Electrical Contact With Dielectric Breakdown of Interfacial Gap

Many applications depend on functioning electrical connects. This includes the growing areas of electric vehicles [1] and alternative energy sources [2,3]. Specifically, there are many cases such as the battery electrodes/current collector bars contact in lithium-ion batteries [4], switches in the electric power industry [5], the lithium metal/solid-state electrolytes contact in all-solid-state lithium metal batteries [6], and pantograph-catenary system in high-speed railways [7]. This includes connectors designed to carry electrical current and also other mechanical components such as gears and bearings carrying the conduction of leakage or unforeseen currents [8,9]. Predictive models of these components depend on a theoretical framework that is often fundamentally based on single asperity contact resistance models [10].

Electrical contact resistance (ECR) is the added resistance that occurs between contacting conductive surfaces in addition to the bulk resistance. In most cases, this refers to the contact of metals. The contact resistance is composed of two main sources: spreading resistance and film resistance. Less conductive oxides and other materials between the surfaces cause film resistance. Since surfaces are practically always rough, the solid contacts are isolated between the peaks or asperities. This results in the electric current (electrons) to contract as they cross between the isolated contacting asperities and then spread again after. The current lines (electron paths) are constricted at the interface so that electrons can only pass through the conductive channel formed by microcontacts. Therefore, the electrical contact resistance is the outcome of multiple resistors in parallel formed by microcontacts. The constricted current with high density further causes the local temperature rise (known as the Joule heating), oxidation and melting of the plating layer, energy loss, and even catastrophic fire.

Holm employed a capacitance calculation method of equipotential surfaces to derive the spreading resistance between two conductive half-spaces connected only by a circular conductive region [5]. More detailed derivation can be found in Ref. [11]. Malucci later employed Holm’s method to calculate the spreading resistance for contact between layered or coated surface asperities [12]. Nakamura [13] also formulated a predictive function for the spreading resistance of square-shaped contact areas. Asperity contact area shapes and sizes can be predicted via contact mechanics theories for single and multiple asperity elastic and elastic–plastic solid contact models [14]. For instance, Shah et al. [15] showed that a sinusoidal-shaped asperity contact would begin circular in shape but would morph into a square shape. When the asperities are elongated, these contact areas would also be oval or rectangular in shape. In reality, however, the asperities are likely to be much more complicated in shape, as suggested by measurements [16,17]. Bowden and Williamson [18] conducted early experiments on electrical contacts and found that the Joule heating in surfaces had a great influence on their softening or failure. Essentially, they found a healing effect that, in many cases, an electrical contact may soften due to Joule heating and increase the contact area in order to carry the load effectively. However, if too much electrical current is applied, the contact area cannot compensate enough, and a thermal run-away occurs, resulting in the contact failing. Electrical load can also accelerate the wear of a contact and effectively increase the roughness of the worn surface [3]. It is also known that contact resistance can vary with the scale or size of the contact areas [19,20]. Once the contact areas approach the micrometer scale or smaller (i.e., the electron mean free path length [21]), the nature of the conduction of electrons can change from a continuum diffusive behavior to a quantum ballistic behavior [22]. However, in the current work, the focus will be on contacts above this length scale, although the gaps between the surfaces at the edge of contact are thin and may allow for significant electron tunneling.

The rough surface electrical contact problem has been solved either deterministically using numerical models [23–27] or analytically based on the electrical contact model with a circular contact area [28–33]. Barber [34] found an analogy between the incremental elastic contact problem and the electrical contact problem, which deduces a linear relation between the interfacial contact stiffness and the contact conductance (reciprocal of ECR). This linear relation is validated by the numerical simulation [24], but it may not be strictly held in cases of elastoplastic contacts [35], especially under high loads. By employing this analogy, the majority of elastic rough surface contact models can be directly utilized to estimate ECR, provided that the interfacial contact stiffness can be determined [36–38]. In addition, when the isolated asperity contacts grow due to higher pressures and come closer, they can influence each other through the electrical field [28,39] and by diverting the current flow from one contact area to another. In other words, each asperity contact will carry part of the electrical current and can be considered as its own constriction from a finite size to the contact area, as modeled by Rosenfeld and Timsit as a cylindrical conductor with a single constricted contact [40]. This effectively lowers the spreading resistance. Following this, Malucci and Ruffino [41] also deduced that the current density was a critical way to determine the probability of a single asperity contact degrading or failing. Therefore, an electrical contact with many asperity contacts is more resilient and stable.

