11institutetext: Yale University, Physics Department 11email:
<EMAIL_ADDRESS>
# Progress in Experimental Measurements of the Surface-Surface Casimir Force:
Electrostatic Calibrations and Limitations to Accuracy

Steve K. Lamoreaux

###### Abstract

Several new experiments have extended studies of the Casimir force into new
and interesting regimes. This recent work will be briefly reviewed. With this
recent progress, new issues with background electrostatic effects have been
uncovered. The myriad of problems associated with both patch potentials and
electrostatic calibrations are discussed and the remaining open questions are
brought forward.

##  1 Introduction

Nowadays, it is unclear what it means to write a review article, or a review
chapter for a book, on a particular subject. This unclarity results simply
from the ease with which modern digital reference and citation resources can
be used; with a mere typing of a keyword or two into a computer hooked up to
the internet, one has an instant review any field of interest. As such, at the
present time, review articles tend to be op-ed pieces that tend to be less
than scientifically enlightening. Rather than continue in the tradition of
collecting up a series of electronic database searches, I will give an
overview of some recent experiments and also describe how anomalous
electrostatic effects might have affected the results of these experiments.
This Chapter is not meant to be a review of every paper in the Casimir force
experimental measurement field, but a review of what I consider are the
credible experiments, that have carried the field forward, that were performed
over the last decade or so. As such, there will be little mention of
experimental studies that have claimed 1% or better agreement, simply because
it is unclear to me what these experiments really mean. If the reader is
interested, a recent review of this 1% level work is presented in mohideenrev
. Of course I admit freely that my review presented here reflects my own
opinions, however I hope the reader accepts or rejects my points based on
verifiable facts and an independent scientific analysis. It must be remembered
that simply because a paper appears in print, in a credible and leading
journal, it is not necessarily scientifically correct or accepted by the
community at large. Neither does the fact that work is funded by the DOE, NSF,
or DARPA (or other funding agencies beyond the realm of the U.S.A.) guarantee
its validity or broad acceptance in the scientific community. An perhaps most
interestingly as a remark on the general history of science, the “consensus
opinion” is not necessarily correct either. In particular, in the surface-
surface Casimir force measurement field, there have been more than a few
“Comments” on various papers; the interested reader would do well to ignore
most, but not all, of these “Comments” as they are confusing, if not bogus,
but certainly inflammatory.

Watching the field develop since my 1997 experimental result skl97prl , which
served as a watershed for new interest in surface-surface Casimir force
measurements, has been fascinating. I had no preconceived notions as to how
large or small the effect should be relative to the case of assumed simple
perfect conductors (e.g., ignoring effects like surface plasmons), but I had
no illusions as to the accuracy of my work, hence the words “Demonstration of
the Casimir force” in the title of my paper. I simply did not have the time or
resources to perform a study of possible systematic effects that likely
limited the accuracy of my result; the precision was at the 5% level, at the
point of closest approach. Again the accuracy of my result was, and remains,
an open question, as it does for any experiment.

At the time the work reported in skl97prl  was performed, there were no
precision calculations of the Casimir force for real materials. Describing the
metal plates with the simplest plasma model, for parallel plates, the
correction to the force compared to the perfect conducting case is schwinger

$\eta(d)=1-{16\over 3}{c\over\omega_{p}d},$ (1)

where $\eta(d)$ is a force correction factor which varies with plate
separation $d$, $c$ is the velocity of light, and $\omega_{p}$ is the plasma
frequency, where the form of the permittivity of the metal is

$\epsilon(\omega)=1-{\omega_{p}^{2}\over\omega^{2}},$ (2)

which is valid at high frequency. As $\omega$ approaches zero, Eq. (2) become
invalid, and in addition the effect of static conductivity must be included
also. Equation (1) can be easily modified for a sphere-plane geometry skl97prl
. However, the magnitude of this correction was certainly outside what was
reasonable based on the precision of my experiment, which appeared to be best
described by plates with perfect conductivity. There was some skepticism
regarding the lack of a finite conductivity correction in my result, and
although several theorists expressed interest in performing a more accurate
calculation, none did. Eventually I attempted the calculations myself, with
mixed results. My calculations were based on published optical properties of
Au and Cu, with the Cu calculations intended as a test case. These
calculations showed roughly 10-15% (for Cu) and 20-30% (for Au) reductions in
force, compared to perfect conductors, for distances of order one micron; I
eventually found an error in the radius of curvature of the spherical-surface
plate used in my experiment sklcalc ; sklerratum  that lowered the
experimentally measured force by 10%, but did not bring the experimental
result into agreement with my Au calculation. Later work showed that Au and Cu
are nearly identical, with my Cu result being the more accurate; the
discrepancy was due to the way I interpolated between data points in the
tabulated optical data lambrecht1 . With the refined calculation, my
experiment and theory appeared to be in agreement, however by this time I was
skeptical of my results, as stated in the ensuing discussion, in lambrecht2 .
Interestingly enough, I had spent considerable effort trying to find
corrections that would bring my experimental result into agreement with my
original inaccurate calculation, so I felt that I was prepared to comment
against a new theoretical result, obtained by Boström and Sernelius bossern ,
that leads to a major correction to the Casimir force between real, non-
superconducting materials. This correction reduces the force by a full factor
of two at large separations. More will be said of this correction later in
this review; in particular, in light of new electrostatic systematic effects
that have recently been discovered, the rhetoric against the result of Boström
and Sernelius no longer appears as certain. In addition, all of the 1% work
that was reported before bossern  does not show the predicted correction, nor
does subsequent 1% level work. So we are faced with the possibility that the
degree of precision isn’t as high as stated in the 1% work, or that the theory
is not at all understood. Instead of questioning experimental accuracy, new
fantastic theoretical suggestions have been made, regarding the low frequency
permittivity of metals, that eliminate the new correction. This remains a
major open topic in the field.

There is a tendency among workers in this field to confuse precision with
accuracy, of which I am guilty myself. Precision relates to the number of
significant figures a measurement device or system provides; lots of digits
can be useful for detecting small changes in some “large” parameter, assuming
that the system is stable. Accuracy is the assignment of meaning to precision,
it is the connection between accepted definitions of, for example, lengths,
voltages, and forces, and the measurements that come out of an experimental
apparatus. As an example, for Casimir force measurements using the sphere-
plane geometry, an essential parameter is the radius of curvature of the
sphere. A radius of curvature accuracy of 0.5% for a sphere of 0.2 mm diameter
corresponds to 1000 nm, a bit larger than the wavelength of visible light.
Thus optical measurements of adequate accuracy appear as hopeless; can
electron microscopy attain this level of precision? The answer is not obvious.
Of course, an experiment can be designed that does not require a high accuracy
radius of curvature measurement, e.g., when the ratio of Casimir to
electrostatic force is measured. Nonetheless, the attention to this problem in
those works reporting 1% or better accuracy does not appear as sufficient to
warrant such accuracy claims. The precision might be that level, but the cross
checks required for accurate work are missing.

In general, to attain a given experimental accuracy, say 1%, requires that the
calibrations and force measurements must be done to much better than 1%
accuracy, particularly for comparisons between theory and experiment with no
adjustable parameters. As there are possibly five or more absolute
measurements that must be made to interpret an experiment, a reasonable
requirement for the average calibration accuracy is 0.5%, assuming that the
uncertainties can be added in quadrature (this point is open to debate; many
precision measurement experts insist that the uncertainties be simply added,
which bring the required average accuracy to the 0.2% level). Some of the
required calibrations are as follows: The optical properties of the surfaces
must be adequately characterized to allow calculation of the force to 0.5%
accuracy; The radius of curvature of the spherical surface (for a sphere-plane
experiment) needs to be measured to 0.5% accuracy; The absolute separation
must be determined to high accuracy. This last point is perhaps the most
difficult, as

$\left|{\delta F\over F}\right|=\left|n{\delta d\over d}\right|,$ (3)

where $n$ is the exponent in the power law. For a sphere-plane geometry where
$n\approx-3$ we see immediately that if we want 0.5% force accuracy as limited
by the distance measurement, at the point of closest approach, say 100 nm,
then the fractional error must be $0.5\%/3$ or about 0.17%, and when $d=100$
nm this corresponds to $\delta d=0.17$ nm $=1.7$ Å. This is at the level
where, in the atomic force microscopy (AFM) community, the definition of the
surface location is agreed as controversial. So we see immediately that it
pointless to include any discussion of experiments that claim 1% accuracy as
the radius measurement is not discussed in sufficient detail in any of the
papers making such claims. My statements here should be considered as a call
for details.

