Stationary nonlinear waves at the surface of a thin liquid layer under inverted gravitation conditions
Instability of the flat surface of a thin liquid layer wetting a solid substrate under inverted gravitation conditions is discovered. The development of this instability leads to formation of a new stationary nonuniform liquid surface state. It looks like a solitary hill with characteristics sens...
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irk-123456789-1191972017-06-06T03:03:50Z Stationary nonlinear waves at the surface of a thin liquid layer under inverted gravitation conditions Kolmakov, G.V. Lebedeva, E.V. Levchenko, A.A. Mezhov-Deglin, L.P. Trusov, A.B. Shikin, V.B. Низкотемпеpатуpный магнетизм Instability of the flat surface of a thin liquid layer wetting a solid substrate under inverted gravitation conditions is discovered. The development of this instability leads to formation of a new stationary nonuniform liquid surface state. It looks like a solitary hill with characteristics sensitive to the liquid film parameters, particularly to the layer thickness at which the instability begins to develop. By application of a variational approach the mechanical stability of such a hill (droplet) in the one-dimensional approximation is proved. A variational picture of the shape evolution for a cylindrical charged droplet in an external electric field is constructed, too. The results obtained are compared with an experiment on liquid hydrogen droplets [A.A. Levchenko, G.V. Kolmakov, L.P. Mezhov-Deglin, M.G. Mikhailov, and A.B. Trusov, Low Temp. Phys. 25, 242 (1999)]. The theory developed is in good agreement with the results of experiments. 2004 Article Stationary nonlinear waves at the surface of a thin liquid layer under inverted gravitation conditions / G.V. Kolmakov, E.V. Lebedeva, A.A. Levchenko, L.P. Mezhov-Deglin, A.B. Trusov, V.B. Shikin // Физика низких температур. — 2004. — Т. 30, № 1. — С. 79-93. — Бібліогр.: 24 назв. — англ. 0132-6414 PACS: 68.15.+e, 05.45.Yv, 67.70.+n http://dspace.nbuv.gov.ua/handle/123456789/119197 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Низкотемпеpатуpный магнетизм Низкотемпеpатуpный магнетизм |
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Низкотемпеpатуpный магнетизм Низкотемпеpатуpный магнетизм Kolmakov, G.V. Lebedeva, E.V. Levchenko, A.A. Mezhov-Deglin, L.P. Trusov, A.B. Shikin, V.B. Stationary nonlinear waves at the surface of a thin liquid layer under inverted gravitation conditions Физика низких температур |
| description |
Instability of the flat surface of a thin liquid layer wetting a solid substrate under inverted
gravitation conditions is discovered. The development of this instability leads to formation of a
new stationary nonuniform liquid surface state. It looks like a solitary hill with characteristics
sensitive to the liquid film parameters, particularly to the layer thickness at which the instability
begins to develop. By application of a variational approach the mechanical stability of such a hill
(droplet) in the one-dimensional approximation is proved. A variational picture of the shape evolution
for a cylindrical charged droplet in an external electric field is constructed, too. The results
obtained are compared with an experiment on liquid hydrogen droplets [A.A. Levchenko, G.V.
Kolmakov, L.P. Mezhov-Deglin, M.G. Mikhailov, and A.B. Trusov, Low Temp. Phys. 25, 242
(1999)]. The theory developed is in good agreement with the results of experiments. |
| format |
Article |
| author |
Kolmakov, G.V. Lebedeva, E.V. Levchenko, A.A. Mezhov-Deglin, L.P. Trusov, A.B. Shikin, V.B. |
| author_facet |
Kolmakov, G.V. Lebedeva, E.V. Levchenko, A.A. Mezhov-Deglin, L.P. Trusov, A.B. Shikin, V.B. |
| author_sort |
Kolmakov, G.V. |
| title |
Stationary nonlinear waves at the surface of a thin liquid layer under inverted gravitation conditions |
| title_short |
Stationary nonlinear waves at the surface of a thin liquid layer under inverted gravitation conditions |
| title_full |
Stationary nonlinear waves at the surface of a thin liquid layer under inverted gravitation conditions |
| title_fullStr |
Stationary nonlinear waves at the surface of a thin liquid layer under inverted gravitation conditions |
| title_full_unstemmed |
Stationary nonlinear waves at the surface of a thin liquid layer under inverted gravitation conditions |
| title_sort |
stationary nonlinear waves at the surface of a thin liquid layer under inverted gravitation conditions |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2004 |
| topic_facet |
Низкотемпеpатуpный магнетизм |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119197 |
| citation_txt |
Stationary nonlinear waves at the surface of a thin liquid layer under inverted gravitation conditions / G.V. Kolmakov, E.V. Lebedeva, A.A. Levchenko, L.P. Mezhov-Deglin, A.B. Trusov, V.B. Shikin // Физика низких температур. — 2004. — Т. 30, № 1. — С. 79-93. — Бібліогр.: 24 назв. — англ. |
| series |
Физика низких температур |
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2025-07-08T15:24:31Z |
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2025-07-08T15:24:31Z |
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| fulltext |
Fizika Nizkikh Temperatur, 2004, v. 30, No. 1, p. 79–93
Stationary nonlinear waves at the surface of a thin liquid
layer under inverted gravitation conditions
G.V. Kolmakov, E.V. Lebedeva, A.A. Levchenko,
L.P. Mezhov-Deglin, A.B. Trusov, and V.B. Shikin
Institute of Solid State Physics RAS, Chernogolovka, Moscow region, 142432, Russia
E-mail: german@issp.ac.ru
Received May 21, 2003, revised July 28, 2003
Instability of the flat surface of a thin liquid layer wetting a solid substrate under inverted
gravitation conditions is discovered. The development of this instability leads to formation of a
new stationary nonuniform liquid surface state. It looks like a solitary hill with characteristics
sensitive to the liquid film parameters, particularly to the layer thickness at which the instability
begins to develop. By application of a variational approach the mechanical stability of such a hill
(droplet) in the one-dimensional approximation is proved. A variational picture of the shape evo-
lution for a cylindrical charged droplet in an external electric field is constructed, too. The results
obtained are compared with an experiment on liquid hydrogen droplets [A.A. Levchenko, G.V.
Kolmakov, L.P. Mezhov-Deglin, M.G. Mikhailov, and A.B. Trusov, Low Temp. Phys. 25, 242
(1999)]. The theory developed is in good agreement with the results of experiments.
PACS: 68.15.+e, 05.45.Yv, 67.70.+n
1. Introduction
A charged or neutral liquid layer is an example of a
two-dimensional (2D) system that can exhibit a me-
chanical instability under inverted gravitation condi-
tions, i.e., under conditions when the layer is sus-
pended at the horizontal surface of a solid substrate.
It is known that a liquid layer suspended on a solid
substrate under inverted gravitation conditions is ab-
solutely unstable if the van der Waals forces are ne-
glected [1]. In the presence of the attractive van der
Waals forces between liquid and substrate, there is a
finite interval of layer thicknesses for which a mecha-
nical equilibrium is possible, and a sufficiently thin
layer covers the substrate uniformly. At thicknesses
larger than some critical value, the van der Waals
forces cannot keep a liquid layer in a flat state and the
instability develops [2].
The mechanical instability of the flat free surface
of a neutral liquid layer of a thickness larger than the
critical value develops at small wave vectors [2],
which is similar to the instability of the charged sur-
face of a thin liquid layer in an external pressing elec-
tric field [3–5]. This situation is quite different from
the known instability phenomena of charged surfaces
of a thick liquid helium layer [6–11], because in this
case the instability develops at a finite wave number
of the order of the inverse capillary wave length. The
scenario of development of the long-wave instability
of thin layer surfaces was not clear till now.
The main goal of this paper is to study the nonuni-
form phenomena that occur on neutral and charged
surfaces of a thin liquid layer. We show that the insta-
bility leads to the formation of a stationary, nonlin-
ear, solitary wave of surface deformations. The wave
amplitude increases with increasing liquid layer thick-
ness or with increasing stretching electric field above
some critical values. This behavior is quite different
from that of a bulk liquid surface, where the final
jump of the deformation wave amplitude takes place
at a critical point.
The computed evolution of the shape of the recon-
structed surface is compared with the data obtained in
earlier experiments [12] on neutral and charged liquid
hydrogen layers.
