Compiled (020.1475) Bose-Einstein condensates; (020.1335) Atom optics;
(110.6760) Talbot and self-imaging effects

# Talbot-enhanced, maximum-visibility imaging of condensate interference

Y. Zhai  Department of Physics, SUPA, University of Strathclyde, Glasgow G4
0NG, United Kingdom  C. H. Carson  Department of Physics, SUPA, University of
Strathclyde, Glasgow G4 0NG, United Kingdom  V. A. Henderson  Department of
Physics, SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom  P.
F. Griffin  Department of Physics, SUPA, University of Strathclyde, Glasgow G4
0NG, United Kingdom  E. Riis  Department of Physics, SUPA, University of
Strathclyde, Glasgow G4 0NG, United Kingdom  A. S. Arnold

###### Abstract

Nearly two centuries ago Talbot first observed the fascinating effect whereby
light propagating through a periodic structure generates a ‘carpet’ of image
revivals in the near field. Here we report the first observation of the
spatial Talbot effect for light interacting with periodic Bose-Einstein
condensate interference fringes. The Talbot effect can lead to dramatic loss
of fringe visibility in images, degrading precision interferometry, however we
demonstrate how the effect can also be used as a tool to enhance visibility,
as well as extend the useful focal range of matter wave detection systems by
orders of magnitude. We show that negative optical densities arise from
matter-wave induced lensing of detuned imaging light – yielding Talbot-
enhanced single-shot interference visibility of $>135\%$ compared to the ideal
visibility for resonant light.

###### doi:

http://dx.doi.org/10.1364/optica.XX.XXXXXX

##  1 introduction

Figure 1: Matter wave interference. Experimental absorption image (a,
$1.4\times 1.4\,$mm2) of a condensate axially split by a blue-detuned optical
dipole beam at $160\,$ms levitation time, scalebar indicates optical density
(OD). Background-subtracted, angle-corrected, row-averaged OD profile (b, red
points) of the red rectangular region in (a). The fitted sine wave has a
parabolic spatial envelope and offset (gray curve) for fringe visibility and
period extraction. Fringe period as a function of levitation time (c):
experimental data (black dots); inferred fringe period evolution (blue curve,
Eq. 2 with $\omega_{z}=13.8\pm 1.4\,$rad/s and $d=45\,\mu$m); ballistic
expansion theory (red line, Eq. 1).

Since the first experimental realisation of gaseous Bose-Einstein condensates
(BECs) [1, 2] the field has grown dramatically and condensates are now used
for a variety of studies including the flow of superfluids [3], studies of
strongly dipolar systems [4, 5], and for producing atom-lasers [6]. Matter-
waves are also ideal for quantum technologies, a key ingredient of which is
atom interferometry. Interferometers yield precise measurements of fundamental
constants [7], rotation [8, 9], acceleration [10], gravity [11, 12] and
gravitational shifts [13]. Interferometry is also a key method for studying
the superfluid states prevalent in quantum degenerate systems [14, 15, 16, 17,
18, 19].

BECs are fully coherent matter wave sources, as first strikingly demonstrated
in Ref. [20]. A cigar-shaped single-well potential was transformed into a
double-well by axial separation with an optical dipole beam. Ballistic
expansion led to wavepacket spatial interference, reminiscent of Young’s
double slits. We use a similar configuration [21], but a simpler levitation-
enhanced imaging technique, achieving improved visibility of $80\%$ (Fig. 1)
above our previous record of $60\%$ by longer levitation and a weaker dipole
beam. BEC interference has also been demonstrated via radial [22, 23, 24, 25,
26] or axial [27, 28] separation using a wide variety of other techniques.

During our BEC interference investigations we observed that the fringe
visibility strongly depended on the camera focal location, which we can now
clearly attribute to the Talbot effect. Talbot’s 1836 observation [29] of
self-imaging of a periodic structure in near-field diffraction arises due to
wave optics [30]. The phenomenon has since been studied in a wide variety of
scientific disciplines including optics, acoustics, electron microscopy,
x-rays and plasmonics [31, 32]. Transversely coherent waves with wavelength
$\lambda$ passing through a regular phase/intensity structure with period
$\lambda_{f}$ produce a self-image with a phase of $\pi$ $(0)$ at odd (even)
integer multiples of the Talbot distance
$\Lambda_{\textrm{T}}={\lambda_{f}}^{2}/\lambda$ [33]. Fractional and fractal
Talbot effects arise if the grating structure contains higher spatial
harmonics [34].

With matter waves the spatial Talbot effect has been observed from hot atomic
beams [35, 36] and the temporal Talbot effect has been seen with BECs in 1D
optical lattices [37, 38]. The effect is also an important trigger for
spontaneous spatial light/matter pattern formation in cold atomic gases [39].
Although the regular period and phase of BEC interference patterns are
extremely important for metrology, the dramatic influence of the Talbot effect
on the propagation of near-resonant light used for imaging condensates has not
been previously seen.

Here we present the first observation of spatially periodic visibility
variation in a BEC interferometer due to the Talbot effect. Imaging light is
absorbed and refracted by $N$ BEC interference fringes, then propagates
through the imaging system. At integer Talbot distances
$i\Lambda_{\textrm{T}}$ ($|i|\lesssim N$) from the central image plane the
diffracted light dramatically re-images to $N-|i|$ fringes (Fig. 2).
Furthermore, by detuning the imaging light, negative optical densities [40]
arise leading to enhanced visibility, nearly doubling the resonant value.

Our results will be of particular relevance to atom interferometry experiments
with several fringes ($N>5$), as opposed to those with only a few fringes or
an overall population amplitude variation. Multi-fringe interferometers may
have been optimised in the past based on high visibility, finding only a local
maximum. In situations where the fringe period is critical, an assumption of
only one image plane could lead to erroneous results (cf. Fig. 2e). In the
limit of short period fringes, with period around one wavelength, the Talbot
distance of order $1\,\mu$m would lead to extreme difficulty to position the
camera focal plane, with corresponding sensitivity to drift. This would be
particularly important if e.g. single-atom absorption imaging [41] were used
in a quantum gas microscope [42, 43].

##  2 experimental setup and fringes

In our experimental setup [21, 44], a few $10^{8}$ 87Rb atoms in the weak-
field-seeking state $|F=2,m_{F}=2\rangle$ are trapped at the top of a magnetic
storage ring [44]. The Ioffe-Pritchard (IP) trap has axial and radial trapping
frequencies of $10\,$Hz and $90\,$Hz, respectively, and a lifetime of $55\,$s.
After $30\,$s RF evaporation, condensates with $5\times 10^{5}$ atoms are
created and observed using standard resonant absorption imaging on the
stretched $D_{2}$ transition
$|F=2,m_{F}=2\rangle\rightarrow|F^{\prime}=3,m_{F}=3\rangle$, with
magnification $\times 2$. A $658\,$nm blue-detuned repulsive dipole beam is
used for axial BEC splitting with $<1\,$mW derived from a diode laser with an
acousto-optic modulator focused to a $10\,\mu$m waist [21].