The classic electrical contact theory predicts an inverted bell-shaped current density distribution within the contact area with an infinite value at the contact periphery. As the stress concentration in the flat-end punch contact problem can lead to yielding at the contact periphery, this current concentration suggests that the electrode material may melt at the same spot irrespective of the electrical loads. Malucci [42] highlighted that the presence of this nonphysical current concentration arises from the assumption that the contact and noncontact areas are not situated on the same plane. According to Malucci [42], the bridge structure (i.e., the three-dimensional geometries of the interfacial gap) adjacent to the contact area can impact how current is distributed at the interface. Malucci [42–44] found the analytical solutions for current density distribution associated with three distinct interfacial gap geometries, all of which do not have infinite current density at the contact periphery.

In addition to the coplanar boundary conditions adopted in the classic electrical contact theory, the infinite current density at the contact edge may result from neglecting discharge just outside the contact area. The current concentration near the contact edge can result in an intensified electric field nearby, potentially leading to the breakdown of the insulating medium between two electrodes. Therefore, electrons can move through the adjacent conductive gap, resulting in a possibly continuous and finite current density on both sides of the contact edge. As mentioned earlier, contacts can degrade due to the Joule heating from the current flow. However, as will be explored in this work, there may be other modes of degradation, such as the electrical discharge machining (EDM). The phenomenon of discharge is prevalent in the rapidly expanding electric vehicle industry, where powertrains are predominantly electrified by inverters utilizing low-amplitude, high-frequency alternating current [45]. For instance, electrically induced bearing damage (EIBD) dominantly occurs at the rolling element’s inner/outer race interface in the electric motor bearing, which is caused by EDM if there is a lack of electrical insulation [46–48]. Jackson et al. [49] recently developed an electrified mixed lubrication model where the probability of a discharge zone for varying operating conditions and properties can be estimated based on the assumption that the lubricant has a constant dielectric strength.

In order to address the initial concern of whether discharge can cancel the current concentration at the contact edge despite the contact and noncontact regions being coplanar, we will revisit the classic electrical contact theory (without discharging) in Sec. 2. In Sec. 3, the numerical model for discharging electrical contact is developed, along with two analytical solutions for low applied voltage cases. Section 4 thoroughly investigates and discusses the impact of discharge on electrical properties at the interface.

Let the potential boundaries of the two contacting bodies far from the contact be $V1$ and $V2(<V1)$⁠. The resistivities of the two half-spaces are $ρ1$ and $ρ2$⁠, respectively. The current is driven by this potential difference (⁠$ΔV=V1−V2$⁠), and all current lines are constricted over the circular contact area if dielectric breakdown does not occur at the interface. A graphical illustration of this electrical contact problem can be seen in Fig. 1(b). The current density vectors (⁠$Jr+(r,z≥g(r))r→+Jz+(r,z≥g(r))z→$ and $Jr−(r,z≤0)r→+$$Jz−(r,z≤0)z→$⁠) and potential (⁠$V+(r,z≥g(r))$ and $V−(r,z≤0)$⁠) of both contacting bodies can be solved either analytically using the Hankel transform [11] or numerically using the conjugate gradient (CG) method and the fast Fourier transform (FFT) [25].

In the discharge zone at the interface, the $V(g)$ relation should strictly follow the Paschen law, i.e., $V(g)=Vb(g)$⁠. With such a complex boundary condition, it is nearly impossible to completely solve the discharging electrical contact problem analytically. Thus, we formulate the numerical model first in Sec. 3.2. Then, two solutions for low voltage applications are respectively derived in closed forms in Sec. 3.3 when two electrodes are in solid–solid contact and complete separation.

The algorithm for solving Eqs. (18)–(20) using the CG method developed by Polonsky and Keer [58] with minor changes.

When two electrodes are completely separated (⁠$a=0$ and $δ=−g(r=0)≤0$⁠), the governing equations of electrodynamics switch from Ampere’s law to Gauss’ law. In the latter electrostatic analysis, the permittivity of the dielectric media between two electrodes (e.g., air and lubricant) greatly influences the local electric field, which is essential to the outburst of discharge events [61]. When the interfacial gap is negligibly small compared with the electrode dimension, we can assume that the interfacial gap at the discharge zone is “artificially” closed. Therefore, Eqs. (15)–(17) remain applicable when $δ≤0$ if air breakdown occurs.