The solution to the problem of stationary surface
shapes of thick liquid layers and droplets placed at the
surface of a solid substrate was proposed many years
ago in Frenkel’s papers [13]. In that consideration the
van der Waals forces between liquid and substrate
© G.V. Kolmakov, E.V. Lebedeva, A.A. Levchenko, L.P. Mezhov-Deglin, A.B. Trusov, and V.B. Shikin, 2004
were neglected, and their interaction was taken into
account by fixing the contact angle between the liquid
and the solid (the wetting angle). Such a formulation
of the problem excludes, by definition, the possibility
of considering a transition from the flat to the nonuni-
form liquid surface shape: in Frenkel’s treatment the
layer is always in a nonuniform state. As we shall see,
the shape of the nonuniform liquid film surface tends
to be that of a droplet, as found in [13], only for a suf-
ficiently large volume of the liquid layer.
To simplify the presentation below we propose to in-
troduce a new term — «reconstruction». Reconstruc-
tion implies the transition from the flat to the new me-
chanically stable nonuniform liquid surface state.
The structure of this paper is as follows. Chapter 2
contains the general mechanical equilibrium equation
(11) for a liquid film under inverted gravitation condi-
tions, with comments related to the instability origin
and corresponding effective energy definition (24).
The one-dimensional version (22) of Eq. (11) is suit-
able for analytical description of the reconstruction de-
tails. We obtain the explicit solution (27), (28) of the
one-dimensional reconstruction problem. In contrast to
the bulk situation [6,7,9] this solution is not unique. It
contains the arbitrary parameter C which should be
physically reasonable. One possibility for defining this
parameter could correspond to a soliton-like represen-
tation of the solution (27). It corresponds to the re-
quirement T � �, where the period T is given by Eq.
(28). But this condition is in contradiction with the
equation of normalization (14). The second possible
picture — the periodic reconstruction — is not valid,
too. This follows from the properties of the second vari-
ation for the functional E defined by Eq. (24). Finally,
the main conclusion of Chapter 2 is the proof of me-
chanically stable reconstruction of the inverted liquid
film in the form of an individual droplet which is sensi-
tive to the boundary conditions.
The one-dimensional speculations from Chapter 2 en-
able us to believe that the main characteristics of the re-
construction phenomena remain the same in the 2D case
(certainly, this assumption cannot be proved). The
variational picture for an individual radial droplet in the
neutral and charged states is constructed (see Chapter
3). The results obtained are shown to be in agreement
with the currently available experimental data. The cor-
responding fitting is presented in Chapter 3, too.
2. Reconstruction of a thin layer
of neutral liquid
2.1. Instability of the flat surface of a liquid layer
We consider first the stability problem for a neu-
tral liquid layer covering a flat horizontal surface of a
solid substrate under inverted gravitation conditions
(i.e., when the gravitational force is opposite to the
attractive van der Waals force between the liquid and
substrate). A schematic picture of the system under
study is shown in Fig. 1. We introduce the following
frame of reference. The (x,y) plane coincides with the
substrate surface. The z axis is directed downward,
parallel to the gravity force. The free liquid surface is
described by the equation z � �( )r , where r is the
two-dimensional vector within the (x,y) plane.
The equilibrium shape of the free surface of the liq-
uid layer is governed by three forces: the gravitation
force, the Laplace capillary force, and the van der
Waals force, which retains the liquid at the substrate.
Under inverted gravitation conditions the pressure at
the free surface is given by the following expression:
P g P
/ w� �
�
� �
� �
��
� �
[ ( ]
( )
1 2 3 2
. (1)
Here � is the surface tension, is the density of the
liquid, g is the free fall acceleration, and
P
f
dw
w
( )�
�
�
� �
�
�
�
�
��
�
3
1
1 (2)
is the pressure of the van der Waals forces (f is the
van der Waals constant and dw is some characteristic
width).
Expression (1) contains two terms conventional for
the capillary problem (the Laplace and the gravita-
tional pressures, see, e.g., [14]). Besides we take into
account the van der Waals (vdW) forces between a
liquid layer and a solid substrate (the last term Pw( )�
of Eq. (1)). The interpolated form (2) for the van der
Waals pressure was proposed in [15]. This expression
describes two dependencies as limiting cases:
P f/ dw w( ) ( )� � �� ��3 at (3)
and
80 Fizika Nizkikh Temperatur, 2004, v. 30, No. 1
G.V. Kolmakov, E.V. Lebedeva, A.A. Levchenko, L.P. Mezhov-Deglin, A.B. Trusov, and V.B. Shikin
0
h
z
g
L
x
� � �x
Fig. 1. Schematic view of a reconstructed liquid layer sus-
pended at a substrate. The liquid layer covers a region of the
surface of a substrate between vertical walls. L is the dis-
tance between the walls, and h is the mean thickness of
layer. The gravitational force is directed downward, along the
z axis. The equation describing the free surface is z x� �( ).
P fd / dw w w( ) ( )� � �� 4 at >> . (4)
The asymptotic expression (3) is applicable in the
situations of «thin» films, where the retarding effects
in the definition of the vdW forces can be neglected,
and expression (4) is valid for «thick» films, for
which the retarding effects play a significant role. The
measurement of the constants f and dw is a special
problem. It is possible to use some original techniques
[15], or, in the case of superfluid helium, to determine
these constants from measurements of the third-sound
propagation [16]. For hydrogen films the values of f
and dw are not yet known.
Generally speaking, the definition of the local vdW
pressure (like in Eq. (1)) is correct for a flat uniform
surface only. It can be used, for example, for estima-
tions of an equilibrium liquid film thickness (see Ref.
17). In the nonuniform case the vdW force has an inte-
gral form [18,19]. The local approximation used in
Eq. (1) is acceptable if
d �� �, (5)
where � is the typical nonuniform scale along the liquid
surface. For the reconstruction problem under consider-
ation the condition (5) is fulfilled.
If the free surface is flat, and the layer thickness is
a constant, � � h, then the Laplace term in Eq. (1)
turns to zero, and the pressure at the surface is
P gh P hw� � � ( ).
One can introduce the effective gravitation accelera-
tion at the liquid surface g P heff � � � � �1( ), which
controls the net force acting on the surface. The pres-
sure P has an extremum at the thickness h� ,
� �P/ h h h| � � � 0, or � �
�
�
�
�
�� �
� �
�
�
g
P h
h
w
h h
( )
0, ,(6)
where the derivative � �P/ h changes sign. So, with in-
creasing layer thickness from h h� � to h h� � , the ef-
fective gravitational acceleration geff changes sign
from positive to negative. This change of the direction
of the effective gravitational accelerational implies
that the flat state of the free surface should be unsta-
ble for thicknesses h h� � , and that small disturbances
of the flat surface should grow with time.
In the limiting cases of large or of small thicknesses of
the film the critical thickness h� is defined as follows:
h fd / g h dw
/
w� �� ��( ) ( )4 1 5 , (7)
h f/ g h d/
w� �� ��( ) ( )3 1 4 . (8)
Note that a solution of Eq. (6) always exists because
the inequality
�
�
P h
h
w( )
� 0 (9)
holds for any values of the constants f and dw .
In the opposite geometry, when the liquid layer
covers the upper horizontal surface of the substrate
(the gravitational and van der Waals forces have the
same direction), the sign of the effective gravitational
force is positive for all thicknesses h. This is in agree-
ment with the fact that the flat free layer surface is al-
ways stable under «normal» gravitation conditions.
As is shown in the next Section, the instability of
flat surface at large h leads to reconstruction, which
consists in the creation of a stationary, nonlinear wave
of deformations of the free surface.
2.2. Equilibrium shape of the reconstructed
liquid surface
In order to describe peculiarities of the reconstruc-
tion at the surface of a liquid layer wetting a horizon-
tal flat substrate, we determine the equilibrium shape
of the free surface of the layer taking into account the
gravitational, capillary, and van der Waals forces.
Generally speaking such a problem reduces to solution
of the equilibrium equation
P( )� � const, (10)
where P( )� is defined by Eqs. (1), (2). Equation (10)
reflects the fact that in a steady state the pressure at
the free surface should be constant over the entire
surface. This gives
� �
�
� �
[ ( ) ]
( )
1
0
2 3 2�
� � � �
/ wg P p . (11)
We denote the constant in the right-hand side of Eq.