The general experimental techniques used in all Casimir experiments to date
are rather straightforward. Many experiments employ AFM or micromechanical
techniques drawn from fields that enjoy tremendous engineering support. The
trick of Casimir force measurements lies in the attainment of very high force
measurement sensitivity subjected to precise and rigorous calibrations, and in
the elimination of long-range background electrostatic effects that can mask
or distort the now-well-studied AFM signals extrapolated to very large
distances. At large distances, the attractive force between two surfaces,
“the” Casimir force, becomes a property of the bulk material(s) that the
plates comprise, and is viewed as a fundamental physical effect arising from
the quantum vacuum, as opposed to AFM signals used to detect surface
roughness, for example. Experimental rigor is required to transform precision
into accuracy on the fundamental vacuum effect.

Because the measurement techniques are largely borrowed from other fields, I
will not give a nuts and bolts discussion of measurements in this review, for
the simple reason that I know nothing about AFM techniques. Nowadays one can
simply buy an AFM system from Veeco, for example, and adapt it to the samples
and longer distance ranges required for Casimir measurement. There are
companies that commercially produce bare cantilevers, and most engineering
schools have fabrication facilities where NEMs and MEMs systems can be
produced with just about any desired properties in configurations limited only
by the imagination. Alternatively, my own work employs torsion balances, and
the interested reader can refer to Cavendish’s experiment for most details of
such systems. An analysis of the force sensitivity of a torsion pendulum can
be found in buttlam .

The principle advantage to AFM type or torsion pendulum type measurements (in
fact there is no fundamental difference between them, it’s a matter of scale)
is elimination of stiction associated with the fulcrum type balances used in
practically all earlier experiments. The proliferation of high accuracy
mechanical and optomechanical translation stages, together with high quality
digital data acquisition systems has made precision Casimir force measurement
possible; the questions of accuracy are now the central theme, not the simple
detection of the force.

This is not to say that the experiments are easy or simple; again, the art of
the experiments lies in the attainment of high force measurement sensitivity,
reliable calibrations, the production of well-characterized optical surfaces,
and the elimination of background effects due to, for example, electrostatic
effects. The electrostatic effects are common to all experiments, either in
regard to system calibrations or systematic background effect, or both. Given
the importance of electrostatic effects, I will discuss them at length in this
review.

It is often said that the Casimir force is simply the retarded van der Waals
potential. This view strikes me as fundamentally flawed, as the Casimir force
does not depend on the properties of the individual atoms of the plates, but
on their bulk properties. Indeed, the non-additivity of the van der Waals
effect has been discussed at length in the literature (see milonni  for a
discussion and references). It is more profitable to think of the Casimir
force as the zero point electromagnetic field stress on a parallel plate
waveguide. This force is apparently largest when the waveguide is constructed
from perfectly conducting material(s). The effects of imperfect conductivity
can be calculated provided the optical constants of the material(s) are known
over an adequate wavelength range. Furthermore, most of the surface-surface
Casimir effect is due to conduction electrons. It is meaningless to assign a
retarded van der Waals force between the individual electrons in a conductor.
Likewise, if the Casimir force was simply the retarded van der Waals force, it
would make little sense consider modifying the Casimir force, in a fundamental
way, by altering the mode structure imposed by specially tailored boundaries.

##  2 Motivation for the Experimental Study of the Casimir Force: Some Recent
Results

The Casimir force is of fundamental interest in that it is taken as evidence
for the existence of the fluctuations associated with the quantum vacuum cas .
One can almost as easily derive the Casimir force by treating the
electromagnetic field classically, with the field fluctuation due to
dissipation in the material bodies; this is the Lifshitz approach lifshitz . A
principal controversy associated with the quantum vacuum interpretation lies
in the fact that the zero point electromagnetic field energy, when integrated
to the Planck scale (which is the natural cutoff), leads to a cosmological
energy density some 130 orders of magnitude larger than observed. This is an
open problem in modern physics.

There are three principal motivations for studying the Casimir force. One
question is how well do we understand the basic underlying physics? This
relates to second motivation which lies in the testing for the existence of
short range corrections to gravity, or a new force associated with axion
exchange, for example. For such tests, the Casimir force represents a
systematic background effect that must be characterized or physically
eliminated by employing a shield. The third motivation comes from interest in
modifying the Casimir force to eliminate stiction, for example, or make it
useful in nanodevices. These categories are not mutually exclusive, and of
course overlap considerably as the questions all have a fundamental element.

###  2.1 Progress in Understanding the Fundamental Casimir Force

In 2000, Bostöm and Sernelius bossern  put forward the first fundamentally new
idea relating to the surface-surface Casimir effect in over 40 years, since
Lifshitz’s paper lifshitz , which lies in the treatment of material
permittivities in the zero-frequency limit. The problem of finite conductivity
was addressed earlier by Hargreaves and later by Schwinger et al. schwinger
who proposed a possible means to deal with it, that is, to let the surface
material permittivity diverge before setting the frequency to zero. The point
is that in calculating the Casimir force at finite temperature, the integral
includes a Boltzmann’s factor which accounts for the thermal population of the
electromagnetic modes,

$N(\omega)+{1\over 2}={1\over e^{\hbar\omega/k_{b}T}-1}+{1\over 2}={1\over
2}\coth{\hbar\omega\over 2k_{b}T},$ (4)

where $\hbar\omega$ is the energy of a photon, $k_{b}$ is Boltzmann’s
constant, and $T$ is the absolute temperature. Because $\coth x$ has simple
poles at $x=\pm in\pi$, the integral over frequency in calculating the Casimir
force can be replaced by a sum of the residues at the poles of Eq. (4), or
Matsubara frequencies,

$\omega_{n}={n\pi k_{b}T\over\hbar}.$ (5)

Analytic continuation of the permittivity function allows the transformation
of the integral from over real frequencies to a contour integral on the
complex frequency plane, and it is valid to replace the integral over
frequency with a sum over the poles. The upshot is that the transverse
electric ($TE$) mode with $n=0$ does not contribute to the force at all if the
permittivity diverges slower than $\omega^{-2}$ in the limit as $\omega$ goes
to zero. It is generally assumed that for metals with a finite conductivity,
at zero frequency the permittivity goes as

$\epsilon(\omega)={4\pi i\sigma\over c\omega},$ (6)

in which case the $TE$ $n=0$ mode does not contribute at all to the force.
This is important because at room temperature, at distances greater than about
10 microns, this mode accounts for roughly half of the force. The implied
correction at separations of 1 micron is about 30%. This appears to be at odds
with a number of experiments, including my own. In particular, I had spent
much effort in finding a correction to my experiment that would bring the
results into agreement with my own incorrect calculation for Au. Thus I was
well-equipped to reject this result outright, as did a number of others.

One possible solution is that the permittivity diverges as $\omega^{-2}$ as
the frequency goes to zero. This has led to the proposal of a generalized
plasma model mosplas ,

$\epsilon_{gp}(i\xi)=\epsilon(i\xi)+{\omega_{p}^{2}\over\xi^{2}},$ (7)

where $i\xi$ represents the frequency along the imaginary axis, $\epsilon$ is
the usual Drude model permittivity, for example, and $\omega_{p}$ is the so-
called plasma frequency due to free electrons. Normally this expansion is
assumed to be valid at very high frequencies, much above the resonances in the
system of atoms and charges that comprise the plates. However assuming the
permittivity of this form brings back the contribution of the $TE$ $n=0$ mode,
and apparently improves the agreement between theory and experiment.

