The Emergence of Small-scale Self-affine Roughness from Deformation

**Principal Investigator:**

Prof. Dr. Lars Pastewka

**Affiliation:**

IMTEK – Department of Microsystems Engineering, University of Freiburg

**Local Project ID:**

chka18

**HPC Platform used:**

JUQUEEN and JUWELS of JSC

**Date published:**

Rough surfaces are everywhere in our environment. While we may perceive some as smooth when we look at them or touch them, there is hardly any natural or artificial surface that is not rough at some scale. The size of features varies drastically, from micrometer-sized asperities and cracks on polished surfaces of household devices, through millimeter scale bumps and pores on construction materials, up to several hundred meters high crags and crevasses in mountain ranges on the surface of the earth, see Fig. 1.

A remarkable feature is that rough surfaces are often *self-affine*, meaning that if we zoom in on a part of the surface, we see similar features as on the full surface, but at a different scale [3] [4] [5] [6]. Let *h(x,y)* be the surface height *h* at a point described by the coordinates *x* and *y* in the plane. A common measure for the roughness of the surface is the *root mean square height* *h*_{rms}. To estimate *h*_{rms} , we could measure *h(x,y)* at *N* points *(**x*_{i}*,**y*_{i})*(i=1…N)* and calculate the root mean square of *h(**x*_{i,}*y*_{i}*)* ,

*h*_{rms} may depend on the side length *L* of the surface, so we write *h*_{rms}*=**h*_{rms}*(L)* . In the case of a self-affine surface, *h*_{rms} scales with *L* as a *power law*. Consider a square surface with side length *L* , e.g. Fig. 1(b). If we zoom in on a square part of the surface with side length *L/**ζ* (where *ζ* is the level of magnification) and measure *h*_{rms} of this part, then we find

where *H* is a constant independent of *ζ* . This number is called the *Hurst exponent*.

Self-affine scaling is observed from the microscopic to the geological scale [7] [8]. How does it emerge? One plausible origin of self-affine roughness is plastic (irreversible) deformation during formation and processing of the body. Necessarily, the surface is deformed, too, and therefore stores the spatial signature of the microscopic mechanisms of deformation. It could then be spatial correlations in these deformation mechanisms which lead to a self-affine topography.

One of the goals of project chka18 was to investigate the connection between plastic deformation and roughening. Computer simulations were carried out to determine whether plastic deformation alone could cause self-affine roughening of a body with an initially flat surface. The Molecular Dynamics (MD) simulation technique was used for this purpose. In MD, all atoms in the body are simulated. Quantum mechanical effects are neglected. Instead, the motion of the atoms is assumed to follow classical mechanics, like the motion of Billiard balls. This approximation allows to simulate systems with millions of atoms. Indeed, such a large number of atoms was required to analyze the variation of *h*_{rms}*(L/ζ)* over at least one order of magnitude in *ζ* .

The starting configurations for MD simulations were cubes with side length of *L**=100 *nm , see Fig. 2. Depending on the material (more on that later), the cubes contained 58 to 83 Mio. atoms. The surfaces along the *z* -direction were initially atomically flat. Furthermore, periodic boundary conditions were applied in the *x* - and *y* -directions (normal to the surface), meaning that atoms with coordinates *x≈0* and *x≈L* , for example, interact with each other. Hence the surfaces of the cube in *x* - and *y* -directions do not behave like free surfaces, but like sections in the interior of the body. Effectively, an infinitely wide plate is simulated. The cubes were then compressed simultaneously in *x* - and *y* -direction until the side length in these directions had shrunk by 30%.

Such compression tests were carried out with three different materials: (i) a perfect Gold crystal, where all atoms are arranged on a lattice; (ii) a complex alloy composed of 36.37% Nickel, 36% Cobalt, 16.67% Iron, and 36.67 % Titanium (in percent of the total number of atoms), where the atoms occupy a lattice but the distribution of elements is random; and (iii) a “metallic glass” composed of 50% Copper and Zirconium atoms, respectively, where there is no long-range order of the atoms at all. Fig. 3 illustrates the atomic arrangement in the three materials.