(10) as p. The value of the constant p should depend
on the shape of the liquid surface and should be deter-
mined from «the normalization condition»
d V2r r� ��( ) , (12)
where V is the total volume of the liquid in the layer,
and the integration is over the area of the substrate
covered by liquid.
As is shown below, taking the boundary conditions
(wetting the substrate by the liquid) into account is of
fundamental importance for consideration of insta-
bility and reconstruction phenomena. We discuss here
a model case in which the layer is restricted in the
horizontal direction by vertical walls (see Fig. 1). The
effective wetting angle (the angle of slope of the li-
quid surface at the point of contact between the li-
quid and the solid surfaces) we denote as �. The solu-
Stationary nonlinear waves at the surface of a thin liquid layer under inverted gravitation conditions
Fizika Nizkikh Temperatur, 2004, v. 30, No. 1 81
tion to the set of equations (11), (12) with the
following boundary conditions at the vertical walls,
| | tan
� �� , (13)
will uniquely determine the stationary shape of the
liquid surface.
In order to describe the main qualitative features of
the reconstruction phenomena we restrict our consi-
deration in this Section to the model case where a
one-dimensional (1D) wave of deformation appears at
the surface. In this case the function � depends only on
one horizontal coordinate, x.
The normalization condition (12) in the one-dimen-
sional case reads
dx x S� �� ( ) , (14)
where S is the «two-dimensional volume» (area) of
the liquid layer.
We suppose, also, that the angle of slope of the free
surface with respect to the horizontal plane is small,
| |� ��� 1. As is shown below, this condition holds to a
high accuracy when h h� �103 , which is true for typi-
cal conditions of experiment. In this case one can ex-
pand Eq. (11) in the gradient �� ; as a first approxima-
tion this equation will read
�� � ��� � � � �g P pw( ) 0. (15)
Note that the deformation of the surface is not as-
sumed to be small in the general case, so we keep the
term Pw( )� in this equation.
One can show that the development of the instability
of the flat surface at the initial stage of the reconstruc-
tion, i.e., if
( ) ( )h h /h h h� �� �� � �1 , (16)
does not depend on the model behavior (Eq. (4) or Eq.
(3)) assumed for the vdW forces. It is shown below
that if the inequality (16) holds, then the stationary
nonuniform deviation of the reconstructed surface
from the initial flat state is small with respect to h� . In
this case the surface shape can be calculated using a
perturbation approach. For this purpose it is
convenient to present the deformation of the surface as
� �( ) ( )x h x� �� , (17)
where
| ( )|� x
h�
�� 1. (18)
By using the representation (17) the equation (15)
can be expanded in a series in the small parameter
(18). In this way the deviation � ( )x should be repre-
sented as a series
� � �( ) ( ) ( ) ...( ) ( )x x x� � �1 2
(19)
The term �( )( )1 x is a small function of the first order,
the term �( )( )2 x is a small function of the second or-
der, etc. Equation (15) reads as a linear approximation
� �( )( ) ( )1 1 0�� � �p . (20)
Here p( )1 is a first-order correction to the p constant.
The second (quadratic) approximation reads
� �
�
�
�( ) ( )( ) ( ) ( )2
2
2
1 2 21
2
0�� �
�
�
�
�
�
�
� �
�
P
h
pw
h
. (21)
The higher-order equations can be written down in
straightforward way. The values of p( )1 , p( )2 should
be found from the normalization condition (14).
It can be seen that the expression Pw given by Eq.
(2) is dropped from the linearized Eq. (20) due to con-
dition (6), and the derivative of Pw (taken at h h� �)
is a factor in the quadratic term of the second-order
equation (21), only. This means that the choice of the
explicit functional dependence of the vdW pressure
Pw on the layer thichness P P hw w� ( ) from model (3)
or from model (4) is not of importance for analyses of
the reconstruction transition at h close to h� . The de-
rivative � �2 2P / hw h h| � �
(which can be evaluated eas-
ily in both models) controls only the «nonlinearity
level» in the system at given h.
As we show in this paper, in the opposite case
h h�� � the main part of liquid in the layer should be
accumulated in a droplet of macroscopic dimension
formed at the substrate surface. The thickness of the
vdW liquid layer �min wetting the substrate outside
the droplet should remain small. Moreover, this thick-
ness is �min � �h and it decreases further with increas-
ing total volume V of the liquid. For very large (mac-
roscopic) V the thickness �min of the vdW wetting
layer should become comparable to or even less than
dw , and in this case the model (3) is more appropriate
(see the corresponding estimations in Section 2.3).
Keeping in mind that the goal of this paper is not
only to analyze the peculiarities of the reconstruction
picture itself but also to compare the results of calcula-
tions with the experimental data obtained in [12] for a
macroscopic droplet, we will not restrict ourselves to
the expansion provided by Eqs. (20), (21) but will
consider the more general equation (15). Unfortu-
nately, integration of the Eq. (15) in analytical form
with the general expression for Pw( )� from (2) is a
complicated problem. But this integraton can be done
easily if Pw( )� is taken in a power-like form (3) or (4).
In this paper we restrict our consideration to the limit-
ing case (3) for Pw( )� . It should be taken into account
that the results obtained are valid quantitatively in the
82 Fizika Nizkikh Temperatur, 2004, v. 30, No. 1
G.V. Kolmakov, E.V. Lebedeva, A.A. Levchenko, L.P. Mezhov-Deglin, A.B. Trusov, and V.B. Shikin
macroscopic case h h�� � (large V), only. But the qua-
litative picture of the reconstruction does not depend on
the model assumed for the vdW pressure.
Under the above-stated assumptions Eq. (15) reads
�� �
�
�� � � � �g
f
p
3
0. (22)
In the rest of this chapter the solutions of Eq. (22)
determined under the additional constraint (14) are
analyzed.
It is useful for the next consideration to note here
that Eq. (11) with Pw( )� from (3) can be derived
from the following variational principle: the total
energy of the liquid at equilibrium must reach a mini-
mum, which is found under the additional restriction
of Eq. (12), i.e.,
� �[ ( )]E � ��p d2 0r r . (23)
The total energy of the liquid, which includes contri-
butions from the surface tension, the gravitational
force, and the van der Waals force, is
E � � � �
�
�
�
�
!
"
"�d g
f2 2
2
1
1
2 2
r � � �
�
( )
. (24)
The normalization condition (12) is taken into ac-
count in Eq. (23) using the Lagrange undetermined
multiplier method, the pressure p at surface of liquid
playing the role of that undetermined multiplier.
The first integral of the one-dimensional equation
(22) is
1
2
2� � �� �U C( ) , (25)
whereU g p f /( ) ( )� � � ��� � �4 3 22 2 .
To analyze the above set of equations, the follo-
wing mechanical analogy is useful. Equation (25) can
be considered as the energy conservation law for an ef-
fective point-like particle with mass m = 1 moving in
an external potential, U( )� . The variable x plays the
role of time, and � is the coordinate of the particle.
The first term in Eq. (25) is the kinetic energy of the
particle, and C is the total energy.
Figure 2 shows the evolution of the effective poten-
tial U( )� with increasing layer thickness. Figure 2,a
illustrates the case of small layer thicknesses h h� � ,
where h� is defined by Eq. (6). The point of the maxi-
mum � � h on the U( )� curve corresponds to a stable
flat surface of the thin liquid layer. The «velocity» of
the particle at this point is � �� 0, and C U h� ( ).
With increasing h up to the critical value h� the
curveU( )� transforms into a curve with an inflection
point at h h� � (see Fig. 2,b). If h h� � , the dip on the
curve develops again (Fig. 2,c). The point of the mini-
mum of the functionU( )� at some � � �h corresponds
to an unstable flat state of the liquid surface. Forma-
tion of the possible stationary surface deformation
wave is described by periodic motion of the particle in
the potential well around the minimum in the region
whereU C( )� � � 0.
The coordinates of the turning points that restrict
the motion of the particle can be found from the equa-
tion U C( )� � � 0. In the general case this equation
has four different solutions, � � zi , i � 1 4, ..., , which
are the roots of the polynomial
� � � �g p C f4 3 22 2� � – .
We will enumerate these roots in the following order:
z z z z4 3 2 1� � � .
