Line tension at wetting: finite-size effects and scaling functions

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Line tension at wetting: finite-size effects and scaling functions

J. Indekeu, H. Dobbs

To cite this version:

J. Indekeu, H. Dobbs. Line tension at wetting: finite-size effects and scaling functions. Journal de

Physique I, EDP Sciences, 1994, 4 (1), pp.77-85. �10.1051/jp1:1994121�. �jpa-00246892�

Classification Physics Abstracts

68.10 68.42 64.70

Line tension _at wetting: finite-size effects and scafing functions

J. O. Indekeu and H. T. Dobbs (*>**)

Laboratorium _voor Vaste Stof~fysika _en Magnetisme, Katholieke Universiteit Leuven, 8~3001 Leuven, Belgium

(Received

23 June 1993, accepted in final form 8 October

1993)

Abstract. We study trie effect of limite system size on trie tension _r of

a three-phase contact fine. This _is especially important in systems with forces that _are sufliciently long-ranged for _r to

diverge in trie thermodynamic Iimit. Near wetting phase transitions _we propose an anisotropic finite-size scaling hypothesis, which takes into account trie elfect of thermal capillary-wave fluc- tuations. Trie singular behaviour of _{r near a} first-order wettiJ~g transition is aJ~alyzed. Scaling functions _are calculated explicitly in the mean-field _regime, which includes trie physically most

common systems ^with van der Waals forces _m dimension d

= 3.

There bas

recently

been considerable work _on trie fine tension _r associated with _a

three-phase

contact fine. An

exciting development

_concerns the

question

whether _or not the contact fine and

its tension

disappear

when _a

wetting phase

transition is

approached.

That is, when trie contact

angle

tends to _zero. There is strong ^evidence

that,

except ^for continuons

wetting

transitions,

trie _answer is

negative

_[1-6]. When trie

wetting

transition is of

jirst order,

_as is

usually

trie

case, trie contact fine becomes _a

macroscopic inhomogeneity

_or transition _zone,

connecting

_a

thin film to _an

'infinitely'

^thick

wetting layer.

Its tension _r increases

(with diverging slope)

towards _a

positive limit,

which _may be finite _or infinite

depending

_on trie _range of the forces

(the

borderline _case

being

van der Waals

forces) il,

_4].

Furthermore,

for

sufficiently long-ranged

forces _{r may}

diverge, regardless

of any

wetting transition,

_m the

thermodynamic

limit

il, 4].

In order to understand

this,

_r must be reformulated _{as a} function of the system size L, To

appreciate

the

generality

of this

problem,

consider _a system contained in _a cubic volume of size

L~.

_{The total} free _energy F _can thon be

decomposed

into

bulk, surface, edge,

_corner,

etc.

contributions,

F =

fL~

₊

~L~~~

+

rL~~~

₊

~L~~~

₊

(i)

This _is what _one expects. However,

(i)

presupposes that trie forces _are

slljficiently

short~

rangea. Indeed,

_as is well

known,

for intermolecular

potentials çi(r) decaying

_as

^r~(~+°)

in d

(*) PTesent addTess: Department of Theoretical Physics, 1Keble Road, Oxford OXI 3NP, England.

(**) Future addTess: IFF, Forschungszentrum Jübch, D~5170 Jülich, Germany.

78 JOURNAL DE PHYSIQUE I N°1

li

o _x

Fig.

1. Interface displacement profile for _a three-phase contact fine at partial wetting. We _inves- tigate the effect _on the fine tension of the truncation of the wedge at _a maximum width _xm~x and _a

corresponding maximum height 1(xm~x).

dimensions, the bulk free _energy

density f

exists if and

only

if _a > 0. The interaction energy

e of _a

particle

with

surrounding

bulk matter with

density n(r)

behaves _as

L

e m

/

dT

T~~~çi(T)n(r), (2)

ao

with _{ao a}

microscopic length. Consequently

trie energy

density

_e is finite for _{a >} 0 but

diverges

as In L for _a

= 0 and _as L~° for _{a <} 0.

Similarly,

trie surface free _{energy ~} in

(1) only

exists

provided

_a > and trie fine free _{energy r}

provided

_{a >} 2

il,

_4].

Nevertheless,

for _a < 2, ^when

r

does, strictly speaking,

not exist, it is

important

to find ont how _r

diverges

with L.

We _are concerned with the elfect of finite system size _on the fine tension, and make _a

general scaling hypothesis, applicable

to first-order _as well _as continuons

wetting

transitions.