Fig. 4 shows the topography after 20% compression. The materials roughened to a different degree and different patterns have formed. In the case of the crystals Au and Ni_{36.67}Co_{30}Fe_{16.67}Ti_{16.67} triangular structures can be identified. They were formed by the intersection of lattice dislocations, which are line-like imperfections in the crystal lattice that cause plastic deformation.

The scaling of *h*_{rms}*(L/ζ)* with *ζ* was examined to determine whether the surfaces are self-affine. *h*_{rms}*(L/ζ)* was first calculated for the whole surface (*ζ=1* ). Then the surface was subdivided into 4 segments (*ζ=2* ), and *h*_{rms} was calculated in each sub-segment. This procedure was repeated for higher magnifications *ζ=4…128* . In this way, curves of *h*_{rms} versus *ζ* were obtained, and it could be determined that *h*_{rms} is indeed self-affine over roughly one order of magnitude in *ζ* .

A similar scaling analysis was performed for atomic displacements in the interior of the configuration. It was shown that these bulk displacements are self-affine as well. The simulations thus indicate that self-affine roughness may emerge naturally from plastic deformation even on initially flat surfaces. Further details of these simulations, such as the evolution of *h*_{rms} with deformation and the effect of temperature, can be found in a recent publication:

*Hinkle, A. R., Nöhring, W. G., Leute, R., Junge, T. & Pastewka, L. The emergence of small-scale self-affine surface roughness from deformation. Science Advances 6, eaax0847 (2020) *https://doi.org/10.1126/sciadv.aax0847

**Who might be interested in these results? **The consequences of surface roughness are felt in many ways, even though the connection between a specific effect and the roughness properties of the surface is not always intuitive. For example, surface roughness determines the contact area between surfaces, and therefore surface forces, which in turn affects contact stiffness [10] [11], adhesion [12], and friction [13]. Moreover, roughness affects electrical and thermal transport across interfaces [14] [15]. Besides addressing a topic of general scientific interest, the present study may therefore be useful for other researchers investigating surface-controlled phenomena.

Gola, A. & Pastewka, L. Scratching Cu|Au Nanolaminates. Lubricants7, 44(2019). https://doi.org/10.3390/lubricants7050044

Gola, A., Zhang, G.-P., Pastewka, L. & Schwaiger, R. Surface flaws control strain localization in the deformation of Cu|Au nanolaminate pillars. MRS Communications 9, 1067–1071 (2019). https://doi.org/10.1557/mrc.2019.93

Gola, A., Schwaiger, R., Gumbsch, P. & Pastewka, L. Pattern formation during deformation of metallic nanolaminates. Physical Review Materials 4, 013603(2020) https://doi.org/10.1103/PhysRevMaterials.4.013603

- Topography data from the NASA Shuttle Radar Topography Mission, https://dds.cr.usgs.gov/srtm/version2_1/SRTM3/.
- Topography data from T. Jacobs, private communication.
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- Gujrati, A., Khanal, S. R., Pastewka, L. & Jacobs, T. D. B. Combining TEM, AFM, and profilometry for quantitative topography characterization across all scales. ACS Applied Materials & Interfaces 10, 29169–29178 (2018).
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Adam Hinkle^{1,2}, Wolfram Nöhring^{2}, Richard Leute^{2}, Till Junge^{3}, Jan Grießer^{2}, Adrien Gola^{4}

^{1} Material, Physical and Chemical Sciences Center, Sandia National Laboratories, Albuquerque, USA^{2} IMTEK - Department of Microsystems Engineering, University of Freiburg^{3} École Polytechnique Fédérale de Lausanne, Switzerland^{4} Institute for Applied Materials, Karlsruhe Institute of Technology

**Scientific Contact**

Prof. Dr. Lars Pastewka

IMTEK - Department of Microsystems Engineering

University of Freiburg

Laboratory for Simulation

Georges-Koehler-Allee 103, D-79110 Freiburg (Germany)

e-mail: lars.pastewka [@] imtek.uni-freiburg.de

*JSC project ID: chka18*

*June 2020*