Now it is reasonable to remind ourselves that we
are trying to find a possible nonuniform solutions of
the problem (22). In the case of an infinite film area
� � � � � �x the nonuniform shape of the surface
� ( )x can be periodic, or, under some special condi-
tions, a soliton-like one. Therefore we should obtain
the periodical solutions of Eq. (22) and investigate
their properties, first.
Stationary nonlinear waves at the surface of a thin liquid layer under inverted gravitation conditions
Fizika Nizkikh Temperatur, 2004, v. 30, No. 1 83
U
0
h
C
h�
�
h < h�a b h�
h�h�
h =
C
0 ��
U
c
U
0
C
h
h�h >
z1z2z 3
z4
Fig. 2. Evolution of the effective potential U( )� with increasing mean layer thickness h. a — small thickness, h h� �;
b — the thickness of layer is equal to the critical value h� ; c — the thickness h h� �. The points zi , i = 1, ..., 4 corre-
spond to solutions of the equation U z C( ) � .
The general periodic solution to Eq. (25) in the region
z z2 1� �� can be written in integral form as follows:
d
C U
x x
z
�
�
�
2
2
0
[ ( )]�
� �� . (26)
The arbitrary parameter x0 in the right-hand side of
Eq. (26) appears due to the invariance of the initial
equation (11) with respect to translations in the hori-
zontal plane.
The integral in the left-hand side of Eq. (26) can be
expressed via the elliptic integrals of the first and
third kinds, F r( , )# and $( , , )# n r , as follows:
2
1 3 2 4
2 3
1 2
1 3( )( )
( ) , ,
z z z z
z z
z z
z z
r
� �
�
�
�
�
�
�
�
�� �
�
�
� $ �
�
!
" �
�
z F r
x x
a3
0( , ) ,# (27)
where
�
�
�
�
� �
� �
arcsin
( )( )
( )( )
,
z z z
z z z
1 3 2
1 3 3
r
z z z z
z z z z
�
� �
� �
( )( )
( )( )
1 2 3 4
1 3 2 4
,
and a g /� ( )� � 1 2 is the capillary length.
Equation (27) defines in implicit form the recon-
structed shape of the free surface of the liquid layer
covering the solid substrate under inverted gravitation
conditions. The period T of the surface deformation
wave defined by Eq. (27) is equal to
T
a
z z z z
�
� �
%
4
1 3 2 4( )( )
% �
�
�
�
�
�
�
�� � �
�
�
�
�
�
�
�
( ) , , ,z z
z z
z z
r z F r2 3
1 2
1 3
32 2
$
& &
!
". (28)
We should mention that the roots zi depend on two
free parameters, i.e., on the pressure p at the surface
and on the constant of integration C (in addition to
the dependence on the constants characterizing the
properties of the liquid and its interaction with the
substrate). The solution defined by Eq. (27) should
obey the normalization condition (14), too. This con-
dition provides an additional relation between param-
eters p and C. Thus one can conclude that the function
� depends on one free parameter, in addition to the de-
pendence on the x coordinate. This parameter can be
chosen arbitrarily. In the subsequent analysis we sup-
pose that the constant C plays the role of this parame-
ter. The period T of the stationary wave depends on
this parameter, as well.
One possibility of avoiding the problem nonu-
niqueness of the parameter C could be the realization
of a soliton-like solution for � ( )x . Following the con-
ventional recommendation (see, e.g., the problem of
steady-state wave propagation at the free liquid sur-
face in Ref. 20) we have to use for this purpose the ad-
ditional requirement
T � �. (29)
However, this requirement turns out to be incom-
patible with the normalization condition (14), be-
cause in this case the integral
dx x h� �( ( ) )� (30)
diverges. Therefore the conventional soliton-like solu-
tion is not valid.
Now, turning back to the general periodic picture of
reconstruction, we have only one possibility for resol-
ving the problem of nonuniqueness of the parameter C.
Here it is necessary to use stability arguments. Using the
results obtained below (see paragraph 2.3) as a veri-
fication, we can show that the deformation �( )x with the
maximal period T is the most favorable energetically. For
the case of unbounded geometry �� � � ��x this
means that the period of the structure on the recon-
structed surface should tend to infinity. Therefore ener-
getically stable periodic reconstruction of the un-
bounded thin helium film is impossible, too.
Nevertheless, we can bring the factor of stability
into the formulation of the problem artificially, by
considering a film of limited area. Such a scenario cor-
responds to the experimental situation. So, let us as-
sume that
� ' 'L/ x L/2 2 (31)
and, in addition, that the following boundary condi-
tions are fulfilled:
� �(� � )L 0. (32)
The simplest boundary conditions (32) with the con-
tact angle at the walls � � 0 are necessary to save the
basic instability indicator h� in the form (6).
After reformulation (31), (32) the dependence of
the function �( )x defined by Eqs. (27), (28) on the
parameter C changes its character. Under the new
boundary conditions C can take only a definite discrete
set of values. Now our objective is to demonstrate that
the most energetically preferable solution can be found
among the solutions corresponding to these values.
84 Fizika Nizkikh Temperatur, 2004, v. 30, No. 1
G.V. Kolmakov, E.V. Lebedeva, A.A. Levchenko, L.P. Mezhov-Deglin, A.B. Trusov, and V.B. Shikin
2.3. Reconstruction of the free surface of a thin
layer in restricted geometry
Following the speculations above, we have to in-
vestigate the solutions (27) under the boundary condi-
tions (32).
The values of the parameter C at which the bound-
ary conditions (32) are satisfied can be found from the
requirement that the ratio L/T is equal to some natu-
ral number N. The number N coincides with the num-
ber of maxima of the function �( )x in the region
� � �L/ x L/2 2. So, such a solution describes the
formation of N humps, or «droplets», at the liquid
surface. These droplets are connected between each
other by the liquid layer covering the substrate.
Detailed numerical analysis of the stability prob-
lem for the solution with N droplets at the surface
shows that only the solution with N = 1 can be real-
ized in experiment; see below.
First, the energy of the liquid layer with N droplets
at the surface («N droplet configuration») is estimated
numerically. The results of estimations show that the
configuration with N = 1 corresponds to a minimum of
the energy, while the energy of configurations with
N � 1 is higher. Figure 3 shows an example of two pos-
sible stationary profiles of the reconstructed surface
with N � 1 (curve 1) and N � 2 (curve 2) calculated for
case h � 1.78 h� . The horizontal coordinate x is expressed
in units of the capillary length a, and the vertical coordi-
nate is expressed in units of the critical thickness h� . The
distance between the vertical walls is L � 13.3 a. The line
with zero vertical coordinate is the substrate surface. The
horizontal dashed line corresponds to the flat layer of
thickness h h� � . The estimated energy of the configura-
tions is expressed in energy units E g h //
0
1 2 2 4� ( )� .
The energy of the configuration with N � 1 (the profile
has one maximum) is equal to E1 � � 3.80 E0, while the
energy of the configuration with two maxima is equal to
E2 � � 2.86 E0. So, the configuration with N � 1 is
energetically preferable and should be realized in
experiment.
Now we study the stability of the solution with a
single maximum against small perturbations of the sur-
face. For this purpose the sign of the second variation
of the energy �( )2 E at small perturbations of the shape
of the liquid surface is determined. For a stable solu-
tion the second variation is positive definite (a solution
should correspond to a minimum of the total energy).
The variation �( )2 E is represented by the following
functional, which is quadratic in the small perturbation
� �( )x of the steady-state surface profile �( )x :
� �� � �( ) ( ) � ( )2
2
2
E dx x x
L/
L/
�
�
� * ,
where
�
( )
* � � � �
� �
�
�2 2
32
2 4x
g f
x
is a linear differential operator. The positive definite-
ness of the second variation of energy is equivalent to
the fact that the operator �* has only positive
eigenvalues � n � 0. The problem on finding the spec-
trum � n of the operator �* can be mapped onto the
well-known problem of the energy spectrum of a quan-
tum particle with mass m /� 1 � moving in the poten-
tial wellU g/ f/ x� � � �2 3 4( ). We study the stabil-
ity problem for the case when the perturbation does
not affect the wetting conditions at the walls, so the
boundary conditions for the «wave function» are
� �� �( )x 0 at x L/� ( 2. It is clear from this consider-
ation that the perturbation that corresponds to the
lowest eigenvalue of the «Hamiltonian» �* (i.e., to the
«ground state level») is the most dangerous for de-
stroying the stability of the steady-state surface shape.
