing the scaling
forms for field-driven critical wetting in three dimen- sions, we adapted the formalism of BHL to the case at hand and discovered agreement with the
phenomenological predictions, apart from logarith-
mic corrections in systems exhibiting model II, III behaviour. We emphasize that there is no logarith-
mic correction associated with model I. Of course,
by agreement we mean that the phenomenological scaling form Am, - h 1 - 1/2 ’ is confirmed as long as
one supplies the correlation length exponent
v = v (w ) appropriate to the model of interest. Why
Binder et al. see only mean-field exponents in their computer experiments on field-driven critical wetting
is unclear. The scaling form for Am, that they employ to determine v is consistent with those derived in this paper, aside from the In corrections, which, in any case, would cause a small, systematical- ly observable, and separable effect. Nevertheless, they find v = 1. One worry arises from the fact that, since ZO - 6b In I/A for the model II Ising system, the interface never really gets far from the wall in the simulations of Binder et al., indeed, the plot they use
to extract v involves h:> 10- 3, which implies
Zo : 5 lattice spacings. This might raise some ques- tions regarding the applicability of a coarse-grained
continuum theory. However, they do retrieve the mean-field behaviour of the aforementioned con-
tinuum theory. Another more likely possibility is
that they’ve not yet reached the critical regime. They
argue that a nice linear plot of the surface excess
(Fig. la of Ref. [7]) insures that their range of fields h is appropriate for studying the critical behaviour of the wetting transition. We’re troubled by this asser-
tion since the surface excess is proportional In 1/h
even at the mean-field level. Moreover, recent preliminary work by us, involving the formulation of
a Ginzburg criterion for the critical wetting prob-
lem [10], suggests that one needs substantially smal-
ler fields to be confident that one is well within the critical regime.
Acknowledgments.
T. H.-H. would like to express his sincere gratitude
for the support provided by the French Government
(Bourse Chateaubriand), Harvard University (NSF
Grant No. DMR-85-14638), and the physics depart-
ment of the ENS.
References
[1] PANDIT, R., SCHICK, M. and WORTIS, M., Phys.
Rev. B 26 (1982) 5112.
[2] NAKANISHI, H. and FISHER, M.-E., Phys. Rev. Lett.
49 (1982) 1565.
[3] BRÉZIN, E., HALPERIN, B. I., LEIBLER, S., J. Physi-
que 44 (1983) 775.
[4] LIPOWSKY, R., KROLL, D. M. and ZIA, R. K. P., Phys. Rev. B 27 (1983) 4499.
[5] BREZIN, E., HALPERIN, B. I., LEIBLER, S., Phys.
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[6] FISHER, D. S. and HUSE, D. A., Phys. Rev. B 32 (1985) 247.
[7] BINDER, K., LANDAU, D. P. and KROLL, D. M., Phys. Rev. Lett. 56 (1986) 2272.
[8] MOLDOVER, M. R., Phys. Rev. A 31 (1985) 1022.
[9] That 03BD = (1 - 03C9)-1 for model I in the
present context follows simply from equations (11a, b) with h = 0. Recall that the correlation
length exponent is defined on the coexistence
curve. The exponents associated with models I and II are obtained in the same manner.
[10] HALPIN-HEALY, T. and BRÉZIN, E., Phys. Rev. Lett.
to be published.