There are consequences in a broader complex of phenomena when this generalized
plasma model is introduced. In particular, if we consider the interaction of a
low-frequency magnetic field with a material surface, by use of Maxwell’s
equation, it is straightforward to show that jackson

$-\nabla^{2}\vec{H}={\omega^{2}\over c}\epsilon(\omega)\vec{H},$ (8)

which represents so-called eddy current effects, and can be easily extended to
the complex frequency plane. We see immediately that if $\epsilon$ diverges as
$\omega^{-2}$ that at zero frequency,

$-\nabla^{2}\vec{H}\propto\vec{H},$ (9)

which predicts that a static magnetic field will interact with an ordinary
conductor in a manner different from universal diamagnetism. Such an extra
effect is not experimentally observed, as Eq. (8) together with Eq. (6) is
known to describe the non-diamagnetic interaction of low frequency fields with
conductors. So we are faced with discarding over a century of electrical
engineering knowledge in order to explain a few 1% level Casimir force
experiments of questionable accuracy, and my own. This is not acceptable.

The crux of the problem lies in the fact that at equilibrium, all electric
fields at a surface of a conductor must terminate normal to the surface landl
. An electric field parallel to a surface implies a flowing current; such
currents can exist in a transitory fashion as associated with a fluctuation as
required for generating the Casimir force, but such fluctuations cannot occur
with zero frequency. For the $TE$ modes, the electric field is parallel to the
surface, so at zero frequency $TE$ modes simply cannot be supported, assuming
that equilibrium and zero frequency are equivalent. We will return to this
problem later in this review in relation to electrostatic calibrations.

This issue is, however, not yet settled as new precise experiments are
required. It is interesting that this effect becomes less pronounced at
smaller separations, simply because the $n=0$ modes contribute a relatively
smaller fraction to the total force. For my own experiment skl97prl  the
possibility of a systematic error is becoming more and more apparent. It
should be emphasized, however, that AFM type experiments probe an order of
magnitude smaller distance scale that the torsion pendulum experiments, and
the relative contributions of various effects are rapidly varying.

Work with AFMs and MEM type systems have demonstrated the difficulty of
producing metal and other films, together with their characterization, that
allows a comparison between experiment and theory at a level of better than
10%. For example, Svetovoy et al. optprops  show that the prediction of the
Casimir force between metals with a precision better than 10% must be based on
the material optical response measured from visible to mid-infrared range,
that the tabulated data is generally not good enough for precision work better
than 10% accuracy. The issues of roughness are well-discussed in mohideenrev ,
however, additional new work by de Zwol et al. dezwol  amplifies the problems
of surface roughness particularly in determining the absolute separation. It
appears that the best prospect for determining the correct form of the
permittivity function at zero frequency is to do a measurement at very large
separations. Indeed, problems of surface roughness correction virtually
disappear for typical optical finishes at distances about 500 nm. Above 2-3
microns, the difference between the force with and without the $TE$ $n=0$ mode
approaches a factor of two. Recent experimental work on Au films at Yale show
that the Boström-Sernelius analysis is likely correct, but this work is at a
very preliminary stage.

###  2.2 The Detection of New Long Range Forces

In the mid-1980’s, the question of the possible existence of a new so-called
fifth force was suggested based on data from Eötvos-type experiments fishbach
. Presently, interest in such forces is greater than ever due to possible
modification of gravity as allowed by String Theory, and due to the
observation of dark energy in the Universe which might be due to particles
associated with new long range forces that could manifest themselves on many
different length scales mostepdark . The basic idea is that our four
dimensional Universe is embedded in a space of more than 10 dimensions.
Leakage of lines of force between the larger space and our four dimensional
world could lead to a modification of the inverse square law, for example.
Although there is no specific prediction from a String theory, the possibility
does exist in its context.

With the publication of my 1997 experimental result, I received many
suggestions to analyze my experiment in light of an additional force that
would appear along with the Casimir force, however I rejected these
suggestions because my experiment was intended as a demonstration and any
limit would be at the level of 100% of the Casimir force. Taken as a fraction
of the gravitational field, my result was not particularly spectacular.
Nonetheless, other analyzed my experiment. Among the first to do so, in the
context of a general review of limits on sub-centimeter forces, was Long et
al. longprice  and earlier, with a more detailed analysis, was Klimchitskaya
et al. klimfifth .

The most ambitious recent work on this subject is by Decca et al. deccafifth
who achieved an astounding accuracy without observing any anomalous effects.
Use of the proximity force theorem, to be discussed later in this review, to
calculate the limits on a possible new force has been criticized. The issue is
that the proximity force theorem really only applies to a force that depends
on the location of the body surfaces; the approximation is not valid for the
volume integral required for calculating the anomalous force. The
applicability is addressed by Dalvit and Onofrio dalono  where corrections to
the calculation in deccacalc  are pointed out.

Earlier work by Decca et al. deccaiso  appears as more reliable at
constraining new forces. The technique developed here, a so-called
isoelectronic method, relied on the properties of an Au film being independent
of the substrate. For different materials coated with Au films of identical
optical characteristics and of sufficient thickness, the Casimir force should
be the same. In this work Au/Au and Au/Ge composites are compared, and the
result is “Casimir-less.” Techniques such as this appear as the most likely
way to achieve the best sensitivity to new forces, however, unfortunately the
minimum separation is limited by the Au film thickness, hence the later work
deccafifth . It should be noted that use of a screening film to eliminate
electrostatic forces and other background effect have been used in other
“fifth force” experiments for separations at the mm scale, but clearly the
trick can be scaled down to distances limited only by the skill of the
experimenter luther .

###  2.3 Modification of the Casimir Force

The possibility of modification of the Casimir force is a topic of current
great interest. With the rising of nanotechnology, the need to control,
modify, or make good use of the Casimir force is imperative as it is among the
dominant forces affecting MEMs and NEMs. At very short distances, at the
atomic scale, the large-scale geometrical aspects of the surfaces become
irrelevant, and the force becomes dominated by the van der Waals force between
atoms comprising the plates; the atom-atom force along with roughness leads to
stiction and friction. At such short distances, the treatment of the plates in
a continuum fashion fails. Any possibility to control either the short range
or long range force can have enormous technological benefits. These issues
have generated renewed interest in measuring the Casimir force with improved
precision, in applying it to nano-mechanical devices, and in controlling it.
In many instances, the attractive nature of the force leads to more problems
that to solutions because, for example, it leads to irreversible sticking of
the components in a nano-device. There have been proposals to develop
“metamaterials” which provide a boundary condition that makes the force
repulsive, but the extremely large frequency range of electromagnetic field
modes that contribute to the force suggests that this is not possible
dalvitmilonni .

The internal sticking problem of MEMs, however, might be slightly overstated.
Recent commentary relating to this possible problem has been based on the work
of Buks and Roukes bandr  where irreversible stiction was observed in MEMs
devices. In this work, the mechanical motion was monitored by use of an
electron beam which caused the components of the MEMs to become highly
charged. Whether the irreversibility is really due to the Casimir force, or if
it is due to charge surface interactions, remains an open question.
Nonetheless, it is agreed that a full understanding of the Casimir force, and
its possible control, are central to the future of MEMs and NEMs engineering.

The prospects of engineering a coating that can significantly modify the
Casimir force appear as dismal. This is because the Casimir force is a “broad-
band” phenomenon. Use of magnetic films has been suggested, but unfortunately
ferromagnetic response does not extended into the near-infrared and visible
spectrum that would be required to modify the Casimir force.

Recently, it has been demonstrated experimentally that a conductive oxide
film, Indium-Tin Oxide (ITO) produces a Casimir force about half of that due
to metals ito . ITO has a number of interesting features, including
transparency over the optical spectrum and chemical inertness. Thus it appears
as an interesting material from a nanoengineering viewpoint.

Casimir himself attempted to apply his namesake force to the electron,
specifically to calculate the fine structure constant. Casimir modelled the
electron as a conducting ball of uniform charge that would contract due to the
zero point energy of the external electromagnetic modes. This force would be
balanced by the space charge repulsion of the uniform charge density, when the
conducting sphere of constant total charge was just the right diameter. The
fine structure constant $\alpha\approx 1/137$ constant, which relates to the
electron diameter, could then be determined from fundamental parameters along
with a calculation of how the electromagnetic mode zero point energy changes
as the sphere contracts milonni . However, Boyer subsequently found that the
exterior spherical modes cause the sphere to expand boyer . Boyer’s result was
interesting enough that it led to the exploration of the effects of geometry
on the Casimir force.

