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Paper
 Superfluid helium
has unusual thermal and mechanical properties[1] and is well known for
it ability to spread over surfaces and to flow without dissipation through
even microscopic holes. Virtually all of the walls and surfaces used in
previous dissipationless flow experiments are wet by superfluid helium.
This effect means that droplets on these substrates are unstable and immediately
spread to form a smooth continuous film over the entire surface so that
vapor and substrate are never in contact. Recent work [2] has shown that
alkali metals are a special class of materials not completely wetted by
superfluid helium. In particular, cesium substrates can be used to prepare
superfluid samples with a distinctly different topology consisting of a
droplet with an edge where substrate, superfluid, and vapor meet at a three-phase
contact line [3]. We present here direct observations of isolated droplets
of superfluid on a substrate [4]. Both the static and dynamic behavior
of the droplets are unusual. We found that the contact angle was an extremely
hysteretic function of the volume of the drop. Perhaps the most remarkable
property of superfluid droplets is that they would not move across the
surface until a considerable force was applied to them. This result is
surprising because solid surfaces are well known not to exert transverse
forces on bulk superfluid or superfluid films without edges.
Our apparatus consisted
of a substrate that can be rotated about a horizontal axis mounted in an
optical dewar with windows that provide an edge-on view of the substrate
as well as a view from above at an angle of 60° from the normal. The
substrate is a quartz microbalance with gold electrodes similar to those
used in our previous thermodynamic studies[5,6]. Fifty atomic layers of
cesium were vapor deposited onto the quartz and gold surfaces of the microbalance
at a rate of 0.01 layers/sec. During the evaporation, the temperature of
the substrate and the walls of the container were maintained below 6 K
to maintain ultra high vacuum conditions. The microbalance was used to
monitor the deposition and to perform thermodynamic characterizations of
the surface; the wetting temperature was measured to be Tw=2.04
K. A capillary tube (0.04 cm, outside diameter) attached to a source of
room-temperature gas through a mass flow controller provided a means of
putting drops of superfluid on the surface. The system was maintained at
liquid-vapor coexistence by filling the bottom of the container with bulk
liquid 4He. The drops were observed with a long focal distance
microscope that provided a magnification of approximately 30.
Figures 1 and 2 show
a sequence of photographs of superfluid drops on a cesiated substrate at
T=1.16K. Figure 1 shows pictures taken with the microscope looking down
on the substrate at an angle of 30° above the horizontal. The dark
bar at the top of the pictures is the capillary tube, and the lower bar
is its shadow. The tube was left in contact with the superfluid drop so
that fluid could be added and withdrawn. This geometry is conventional
for contact angle measurements and typically yields the advancing and receding
contact angle[7]. The drops appear oval because of the viewing angle. The
edge of the gold electrode can be seen in the extreme and upper left; the
cesium film is too thin to provide appreciable optical contrast. Figure
2 shows edge-on views of the same droplets as shown in Figure1 illuminated
from the back. The optical axis is a fraction of a degree above horizontal,
so that both the free surface of the drop and its reflection in the substrate
are visible. The capillary tube is clearly visible protruding from the
top of the drop. The focus was adjusted so that a diameter of the drop
lies in the focal plane.
The plane of the substrate
can be located by drawing a line between the two points where the profile
of the drop meets its mirror image. The contact angle is the angle between
the tangent to the drop profile and the substrate at the point of contact.
Despite the superfluidity of the drops, the value of the contact angle
which we observed depended critically on the way the drop was prepared.
When the volume of the drop was increasing, the contact angle was approximately
32° [8] and independent of the volume of the drop; Figure 2A and B,
show snapshots of the same drop as fluid was added (the corresponding top
view of the growing drop is shown in Figure 1A and B). When fluid was withdrawn,
the contact line remained stationary and the receding contact angle approached
zero, (Figures 1C and 2C). The bulk fluid could be completely removed,
but apparently a thin film remained in the region bounded by the original
contact line, since when fluid is added, the drop immediately obtained
its previous diameter and only slowly increased its volume and contact
angle. The only way to prepare a smaller drop (such as Figure 1A) in the
same area was to remove the microscopic film by briefly heating the substrate
with a flash of light.
