# Repulsive Casimir effect

## Main Question or Discussion Point

It has been proposed and demonstrated that two materials with dissimilar permittivities immersed in a fluid with a permittivity value in-between the two solid materials will result in a repulsive Casimir force generated between the surfaces of the solid materials.

Why?

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It has been proposed and demonstrated that two materials with dissimilar permittivities immersed in a fluid with a permittivity value in-between the two solid materials will result in a repulsive Casimir force generated between the surfaces of the solid materials. Why?
Can you provide a link to some article? Sounds more like a van der Waals interaction type of thing.

Can you provide a link to some article? Sounds more like a van der Waals interaction type of thing.
There is an article published in Nature, which you have to pay for or have access to a subscription for:
http://www.nature.com/nature/journal/v457/n7226/abs/nature07610.html

The research was done by the Capasso group headed by Fredrico Capasso. There are multiple publications on the capasso group website about this research which can be downloaded from the 'publicatons' section free of charge:

http://www.seas.harvard.edu/capasso/index.htm[/URL]

You have to troll through by year, but if you target the years 2009, 2008, and 2007 and use your browser's 'find' feature to search for 'Munday' (the main researcher) you will find the papers about this. I'm giving links to the PDFs below but the above information is in case you don't want to download something directly from my post.

this is a direct link to the nature article available free of charge from Capasso group website -> publications -> 2009 -> 4th from bottom:
[PLAIN]http://www.seas.harvard.edu/capasso/publications/Munday_Nature_457_170_2009.pdf[/URL]

this capasso group publication gives more detail and some better formulas about the research:
[PLAIN]http://www.seas.harvard.edu/capasso/publications/Munday_PRA_78_032109_2008.pdf[/URL]

I see the liftgarbagez formula and their calculations, etc. but I still don't understand how it works. If someone can explain it in plain english I'd love it.

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Thanks for the links. Can't think of any simple explanation (eg. in terms of change of wavelength of ZPF fluctuations from one media to another). But I'm a layman in this subject. This article has a somewhat different take on it and may help:
Physics of Repulsive Van der Waals forces.
L. P. Pitaevskii
http://cnls.lanl.gov/casimir/PresentationsSF/Repulsive_force09.pdf

alxm
You realize that the Casimir effect is the van der Waals (London) force, at the short-range limit where the finite speed of light comes into play?

You realize that the Casimir effect is the van der Waals (London) force, at the short-range limit where the finite speed of light comes into play?
My very limited understanding is that 'standard' Casimir force is interaction between idealized perfectly conducting surfaces and vacuum fields. And that this is modified for real materials by detailed treatment involving consideration of van der Waals interactions between media, but not primarily due to such.
From http://en.wikipedia.org/wiki/Casimir_effect
"..This force has been measured, and is a striking example of an effect purely due to second quantization.[3][4] However, the treatment of boundary conditions in these calculations has led to some controversy. In fact "Casimir's original goal was to compute the van der Waals force between polarizable molecules" of the metallic plates. Thus it can be interpreted without any reference to the zero-point energy (vacuum energy) or virtual particles of quantum fields.[5].." Further down, under 'More recent theory':

"A very complete analysis of the Casimir effect at short distances is based upon a detailed analysis of the van der Waals force by Lifgarbagez.[11][12] Using this approach, complications of the bounding surfaces, such as the modifications to the Casimir force due to finite conductivity, can be calculated numerically using the tabulated complex dielectric functions of the bounding materials. In addition to these factors, complications arise due to surface roughness of the boundary and to geometry effects such as degree of parallelism of bounding plates.
For boundaries at large separations, retardation effects give rise to a long-range interaction. For the case of two parallel plates composed of ideal metals in vacuum, the results reduce to Casimir’s.[13]"

But will stand corrected on this.

alxm
The London dispersion force is the attraction between two polarizable object (such as neutral atoms) at a distance, treated quantum mechanically but where the electrical field is treated using a classical, Coulomb gauge, where the interaction is instantaneous. In other words, as if light moved at infinite speed. Another way of looking at it that doesn't invoke the relatively-macroscopic concept of polarizability, is to view it as the long-range correlation of electronic motion.