The change in boundary conditions that had been considered cannot be realized
experimentally; for example, if one cuts a conducting sphere in half and tries
to measure the force between the hemispheres, the force is different from the
stress outside the continuous conducting sphere– simply because the two halves
are now separated by a vacuum gap and there will be an attraction there, and
because the structure of the surface modes is altered by the gap. Nonetheless,
several experiments aimed at directly modifying the Casimir force have been
performed in the last decade or so, and are continuing.

###  2.4 Hydrogen Switchable Mirror

An experiment with a surprising result employed a hydrogen switchable mirror,
and a change in the Casimir force was sought when the mirror was switched
between its low reflectivity and high reflectivity states capasso . The
surprise was that no significant change in the Casimir force was observed with
the switching, despite the rather dramatic change in the mirror from nearly
transparent to highly reflecting.

The explanation of the null result likely lies in the construction of the
mirror which has a very thin (5 nm) palladium layer to protect the underlying
sensitive structure. This layer tends to dominate the Casimir effect, even
though the layer is about one-half of a skin depth for the frequencies that
are affect by the hydrogen switching. Other complications include the narrow
spectral width of the mirror state which reduces the effect further, and the
layered structure of the mirror–it is possible that the principal activity
occurs in the deeper layers. In spite of these problems, hope remains that an
effect on the Casimir force will be detectable demanh .

###  2.5 Geometrical Boundary Effects

Until now, no significant or non-trivial corrections to the Casimir force due
to boundary modifications have been observed experimentally. As mentioned
above, for the systems that had previously been considered such as the
conducting sphere, it is not clear that an experimental measurement of the
external stress is even possible. Cutting a sphere in half clearly changes the
boundary value problem; it is unlikely that the two halves of such a sliced
sphere will be repelled with a force that is given by the external stress on
the sphere.

However, there are other possible ways to generate a geometrical influence on
the Casimir force. A conceptually straightforward way is to contour the
surfaces of the plates at a length scale comparable to the mode wavelengths
that contribute most to the net Casimir force. For a plate separation $d$, the
wavelengths that contribute most are $\approx\pi d$. This means that a surface
nano-patterned at 400 nm length scale should show a significant geometrical
effects for separations below 1 $\mu$m. Using such a system, Chan et al. have
produced a convincing measurement of a non-trivial geometrical influence on
the Casimir force chan .

[scale=.65]chanexpcopy.eps

Figure 1: An approximately scaled schematic representation of the experiment
of Chan et. al. The trench arrays, of varying width and depth, were made from
the same doped p-type Si substrate. (Public Domain, by S.K. Lamoreaux)

These measurements, between a nanostructured silcon surface and a Au coated
sphere, were made using a micromechanical torsional oscillator. The change in
resonant frequency of the oscillator, as a function of separation between the
Au sphere and the surface, provided a measure of the gradient of the Casimir
force. The sphere, of radius 50 $\mu$m coated with 400 nm of gold, was
attached to one side of the oscillator that comprised a 3.5 $\mu$m thick, 500
$\mu$m square silicon plate suspended by two tiny torsion rods. The sphere and
oscillator was moved toward the nanostructured surface by use of a
piezoelectric actuator.

Two different nanostructured plates, compared with a smooth plate, were
measured in this work. The geometry of the nanostructures, rectangular
trenches etched in the surface of highly p-doped silicon, were chosen because
the effects are expected to be large in such a geometry. Emig and Büscher had
previously calculated the effective modification of the Casimir force due to
such a geometry, but for the case of perfect conductors emig . Even though the
calculations were not for real materials, these theoretical results appeared
as a reasonable starting point for a comparison with an experiment.

Although much progress has recently been made toward a realistic and
believable accuracy and precision with which the Casimir force can be
calculated for real materials optprops , the problems associated with the
well-known experimental variability of sputtered or evaporated films were
avoided in the work of Chan et al. by comparing two different nanostructured
plates with a smooth plate, all made from the same silicon substrate, and all
using the same Au coated sphere. The trick is comparable to the Isoelectronic
method described in Sec. 2.2. So even though ab initio calculations of the
Casimir force for real material using tabulated optical properties cannot be
accurate to better than 10%, this problem was simply circumvented by the
comparison technique.

The geometric modification of the Casimir force was detected by measuring a
deviation from that expected by use of the Proximity Force Approximation
(PFA), or the Pairwise Additive Approximation (PAA), both of which will be
described later in this review. The success of the PFA is so good that it
suggests a means of detecting a geometrical effect. Basically, the surface is
divided into infinitesimal units, and it is assumed that the total Casimir
force can be determined by adding the Casmir force, appropriately scaled by
area, between surface unit pairs in opposite surfaces; this is the PAA. Thus,
for the nanostructured surfaces, a 50% reduction in force would be expected by
the PAA, because the very deep trenches (depth $t=2a\approx 1\mu$m), etched as
a regular array, were designed to remove half of the surface. As mentioned,
two different trench spacings $\lambda$ were fabricated and measured, such
that $\lambda/a=1.87$ (sample A) and $0.82$ (sample B), and compared to a
smooth surface. The Casimir force between the gold sphere and the smooth
plate, as calculated from the tabulated properties of gold and silicon, taking
into account the conductivity due to the doping, agree with the experimental
results to about 10% accuracy. For sample A, the force is 10% larger than
expected by the PAA, using the measured smooth surface force, and for sample
B, it is 20% larger, in the range $150<z<250$ nm. The deviation increases as
$\lambda/a$ decreases, as expected.

The theory of Emig and Büscher predicts deviations from the PAA twice as large
as were observed. Nonetheless, the results of Chan et al. indicate a clear
effect of geometry on the Casimir force. However, much theoretical work
remains to be done toward gaining a complete understanding of the experimental
observations. The already difficult calculations are made more so by the
finite conductivity effects of the plates, and the sharp features of the
trenches as opposed to the smooth simple sinusoidal corrugations. New
calculational techniques have been developed that will allow reasonable
accuracy calculations. Also a number of possible systematics associated with
electrostatic effects were not fully investigated.

###  2.6 Repulsive Casimir Effect

The generalized Liftshitz formulation of the Casimir force allows for a
material between the plates. The force is thus altered from the case of a
vacuum between the plates, and the effect can be calculated. It is easy to
envision filling the space between the sphere and plate of a Casimir setup
with a liquid and measuring the effects of replacing the vacuum. A first
experiment using alcohol between the plates was done by Munday et al. alcohol
where a substantial reduction in the force was observed compared to what is
expected with vacuum between the plates. The effects of Debye screening and
other electrostatic effects were also thoroughly studied screening .

Munday et al. extended their studies to a very interesting situation where the
Casimir force becomes repulsive, by suitably choosing the permittivities of
the plates and liquids. If the plates’ material dielectric permittivities are
$\epsilon_{1}$ and $\epsilon_{2}$, and the liquid between has $\epsilon_{3}$,
the force will be repulsive when $\epsilon_{1}>\epsilon_{3}>\epsilon_{2}$. Of
course, the permittivities are frequency dependent, so this relationship must
hold over a sufficiently broad range of frequencies.

Perhaps a more familiar problem is the wetting of a material surface by a
liquid. In this case, one plate is replaced by air or vacuum so
$\epsilon_{2}=1$, and if the liquid permittivity is less than that of the
remaining plate, the liquid spreads out in a thin film rather than forming
droplets. For example, liquid helium, which has a very small permittivity,
readily forms a thin film because it is “repelled” by the vacuum
$(\epsilon_{1}>\epsilon_{3}>\epsilon_{2}=1)$, and we say that the liquid wets
the surface. On the other hand, liquid mercury which has a high effective
permittivity does not wet glass $(\epsilon_{1}<\epsilon_{3}>\epsilon_{2}=1)$.

Although there are many liquids that wet glass or fused silica, there are only
a few sets of materials that will satisfy the requirement for a repulsive
force between material plates. The set employed by Munday et al. was fused
silica and gold, with bromobenzene as the liquid. The experimental setup was
based on an atomic force microscope (AFM) that was modified slightly for the
detection of average surface forces rather than atomic-scale point forces. For
measuring the Casimir force, the sharp tip was replaced by a gold coated
microsphere (diameter = 39.8 microns) which serves as the gold plate. Using a
spherical surface for one plate simplifies the system geometry, which is
completely defined by the sphere radius and distance of closest approach from
the flat fused silica plate.