The accuracy of the numerical model developed in Sec. 3.2 is highly sensitive to the local mesh density. This mesh-dependent feature complicates its usage unless an adaptive mesh is used so that the element size is adaptively refined at a high current density gradient region. If we restrict the electrical contact problems to those low applied voltage cases, where $ΔV≪min(Vb)≈244.8$ V, we can obtain two sets of closed-form solutions, respectively, when two surfaces are in solid–solid contact (contact phase) and complete separation (separation phase).

Equations (24) and (25) are not the final solutions but rather auxiliary solutions of $V′(r)$ due to a Hertzian pressure-like current density distribution. These auxiliary solutions can serve as building blocks to achieve the final closed-form solutions.

Then, all nontrivial electrical interfacial solutions in dimensionless forms are tabulated as follows:

• Nondischarging state (⁠$δ*>0$⁠):

  $V*(r*>1)=1−2πarcsin(1/r*)$

  (46a)

  $J*(r*)=12(δ*)−1/2/Re1−r*2$

  (46b)

  $Rc*=23(δ*)−1/2$

  (46c)
• Discharging state at contact phase (⁠$δ*>0$⁠):

  $V*(r*∈(1,c*])=δ*{1πr*2−1+(2−r*2)[1πarcsin(1/r*)−12]}$

  (47a)

  $V*(r*>c*)=1−δ*π{[(2c*2−r*2)arcsin(c*/r*)+c*r*2−c*2]−[(2−r*2)arcsin(1/r*)+r*2−1]}$

  (47b)

  $J*(r*)=δ*[Rec*2−r*2−Re1−r*2]$

  (47c)

  $Rc*=[(1+δ*)3/2−(δ*)3/2]−1$

  (47d)

  where $c*=1+1/δ*$⁠. By setting the right-hand side of Eq. (46c) equal to the right-hand side of Eq. (47d), we find no real root for $δ*$⁠. Thus, it is easy to prove that discharging $Rc*$ is always less than nondischarging $Rc*$⁠, $∀δ*≥0$⁠. A graphical illustration of this inequality can be found in Fig. 6.
• Discharging state at separation phase (⁠$δ*≤0$⁠):

  $V*(r*≤1)=−δ*+12(1+δ*)(r*)2$

  (48a)

  $V*(r*>1)=1−1π(1+δ*)[(2−r*2)arcsin(1/r*)+r*2−1]$

  (48b)

  $J*(r*)=1+δ*Re1−r*2$

  (48c)

  $Rc*=(1+δ*)−3/2.$

  (48d)

Consider a typical electrical contact between an elastic, parabolic, copper electrode of radius $R=0.5$ mm and a flat copper electrode. The resistivity of copper is $1.68×10−5$$Ω⋅m$⁠. Figure 3(a) shows that the analytical discharging solutions at the contact phase (solid lines), $J*(r*)$ and $V*(r*)$⁠, are nearly identical to the corresponding numerical solutions (hollow circles) with a visually indistinguishable difference. Conventional electrical contact theory predicts an inverse bell-shaped current density distribution (dashed line), where $J*$ is fairly constant at the contact center and grows rapidly toward the contact periphery (dash-dotted line) until it becomes singular at $r*=1$ (see Fig. 3(a)). Since the discharge is more likely to occur adjacent to the contact periphery with a vanishing gap, the gas medium over an annulus region immediately outside the contact area becomes conductive and allows the current to flow through. The corresponding $J*(r*)$ distribution shown in Fig. 3(a) implies that the singular current density is canceled by discharging. Physically speaking, the conductive path induced by discharging relieves the “traffic jam” of the current flow at the contact edge. The cancellation of the singular current density is similar to that in fracture mechanics and contact mechanics, where stress singularities at the tip of the Griffith crack [63] and the edge of the adhesive contact [51] are relieved, respectively, by the introduction of the yield zone [64] and the cohesive zone [65]. As the current concentration is resolved, $J*(r*)$ inside the contact area is lower than that of the nondischarging solution. As $r*→1−$⁠, $J*(r*)$ grows more rapidly toward a finite peak and is followed by a monotonic drop to zero at $r*=c/a$⁠. The nondischarging $V*$ monotonically increases from zero at $r*=1$ with a decreasing rate as $r*(>1)$ increases (see dashed line in Fig. 3(a)). The nondischarging $V*$ eventually converges to unity at the distant radial location, which is not illustrated in Fig. 3(a). The discharging $V*(r*)$ behaves similarly to the nondischarging solution. It is relatively lower than the nondischarging potential with a finite slope at $r*=1$⁠. The intersection between the dashed line and the modified Paschen law shown in Fig. 3(b) indicates that the potential drop associated with the small gap when a discharge is not considered is larger than the breakdown voltage governed by the modified Paschen law. This is a strong signal that discharge inevitably occurs at small gap ranges, where the $V(g)$ relation considering discharge follows almost exactly the modified Paschen law within the discharge zone. When the gap is larger than a threshold value, discharge is extinguished, and $V(r)<Vb(g(r))$⁠.

