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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 by the classical theory of Dirichlet problems for elliptic linear PDE’s (cf. [6], for

instance) we will have a continuous solution V of problem (1.3)-(1.5) such that

will be continuous everywhere except for the contact line. There∂V

since V has the asymptotic behavior

∂V

∂n

∂nwill be singular

V = V0+ Aραsin(αθ) ,

where (ρ,θ) are polar coordinates about (r,z) = (L,0) and α is such that sin(α0) =

sin(α(2π − θY)) = 0 and∂V

line. Hence α =

∂ρis square integrable in the neighborhood of the contact

2−θY/πand

σ = −∂V

1

∂n∼ A?ρ

1

2−θY/π−1as ρ → 0 .

Notice that σ2is integrable with respect to ρ provided θY > 0 and, therefore, is

integrable over the whole ∂Ω.

Let us assume that V0= ε ? 1 so that we can write the surface charge distribution

of a unit volume spherical cap with contact angle θY as εΣ. Then equation (1.7) and

(2.1) yield the equation

?

By integrating once we get

1

r

d

dr

r

hr

(1 + h2

r)

1

2

?

= −ε2Σ2

2

+ p .

sinψ =1

2pr − ε2a1(r) ,

where

a1(r) =1

r

?r

0

Σ2

2r?dr?.

Notice that

−sinθY =1

2pL − ε2a1(L) .

(2.2)

In order to find the profile we use

hr= tanψ =

1

2pr − ε2a1(r)

1 −?1

−

?1 −1

?

?

2pr − ε2a1(r)?2

1

4p2r2?3

=

1

2pr

?

1 −1

4p2r2

2ε2a1(r) + O(ε4) ,

and then

h(r) = h0+2

p−2

p

1 −1

4p2r2− ε2a2(r,p) ,

where

a2(r,p) =

?r

0

1

?1 −1

4p2r?2?3

2a1(r?)dr .

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