Prewetting transition in supercritical argon

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Submitted on 1 Jan 1990

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Prewetting transition in supercritical argon

G. Ascarelli, H. Nakanishi

To cite this version:

Prewetting

transition

in

_{supercritical}

_argon

G. Ascarelli and H. Nakanishi

Physics Department,

Purdue

_University,

W.

Lafayette,

In 47907, U.S.A.

(Reçu

le 20

_juin

1989,

accepté

sous

_{forme définitive}

le 10 octobre

₁₉₈₉₎

Résumé. 2014 Nous

avons obtenu la densité de

_l’argon

au

_voisinage

du

_point

_critique

à

_partir

de

mesures de la constante

_{diélectrique à}

basse

_fréquence.

Nous avons observé une discontinuité de la densité le

_long

de l’isobare

_{correspondante à}

50 atm à une

_température

un _{peu au-dessus de la}

température

critique.

Cette discontinuité

_suggère

une transition de

_{pré-mouillage.}

Un calcul

simple,

selon le modèle du

_champ

_{moyen, réussit à rendre compte} de

_quelques

_aspects des résultats des mesures.

Abstract. 2014

From the low

_frequency

measurements of the dielectric constant of ultra low

impurity liquid

argon, we obtain the

_density

of the

_liquid

in the

_vicinity

of the critical

_{point. Along}

the 50 atm isobar

_slightly

above the critical _temperature, we find _apparent

_{discontinuity}

in the

density, suggesting proximity

to a

_prewetting

transition. A

_simple

mean field theoretical

_analysis

provides

a

_{qualitative description}

of some _aspects of this behavior. Classification

Physics

Abstracts 64.70F 2013 64.80 2013 64.90

1. Introduction.

In _{the process of}

_measuring

the Hall

_mobility

of electrons

_injected

_{in argon} we also measured

the low

_frequency

dielectric constant of ultra

_{high purity}

_argon

_along

two

_{supercritical}

isobars.

During

these measurements we observed anomalies that

_suggest

the existence of a so called

prewetting

transition

_{[1, 2].}

This is the first

_example

of such a transition in a _{pure substance.}

The

_{only previously}

known

_example

was in a

_binary

fluid

_system

_[3].

The fact that the state of an electron emitted from a

_photocathode

into

_liquid

NH3 might

have an _{energy that is different from what it would have in the bulk of the}

_fluid,

was

_initially

_{suggested by}

Krohn and

_Thompson

_[4]

and later

_{experimentally}

confirmed

_by

Bennet et al.

_[5].

Such an observation may be

_interpreted

as an indication that the molecules in the immediate

_vicinity

of the

_photocathode

are in a somewhat « ordered » state. Such a

state, if it

_exists,

_{may be}

_{produced by}

the interaction of the

_dipole

moment of the molecule with its

_image

in the metal.

_Intuitively

one

_might

_expect

that the

_resulting

fluid would then have a

_{larger density}

near the

_{photocathode.}

In the case of rare _gas

_fluids,

the individual atoms have no

_permanent

_dipole

moment but

only

a

_dipole

moment that fluctuates

_randomly

both in

_magnitude

and direction. The van der Waals interaction between two atoms A and B is

_{normally pictured}

as the interaction of this

random

_dipole

on atom A with the electric

_dipole

it induces on atom B.

In the

_vicinity

of a metal

surface,

this

_randomly

_fluctuating

electric

_dipole

moment will interact with its

_image

in the metal. If this interaction is

_{sufficiently large,}

we _may

_expect

that

the fluid

_density

near the metal will be

_larger

than the

_density

in the bulk. In

_general

this deviation of the

_density

near the surface from the bulk

_density

will have a short range, of the

order of the correlation

_length.

_However,

as we discuss

below,

even above the so called

prewetting

critical

_point,

remnants of this

_density

deviations _{may form} a

_macroscopic

film.

The remainder of this paper has four sections : the

_description

_{of the gas}

_purification

procedure,

the

_experimental

_setup,

the

_experimental

results and

_finally

an

_Ising

model

description

of the

_wetting

transition which

_suggests

that the

_experimental

result may be a

reflection of the remnants of the

_{density discontinuity}

associated with the

_prewetting

transition.

2. Gas

purification

procedure.

