^{1}

^{2}

^{*}

^{†}

^{1}

^{2}

^{†}

^{1}

^{2}

Edited by: Yang Xu, University of Glasgow, United Kingdom

Reviewed by: Mehmet Ayyildiz, Istanbul Bilgi University, Turkey; Witold Franciszek Habrat, Rzeszów University of Technology, Poland

This article was submitted to Tribology, a section of the journal Frontiers in Mechanical Engineering

†These authors have contributed equally to this work

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Electroadhesion is a very hot topic in tribology due to its implications for the science of surface haptics. Building on theories of Persson, we develop a simpler theory for electroadhesion between rough surfaces using the bearing area model of Ciavarella. The theory is derived for the cases of conducting and insulating bodies and shows that pull-off forces depend mostly on the applied voltage, the surface root mean square (RMS) height, and the longest wavelength in the surface representation. However, the real contact area and friction additionally depend on rms slopes and hence on truncation of the roughness spectrum. Two stickiness criteria are derived based on the present theory and on the energy balance proposed by Persson and Tosatti. The coefficient of friction decays with the normal pressure, tending to a plateau in a manner consistent with existing experiments.

An electric voltage applied between two contacting rough elastic bodies leads to accumulation of charges of opposite signs on the surfaces, which obviously produces electrostatic attraction. The electrostatic load adds to the external repulsive loads and possibly to van der Waals adhesive interactions. Also, it influences the intimate contact area and therefore the frictional force when the solids are sliding, which is of great interest for new touch screen applications (Vardar et al.,

Persson (

It is, however, unclear from the full Persson theories which are the main parameters governing the problem, since Persson's theories were developed in very general settings, particularly in terms of the full spectrum of roughness and the complex details of the gap between the surfaces. This results in quite elaborate theories which may not be easy to apply or interpret. A simpler theory was proposed for van der Waals adhesion by Ciavarella (

As suggested in a recent paper by Dalvi et al. (

Let us consider the contact configuration shown in _{1} and _{2}, respectively, and relative dielectric constants ε_{1} and ε_{2} (see

Sketch of the geometry considered: A rough elastic solid is squeezed against a flat rigid solid. Both solids are made of conducting materials, with insulating coatings of thicknesses _{1} and _{2} and dielectric constants ε_{1} and ε_{2}. An electric voltage difference

By writing the variation of the electric potential through the contact pair, Persson (^{2} is the vacuum permittivity. Hence, we have an adhesive mean pressure
_{0} is the nominal contact area, and _{1}(_{ext} at a given mean separation _{ad} (pressure is negative when compressive), which results in

In what follows, we shall assume for simplicity self-affine fractal rough surfaces that have a typical power-law PSD
_{1}, and _{0} are respectively the largest and smallest wavenumbers in the surface representation, and _{rms} being the root mean square (rms) amplitude of roughness (see

Typical power-law PSD of a self-affine randomly rough surface. In this figure, ^{−8} m^{0.6}, with

Determining the external load _{ext} by Equation (7) requires in general that the probability distribution _{1}(

Typical electrostatic attraction force law replaced by a Maugis step function law, plotted in dimensionless form with

For a nominally flat surface with Gaussian distribution of heights, the main idea is to estimate the adhesive area^{1}_{rms} is the rms amplitude of roughness.

For intermediate mean separations, the repulsive pressure can be obtained by using Persson's theory (Persson, ^{*} is the composite elastic modulus of the material.

While for the spherical problem the bearing area estimate above excludes the contact area itself, for Gaussian roughness the expression (12) tends to 1 in the limit, so it would be inconsistent to add separately the true contact area _{0}: hence, in our theory there will be no effect of rms slopes, which in Persson's (_{L} is the longest wavelength in the surface representation, i.e., _{0} = 2π/λ_{L}. Notice that in the standard case of van der Waals forces, _{a}/ϵ, and by changing the voltage it can be varied by many orders of magnitude. Hence, adhesion is favored at high applied voltage ^{*}, and thin insulating layer thickness. Clearly, if both the van der Waals adhesion forces and the electroadhesive tractions are present, they should be summed.

If an electric voltage is applied between two contacting solids with certain electric conductivity, then one has to account for the electric current flow through the contact patches; hence, there would be attraction only outside of the contact areas. However, the voltage drop _{1} and κ_{2} of the solids. Barber's (^{2}_{0} is the adhesive strength for insulating bodies. Hence, for a fixed separation one obtains

As noted by Persson, adhesion increases very slowly as the voltage increases, but above a certain threshold the increase becomes very rapid as the surfaces approach each other. It may be useful in certain applications to have an estimate of the voltage at which the rough surfaces would be near full contact. Assuming in our simplified theory that this occurs already at _{ext} results in the following condition for υ:
_{full} depends are _{0} and _{rms}.