So the London force (I avoid the term vdW, because it doesn't strictly refer to London forces alone) which scales as 1/r6, is only a valid approximation at large distances where the effects of the retardation of the field are small, and the distance is significantly larger than the interaction wavelength so you don't have resonant coupling. This was first pointed out by Wheeler in 1941 (meeting abstract in Phys Rev, v59, p928), before Casimir's work, but Wheeler never published any work on it. Anyway, so at short range you have two additional effects that come into play; the finite speed of light (retarded Coulomb potential) and resonant coupling. These aren't unrelated, since you need a full quantum-mechanical treatment of the field (QED) to calculate them. Which, contrary to most popular-scientific depictions, isn't to say you necessarily need virtual particles, 'vacuum energy' and perturbation theory to do so. (See, e.g. http://pra.aps.org/abstract/PRA/v48/i6/p4761_1" [Broken] treatment)

The fact that they usually use idealized metal plates and such isn't really related to the force itself, it's because it makes the calculations easier, since the physical shape is easier if they're flat, and the polarizability is more easily described for a metal. The Casimir(-Polder) effect is basically, by definition, the deviation from the London force when you have a proper QED description of the field. It's really got more to do with our own limitations than with the physics; if we'd always had quantized fields and it hadn't been much more difficult mathematically to describe that situation, they'd probably be considered to be the same force. I mean, it's not like there are two different kinds of electromagnetic fields in reality.

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Thanks for that more detailed explanation. Puzzled by one feature, which I have seen mentioned before but could never figure:

"So the London force ... which scales as 1/r6, is only a valid approximation at large distances where the effects of the retardation of the field are small, and the distance is significantly larger than the interaction wavelength so you don't have resonant coupling. Anyway, so at short range you have two additional effects that come into play; the finite speed of light (retarded Coulomb potential) and resonant coupling."

My initial expectation would be the opposite - retardation effect would grow with r (or more generally be a cyclical function of r). Would it be fair to say the opposite is true owing to the fact that a point on one surface is at larger r sampling a larger stochastic field emanating from the other surface and so nearly complete phase cancellation from 'point' oscillators occurs? Which implies that retardation effect is relatively much more important between say two molecules than between two surfaces?

My very limited understanding is that 'standard' Casimir force is interaction between idealized perfectly conducting surfaces and vacuum fields. And that this is modified for real materials by detailed treatment involving consideration of van der Waals interactions between media, but not primarily due to such.
From http://en.wikipedia.org/wiki/Casimir_effect
"..This force has been measured, and is a striking example of an effect purely due to second quantization.[3][4] However, the treatment of boundary conditions in these calculations has led to some controversy. In fact "Casimir's original goal was to compute the van der Waals force between polarizable molecules" of the metallic plates. Thus it can be interpreted without any reference to the zero-point energy (vacuum energy) or virtual particles of quantum fields.[5].." Further down, under 'More recent theory':

"A very complete analysis of the Casimir effect at short distances is based upon a detailed analysis of the van der Waals force by Lifgarbagez.[11][12] Using this approach, complications of the bounding surfaces, such as the modifications to the Casimir force due to finite conductivity, can be calculated numerically using the tabulated complex dielectric functions of the bounding materials. In addition to these factors, complications arise due to surface roughness of the boundary and to geometry effects such as degree of parallelism of bounding plates.
For boundaries at large separations, retardation effects give rise to a long-range interaction. For the case of two parallel plates composed of ideal metals in vacuum, the results reduce to Casimir’s.[13]"

But will stand corrected on this.
well, my understanding in this is limited too, which is why it is good to ask questions.

Honestly I don't know the 'boundaries' of the definition of Casimir. I am aware of its relation to the van der Waals forces. The experiments being conducted using cantilevers and gold-coated spheres etc. to measure these forces still involve the term 'Casimir' although these don't involve calculations of idealized infinately conductive parallel plates. The Casimir force is also being computed for various types of surface geometries; there are many papers involving the mathematics of calculating this force for arbitrary surfaces and there is even a computer program now for doing this:

I don't think it necessarily matters where 'Casimir' ends and 'van der Waals' begins... what I'd like to understand is how the differences in permittivities 'reverse' the sign of the Casimir/van der Walls force. does this have something to do with the index of refraction?

Dr. Leonhardt and Dr. Philbin of St. Andrews University released a paper on using metamaterials to reverse the casimir effect. this is done by creating a negative index of refraction. Their paper can be found through google and it has a diagram that helps to visualize this successfully. The problem is that the metamaterial proposed is hypothetical and does not yet exist. However, the Capasso group seems to have reversed the casimir effect without metamaterials using only a fluid in-between the two different surfaces, each with different permittivities, the fluid with a permittivity in-between each of the solid materials whose surfaces repel.

it does have to be recognized that the vacuum expectation value is not neglidgable but depends on some minimum quantity due to the heizenburg principle. Vacuum energy must exist, and I don't think that it is necessarily a stretch to think of the Casimir/van der Waals forces in terms of virtual particles, since it does simplify things.

