The Casimir Force – Neutral or Electrostatic?

T. V. Prevenslik
14B, Brilliance Court, Discovery Bay, Hong Kong

Abstract. Since Casimir, the attractive force between uncharged metal surfaces separated by an evacuated gap has been attributed to the zero point energy (ZPE) of quantum mechanics. But the Casimir theory has difficulty in explaining how contact of otherwise neutral surfaces in micro electro-mechanical systems (MEMS) devices can cause permanent adhesion without fusion of surfaces by electrostatic discharge. A modified Casimir theory is proposed to explain how charge and ZPE is produced from the gap surfaces. By this theory, the ZPE is electromagnetic (EM) radiation in the gap at vacuum ultraviolet (VUV) frequencies, the source of ZPE at ambient temperature being the thermal kT energy in the surface atoms emitted at infrared (IR) frequencies. Gap surfaces always fluctuate about the nominal gap dimension by atomic dimensions, and if surface atoms having thermal kT energy move into the gap having a VUV resonance, the IR radiation from the atoms is momentarily suppressed because of cavity quantum electrodynamics (QED). To conserve EM energy, the suppressed thermal kT energy of the atoms is spontaneously emitted as IR radiation into the gap which is resonant at VUV frequencies.  Surface atoms undergo field ionization and depending on the relative electron yields, the surface losing more electrons acquires a positive charge, the other gaining the electrons charging negative. The Casimir force is therefore one of electrostatic attraction, the electrical charge produced by the steady conversion of heat from the thermal surroundings mediated by the electrical grounding of the surfaces. Electrical breakdown may induce static discharge causing the permanent adhesion observed in the MEMS device. Because of the dependence of the ZPE on thermal kT energy, neither the Casimir force nor the ZPE in the modified theory exist at absolute zero.

Keywords: Casimir, zero point energy, cavity QED, MEMS

1   INTRODUCTION

For over 50 years, the Casimir effect [1] has described the interaction between electrically neutral conductive bodies separated from each other by an evacuated space, although Casimir first studied the interaction between a pair of perfectly reflecting plane and parallel mirrors. Fundamental in the Casimir effect is the notion of ZPE more commonly called the energy of the vacuum thought to pervade all of space. However, even the ZPE needs a source, and here the ZPE is proposed to have a thermal origin.

The most recent experiment [2] to duplicate Casimir’s pair of plane mirrors comprised a chromium-coated plate and a flat surface of a cantilever beam of the same material separated by a gap form 0.5-3 mm. Between parallel plates, the Casimir force F_c is attractive,

                                                                 (1)

where, c is the speed of light, h is Planck’s constant, A is the area of the plates, and z is the separation between the plates. The comparison of experiment with the theoretical Casimir force was within 15%. However, flat plates are normally not used in Casimir experiments because of the difficulty in maintaining alignment. Instead, the interacting surfaces are usually [3-6] taken as a sphere and a flat plate having an attractive Casimir force F_c given by,

                                                                    (2)

where, R is the radius of the sphere, and  is the z distance between the sphere and the flat plate.

One of the early Casimir experiments [3] using the sphere and flat plate geometry measured the Casimir force in the 0.6 –6 mm range. The sphere was a 4 cm diameter spherical lens and the flat plate was a 2.5 cm diameter optical flat, the optical surfaces Cu coated with a top Au coating. A noticeable change in the Casimir force was not found until the gap between the sphere and flat plate reached the 0.6 mm lower limit. More recently, the Casimir force was determined [4] with an atomic force microscope using an Au coated sphere about 200 mm in diameter and a flat plate. The Casimir force F_c was measured from 0.1 to 0.9 mm and corrected for plasmon frequency, roughness of the surface, and finite temperature. The Casimir force was found to only noticeably change at separations below about 0.2 mm.  In [5] a Cr coated polysilicon plate topped with Au coating was suspended on a torsion rod. A similarly coated 200 mm diameter sphere was placed off the torsion axis, the Casimir force between the sphere and plate causing a torque to rotate the plate. The Casimir force was measured over a range 0.1 – 1 mm with the force noticeably beginning to increase at about 0.2 mm.  Generally, the gap d < 0.2 mm for the Casimir force to be significant.

In a micro-mechanical systems (MEMS) device, a doubly clamped Au coated beam [6] having a rectangular protrusion at mid-span was positioned to form a gap between the protrusion and flat electrode. Upon the evacuation of the gap, the Casimir force loaded the beam, and after a period of time the beam collapsed against the electrode, the contact causing the permanent adhesion of the beam to the electrode. Since contact of neutral and otherwise chemically non-reactive surfaces under the attractive Casimir force is unlikely to cause permanent adhesion unless fusion bonding has occurred, the conclusion here is that electric charge is somehow produced in the Casimir effect, the permanent adhesion caused by static discharge. But in the standard formulation of the Casimir theory, the surfaces are assumed neutral and uncharged surfaces.