Hereby

we extend _a

recently proposed scaling theory iii

to systems ^{of finite} size. For concreteness, consider _a

liquid wedge

adsorbed at a

planar watt,

as shown in

figure

i. The

displacement 1(~j

of the

liquid-vapour

interface makes _{a contact}

angle

with the watt. We

neglect gravity,

and _{assume a} finite

wedge

size

parallel

to the

watt,

_{Ljj e ~n~ax,} and

perpendicular

to

it,

L _{i e} 1(~n~~x). ^{Note that} Li depends on

Ljj.

^Due to the intrinsic anisotropy these two

lengths

acquire ^dilferent

scaling properties.

For the

singlllar

part ^of _r we

postulate

r~;ng(t, Ljj)

= À/~~~~~ r~ing(À(/~" t,

À[~Lii) (3) Tsinglt, L1)

# ~

j~~~~)/~ TsinglÀi~~

t, ~

j~L1), _j4)

where is _a positive

length-rescaling factor,

t

=(

T Tw

/Tw,

Tw trie

wetting

temperature, and _vii

(vi

trie interface

correlation-length

exponent for fluctuations

parallel (perpendicular)

to the interface. For

jirst-order wetting

transitions, to which _we

presently

restrict ourselves, these exponents pertain to the critical

phenomenon

of

compiete wetting

_[8]. The exponent

(

is the

roughness

_or interface

wandering

exponent, and _{vi =}

(vjj.

Introducing

the correlations

lengths,

_{iii cc} t~~" and

ii

cc

i)

_cc t~~i,

(3)

and

(4) imply

Tsinglt>

Li)

₌

^t~~°'

_Xi

IL Il là)

r~jng(t,Li)

=

t~~"i _Xi (Li Iii) (6)

Here, _ai ^is the fine

specific

heat exponent [5,

ii,

and

Àfii(xii ^and

Àfi(~i)

_are

scaling

functions, with umversal properties. ^We emphasize that

iii

^and ii pertain to trie

injinite

system and

diverge

at Tw.

We

distinguish

finite-size corrections associated with trie

thermodynamic

limit Ljj »

(jj,

^and

the critical limit (jj ^»

Ljj.

^Trie cross-over occurs for Ljj

/ijj

m i, _or,

equivalently,

for _{a contact}

angle

^Ùt which satisfies

Ùt _m

Lj~/~~"

_,

(7)

since _cc

t(~~"8)/~,

where _a~ is trie surface

specific

heat exponent, ^with _{o~ =} ^for

jirst-order wetting.

Explicit

finite-size

dependences

_{are apparent} when rsjng ^is

rewritten, using (3)

aJ~d

(4),

_as

Ging(t, Lll)

" Ljj ^{~~ ~~} VII

(Lll/fil)> (8) r~;~~jt,

L

i ⁼ L

j(d-2)/( vi jL

i

Ii

_i

jg)

For trie _common thermal

capillary-wave fluctuations, (

takes trie universal value _[8]

çjà)

=

j3 à) /2,

for i 1 à 1 3

jio)

((d)

=

0,

for d 2 3

iii)

It is convenient to introduce _a mean-field

(MF) anisotropy

exponent

(~~

_e

_{((du ),}

_where

_du

_is

trie _Upper critical

dimension,

above which fluctuations _are irrelevant. Trie

scaling

relations of

hyperscaling

type

(those

in which d enters

explicitly)

remain valid in trie MF

regime

d >

du, provided

d is set to du.

Thus, ((d)

is then

replaced by (~~ ii, _8].

In

physical

terms, trie fluctuation-induced interfacial

roughness

at d _>

du

is weak

compared

with trie

inhomogeneity imposed by

trie mean-field

anisotropy

exponent.

Similarly,

in trie MF

regime theTmodynamic lengths (f~

and

( f~,

with critical exponents

vjf~

^and

vf~, replace

trie

corresponding

_carre-

lation

lengths.

We _now ask how Li

depends

_on

Ljj.

^We

^infer,

Li ₌

Lj Zjj(Ljj /ijj), (12)

~'~~~~

i/2vjj

~ll(zjj)

_+~ ^{~ll ' Zll ~ "}

~°~~~°~~~~>

Z]]~0

^'

⁽¹³⁾

so

that,

in trie

thermodynamic limit,

Li _cc ÙLjj, ^{since for} first-order

wetting,

_vjj

=

[2(1- ci

^~~

iii.