A problem that all Casimir force experiments face is the system force
calibration. For this work and related work, a most clever calibration
technique was devised. Because the fluid produces a hydrodynamic force when
the sphere/plate separation is changed, and this force is linear with
velocity, subtracting the force when the separation is changed at two
different speeds produces the hydrodynamic force without any contribution from
the Casimir force. The hydrodynamic force thus measured, which can be
calculated to high accuracy, provided the calibration. In addition, this
force, scaled to the appropriate velocity, was then subtracted from the force
vs. distance measurement, yielding a clean measurement of the Casimir force.
The measurements spanned a range of 20 nm to several hundred nm, with the
minimum distance limited by surface roughness, and the maximum distance
limited by system sensitivity. Various spurious effects were accounted for and
shown to have no significant contribution within the statistical accuracy of
the measurement.

Showing that it is indeed possible to produce and measure a repulsive Casimir
force is important to both fundamental physics and to nanodevice engineering.
There has been much discussion of such forces as they will provide a means of
quantum levitation of one material above another. Even in a fluid, it will be
possible to suppress mechanical stiction and make ultra-low friction sensors
and devices. It might be possible to “tune” the liquid (e.g., by use of a
mixture) so that at sufficiently large distances, the force becomes
attractive, while being repulsive at short distances. This would allow objects
to levitate above a liquid covered surface, for example.

##  3 Approximations, Electrostatic Calibrations, and Background Effects

Wittingly or unwittingly, many approximations have been included in all
Casimir force experiments to date. For example, most experiments employ the
use an electrostatic force from accurately measured applied voltage for
calibrations and the detection of spurious contact potentials between the
plates. The force is assumed to follow the form

$F(d)={1\over 2}{\partial C(d)\over\partial d}V^{2},$ (10)

where $C(d)$ is the capacitance between the Casimir plates, as a function of
distance $d$ between them. An exact calculation exists between a sphere and a
plane, however, for most situations the so-called Proximity Force
Approximation (PFA) can be used. In the case of a plate with spherical surface
with curvature $R$, with a distance $d$ at the point of closest approach to a
plane surface, the force between the two plates is

$F(d)=2\pi R{\cal E}(d),$ (11)

where ${\cal E}(d)$ is the energy per unit area between plane parallel
surfaces that leads to the attractive force.

Briefly, the PFA was introduced by Deryagiun Deryagiun  to describe the
Casimir force between a curved surfaces, and this approximation is known to be
extremely accurate when the curvature is much less than the separation between
the surfaces. The PFA can be used beyond the Casimir force and has quite
general applicability PFA . The PFA is a special case of the Pairwise Additive
Approximation (PAA) where the plate surfaces are divided into infinitesimal
area elements, and the force is determined through a pairwise addition of
corresponding elements. The PFA and PAA work very well for electrostatic
effects because, for a conductor (even poor) in equilibrium, the electric
lines of force must be normal to the surface, otherwise currents would flow in
contradiction to the assumption that the system is in equilibrium.

The use of a sphere and a flat plate vastly simplifies an experiment because
the system is fully mechanically defined in terms of the point of closest
approach and the radius of curvature of the sphere. For two flat plates the
system is specified by two tilt angles, the areas, long-scale smoothness, and
a separation, which all need to be defined, measured, and controlled. It is
interesting to note that if the force is measured as a function of applied
voltage in the sphere-plane configuration that the result should be

$F(d)={\pi\epsilon_{0}R\over d}V^{2}=\alpha V^{2},$ (12)

where $\epsilon_{0}$ is the permittivity of free space, and $R$ is the radius
of curvature of the spherical surface. The absolute distance between the
sphere and the plane surface is proportional to $\alpha^{-1}$ and this
provides a means of determining the distance.

Even when the full form of the sphere-plane capacitance is used in Eq. (10),
approximations still exist. Specifically, there are additional terms to the
force given by Eq. (10) because the capacitance is in fact a tensor. This can
be easily seen, as when a charged sphere is bisected, the two halves repel
each other, with a force

$F={q^{2}\over 8R},$

where $q$ is the charge on the sphere landl  (Prob. 2, Sec. 5). Note that this
is the force for a fixed charge, which must be modified for a fixed voltage.
The point is that the two halves experience a force, even though their
potential difference is zero; there are apparently additional terms that need
to be added to Eq. (10). As the geometry is not critical in this argument, we
can conclude that if the two plates of a Casimir experiment are at the same
non-zero potential, there will be an additional force repulsive force between
them. This sort of effect has not been considered at all.

The other problem that has received significant attention only recently is the
effect of patch potentials on a conducting surface. The effect is well-known,
and is largest with clean samples because when dirt is present, ions tend to
accumulate at the boundaries between the patches, shielding the effect landl
(Sec. 23).

To date, every Casimir effect that has bothered measuring the contact
potential as a function of distance has shown an apparent distance dependence
of that potential. Various experiments are nicely reviewed in staticsurvey .
The basic essential problem manifests itself in anomalous behavior in the
electrostatic calibration of an experiment, for example, as experienced in
kimanom . It was suggested that the anomalous effects that were observed are
due to irregularities of the spherical surface. Roughness effects rough
certainly can cause problems at short distances, but the possibility that the
anomalous effects are due to simple geometrical effects is credibly discarded
in staticsurvey .

The contact potential is simply measured by finding a voltage potential
difference $V_{m}$ between the two plates that minimizes the force given by
Eq. (10). $V_{m}$ is manifest as an asymmetry in the force between $\pm V$
applied between the plates.

For our measurements using Gemanium (Ge) plates ourge , we were initially
confused because a $1/d^{1.2}$ to $1/d^{1.5}$ force persisted when the
electrostatic force was minimized at each distance. Our initial conclusion was
that there was a distance offset, as described in the next section, together
with an uncompensated voltage offset. de Man et al. iannuzzi2009  have also
observed a distance dependence of the contact potential, and concluded that it
did not lead to any anomalies in their electrostatic calibrations, however,
the measurements are at shorter distances than were used in the Ge experiment.
In general, the relative electrostatic effect, compared to the Casimir force,
should scale roughly as $(1/d)/(1/d^{3})=d^{2}$. I will now tell the story of
how we came to understand the results of our measurements using Ge plates.

[scale=.65]casphoto.eps

Figure 2: A photograph of the apparatus, in operation, used to measure the
attractive force between Ge plates. The glass bell jar introduces some
distortion; visible are the “compensating plates” on the left of the torsion
pendulum, and the plates (2.54 cm diameter) between which the Casimir force is
measured, on the right. A ThorLab T25XYZ translation stage is used to position
the “fixed” plate. The fine tungsten torsion wire is not visible. (Public
Domain, by S.K. Lamoreaux)

###  3.1 Inclusion of the Debye Screening Length?

In the early calibrations of our Ge plate Casimir experiment ourge , we had a
long-range background force that depended on distance not quite as $1/d$, as
described above. Our initial guess was that there was a distance offset in our
calibrations due to penetration into the plates of the calibration electric
field. The problem is that a quasi-static electric field can propagate a
finite distance into a semiconductor (see, e.g., 5 ); this distance is
determined by the combined consideration of diffusion and field driven
electric currents, leading to an effective field penetration length (Debye-
Hückel length)

$\lambda=\sqrt{\epsilon\epsilon_{0}kT\over e^{2}c_{t}},$ (13)

where $c_{t}=c_{h}+c_{e}$ is the total carrier concentration, which for an
intrinsic semiconductor, $c_{e}=c_{h}$. For intrinsic Ge $\lambda\approx 0.6\
\mu$m, while for a good conductor, it is less than 1 nm. $\lambda$ is
independent of the applied field so long as the applied field $E$ times
$\lambda$ is less than the thermal energy, $k_{b}T$ where $k_{b}$ is
Boltzmann’s constant. In this limit, and at sufficiently low frequencies and
wavenumbers, thermal diffusion dominates the field penetration into the
material. A sufficiently low frequency for Ge would be $v_{c}/\lambda\sim 10$
GHz, where $v_{c}$ is a typical thermal velocity of a carrier.