The validity of the analytical discharging solutions at the separation phase is confirmed by the numerical model in Fig. 4(a). The current density distribution $J*(r*)$ has a global maximum $r*=0$ and monotonically drops to zero as $r*→1−$⁠. The corresponding $V*(r*)$ has a local minimum at $r*=0$⁠, followed by a parabolically increasing trend within the discharge zone. The only difference between the $V(g)$ relations in the contact and separation phases is that the latter starts with a finite interfacial gap (Fig. 4(b)). Comparing the current density distribution in Figs. 3(a) and 4(a), we can expect that, as indentation transits from positive (contact phase) to negative (separation phase), the central convex current distribution gradually shrinks till vanishing. The outer concave current distribution eventually dominates the whole interface and evolves self-similarly as $−δ$ increases.

Figure 5 shows how $ΔV$ influences $J*(r*)$ and $V*(r*)$⁠. As $ΔV$ increases, $J*$ at the contact center is barely influenced by $ΔV$⁠, while $J*$ greatly drops with $ΔV$ at the contact edge. This implies that the peak-to-center ratio of the current density drops as $ΔV$ increases. The discharge zone at the noncontact area (⁠$r*>1$⁠) gradually grows with $ΔV$⁠. As $ΔV>500$ V, the $V(g)$ relation interacts with the modified Paschen law within two distinct gap ranges (see the inset of Fig. 5(a)), and the discharge zone is split into two annulus regions (⁠$J*(r*)$ with $750$ V). As $ΔV$ further increases, two annulus regions are expected to coalesce into one. This surprising dual discharge zone is related to the nonmonotonic nature of the modified Paschen law and makes it nearly impossible to obtain analytical solutions at high $ΔV$⁠. The piece-wise Paschen law has been confirmed both experimentally and numerically using the micro-gap breakdown tests [56] and Particle-in-cell with the Monte Carlo collision (PIC/MCC) models [52], respectively. Thus, the dual discharge region is valid under the assumption that the micro-gaps within the discharge zone can be “artificially” closed. Since there is a lack of physical evidence to prove the existence of this second annulus zone, future studies should focus on accurately imaging the discharge zone. Similar to the evolution of $J*(r*)$⁠, $V*(r*)$ and its initial slope at $r*=1$ monotonically drop with an increasing $ΔV$ (see Fig. 5(b)). The dimensionless potential, $V*$⁠, nonmonotonically grows with $r*$ temporarily within the second discharge region (see the curve with $ΔV=750$ V in Fig. 5(b)), and it quickly gets back to the monotonic trend as $r*$ proceeds.

The reciprocal of various $Rc*$ (also known as the contact conductance) are shown in Fig. 6. The discharging electrical contact model provides closed-form solutions of $Rc*$ at both contact and separation phases (see Eqs. (47d) and (48d), and $Rc*$ is continuous at $δ*=0$⁠. The discharge not only resolves the current concentration but also extends the nonsingular $Rc*$ to negative indentation ranges at the separation phase, and further postpones the divergence of $Rc*$ from $δ*→0+$ to $δ*→−1+$⁠.