Argon

gas whose nominal

impurity

content was 5 ppm was further

_{purified by initially}

condensing

it into a vacsorb pump

_containing

4A molecular sieve

[6]

that had been activated

in vacuum

_by

_heating

_up to 350 °C. After the

_liquid

_{argon remained 12 hours in} contact with the molecular sieve the gas was transferred into an all metal

_system

and

_passed

3 times over Ti

shavings

held at .-- 700

°C,

then it was

_passed

two more times over the same Ti

_shavings

held

at room

_temperature.

The

_system

_containing

the Ti

_shavings

had been

_previously

_pumped

and baked so as to reach a _{pressure --}

10- 9

Torr when the Ti was held at ~ 750 °C and the

remaining

system

was between 200 °C and 300 °C.

Finally

the gas was condensed in a stainless steel

_sample

cell

suspended

at the end of

a = 1 m

_long

3/4" OD stainless steel tube. The cell and its

connecting

tube had

previously

been baked out so that the

_limiting

_{pressure before the introduction of the argon} was

-w

10- 10

Torr when the

_system

_{was at room}

_{température.}

To maintain the

purity

of the

sample

over the 9 months

_during

which the data

_reported

here as well as other measurements were

taken,

non-evaporable

getters

[7]

were in the

_system

near the

_high

_{pressure valve that isolated}

the

_sample

cell from the rest of the vacuum

_system.

This

_sample

_{purification procedure}

was _{necessary in order} to

_get

_long

electron lifetimes for

a

_separate

_experiment

in which the Hall

_mobility

measurements were made

_concurrently

with

the

_present

_experiment.

The

_major

electron

_{trapping impurity}

that is

_normally

_present

in Ar is

₀₂

with a cross

section --.

10 - 16 CM2

_[8].

The other

_component

of

air,

N2

has a

_trapping

cross section at least 200 times smaller

_[8].

=

10-4

_S,

The fact that the

_resulting

electron lifetime

was = 10-4

s, indicates that the oxygen concentration was _{in the range of 1}

_ppb,

_{well below the range estimated}

_by

the Ar

_supplier.

This electron lifetime did not

_change

over a

_period

of 9 months

_indicating

no detectable

change

of the

_impurity

content of the fluid. 3.

_{Expérimental}

_apparatus.

The

_sample

cell was _{press fitted into} a _copper

_cylinder

with 3 mm thick wall and a heater was

wound over it. Inside the cell a 0.13 mm thick copper liner was introduced to further

_help

maintain a uniform

_temperature.

About 4 cm above the end of the

_sample

cell,

a

_piece

of

copper sheet 0.13 mm thick and about 25 cm

_long

was wound around the stainless steel tube

and another heater was wound over it. When the fluid was at the

_triple

_point,

its level reached well into this

_region

of the tube.

_Finally,

about 20 cm above the end of the second

_heater,

a

third heater was wound over another smaller

_piece

_{of copper (about} 4 cm

_wide)

and over

about 15 cm of stainless steel tube. The function of this third heater was to assure that there would be no

_temperature

inversion between the

_sample

cell and the valve held at room

temperature

that closed the

_sample

cell. It was also used to

_adjust

_{the pressure in the}

_system

Fig.

1. -

Sample

cell and a small

_portion

of the tube

_connecting

it to the

_high

_{pressure valve and} to the pressure gauge. The electrode

assembly pictured

in the cut _{away is}

_suspended

in the

_sample

cell. CE is the 3 mm _{thick wall copper tube into which the}

_sample

_{cell is press fitted. THERM} are thermistors used for the _{temperature control,} TC is a

_{thermocouple.}

HTR are heaters, the _top one wound around the

copper foil CF that is

wrapped

around the stainless steel tube

_supporting

the cell. A _{similar copper foil}

lines the walls of the

_sample

cell. The

capacitor

is made out of the electrodes E, E’

_(platinum

on

The

_sample

cell and its

_supporting

tube were inserted into another stainless steel tube surrounded

_by

_liquid

_nitrogen.

The inside of this second tube was

_pumped

to about

10- 2

Torr so that the energy transfer between the

_sample

_cell,

its

_supporting

tube and the

stainless steel tube immersed into the

_{liquid nitrogen}

was

_primarily

due to radiation when T 2:: 130 K.

The function of both the teflon 0

_ring

and of the teflon disc at the bottom of

_figure

1 is to

avoid thermal shorts between the

_sample

cell and the walls of the tube

_containing

the heat

exchange

gas.