Split BECs are released by rapidly switching off the optical and magnetic
potential and they expand in space for sufficiently long that the final atomic
cloud size is much larger than the initial distribution. Thus even initial
spatial distributions with some broken symmetry behave like two matter wave
point sources. This leads to clear ‘Young’s-slit’ type planar interference
patterns in 3D, (Fig. 1a,b), obviating the need for specialised tomographic
imaging [20].

The BEC fringe period $\lambda_{f}$ comes from the de Broglie wavelength of
the two condensates and can be expressed as,

$\lambda_{f}=\frac{ht}{md}$ (1)

where $h$ is Planck’s constant, $t$ the ballistic expansion time, $m$ the
atomic mass and $d$ is the initial BEC centre-of-mass separation. The free
fall time is limited to $0-30\,$ms by the size of the imaging area, and times
$<60\,$ms by the physical extent of the vacuum cell. To overcome these
limitations we use a magnetic levitation field [1, 45, 46] derived from a
toroidal quadrupole [44] offset by an additional vertical constant field. BECs
therefore experience a weak inverted parabola potential
($U_{z}=-m{\omega_{z}}^{2}z^{2}/2$ with $\omega_{z}$ the angular velocity
along the $z$ direction i.e. the BEC axis) due to the circular nature of the
levitation coils, which we showed in Ref. [21] modifies $\lambda_{f}$ in Eq. 1
to:

${\lambda_{f}}^{\prime}=\lambda_{f}\frac{\sinh({\omega_{z}}t)}{{\omega_{z}}t}.$
(2)

Although our initial BEC separation $d$ is large in comparison to other
experiments, the levitation potential we apply magnifies the spatial scale of
our fringes [21, 17] (Fig. 1c). To achieve the equivalent maximum fringe
period by ballistic expansion we would need $800\,$ms i.e. $3\,$m of free-
fall. To resolve the fringes levitation times $\geq 70\,$ms are needed until
the period corresponds to two or more camera pixels ($\geq 10\,\mu$m).

Using resonant light absorption imaging fringe periods of $37.5(0.8)\,\mu$m
are observed, with the highest visibility seen in this type of interferometer
- 80% (Fig. 1a,b) - close to the theoretical limit due to camera pixellation
of the fringes. Depending on the timing and power of the dipole beam
application either two phase-separate condensates are created or a single
condensate can be split into two phase-coherently [47, 22]. The size of our
interference pattern in comparison to our centre-of-mass uncertainty currently
prevents us from ascertaining whether the relative phases of our split
condensates are random or correlated, however the main results are unaffected.

##  3 talbot effect: theory and experiment

The effect of imaging light interacting with periodic BEC fringes can be
modelled by Fourier-propagating the light waves, based on the initial
conditions of the field immediately after the condensate, then performing the
inverse Fourier transform to convert back to real space [48, 49, 50, 51].
Propagation effects inside the BEC are neglected as the condensate size in the
beam propagation direction is considerably smaller than the Talbot period.

Figure 2: Theoretical light propagation through BEC fringes. a-c Ten BEC
fringes along $x=0$ of peak optical density OD${}_{0}=0.4$ and visibility
$80\%$ interact with resonant light leading, on propagation $(x>0)$, to
periodic phase (a in radians) and OD (b) profile revivals. Antinodes of OD
(red points) correspond to nodes of the phase, and vice versa. The central
fringe at $z=0$ is displayed (c) for red, resonant (black) and blue laser
detunings of -$\Gamma/2$, 0, and $\Gamma/2$, illustrating how detuning not
only phase-shifts the fringes in $x$, but can enhance visibility relative to
the initial optical density. After the light in images a, b and c propagates
through a $300\,$mm long $\times 2$ magnification imaging system, the
corresponding phase, OD and visibility is shown in d, e and f respectively.
Note the chirp in the spatial period about the image plane (white dashed line
in d,e). Figure 3: a Single-shot 1D BEC OD profile (21 row average,
$\Delta=+4\,$MHz), with visibility $C=1.5$ from fitting data points (black)
with Eq. 4 (green curve). b Experimental (dots) and theoretical (curves from
Fig. 2e) visibility as a function of camera position for detunings
$-4,0,+4\,$MHz (red, black and blue, respectively). All error bars in this
paper represent the standard deviation of results from three separate images.
The central Talbot period was fit to be $6.1(0.1)\,$mm, which agrees well with
the theoretical Talbot period of $6.4(0.3)\,$mm predicted from the fringe
period and Eq. 5.

When light passes through the BEC fringes, the light attenuation and phase
shift depend on the light detuning and the integrated atomic density. In the
standard absorption imaging we use, the light passing through the atomic cloud
indicates a transversely spatially dependent optical density

$\textrm{OD}=\ln\left(\frac{I_{i}-I_{b}}{I_{e}-I_{b}}\right),$ (3)

where $I_{i}$, $I_{e}$ and $I_{b}$ are the light intensity distributions
incident on the atoms, exiting the atoms, and due to background light,
respectively. For low intensity light the optical density in the BEC pattern
varies with detuning as
$\textrm{OD}=\textrm{OD}_{0}/(1+4(\Delta/\Gamma)^{2})$, where $\Delta$ is the
light detuning from the atomic resonance, OD0 the (resonant) peak optical
density and $\Gamma$ the natural linewidth. In our system $\Gamma$ is power-
broadened from $6$ to 8 MHz. The light phase shift due to the BEC is given via
the Kramers-Kronig relation, i.e. $\delta\phi=-\textrm{OD}\,\Delta/\Gamma,$
with a maximum phase shift at $\Delta=\pm\Gamma/2.$ Under conditions similar
to Fig. 1b, with $\textrm{OD}_{0}=0.4$ and $\Delta=0\,$MHz, in Fig. 2 we
investigate the theoretical fringe behaviour for ten BEC fringes with
$\textrm{OD}_{0}=0.4$ and visibility $80\%$ observed with on-resonant imaging,
as well as the key effect of changing the imaging beam detuning (Fig. 2).