Contact angle hysteresis
is a common phenomenon in wetting measurements of conventional liquids
and substrates that is typically attributed to substrate heterogeneity,
or kinetic effects associated with the viscosity of the liquid, or both
[7,9,10]. Because our experiment utilized superfluid and the time scale
of the observations was several minutes, kinetics cannot be invoked to
explain the hysteresis we observed. Similarly, it is difficult to find
a plausible source of surface heterogeneity, because the substrate exhibits
the same sharp thermodynamic signatures of the wetting transition that
we have explored in previous work[5,6]. In order to explain a receding
contact angle of zero, standard models based on consideration of metastable
states on a heterogeneous substrate would require that more than half of
the surface be covered with patches where the local contact angle is zero,
that is, where the liquid wets[7,10]. This possibility is ruled out by
microbalance measurements which show that the average equilibrium coverage
at liquid-vapor coexistence on our substrate at temperatures far below
the wetting temperature is less than 2 monolayers, which implies that the
fraction of the surface that was wetted is less than 4%. The possibility
of point-like pinning centers also seems unlikely because the contact line
(Fig 1) appears perfectly smooth on length scales of a few micrometers.
Although the conventional
explanations of contact angle hysteresis due to surface heterogeneity do
not apply to our experiment, there must nevertheless be some mechanism
which provides metastable states that can trap the droplet in configurations
with a continuous range of contact angles. The fact that the contact line
did not move even when the apparent contact angle was reduced to zero when
the drop was deflated allows us to place a lower limit on the pinning force
per unit length which these metastable states can sustain. We assume as
is customary that the advancing contact angle is equal to the thermodynamic
equilibrium contact angle q [8] which satisfies
Young’s equation , slv cosq
= ssv- ssl
where the sij are surface tensions,
and the l,s,v subscripts denote liquid, substrate, and vapor, respectively.
In this case, the force per unit length on the contact line due to surface
tension when the contact angle was reduced to zero is slv
(1- cosq) = 46 mdyne/cm. The maximum pinning
force must be at least as large.
Another manifestation
of forces on the contact line can be seen when a superfluid drop was placed
on an inclined surface. Figure 3 shows an edge-on view of a drop on a cesiated
surface inclined at ~ 10° to the horizontal; a pendant drop of fluid
formed by forcing helium down the capillary faster than the superfluid
film on the outer surface could drain it, can also be seen in the upper
right corner. The most remarkable feature of the drop on the substrate
is that it is stationary. Even vigorous shaking of the apparatus, which
caused easily discernible waves in the drop, did not cause it to flow down
the incline. The downhill edge of the drop had the same contact angle as
the advancing edge of a growing drop, while the uphill edge had a vanishing
contact angle. As more fluid was added to the drop, it eventually rolled
down the incline, often with a jerky stick-slip motion. Subsequent drops
immediately spread out across the path of the previous drop and rapidly
flowed downhill. It seems as if the first drop, which moves across a dry
substrate left a trailing film that “lubricates” the motion of subsequent
drops. This film, which persisted for hours, may be related to the metastable
thick films we have observed in previous experiments[6]. The trailing film
had submicroscopic thickness and was invisible in an edge-on view. It could
be detected ellipsometrically and was superfluid because locally heating
a spot with a laser beam produced a thermo-mechanically driven bump in
the film profile.
Superfluid droplets on
cesiated substrates have remarkable spreading and flow properties that
are not simple consequences of bulk superfluid behavior. Liquid helium
has exceptional chemical purity and the heterogeneity of our cesium surface
is constrained by thermodynamic adsorption measurements. For these reasons,
helium on cesium would naively be expected to display nearly ideal reversible
spreading behavior, because even the complications due to viscosity are
negligible. In contrast, superfluid contact angles are found to be even
more hysteretic than typical classical fluid drops on macroscopically heterogeneous
surfaces. The hysteresis is so extreme that the superfluid contact line
appears to move in only one direction, that is so as to increase the wetted
area.
It is difficult to reconcile
these observations with standard models of contact angle hysteresis. Regarded
as a superfluid, droplets are remarkable because they can resist flow against
a substantial chemical potential gradient. Both of these effects are presumably
due to metastable configurations of the superfluid contact line, which
have been inaccessible to experimental observation until very recently.
In order to attribute the metastability to extrinsic defects, a mechanism
allowing small defect concentrations to cause extremely large hysteresis
would need to be identified.
This work supported by NSF grant DMR 9623976.
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