The potential in a plane semiconductor, if the potential is defined on a
surface $x=0$ is

$V(x)=V(0)e^{-|x|/\lambda},$ (14)

where $\lambda$ is the Debye-Hückel screening length, defined previously.

We are interested in finding the electrostatic energy between two thick Ge
plates separated by a distance $d$, with a voltages $+V/2$ and $-V/2$ applied
to the back surfaces of the plates. In this case, the field is normal to the
surface. After we find the energy per unit area, we can use the proximity
force approximation to get the attractive force between a spherical and flat
plate.

Let $x=0$ refer to the surface of the plate 1, and $x=d$ refer to the surface
of plate 2. By symmetry, the potential at the center position between the
plates is zero. The potential in plate 1 can be written as

$V_{1}(x)=V/2-(V/2-V_{s})e^{-|x|/\lambda},$ (15)

and for the space between the plates

$V_{0}(x)=-2V_{s}x/d+V_{s},$

where we assume the field is uniform. $V_{s}$, the surface potential, is to be
determined.

We need only consider the boundary conditions in plate 1, which are

$V_{1}(-\infty)=V/2,$ $V_{0}(0)=V_{1}(0),$

(which has already been used)

$\epsilon{dV_{1}(x)\over dx}|_{x=0}={dV_{0}(x)\over dx}|_{x=0},$

where the last two imply that $D=\epsilon E$ is continuous across the
boundary.

The solution is

$V_{s}={V\over 2}\left({1\over 1+2\lambda/\epsilon d}\right).$ (16)

With this result, it is straightforward to calculate the total field energy
per unit area in both plates and in the space between the plates. The result
is

${\cal E}={1\over 2}{\epsilon_{0}V^{2}\over
d}\left[{y+y^{2}\over(y+2)^{2}}\right],$ (17)

where the dimensionless length $y=\epsilon d/\lambda$ has been introduced. By
expanding this result for small $y$, it can be easily seen that the effects
appears as an apparent offset in the distance that is determined by measuring
the capacitance between the plates. For small voltages, this offset is
approximately $\lambda/\epsilon=0.68/16\approx 0.05\ \mu$m.

If $V-V_{s}$ is large compared to $k_{b}T$, the effective penetration depth
increases because the charge density is modified in the vicinity of the
surface. The potential in the plates is no longer a simple exponential,
however one can define an effective shielding length 5

${\lambda^{\prime}\over\lambda}={|\Phi|\over\sqrt{e^{\Phi}+e^{-\Phi}-2}},$
(18)

where

$\Phi={V-V_{s}\over k_{b}T}.$ (19)

Given that $k_{b}T=30$ meV, at plate separations of order 1 $\mu$m for Ge this
begins to be a large correction when voltages larger than 60 mV are applied
between the plates, however, the potentials used in our experiment were far
smaller.

We eventually realized that this effect is not present at very low
frequencies; the lifetime of Ge surface states is on the order of
milliseconds. The lack of penetration of quasi-static fields into
semiconductors was first observed in the development of the field effect
transistor, and explained by Bardeen bardeen  as shielding due to surface
states. Again, in equilibrium, the electric field must enter normal to the
plate surfaces, otherwise a current would be flowing in contradiction to the
assumption of equilibrium. Therefore, even on very poor conductors, charges
rearrange to force any applied field to be perpendicular to the surface; when
this situation is attained, the electric field terminates at the surface. The
boundary condition is that of a perfect conductor.

The presence of time-dependent surface states might be responsible for some of
the anomalous electrostatic calibration effects observed by Kim et al. kimanom
. Particulary if there is a slight oxide coating on a metal surface, the
surface states might not have enough time to reach equilibrium in the dynamic
measurement system that was employed. The relaxation times for trapped surface
states can be many milliseconds. However, the possibility that these sorts of
states contribute to the anomalous effect is very speculative, and it is
difficult to come up with an experiment to check this hypothesis.

As an aside, our consideration of this effect led us to the realization that
the usual permittivity treatment of materials with non-degenerate conduction
electrons is not correct, but must be solved in a different way than simply
assigning a conductivity to the material carrierpaper . The discussion of this
theoretical point is beyond the scope of this review.

###  3.2 Variable Contact Potential

It was recognized that a distance dependence of the minimizing potential would
lead to extra electrostatic forces that are not necessarily zero at the
minimizing potential myarxiv . The force at the voltage which minimizes the
force at each separation was thought to represent the pure “Casimir” force
between the plates. However, the applied voltage $V_{a}(d)$ required to
minimize the (electrostatic) force is observed to depend on $d$, and is of the
form (in the 1-50 $\mu$m range)

$V_{a}(d)=a\log d+b,$ (20)

where $a$ and $b$ are constants with magnitude of a few mV. This variation
leads to a long-range $1/d$-like potential for the minimized force. An
analysis suggests that this force is better described as $1/d^{m}$ where
$m\approx 1.2-1.4$.

As we show here, the variation in $V_{a}(d)$ implies an additional force that
increases as $1/d^{1.25}$, assuming that the voltage variation is due to the
potential of the plates actually changing with distance. Such changes could
come about due to external fixed fields or potential variations associated
with the plate translation mechanism, and is equivalent to having an
adjustable battery in series with the plates. We were unable to come up with a
model that can give a sufficiently large effect based on interactions between,
for example, the charge carriers. in the plates. However, at sufficient
sensitivity, it is likely that such effects will be important.

This analysis, while it predicts the correct form of the extra force, predicts
that this force is negative or repulsive. However, it is enlightening to go
through the analysis, and this work will never be published elsewhere. An
understanding of the specific origin of the variation of applied minimizing
potential $V_{a}(d)$ is not necessary to correct for the additional force that
it causes, we simply need the experimentally determined $V_{a}(d)$, and assume
it is tied to the plate positions.

We note further that $V_{a}(d)$ is not a measure of the contact potential, but
the voltage which minimizes the force. We call the “true” contact potential
$V_{c}(d)$, which might depend on distance.

In performing our experiment, at each separation $d$, $V_{a}$ is varied and
its value that minimizes the attractive force is determined.

${\cal E}(d)={1\over 2}C(d)(V_{a}+V_{c}(d))^{2},$ (21)

where $C(d)$ is the capacitance between the plates, $V_{a}$ is the applied
potential and is an independent variable, and $V_{c}(d)$ is the average
weighted contact potential between the plates.

The force between the plates is given by the derivative of $\cal E$,

$F(d)={\partial{\cal E}(d)\over\partial d}={1\over 2}{\partial
C(d)\over\partial d}(V_{a}+V_{c}(d))^{2}+C(d)(V_{a}+V_{c}(d)){\partial
V_{c}(d)\over\partial d},$ (22)

Now the minimum in the force is determined by the derivative with $V_{a}$:

${\partial F(d)\over\partial V_{a}}={\partial C(d)\over\partial
d}(V_{a}+V_{c}(d))+C(d){\partial V_{c}(d)\over\partial d}=0,$ (23)

which determines $V_{a}(d)$, no longer an independent variable. Thus,

${\partial V_{c}(d)\over\partial d}=-{1\over C(d)}{\partial C(d)\over\partial
d}(V_{a}(d)+V_{c}(d)),$ (24)

which allows the determination of $V_{c}(d)$ when $V_{a}(d)$ is known. The
differential equation can be solved numerically, noting that at long distances
$V_{a}(d)=-V_{c}(d)$, and that $V_{c}(d)$ become constant.

The electrostatic force between the plates at the minimized potential is given
by

$F(d)=-{1\over 2}{\partial\over\partial
d}\left[C(d)(V_{a}(d)+V_{c}(d))^{2}\right].$ (25)

There are some nice features to this result. First, if we apply a constant
offset $V_{0}$ to $V_{c}(d)$, this effect is compensated by $V_{a}(d)-V_{0}$
which is easily seen as the relationship is linear.

Unfortunately, the sign of the effect indicates that it is repulsive, and thus
is not the explanation of the long range force that persists at the minimizing
potential as observed in our Ge experiment.

However, it should be emphasized that any precision measurement of the Casimir
force requires verification that the contact potential is not changing as a
function of distance, and if it is, a correction to the force as described
here might very well exist.