By taking $z$ slices through the Talbot optical density ‘carpet’ in Fig. 2b,e
the visibility as a function of $x$ for imaging detuning $\Delta=0\,$MHz can
be obtained, and compared to the behaviour for detunings $\Delta=\pm 4\,$MHz
(Fig. 2 c,f). The model we use to fit fringes at a given $z$ is a sine wave of
visibility $C$, modulated by a parabolic amplitude to incorporate the typical
experimental behaviour of the fringes due to their origin from the overlap of
two initially spatially separated clouds:

$(A_{0}+A_{2}(z-z_{0})^{2})(C\sin(kz+\phi)+1).$ (4)

Visibilities $C>1$ occur when the absorption imaging beam shows regions of
negative optical density – where more light is in the beam after it passes
through the BEC fringes than before – a single-shot experimental example of
which is shown in Fig. 3a with $C\approx 150\%$. Fig. 3b presents the
corresponding spatial Talbot experimental visibility data to Fig. 2e, at the
same detunings, for imaging light interacting with periodic BEC fringes. For
both positive and negative detuned light the experimental and theoretical
visibility agree well and are enhanced from $\leavevmode\nobreak\ 80\,$% to
$135\,$% (Fig. 3b). This visibility enhancement for detuned light is due to
the phase variation in the initial conditions converting reversibly into
intensity variation upon propagation. The enhancement can also be interpreted
as focusing since the sinusoidal spatial phase variation is like a collection
of alternating positive and negative lenses.

Figure 4: Talbot period variation with fringe period. Visibility is observed
as a function of camera position for $\Delta=+0.5\,$MHz imaging light after
140, 160 and $170\,$ms levitation. From fits to these data Talbot periods were
measured to be $3.0(0.1)\,$mm, $6.1(0.2)\,$mm and $7.6(0.2)\,$mm,
respectively, which agree with the theoretical Talbot periods of
$3.4(0.1)\,$mm, $6.3(0.2)\,$mm and $8.1(0.8)\,$mm determined from the
experimental fringe periods associated with the $140$, $160$ and $170\,$ms
levitation times. Inset optical density plots ($320\times 280\,\mu$m2) show
the observed visibility changing through the Talbot period.

The variation of the observed fringe visibility as a function of the camera
position can be attributed to the Talbot effect. The magnification of the
imaging system modifies the Talbot period, which is therefore given by

$\Lambda=\frac{(M\lambda_{f})^{2}}{\lambda}\leavevmode\nobreak\ ,$ (5)

for light of wavelength $\lambda$ incident on a periodic density/phase
modulation of period $\lambda_{f}$ where $M$ is the camera magnification. With
the camera position at $13\,$mm, the observed fringes periods on the CCD for
on-resonance light in Fig. 3 were $70.9(1.5)\,\mu$m. The corresponding Talbot
periods where measured to be $6.1(0.1)\,$mm, which is in good agreement with
the theoretically predicted Talbot period of $6.4(0.3)\,$mm.

To verify that the Talbot effect was at work we changed the period of the
interference pattern, thus changing the Talbot period. The fringe period was
altered by using different levitation times (Fig. 1c), $140$ and $170\,$ms
were chosen as they would result in a notable change in the observed Talbot
period. The fringe period under the experimental conditions in Fig. 4, with
the camera at $13\,$mm and $\Delta=+0.5\,$MHz, were measured to be
$51.2(0.3)\,\mu$m and $79.5(3.7)\,\mu$m for 140 and $170\,$ms respectively.
The measured Talbot periods were found to be $3.0(0.1)\,$mm for $140\,$ms and
$7.6(0.2)\,$mm for $170\,$ms, which are also in good agreement with the
theoretical Talbot periods of $3.4(0.1)\,$mm and $8.1(0.8)\,$mm for 140 and
$170\,$ms respectively. Fig. 4 also presents density plots of the interference
fringes at the peak, middle and trough of a Talbot period, showing the drastic
effect the spatial Talbot effect can have on the interference signal.

##  4 optimal visibility

Figure 5: Interference visibility as a function of levitation time and imaging
detuning. Red circles, black squares and blue triangles denote detunings
$\Delta=-4\,$MHz, $0\,$MHz and $+4\,$MHz, respectively. For comparison theory
curves are provided of optimal visibility using Eq. 6 with pixellation-free
final visibility values of $C=0.8,$ 1.0 and 1.3, shown by the black, green and
cyan curves, respectively. The $C$ values 0.8 and 1.3 are chosen to
approximately match the theoretical and experimental values from Figs. 2 and
3, where $\textrm{OD}_{0}=0.4$.

In Fig. 5 fringe visibility vs. expansion time is shown for both experiment
and theory, with experimental data reaching the highest visibility reported in
this type of interferometer. Note that for each detuning in Fig. 5 the data at
different levitation time were selected from the peak value within one Talbot
period. For comparison to our experimental visibility vs. levitation time
results shown in Fig. 5, we first consider the theoretical maximum visibility
that can be reached. Assuming two perfect matter wave point sources,
sinusoidal interference fringes result of the form $F(x)=1+C\sin(2\pi
x/\lambda_{f})$, where $\lambda_{f}$ is the fringe period, and $C$ is the
visibility. When these fringes hit the $j^{\textrm{th}}$ pixel of a perfect
CCD camera, with pixel size $l$, the fringe signal averaged over the range
$(j-1/2)l\leq x\leq(j+1/2)l$ is

$F_{\textrm{CCD}}=1+\textrm{sinc}(\pi l/\lambda_{f})C\sin(2\pi
jl/\lambda_{f}),$ (6)

i.e. the original fringe visibility $C$ reduces to
$C^{\prime}=\textrm{sinc}(\pi l/\lambda_{f})C$. In our experiment $l=5\,\mu$m
is constant, and the visibility only depends on the time-dependent fringe
period – which we know from the fit to Eq. 2 in Fig. 1c. This allows us to
determine the theoretical maximum visibility sinc curves shown in Fig. 5,
which for levitation times after 120 ms reaches visibilities more than $0.95C$
and is surprisingly similar to experimental data. This is quite remarkable as
in the theory we assumed perfect matter wave point sources without any
allowance for our non-tomographic imaging which is susceptible to any spatial
asymmetry in the source BECs and hence 3D curvature of the fringes (cf. Ref.
[20]). The reason for a resonant contrast $<100\%$ is attributed to either a
slight angle between the fringe planes and the imaging beam k-vector, or
population asymmetry in the split BECs – however it is still the highest
resonant visibility seen in this type of interferometer.