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I don't think it necessarily matters where 'Casimir' ends and 'van der Waals' begins... what I'd like to understand is how the differences in permittivities 'reverse' the sign of the Casimir/van der Walls force. does this have something to do with the index of refraction? ...However, the Capasso group seems to have reversed the casimir effect without metamaterials using only a fluid in-between the two different surfaces, each with different permittivities, the fluid with a permittivity in-between each of the solid materials whose surfaces repel...
OrigamiNinja, the article linked in #4 is I think suggesting something analogous to an interesting classical 'dilemma' posed in section 6-7 of 'Classical Electricity and Magnetism' - 2nd Ed'n (Panofsky & Phillips). Two closely spaced charged parallel capacitor plates separated by a solid dielectric are attracted to each other by a force that does not depend on the intervening dielectric. However if the solid dielectric is replaced by a liquid of the same relative permittivity er, the force between the plates drops by the factor 1/er. The explanation involves the positive hydrostatic pressure induced in the liquid by the electric fringing fields at the plate edges. While somewhat more involved, this seems to be basically what Pitaevskii is claiming - field nonuniformities are the key. I can't really say - there are so many subtle contributions.:zzz:

alxm
My initial expectation would be the opposite - retardation effect would grow with r (or more generally be a cyclical function of r).
Well, if you think of it in terms of electronic correlation, it's somewhat clearer. Although correlation itself is also one of those things that's more a mathematical convenience than a genuine physical effect. But I'm a quantum chemist and we tend to think about stuff in that way. So anyway, correlation is itself defined as the deviation in kinetic energy due to the instantaneous interactions of the electrons. So the less instantaneous that interaction is, the smaller the effect on correlation.

Or in other words, if the interactions were infinitely fast, and electrons moved infinitely fast, the correlation energy would be zero.

alxm
I don't think it necessarily matters where 'Casimir' ends and 'van der Waals' begins... what I'd like to understand is how the differences in permittivities 'reverse' the sign of the Casimir/van der Walls force. does this have something to do with the index of refraction?
Yes, since the index of refraction is related to the polarizability and permittivity. There's a deep relationship between these things. It's not without reason the 'dispersion' force is named as such.

Anyway, this might be of interest: http://prola.aps.org/pdf/PR/v73/i4/p360_1" [Broken] a more recent paper outlining the theoretical details (in terms of polarizability/permittivity) for attractive Casimir forces.

The basic reason why you can have repulsive Casimir and vdW forces is because neither of them are actually additive, even if we can approximate them as such to a decent extent.
Which shouldn't really come as a surprise, since they have their origins in what's fundamentally a many-body effect (correlation), and thus non-linear.

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Ok, so say I want to calculate the repulsive Casimir force. The best equations I've found for doing this are provided in one of the papers released by the Capasso group. This is the Liftgarbagez equation for two solids of different permittivities immersed in a fluid of a third permittivity. The equation requires the dielectric function of each of these three materials.

The dielectric function can be calculated using optical data for the materials (the plasma frequency and the relaxation frequency) and is a function of the frequency of the material.

$$F_{\mathrm{CL}}|_{m=0}=\frac{k_{B}TR}{2}\int_{k=0}^{\infty} \! k \ln\left[1-\left(\frac{\varepsilon_{1}k-\varepsilon_{3}\sqrt{k^2+\kappa^2}}{\varepsilon_{1}k+\varepsilon_{3}\sqrt{k^2+\kappa^2}}\right)\times\left(\frac{\varepsilon_{2}k-\varepsilon_{3}\sqrt{k^2+\kappa^2}}{\varepsilon_{2}k+\varepsilon_{3}\sqrt{k^2+\kappa^2}}\left)e^{-2d\sqrt{k^2+\kappa^2}}\right]\,\mathrm{d}k$$

where

$$\varepsilon_{1} , \varepsilon_{2}$$

are the dielectric functions of the solid materials with the repulsively interacting surfaces, and

$$\varepsilon_{3}$$

is the dielectric function for the fluid (ethanol)

and

$$\kappa$$

is 1/(Debye screening length).

here is the equation for the drude model in latex:

$$\varepsilon(\omega) = 1-\frac{\omega_{p}^2}{\omega(\omega+i\omega_{r})}$$

where

$$\varepsilon(\omega)$$

is the dielectric function of frequency of the material for one of the two repulsively interacting solid surfaces, given the parameters

$$\omega_{p}$$

which is the plasma frequency, and

$$\omega_{r}$$

which is the relaxation frequency.

I have questions though about how to use these equations because I'm not familiar with the physics. When calculating the dielectric function of one of the materials, should I solve it for one frequency or for a range of frequencies? in the paper, Munday says that the dielectric function for gold is solved from 0.125eV to 9184eV.

I'm confused about how to solve it across this range.

What is the role of eigen vectors in d formation of matrix mechanics?

