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Research Overview
Research Overview
Derek E. Moulton
October 6, 2007
1
Introduction
Surface tension and electrostatic forces are quite common in everyday experience. Really, these
forces are common in two senses of the word. They are common in the sense that they are
regular; we encounter them a lot. But, they are also common in the sense that they often seen as
being undistinguished, unimportant. Indeed, few phenomena that we observe in the macro world
have either of these as their driving force. However, when we go down to the micro world, things
become very much dierent. Due to favorable scaling laws, surface tension and electrostatic forces
suddenly become dominant, and many phenomena that exist on these scales occur due to these
forces. Hence, it is of great value to understand them, so that we might combine them in clever
ways and manipulate or control these phenomena.
My research involves the mathematical modeling, analysis, and experiment of systems of elec-
trostatic forces interacting with surface tension and elastic membranes. The exploration of such
systems largely began with the pioneering work of G.I. Taylor in the late 1960s. In particular,
Taylor studied the electrostatic deection of liquid drops and planar soap lms [10]. Today, such
systems sit at the conuence of several elds of application. Most noticeably, electrostatic actuation
of elastic membranes is of key importance for micro and nanoelectromechanical systems (MEMS
and NEMS) [6]. The operation of such systems is based on the same ideas of electrostatic deection
explored by Taylor - a potential dierence is applied between mechanical components of a device,
at least one of which has elastic properties, a Coulomb force between the parts causes the elastic
component to move, and this actuation is manipulated to perform some task. This is the key idea
underlying such devices as grating light valves, micro-mirrors, comb drives, and micro-pumps, to
name a few. The voltages required to produce actuation in these devices tend to be very small, and
so it stands to reason that electrostatics will continue to play an important role in these technologies
for years to come. There are, however, many important issues that arise in the proper functioning
of these devices, and consequently there is a dire need to understand the mathematics behind them.
At the heart of this understanding is the interaction between electrostatic and elastic systems.
Concurrently, the systems we describe below may also be traced to the theory of capillary
surfaces, which are liquid/gas or liquid/liquid interfaces whose shape is governed primarily by
interfacial tension. Note that if we go back to the original experiments of Taylor, i.e. deecting
soap lms and drops, and remove the electric eld, we have a capillary surface. Capillary surfaces
and capillary forces are found in many places. Such phenomena as bugs walking on water, liquid
rising up a capillary tube, and liquid columns breaking up into drops, are all due to capillary forces.
Hence, they comprise a vast eld of study, and have applicability in a wide range of engineering
systems, including micro and nano technology.
The value in studying systems that lie at the intersection of electrostatic actuation and capillary
phenomena is well demonstrated by its potential in the eld of self-assembly. In this quickly
1 Figure 1:
Sketch of the setup for the outer and inner cylinder electrostatically deected catenoids, as well
as a still photo of the inner cylinder setup. Photo courtesy of the MEC Lab.
growing eld, capillary forces provide one of the most common and basic driving forces behind
self assembling systems. One example of note is the work by the George Whitesides group [3].
In attempting to develop new fabrication technologies for MEMS and NEMS, Whitesides placed
drops of polydimethylsiloxane (PDMS) between two rigid plates. Driven by surface tension, these
droplets naturally form a liquid bridge. By adjusting the gap between the plates and the relative
orientation of the plates, a variety of structures can be formed. In the PDMS system, the polymer
can then be cross-linked, solidifying the liquid bridge and hence leading to the production of small
components with a variety of shapes. An interesting next step, as noted by Whitesides, would be to
manipulate the bridges with an electric eld, thereby greatly extending the range of possible shapes.
Another example occurs in microfabrication. In [9], a technique is described in which components
are assembled by surface tension powered self assembly, whereby a drop minimizing its surface
area pulls components into place. By combining the surface tension forces with electrostatic forces,
the potential is there to speed up these processes as well as to achieve congurations otherwise
unattainable.
If nothing else, these examples suggest that there is potentially much to be gained by interacting
electric elds with 3-dimensional capillary surfaces. My research has largely focused on developing
mathematical models that investigate these interactions and exploring such things as how we can
alter, control, and/or manipulate these surfaces with a eld.
2
Catenoid in an electric eld
The rst system we consider is depicted in Figure 1. An elastic membrane is suspended between
two circular parallel rings. The main force acting on the membrane is surface tension, and in the
absence of external forces, the membrane will take the shape of a minimal surface catenoid. We
add to this system an electric eld. Two geometries are considered. In one, we place an outer
cylindrical electrode around the catenoid, and in the other, the cylindrical electrode is inside the
catenoid along the radial axis. In both cases, we impose a potential dierence between the electrode
and the membrane.
There are several issues we address. One is the structure of the equilibrium solution set, in-
cluding multiplicity, evolution, and stability. In deriving the model, several key parameters arise,
and we seek the dependence of solutions on the dierent parameters as well as any critical values
at which the solution structure changes. This last issue tends to be very important in mathe-
matical models for MEMS. In particular, when designing devices that operate under electrostatic
actuation, one must be conscientious of a critical voltage which is found to exist in such systems,
called the pull-in voltage, at which stable equilibria is lost. In relation to capillary phenomena, we
explore the general eect of adding the electric eld. Indeed, the catenoid is a well studied and
2 understood surface, whose stability limits and shape change dynamics [7] are well documented. Its
value in capillary phenomena is not to be understated, in large part because it is easily realized
experimentally with the use of soap lm and may be related to liquid bridge breakup.
The systems of Figure 1 are described by the general equation
Hu = f (u)
(1)
where u gives the radial coordinate of the surface of revolution, H is the mean curvature operator,
and f captures the Coulomb force of the electric eld. This equation clearly demonstrates the
competition between elastic and electrostatic forces. Indeed, in the absence of the electric eld, we
set f = 0 and we have the equation for a minimal surface, i.e. H = 0. Hence, one way to view
Equation (1) is that the mean curvature of the surface is being driven by the function f . In light
of this interpretation, we may classify surfaces which satisfy Equation (1) as Field Driven Mean
Curvature (F.D.M.C.) surfaces.
It is worth noting that in typical models of electrostatic actuation, the general form of the
governing equation is
u = g(u)
(2)
where u describes the shape of the deected membrane, is the Laplacian operator, and g(u)
contains the eect of the electric eld. The only real dierence between these two forms is in
geometry. Equation (2) arises when working in the parallel plate regime. That is, when the
components of the system may be treated as being parallel plates, linear elasticity is sucient to
describe the elastic force, which leads to the Laplacian operator. For the most part, this linearization
has been perfectly justied, as most MEMS devices fall into the parallel plate category. In an eort
to extend the current theory, we explore more complicated geometries, where linear elasticity is no
longer sucient. For the catenoid bridge geometries we consider presently, we arrive instead at the
mean curvature operator.
The governing equations for these systems are derived through the minimization of an energy
functional. For the outer cylinder geometry, the energy in the system is given in non-dimensional
coordinates by
E
[u] =
1
/2
1/2
u
1 +
2
u
2
ln (
1
/u) dz .
(3)
Here, r = u(z) is the radius of the deected membrane surface of revolution, and = a/L