##  5 conclusions

We report the first observation of the spatial Talbot effect of imaging light
interacting with periodic BEC interference fringes. This interpretation is
confirmed by the strong agreement between the Talbot lengths predicted by the
fringe period and the measured Talbot periods from the experimental data. We
have shown that the spatial Talbot effect can have drastic effects on the
interference signal, and is therefore relevant for all BEC interferometers.
More importantly, this effect can then be used as a tool, or diagnostic, to
focus interference experiment imaging systems and help obtain the maximum
possible visibility. By using detuned imaging light ($\pm 4\,$MHz) the Talbot
effect converts the imaging light periodic phase shift from the BEC fringes
into an intensity ripple, which results in an increase in the observed fringe
visibility. This enhancement in visibility allows us to observe single-shot
interference visibility (as defined in Eq. 4) of $C>135\,$%, which is the
highest ever reported in this type of interferometer. Such a contrast is not
unphysical because the Talbot effect redistributes light into different parts
of the absorption imaging beam – leading to some beam regions with increased
power, i.e. negative optical densities. The limit to attaining even higher
visibility is probably due to the underlying $80\%$ resonant visibility. This
is either due to a slight angle between the imaging beam and the fringes, or
population imbalance in the split BEC – we have shown that it is unlikely to
be due to CCD camera pixellation.

Talbot-enhanced interferometers will be particularly advantageous in
situations where the fringe period/camera pixel size ratio is small (or even
less than 1), as the longer Talbot period can be used to accurately infer the
matter wave fringe period. Moreover, by appropriately tailoring the incident
imaging light to have variable focal distance (e.g. with a variable focus lens
[52]), the entire 2D Talbot carpet (Fig. 2e) can be mapped out automatically.
With the modification of varying imaging light propagation direction (e.g.
with crossed AOMs [53, 54, 55]) and a single-pixel detector [56], this would
extend the scheme to short imaging wavelength detection and enable single shot
2D Talbot mapping. Access to the full 2D Talbot carpet would enable more
accurate phase determination of the fringes.

An advantage of our axial inverted parabolic levitation potential is that even
in a compact vacuum system, with large initial condensate separation, it
exponentially magnifies fringe periods to the same level seen in drop-tower
[12] and $10\,$m vacuum experiments [57], however a caveat is that samples in
the potential are also exponentially sensitive to initial position and
velocity. In future we will enable accurate metrology with the interferometer
by reducing our rms centre-of-mass noise. To further increase future
interferometer phase accumulation time, the split BEC matter waves could be
guided in a waveguide [58] or ring trap [59, 44, 60, 61, 62], and delta
kicking could be used to minimise atomic velocity spread [57, 63, 64].

The dataset for this paper is available online [65].

## Funding Information

The Leverhulme Trust (RPG-2013-074), DSTL (DSTLX-1000095638R), and EPSRC
(EP/M013294/1).

## Acknowledgments

We greatly appreciate valuable discussions with T. Ackemann, G.-L. Oppo, G. R.
M. Robb and W. J. Firth.

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010001 2009 S. A. Cannas 010001

The adsorption of Ar on substrates of Li is investigated within the framework
of a density functional theory which includes an effective pair potential
recently proposed. This approach yields good results for the surface tension
of the liquid-vapor interface over the entire range of temperatures, $T$, from
the triple point, $T_{t}$, to the critical point, $T_{c}$. The behavior of the
adsorbate in the cases of a single planar wall and a slit geometry is analyzed
as a function of temperature. Asymmetric density profiles are found for fluid
confined in a slit built up of two identical planar walls leading to the
spontaneous symmetry breaking (SSB) effect. We found that the asymmetric
solutions occur even above the wetting temperature $T_{w}$ in a range of
average densities $\rho^{*}_{ssb1}\leq\rho^{*}_{av}\leq\rho^{*}_{ssb2}$, which
diminishes with increasing temperatures until its disappearance at the
critical prewetting point $T_{cpw}$. In this way a correlation between the
disappearance of the SSB effect and the end of prewetting lines observed in
the adsorption on a one-wall planar substrate is established. In addition, it
is shown that a value for $T_{cpw}$ can be precisely determined by analyzing
the asymmetry coefficients.

# Correlation between asymmetric profiles in slits and standard prewetting
lines

Salvador A. Sartarelli  [inst1] Leszek Szybisz[inst2]-[inst4] E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(17 June 2009; 28 August 2009)

††volume: 1

99 inst1 Instituto de Desarrollo Humano, Universidad Nacional de General
Sarmiento, Gutierrez 1150, RA–1663 San Miguel, Argentina. inst2 Laboratorio
TANDAR, Departamento de Física, Comisión Nacional de Energía Atómica, Av. del
Libertador 8250, RA–1429 Buenos Aires, Argentina. inst3 Departamento de
Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires,
Ciudad Universitaria, RA–1428 Buenos Aires, Argentina. inst4 Consejo Nacional
de Investigaciones Científicas y Técnicas, Av. Rivadavia 1917, RA–1033 Buenos
Aires, Argentina.

##  1 Introduction

The study of physisorption of fluids on solid substrates had led to very
fascinating phenomena mainly determined by the relative strengths of fluid-
fluid ($f$-$f$) and substrate-fluid ($s$-$f$) attractions. In the present work
we shall refer to two of such features. One is the prewetting curve identified
in the study of fluids adsorbed on planar surfaces above the wetting
temperature $T_{w}$ (see, e.g., Pandit, Schick, and Wortis [1]) and the other
is the occurrence of asymmetric profiles of fluids confined in a slit of
identical walls found by van Leeuwen and collaborators in molecular dynamics
calculations [2, 3]. It is known that for a strong substrate (i.e., when the
$s$-$f$ attraction dominates over the $f$-$f$ one) the adsorbed film builds up
continuously showing a complete wetting.In such a case, neither prewetting
transitions nor spontaneous symmetry breaking (SSB) of the profiles are
observed, both these phenomena appear for substrates of moderate strength.

The prewetting has been widely analyzed for adsorption of quantum as well as
classical fluids. A summary of experimental data and theoretical calculations
for 4He may be found in Ref. [4]. Studies of other fluids are mentioned in
Ref. [5]. These investigations indicated that prewetting is present in real
systems such as 4He, H2, and inert gases adsorbed on alkali metals.

On the other hand, after a recent work of Berim and Ruckenstein [6] there is a
renewal of the interest in searching for the SSB effect in real systems. These
authors utilized a density functional (DF) theory to study the confinement of
Ar in a slit composed of two identical walls of CO2 and concluded that SSB
occurs in a certain domain of temperatures. In a revised analysis of this
case, reported in Ref. [7], we found that the conditions for the SSB were
fulfilled because the authors of Ref. [6] had diminished the $s$-$f$
attraction by locating an extra hard-wall repulsion. However, it was found
that inert gases adsorbed on alkali metals exhibit SSB. Results for Ne
confined by such substrates were recently reported [8].

