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THE SHAPE OF CHARGED DROPS OVER A SOLID SURFACE AND
SYMMETRY-BREAKING INSTABILITIES∗
M. A. FONTELOS†AND U. KINDEL´AN‡
Abstract.
solid substrate, surrounded by a gas, and in absence of gravitational forces. The question can be
formulated as a variational problem where a certain energy involving the areas of the solid-liquid
interface, of the liquid-gas interface and the electric capacity of the drop, has to be minimized. As
a function of two parameters, Young’s angle θY and the potential at the drop’s surface V0, we find
the axisymmetric minimizers of the energy and describe their shape. We also discuss the existence
of symmetry-breaking bifurcations such that, for given values of θY and V0, configurations for which
the axial symmetry is lost are energetically more favorable than axially-symmetric configurations.
We prove the existence of such bifurcations in the limits of very flat and almost spherical equilibrium
shapes. All other cases are studied numerically with a boundary integral method. One conclusion
of this study is that axisymmetric drops cannot spread indefinitely by introducing sufficient amount
of electric charges, but only can reach a limiting (saturation) size and after that the axial symmetry
would be lost and finger-like shapes are energetically preferred.
We study the static shape of charged drops of a conducting fluid placed over a
1. Introduction. The determination of the stationary shapes of liquid drops
surrounded by a vapor phase and in contact with a solid surface is an old problem
both in fluid mechanics and in the theory of partial differential equations (see [7] and
references therein). The problem can be posed, since Gauss, in a variational setting
consisting of obtaining the configurations of a given mass of fluid that minimize (or
in general, make extremal) an energy defined by
E = γlvAlv− (γsv− γsl)Asl+ EF ,
(1.1)
where γlv, γsv and γsl denote the liquid-vapor, solid-vapor and solid-liquid surface
tensions respectively; Alvand Asldenote the area of the liquid-vapor and solid-liquid
interfaces respectively (see Figure 1.1). EF is the contribution of external forces to
the total energy. If the drop is affected by gravity, then EF =?
forces, the configurations that minimize the energy (1.1) are spherical caps such that
the contact angle θY, called Young’s angle, between the liquid-vapor and solid-liquid
interfaces satisfies
Ωg · xdx where g is
the gravitational force and Ω the domain occupied by the fluid. In absence of external
cosθY =γsv− γsl
γlv
.
When the volume of fluid under consideration is sufficiently small, the contribu-
tion of gravitational forces to the energy is negligible in comparison with interfacial
energies. A consistent approximation is then to ignore gravity in (1.1), as it is done
systematically in the study of multiphase flows in microfluidic applications, for in-
stance (see [18]). It is precisely in connection with such microfluidic applications that
electric fields are incorporated with the purpose of controlling the shape and motion
of small masses of fluid. This is the case, for instance, of electrowetting applications in
∗The authors thankfully acknowledge the computer resources provided by the Centro de Super-
computaci´ on y Visualizaci´ on de Madrid (CeSViMa) and the Spanish Supercomputing Network.
†Instituto de Ciencias Matem´ aticas, (ICMAT, CSIC-UAM-UCM-UC3M), C/ Serrano 123, 28006
Madrid, Spain
‡Departamento de Matem´ atica Aplicada y M´ etodos Inform´ aticos, Universidad Polit´ ecnica de
Madrid, C/ R´ ıos Rosas 21, 28003 Madrid, Spain.
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which the shape of a mass of electrically conducting fluid is controlled via the addition
of electric charges and application of external electric fields (see [12] and references
therein).
The simplest situation corresponds to a drop of perfectly conducting fluid with a
total charge Q. In this case, the energy would be given (cf. [12]) by
E = γlv[Alv− (cosθY)Asl] −1
2ε0
?
R3\Ω
|E|2dx ,
(1.2)
where E = −∇V is the electric field (V is the electric potential) created by the
charges, concentrated in ∂Ω with a density σ = −ε0∂V
is the dielectric constant of the medium surrounding the drop and we shall consider
it as a constant. The potential V at the surface of a conductor is constant and, in
absence of charges in R3\ Ω is harmonic in the exterior of Ω. Hence, V is solution of
the boundary value problem
∂nin a perfect conductor. ε0
∆V = 0 at R3\ Ω ,
V = V0 at ∂Ω ,
V = O(|r|−1) as |r| → ∞ .
(1.3)
(1.4)
(1.5)
The electrical energy term in (1.2) can be written, after integration by parts, in the
following equivalent forms
?
where C is the capacity of Ω defined as
?