In the numerical and analytical models (Secs. 3.2 and 3.3), the usage of the interfacial gap is conflicting: The interfacial gap is “artificially” closed within the discharge zone so that electrons, driven by discharge events, can pass through it. However, the nonzero gap value is used to estimate the dielectric breakdown voltage based on the modified Paschen law. In practice, the discharge between two surfaces is a complex phenomenon of electron flow driven by electron impact ionization, secondary emission, field emission, and so on. PIC/MCC technique is often used in plasma dynamics to simulate the motion and interactions of the charged particles [52]. Like the other molecular/atomic dynamics models, finite computational resources restrict the application of the PIC/MCC model to a microscopic computational domain that is impractical for a macroscopic electrical contact simulation. To overcome this limitation, the PIC/MCC model may be applied locally at those discrete discharge regions to predict the outbreak of the discharge events without referring to the modified Paschen law, as well as the current density spatial distribution within the conductive channel. To further improve the efficiency of the discharging electrical contact model, future studies should focus on curve fitting an empirical law from the numerical results of the PIC/MCC model to characterize the interrelation between the dielectric breakdown voltage, interfacial gap, and current density within the discharge zone. Therefore, the conflicting usage of the interfacial gap is a compromising approach toward reducing the complexity of the plasma dynamics model. We can expect that, as the discharging gap is larger than the threshold value, the current density at the interface estimated by the present work gradually deviates from the numerical solutions of the PIC/MCC model.

There is a lack of direct evidence to support the existence of single and double annulus discharge regions outside the contact area of a sphere/flat electrical contact. The EIBD in electric powertrains (e.g., rolling element bearings in electric motors, and gears), however, can serve as indirect evidence supporting the existence of the circular discharge zone when a parabolic electrode is completely separated from its mating electrode. For instance, numerous researchers [46,47] have experimentally observed the randomly distributed pits with various sizes and depths on the raceway of the electrified rolling element bearing. Those pits are due to the electrical corrosion induced by multiple discharge events emitted from the peaks of the asperities on the cathode, which is completely separated from the anode with a thin film lubricant. In future studies, an electrical contact test may be conducted between a cathode and a transparent anode in nitrogen gas, and discharge events may emit faint flashes of visible light through nitrogen luminescence. The shape of the discharge zone on the contact interface can be captured using a high-sensitivity electron-multiplying charge-coupled device camera [67]. The image of the electrified interface shows a strong contrast between discharging and nondischarging regions, which can be efficiently segmented and binaried through image thresholding.

The numerical model proposed in the present work can be used in conjunction with the existing boundary/mixed/elastohydrodynamic/hydrodynamic lubrication models to simulate the interfacial current density within the discharge zone, which can be further used to predict the topographical change of the lubricated surfaces caused by electrical discharge machining-like damage (e.g., frosting, pitting, fluting) [45]. In an electrified rolling element bearing, for instance, each lubricated interface can be simplified as a resistor and a capacitor in parallel. The analytical solution proposed in the present work can be used to build the electrical contact resistance and capacitance of an electrified lubricated rough interface, which serve as the foundation of the circuit model of an electrified rolling element bearing [68]. This work also suggests that it might be extremely difficult to completely eliminate leakage current discharge damage in rolling element bearing contacts, even if they have solid conductive contact between them. Other works have suggested that the damage only occurs when there is a non-conductive film of lubricant completely separating the surfaces.

In this work, we developed a numerical model for the axisymmetric electrical contact problem where insulating air breaks down adjacent to the contact area. We derived analytical solutions for discharging electrical contacts subjected to low applied voltage in two respective scenarios: two electrodes are in contact and completely separated. We show that the singular current density at the contact periphery becomes finite once the conductive channel induced by discharging is formed adjacent to the contact area. When two electrodes are in solid–solid contact, discharge reduces the current density inside the contact area and allows electrons to flow through a much larger region. The corresponding current and ECR increases and decreases, respectively. Moreover, discharge extends the electrical contact theory to predict the current density and the potential distribution between two separate electrodes. The present work theoretically proves that the ignorance of the dielectric breakdown of air can lead to singular current density at the contact periphery and ECR overestimation. It serves as a foundation for modeling discharging rough surface electrical contact and EIBD.

The authors would like to thank anonymous reviewers and Dr. Haibo Zhang for their constructive comments.

• The National Natural Science Foundation of China (No. 52105179).

• The Fundamental Research Funds for the Central Universities (Nos. JZ2023HGTB0252 and PA2024GDSK0044).