The argon pressure was measured with a 16" diameter Bourdon tube gauge (Heise) to an

estimated accuracy of ± 0.3 atm. The

_temperature

of the fluid inside the

_sample

cell was

measured with a

_platinum

resistor in the 4 wire

configuration

that was calibrated

_against

a

GaAs diode whose calibration could be traced to the NBS. This calibration was checked at the

triple point

of argon. The current

_through

the

_platinum

resistor was

_kept

at 0.1 ma to assure

minimal heat

_{dissipation ;}

near the critical

_point

_{the power}

_dissipated

was = 0.5

_uW.

The

temperature

accuracy is estimated at ± 0.1 K.

The dielectric constant was measured

_{by measuring}

the value of the

_capacitance

of a

parallel

plate capacitor

whose

_plate

_separation

was = 0.12 mm and whose area was

1 cm x 5 cm. Such a

_{plate separation}

is

_{clearly macroscopic}

so that any measured

change

of

capacity

due to condensation of the fluid does not reflect

_{changes taking place}

over

_just

a few

monolayers.

This

_capacitor

was located next to the electrodes used for the Hall

_mobility

measurements ; these measurements are the

_subject

of a

_separate

_publication

[9].

The

capacitor

was made

_by

_baking

an

_organic

Pt film

[10]

over 0.6 mm thick alumina

plates.

In the

vicinity

of 150 K the thermal

_conductivity

of alumina is = 0.7 W/cmK

[11].

Spacers

were made

using

microscope

cover slides. The

_capacitance

was measured

_using

the three wire

configuration

of a General Radio model 1 615 A transformer ratio

_bridge.

The 1 kHz

voltage

applied

to the

_bridge

was

_{normally kept}

at 1.3 V r.m.s. ;

decreasing

it 10 times did not have any effect on the measured

_capacitance.

This indicates that the

_anomaly

to be described

below,

is not caused

_by

the

_applied

electric field. The

_output

of the

_{capacitance bridge}

was

measured

_using

a

_phase

sensitive detector so that the value of the

_capacitance

could be read to

1 x

10- 4

_pF.

Random

_changes

of the

_capacitance

associated with vibrations limited the

long

term

_{reproducibility}

of the

_capacitance

to about 5 x

10-3 pF.

4.

_Experimental

results.

From the measurement of the dielectric constant

_(E),

the

_density

_{(n )}

of the fluid can be calculated

_using

Clausius Mossotti relation :

M is the Clausius Mossotti constant : M = 4 Tra

_/3,

and a is the

_{polarizability.}

In the

_optical

_{range, the}

_{corresponding}

_{equation (called}

the Lorentz Lorentz

_relation)

was

used

_{by Teague}

and

_Pings

_[12]

who made a detailed

_study

of its

_{applicability}

in the critical

region

of argon. An excellent

agreement

was found between the measured

_temperature

dependence

of the difference of the

_density

of the

_liquid

and the

_density

_{of the gas}

(ne -

ng)

and the

_predictions

of the

_theory

of critical

_{phenomena. Teague}

and

_Pings

indicate that the Clausius Mossotti constant

_{changes approximately}

1 % between the low

_density

_gas and the saturated

_liquid.

Garside et al.

_[13]

made similar measurements in the case of xenon.

_They

find that the

constant M varies

_{approximately}

4 % between the saturated

_liquid

and the low

_density

_vapor.

Thoen and Garland

_[14]

measured the low

_frequency

dielectric constant of xenon near the

critical

_{point using}

a series of

_capacitors

whose

_{plate separation}

was similar to what was used in the

present

experiment.

Some of these

_capacitors

were above and some were below the

liquid

vapor

separation

surface. These authors found no

_anomaly

in the constant M near the

critical

_point.

Initially

we measured the dielectric constant of the saturated

_liquid

between the

_{triple point}

and 147 K. The value of

_Tc

measured

_by

others is between 150.7 K and 150.8 K. Since the value of the

_density

could be

_{independently}

calculated with the

_approximate

_equation

of state

[15]

using

as

_inputs

_{the pressure and}

_temperature

of the

fluid,

the Clausius-Mossotti constant

could be calculated. A

_least-square

fit to a series of 98 measurements _{gave M} =

(4.110 ±

0.001 )

10-3

1/mol. This is to be

_compared

with the result of

Amey

and Cole

_[16]

at

the

_{triple point}

_{(4.097 ±0.005)}

10-3

_1/mol,

and

_{(4.106 ± 0.005 )}

10-3

1/mol at the normal

boiling point.