The low-energy part of the spectrum of the operator
�* is found numerically, by direct diagonalization of
the matrix corresponding to the operator �*. The ma-
trix elements *mn are calculated in an orthogonal nor-
malized basis presented by a set of symmetric (g) and
antisymmetric (u) functions
2 2/L q x /L q xn
g
n
ucos sin( ), ( )( ) ( ) ,
Stationary nonlinear waves at the surface of a thin liquid layer under inverted gravitation conditions
Fizika Nizkikh Temperatur, 2004, v. 30, No. 1 85
6 4 2 0 2 4 6
6
4
2
0
�
/ h
�
x/a
h = 1.78 h�
E1= 3.80 E0
E2= 2.86 E 0
2
1
Fig. 3. Results of a calculations of the stationary shape of
the reconstructed layer with a single droplet (curve 1) and
with two droplets (curve 2) formed at the surface. The x co-
ordinate is normalized to the capillary length a, and the
layer thickness is normalized to the critical thickness h�. The
distance between the vertical walls is equal to L �13.3 a.
The mean thickness of the layer is h � 1.78 h�. The dashed
line corresponds to the flat surface of a layer with h h� �. The
estimated energy of the single- and two-droplet solutions are
E1 � �3.80 E0, and E2 � �2.86 E0, where E gh /0
2 4� � .
where q n /Ln
g( ) ( ) ,� �2 1&
and ( ) , , , , ...( )q n /L nn
u � � �& 2 1 0 1 2
The function �( )x is symmetric with respect to in-
version of the x coordinate, � �( ) ( )� �x x . This means
that the matrix is diagonal with respect to the index of
the perturbation parity, i.e., only the matrix elements
with two u or two g indices are nonzero. This enables
examination of the stability problem against symmet-
ric and antisymmetric perturbations separately. The
matrix element calculated for two basis functions of
the same parity corresponding to excitation levels n
and m is
*nm mn
nm
g
f
x
� ��
��
!"
�
�
�
�
�
�
�
�
�
�2 2
3
1
4( )
,
where � nm is the Kroneker delta, and the index u or g
is omitted. The elements ( )1 4/ nm� are calculated nu-
merically for the lowest 10 harmonics, n m, ' 10. Nu-
merical estimations show that taking higher harmonics
( , )n m � 10 into account does not change noticeably the
character of the lower part of the spectrum of the oper-
ator �*. We study the case where L a� 15 and h h� �10 .
Analysis show that all eigenvalues are positive, � n � 0.
This means that small perturbations of the recon-
structed surface with N � 1 increase the total energy,
and, hence, the shape of the steady state of the surface
is stable.
Such a direct stability analysis can not be done suc-
cessfully by numerical reasoning if the mean thickness
of the liquid layer h h� �10 or the dimension L a� 15 ,
due to losses of accuracy in the numerical calculations
of the values of roots zi .
Meanwhile, at high h one can use the following qual-
itative considerations to demonstrate the fact that the
solution with one droplet at the reconstructed surface is
energetically preferable, and only this configuration can
be formed [21]. In the general situation, the profile of
the steady state of the reconstructed surface corresponds
to the formation of N droplets at the surface, which are
connected by the liquid layer covering the substrate. In
the limiting case where the volume of liquid is macro-
scopically large, the thickness of such a layer is small
with respect to the height of the droplet. The main part
of the liquid is accumulated in droplets, and the volume
of liquid contained in the layer covering the substrate
between the droplets is negligibly small. As a first ap-
proximation the shape of each droplet can be described
by the 1D Frenkel solution (see Ref. 13)
�( ) ( cos )x
A
x/a� �0
2
1 , (33)
where the height of the droplet is
A S/ a0 � & , (34)
and S is the «two-dimensional volume» (area) of the
droplet. The energy of a single droplet whose shape is
described by Eq. (33) can be calculated directly. This
gives
E S
a
S1 3
2
2
( ) � �
�
&
. (35)
The total energy of the liquid can be estimated as a
sum of energies of each «isolated» droplet. If total
volume S of liquid is distributed among N identical
droplets, than the energy of such a configuration can
be estimated as follows:
E NE S/N
a
S
N
� � �1 3
2
2
( )
�
&
. (36)
So, the energy of the liquid is increased if the liquid
is redistributed from one droplet into a number of
droplets. In the macroscopic case ( )h h�� � the solu-
tion with one droplet formed inside a cell is also
preferable from energy considerations.
From the results obtained it follows that the con-
sideration of the reconstruction process can be re-
stricted to the case where a single droplet is formed at
the surface. Below we present results of numerical
computations of the evolution of the shape of the re-
constructed layer with one droplet formed at the sur-
face with increasing mean layer thickness h.
Figure 4 shows the shape of the reconstructed sur-
face obtained by numerical integration of Eq. (22) in
cases where the mean liquid layer thickness is equal to
h h� �15. (curve 1), 2 2. h� (curve 2), and 7h� (curve 3).
The x coordinate at the graph is expressed in capillary
86 Fizika Nizkikh Temperatur, 2004, v. 30, No. 1
G.V. Kolmakov, E.V. Lebedeva, A.A. Levchenko, L.P. Mezhov-Deglin, A.B. Trusov, and V.B. Shikin
4 2 0 2 4
�
/h
�
3
2
1
6
4
2
0
x/a
Fig. 4. Evolution of the shape of the reconstructed liquid
layer with a single droplet at the center with increasing
mean thickness h. Curve 1: h � 1.5 h�, curve 2: h � 2.2 h�,
curve 3: h h� �7 . The thickness of the layer is expressed in
units of the critical thickness. The distance between the
vertical walls is L a� 9 .
length units a, and the z coordinate is expressed in the
critical thickness units h� . The distance L between the
vertical walls is equal to 9a. The z coordinate of the
substrate surface is equal to zero.
It can be seen from Fig. 4 that the amplitude of the
wave of surface deformation increases gradually with
increasing h above h� . The maximum of the deforma-
tion is situated at the center of the cell. The wave ac-
quires a droplet-like shape at sufficiently large h. The
characteristic horizontal size of the droplet is of the
order of a few capillary lengths, and it depends
weakly on the mean thickness h. The thickness
� �min ( )� (L/2 of the liquid layer covering the sub-
strate outside the droplet is less than the critical
thickness h� , and it decreases with increasing mean
thickness h. In case h h�� � the main part of the liquid
is concentrated in the droplet, and the volume of
liquid in the thin layer wetting the substrate outside
the droplet is much smaller than the total liquid
volume.
As was shown in [21]
�min
h�
�
�
�
�
��
3
�
h�
3�max
, �max � A0 � �min. (37)
Here A0 is given from Eq. (34). The combination of
Eq. (37) with the requirement �min � dw leads to es-
timations of the critical Smin, which is necessary for
simplification (3). So, it should be
S S� min, where S ah / dwmin � �& 4 33 . (38)
Estimate (38) confirms the propositions made in the
preceding qualitative analysis.
Figure 5 shows the dependence of the droplet height
A L/� �� �( ) ( )0 2 on the parameter � � � � �( )h h /h
(the overcriticality parameter). For comparison of our
results with the results of macroscopic considerations
[13], the droplet height A is normalized to the height
A0 given by Eq. (34) for Frenkel’s droplet, i.e., on the
height of the droplet formed under inverted gravitation
conditions from a liquid of the same volume S Lh� and
calculated in case where the angle of wetting of the
substrate (not the walls) by the liquid is � � 0. In these
calculations the horizontal dimension of the cell is
L a� 9 , which is the same as in Fig. 4.
It follows from Fig. 5 that at high thickness h h�� �
(i.e., at high overcriticality level, � �� 1) the height of
the droplet formed at the center of the cell is close to
the height of Frenkel’s droplet of the same volume.