The aim of the present investigation is to study the relation between the
range of temperatures where the SSB occurs and the temperature dependence of
the wetting properties. In this paper we illustrate our findings describing
the results for Ar adsorbed on Li. Previous DF calculations of Ancilotto and
Toigo [9] as well as Grand Canonical Monte Carlo (GCMC) simulations carried
out by Curtarolo et al. [10] suggest that Ar wets Li at a temperature
significantly below $T_{c}$. So, this system should exhibit a large locus of
the prewetting line and this feature makes it very convenient for our study as
it was already communicated during a recent workshop [11].

The paper is organized in the following way. The theoretical background is
summarized in Sec. 2. The results, together with their analysis, are given in
Sec. 3. Sec. 4 is devoted to the conclusions.

##  2 Theoretical background

In a DF theory, the Helmholtz free energy $F_{\rm DF}[\rho({\bf r})]$ of an
inhomogeneous fluid embedded in an external potential $U_{sf}({\bf r})$ is
expressed as a functional of the local density $\rho({\bf r})$ (see, e.g.,
Ref. [12])

$\displaystyle F_{\rm DF}[\rho({\bf r})]$ (1) $\displaystyle=$
$\displaystyle\nu_{\rm id}\,k_{B}\,T\int d{\bf r}\,\rho({\bf
r})\,\\{\ln[\Lambda^{3}\rho({\bf r})]-1\\}$ $\displaystyle+$
$\displaystyle\int d{\bf r}\,\rho({\bf r})\,f_{\rm HS}[\bar{\rho}({\bf
r});d_{\rm HS}]$ $\displaystyle+$ $\displaystyle\frac{1}{2}\int\int d{\bf
r}\,d{\bf r\prime}\,\rho({\bf r})\,\rho({\bf r\prime\prime})\,\Phi_{\rm
attr}(\mid{\bf r}-{\bf r\prime}\mid)$ $\displaystyle+$ $\displaystyle\int
d{\bf r}\,\rho({\bf r})\,U_{sf}({\bf r})\;.$

The first term is the ideal gas free energy, where $k_{B}$ is the Boltzmann
constant and $\Lambda=\sqrt{2\,\pi\,\hbar^{2}/m\,k_{B}\,T}$ the de Broglie
thermal wavelength of the molecule of mass $m$. Quantity $\nu_{\rm id}$ is a
parameter introduced in Eq. (2) of [13] (in the standard theory it is equal
unity). The second term accounts for the repulsive $f$-$f$ interaction
approximated by a hard-sphere (HS) functional with a certain choice for the HS
diameter $d_{\rm HS}$. In the present work we have used for $f_{\rm
HS}[\bar{\rho}({\bf r});d_{\rm HS}]$ the expression provided by the nonlocal
DF (NLDF) formalism developed by Kierlik and Rosinberg [14] (KR), where
$\bar{\rho}({\bf r})$ is a properly averaged density. The third term is the
attractive $f$-$f$ interactions treated in a mean field approximation (MFA).
Finally, the last integral represents the effect of the external potential
$U_{sf}({\bf r})$ exerted on the fluid.

In the present work, for the analysis of physisorption we adopted the ab
initio potential of Chismeshya, Cole, and Zaremba (CCZ) [15] with the
parameters listed in Table 1 therein.

###  2.1 Effective pair attraction

The attractive part of the $f$-$f$ interaction was described by an effective
pair interaction devised in Ref. [5], where the separation of the Lennard-
Jones (LJ) potential introduced by Weeks, Chandler and Andersen (WCA) [16] is
adopted

$\displaystyle\Phi^{\rm WCA}_{\rm attr}(r)$ (4) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}-\tilde{\varepsilon}_{ff}\;,&r\leq
r_{m}\\\
4\tilde{\varepsilon}_{ff}\biggr{[}\left(\frac{\tilde{\sigma}_{ff}}{r}\right)^{12}-\left(\frac{\tilde{\sigma}_{ff}}{r}\right)^{6}\biggr{]}\;,&r>r_{m}\;.\end{array}\right.$

Here $r_{m}=2^{1/6}\tilde{\sigma}_{ff}$ is the position of the LJ minimum. No
cutoff for the pair potential was introduced. The well depth
$\tilde{\varepsilon}_{ff}$ and the interaction size $\tilde{\sigma}_{ff}$ are
considered as free parameters because the use of the bare values
$\varepsilon_{ff}/k_{B}=119.76$ K and $\sigma_{ff}=3.405$ Å overestimates
$T_{c}$.

So, the complete DF formalism has three adjustable parameters (namely,
$\nu_{id}$, $\tilde{\varepsilon}_{ff}$, and $\tilde{\sigma}_{ff}$), which were
determined by imposing that at $l$-$v$ coexistence, the pressure as well as
the chemical potential of the bulk $l$ and $v$ phases should be equal [i.e.,
$P(\rho_{l})=P(\rho_{v})$ and $\mu(\rho_{l})=\mu(\rho_{v})$]. The procedure is
described in Ref. [5]. In practice, we set $d_{\rm HS}=\tilde{\sigma}_{ff}$
and imposed the coexistence data of $\rho_{l}$, $\rho_{v}$, and
$P(\rho_{l})=P(\rho_{v})=P_{0}$ for Ar quoted in Table X of Ref. [17] to be
reproduced in the entire range of temperatures $T$ between $T_{t}=83.78$ K and
$T_{c}=150.86$ K.

###  2.2 Euler-Lagrange equation

The equilibrium density profile $\rho({\bf r})$ of the adsorbed fluid is
determined by a minimization of the free energy with respect to density
variations with the constraint of a fixed number of particles $N$

$\frac{\delta}{\delta\rho({\bf r})}\biggr{[}F_{\rm DF}[\rho({\bf r})]-\mu\int
d{\bf r}\,\rho({\bf r})\biggr{]}=0\;.$ (6)

Here the Lagrange multiplier $\mu$ is the chemical potential of the system. In
the case of a planar symmetry where the flat walls exhibit an infinite extent
in the $x$ and $y$ directions, the profile depends only on the coordinate $z$
perpendicular to the substrate. For this geometry, the variation of Eq. (6)
yields the following Euler-Lagrange (E-L) equation

$\displaystyle\frac{\delta[(F_{\rm id}+F_{\rm HS})/A]}{\delta\rho(z)}$
$\displaystyle+$
$\displaystyle\int^{L}_{0}dz\prime\rho(z\prime)\bar{\Phi}_{\rm attr}(\mid
z-z\prime\mid)$ (7) $\displaystyle+$ $\displaystyle U_{sf}(z)=\mu\;,$

where

$\frac{\delta(F_{\rm id}/A)}{\delta\rho(z)}=\nu_{\rm
id}\,k_{B}\,T\,\ln{[\Lambda^{3}\,\rho(z)]}\;,$ (8)

and

$\displaystyle\frac{\delta(F_{\rm HS}/A)}{\delta\rho(z)}=f_{\rm
HS}[\bar{\rho}(z);d_{\rm HS}]$ (9) $\displaystyle+$
$\displaystyle\int^{L}_{0}dz\prime\,\rho(z\prime)\,\frac{\delta f_{\rm
HS}[\bar{\rho}(z\prime);d_{\rm
HS}]}{\delta\bar{\rho}(z\prime)}\,\frac{\delta\bar{\rho}(z\prime)}{\delta\rho(z)}\;.$

