Contact electrification by cavity QED

1. Introduction

Over 200 years ago, Volta discovered metals could be ordered by contact potential, but did not explain why this was so. Today, electrons occupying energy states up to the Fermi level explain the contact potential. Contact transfers electrons from higher to lower Fermi levels and ceases as the levels equilibrate. A single contact is all that is necessary for contact charging of metals.

In contrast, charging of insulators by metals requires many contacts for saturation, and remains unexplained [1] even though studied extensively since Lord Kelvin [2]. In an insulator, the forbidden gap between the valence and conduction bands is usually in excess of 5 eV so that the electrons only occupy the valence band. Contact charging of insulators by the transfer of electrons from the metal occurs if the Fermi level of the metal is higher than the bottom of the conduction band of the insulator. But metal to insulator charging usually requires some mechanism to raise the Fermi level of electrons in the metal.

Today, quantum mechanical tunneling [3] during contact is usually offered as the explanation of how the electrons overcome the forbidden gap in insulators. Tunneling between the charged solid and gas molecules [4] has even been proposed to explain the gas breakdown observed in contact charging. But a more tenable proposition [5] is that after tunneling of electrons in the metal, a charged metal surface is produced by which contacting gases break down by Paschen's law. However, tunneling explanations exclude separation. Recently, a Two-step model [1] was proposed for contact electrification, the model asserting charge accumulation occurs by the combined sequence of contact and separation, the electron energy levels "jacked-up" during separation. But the mechanism by which the electrons are raised to higher energy states during separation is not identified.

The purpose of this paper is to propose a cavity QED mechanism in the Two-step model of contact electrification by which the electron energy level is raised above the forbidden gap of the insulator. Electron tunneling arguments are not invoked.

2. Theoretical background

2.1 Description

Contact electrification in the interface between metals and insulators is similar to the nucleation and collapse of bubbles in water, in that both the interface and bubble form high frequency QED cavities. During bubble nucleation, water molecules having low frequency EM radiation are displaced in the continuum to form the bubble cavity surface. Since the bubble cavity has a high resonant frequency, the EM radiation from the molecules is inhibited by QED as they reach the bubble surface, the inhibited EM radiation conserved by an increase in Planck energy of the bubble surface molecules. Bubble collapse returns the cavity to the continuum. In both nucleation and collapse, the Planck energy at VUV frequencies is sufficient to dissociate the surface water molecules into hydronium and hydroxyl ions and raise the radicals to higher energy states. Nucleation and collapse of bubbles in water is discussed in Appendix A.

In contact electrification, inhibited EM radiation from atoms in the interface is compensated by an increase in the Planck energy that by the photoelectric effect produces electrons in the metal and insulator surfaces. Planck energy E_G produced in contact electrification by the Two-step model depends on the structure of the insulator and metal. Liquid mercury metal and a solid polymer insulator, and solid metal and solid or powder insulators are illustrated in Fig. 1 (a) and (b), respectively.

Fig. 1 Contact electrification: Two-step model

(a) Liquid metal and solid polymer (b) Solid metal and insulator with powder.

Liquid mercury [1] offers the advantage of good electrical contact while avoiding the difficulties of contact deformation and shifts in contact area. Fig. 1(a) shows that during contact the interface only contains liquid mercury. During separation, clusters of mercury atoms separate from the liquid and are briefly suspended in the free space between the mercury and the polymer insulator surfaces. Each mercury atom in these suspended clusters has low frequency EM energy of magnitude 3 x ½ kT. At the instant of expansion, the interface is a high frequency QED cavity, and therefore the EM radiation of each atom in the cluster is promptly inhibited. To conserve energy, the inhibited EM radiation within the interface is compensated by the prompt absorption of Planck energy in the insulator and liquid mercury surfaces. By the photoelectric effect, the Planck energy from the inhibited EM radiation raises electron energy levels in the insulator from the valence to conduction band while liberating electrons from the liquid mercury. Subsequently, contact transfers the freed electrons from the mercury to the insulator, the mercury acquiring a positive charge because of electron loss while the insulator with an electron gain assumes a negative charge.