This agrees with trie

simple geometrical

consideration Li _oc tan

ÙLji,

^for - 0. On

trie other

hand,

in trie critical

limit,

Li _cc

L(.

^{We also note that}

ⁱⁱ

cc

i(

_{cc Ù(jj,} since

vi =

(vii

= vij

1/2.

We _now illustrate these

general

considerations in _an

approximation

_m which

expirait scaling

functions _can be calculated: mean-field

(MF) theory.

Trie free _{energy excess per} unit

length,

r, associated with _a surface

inhomogeneity,

is

expressed

_{as a} functional f of trie

displacement 1(x)

^{of trie} interface in trie _z

direction, perpendicular

to trie watt _11,

4],

+1

₌

j~"

^dz

j~~ jwn _ij

_{+ v}

_jijx))

₊

_cjx)j

,

j14)

where trie

subscript

_x indicates trie usual dilferentiation. We _assume translational invariance

m the

remaining

d 2 of trie d i directions

parallel

to the waII. Trie _square root in

(14),

which is

multiplied by

the surface tension of trie

interface,

_~o, is

frequently approximated by

a

gradient-squared

term,

although keeping

trie fuit fine element ds

=

@@

dz does not

80 JOURNAL DE PHYSIQUE I N°i

complicate

trie calculations much [9].

V(1)

is trie interface

potential,

which for trie _case of

partial wetting

is normalised _{so as} to take the value 0 for _a uniform thin

film,

of thickness

( (see Fig. i).

The

limiting

value E of

V(1),

for

~ cc, is related to the

spreading

coefficient S

by

E

=

-S,

and to the contact

angle by

cos = i

E/~o (là)

The

piecewise

constant

c(x)

in

(14)

must be chosen _{so as} to make the

integrand

tend to _zero for _x

(~

_{cc, away} from the surface

inhomogeneity

[4].

Trie result for trie fine tension is [9],

r(E, Li)

=

2l~oÎb /~~

^dl

^{Ù(1)~/~ (i (1)/2)~~~ Ê~/~ (i /2)~~~j}

,

(161

ii

where, for convenience, and Li bave been scaled with trie blllk correlation

Iength (~

so as to become

dimensionless,

and V _e

V/~o,

etc. This

expression

reduces to _a

simpler

form when the

gradient-squared approximation

is made [4]. The results _we will describe _are insensitive to this

approximation,

as

expected

_on the basis of

universality.

1.

Long~range

forces.

We first discuss

long~range ^forces,

^{for which} finite-size elfects _are of utmost

importance.

^We

assume intermolecular

potentials çi(r)

_cc

r~(~+°l,

for _{T ~ cc.} Trie exponent

(~~

_{takes a}

non-trivial

value,

because du _< 3. For first~order

wetting

du

= 3

4/(a

+

i),

and nonce

(~~

=

((du

=

2/(a

+

i).

In trie critical

limit, (

controls trie

anisotropy

of

length scaling.

This _is

dearly

_seen in trie

asymptotic

behaviour

of1(~),

for

large

_~, ut first~order wetting. In mean-field

theory, 1(x)

cc

x~/(°+~) ii,

_4].

We consider _an interface

potential

^{of trie form}

Vil)

= E +

Al~(°~~)

_{+ Bl~° +}

Cl~("+~) (ii)

where

A(> 0), B(< 0),

and

C(> 0)

determine trie value of E, ^and A takes the value Ao at

wetting.

Turning

^first to a > 3

(induding

retarded _van der Waals

forces),

_we define the finite-size correction to r near first~order

wetting

as

ôr(E, L)

_e

r(E, L) r(E, cc), (18)

where L may stand for Ljj or

Li,

and _[9]

riE,cc)

₌

rie, cc)-21/2~ioià [iAo/~o)1/~«-IiKia)iE/~o)~«-31/2~«~I)

⁺

oiiE/~o)1/2)j

,

i19)

where

r(0, cc)(> 0)

is finite and

non~universal,

and

K(a)(> 0)

is

given

m equation

(4.30)

of [4].

Calculating r(E, L) using (16),

_we

find,

for L

~ cc and E

~ 0,

ô~j

E L

~)

~

-21/2 ~~j~j ~~ /~~)i/(«-i)

~ç

/~~)(«-3)/2(«-1)

~~

~~j

~ç

/ ~~)i/(«-i) _j~~)

'

Now trie argument xi +

Li(E/Ao)~/~"~~~

_can be written _as

Li/Î Î~,

_smce for first-order

wetting

E _cc t~~"S

c~ t and since _v

f~

=

il la i).