1
2ε0
R3\Ω
|E|2dx =1
2ε0
?
∂Ω
V∂V
∂ndS =1
2
?
∂Ω
V0σdS =1
2QV0=1
2CV2
0,
C = −ε0
V0
Ω
∂V
∂ndS .
Fig. 1.1. Sketch of the problem.
The determination of the capacity of a given set is, in general, a difficult problem.
There are explicit expressions only for a few configurations such as spheres and discs
(see [10]). The best source concerning estimation of the capacity of arbitrary sets is
[14] and the related article [15]. More complex configurations, such as spherical caps,
have a capacity that can be estimated from above and below but no explicit formulae.
This is the main difficulty in the deduction of minimizers of (1.2).
A particular case of the problem corresponds to θY = 2π, that would correspond
to absence of contact with solids. This is the case of levitating droplets which the
energy is, instead of (1.2),
E = γlvAlv−1
2CV2
0,
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and extremal values are reached by spheres as Lord Rayleigh showed in 1882 [17]. This
particular situation points out to the possibility of appearance of instabilities breaking
the axial symmetry and leading to singularities at the drop’s surface. When the elec-
tric charge (and V0therefore) is large enough, those spheres cannot be stable. Two
scenarios are possible then: evolution towards nonspherical stationary configurations
that accommodate such amount of charge or destabilization leading to singularities at
the interface in finite time. The existence of nonspherical equilibrium configurations
in the form of spheroids was proved in [8] and they were characterized as branches of
solutions bifurcating, via Crandall-Rabinowitz’s theorem, from the branch of spher-
ical solutions. The appearance of singularities in the evolution of initially spherical
levitating drops with large charge was shown in [3]. The drop evolves initially into a
prolate spheroid with its poles increasing in curvature and becoming conical tips in
finite time.
By computing the first variation of the functional (1.2) one arrives at the following
equation:
γκ −σ2
2ε0
= −p ,
(1.6)
where γ = γlv, κ is the mean curvature (sum of the principal curvatures) of the
liquid-vapor interface at a point x, σ = −ε0∂V
same point x and p is a constant to be determined through the constraint that the
drop has a given volume. In the fluid dynamics context, p is the difference of pressure
across the interface.Equation (1.6) has to be complemented with the boundary
condition stating that the normal vector to the interface forms a constant angle with
the normal to the solid substrate at any point of the contact line Γ, the set where
the liquid-vapor and the solid-liquid interfaces meet. This angle has to be, exactly,
θY (cf. [13]). Finally, we introduce characteristic length (V ol.)
volume of the drop, characteristic potential (V ol.)
charge density (V ol.)−1
we change variables and unknowns in the form x → (V ol.)
σ → (V ol.)−1
charge density and pressure are now dimensionless. We end up with the following
dimensionless version of (1.6):
∂nis the surface charge density at the
1
3 where V ol. is the
2, characteristic surface
3. Accordingly,
3x, V → (V ol.)
1
6(γε−1
0)
1
6(γε0)
1
2 and characteristic pressure γ(V ol.)−1
11
6(γε−1
0)
1
2V ,
6(γε0)
1
2σ, p → γ(V ol.)−1
3p so that space coordinates, potential, surface
κ −σ2
2
= −p ,
(1.7)
and the variational problem associated to the functional
E = [Alv− (cosθY)Asl] −1
2CV2
0,
(1.8)
where C = −1
of unit volume.
One important motivation for this work is its relation with the phenomenon of
electrowetting. This consists in the control of the wetting properties of fluids by means
of electric fields. The simplest situation is that of a drop of conducting fluid connected
to a battery and, therefore, kept at a given difference of potential with respect to an
electrode placed at some distance below the solid substrate. In the situation studied
in the present paper, such electrode would be placed at infinity, so that we establish
V0
?
Ω
∂V
∂ndS being V the solution to (1.3)-(1.5) and Ω is now a domain
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a given potential on the drop’s surface and assume the potential to decay at infinity.