Moreover, the standard Casimir theory assumes perfectly reflecting surfaces. However, the measurements of Casimir force are made with real materials that are not perfectly reflective, and further have reflectivity that depends on the frequency of the EM radiation in the gap, the latter suggesting the Casimir force may vanish at VUV frequencies as the surfaces contact. Indeed, it has been shown [7] that the Casimir theory based on perfect reflectivity is only valid for gaps > 0.5 mm. For an EM wave standing in a gap d, the resonant wavelength l = 2 d , and therefore the validity of the Casimir theory corresponds to wavelengths l > 1 mm in the IR where most metals are highly reflective. But experiments [4,5] show the measured Casimir forces are negligible at IR wavelengths where the Casimir theory is valid. Indeed, Casimir forces of any significance are only found at gaps less than about 0.2 mm corresponding to VUV and not IR wavelengths.

What this means is that the standard Casimir theory may need to be modified. The purpose of this paper is to propose a modified Casimir theory that includes electrical charge, at least for gaps less than about 0.2 mm where the Casimir forces are known to be significant. Calculations and comparisons with experiment are presented to illustrate the modified Casimir theory. 

2  THEORETICAL BACKGROUND

The source of ZPE in the modified Casimir theory is proposed to be the thermal kT energy available from the atoms in the surfaces of the gaps between the interacting bodies. In Fig. 1, a pair of electrically conductive and neutral bodies is shown to attract each other through a gap d as the ZPE at VUV frequencies produces electrons from gap surfaces by multi-IR photon field ionization. The ZPE is produced from the cavity QED induced spontaneous emission of IR radiation from surface atoms that undergo fluctuations in the gap of order O(d) of atomic dimensions . The modified Casimir force F_c is electrostatic with the charge build-up mediated by electrical grounding. 

Since Casimir, experiments performed at ambient temperature have generally confirmed the Casimir force at gaps less than about 0.2 mm, and since the tests were performed absent pressure in a vacuum, it can only be concluded that the Casimir force has a thermal origin. Conversely, a vacuum origin to the Casimir force would be indicated, if the Casimir force were confirmed [7] in tests near absolute zero. But tests near absolute zero have never been performed, and therefore the claim that the Casimir effect produces a force from the ‘nothing’ of the vacuum is speculative, and at the very least violates the conservation of energy. In contrast, the modified Casmir theory asserts the ZPE finds its origin in the thermal kT energy of the atoms on the surfaces of the gaps.

2.1  Thermal Origin of the Casimir Force

The Casimir force in the gap is produced by the ZPE that finds origin [8] in the Planck energy E representing the thermal kT energy of the atoms on gap surfaces by the harmonic oscillator. The Planck energy E of the harmonic oscillator [9] is given,

                                                            (3)

Fig. 2 Thermal Planck Energy E - Harmonic Oscillator at Ambient Temperature

2.2     Source of ZPE Energy

The ZPE is proposed to find its origin in the cavity QED induced spontaneous emission of momentarily suppressed IR radiation from groups of atoms that move into [10] the high frequency VUV cavity. Similarly, static electricity may be explained [11] by the cavity QED induced spontaneous emission of suppressed IR radiation from micron size particles trapped in the gap between contacting surfaces.

In the Casimir effect, the surfaces separated by gap d are comprised of atoms having thermal kT energy and emit IR radiation at wavelength l _IR. If d / l _IR < 0.1, spontaneous emission of atoms between conducting parallel plates separated by a gap d is fully suppressed (Fig. 1 of [12]). Since Fig. 2 shows most of the thermal kT energy at ambient temperature resides at l _IR > 10 mm, and since the Casimir force is only significant [4] in the VUV at d < 0.10 mm, the IR radiation is fully suppressed by cavity QED as d  / l _IR << 0.01.  The IR radiation from surface area A comprised of atoms having thermal kT energy is suppressed because the gap always undergoes fluctuations O(d) on the order of atomic dimensions, the fluctuations intermittently moving a number N_A  of atoms into the VUV resonant gap, i.e., N_A = A O(d) / D³, where D is the spacing between atoms in the surface at solid density. Since the atomic emission occurs at half IR wavelengths ½ l _IR > d , the  EM energy U_EM suppressed by cavity QED,

                                                    (4)

Typically, D ~ 0.3 nm. To conserve EM energy, the sub-surface atoms spontaneously emit at least half of the VUV radiation into the gap. The number N_P of VUV photons spontaneously emitted having Planck energy E_VUV,

                                                   (5)

To illustrate the modified Casimir theory, assume a fluctuation of N_A = 10¹⁰ atoms.  For a single layer of atoms, O(d)  = D, the N_A atoms cover a circular surface area A = 3.1x10^-10 m² having a diameter of about 20 mm. Prior to the fluctuation, the atoms have thermal kT energy of 0.0258 eV while the boundary conditions on the gap d = 0.1 mm require a ZPE having an EM resonant wavelength l_c = 0.2 and E_VUV  = 6.2 eV. Steady heat transfer from the surroundings provides a continuous supply of thermal kT energy to compensate for the cavity QED induced spontaneous emission of  suppressed EM energy. From equations 4 and 5, U_EM = 62 pJ and the number N_P  = 3.1x10⁷ of VUV photons.