Here $F_{\rm id}/A$ and $F_{\rm HS}/A$ are free energies per unit of one wall
area $A$. $L$ is the size of the box adopted for solving the E-L equations.
The boundary conditions for the one-wall and slit systems are different and
will be given below. The final E-L equation may cast into the form

$\displaystyle\nu_{\rm id}\,k_{B}\,T\,\ln{[\Lambda^{3}\,\rho(z)]}+Q(z)=\mu\;,$
(10)

where

$\displaystyle Q(z)$ $\displaystyle=$ $\displaystyle f_{\rm
HS}[\bar{\rho}(z);d_{\rm HS}]$ (11) $\displaystyle+$
$\displaystyle\int^{L}_{0}dz\prime\,\rho(z\prime)\,\frac{\delta f_{\rm
HS}[\bar{\rho}(z\prime);d_{\rm
HS}]}{\delta\bar{\rho}(z\prime)}\,\frac{\delta\bar{\rho}(z\prime)}{\delta\rho(z)}$
$\displaystyle+$
$\displaystyle\int^{L}_{0}dz\prime\,\rho(z\prime)\,\bar{\Phi}_{\rm attr}(\mid
z-z\prime\mid)$ $\displaystyle+$ $\displaystyle U_{sf}(z)\;.$

The number of particles $N_{s}$ per unit area, $A$, of the wall is

$N_{s}=\frac{N}{A}=\int^{L}_{0}\rho(z)\,dz\;.$ (12)

In order to get solutions for $\rho(z)$, it is useful to rewrite Eq. (10) as

$\rho(z)=\rho_{0}\,\exp{\left(-\frac{Q(z)}{\nu_{\rm id}\,k_{B}\,T}\right)}\;,$
(13)

with

$\rho_{0}=\frac{1}{\Lambda^{3}}\exp{\left(\frac{\mu}{\nu_{\rm
id}\,k_{B}\,T}\right)}\;.$ (14)

The relation between $\mu$ and $N_{s}$ is obtained by substituting Eq. (13)
into the constraint of Eq. (12)

$\displaystyle\mu$ $\displaystyle=$ $\displaystyle-\nu_{\rm id}\,k_{B}\,T$
(15) $\displaystyle\times$
$\displaystyle\ln{\biggr{[}\frac{1}{N_{s}\Lambda^{3}}\int^{L}_{0}dz\exp{\left(-\frac{Q(z)}{\nu_{\rm
id}k_{B}T}\right)\biggr{]}}}\;.$

When solving this kind of systems, it is usual to define dimensionless
variables $z^{*}=z/{\tilde{\sigma}}_{ff}$ for the distance and
$\rho^{*}=\rho\,{\tilde{\sigma}}^{3}_{ff}$ for the densities. In these units
the box size becomes $L^{*}=L/{\tilde{\sigma}}_{ff}$.

##  3 Results and Analysis

In order to quantitatively study the adsorption of fluids within any
theoretical approach,one must require the experimental surface tension of the
bulk liquid-vapor interface, $\gamma_{lv}$, to be reproduced satisfactorily
over the entire $T_{t}\leq T\leq T_{c}$ temperature range. Therefore, we shall
first examine the prediction for this observable before studying the
adsorption phenomena.

###  3.1 Surface tension of the bulk liquid-vapor interface

Figure 1 shows the experimental data of $\gamma_{lv}$ taken from Table II of
Ref. [18]. In order to theoretically evaluate this quantity the E-L equations
for free slabs of Ar, i.e. setting

$U_{sf}(z)=0\;,$ (16)

were solved imposing periodic boundary conditions $\rho(z=0)=\rho(z=L)$. At a
given temperature $T$, for a sufficiently large system one must obtain a wide
central region with $\rho(z\simeq L/2)=\rho_{l}(T)$ and tails with density
$\rho_{v}(T)$, where the values of $\rho_{l}(T)$ and $\rho_{v}(T)$ should be
those of the liquid-vapor coexistence curve. The surface tension of the
liquid-vapor interface is calculated according to the thermodynamic definition

$\gamma_{lv}=(\Omega+P_{0}\,V)/A=\Omega/A+P_{0}\,L\;,$ (17)

where $\Omega=F_{\rm DF}-\mu\,N$ is the grand potential of the system and
$P_{0}$ the pressure at liquid-vapor coexistence previously introduced. We
solved a box with $L^{*}=40$. The obtained results are plotted in Fig. 1
together with the prediction of the fluctuation theory of critical phenomena
$\gamma_{lv}=\gamma^{0}_{lv}(1-T/T_{c})^{1.26}$ with $\gamma^{0}_{lv}=17.4$
K/Å2 (see, e.g., [19]). One may realize that our values are in satisfactory
agreement with experimental data and the renormalization theory over the
entire range of temperatures $T_{t}\leq T\leq T_{c}$, showing a small
deviation near $T_{t}$.

Figure 1:  Surface tension of Ar as a function of temperature. Squares are
experimental data taken from Table II of Ref. [18]. The solid curve
corresponds to the fluctuation theory of critical phenomena and the circles
are present DF results. Figure 2:  Adsorption isotherms for the Ar/Li system,
i.e., $\Delta\mu$ as a function of coverage $\Gamma_{\ell}$. Up-triangles
correspond to $T=119$ K; circles to $T=118$ K; diamonds to $T=117$ K; squares
to $T=116$ K; down-triangles to $T=114$ K and stars to $T=112$ K.