The difference between the heights of the «van der
Waals droplet» and Frenkel’s droplet is about 10% at
� + 6 and decreases with increasing �. Thus, at a layer
thickness one order of magnitude higher than the
critical thickness h� the shape of the droplet formed
under inverted gravitation conditions can be described
to rather good accuracy by the «macroscopic» solution
(33). The effective angle of wetting of the substrate
by liquid at the edges of the macroscopic droplet is
�eff � 0. (39)
The characteristic horizontal dimension of the droplet
is of the order of capillary length at � � 1. So, if the
overcriticality level is not very small, and the dimen-
sions of the cell are much larger than a, then the
boundary conditions should have a weak effect on the
equilibrium shape of the droplet.
At h ∼ h� the dependence of the droplet height
A A h� ( ) differs significantly from that obtained in
the macroscopic approximations, and at h h� � (at the
overcriticality � � 0) the height of the droplet tends to
zero. At thicknesses smaller than the critical thickness
the flat liquid surface is stable, and the layer is in the
nonreconstructed state (no deformation wave appears
at the surface). Details of the behavior of the surface
shape at h close to h� depend on the value of the wet-
ting angle �. The corresponding analysis will be pub-
lished elsewhere.
Let us consider in addition to (38) the limitations
for applicability of the results obtained. The main as-
sumption in the treatments presented is the small
angle approximation, | ( )|� ��� x 1. The characteristic
length scales in the horizontal and vertical directions
differ drastically. The horizontal size of the deforma-
tion wave is of the order of the capillary length. For
liquid hydrogen it is equal to a + 0.2 cm. The charac-
teristic scale of thickness is presented by the critical
thickness h� , and in the case of a hydrogen film
covering a dielectric substrate it is h�
�+ 10 4 cm. The
condition at which our treatment fails can be written
Stationary nonlinear waves at the surface of a thin liquid layer under inverted gravitation conditions
Fizika Nizkikh Temperatur, 2004, v. 30, No. 1 87
0 1 2 3 4 5 6 7
A
/A
0
�� = h/h 1
1.0
0.8
0.6
0.4
0.2
Fig. 5. Dependence of the amplitude of the wave of sur-
face deformation on the overcriticality parameter
� � � �h/h 1. Amplitudes are measured in units of A0,
where A S/ a0 4� & is the amplitude of Frenkel’s droplet,
and S is the total area of one-dimensional layer.
as A/a ∼ 1, or, in equivalent form, h∼ a /L2 . One can
conclude from these estimations that the results of our
considerations are valid (from the side of large thick-
ness) if � � �a/h ∼ 103, or h < 10–1 cm, for L ∼ a.
From the side of small thickness the applicability of the
results is restricted to the possibility of considering the
liquid as a continuous medium, i.e., h � �10 6 cm. So,
there is a macroscopically wide interval of layer thick-
ness where the results obtained are valid.
The problem of the stationary shape of a liquid
droplet can be solved without the assumption of
smallness of the gradient of the surface deformation if
the capillary forces are taken into account (see, e.g.,
[13]). This assumption is of importance for the pre-
sentation of the van der Waals pressure as a power law
of the layer thickness in Eq. (1). The condition
| ( )|� ��� x 1 is of crucial importance for calculations of
the electric pressure at the surface of charged liquids,
too (see the next Section).
3. Macroscopic charged droplet formed
at a solid substrate
3.1. Equipotentially charged surface of
a liquid in an electric field
In this Section the evolution of the shape of a
charged droplet with increasing mass of the droplet
and external electric field strength is studied. Calcula-
tions are carried out for two cases, where a droplet is
suspended at a metallic substrate or it is lying on the
substrate. In the first case the direction of the stretch-
ing electric force acting on the liquid surface is the
same as the direction of the gravity force, and in the
second case the forces have opposite directions. These
situations correspond to conditions of experiments
[12] with liquid hydrogen droplets. In the experi-
ments the upper or the lower plate of a horizontally
arranged flat capacitor play the role of the substrates.
It is supposed that the charges are localized under
the free surface of the liquid and they create a
quasi-two-dimensional layer, which totally screens the
electric field in the bulk of the liquid (i.e., the electric
field and the density of charges in the bulk of the liq-
uid are equal to zero). The electric potential at the liq-
uid surface is maintained equal to some constant value
by an external battery, and the total number of posi-
tive charges localized under the surface can be varied
due to a continuously working source of charges,
which is placed in the bulk of the liquid.
A macroscopic situation will be considered here,
i.e., the height of the droplet is supposed to be much
larger than the critical thickness h� . As was pointed
out above, a typical value of the critical thickness is of
the order of 10 4� cm, and it is smaller than droplet
heights suitable for optical observations. The influ-
ence of the van der Waals forces on the shape of the
macroscopic droplet is negligibly small, and one can
exclude it from considerations. Meanwhile, the van
der Waals forces are comparable with the electric and
gravitational forces at the edges of the droplet, where
the liquid depth becomes the same order as the thick-
ness of thin liquid layer wetting the substrate, i.e., of
the order of the critical thickness. As is shown in the
previous Section, the influence of the wetting layer at
the substrate surface on the shape of the droplet could
be taken into account by using the effective boundary
conditions (39) that read � �� 0 at the edges of the
droplet. The thin liquid layer wetting the substrate
outside the droplet should be considered to have zero
thickness ( )� � 0 in the macroscopic consideration
used in this Section.
It was shown above that almost all the volume of a
neutral liquid covering a substrate is concentrated in a
single droplet at the substrate surface. It is natural to
assume that this property also holds for a charged
liquid, and a single droplet should appear at the
equipotentially charged surface of the liquid.
Here we consider a real «two-dimensional» situa-
tion in which the shape of the liquid surface depends
on two coordinates in the ( , )x y plane. We also assume
that the droplet formed at the surface has an axially
symmetric shape, i.e., the liquid surface is described
by an equation z � � (| | )r .
Calculations of the equilibrium shape of the drop-
let are carried out in two steps. First, the effective en-
ergy of the charged droplet formed at the capacitor
plate is calculated as a function of the shape of the
droplet and of the voltage U applied to the capacitor.
After that, the dependence of the height A and the ef-
fective radius R of the droplet is found using the vari-
ation approach, from the condition that the effective
energy acquires a minimal value at the equilibrium
state.
Let us consider a droplet suspended at the upper plate
of the flat horizontal capacitor (i.e., the droplet under
inverted gravitation conditions). The equilibrium shape
of the liquid droplet with an equipotentially charged
surface corresponds to a minimum of the effective energy
(see Ref. 22)
E E� � ��mech
1
8
2 2
&
�
d dzr E
d
. (40)
Here Emech is the mechanical energy of the liquid (a
sum of the surface energy and the energy of the liquid
in the gravitational field), the last term in the
right-hand side presents the energy of the electric field
in the capacitor, E is the electric field strength in the
88 Fizika Nizkikh Temperatur, 2004, v. 30, No. 1
G.V. Kolmakov, E.V. Lebedeva, A.A. Levchenko, L.P. Mezhov-Deglin, A.B. Trusov, and V.B. Shikin
gas phase, and d is the distance between the capacitor
plates. The frame of reference is chosen the same as
used in the previous Section: the (x,y) plane coin-
cides with the surface of the upper capacitor plate, at
which the droplet is suspended, and the z axis is di-
rected along the gravitational force.
The form of the expression for the electric field en-
ergy in a capacitor depends on the ratio between the
horizontal size of the droplet (its effective radius R)
and the distance d between the capacitor plates. In
the calculations presented below we assume that the
condition d/R �� 1 holds.
Also, in order to simplify the calculations it is as-
sumed that the height of the droplet A is small with
respect to the distance d, A << d.
From this it follows that the angle of the slope of
the surface with respect to the horizon is given in or-
der of magnitude by a product of two small quantities,
| |,� ∼ �/R∼ ( ) ( )�/d d/R% , and it is a small quantity
of high order. Restrictions for applicability of the re-
sults obtained, which follow from these assumptions,
will be discussed later.
In order to determine the contribution of the elec-
tric field energy to the total effective energy (40) one
should calculate the electric potential - inside the ca-
pacitor. The electric potential obeys Laplace’s equa-
tion and the boundary conditions at the upper and
lower plates
- - -�� � �� �0 0, | , |( )z z d Ur . (41)
The value of the electric potential at the charged sur-
face of a droplet suspended at the upper plate is as-
sumed to be equal to the value of the potential at the
plate. This value is chosen equal to zero. The value of
the potential at the lower plate is denoted as U.