Solid insulators [3] in repetitive contact with solid metals break down to form a layer of fine powder as shown in Fig. 1(b). The powder may be produced from a smooth insulator surface such as an epoxy-resin, or from powders of silicon or coal applied to the epoxy-resin surface. Each atom in the powder has 3 x ½ kT of low frequency EM energy. During separation, the EM radiation from atoms in the suspended powder is inhibited because the interface between metal and insulator is a high frequency QED cavity. The EM radiation inhibited from the interface is compensated by the absorption of Planck energy at VUV frequencies, the photoelectric effect in the VUV producing a high quantum yield [5] of electrons for most metals. Metals acquire a positive charge upon electron loss while the insulator is charged negative upon electron gain. Subsequent contact transfers free electrons to the solid insulator.

2.2 Available EM energy

Like bubble nucleation, separation in the Two-step model of contact electrification produces Planck energy that may be quantified by the energy density Y of the liquid or powder in the QED cavity. During contact, the density of the liquid is not changed while compaction of the powder is assumed to produce a near solid density continuum. The available EM energy U_EM is,

where, d_o is the compacted interface thickness, and A is the contact area. The EM energy density is, Y ~ 3 x ½ kT / D³, where D is the spacing between liquid or powder atoms.

During separation, the interface thickness increases from d_o to d. The Planck energy E_G in the interface surfaces may be represented by EM waves standing in the space between pairs of atoms on opposing insulator and metal surfaces, E_G ~ hc / 2d. Assuming all pairs are active over the full contact area A, the total Planck energy U_Planck,

where, d is the atomic spacing in the metal and insulator. If all the available EM energy U_EM inhibited during expansion is conserved with the Planck energy U_Planck of the surface atoms,

and

For illustration, consider a liquid metal or powder of thickness d_o ~ 70 nm. Taking the atomic spacing D = d ~ 0.292 nm and T ~ 300 K, gives a separation d ~ 67 nm. The Planck energy E_G ~ 9.3 eV is in the VUV having a wavelength l ~ 2d ~ 134 nm. The Planck energy E_G would require EM radiation to be inhibited from ( d_o / D ) ~ 238 atoms.

2.3 Gas breakdown

Gas breakdown in contact electrification [3] occurs as the insulator and metal separate. It is generally thought gas breakdown occurs because of the Paschen discharge of the accumulated charge in the space between the insulator and metal surfaces. However, it is unlikely that Paschen discharge is the cause of gas breakdown for the following reasons:

(1) Gas breakdown is observed at atmospheric pressure and not at low pressure - opposite to what would be expected by Paschen's law. Indeed, breakdown is not observed at pressures less than 1 mbar, but rather at pressures from 260 to 1000 mbar.

(2) Atmospheric air is known to break down at an electrical field of about 3 V/mm. But high levels of charge accumulate in contact electrification. For smooth epoxy resin in contact with brass in nitrogen at a pressure of 260 mbar, the accumulated charge is about 1.4 nC. For a 2.5 mm diameter contact area, the surface charge density is about 3x10^-4C m^-2 giving a surface electric field of about 30 V/mm. But this is an order of magnitude greater than that expected for breakdown in air.

Instead, it is likely that the Planck energy at VUV frequencies produced by inhibited EM radiation ionizes the gas in the interface, the ionized gas increasing the electrical conductivity of the interface, the observed breakdown caused by leakage of charge across the interface. Indeed, the conductivity of ionized air was used [2] by Lord Kelvin to measure the contact potential of metals. Leakage breakdown is consistent with [3] observations. Little gas is present at low pressure, the Planck energy absorbed solely by the metal and insulator, the charge electrification increasing to saturation without breakdown. At higher pressure, the gas absorbs more of the Planck energy and ionizes, the ionized gas increasing the electrical conductivity across the interface and causing leakage breakdown.

3. Summary and conclusions

In the Two-step model, contact electrification of an insulator with metals by the photoelectric process is tenable. The photoelectric process relies on the fact that the energy loss from the inhibited EM radiation of the liquid and powder atoms in the interface is compensated by an increase in Planck energy of the metal and insulator surfaces. The Planck energy may be estimated by 3 x ½ kT of low frequency EM energy for every atom that separates from the metal and insulator surfaces. The Planck energy in the VUV is sufficient to raise electron energy levels in interface surfaces and produce free electrons that are subsequently transferred to the insulator upon contact. Tunneling arguments are not invoked.

Gas breakdown in contact electrification is most likely not caused by Paschen discharge of accumulated charge. Instead, the Planck energy produced from the EM radiation inhibited from the interface cavity ionizes the interface gases adjacent to the metal and insulator. Gas ionization increases the electrical conductivity across the interface, the increased conductivity leading to the leakage of charge accumulated by contact electrification.