###  3.2 Adsorption on one planar wall

It is assumed that the physisorption of Ar on a one wall substrate of Li is
driven by the CCZ potential, i.e.,

$U_{sf}(z)=U_{\rm CCZ}(z)\;.$ (18)

The E-L equations were solved in a box of size $L^{*}=40$ by imposing
$\rho(z>L)=\rho(z=L)$. The solution gives a density profile $\rho(z)$ and the
corresponding chemical potential $\mu$. Adsorption isotherms at a given
temperature were calculated as function of the excess surface density. This
quantity, also termed coverage, is often expressed in nominal layers $\ell$

$\Gamma_{\ell}=(1/\rho^{2/3}_{l})\int_{0}^{\infty}dz[\rho(z)-\rho_{B}]\;,$
(19)

where $\rho_{B}=\rho(z\to\infty)$ is the asymptotic bulk density and
$\rho_{l}$ the liquid density at saturation for a given temperature. By
utilizing the results for $\mu$ obtained from the E-L equation and the value
$\mu_{0}$ corresponding to saturation at a given temperature $T$, the
difference $\Delta\mu=\mu-\mu_{0}$ was evaluated. Figure 2 shows the
adsorption isotherms for temperatures above $T_{w}$, where an equal area
Maxwell construction is feasible. This is just the prewetting region
characterized by a jump in coverage $\Gamma_{\ell}$. The size of this jump
depends on temperate. The largest jump occurs at $T_{w}$ and diminishes for
increasing $T$ until its disappearance at $T_{cpw}$. Density profiles just
below and above the coverage jump for $T=114$ K are displayed in Fig. 3, in
that case $\Gamma_{\ell}$ jumps from 0.5 to 3.6. Therefore, the formation of
the fourth layer may be observed in the plot.

Figure 3:  Examples of density profiles of Ar adsorbed on a surface of Li at
$T=114$ K displayed as a function of the distance from the wall located at
$z^{*}=0$. Dashed curves are profiles for $\Gamma_{\ell}$ below the coverage
jump, while solid curves are stable films above this jump. Figure 4:
Prewetting line for Ar adsorbed on Li. The solid curve is the fit to Eq. (20)
and reaches the $\Delta\mu_{pw}/k_{B}=0$ line at $T_{w}=110.1$ K.

The wetting temperature $T_{w}$ can be obtained from the analysis of the
values of $\Delta\mu/k_{B}$ at which the jump in coverage occurs at each
considered temperature. The behavior $\Delta\mu_{pw}/k_{B}\,\rm vs\,T$ is
displayed in Fig. 4. A useful form for determining the temperature $T_{w}$ was
derived from thermodynamic arguments [20]

$\displaystyle\Delta\mu_{pw}(T)$ $\displaystyle=$
$\displaystyle\mu_{pw}(T)-\mu_{0}(T)$ (20) $\displaystyle=$ $\displaystyle
a_{pw}\,(T-T_{w})^{3/2}\;.$

Here $a_{pw}$ is a model parameter and the exponent $3/2$ is fixed by the
power of the van der Walls tail of the adsorption potential $U_{sf}(z)\simeq-
C_{3}/z^{3}$. The fit of the data of $\Delta\mu/k_{B}$ to Eq. (20) yielded
$T_{w}=110.1$ K and $a_{pw}/k_{B}=-0.16$ K-1/2.

On the other hand, according to Fig. 2, the critical prewetting point
$T_{cpw}$ lies between $T=118$ and 119 K. At the latter temperature, the film
already presents a continuous growth.

Our values of $T_{w}$ and $T_{cpw}$ are smaller than those obtained from prior
DF calculations [9] ($T_{w}=123$ K and $T_{cpw}\simeq 130$ K) and GCMC
simulations [10] ($T_{w}=130$ K). The difference with the DF evaluation of
Ref. [9] is due to the use of different effective pair potentials as we
explain in Ref. [5], where the adsorption of Ne is studied. The present
approach gives a reasonable $\gamma_{lv}$, while that of Ref. [9] fails
dramatically close to $T_{t}$. The difference with the GCMC results cannot be
interpreted in a straightforward way.

###  3.3 Confinement in a planar slit

In the slit geometry, where the Ar atoms are confined by two identical walls
of Li the $s$-$f$ potential becomes

$U_{sf}(z)=U_{\rm CCZ}(z)+U_{\rm CCZ}(L-z)\;.$ (21)

The walls were located at a distance $L^{*}=40$, this width guarantees that
the pair interaction between two atoms located at different walls is
negligible. In fact, this width is wider than $L^{*}=29.1$, which was utilized
in the pioneering molecular dynamics calculations [2, 3]. Accordingly, the E-L
equations were solved in a box of size $L^{*}=40$. In this geometry, the
repulsion at the walls causes the profiles $\rho(z=0)$ and $\rho(z=L)$ to be
equal to zero. The solutions were obtained at a fixed dimensionless average
density defined in terms of $N$, $A$, and $L$ as
$\rho^{*}_{av}=N\,{\tilde{\sigma}}^{3}_{ff}/A\,L=N^{*}_{s}/L^{*}$.

Figure 5:  Free energy per particle (in units of $k_{B}\,T$) for Ar confined
in a slit of Li with $L^{*}=40$ at $T=115$ K displayed as a function of the
average density. The curve labeled by circles corresponds to symmetric
solutions, while that labeled by triangles corresponds to asymmetric ones. The
SSB occurs in a certain range of average density
$\rho^{*}_{ssb1}\leq\rho^{*}_{av}\leq\rho^{*}_{ssb2}$.

For temperatures below $T_{w}=110.1$ K, we obtained large ranges of
$\rho^{*}_{av}$ where the asymmetric solutions exhibit a lower free energy
than the corresponding symmetric ones. In spite of the fact that there is a
general idea that a connection exists between the SSB effect and nonwetting,
we have found, by contrast, that SSB behavior extends above the wetting
temperature. Furthermore, we have also found a relation between prewetting and
SSB.

Figure 5 shows the free energy per particle, $f_{\rm DF}=F_{\rm DF}/N$, for
both symmetric and asymmetric solutions for the Ar/Li system at $T=115$
K$\,>T_{w}$ as a function of the average density. According to this picture,
the ground state (g-s) exhibits asymmetric profiles between a lower and an
upper limit $\rho^{*}_{ssb1}=0.057\leq\rho^{*}_{av}\leq\rho^{*}_{ssb2}=0.192$.
Out of this range no asymmetric solutions were obtained form the set of Eqs.
(10)-(15). Similar features were obtained for higher temperatures until
$T=118$ K, above this value the profiles corresponding to the g-s are always
symmetric. Figure 6 shows three examples of solutions determined at $T=115$ K.
The result labeled 1 is a small asymmetric profile, that labeled 2 is the
largest asymmetric solution at this temperature. So, by further increasing
$\rho^{*}_{av}$, the SSB effect disappears and the g-s becomes symmetric, as
indicated by the curve labeled 3. When the asymmetric profiles occur, the
situation is denoted as partial (or one wall) wetting. The symmetric solutions
account for a complete (two wall) wetting. These different situations can be
interpreted in terms of the balance of $\gamma_{sl}$, $\gamma_{sv}$ and
$\gamma_{lv}$ surface tensions, carefully discussed in previous works [2, 3,
7]. Here we shall restrict ourselves to briefly outline the main features.
When the liquid is adsorbed symmetrically like in the case of profile 3 in
Fig. 6, there are two $s$-$l$ and two $l$-$v$ interfaces. Hence, the total
surface excess energy may be written as