2.3     Field Ionization

The electric field induced in the gap surface by the spontaneous emission of far IR photons induces the ionization of surface atoms to produce electrons and charged atomic states. Dependent solely on the power P produced in the particle while avoiding arguments of coherency of multi-IR photons, field ionization is significant because the thermal kT energy is induced by cavity QED to undergo spontaneous emission over very short times.

Laser induced field relations [13] may be used to quantify field ionization induced by cavity QED. Consider the gap surface atoms as a laser spontaneously emitting a short pulse of coherent thermal kT energy at far IR frequencies.  The power P, 

                                                                                                                                              (6)

where, t is the time of spontaneous emission. At the gap surface area A, the laser intensity I is,

                                                                            (7)

where, m₀ is the permeability of free space. The electric field E_f is,

                                                  (8)

Assuming a spontaneous emission time t < 100 ps, equations (6) and (7) give the laser power P > 620 mW and intensity I  > 2 GW×m^-2. In equation 8, taking O(d)  = D gives the surface field E_f  > 8.7 x10⁵ V×m^-1 suggesting electrons are liberated from the Au surface atoms by field ionization.

2.4  Photoelectric Yield

Photoelectric yields g_P are generally thought to be a function of the surface material, but may also depend on geometry. Indeed, microscopic particles are known [14] to have significantly higher yields than for the bulk surface. In Casimir force [4,5] experiments, microspheres and plates are common, and although both are usually Au coated, may not have the same yield.

Generally, the Casimir force only begins to be come significant below gaps d  < 0.1 mm, or at EM resonant wavelengths l_c < 0.2 mm. This is consistent with the work function W of most metals that gives the Planck energy E_VUV below which electrons are not produced, i.e., electrons are not produced if E_VUV < W. For example, the work functions W of Au and Al are about 4 eV that is equivalent to an EM resonant wavelength l_c ~  0.3 mm, or a gap d ~ 0.15 mm. For Al and Au coated gap surfaces, the photoelectron yield g_P data ( Fig. 2 of [15] ) for E_VUV from 4-12 eV is approximated by,

                                                 (9)

where, E_VUV  is in eV.  For E_VUV = 6.2 eV, the photoelectric yield g_P = 4.5x10-6.

2.5     Number of Electrons and Electrical Charge

The number N_e of electrons produced depends on the number N_P of VUV photons and the photoelectric yield g_P of the gap surfaces in electrons per photon,

                                                                     (10)

In the Casimir force experiments, both the sphere and flat plate are usually Au coated, and therefore there may not be any difference in the photoelectric yield g_P. If so, the net charge Q produced is zero and there is no Casimir force in the modified theory. But the coatings are not likely identical, and geometrical effects may be significant. Hence, a parameter h < 1 is used to characterize the mismatch of otherwise identical materials, the charge Q produced,

                                                                         (11)

Here the charge Q is upper bound by taking h = 1, or a perfect mismatch of photoelectric yields. For the photoelectric yield g_P = 4.5x10^-6 and number N_P = 3.1x10⁷ of VUV photons, the number of electrons N_e  = 140 and charge Q = 2.2x10^-17 C.

2.6  Casimir Force

The VUV radiation in the gap in combination with the photoelectric yields of gap surfaces means electrical charge Q is produced, and therefore the modified Casimir force F_c is attractive finding its origin in electrostatics,

                                                            (12)

where, e_o is the permittivity of free space. For the N_e  = 140 electrons and gap d = 0.1 mm, the modified Casimir force F_c = 4.5x10^-10 N. For comparison, a 0.1 mm gap gives an experimental [4] Casimir force F_exp = 1.4x10^-10 N. 

The Casimir force F_c for the standard and modified theory normalized to the force F_c,0.1 at gap d = 0.1 mm are compared by maintaining the same heat flow,

                                                          (13)

Combining,                                   

                                                  (14)

Similarly, the number N_e of electrons is,

                                                      (15)

In contrast, the Casimir force in the Standard Theory,

                                                                    (16)

where, the gap d is in microns. The range on d is from 0.1 to 0.05 mm corresponding to wavelength l_c range form 0.2 to 0.1 mm, or Planck energy E_VUV from 6.2 to about 12 eV.

3   CONCLUSIONS

A modified Casimir theory is presented to explain the interaction between bodies separated by gaps less than about 0.1 mm, the conclusions of which are summarized as follows:

The modified Casimir force between bodies separated by an evacuated gap is attractive and finds its origin in electrostatics because of electrical charge produced by the bodies. The electrical charge is produced by the photoelectric effect from the field ionization of surface atoms induced by the spontaneous emission of thermal kT energy by cavity QED.

In the MEMS device, the permanent adhesion observed upon contact cannot be caused by neutral surfaces. Electrostatic discharge is a more likely explanation, but the standard Casimir theory is based on an attractive force between neutral bodies.

The modified Casimir force based on electrical charge build-up may occur even as a current flowing in the ground between the bodies dissipates the charge. But the rapid increase in charge for gaps less than 0.1 mm suggests that measurements of current rather than force may be a meaningful measure of the Casimir force.

In the modified Casimir theory, the ZPE finds its origin in the thermal kT energy of the atoms, and therefore the ZPE and the Casimir force cannot exist at absolute zero.          

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