The Two-step model extended by the photoelectric effect during separation may find application in understanding electrostatic discharge in powders and liquids. Tribo-electrification may be explained without invoking tunneling and friction.

References

[1] Z.Z. Yu, K. Watson, Two-step model for contact charge accumulation, J. Electrostat. 51-52 (2001) 319-325.

[2] Lord Kelvin, Phil. Mag. V (1898) 82.

[3] B.A. Kwetkus, K. Sattler, H.C. Siegmann, Gas breakdown in contact electrification,

J. Phys. D: Appl. Phys. 25 (1992) 139-146.

[4] E. Nasser, Fundamentals of Gaseous Ionisation and Plasma Electronics, Wiley Interscience, New York, 1971.

[5] E.W. McDaniel, Collision phenomena in ionized gases, Wiley, New York, 1965.

Appendix A

EM energy in bubble nucleation and collapse

The EM energy in bubble nucleation and collapse finds basis in the phenomenon of sonoluminescence (SL). SL may be described [A1] by the emission of ultraviolet (UV) and visible (VIS) photons during of the cavitation of liquid water, but is also known to dissociate water molecules and produce hydroxyl ions [A2].

The Planck theory of SL [A3] postulates the SL photons are produced from the concentration of Planck energy E in the bubble wall surface molecules because of the EM energy produced in the bubble cavity during nucleation or collapse. The Planck energy E of the EM radiation is,

where, h is Planck's constant, u = c / l is the bubble resonant frequency, c is the speed of light, and l is the wavelength of the bubble resonance. In a spherical bubble of radius R, the bubble resonance may be considered to have a wavelength l ~ 4R and frequency u ~ c / 4R.

Harmonic oscillators and ZPE

In the Planck theory of SL, the bubble wall surface water molecules may be considered to produce EM radiation from vacuum ultraviolet (VUV) to soft X-ray frequencies even though the bubble wall is at ambient temperature. This is consistent with the zero point energy (ZPE) included in the original formulation [A4] of black body radiation by Planck and for whom the Planck theory of SL is named.

The Planck theory of SL treats each surface molecule on the bubble wall as a harmonic-oscillator, the normal modes of which correspond to the field modes of the bubble cavity that include the ZPE. Planck’s derivation of ZPE was based on the principle of least action that relates Planck's constant h to areas in the amplitude-velocity space of harmonic oscillator solutions, but the physical rationale are obscure. In the Planck theory of SL, the derivation of ZPE follows as the logical consequence of the bubble cavity containing temperature independent Planck energy E_G. The Planck energy E in the bubble cavity,

where, E_T = hu / ( exp (hu / kT ) - 1) is the usual temperature dependent Planck energy, u is frequency, k is Boltzmann's constant, and T is absolute temperature. E_G is the temperature independent Planck energy described by EM waves or cavity field modes, the standing waves depending on the bubble geometry G.

The cavity field modes correspond to standing EM waves having a Planck energy E_G = hu_f, where u_f is the fundamental resonant frequency of the bubble cavity. Since the Planck energy E_G is formed by pairs of harmonic-oscillators on opposing bubble wall surfaces, the ZPE of each harmonic-oscillator in the pair is half of the full Planck energy E_G,

ZPE = ½ E_G = ½ h u_f

The ZPE is restricted by cavity QED. Since the bubble resonant frequency u_f varies from VUV to soft X-rays, low frequency ZPE is inhibited by QED from the bubble cavity, u< u_f. Only high frequency ZPE may exist in the bubble cavity, u > u_f .

Thermal equilibrium of EM radiation

In the Planck theory of SL, the surface water molecule VUV emission is not in equilibrium with the temperature of the bubble wall. Stimulation of VUV states of the surface molecules at ambient temperature occurs through the ZPE. Consistency [A4] is found with Planck's general blackbody spectrum density r (u,T ) restricted here for cavity QED by,

where, u > u_f

Boyer's random electrodynamics [A5] is consistent with Planck, but the ZPE in both Planck and Boyer formulations differs from that by Einstein and Hopf [A6] who excluded the ZPE because they neglected the interaction of radiation with the walls of a cavity.