It is convenient to introduce a new potential. defined
by the equation . -� � E z0 , where E U/d0 � � . As
follows from Eq. (41), the boundary conditions for
the . potential are the following:
. ��| ( )z E� �r 0 , . | z d� � 0. (42)
In the case under study, � �� ��d R, the potential .
could be calculated using perturbation theory, with the
liquid layer thickness � considered as a small function,
. . .� � �( ) ( ) ...,1 2
(43)
where . ( )1 ∼ � , . ( )2 ∼ �2, etc. In order to determine
the boundary conditions for the functions . ( )n one
should substitute the expansion (43) into Eq. (42)
and equate the terms of the same order. For the
first-order terms the conditions read
. . ��/�( ) | , |1
0 00z d z E� �� � . (44)
We assume that the droplet has an axially symmetric
shape, i.e., � �� ( )r , where r � | |r . The solution of the
problem of the potential distribution with boundary
conditions (44) is known from potential theory. In
the region � ' 'z d, where the electric field is not
equal to zero, the solution can be found using a
Bessel transformation. It reads
. ( )( , ) ( ) ( )
( )1
0
0
0r z E dt J rt F t t
t d z
td
�
�
�
�
sh
sh
, (45)
where F t dr r J rt r( ) ( ) ( )�
�
�
0
0 � ,
and J t0( ) is the Bessel function of order zero.
The Bessel transform F t( ) of the function �( )r is es-
sentially nonzero if its argument t /R' 1 . Thus the ar-
guments of the hyperbolic functions in the right-hand
side of Eq. (45) are of order d/R �� 1, and these func-
tions can be expanded in series in their arguments. Be-
low we need the expression for the derivative of the
. ( )1 function. Calculations give the following relation
�. � �( ) |1
0 0/ z E /dz � � � . (46)
Here we use the inverse Bessel transformation in the
calculations.
We will calculate the dependence of the electric
field energy on the shape of the droplet up to third-or-
der accuracy in the ratio �/d. The energy of the elec-
tric field in the capacitor is expressed via the . poten-
tial as
Eel � ,��
1
8
2 2
&
.
�
d dzr
d
( ) , (47)
where we have neglected an inessential constant in
the right-hand side of (47). In the same third-order
approximation this expression can be reduced to an
integral over the surface of the upper plate as follows:
Eel � �
�
�
�
�
!
"
"
�
�
�
�
�
!
"
�
E
d
z zz
0 2
1
0
2
1
28&
�
�.
�
�
� .
�
)
r
( ) ( )
"
�
�
�
�
�
�
�
�
z 0
.
(48)
It follows from the relation obtained that the energy
of the electric field written up to the third-order
terms is expressed via the first order term of the . po-
tential and via the � function. One can exclude the
. ( )1 function from the expression (48) using Eq. (46)
and a similar equation for the second derivative of the
potential obtained from the integral representation
(45). This gives the following dependence of the en-
Stationary nonlinear waves at the surface of a thin liquid layer under inverted gravitation conditions
Fizika Nizkikh Temperatur, 2004, v. 30, No. 1 89
ergy of electric field in the capacitor on the droplet
shape:
Eel � �
�
�
�
�
�
��
U
d
d
d d
2
2
2
3
38&
� �)
r .
One can see from this expression that the energy of
the electric field depends locally on the thickness of
the liquid layer �( )r . This results from the assumption
that the effective radius of the droplet is large with
respect to the distance between the capacitor plates.
In calculating the contribution from the surface en-
ergy to the mechanical energy of the liquid to the same
accuracy, one should take into account only the
lowest-order terms in the angle of slope of the surface
in the expression for the capillary energy.
The total effective energy of the system (40) calcu-
lated in the approximation adopted is given by the fol-
lowing expression:
E � , � �� �
�
� �
2
1
2
2 2 2 2d g dr r( )
� �
�
�
�
�
�
��
U
d
d
d d
2
2
2
3
38&
� �)
r . (50)
Note that the electric energy in (40) has the oppo-
site sign to the mechanical energy.
3.2. Equilibrium shape of a charged droplet
Let us determine the dependence of a shape of a
charged droplet on the voltage U applied to the capac-
itor. A nonlinear equation describing the shape of the
droplet can be derived using a variational principle
from expression (50) for the effective energy of the
system in the same way as was done for Eq. (1). But
the solution of the equation obtained in the 2D situa-
tion meets difficulties because of the absence of a first
integral of the equation. In this work we determine
the height and the radius of the droplet using a varia-
tional approach in which the shape of the droplet is
approximated by some trial function.
We seek the coordinate dependence of the trial
function � in a general axially symmetric form
�( )r � �
�
�
�
�Af
r
R
, (51)
where A is the droplet height and R is the effective
radius of the droplet. Parameters A and R are positive
variables playing the role of variational parameters in
the problem. The function f x( ) is a bell-shaped func-
tion of the variable x r/R� . It is normalized by the
conditions f( )0 1� , and f x( ) � 0 at x >> 1 (recall
that the effective wetting angle at the edges of the
droplet is equal to zero).
The dependence of the total energy of the liquid on
the variational parameters A and R can be calculated
by substituting the function (51) into expression
(50). This gives the following expression
E � � �
�
�
�
�
�
�
�
�
& &2
1
2 8 8
0
2
2
3 2
2 2
2
4 3
3 2c A g
U
d
c A R
U
d
c A R ,
(52)
where the constants cn are defined as follows:
c dxx f x0
0
22� �
�
�& ( ),
c dx x f x nn
n� �
�
�2 0
0
& ( ) ( )at .
(53)
The region of integration in (53) is enlarged to infin-
ity due to convergence of the integrals.
The values of the variational parameters in the
equilibrium state can be determined from the condi-
tion that the effective energy of the liquid acquires a
minimal value under the additional normalization con-
dition (12). The normalization condition can be ex-
pressed via the variational parameters as
c AR V1
2 � . (54)
The condition (54) can be taken into account using
the Lagrange multipliers method, as was done earlier
in calculations of the stationary shape of the neutral
liquid surface. Thus the values of the parameters in
the equilibrium state can be determined from the
equations
�
�
�
�
~
,
~E E
A R
� �0 0, (55)
where ~E E� � pAR2 and p is the Lagrange multiplier.
The solution of Eqs. (55) gives the following de-
pendence of the droplet height A on the voltage U:
A k V
U
d
g
k U V
d
� �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
1
2
3
2
2
4
1
4 4&
�
&
. (56)
The constants in (56) are equal to k c / c c1 2 0 12� ,
k c /c c2 3 0 1� .
Note that the function f x( ) enters into expressions
(54), (56) only as an integrand in the constants k k1 2,
and c1. Thus the choice of the explicit form of the
function f x( ) has only a weak effect on the calculated
values of the droplet sizes.
In order to estimate the values of the constants in
(56) we choose the function f x( ) in the form
90 Fizika Nizkikh Temperatur, 2004, v. 30, No. 1
G.V. Kolmakov, E.V. Lebedeva, A.A. Levchenko, L.P. Mezhov-Deglin, A.B. Trusov, and V.B. Shikin
f x q J x J x( ) ( ( ) ( )), ,� � � �0 0 1 100 0
f x x( ) , .� �0 10 (57)
Here 01 3 83+ . is the minimal value of the argument of
the Bessel function of the first order, at which the func-
tion goes to zero, J1 1 0( )0 � , and q � ( ( ))1 0 1
1� +�J 0
+ 0 71. is a normalization constant. The explicit form
(57) of the f x( ) function is chosen from the following
considerations. The function (57) is a solution of the
equation which describes the liquid surface shape in a
linear approximation over the droplet’s height. The
exact shape of the droplet should be close to that
function at a small nonlinearity level in the system,
i.e., when the inequalities | | ,, �� ��� 1 1A/d hold.
Values of the constants estimated from definition
(53) using the function (57) are k1 � 0.0754,
k2 � 0.106, c1 � 18.5.
The dependence of the effective radius of the drop-
let on the voltage U can be calculated from Eq. (54)
as R V/c A /� ( )1
1 2, where A is given by Eq. (56). The
diameter of the droplet D (i.e., the diameter of the cir-
cle basement of the droplet) can be estimated as
D R+ 7 66. .