In the

_present

measurements the deviations between the individual densities calculated from the

_equation

of state and those inferred from the dielectric constant measurements

_(using

the overall fitted value of M

_quoted

_above)

were under 2 %.

Fig.

2. -

Projection

of

_portion

of the nPT _{surface of argon in the nT}

_plane.

_Pt

indicates the

_position

of the critical

_point

in the bulk

sample.

The broken line in it _represents the

_region

where the

_equation

of

state

_[12]

_gives

unreliable results. The solid lines

_terminating

on this broken line _represent the saturated

liquid

and saturated vapor lines. The other broken line,

starting

at the

_wetting

_point

_T,,

and

_terminating

at

_Pc,

is a

_{speculative representation}

of the

_prewetting

transition line. Two _isobars,

_{corresponding}

to

50 atm and 55 atm are shown as well. It is not clear from the

_experimental

data whether

Larger

deviations between the densities obtained from the

_equation

of state and those obtained from the measurement of the dielectric constant

_{(= 5 %)}

are obtained between

147.0 K and 150.0 K. These deviations were

_assigned

to the

_acknowledged

deficiencies of the

equation

of state

_along

the saturation line near the critical

_point

[15].

Between 150 K and the

critical

_point

these deviations become as

_large

as 30 %.

In

_figure

2 we

_present

a

_projection

of the nPT surface into the nT

_plane.

The

_liquid

_vapor coexistence curve is shown as well as the 50 atm and the 55 atm isobars. The

_figure

is calculated from the

_equation

of state

_given

in reference

_[15].

Experimental

data do not bear out the

_expectation

that the deviations between the

_density

obtained from the measurement of the dielectric constant and that obtained from the bulk

equation

of state decrease when

_moving

_{away from the} critical

_point

_{(Fig. 3).}

Such a decrease

would be

_expected

if the deviations arose either on account of inaccuracies of the

_equation

of

state near the critical

_point

or due to the

_{extremely large}

_{compressibility}

of the fluid near

Tc.

Fig.

3. -

Comparison

of the

_{density n}

calculated from the

_equation

of state

_[15]

and the

_density

nk calculated from the measured dielectric constant

_along

the 50 atm

₍₀₎

and

_along

the 55 atm

_(A)

isobars.

A

_plot

of the

_temperature

_dependence

of the

_density

obtained from measurements of the dielectric constant

_along

the 50 atm and the 55 atm isobars are

_given

in

_figure

4. The

_sharp

break observed in the 50 atm data near 152 K is reminiscent of a

_phase

transition.

In

_figure

5 we

_plot

the

_temperature

_dependence

of the difference of the

_density

calculated

from the measured

_capacitance

and the

_density

calculated from the

_equation

of state

_[15].

As

_pointed

out

_before,

_optical

data indicate no

_anomaly

of the Clausius-Mossotti constant.

Thus the

_present

observations indicate that the

_density

of the fluid between the

_capacitor

plates

is

_larger

than in the bulk of the fluid over a small

_region

of the nP T surface above the

Fig.

4. -

Temperature dependence

of the

_density

obtained from the measured dielectric constant

_along

the 50 atm

_(e)

and the 55 atm

_(A)

isobars. Both the full and the broken lines are

_only

a

_guide

to _{the eye.}

Fig.

5. -

film that makes up the

capacitor

plates

is

_sufficiently

_strong

to cause a

_buildup

of the

_density

near the surface.

One must

_enquire

whether the

_present

_{result may be due either} to a thermal inversion or an

effect due to

_{gravity. The latter appears}

_unlikely

because the detected effect increases when

one

_gets

further from the critical

_point.

It is a maximum

_along

the 50 atm isobar about 1.2 K

above

_T,,

while it is a maximum

_along

the 55 atm isobar about 4 K above

_Tc.

A

_temperature

inversion does not seem

likely

either insofar as the

_temperature

measured

_by

a

_thermocouple

tucked under the copper foil 4 cm above the

_sample

chamber measured a

_température

less

than about 1 K above that of the

_platinum

thermometer located below all the electrodes used for the Hall

_mobility

measurements. The

_separation

between both sensors was about 20 cm.

The data were not taken with either

_{monotonically increasing}

or

_{monotonically decreasing}

temperatures.