The simplest theories on electrowetting follow the original ideas of Lippmann, who
developed a formula (cf. [11]) that predicts an unlimited spreading of droplets through
application of sufficiently strong electrostatic potentials. Nevertheless, the physical
observation is that drops do not spread infinitely but reach a saturation regime such
that an increase of the potential does not produce any additional spreading but the
appearance of instabilities at the contact line with subsequent emission of a varying
number of satellite filaments (see [12] and references therein). The demonstration of
such facts, also appearing when the electrode is at a fix distance to the drop, requires
a somewhat different analysis and will be published elsewhere
In this paper we shall study, as a function θY and V0, the equilibrium config-
urations. We will present explicit formulae for the geometry of axially symmetric
profiles in certain limits of θY and V0. We will deduce a first constraint to indefinite
spreading due to the fact that static solutions develop dewetted cores with the fluid
concentrated in a rim around these cores. The second constraint concerns stabil-
ity. By analyzing the energy functional (1.8) we will conclude that axially symmetric
solutions must become unstable under non axially-symmetric perturbations with n
undulations (n = 2,3...) provided V0is large enough. We shall determine numerically
(and analytically in some limiting cases) for what values of V0such instabilities do
develop as a function of θY. This offers a possible explanation to the saturation effect
explained above and the contact line instabilities observed in experiments.
The paper is organized as follows. In Section 2 we study the radially symmetric
configurations. We perform an analysis in the limiting cases of almost spherical drops
and almost flat drops and then develop a numerical code to compute the profiles in
all intermediate cases. This allows us to represent in a phase diagram the radius of
spreading of a drop as a function of V0for arbitrary θY. In Section 3 we study the
stability of the radially symmetric solutions under symmetry-breaking perturbations.
Again, we focus the theoretical discussion in the limiting cases of almost spherical and
flat drops, but end with a numerical study of all cases. Finally, Section 4 is devoted to
the study of the capacity of axially symmetric configurations perturbed in the radial
direction and it also includes the proof of several results used in previous sections.
2. Radially symmetric configurations. These are solutions of (1.6) which are
invariant under rotations about an axis normal to the solid surface. We can describe
the height of each point of the fluid-vapor interface by a function h(r) where r is the
radial coordinate in a cylindrical coordinate system about the axis of symmetry. The
mean curvature is then (see [7])
?
where ψ is the angle of inclination of the solution curve h(r) with respect to the r-axis.
Notice that tanψ =dh
in two limits for which the analysis simplifies: 1) the limit of small potential V0at ∂Ω
so that drops are almost spherical caps, and 2) the limit of large potential V0at ∂Ω
or small contact angle so that drops are almost flat discs. The profiles between these
two situations will be found numerically.
κ = −1
r
d
dr
r
hr
(1 + h2
r)
1
2
?
= −1
r(rsinψ)r
(2.1)
dr. We shall study in this section the radially symmetric profiles
2.1. Almost spherical shapes. If we assume ∂Ω to be continuously differen-
tiable except for the contact line, that we consider located at r = L, where h(r)
is only Lipschitz continuous (due to the existence of a corner of opening angle θY),
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Then, as long as nε ? 1 we can expand
?
C =
R3?Ω0
|∇V0|2dx?+ 2ε
?
R3?Ω0
∇V0· ∇V1dx?+ ε2
?
R3?Ω0
|∇V1|2dx?
+ε2
?
R3?Ω0
?
2
r?
∂V1
∂θ?
∂V0
∂r?nsin(nθ?) +
?∂V0
∂r?
?2
n2sin2(nθ?)
+1
2cos(2nθ?)
?∂V0
∂z?
?2
+ 4cos(nθ?)∂V1
∂z?
∂V0
∂z?
?
r?dr?dθ?dz?+ O(ε3) ,
and since?
R3?Ω0∇V0· ∇V1dx?= 0 (as one can show after integration by parts and
using ∆V0= 0 outside Ω0and V1= 0 at ∂Ω0) we can write
C = C0+ ε2Cn,1+ O(ε3) ,
where
Cn,1=
?
R3?Ω0
|∇V1|2dx?+
?
R3?Ω0
??∂V0
∂r?
?2
n2sin2(nθ?)
+2
r?
∂V1
∂θ?
∂V0
∂r?nsin(nθ?) + 4cos(nθ?)∂V1
∂z?
∂V0
∂z?
?
r?dr?dθ?dz?.
By Dirichlet’s principle, V1can also be characterized as the function W for which the
minimum of
?
is achieved or, equivalently, the solution to the boundary value problem
R3?Ω0
?
|∇W|2+2
r?
∂W
∂θ?
∂V0
∂r?nsin(nθ?) + 4cos(nθ?)∂W
∂z?
∂V0
∂z?
?
dx?,
2∆V1= −2
V1= 0 at ∂Ω0.
r?
