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# Surface Roughness and Electrical Contact Resistance

J.R.Barber The contact of rough surfaces Surfaces are rough on the microscopic scale, so contact is restricted to a few `actual contact areas'. If a current flows between two contacting bodies, it has to pass through these areas, causing an electrical contact resistance. The problem can be seen as analogous to a large number of people trying to get out of a hall through a small number of doors.

Classical treatments of the problem are mostly based on the approximation of the surfaces as a set of `asperities' of idealized shape. The real surfaces are represented as a statistical distribution of such asperities with height above some datum surface. However, modern measurement techniques have shown surfaces have multiscale, quasi-fractal characteristics over a wide range of length scales. This makes it difficult to decide on what scale to define the asperities.

A theorem bounding electrical contact resistance A recent theorem places upper and lower bounds on the electrical contact resistance due to surface roughness, using only properties of the 'determinate' shape of the contacting bodies and the maximum peak-to-valley roughness. First an analogy is established between the mathematical descriptions of the electrical conduction problem and the elastic contact problem. The electrical potential is governed by Laplace's equation and the interfacial plane is an equipotential surface. The elastic problem can also be cast in terms of a harmonic potential function and the analogous problem is that in which the normal displacement in the contact region is uniform, whilst the normal traction is zero outside the contact region. These boundary conditions define the * incremental * contact problem - i.e. the response of the system to a small increment of normal load. It follows that there is a strict linear relation between the contact conductance (reciprocal of the resistance) and the incremental normal contact stiffness that depends only on the electrical and elastic properties of the contacting materials.

The reciprocal theorem is then used to bound the incremental stiffness of the elastic contact between the solutions of two smooth contact problems. Combination of these two results yields the required bounds.

Figure 1: Two bounding contact problems

Suppose a rigid smooth indenter is pressed to a prescribed depth into a half space with a rough surface, identified as *(a)* in the figure. We compare this state with two limiting smooth contact problems in which the rough half space is replaced by the smooth half-spaces *(b)* and *(c)* respectively. Surface *(b)* passes through the highest point of the surface and *(c)* through the lowest point, so that the real rough surface *(a)* is completely contained between *(b)* and *(c)*. The vertical distance between these planes *s* is also the maximum peak-to-valley roughness of the surface Rt.

Common sense suggests that *F(b)>F(a)>F(c)* on the grounds that adding material to the half space above plane *(c)* can only make it harder to press in the indenter to the specified depth. The proof of this result is given by Barber (2003). However, notice that the roughness of surface *(a)* will generally cause the contact area *A(a)* not to be completely enclosed within *A(b)* or *A(c)*.

The load-displacement relation for the smooth bounding surfaces *(b)* and *(c)* are similar but separated by a distance *s*. The curve for the rough surface *A(a)* must lie between these extremes.

It can be shown that the contact area *A* and the incremental stiffness must be non-decreasing functions of the indentation ζ, so the load displacement curve for both smooth and rough surfaces must be concave upwards. The limiting slope (incremental stiffness) at a given force *F* can therefore be bounded between two tangent lines. For more information on this procedure, please download the file Bounds on the electrical resistance between contacting elastic rough bodies.

In Figure 1, we show the bounding surfaces as planes because this simplifies the solution of the bounding contact problems, which then become Hertzian. However, the argument does not depend on the bounding surfaces being planes. For example, if we have a solution for the rough surface contact problem at some finite level of resolution, the effect of features below the resolution truncation can be assessed by defining two parallel surfaces separated by the maximum peak-to-valley variance of the scales neglected. The finite roughness problem might be solved by direct numerical methods such as finite element (Hyun * et al.* 2004), or by an asperity model theory with the asperities defined at some finite scale. This provides a means to tighten the bounds in practical cases. It also demonstrates that the 'infinite tail' (e.g. the terms beyond some fairly large number in a Weierstrass series) of a theoretical fractal distribution have negigible effect on the electrical resistance. Put another way, the resistance is largely determined by the coarse scale properties of the rough surface, which supports more recent arguments about asperity models due to Greenwood & Wu (2001).

References

- J.R.Barber, Bounds on the electrical resistance between contacting elastic rough bodies,
*Proc.Roy.Soc. (London)*, Vol. A 459 (2003), pp. 53-66. - M.Ciavarella, G.Demelio, J.R.Barber and Yong Hoon Jang, Linear elastic contact of the Weierstrass profile,
*Proc.Roy.Soc. (London)*, Vol. A 456 (2000), pp. 387-405. - J.A.Greenwood and J.J.Wu, Surface roughness and contact: an apology,
*Meccanica*Vol.36 (2001), pp.617-630. - S. Hyun, L. Pei, J.-F. Molinari, and M. O. Robbins, Finite-element analysis of contact between elastic self-affine surfaces,
*Physical Review E*, Vol. 70, (2004), art. no. 026117.

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## Comments

## Asperities

Allow me to apologize in advance for asking such a basic question -- I am a senior in high school attempting to develop an understanding of electrical contacts. Down to the atomic scale, what is the reason for asperity formation? I understand their significance and by no means question their existence, but it seems that the nature of metallic bonding would not be conducive to such formations. What, then, is the mechanism for their existence? Thank you for any information.

## this problem in the meanwhile was solved by Persson in 2007.

See e.g. our paper on load-separation curves, which, by differentiation, lead to linearity between contact stiffness and load, and hence also conductance and load https://www.nature.com/articles/s41598-017-07234-4