No

_hysteresis,

nor _{any effects that}

_might

be associated with the

_sample

_history

were detected.

It must also be remarked

_that,

if there would have been a

_significant

_{temperature inversion,}

the fluid in the coldest

_portion

of the

_system

would have been denser than in the

_region

where the

_capacitor

was located.

Therefore,

_contrary

to

_observation,

since the total

_quantity

of the argon in the

system

is constant, a

_{density anomaly corresponding}

to a

_density

lower than the

bulk

_density

should have been observed.

Finally,

if the

_present

observation would have been associated with a bulk

_property,

some

anomaly

would have been seen in the measurement of the electron

_mobility

that has a

_sharp

maximum centered at a

_density

near 1.28 x

1022 cm- 3.

This maximum has been seen

_by

several authors

_{[9, 17-19] and appears}

to be

_{present (at}

different

_densities)

in several

_high

electron

_mobility

_liquids.

Below we

_suggest

a mechanism which would

_produce

the

strong

interaction necessary for the

_growth

of a

_macroscopic

film.

5.

_Ising

model

description.

A realistic calculation of the

_wetting

transition for a fluid is

_beyond

_{the scope of this work.}

Instead,

assuming

that the

_anomaly

observed is indeed caused

_by

_prewetting,

we consider a

simple Ising

model

_description

which includes a

_strong

surface interaction and

_study

some of

its features associated with

_prewetting.

Our modest aim is to find out if the

_experimental

data indicates that the 50 atm isobar intersects the line of

_prewetting

transitions,

whether we can

expect

the

_prewetting

critical

_point

to be above the bulk critical

_point

for the

fluid,

and whether one

_might

_expect

remnants of the

_wetting

transition to _appear as anomalies in the

temperature

dependence

of the

_density

in the

_vicinity

of a metal surface.

Although

the

_Ising

model

_{oversimplifies}

the

_properties

of the fluid and its

_{phase diagram}

has a

_symmetry

that is different from that of a real

_fluid,

it has served well to understand the bulk critical behavior of fluids

_[20].

In

addition,

it has been a model that has shed

_light

on

several

aspects

of surface

_phase

transitions

_{[21, 22].}

We

_picture

the van der Waals interaction between two atoms A and B

_{by envisaging}

a

randomly

fluctuating

dipole

moment p on atom A that induces a

_dipole

moment

pl = a E _{on atom}

B ; a

is the atomic

polarizability.

The average energy of pl in the field

created

_by

_p is

where

_{(... )}

_{denotes average} over the random

angles

of p and r is the distance between A and

In order to calculate the

_binding

of atom A to a metallic surface when A is next to the

surface,

the induced

_dipole

moment _{pI on B used in}

_{obtaining equation}

₍₂₎

must be

_replaced

by

the

_{image dipole}

p’,

which is much

_larger

_{than pj,} at the mirror

_symmetric

_position

A’. In this case, _{the average energy is}

_{given by}

where x is the distance between A and the metallic surface. If we let r = 2 x = a in

equations

(2)

and

_(3),

where a is the hard core radius of the atoms, and

_appropriate

values

[23]

for Ar are

_substituted,

the

ratio U AA’ ) / U AB)

can be as

_large

as 16. It is

appropriate

to

point

out,

however,

that in the solid at 0 K the nearest

_neighbor

distance is 1.11 u. As is seen

in

_figure

2,

the

_density

of the fluid is

_actually

near half of the

_density

at the

_{triple point.}

_{In any}

case,

_UAA-

is

_always

much

_larger

than

_{UAB independent}

of the orientation of the

_dipole

moment p on A. This

_large

surface-atom interaction will be seen below to

_yield

_strong

surface induced transitions. In

_addition,

_higher

order induced

_dipolar

interactions decrease

_by

_powers of

_{a / u 3}

which is about 0.04 for Ar.

In

_formulating

a

_{lattice-gas Ising}

model to

_{approximately}

_represent

this

situation,

we

truncate both the atom-atom and atom-surface interactions at the nearest

_neighbor

distance

cr. As in

_{usual lattice-gas}

models for

studying

the

liquid-gas

critical

behavior,

we

_partition

the

space into

regular

array of cells of size

equal

to the hard core diameter of the atoms and

represent

a cell with an atom

_by

an _{up (+}

_{1 )}

_{Ising spin}

and one without an atom

_by

a down

(- 1 )

spin.