The Planck theory of SL is consistent with Planck and Boyer in the assertion that VUV emission may be stimulated by ZPE at ambient temperature in the same way as if the surface molecules were irradiated with a VUV laser. In this regard, Planck stated that the ZPE provides an explanation of atomic vibrations that are independent of temperature, specifically citing as an example the temperature independence of electrons liberated by the photoelectric effect. In contrast, the Einstein and Hopf formulation of black body radiation requires for the stimulation of VUV emission (~ 10 eV) an unrealistic temperature of about 100,000 K.

Available EM energy

In the Planck theory of SL, the source of Planck energy is the EM radiation in the water molecules of the bubble wall that during nucleation and collapse is inhibited from the bubble cavity by QED. The water molecule has 6 DOF, and therefore every molecule in the continuum has an EM energy of 6 x ½ kT = 3 kT at low frequencies, u < kT / h. But high frequencies from soft X-rays to the VUV characterize the bubble cavity resonance u_f. QED inhibits the EM radiation from any water molecule within the bubble cavity as u < u_f for microscopic bubbles. Nucleation and collapse are shown in Fig. A-1 (a).

In nucleation, bubbles grow as water molecules are displaced in the continuum to form the bubble surface, the EM radiation inhibited as soon as the molecule reaches the surface. Hence, the inhibited EM radiation for the full bubble is the EM energy of all water molecules in the continuum that after nucleation are excluded from the bubble cavity. For a spherical bubble of radius R, the inhibited EM radiation U_EM,

where, Y = 3 kT / D³ is the EM energy density of the continuum and D is the spacing between water molecules at liquid density, D ~ 0.31 nm. Bubble collapse returns the bubble cavity to the continuum. In the collapse of a spherical bubble of radius R by an increment d, the inhibited EM radiation U_EM,

The inhibited EM radiation does not increase the temperature of the bubble wall molecules because bubble nucleation is a rapid process compared to the slow thermal response of the massive bubble wall. Instead, the EM energy increases the Planck energy U_Planck of the temperature independent state of the bubble wall surface molecules by,

where, ½ N_S corresponds to the number of diametrically opposite water molecule pairs on the bubble surface. Conservation of U_EM with U_Planck upper bounds the number N_S of molecules and Planck energy E_G available in bubble nucleation,

and

At T ~ 300 K, the number N_S of surface molecules having Planck energy E_G as a function of bubble radius R is shown in Fig. A-1(b).

Fig. A-1 (a) EM energy in bubble nucleation and collapse

(b) Bubble nucleation : Planck energy and number of surface molecules.

The standard unit of SL giving the number N_S ~ 2x10⁵of photons observed from a collapsing air bubble in water [A7] corresponds to a bubble having radius R ~ 39.1 nm with a wavelength of l ~ 156.4 nm in the VUV and a Planck energy E_G ~ 8 eV. The number of N_S of photons observed corresponds to the number of water molecules dissociated, but after recombination only about 0.2 N_S ~ 4x10⁴ ions and 6.4 fC of charge are available for electrification. This is consistent with the assertion [A8] that the Lenard effect in waterfall electricity is caused by the nucleation of bubbles in the splash of the waterfall.

References

[A1] H. Frenzel, H. Schultes, Ultrasonic vibration of water. Z. Phys. Chem., 27B, (1934) 421-424.

[A2] Y.T. Didenko, S.P. Pugach, Spectra of sonoluminescence. J. Phys. Chem., (1992) 9742-49.

[A3] T.V. Prevenslik, Dielectric polarization in the Planck theory of sonoluminescence. Ultrasonics-Sonochemistry, 5 (1998) 93-105.

[A4] M. Planck, Theory of Heat radiation, Translated by M. Masius, Dover 1956.

[A5] T.H. Boyer, Classical statistical thermodynamics and electromagnetic zero-point radiation. Phys. Rev., 1969, 186:1304-1318.

[A6] A. Einstein, L. Hopf, Further investigations of resonators in radiation fields. Ann. Physik.,1916, 33: 1105- 1115.

[A7] R.T. Hiller, K. Weninger, S.J. Putterman, B.P. Barber, Effect of noble gas doping on single-bubble sonoluminescence. Science, 1994, 266: 248-250.

[A8] T.V. Prevenslik, Niagara falls: ion emission and sonoluminescence. ESA 2000, Brock University Niagara Falls Ontario June 18-21 2000.