Aise, for first~order

wetting

_ai

= a~ + ujj ⁼

1 ₊ vjj

iii,

and

vf~

₌

la

+

1) /[2(a i)].

Dur result thus satisfies standard

scaling

in trie form

(6).

For trie

scaling

function

Àfi (xi

_we derive trie

following asymptotic behaviour,

~~~~~~'~ IÎÎÎa ^{Î)] ~~~~~~,} _~ÎÎi~~Î

^~~~~

The finite-size correction ut

wetting

is obtained in trie limit _{xi ~} 0, which

gives ôT(°, Li)

+~

-2~/~Î2/(° 3)]~o(b(Ao/~o)~/~L

~~"~~~/~

(22)

We note that

la 3)/2

=

(du 2)/((du),

_so that

(22)

_agrees with

(9).

Trie _{case a =} 3

(non~retarded

Van der Waals

forces)

merits

special

attention. The fine

tension in _an infinite system

diverges (towards +cc)

at the

wetting

transition. We found _[9]

IIE> C°)

_"

-2~~~~o(b (lAo/~o)~~~ 'nlE/~o)~~~

+

°ll)j 123) Adopting (18)

_we

obtain,

for L _{~ cc} and E _~ 0,

ôr(E, Li)

+~

-2~/~~olb(Ao/~o)~/~Xi (Li(E/Ao)~/~)

,

(24)

where _{xi +}

Li(E/Ao)~/~

_can be identified with Li

If Î~,

and

~~~~~~

'~

În(~ÎÎ),~ÎÎÎÎ

_~~~~

This is _consistent with

(6),

but

Iogarithmic

corrections _are present. In

particular,

for _{xi ~} 0,

approaching Tw,

the

Iogarithm

in the

scaling

function cancels the

leading logarithm

in

(23)

to

produce

the

infinite~size diveTgence r(0, Li)

+~

2~/~~o(b(Ao/~o)~~~

ln

Li (26)

The

foregoing

_can be

readily

extended _to 2 _{< a <} 3. We found _[9]

IIE, ce)

=

2~/~~oib (lAo/~o)~/~"-~~Lla)lE/~o)-~~-"~/~~"-~~

⁺

Cil)]

,

127)

where

Lia) (> 0)

is

given

in

equation (4.35)

of [4]. ^{We recall that}

Lia)

is itself

divergent

^for

a < 2. We

obtain,

for L _{~ cc} and E _~ 0, and _now in terms of _r rather thon

ôr, IIE, Li)

+~

2~/~~ofblAo/~o)~/~"~~~lE/~o)~~~~"~/~~"~~~Xi (LilE/Ao)~/~"~~~)

_,

¹²⁸⁾

where _{xi +}

Li(E/Ao)~/~"~~~

_can be identified with

Li If Î~,

and

l~j~) ^j~j ~)j-I~-(OE-2)

_{~ ~ ~}

x

j~

~

° l 1

j~g)

~ ~

[2/(3 a)] xf~"~/~,

_{xi ~} 0

This is consistent with

(6).

An infinite-size

divergence

ut

wetting

results for _{xi ~}

0, rie, Li)

~w

21/212/13 a)i~oi~iAo/~o)1/2Lf~°~/~

130)

82 JOURNAL DE PHYSIQUE I N°1

The _{case a}

= 2 is relevant to nematic

liquid crystals

_[10], where it arises from elastic

long-

range forces rather thon intermolecular forces. We

find,

for L _{~ cc} and E

~ 0,

rjE,

Li _~

21/2~oiblAo/~o)lE/~o)~~/~xilLiE/Ao),

₁₃₁₎

where _{xi +}

LIE/Ao

_con be identified with

Li/Î Î~,

and

~~~~~~

~ 2x

)/~, ÎÎ ^~0

~~~~

This is consistent with

(6).

Note that, _{even away} from the

wetting

transition, at finite E, there is _a weak

(logarithmic) divergence

in the

thermodynamic

hmit. A stronger

divergence

is obtained in the critical

limit,

for _{xi ~} 0,

resulting

in the

following

infinite~size

divergence

at

wetting,

r(0, Li

~

2~/~2~oÎb(Ao/~o)~/~L )/~ (33)

Further extension to 1 _{< a} < 2 poses no

problems.