Since we must have a surface in the

_model,

we use the semi-infinite

_simple-cubic

lattice. The crudeness of a

_lattice-gas

model is irrelevant for the so-called universal

_aspects

of

the bulk critical effects and the

_{approximation}

is also

_expected

to be useful in

_giving

qualitative descriptions

of surface induced

_phase

transitions

_[22].

Some of the well known defects of this

_approach

are : an atom

_(s

=

1 )

and _a hole

(s

= -

1 )

are treated

_{symmetrically,}

and pressure is not well

_{represented by}

the

_parameters

of the model. In

addition,

a

_potentially

more serious

_problem

is the truncation of the

_long-range

atom-surface interaction

_{[22, 23]}

this

aspect

will be

_pursued

further in a

_{subsequent study.}

Thus,

our initial model Hamiltonian is

where

and ni = (si + 1 )/2

is the

_occupation

variable

₍₁

or

₀₎

at site i. The bulk term

hB

represents

the chemical

_potential

of the atoms in this

_grand

canonical ensemble. We further

_approximate

this

_{by taking}

the

_{exchange integral}

J in the bulk to

_correspond

to the average energy

_{(- UAB)}

between

_neighboring

atoms and the surface term

_hs

to

_represent

the

dipole-image-dipole average energy

(- Upp-)

for an atom A in the surface

_layer

_(next

to the metallic

_surface).

This is

_already

a kind of mean field

_{approximation}

for the

_fluctuating

dipoles,

but below we will further use the results of a mean field

_{approximation}

on the

_density

variable

_(i.e.

the

_spin

_variables)

in H.

This

_type

of model is well studied

_{theoretically [21, 22]}

and is known to

_give

surface induced

phase

transitions called

_wetting

and

_prewetting

transitions _{for wide range of}

_parameters.

In

layer

of

_{liquid phase}

forms on the surface even

_though

_{the bulk may be in the gas}

_phase

-

a

surface induced

_{phase separation.}

A series of

_prewetting

transitions can occur

_slightly

off

coexistence,

leading

up to the

_wetting

_transition,

_whereby

the surface

_acquires

a

_{high density}

layer suddenly

(the

thickness of which varies

_according

to the distance from the

_wetting

transition and other

_factors).

The

_larger

the surface energy

hs

the

_longer

the line of

_prewetting

transitions between the

_{wetting point}

and the so-called

_prewetting

critical

_{point P,.}

Now consider a

_dipole

_p at A on the surface

_layer.

Then the interaction induced

_by

p between

A

and another atom in the

_layer

is

_{represented by}

a

surface exchange

JI. By considering only

two

_{nearest-neighbor}

sites on the surface

_layer

and

_identifying

the

energies

of the various combinations of the sites

_{being occupied}

or

_empty

from the Hamiltonian

₍₄₎

and those obtained

_directly

from the van der Waals

interactions,

we obtain

where

_primes

on A and B denote the

_image

locations and those on U the fact that we are

_only

considering

interactions

_arising

from the source

_dipole

p on A. Note here that we are

neglecting

all

_higher

order

effects,

including higher multipoles, higher

order induced

_dipoles

such as those

_through

a third atom

_C,

and direct further

_neighbor

interactions. We need

_only

to consider one source

_dipole

_(together

with its

_image,

the moments induced

_by

them,

and their

_images

in

_turn)

since interactions between two

_{independent dipoles}

will

always

average

to zero. We can then write

where,

in

_{principle, ¿;1 depends}

on the orientation of the

_dipole

moment _p. Within the

present

approximation,

however,

it is

_appropriate

to evaluate it

_{by using}

J and

_J,

obtained

_by

averaging

over the orientations of p. This

parameter A

is related to the enhancement

_{factor g}

discussed,

e.g., in reference [21]. Within mean field

_theory

_(and

for a

_simple

cubic

_lattice),

this

_relationship

is :

where J

_depends

on the

_magnitude

of the

_{fluctuating dipole}

moment _p as discussed.

Hence,

within this

_theory

_[22],

if

_A,

= 1/4 the enhancement is sufficient to extend the

_prewetting

transitions above

_7c

for small

_hs.