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

$a$ =

radius of contact area

$c$ =

outer radius of discharge zone

$d$ =

vertical separation between the rigid flat and the mean level of the undeformed rough surface

$g$ =

interfacial gap between the deformed surface and the rigid flat

$h$ =

surface height of the undeformed parabolic indenter

$n$ =

number of nodes on the computational domain

$p$ =

contact pressure

$r$ =

radial coordinate

$z$ =

coordinate on the $z$-axis

$r$ =

column vector of discrete radial coordinates

$r→$ =

unit vector pointing in the $r$ direction

$z→$ =

unit vector pointing in the $z$ direction

$A$ =

constant in Paschen law under the ambient pressure, $A=1.13×103$ mm⁻¹

$B$ =

constant in Paschen law under ambient pressure, $B=2.74×104$ V/mm

$E$ =

Young’s modulus

$F$ =

compressive normal force

$J$ =

interfacial current density perpendicular to the interface, $J(r)=Jz+(r,z=g(r))=Jz−(r,z=0)$

$K$ =

dielectric strength of air at microgaps

$N$ =

number of asperities over the nominal contact area

$R$ =

radius of curvature at the apex of the parabolic indenter

$V$ =

potential drop across the interface, $V(r)=V+(r,z=g(r))−V−(r,z=0)$

$J$ =

column vector of $J$ defined over $r$

$K$ =

influence coefficient matrix

$V~$ =

potential $V~(r)=Vb(g(r))−V(r)$

$V~$ =

column vector of $V~$ defined over $r$

$g0$ =

interfacial gap associated with the minimum breakdown voltage in the Paschen law

$gc$ =

transitional gap between field emission process and Townsend process portions in the modified Paschen law

$rmax$ =

maximum radius of the computational domain

$u¯z$ =

surface displacement of a half-space in the $z$ direction

$Icon$ =

nodal index set associated with the conductive region

$Incon$ =

nodal index set associated with the nonconductive region

$K⊥$ =

normal contact stiffness

$Kij$ =

influence coefficient, see Eq. (21)

$Rc$ =

electrical contact resistance

$V1$ =

potential applied on the parabolic surface far from the contact

$V2$ =

potential applied on the rigid flat far from the contact

$Vb$ =

breakdown voltage

$d*$ =

dimensionless $d$⁠, $d*=dK/ΔV$

$g*$ =

dimensionless interfacial gap, $g*=g/δ$

$h*$ =

dimensionless $h$⁠, $h*=hK/ΔV$

$r*$ =

dimensionless radial coordinate, $r*=r/a$ at contact phase and $r*=r/c$ at separation phase

$E*$ =

plane strain modulus $1/E*=(1−ν12)/E1+(1−ν22)/E2$⁠, where $Ei$ and $νi$⁠, $i=1,2$⁠, are Young’s modulus and Poisson’s ratio of two contacting bodies

$J*$ =

dimensionless current density, $J*=J/(4πρKΔVR)$

$V′$ =

potential, $V′=ΔV−V$

$V*$ =

dimensionless potential drop, $V*=V/ΔV$

$V+$ =

potential of the parabolic indenter

$V−$ =

potential of the rigid flat

$Jr+$ =

current density of the parabolic indenter in the $r$ direction

$Jr−$ =

current density of the rigid flat in the $r$ direction

$Jz+$ =

current density of the parabolic indenter in the $z$ direction

$Jz−$ =

current density of the rigid flat in the $z$ direction

$Rc*$ =

dimensionless electrical contact resistance, $Rc*=Rc/(3ρ8KRΔV)$

$γse$ =

secondary electron emission coefficient

$δ$ =

indentation, $δ>0$⁠: contact phase; $δ≤0$⁠: separation phase

$δ*$ =

dimensionless indentation, $δ*=δK/ΔV$

$Δu¯z$ =

incremental change of the surface displacement of a half-space in the $z$ direction in response to $Δp$

$Δδ$ =

infinitesimally small indentation

$Δp$ =

incremental change of the contact pressure acting on the boundary of a half-space

$ΔV$ =

potential drop across the interface far from the contact

$⟨|∇h|2⟩$ =

root-mean-square slope of the surface topography

$⟨|∇h|2⟩$ =

root-mean-square slope of the surface topography

$ν$ =

Poisson’s ratio

$ρ$ =

composite resistivity, $ρ=ρ1+ρ2$

$ρ1$ =

resistivity of the parabolic surface

$ρ2$ =

resistivity of the rigid flat

$Φ$ =

probability density

$Ωcon$ =

conductive region over the interface

$Ωc$ =

contact region over the interface

$Ωncon$ =

nonconductive region over the interface

$Ωnc$ =

noncontact region over the interface