We

find,

for L _{~ cc} and E

~ o,

IIE, Li)

_m~

21/2~oj~jAo /~o)i/(a-i)jE/~o)-(3-a)/2(a-I)xi (LijE/Ao)I/(a-1)), ^j34)

where _{xi +}

Li(E/Ao)~/~°~~~

_can be identified with Li

Il Î~,

and

~~~~~~

~

[2/(3 a)]

_~

~~"~/~,

_{xi ~} 0 ~~~~

This is consistent with

(6).

Now, _away from trie

wetting

transition, at finite E, there is _an

algebraic divergence (proportional

to

L)~

_{in trie}

thermodynamic

limit. A stronger

divergence

is obtained in the critical

hmit,

for _~i

~ 0,

resulting

_m the

following

infinite-size

divergence

at

wetting,

r(0, Li)

~

2~/~[2/(3 a)]~olb(Ao/~o)~~~Lf~"1/~ (36)

We remark that the

leading

finite~size correction

(22)

for _{a >} 3, _as well _as the

leading

finite- size behaviour

(26), (30), (33),

and

(36)

for _a < 3, _can be summarized in _a

single equation

for the finite-size behaviour of trie

singular

part ^of r, valid for ail _a > i. This

equation

reads

Ts>ng(o,

LÀ)

~

~~/~~01blA0/~0)~/~l~/13 °)ILÎ~~~~~'

f°~ _{° ~} 137) Note that the

logarithmic divergence

featured in

(26)

is also contained in

(37),

_m trie hmit

a ~ 3.

In

dosing

this part we mention that infinite-size

divergences

_were also obtained for the

boundary

_energy of _an interface between _two two-dimensional domains

ii ii.

The connection between that situation and _{ours is} the

followmg.

In _a d-dimensional system ^the interfacial tension

(of

_an interface with

dimensionality

d

i) diverges

if

a < i. This follows from

simple

dimensional

analysis,

_m the _{same way as} the

divergence

of the bulk _energy

density

_is obtained for _a < 0. In the _case considered

by

Flament and Gallet

iii]

the interface is one-dimensional and the bulk _space in which summation of

pair potentials

is

donc,

_{is a} two-dimensional

plane.

Since then d

=

2,

the borderline _{case a}

= i

corresponds

to a

potential decaying

_as

^r~~,

which agrees with their results.

Divergences

of the type encountered in

iii]

_are not contained in _our description, ^due to _our restriction to _a > i. The

divergences

related to the contact fine already

occur at _a > i. If the bulk dimension

equals

d

= 3

las

_we

normally assume),

the

potentials

_we

allow

decay

faster than r~~

2.

Short-range

forces.

For first-order

wetting

and

short~range

forces,

du

= 3 and

((du)

= 0, _so that

logarithmic singularities

_can

play

_an

important rote, already

in the mean~field

theory.

Trie

profile

at first~

order

wetting

satisfies

dl/dx

_cc

V(1)Q~,

_so that

1(~)

_+~

21nx,

for _{x ~ cc} [4]. ^It follows that

e~i

+~

L(,

^{which will be useful below. Note}

^that,

^{like for}

long~range forces, Li

_{cc ÙLjj} is valid in trie

thermodynamic limit,

and _vjj

=

[2(1 ()]~~

for first~order

wetting.

We consider forces

leading

to

exponentially decaying

interface

potentials

like

Vil)

₌ E + A

expj-1)

+ B

expj-21)

+

Cexpj-31), j38)

where

A(> 0), B(< 0)

and

C(> 0)

determine the value of

E,

and A takes trie value Ao at

wetting.

We

adopt (18),

with [9]

IIE, ce)

= T10,

ce)

+

21~oib (lE/~o)1'nlE/~o)

^{+ O}

(lE/~o)~/~)j

,

139) andfind,forL~ccandE~0,

ôT(E, Li)

_"

~2~/~~olb(E/~o)~~~Xi(e~~E/Ao ), (4°)

where _xi +

e~iE/Ao

_can be identified with

e~i lefl~,

_and

~~~~~~

~

Îx ^~,~ JÎ~~Î

_~~~~

This result is non-standard because _xi is here _a function of trie dilference

Li -Î Î~

rather than trie ratio Li

Il Î~. _However,

in trie critical limit it _can be _rewritten in trie standard form

là), using

_{xi "}

e~iE/Ao

"

L(E/Ao

" (Ljj

/ijj

)~ +

x(,

and

replacing

Xi

(xi by

Àfjj(xjj

).

We recall that E _cc t~~°8 _{cc t,} and

vf~

=

1/2

for

short~range

forces

iii. Also, of~

= 1+

vf~

₌

3/2.