Strictly

within this

_{approximation,}

it is not difficult to evaluate the value of à for Ar on a

metallic surface. As shown in the

_Appendix,

_however,

the result to the first order in

a

/r3

is :

Thus there is no net enhancement to this order and we cannot _argue on this basis the

possibility

of

_{Pc being}

above

_Te.

It remains to be seen whether

_{including long-range}

effects or

otherwise

_improving

the calculations

_(such

as to include second order terms in

_{a /r3)}

would allow this or not.

From the data in

_figure

_3,

it is not clear whether the lack of

_sharpness

of the

_phase

transition is due to intrinsic

_properties

of the

_prewetting

_transition,

to the

_possibility

that the

_point

Pc

may be below the 50 atm isobar or to

_experimental

errors. As is seen

_comparing

either the data

_{corresponding}

to the 50 atm and the 55 atm

_isobars,

_{the further away} one _{goes from}

TW

the smaller is the difference between the

_density

of the film and the

_density

of the bulk fluid. This trend is also illustrated in the model calculation of Nakanishi and Pincus

_[24]

thickness much

_greater

than

_expected

in such calculations at a

_comparable

distance from the

coexistence curve.

However,

broadened and

_attenuated,

remnants of the transition remain at

_temperatures

above that

_{corresponding}

to

_Pc,

_{e.g. what is} seen

_along

the 55 atm isobar. This effect can be

demonstrated

_by

calculations based on a

_{simple Ising}

model described

_{above ;}

an illustration

is

_provided

in

_figure

6. Shown in this

_figure

are the results from the continuum version

[22, 24]

of the

_lattice-gas

model described earlier. The vertical axis is the surface excess

_Tper

unit area

of surface dx where m is the continuum version of the

_{Ising spin s}

(m

= 1

corresponding

to the nominal saturation

density)

and where the unit of

length

x is the

nearest-neighbor

distance. The horizontal axis is the reduced

temperature t

=

(T -

Tc)/Tc.

The

_geometrical

parameters

chosen for this

_plot

are

_appropriate

for a

_simple

cubic lattice and

the surface dimensionless

_{parameters g}

and

_{hl (see}

Refs.

_[22]

and

_[24]

for

_notation)

are

ârbitrarily

chosen to be

_1/6,

so that the

_prewetting

critical

_{point Pc}

occurs at about

m = -

0.7, t

=

0.1,

and h = - 0.18.

(The

terms

hl

and h _are

essentially

like

hs/2 kB T

and

hB/2

kB T.)

Since the bulk field

_parameter

for this

_figure

of h = - 0.185 is

slightly

less than

that for

_Pc,

there is no

_{sharp prewetting}

transition in this

figure ;

_yet

there is

clearly

a remnant

of the transition in the form of a broad maximum.

Fig.

6. -

Temperature

dependence

of surface excess

_density

_{theoretically}

calculated from a continuum Landau theoretical version

_(see

Refs.

_{[22, 24])}

of the

_{short-range lattice-gas}

model described in this paper. The format of this

figure

is

_analogous

to that of

_figure

5. The _parameters chosen for this

plot

are

described in text. The bulk field value chosen makes this

_graph

not to cross the line of

_prewetting

transitions. Note that the

_{high adsorption}

occurs on the

_high

_temperature side in this calculation in

contrast to the

_experimental

case where it should occur at the low temperature side. This difference is attributed to the difference in the theoretical

_path

_(h

=

constant )

and the

expérimental path

(pressure

=

constant)

and the defects associated with the forced _symmetry of the

_Ising

model.

6.

_Summary.

In

_conclusion,

the

_experimental

data indicates the existence of some sort of

_phase

transition

slightly

above the critical

_point

that is reminiscent of the theoretical

_expectations

of the

between an _argon atom and a metal surface

_compared

with the interaction between two _argon atoms in the bulk has been

_proposed.

However,

a

_{corresponding plausible}

mechanism for an

enhancement of the interaction between two atoms on the metallic surface does not _appear to

arise from the

_short-range,

_lattice-gas

model described here.

Acknowledgments.

This work has been

_supported

in

_part

_by

_grant

DE-FG02-84ER13212 from the United States

Department

of

_Energy.

_{Acknowledgment}

is made also to the Donors of The Petroleum Research

Fund,

administered

_by

American Chemical

_Society,

for the

_partial

_support

of this

research. The authors are

_grateful

to Mr. John Koenitzer for the calibration of the Pt resistance thermometer.

Appendix.

