Simulation and measurement of the time-varying Casimir force (online version)

John Kohoutek, Ivy Yoke Leng Wan, Hooman Mohseni

Bio-Inspired Sensors and Optoelectronics Laboratory (BISOL), EECS, Northwestern University

Introduction

The Casimir Force

The Casimir force is the force manifested between two bodies which are very close together and is caused by the quantum mechanical fluctuations of the vacuum in the space between those two bodies.

http://en.wikipedia.org/wiki/Casimir_force

Motivation

What is motivating study of the Casimir force in this day and age?

Shifeng Li and Chang Liu, Electrochemical Society Transactions, 3(10), pp. 463-472, 2006. 

http://en.wikipedia.org/wiki/Kelvin_probe_force_microscope

Answer - MEMS (micro-electro-mechanical systems), now becoming NEMS (nano), and scanning probe microscopy.

History

Flat metal plates

Hendrik Casimir first predicted the Casimir effect for 2 perfectly conducting metal plates in a vaccum in 1948:

F(d)=((ℏ c (π^2))/(240 (d^4)))

Sphere/plate combinations

Since then people have realized that it is much easier to materialize a sphere/plate combination than 2 perfectly parallel plates.

U. Mohideen and Anushree Roy, Physical Review Letters 81(21), pp. 4549-4552, 1998.

Enter the atomic force microscope

Around 1998 a group led by U. Mohideen at UC-Riverside began using an atomic force microscope with the above pictured setup to measure the Casimir force in a static fashion.

Our method

Scanning electron micrograph of AFM tip

A dynamic method

Our method aims to measure the Casimir force in a dynamic fashion - with the Atomic Force Microscope in tapping mode.  This lends itself to to be modeled by the harmonic oscillator.  We start with the equation for the forced, damped harmonic oscillator with the force terms acting on the tip on the right hand side :

m (((d^2)x)/(d (t^2)))+r ((d x)/(d t))+k x=(Subscript[F, driving])+(Subscript[F, electrostatic])+(Subscript[F, Casimir])

In the above equation, x(t) is the position of the tip as a function of time, m is the mass parameter, r is the damping parameter, and k is the spring constant.  k is actually given by the manufacturer of the AFM tip to be 3 N/m.  But how do we get the m and r parameters?

Answer:  We fit the amplitude response of the tip with an ideal amplitude response given by that of a forced, damped harmonic oscillator.

Here they are on top of each other:

Our experiment

Setup

To measure the Casimir force, we have to excite the tip with some externally known force - in this case the electrostatic force.  For our calculations we assume the tip has the form of a sphere with radius R above a flat surface.  The form of the electrostatic force is then given below:

(Subscript[F, electrostatic])=-2π (Subscript[ϵ, 0]) (V^2)(∑_(n=1)^∞)Csch[n α[a]](Coth[ α[a]]-n Coth[n  α[a]])
α[a]=ArcCosh[1+(a/R)]

Vdc and Vac are the dc and ac voltages we will apply to the tip in V, R is the radius of curvature of the tip in nm (which we have measured with a scanning electron microscope), and a is the distance from the tip to the sample in nm, which is changing because the tip is vibrating.

We can't forget about the Casimir force.  The Casimir force is given for a sphere above a flat plane by the formula below:

(Subscript[F, Casimir])=(((Subscript[H, ts])R)/(6(a^2)))

Here (Subscript[H, ts]) is the interaction constant between the 2 materials (called the Hamaker constant), R is the radius of the sphere, and a is the distance between the sphere and the surface.  We have used many important references to calculate  (Subscript[H, ts]) for our setup (diamond tip and steel plate).  

We also have a roughness correction to the Casimir force.  This is just a 4th order correction that accounts for the surface roughness of plate.  A is the rms surface roughness of the plate in nm and a is the distance from the tip to the sample.  A was measured by doing an AFM scan on the surface of the plate.

roughness correction[a] = 1+ 6((((A/a)))^2)+10((((A/a)))^3)+15((((A/a)))^4)

We can't forget about the force that is actually driving the vibration of the tip, the driving force.  It is a simple Cosine wave.

(Subscript[F, driving])=(Subscript[F, 0])Cos[(Subscript[ω, 0])t]

The amplitude (Subscript[F, 0]) is determined by solving the below equation at a far distance where the above two forces that depend on the distance are minimal and matching the amplitude to the known amplitude at that distance.  The frequency is the driving frequency that we choose (250 Hz above the resonant frequency for reasons that we can discuss later).

Solving

Finally, we group our equations together and NDSolve!  We actually will NDSolve twice.  Once WITH the electrostatic force caused by the AC voltage, and once WITHOUT.  This is because we want to calculate the DIFFERENCES caused in the solution to the equation by this electrostatic force.  For that, we will need both solutions.  First, WITHOUT:

Now, with:

Now, we can do some interesting things with our solutions.  We can plot them:

But theres not much to see.  If there's a small phase difference we should be able to detect it by detecting the change in the peaks with FindRoot:

Bingo.  This phase shift looks very similar to the phase shift being output by our AFM controller:

There are a couple of other things we can do with these solutions.  First, we can check their amplitude (again with FindRoot):

This is a single number for a given set of input parameters.  But we can check the amplitude at varying distances to do a theoretical amplitude versus distance curve:

But we can also measure this with the AFM:

Finally, if we believe our model to be true, we can use it to plot the forces acting on the tip:

Extras

Sensitivity to experimental parameters

We agree that this model is very sensitive to experimental parameters.  Thats why we have performed an extensive error analysis on the two (we feel) most important parameters - Hamaker constant and surface roughness.
To do this, we have varied these parameters and calculated the root mean square deviation between our actual phase shift and our theoretical phase shift.  But the actual phase shift and theoretical phase shift are both vectors of different length - so we have interpolated them and sampled along this interpolation.

Future Work

Right now our model fails when the surface attracts the tip too closely - but really we can model this interaction with a repulsive force.  For instance, the interaction energy of an Argon dimer is shown below with the formula:

V[r]=(A/(r^12))-(B/(r^6))

Our force-distance relation is shown below:

There should be a repulsive force representing the atoms of the surface but there isn't!  This repulsive term is easy to represent for a single atom but how do you represent it for our geometry?