External pressure

If max[

Pull-off mean pressure

Pull-off mean pressure

The above results show that the pull-off traction is principally determined by _{rms}, _{0}, and υ. Indeed, for a low contact area, neither the repulsive nor the attractive tractions depend on the surface “magnification” ζ = _{1}/_{0} as in the adhesionless load-separation relation (13). By adopting the BAM approach to estimate the adhesive contact area and using the results in Ciavarella (

Alternatively, a stickiness criterion could be defined using the energy balance proposed by Persson and Tosatti's theory for classical van der Waals adhesion. Persson and Tosatti (_{eff} available at pull-off is
_{0} and is greater than 1 because of a roughness-induced effect, and _{el} is the elastic strain energy stored to squeeze the roughness, with isotropic power spectrum ^{3}_{0} ≃ 1)
_{eff} = 0 in (28), which can be cast in terms of the roughness rms amplitude as

We can rewrite the stickiness criteria in dimensionless form:

Pull-off mean pressure

Finally, in

Friction coefficient _{ext} for

In this section we provide a possible example with dimensional results based on a typical rough surface with the PSD shown in ^{−8} m^{0.6} with _{rms} = 8.43 μm and the surface rms slope is _{ad} = _{ext} − _{0} as a function of the applied voltage for ^{*} = [1, 10, 100] MPa and for both insulating (solid lines) and conducting (with _{0} = 10 mm, dashed lines) solids. All the plots are for a constant external pressure _{ext} = 100 Pa, and the normalized contact area is computed according to Equation (34). For harder solids we get lower adhesive contact pressure, higher mean separation, and smaller contact area. _{0} remains very small over the entire range of applied voltage, indicating that the approximation we used is correct. In general, for a given voltage, conducting bodies show less adhesion, higher average separation, and smaller contact area. As in Persson (_{ad} is observed and which can be estimated using Equations (23) and (24); see the black circles and squares in

_{ad} = _{ext} − _{0} as functions of the applied voltage ^{*} = [1, 10, 100] MPa and for both insulating (solid lines) and conducting (with _{0} = 10 mm, dashed lines) solids. All plots are for a constant external pressure _{ext} = 100 Pa. In _{0}. In all panels the black circles and squares represent estimates of the critical voltage υ_{full} using Equations (23) and (24).

Electroadhesion with direct current is not as easy to detect as it is when alternating current (AC) is applied. However, in terms of the mathematical model, very few changes are needed to translate the above results to the case of AC voltage. Indeed, when the applied voltage oscillates in time, _{0} in Equation (2) with a function
_{i}(ω) are the dielectric functions of the two media. In the classical case of sinusoidal voltage, _{0}cos(ω_{0}_{zz}(

Using the bearing area model of Ciavarella, we have developed a simple theory for electroadhesion between two hard rough surfaces. The model is derived for the cases of insulating and conducting bodies, where in the latter case the voltage drop near the interface needs to be evaluated by taking into account the current flowing through the asperity contact regions. We have shown that the friction coefficient increases with increasing voltage and can reach very high values for low normal forces, in agreement with experimental results. We have introduced a new dimensionless parameter for electroadhesion, which governs the behavior of the proposed model. Pull-off forces depend mostly on well-defined quantities, such as the applied voltage, the surface rms height, and the longest wavelength in the surface representation. In the limit of small contact area, two stickiness criteria have been derived based on the present theory and on the energy balance proposed by Persson and Tosatti, which turns out to give very similar results (except for a prefactor) for pure-power-law power spectral densities.

All datasets generated for this study are included in the article/supplementary material.

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

MC and AP acknowledge support from the Italian Ministry of Education, University and Research under the Programme Department of Excellence Legge 232/2016 (grant no. CUP-D94I18000260001). AP is thankful to the DFG (German Research Foundation) for funding the project PA 3303/1-1, and acknowledges support from PON Ricerca e Innovazione 2014-2020-Azione I.2 - D.D. n. 407, 27/02/2018, bando AIM (grant no. AIM1895471).

^{1}The adhesive area is defined as the portion of surface on which tensile stress is applied.

^{2}Notice that we are following a DMT-like approach, so from Equation (13) we have written the contact stiffness _{⊥} = _{rms}); see Persson (

^{3}Notice that we use the original convention and notation for ^{iso}(^{2}