###  3.3 Patch Potential Effects

It is often assumed that the surface of a conductor is an equipotential. While
this would be true for a perfectly clean surface of a homogeneous conductor
cut along one of its crystalline planes, it is not the case for any real
surface which can be polycrystalline, stressed, or chemically contaminated.
Experiments show that even with precautions for extreme cleanliness, typical
surface potential variations are on the order of at least a few millivolts
LIGO . This is most likely due to local variations in surface crystalline
structure, giving rise to varying work functions and hence varying-potential
patches. It is well known that the work function of a metal surface depends on
the crystallographic plane along which it lies; as an example, for gold the
work functions are 5.47 eV, 5.37 eV, and 5.31 eV for surfaces in the $\langle
100\rangle$, $\langle 110\rangle$, and $\langle 111\rangle$ directions
respectively. This variation is most likely due to different effective
electron masses, hence Fermi energies, for the different axes.

The means by which surface potential patches form is described in landl , Sec.
22. Briefly, When two conductors, A and B, of different work functions are
brought into contact, electrons flow until the chemical potential (i.e., the
Fermi energy) in both conductors equalizes. If we consider moving an electron
in a closed path that moves from inside conductor A, across the boundary to
inside conductor B, through the surface of B into the vacuum, back though
surface A, and to the starting point, the total work must be zero in
equilibrium. If we take the contact potential difference between the
conductors as $\phi_{ab}$, and the surface work functions as $W_{a}$ and
$W_{b}$, for the total work to be zero we must have

$\phi_{ab}=W_{b}-W_{a},$

implying that the contact potential is simply the difference in the surface
work functions.

It is straightforward to calculate the electric field energy of random
patches, as has been done by Speake and Trenkel Speake2003 . Consider two
plane and parallel surfaces separated by a distance $d$. Assume a potential
$V=0$ at $x=0$, while at $x=d$, $V=V_{0}\cos ky$. It is easy to show that, in
the region between the plates,

$V(x,y)=V_{0}\cos ky{e^{kx}-e^{-kx}\over e^{kd}-e^{-kd}}.$

The field energy, per unit area is given by

${\cal E}=\int_{0}^{d}\left[\left({\partial V\over\partial
x}\right)^{2}+\left({\partial V\over\partial y}\right)^{2}\right]dx,$

where we have used the fact that
$\langle\cos^{2}ky\rangle=\langle\sin^{2}ky\rangle=1/2$ to do the $y$
integral. Letting

$u=e^{kx}-e^{-kx}\ \ \ \ dv=e^{kx}-e^{-kx},$

so

$du=k[e^{kx}+e^{-kx}]\ \ \ \ \ v={1\over k}[e^{kx}+e^{-kx}],$

and integrating by parts

$\int_{0}^{d}[e^{kx}-e^{-kx}]^{2}dx={1\over
k}[e^{2kx}-e^{-2kx}]|_{0}^{d}-\int_{0}^{d}[e^{kx}+e^{-kx}]^{2}dx.$

The LHS is proportional to the field energy for $E_{y}$ while the last term on
the RHS is proportional to (minus) the field energy for $E_{y}$. We thus have

${\cal E}=k{V_{0}^{2}\over 2}{e^{2kd}-e^{-2kd}\over[e^{kd}-e^{-kd}]^{2}}.$

By use of the proximity force approximation, the (attractive) force between a
flat surface and spherical surface is $F(d)=2\pi R{\cal E}(d)$ where $R$ is
the radius of curvature, where $d$ is the point of closest approach between
the surfaces. In the limit $kd\rightarrow 0$,

$F=2\pi R{V_{0}^{2}\over 4d}\propto{1\over d}.$

This shows that when $kd\ll 1$ or $d\ll\lambda/2\pi$ where $\lambda$ is a
characteristic length of a potential patch, the force goes as $1/d$. This is
what we expect from the PAA when the surfaces are very close.

There is an intermediate range where the force transforms from $1/d$ to
exponential variation; at further distances, the force becomes a constant, as
$\cal E$ does not vary with $d$. Between parallel plates, at long distances,
the force is zero because the field energy does not change with separation. It
is interesting to note this significant difference between the PFA result for
a spherical surface and the result for parallel plates. As a constant force is
in reality unobservable, this long distance force should be subtracted from
the PFA result.

It should be noted that the field equations are linear, so we can add other
$\cos(k^{\prime}y),\ \cos(k^{\prime}z)$ fluctuations, and the integral over
$z,y$ leads to delta functions of $k-k^{\prime}$. We can therefore rewrite the
attractive force as an integral over $k_{y},k_{z}$ where we have
$V_{k_{y}}(y)+V_{k_{z}}(z)$ representing the amplitude spectrum in $k$ space
of the surface fluctuations. If we take $V_{k_{y}}(y)\sim V_{k_{z}}(z)$ and
assume they are uncorrelated, the integral over $k_{y},k_{z}$ leads to

$F=\pi RV_{rms}^{2}\int_{0}^{\infty}(2\pi k\
dk)(kS(k)){e^{2kd}-e^{-2kd}\over[e^{kd}-e^{-kd}]^{2}},$

where, by use of the Wiener-Khinchine theorem, $S(k)$ is the normalized cosine
Fourier transform (in polar coordinates) of the surface potential spatial
correlation function.

In order to compute the patch effect on the force in the sphere-plane
configuration we make use of the proximity force approximation. Just as in the
case of roughness in Casimir physics rough , one must distinguish between two
PFAs: one is for the treatment of the curvature of the sphere (valid when
$d\ll R$, where $R$ is the radius of curvature), and the other one is the PFA
applied to the surface patch distribution (valid when $kd\ll 1$). We assume
that we are in the conditions for PFA for the curvature, but we keep $kd$
arbitrary. Then, the electrostatic force in the sphere-plane case is
$F_{sp}(d)=2\pi R{\cal E}(d)$, implying

$F_{sp}=2\pi\epsilon_{0}R\int_{0}^{\infty}dk\frac{k^{2}e^{-kd}}{\sinh(kd)}S(k).$
(26)

There are a number of models that can be used to describe the surface
fluctuations. The simplest is to say that the potential autocorrelation
function is, for a distance $r$ along a plate surface,

${\cal R}(r)=\left\\{\begin{array}[]{ll}{V_{0}^{2}}&{\rm for}\
r\leq\lambda,\\\ 0&{\rm for}\ r>\lambda.\end{array}\right.$ (27)

Then, by the Wiener-Khinchin theorem, the power spectral density $S(k)$ can be
evaluated as the cosine two-dimensional Fourier transform of the
autocorrelation function, which in our notation is stein

$S(k)={V_{0}^{2}}\lambda^{2}{J_{1}(\lambda k)\over\lambda k},$ (28)

with $J_{1}$ the Bessel function of first kind. The plane-sphere force is then
given by, using $k=u/\lambda$,

$F_{sp}=2\pi\epsilon_{0}R\int_{0}^{\infty}du\ u{J_{1}(u)\over
e^{2ud/\lambda}-1}.$ (29)

A numerical calculation shows that, for $d<.01\lambda$,

$F_{sp}\approx{\pi\epsilon_{0}RV_{0}^{2}\over d},$ (30)

suggesting that $V^{2}_{\rm rms}=V_{0}^{2}$, as expected. For
$50\lambda>d>\lambda$, the force falls with distance as $1/d^{3}$.

We see immediately that at short distances, there is a residual force due to
patches that varies as $1/d$, and there is no minimizing potential that can
compensate this effect. It is, in a restricted sense, equivalent to having an
oscillating potential between the plates; there is no way for a static field
to compensate the oscillating field energy.

[scale=.65]patchmodelcopy.eps

Figure 3: A toy model illustrating the mechanism for the generation of a
distance-dependent minimizing electrostatic potential $V_{m}(d)$ and
electrostatic residual force $F^{\rm el}_{\rm res}(d)$. (Public Domain, by
S.K. Lamoreaux)

As described in the last Section, in our own work ourge  and in a number of
other experiments kimanom ; iannuzzi2009 , a distance-variation in the
electrical potential minimizing the force between the plates has been
observed. It had been suggested already that this variation in contact
potential can cause an additional electrostatic force, and an estimate was
made for the possible size of the effect myarxiv . However, further
experimental work shows that the model used in myarxiv , where the varying
contact potential is considered to be a varying voltage in series with the
plates, does not reproduce our experimental results ourwork2009 .