$\gamma^{sym}_{tot}=2\,\gamma_{sl}+2\,\gamma_{lv}\;.$ (22)

On the other hand, for a asymmetric profile $\gamma^{asy}_{tot}$ becomes

$\gamma^{asy}_{tot}=\gamma_{sl}+\gamma_{lv}+\gamma_{sv}\;.$ (23)

The three quantities of the r.h.s. of this equation are related by Young’s law
(see, e.g., Eq. (2.1) in Ref. [21])

$\gamma_{sv}=\gamma_{sl}+\gamma_{lv}\,\cos{\theta}\>,$ (24)

where $\theta$ is the contact angle defined as the angle between the wall and
the interface between the liquid and the vapor (see Fig. 1 in Ref. [21]). By
using Young’s law, the Eq. (23) may be rewritten as

$\gamma^{asy}_{tot}=2\,\gamma_{sl}+\gamma_{lv}\,(1+\cos{\theta})\;,$ (25)

with $\cos{\theta}=(\gamma_{sv}-\gamma_{sl})/\gamma_{lv}<1$. If one changes
$\gamma_{sl}$ by increasing enough $N_{s}$ (as shown in Fig. 5), and/or $T$,
and/or the strength of $U_{sf}(z)$, eventually the equality
$\gamma_{sv}-\gamma_{sl}=\gamma_{lv}$ may be reached yielding
$\cos{\theta}=1$. Then, the system would undergo a transition to a symmetric
profile where both walls of the slit are wet.

Figure 6:  Density profiles of Ar confined in a slit of Li with $L^{*}=40$ at
$T=115$ K. The displayed spectra denoted by 1, 2 and 3 correspond to average
densities $\rho^{*}_{av}=0.074,0.192$ and $0.218$, respectively.

It is important to remark that, indeed, there are two degenerate asymmetric
solutions. Besides that one shown in Fig. 6 where the profiles exhibit the
thicker film adsorbed on the left wall (left asymmetric solutions - LAS),
there is an asymmetric solution with exactly the same free energy but where
the thicker film is located near the right wall (right asymmetric solutions -
RAS).

The asymmetry of density profiles may be measured by the quantity

$\Delta_{N}=\frac{1}{N_{s}}\int^{L/2}_{0}dz\,[\rho(z)-\rho(L-z)]\;.$ (26)

According to this definition, if the profile is completely asymmetrical about
the middle of the slit, i.e. for: (i) $\rho(z<L/2)\neq 0$ and $\rho(z\geq
L/2)=0$; or (ii) $\rho(z<L/2)=0$ and $\rho(z\geq L/2)\neq 0$ this quantity
becomes $+1$ or $-1$, respectively, while for symmetric solutions it vanishes.

Figure 7:  Asymmetry parameter for Ar confined by two Li walls separated by a
distance of $L^{*}=40$ as a function of average density. From outside to
inside the curves correspond to temperatures $T=112,114,115,116,117$ and 118
K. The asymmetric solutions occur for different ranges
$\rho^{*}_{ssb1}\leq\rho^{*}_{av}\leq\rho^{*}_{ssb2}$. Figure 8:  Circles
stand for both branches of the asymmetry parameter for Ar confined in an
$L^{*}=40$ slit of Li walls for temperatures between $T_{w}$ and $T_{cpw}$.
The solid curve is the fit to Eq. (27) used to determine $T_{cpw}$.

We evaluated the asymmetry coefficients of solutions obtained for increasing
temperatures up to $T=118$ K. The results for LAS profiles at temperatures
larger that $T_{w}$ are displayed in Fig. 7 as a function of the average
density. One may observe how the range
$\rho^{*}_{ssb1}\leq\rho^{*}_{av}\leq\rho^{*}_{ssb2}$ diminishes under
increasing temperatures. The SSB effect persists at most for the critical
$\rho^{*}_{av}(crit)=(17/24)\,{\tilde{\sigma}}^{2}_{ff}\times 10^{-2}\simeq
0.074$ with ${\tilde{\sigma}}_{ff}$ expressed in Å.

We shall demonstrate that by analyzing the data of $\Delta_{N}$ for
$\rho^{*}_{av}(crit)$ it is possible to determine the critical prewetting
point. Figure 8 shows these values for both the LAS and RAS profiles,
calculated at different temperatures, suggesting a rather parabolic shape. So,
we propose a fit to the following quartic polynomial

$T=T_{cpw}+a_{2}\Delta^{2}_{N}+a_{4}\Delta^{4}_{N}\;.$ (27)

This procedure yielded $T_{cpw}=118.4$ K, $a_{2}=-14.14$ K, and $a_{4}=-16.63$
K. The obtained value of $T_{cpw}$ is in agreement with the limits established
when analyzing the adsorption isotherms of the one-wall systems displayed in
Fig. 2. These results indicate that the disappearance of the SSB effect
coincides with the end of the prewetting line.

##  4 Conclusions

We have performed a consistent study within the same DF approach of free slabs
of Ar, the adsorption of these atoms on a single planar wall of Li and its
confinement in slits of this alkali metal. Good results were obtained for the
surface tension of the liquid-vapor interface. The analysis of the
physisorption on a planar surface indicates that Ar wets surfaces of Li in
agreement with previous investigations. The isotherms for the adsorption on
one planar wall exhibit a locus of prewetting in the $\mu-T$ plane. A fit of
such data yielded a wetting temperature $T_{w}=110.1$ K. In addition, these
isotherms also show that the critical prewetting point $T_{cpw}$ lies between
$T=118$ and 119 K. These results for $T_{w}$ and $T_{cpw}$ are slightly below
the values obtained in Refs. [9, 10], the discrepancy is discussed in the
text.

On the other hand, this investigation shows that the profiles of Ar confined
in a slit of Li present SSB. This effect occurs in a certain range of average
densities $\rho^{*}_{ssb1}\leq\rho^{*}_{av}\leq\rho^{*}_{ssb2}$, which
diminishes for increasing temperatures. The main output of this work is the
finding that above the wetting temperature the SSB occurs until $T_{cpw}$ is
reached. To the best of our knowledge this is the first time that such a
correlation is reported. Furthermore, it is shown that by examining the
evolution of the asymmetry coefficient one can precisely determine $T_{cpw}$.
The obtained value $T_{cpw}=118.4$ K lies in the interval established when
analyzing the adsorption on a single wall.

###### Acknowledgements.

This work was supported in part by the Grants PICT 31980/5 from Agencia
Nacional de Promoción Científica y Tecnológica, and X099 from Universidad de
Buenos Aires, Argentina.

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