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What is the role of eigen vectors in d formation of matrix mechanics?
I'm sorry. I understand the meaning of each of those words but when you put them together like that you confuse me.

what is the actual question?

:)

A lot of work - you are making this a career?
well, I'm thinking about starting a research project involving the reverse casimir force. It seems to have a lot of potential utility. there's a lot of speculation about using it to counteract friction in MEMS devices, etc. I believe it could be used to do this at a much larger scale.

Once I get this equation into a computer program it'll be ok, I'll never have to solve it again. Just getting there is the problem.

I got it figured out. There were several other equations I needed.

Thanks for looking at it.

I asked Dr. Philbin at the University of St. Andrews about the reverse Casimir effect. He gave an excellent description:

"
One of the problems with trying to have a conceptual understanding of
Casimir forces is that the calculations using the standard Lifgarbagez
theory (which predicts the repulsion you speak of) do not tell you
"what's going on", they just give you the answer. This is not unique
to the Casimir effect, it is true of all quantum physics. But in the
case of repulsion that you mention, an intuitive explanation has been
offered by Capasso, and it is this. First you have to think of one of
the objects in the fluid being a sphere, or a curved object of some
kind. Let's say the plate has the highest permittivity, so the sphere
has the lowest permittivity, and the permittivity of the fluid is in
between. Now the plate attracts both the fluid and the sphere, because
the Casimir force between two dielectrics is always attractive, but
the fluid has a higher permittivity than the sphere, so the plate
attracts the fluid more strongly than the sphere. The fluid particles
surrounding the sphere are therefore trying to get closer to the
plate, more strongly than the sphere is trying to get closer to the
plate. The fluid particles therefore push the sphere out of the way to
get closer to the plate and the sphere is pushed away from the plate -
a repulsive force between the plate and the sphere.
This repulsion has now been measured by the Capasso group "

Thanks Dr. Philbin.

Thanks for the plain English explanation. About what you'd expect from simple electrostatics - in fact I wonder if some MM 'stiction' problems could be fixed simply by applying a like potential to mating components? As from your previous post you have all the info needed, this may or may not be of use:
R.L.Jaffe; seems to be the preeminent authority in this area - has extensive co-authored articles at arXiv.org: http://arxiv.org/find/hep-th/1/au:+Jaffe_R/0/1/0/all/0/1
This one in particular seems to cover all bases: 'Geometry and material effects in Casimir physics - Scattering theory' http://arxiv.org/abs/1007.4355

alxm
Thanks for the plain English explanation. About what you'd expect from simple electrostatics - in fact I wonder if some MM 'stiction' problems could be fixed simply by applying a like potential to mating components?
Well, I'm not an MM guy so I don't know offhand which (if any) of their models take into account repulsive dispersion forces. But this three-body dispersion term is well-known, and called the Axilrod-Teller-Muto term. (http://jcp.aip.org/resource/1/jcpsa6/v11/i6/p299_s1" [Broken] for correcting dispersion in DFT methods. The three-body dispersion term is < 10% of the dispersion energy, and the dispersion energy is of course very small on the scale of intermolecular interactions.

As for the purely Casimir version of that (or contribution to that), it's not really significant at the molecular level. Neither retarded potential (Breit-Pauli) corrections or purely QED effects (Lamb shift, Casimir effects) are normally taken into consideration here, they're so small. IMO, the Casimir effect is more famous due to being one of few directly-measurable QED effects than it is for its significance as an interaction. The Born-Oppenheimer approximation underlying all MM and most QM methods is energetically several orders of magnitude larger than any QED effects.

The Casimir effect only becomes large enough to measure once you're dealing with 'nice', fairly-smooth and fairly-homogenous objects, that are a lot bigger than the molecular or nano scale.

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...As for the purely Casimir version of that (or contribution to that), it's not really significant at the molecular level. Neither retarded potential (Breit-Pauli) corrections or purely QED effects (Lamb shift, Casimir effects) are normally taken into consideration here, they're so small. IMO, the Casimir effect is more famous due to being one of few directly-measurable QED effects, than it is for its significance as an interaction. The Born-Oppenheimer approximation underlying all MM and most QM methods is energetically several orders of magnitude larger than any QED effects...
A sober assessment, but perhaps there is some intermediate size range - maybe on the micro scale, where MEMS devices can materially benefit from repulsive Casimir type forces. If surface roughness between say meshing gear teeth means small actual contact area and hence reduced relative contribution from the much stronger but shorter range 'normal' inter-atomic forces mentioned. But that implies restriction to small applied forces - otherwise contact forces will be overwhelming. Effective repulsive action for really smooth devices at nano scale is therefore probably expecting way too much. More serious a restriction is surely the need to have an intermediate dielectric fluid present. Doesn't seem to dampen efforts though.

Born2bwire