A model that produces a residual electrostatic force consistent with our
observations ourge  is shown in Fig. 3. In this figure, the two capacitors
(short distance, $C_{a}(d)$, long distance $C_{b}(d+\Delta)$) create a net
force on the lower continuous plate (setting $V_{1}=0$ initially),

$F(d,V_{0})=-\frac{1}{2}C_{a}^{\prime}V_{0}^{2}-\frac{1}{2}C_{b}^{\prime}(V_{0}+V_{c})^{2},$
(31)

where

$C_{a}^{\prime}={\partial C_{a}(d)\over\partial d};\ \ \
C_{b}^{\prime}={\partial C_{b}(d+\Delta)\over\partial d},$ (32)

and $V_{0}$ can be varied, with $V_{c}$ a fixed property of the plates. The
force is minimized when

$\displaystyle\left.{\partial F(d,V_{0})\over\partial
V_{0}}\right|_{V_{0}=V_{m}}=0$ $\displaystyle\Rightarrow$ $\displaystyle
V_{m}(d)=-{C_{b}^{\prime}V_{c}\over C_{a}^{\prime}+C_{b}^{\prime}},$ (33)

implying a residual electrostatic force

$\displaystyle F^{\rm el}_{\rm res}(d)$ $\displaystyle=$ $\displaystyle
F(d,V_{0}=V_{m}(d))$ (34) $\displaystyle=$
$\displaystyle-\left[C_{a}^{\prime}+{C_{a}^{\prime 2}\over
C_{b}^{\prime}}\right]{V_{m}^{2}(d)\over
2}=-\left[\frac{C_{a}^{\prime}C_{b}^{\prime}}{C_{a}^{\prime}+C_{b}^{\prime}}\right]{V_{c}^{2}\over
2}.$

It is easy to take a case of parallel plate capacitors
($C_{a}^{\prime}=-\epsilon_{0}A/d^{2}$ and
$C_{b}^{\prime}=-\epsilon_{0}A/(d+\Delta)^{2}$, where $A$ is the area of each
of the upper plates in Fig. 3, assumed to be equal; hence, the lower
continuous plate has area $2A$) and to show that there is a residual
electrostatic force at the minimizing potential. Indeed, in such case,

$\displaystyle V_{m}(d)$ $\displaystyle=$ $\displaystyle-
V_{c}\frac{d^{2}}{d^{2}+(d+\Delta)^{2}},$ (35) $\displaystyle F^{\rm el}_{\rm
res}(d)$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{0}A}{2}\;\frac{V_{c}^{2}}{d^{2}+(d+\Delta)^{2}}.$
(36)

Alternatively, in terms of $V_{m}(d)$ (up to $V_{1}$, see below), the force is

$F^{\rm el}_{\rm
res}(d)=\frac{\epsilon_{0}A}{2}\;\frac{V_{m}^{2}(d)[d^{2}+(d+\Delta)^{2}]}{d^{4}}.$
(37)

Experimentally, $V_{m}(d)$ cannot be directly measured; measurements can only
determine it up to an overall offset $V_{1}$ which arbitrarily depends on the
sum of contact potentials in the complete circuit between the plates.
Therefore the force should be written at proportional to
$(V_{m}(d)+V_{1})^{2}$ instead of simply $V_{m}^{2}(d)$, where $V_{1}$ is
determined by a fit to experimental data, for example. In the limit $\Delta\gg
d$, the residual force is proportional to $1/d^{4}$ in the plane-offset plane
case here considered.

If we now consider the sphere-plane case,
$C_{a}^{\prime}(d)=-2\pi\epsilon_{0}R/d$, and the denominator of Eq. (37)
becomes $d^{2}$. If we further consider the surface divided up into
infinitesimal areas, each with a random potential, and integrate over the
surface to get the net force, there is a further reduction of the power of $d$
in the denominator (just as in the proximity force approximation), leaving the
sphere-plane force proportional to $1/d$. This motivates writing the residual
force as

$F_{res}(d)={\pi\epsilon_{0}R\left[(V_{m}(d)+V_{1})^{2}+V_{rms}^{2}\right]\over
d},$ (38)

where it is understood that $V_{m}(d)$ is experimentally measured, and $V_{1}$
is a fit parameter that represents a sort of surface average potential, plus
circuit offsets (this equation is supported both by numerical studies and by
our experimental results ourge ; ourwork2009 , and is valid when
$|V_{1}|>>|V_{m}(d)|$) as observed. The last term in Eq. (38) is the expected
random (i.e., does not contribute to $V_{m}(d)$) patch potential force, but
here should be thought as a fit parameter that reflects the magnitude of
$V_{rms}$. With this result, the long range force observed in our experiment
could be explained, and our work with Ge was completed. The agreement with
theory is excellent, however, there is very little difference in theoretical
prediction of the force with and without the $TE$ $n=0$ mode, so this work was
not able to help with that controversy.

As a final note, the variations in surface potential could be a simple
function of position on the conducting surface, for example, due to stresses
or impurities within the samples. Alternatively, if there is a slight
roughness to the surface, the peaks could have a different potentials that the
valleys associated with surface irregularities. This latter possibility
appears to be a better model as we were unable to detect a variation in
$V_{m}$ when the plates were moved relative to each other, which might be
expected for positional surface patches. However, the level of the surface
fluctuations is quite small, and for example is beyond the range of state of
the art Kelvin probes kelvinprobe . These issues need further investigation.

##  4 Conclusions and Outlooks

In many respects, we can consider the measurement of the Casimir force between
surfaces as a mature field. However, many open issues remain, particularly in
the limits of accuracy that can be expected. In recent years, we have seen a
number of experiments claiming 1% precision, but many counter claims that such
accuracy is beyond what is possible due to finite knowledge of a plethora of
corrections and required absolute calibrations. Some open issues include the
effects of finite conductivity on the contribution of the $TE$ $n=0$ surface
mode; the usual Drude model of the permittivity of a metal suggests that this
mode does not contribute at all to the force, reducing the force by a factor
of two at large separations. It is unclear whether additional short-range AFM
type measurements will clear this problem up, as at short distances, the
correction is relatively small. Improved measurements at distances above a few
microns would appear to offer the best prospects for bringing these issues to
closure. Recent work with our torsion pendulum system at Yale seems to be in
favor of the no-$TE$ $n=0$ mode, although the precision is not yet sufficient
to make a strong claim. Over the next few months we hope to have new higher
accuracy data analyzed.

The effects of patch potentials has not been fully investigated in all
experiments to date. For example, in my 1997 experiment skl97prl , an
anomalous component to the $1/d$ force would result in an error in the
distance determination, which only needed to be 0.1 micron to bring my
experiment into agreement with the Bostöm and Sernelius calculation. Likewise,
the boundary modification experiment of Chan et al. chan  did not consider in
any obvious way excess forces due to electrostatic patch effects, which might
be expected as substantial due to the sharp features of the etched silicon
trenches, and will vary as $1/d^{3}$ in the limit of the separation much large
than the trench spacing. It is hard to imagine that such an effect is more
than 10% of the Casimir force, but some analysis and additional experiments
are necessary to eliminate the possibility of such a systematic effect.

In any case, a reasonable ultimate experimental goal is the attainment of 1%
agreement between theory and experiment, in terms of true accuracy; this is
not a question of simple precision. Hopefully the readers of this review will
realize the complexity and difficulty of the challenge presented by this goal.

##  5 Acknowledgements

I thank my colleagues and collaborators W.-J. Kim, A.O. Sushkov, H.X. Tang,
and D.A.R. Dalvit for many fruitful discussions that led to the understanding
of our Ge experiment, and to deeper understanding of the Casimir force in
general. I also thank R. Onofrio and S. de Man to a number of discussions over
the last few years that were helpful in clarifying a number of issues. SKL was
supported by the DARPA/MTO Casimir Effect Enhancement project under SPAWAR
contract number N66001-09-1-2071.

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