The emergence of small-scale self-affine surface roughness from deformation

When solids are deformed plastically, they develop surface roughness because plastic flow is not laminar at small scales.


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Section S1. Atomic-scale deformation mechanisms Fig. S1. Detailed analysis of the surface topography of NiCoFeTi. Fig. S2. Detailed analysis of the surface topography of CuZr. Fig. S3. Temperature dependence of the Hurst exponent for CuZr. References (45,46) For the sake of illustration, we here discuss the crystalline systems. In our crystalline fcc systems, deformation occurs by slip on (111) planes. There are three (111) planes, all oriented at the tetrahedral angle (α t = 109.47) with respect to each other. A full dislocation that annihilates at the surface leaves behind a step of height ∆ ⊥ = a 0 / √ 3 where a 0 is the lattice constant of the crystal. During compression, the crystal will shrink by a distance ∆ = ∆/ tan(π − α t ) = a 0 / √ 24 for each surface step. Given a linear dimension L, compression by ε will hence give rise to N = Lε/∆ steps on the surface.
We now regard two limits of this process: In limit A, these steps occur at random positions on the surface. The surface profile then constitutes a random walk (and the Hurst exponent Section S1. Atomic-scale deformation mechanisms would be 0.5). Because the surface remains nominally flat, the walk needs to be self-returning.
This happens either due to lattice rotation or because the stress introduced at the surface when creating a single step makes it more likely that the next step is in the opposite direction (45) .
This self-returning process constitutes a Brownian bridge. Its root-mean square height scales A generalization of process A is the sum of N realizations of a random surface with a given Hurst exponent H. This leads to a progression of surfaces with h rms ∝ N 1/2 , independent of H. Process B is the sum of N identical realizations of a random surface which would trivially lead to h rms ∝ N .
We observe for the surface h rms ∝ ε 1/2 (process A) and for the bulku rms ∝ ε (process B) .
This appears to indicate that the surface and bulk processes discussed here are two limiting scenarios where the surface behaves close to the former and the bulk close to the latter. We believe the reason that the surface behaves differently lies in the fact that it moves perpendicular to the surface normal as the system is deformed. A progression of dislocations nucleating on identical slip systems and positions within the bulk therefore leave the surface at different locations. For CuZr, the process of accommodating deformation is different and this is manifested in smaller values for a h and a z , yet the same scaling with ε. Further work is necessary to quantify the exact nature of the processes described here and derive a model for amorphous materials.
(a) Root-mean-square height h rms as a function of magnification ζ showing self-affine scaling over more than one decade in length. The data collapse in the plastic regime when normalized by ε 1/2 , where epsilon is the strain due to compression. Panel (b) shows the underlying distribution function of heights h at different ε, which collapses upon rescaling heights h by ζ H /(a h ε 1/2 ) with a h = 9 nm and H = 0.77. (c) Root-mean-square amplitude u rms of the z-component of the subsurface displacement field u z as a function of ζ within the bulk. The displacement data collapses when normalized by ε. The bulk displacement field shows self-affine scaling over two decades in magnification. Panel (d) shows the underlying distribution function of the displacements u z , which collapses upon rescaling displacements u z by ζ H /(a z ε) with a z = 42 nm and H = 0.69. Solid and dashed lines in (a) and (c) show perfect self-affine scaling for reference with H = 0.5 and H = 1.0, respectively. The solid lines in panels (b) and (d) show the standard normal distribution.

Fig. S1. Detailed analysis of the surface topography of NiCoFeTi.
(a) Root-mean-square height h rms as a function of magnification ζ showing self-affine scaling over more than one decade in length. The data collapse in the plastic regime when normalized by ε 1/2 , where epsilon is the strain due to compression. Panel (b) shows the underlying distribution function of heights h at different ε, which collapses upon rescaling heights h by ζ H /(a h ε 1/2 ) with a h = 0.8 nm and H = 0.43. (c) Root-mean-square amplitude u rms of the z-component of the subsurface displacement field u z as a function of ζ within the bulk. The displacement data collapse when normalized by ε. The bulk displacement field shows self-affine scaling over two decades in magnification. Panel (d) shows the underlying distribution function of the displacements u z , which collapses upon rescaling displacements u z by ζ H /(a z ε) with a z = 1.6 nm and H = 0.16. Solid and dashed lines in (a) and (c) show perfect self-affine scaling for reference with H = 0.5 and H = 1.0, respectively. The solid lines in panels (b) and (d) show the standard normal distribution.

Fig. S2. Detailed analysis of the surface topography of CuZr.
The figure shows the evolution of the Hurst exponent of the surface (solid lines) and bulk (dashed lines) at the temperatures indicated as a function of applied strain ε. As the temperature approaches the glass transition temperature (around 800 K), the surface roughness and bulk deformation becomes uncorrelated as indicated by a vanishing Hurst exponent.