Calculation of

_Jl

·

To calculate

à,

first note that on account of

_{symmetry UÁB’ > = UÁ’B>.}

To first order in

a

Ir 3,

UÁ’B

is

actually

the sum of two contributions : the interaction

_Ul

between the

image

dipole p’

at A’ and the

_dipole

_(p’l

induced at B

_{by p’,}

and the interaction

_U2

between

p’

and the

_dipole

_p, induced at B

_{directly by}

p. Thus we

_only

need to

_compute

two

_quantities.

We must be

careful, however,

since not all

_dipole

moments in

_question

are

_coplanar

with each

other. That

is,

while p and

p’

are

_coplanar

and so are

_p’

and

(P’)¡,

or p and pI, the

planes

are not common in

_general.

Consider a Cartesian coordinate

_system

with the x-axis

_along

A’A,

and the z-axis

parallel

to

BA. The metal surface is the

_yz-plane.

Denote the

_{polar angle}

of p from the z-axis

by

9,

and

by 0

the azimuthal

_angle

made

_by

the

_projection

of

p into

the

_xy-plane

from the x-axis

_(see

Fig.

7).

Fig.

7. -

System

of coordinates used to calculate the additional surface interaction term. The

_dipole

p at

A

_produces

an

_image

_p’

at A’ and induces a

_dipole

_PI = a E _at B. The

image

of PI is

(pj 1 ’

at B’. The contribution

_U2

to the term

_UÁ’B

is the interaction

_{of p’}

_{and PI}

_{(or equivalently}

that

_{of p}

and

_{(pi )’).}

The contribution

Ul

to

_UÁ’B’

_on the other hand, is the interaction

of p’

and its induced

_dipole

(p’l

at B

_{(or equivalently}

that of p and the

image

of

_{(p 1 ).).}

Both

_(p’l

and its

_image

are left out of the

In this

system,

we have

and

The first contribution

_Ul

is

_simply

in

_analogy

to

_equation

₍₂₎

for the direct interaction. When r = 2 x = _cr, this

gives

The second term

_U2>

is calculated from

where R is

where r is the nearest

_neighbor

distance AB and 2 x is BB’.

_Substituting

(A.2), (A.3)

and

(A.7)

into

_(A.6)

and

_{taking angular}

_averages over 0 and

_cp,

and further

letting

r = 2 x = u, we obtain

Therefore,

from

_(2),

_{(7), (A.4)}

and

_(A.8),

we obtain

Note that the

_sign

of à in this

_{approximation}

is

_negative,

which indicates that there is no net

enhancement in

_JI

and rather there is a 4.5 % decrease.

Thus,

this calculation does not allow for the

_prewetting

critical

_point

_{Pc to lie}

above

_Tc.

This effect is

_largely

due to the

misalignment

on _{average between the}

_image

and the induced

_dipoles

(or

_{equivalently,}

of the

image

of the induced

_dipole

relative to the source

_dipole).

Some of the second order terms in

a

Ir3are

_clearly

_{attractive and may} turn this

_slight

decrease into an enhancement.

However,

it is difficult to

_imagine

an enhancement as

_large

as 25 % as

_required

_by

mean field

theory

described earlier

_{simply by including higher}

order terms.

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We may compare this behaviour with the so called

capillary

condensation. In the latter case, a

liquid phase,

for

example,

fills a

_capillary

rather than

_creating

interfaces between the

_liquid

and the undersaturated gas. The

capillary

condensation in this sense, is the result of a bulk like

term _{in the free energy that competes}

_against

surface like terms. Such a

_description

is

only

appropriate

when the

plate

separation

is

_relatively

small

_(see

_{e.g. EVANS R. and MARCONI U.,}

Phys.

Rev. A 32

_{(1985) 3817).}

In the _present

_experiment

the

separation

of the

plates

is of the order of 0.1 mm ; the

phenomena

are more

_{appropriately}

described

_by

a semi infinite

geometry, and the

_resulting

transition is a

_{prewetting phenomenon.}

Of course, in a _proper

treatment of

_{spatially varying density}

for a finite

_{plate separation,}

it is

_expected

that the

prewetting

transition is no

_longer

a true

_phase

transition. In this case, there remains

only

the shifted

_(and

lower

_dimensional)

bulk

_phase

transition which could be

interpreted

as a

generalized

form of

_capillary

condensation

_(see

NAKANISHI H. and FISHER M. E., J. Chem.