∂V0
∂r?n2cos(nθ?) − 4∂2V0
∂z?2cos(nθ?) in R3?Ω0,
We can find the solution to this problem in the form
V1= cos(nθ?)Φ(r?,z?) ,
which leads to
Cn,1=
?+∞
−∞
?+∞
a(z?)
?
n2
2
????
∂V0
∂r?
????
r?Φ∂V0
2?
r?dr?dz?
+
?+∞
−∞
?+∞
a(z?)
?
1
2
????
∂Φ
∂r?
????
2
+1
2
????
∂Φ
∂z?
????
2
+
n2
2r?2Φ2−n2
∂r?+ 2∂Φ
∂z?
∂V0
∂z?
?
r?dr?dz?.
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Hence, using ∆V0= 0, one finds
−∆(r?,z?)Φ +n2
r?2Φ =n2
Φ = 0 at ∂Ω0.
r?
∂V0
∂r?−2
r?
∂
∂r?
?
r?∂V0
∂r?
?
in R3?Ω0,
(4.1)
(4.2)
We are going to show two important facts: 1) Cn,1is strictly positive and 2) Cn,1is
increasing with n. The first fact follows from the following calculations:
?
∂r?
Cn,1=
?+∞
−∞
?+∞
a(z?)
n2
2
????
∂V0
????
2?
r?dr?dz?+
?+∞
−∞
?+∞
a(z?)
?
1
2
????
∂Φ
∂z?
????
2?
r?dr?dz?
+
?+∞
−∞
?+∞
a(z?)
?
1
2
????
2
∂Φ
∂r?
????
2
+
n2
2r?2Φ2−n2
r?Φ∂V0
∂r?+ 2∂Φ
∂z?
∂V0
∂z?
?
r?dr?dz?
>
?H
0
?+∞
a(z?)
?
n2
2
????
?
∂V0
∂r?
????
????
+1
2
????
+1
∂Φ
∂r?
????
????
2
+
n2
2r?2Φ2−n2
r?Φ∂V0
∂r?− 2∂Φ
∂r?
∂V0
∂r?
?
r?dr?dz?
=
u=log r
?H
0
??+∞
log a(z?)
n2
2
∂V0
∂u
????
2
2
∂Φ
∂u
????
2
+n2
2Φ2− n2Φ∂V0
∂u− 2∂Φ
∂u
∂V0
∂u
?
du
?
dz?.
By performing Fourier transform in u of this last expression we get
?H
0
??+∞
−∞
?n2
2k2????
?H
V0
???
2
+1
2k2????Φ
?n2
???
2
+n2
2
????Φ
???
?1
2
− ikn2?Φ?
V0+ 2k2?Φ?
?????Φ
V0??Φ))
V0
?
dk
?
dz?
=
0
??+∞
−∞
2k2????
V0) + 2k2((??
V0
???
2
+
2k2+ n2
???
2
+ n2k2(??
V0??Φ − ??Φ??
V0??Φ + ??
?
dk
?
dz .
(4.3)
Let
a = (??Φ,??Φ,??
2
V0,??
V0) ,
then the integrand in (4.3) is aT(n,k)atwith
T(n,k) =
k2+n2
2
0
k2
n2k
0
k2
n2k
2
k2
0
k2n2
2
k2+n2
2
−n2k
k2
−n2k
k2n2
2
0
2
2
,
possessing the following two double eigenvalues
λ±=1
4k2n2+1
4k2+1
4n2±1
4
?
k4n4− 2k4n2+ 17k4+ 2k2n4+ 2k2n2+ n4.
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Notice that
λ−=
k4?n2− 4?
k2n2+ k2+ n2+√k4n4− 2k4n2+ 17k4+ 2k2n4+ 2k2n2+ n4>
so that
?H
The capacity is increasing with n:
?
∂r?
k4/2
1+ k2
n2− 4
n2+ 1
Cn,1>n2− 4
n2+ 1
0
??+∞
−∞
1
2
k4
1 + k2
????
V0
???
2
dk
?
dz = c?n2− 4
n2+ 1.
Cn,1=
?+∞
−∞
?+∞
a(z?)
1
2
????
2
∂Φ
????
2
+1
2
????
2
∂Φ
∂z?
????
2
+n2
2
????
????
Φ
r?−∂V0
∂r?
????
????
2
+ 2∂Φ
∂z?
∂V0
∂z?
?
r?dr?dz?
≥
?+∞
−∞
?+∞
a(z?)
?