Let us analyze the result obtained. The droplet
height A estimated from (56) grows infinitely when
the voltage is increased up to the value
U
d
k Vc
/
2
4
2
1 2
4
�
�
�
�
�
�
�
&�
. (58)
At U Uc� 2 Eqs. (55) have no solution. At these
voltages the effective energy of the droplet as a func-
tion of amplitude A decreases with increasing A and,
hence, has no minimum. That means that a suspended
droplet is absolutely unstable at high voltages. Thus
the voltageUc2 given by Eq. (58) can be considered as
an estimate for the second critical voltage, at which
the liquid droplet loses mechanical stability in the ex-
ternal electric field. At voltages higher thanUc2, dis-
charge from the surface accompanied by transfer of
liquid from the suspended droplet to the lower plate
of the capacitor should take place. Expression (58)
should be considered only as an order-of-magnitude
estimate because in the case where the amplitude A is
comparable with the distance d the theory developed
is of a qualitative character.
Dependence (56) can be used to describe the evo-
lution of the shape of the droplet formed from a liquid
layer covering the lower plate of the capacitor, with
increasing voltage. In this case the stretching electric
force, which acts on the equipotentially charged sur-
face of the droplet, is directed oppositely to the gravi-
tational force. To obtain the voltage dependence of
the amplitude of a droplet lying on a substrate one
should change the sign of the constant g in Eq. (56).
This gives the formula
A at U U gdc� � �0 41
3& ,
A k V
U
d
g
k U V
d
U Uc� �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
1
2
3
2
2
4
1
1
4 4&
�
&
at .
(59)
Note that the voltage Uc1 coincides with the critical
voltage at which the reconstruction of a thin equi-
potentially charged layer of liquid takes place, see
Ref. 3. From the dependence (59) one could see that
a droplet is formed at the surface of liquid layer if
the «renormalized» gravitation acceleration geff �
� �g U /Uc
2
1
2 changes its sign from positive to
negative with increasing voltage U. Thus atU Uc� 1
the surface of the liquid layer condensed on the
lower plate of the capacitor could be considered, as a
first approximation, to be under «inverted gravita-
tion conditions», where the effective gravitational
acceleration is directed oppositely to the real gravi-
tational force. This situation is quite similar to the
case of the reconstruction of the thin van der Waals
liquid layer considered in the previous Sections. But
here the renormalization of the gravitational acceler-
ation that acts on the liquid arises from the electric
forces.
As one can see from Eq. (58), the second critical
voltageUc2, at which the droplet surface becomes un-
stable, does not depend on the gravitational accelera-
tion. So, in the «small angle» approximation used the
voltages at which the discharge from the surface
should take place have the same values in the cases of
lying and suspended droplets.
We use the A U( ) dependence given by Eq. (56) for
treatment the results of experiments [12] with
charged hydrogen droplets suspended at a metal
substrate. In these experiments the droplet was cre-
ated by condensation of hydrogen from a gas phase on
a cooled upper plate of a horizontally arranged capac-
itor. This technique allows one to control the layer
thickness over a wide range by using precise control of
the substrate temperature and of the volume of gas in-
troduced into the experimental cell.
In these experiments the evolution of shape of the
suspended charged droplets with increasing voltage
applied to the capacitor plates and with increasing
volume of the droplet was studied.
Figure 6 shows the changes of the profile of the
droplet with increasing voltage U, obtained by analisis
of snapshots of the droplet. The volume of the droplet
was equal to V � 60 mm3 in this experiment. The dis-
tance between the capacitor plates was d � 3 mm. The
experimental data in Figs. 7 and 8 are reproduced from
Stationary nonlinear waves at the surface of a thin liquid layer under inverted gravitation conditions
Fizika Nizkikh Temperatur, 2004, v. 30, No. 1 91
Figs. 8 and 9 of Ref. 12. Figure 7 shows the dependence
of the droplet height A on the voltage U obtained from
plots in Fig. 6. The points represent the results of mea-
surements, and the solid curve the dependence (56).
Note that the voltage at which the discharge of the sur-
face was observed is equal to Uc2 800+ V. Figure 8
demonstrates the evolution of the dependence of the
height of the droplet on the droplet volume with in-
creasing voltage U. The solid circles show the depend-
ence of the amplitude A on the volume V measured at
zero voltageU � 0. The squares show the same depend-
ence measured at U � 630 V. The dashed lines corre-
spond to the theoretical dependencies (56) calculated
at the given voltages.
In the experiments the instability of a droplet at
high voltages, accompanied by a discharge from the
droplet surface, was also observed. The values of the
second critical voltage Uc2 measured in the experi-
ments were approximately three times smaller than
the values estimated from Eq. (58). Such discrepancy
can be attributed to the fact that the theory developed
is of a qualitative character at voltages close to Uc2.
The observed character of the development of the in-
stability is different from that assumed in the calcula-
tions. Observation showed that the discharge was ac-
companied by the creation of a «geyser» from the
surface, similar to that observed in experiments with
the charged surfaces of bulk liquid hydrogen and he-
lium [12,23]. The diameter of the geyser stream was
much less than the capillary length a of the liquid, so
the electric field has high gradients near the stream
surface, and the discharge process can not be consid-
ered in the framework of the small-gradient approxi-
mation used in this paper. The charged surface be-
comes unsteady, and the appearance of electric current
in the system should be taken into account, too. The
mechanism of the geyser stream formation could be
similar to that studied in the paper [24] for singularity
formation at the surface of liquid metal at high elec-
tric field.
4. Conclusions
It follows from our considerations that the flat sur-
face of a thin neutral liquid film suspended at a solid
substrate becomes unstable under inverted gravita-
tion conditions if the thickness of the layer exceeds a
critical value h� . Development of the instability leads
to the formation of a stationary wave of deformation
of the liquid surface, i.e., to reconstruction of the sur-
face. In contrast with the periodic reconstruction of
the charged surface of a thick layer, the stable shape
of the wave corresponds to the formation of a single
hump at the center of a cell. The amplitude of the de-
formation wave increases with increasing the mean
thickness h of a layer. If the mean thickness of the liq-
uid layer exceeds significantly the critical thickness,
92 Fizika Nizkikh Temperatur, 2004, v. 30, No. 1
G.V. Kolmakov, E.V. Lebedeva, A.A. Levchenko, L.P. Mezhov-Deglin, A.B. Trusov, and V.B. Shikin
-10 -8 -6 -4 -2 0 2 4 6 8
U = 0
U = 350 V
U = 560 V
(
)
�
r
,
m
m
r, mm
0
0.5
1.0
1.5
2.0
Fig. 6. Profiles of the droplet obtained by scanning snap-
shots of the suspended charged hydrogen droplet at differ-
ent voltages U applied to the capacitor.
0 200 400 600
A
,
m
m
U, V
2.0
1.5
1.0
Fig. 7. Dependence of the height of the suspended droplet
on the voltage applied to the capacitor. The volume of the
droplet is equal to V � 60 mm3. Points — experiment,
line — theoretical dependence (56).
40 80 1200
1
2
3
U = 360 V
U = 0
A
,
m
m
V, mm3
Fig. 8. Dependence of the droplet height on the volume of
a neutral droplet, U � 0 (circles) and of a charged drop-
let, U � 360 V (squares). Points — experiment, lines —
the dependence (56).
h h� �10 , the shape of the droplet can be described
with good accuracy by Frenkel’s macroscopic model,
in which the effective boundary conditions of full wet-
ting should be used.
The van der Waals forces are responsible for stabili-
zation of the thin liquid background — a film wetting
the substrate outside the droplet. The presence of such
a film is important for transitions between states with
several humps at the substrate to the state with a sin-
gle hump.
The application of an electric field to the equipo-
tential charged surface of liquid layer leads, in a first
approximation, to renormalization of the gravitational
acceleration (to an increase of the effective gravita-
tional acceleration if the droplet is suspended at the
metallic plate, and to a decrease of the effective gravi-
tational acceleration if the droplet lies at the plate).
The results of the theoretical considerations are in
good agreement with experimental observations.
Acknowledgments
This work is supported in part by INTAS grant
2001-0618, RFBR grant 01-02-97037 and RFBR grant
03-02-16865-a. G.K. also thanks the Science Support
Foundation (Russia) for support. We thank M.
Strzhemechny and J. Klier for the helpful discussions
and useful remarks.
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