Trie finite~size correction _at

wetting

is obtained for _{xi ~} 0,

ôT(Ù,Li)

+~

-2~/~2~oÎb(Ao/~o)~~~e

^~~~~ ~~~)

Finally

_{we examine} trie very

interesting

universal behaviour of trie

slope dr/dÙ

_near

wetting.

Its

divergence (towards -cc), generally

observed _on

approach

to _a

jirst~orderwetting transition,

is also curtailed

by

finite-size elfects.

By dilferentiating (16)

with respect to Ù, it is seen

that,

for

short-range

_as well _as for

long~range forces,

ÎÎ

~~~

~°~~'~~

_~~~~

This

remarkably simple

result is consistent with finite~size

scaling

and _we expect it to be valid also

beyond

mean-field

theory,

_on trie

following grounds.

In the critical limit Li <

ii

we

derive,

_on the basis of

(9),

r(Ù, Li)

_" L

j~~~~~/~4l(ÙL )/~~~ ), (44)

where _we used

ii

_cc ^Ù~~~ Now _we make the

plausible assumption that,

since there _can be _no true

wetting singularity

in _a finite system, trie function

4l(çi)

_con be

Taylor expanded

around çi = o.

Now,

since

(

₌

(3 d)/2

and _vi

=

(vjj

=

(/[2(1 ()],

it follows

that,

for _~ 0,

r(Ù, L)

₌

coL j~~~~~/~

+

ciÙLi

+ O(Ù~

(45)

84 JOURNAL DE PHYSIQUE I N°i

This result, which agrees with

(43),

should thus hold both in mean~field

theory

and in the non-dassical thermal fluctuation

regime.

Before

drawing

_a

general conclusion,

_we would like to make _{one more} remark. In the _mean- field

theory

the contact line

inhomogeneity

is described

by

_a

single

coordinate _~

perpendicular

to the contact line and translational invariance is assumed in the direction

(say, y) along

the contact line.

However,

_one should expect

capillary-wave

type fluctuations of the contact

line, dosely

connected to the

capillary

_waves in the interface. In

sufficiently

low

dimensions,

these fluctuations _are relevant and _{one may no}

longer

_assume translational invariance

along

_y.

One _may expect contact line fluctuations to

modify

the

singular

^behaviour of _r. The central

assumption underlying

the

scaling

relations

proposed

in the

beginning

of this _paper, is that

contact line fluctuations _are

governed by

the interface fluctuations. In other words, there is

no new

length

scale besides the

length

scales associated with the interface.

Consequently,

the

typical

extent of _contact hne fluctuations _is

given by

trie interface correlation

length jjj.

It is

through

_(jj that the contact line fluctuations _{enter into} the

singular

behaviour of _r

iii.

Partial support for this

hypothesis

bas

recently

been

provided by

exact results for _{r m} trie two-dimensional

Ising

model at critical

wetting

[12].

In

conclusion,

_we bave

applied

anisotropic finite~size

scaling

to describe trie behaviour of the line tension in systems of

macroscopic

but finite size. Trie

key

variables in this frame-work _are ratios of system size parameters and interfacial correlation

lengths pertaining

to

wetting.

Dur

analysis gives

^detailed

predictions

for trie

leading smgularities

of trie line tension _at

wetting,

m both trie

thermodynamic

hmit and the critical limit. In

particular, divergences

of the line tension _{as a} function of trie system size bave _been scrutinized. Dur results illustrate that finite- size elfects

play

_a

key

rote in systems with forces of

longer

_range than non-retarded _van der

Waals

forces,

for which the

thermodynamic

limit of the line tension does not exist. Extension of these finite~size

scaling

arguments to

prewetting

and to continuous

wetting

transitions is

straightforward,

and will be discussed elsewhere.

Finally,

for _a

comprehensive

review _on the

line tension at

wetting,

the reader is referred _to [13].

Acknowledgements.

We thank Alberto Robledo for his interest in this work. This research has been

supported by

the

Belgian

Incentive

Programme IT/SC /06

for Information

Technology (DPWB),

the

Belgian

Inter~University

Attraction Poles

(IUAP),

the Concerted Action

(GOA)

research programmes, and the

Inter~University

Institute for Nuclear Sciences

(grant

4.0010.92

N).

J.O.I. is Senior

Research Associate of the

Belgian

National Fund for Scientific Research. Additional support for H-T-D- _was

provided by

_a S-E-R-C-

(U.K.)

CASE award with British Gas

plc

and

by

trie

ERASMUS

exchange

_programme.

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