1
2
????
∂Φ
∂r?
????
∂Ψ
∂r?
+1
2
????
∂Φ
∂z?
????
+(n − 1)2
2
Φ
r?−∂V0
∂r?
2
+ 2∂Φ
∂z?
∂V0
∂z?
?
r?dr?dz?
≥1
2min
Ψ
?+∞
−∞
?+∞
a(z?)
?????
????
2
+
????
∂Ψ
∂z?
????
= Cn−1,1,
2
+ (n − 1)2
????
Ψ
r?−∂V0
∂r?
????
2
+ 4∂Ψ
∂z?
∂V0
∂z?
?
r?dr?dz?
with equality if and only if
Φ = r?∂V0
∂r?,
(4.4)
which would imply, by (4.1)
−∆(r?,z?)Φ +2
r?
∂Φ
∂r?= 0 .
(4.5)
Multiplying (4.5) by Φ and integrating by parts using Φ = 0 at ∂Ω, we get
?
which would imply Φ = 0 at almost every point, a fact that is incompatible with (4.4).
Therefore the capacity is strictly increasing with n.
The fact that the capacity is strictly increasing with n is crucial to the proof
of existence of symmetry-breaking bifurcations in Section 3.3. Another important
consequence of the result proved in this section, discussed in section 3.3, is the fact
that perturbations with high order modes (large value of n) may be the most favorable
energetically if the body of revolution that we perturb is sufficiently flat. This may
lead to instabilities of the contact line in the form of numerous fingers.
|∇Φ|2dx?= 0 ,
23
Page 24
REFERENCES
[1] A. A. Ashour, On a transformation of coordinates by inversion and its application to electro-
magnetic induction in a thin perfectly conducting hemispherical shell, Proc. London Math.
Soc. (3) 15 (1965) 557-576.
[2] S. I. Betel´ u and M. A. Fontelos, Spreading of a charged microdroplet, Physica D Nonlinear
Phenomena, 209, Issues 1-4, 15 (2005), 28-35 .
[3] S. I. Betel´ u, M. A. Fontelos, U. Kindel´ an, O. Vantzos, Singularities on charged viscous droplets,
Physics of Fluids 18, 051706 (2006).
[4] S. I. Betel´ u, M. A. Fontelos, U. Kindel´ an, The Shape of Charged Drops: Symmetry-breaking
Bifurcations and Numerical Results, Progress in Nonlinear Differential Equations and Their
Applications, 63 (2005), 51-58.
[5] D. Duft, T. Achtzehn, R. M¨ uller, B. A. Huber and T. Leisner, Rayleigh jets from levitated
microdroplets, Nature, vol. 421, 9 January 2003, pg. 128.
[6] L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998.
[7] R. Finn. Equilibrium capillary surfaces. Springer-Verlag, New York, 1986.
[8] M. A. Fontelos, A. Friedman, Symmetry-breaking bifurcations of charged drops, Arch. Ration.
Mech. Anal. 172,2 (2004), 267-294.
[9] O. D. Kellogg, Foundations of Potential Theory. Dover Publications 1969
[10] N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York, 1972.
[11] G. Lippmann, Relations entre les ph´ enom` enes ´ electriques et capillaires. Ann. Chim. Phys. 5,
494 (1875).
[12] F. Mugele, J. C. Baret, Electrowetting: from basics to applications, J. Phys.: Condens. Matter
17 (2005), R705–R774.
[13] F. Mugele, J. Buehrle, Equilibrium drop surface profiles in electric fields, J. Phys.: Condens.
Matter 19 (2007), 375112 .
[14] G. P´ olya, G. Szeg¨ o, Inequalities for the capacity of a condenser, American Journal of Mathe-
matics, 67-1 (1945), 1-32.
[15] G. P´ olya, G. Szeg¨ o, Isoperimetric inequalities in mathematical physics. (Annals of Mathematics
Studies, no. 27.) Princeton University Press, 1951.
[16] C. Pozrikidis, Boundary integral methods for linearized viscous flow, Cambridge texts in Ap-
plied Mathematics, Cambridge University Press, 1992.
[17] Lord Rayleigh, On the equilibrium of liquid conducting masses charged with electricity, Phil.
Mag. 14 (1882), 184-186..
[18] H. A. Stone, A. D. Stroock, A. Ajdari, Engineering flows in small devices: microfluidics toward
a lab-on-a-chip, Annu. Rev. Fluid Mech. 36 (2004), 381-411.
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