Self-affine Roughness Research Articles

Theoretical study on contact current density in the boundary of random rough surfaces

We theoretically calculate here how random roughness affects the contact current density and equivalently contact resistance by utilizing a perturbative expansion method. Then, By taking into consideration self-affine roughness on the surface of the solids, we find the contact current density. The obtained results illustrate the total contact current exhibits significant dependence on the properties of the self-affine rough surface such as root-mean-square of roughness and Hurst exponent. Assuming a lower value of root-mean-square roughness for a solid in a higher potential compared to another which is connected to a lower potential, results in a decrement of the total contact current compared to the current between solids without roughness. By considering similar rough parameters for both solids, the contribution of the perturbed contact current becomes zero. Moreover, repercussions by considering the conductivity contrast between components are explored.

Modeling the line contact on an elastic half-space with the statistical approach: Self-affine fractal roughness and numerical framework

This paper presents a line contact model for describing the normal contact between rough surfaces with self-affine fractal roughness based on the statistical approach. An iterative algorithm combining the Levenberg-Marquardt (LM) method with penalty function is proposed to overcome the limitation of restricted indentation depth. Results show a competitive relationship between the plasticity index and the fractal dimension on the real contact area. Besides, nonlinear characteristics between force and area appear when the elastic modulus of the substrate is larger than that of the asperity. Furthermore, the wavy machining texture leads to a great increase in the maximum pressure. The current work can be used for modeling the transmission interfaces involving line contact, such as gears and cylindrical rotor bearings.

A permeability model for the fractal tree-like fracture network with self-affine surface roughness in shale gas reservoirs

The complex natural fracture network with self-affine rough surface and branching characteristics significantly impacts the gas transport in shale gas reservoirs. However, its effects on the permeability have not been studied so far. This study proposes an analytical permeability model for the fractal tree-like fracture network with self-affine surface roughness and branching characteristics. Firstly, the self-affine rough profiles of fracture surface are generated at different fractal dimensions by the Weierstrass–Mandelbrot function and a rough fractal tree-like fracture network is constructed with these surface profiles and branching characteristics. Then, an analytical permeability model is proposed to consider the effects of fracture surface roughness and tree-like branching characteristics on gas flow. This analytical model is verified by numerical simulations. Finally, the velocity distribution of the fracture network and the sensitivity of its structure parameters are analyzed. It is found that eddy flow is more easily formed on rougher fracture surfaces with larger fractal dimension when their fracture aperture is at millimeter scale. The eddy flow disappears when the fracture aperture is at micron scale. Bigger gas flow resistance and more energy loss are observed for smaller fracture aperture and rougher fracture surface. The gas velocity in rough fractures decreases by 60% at micron scale, but decreases by 50% at millimeter scale. Gas flow resistance also increases with the increase of branch angle, branch level and length ratio, but decreases with aperture ratio. As a result, permeability decreases with fractal dimension, branch angle, branch level and length ratio, but increases with aperture ratio.

Developmental framework of line edge roughness reference standards for next-generation functional micro-/nanostructures

The edge of a micro-/nanoscale line pattern fabricated by lithography is not a straight line as in the design but has a roughness. This is called the line edge roughness (LER) and is important dimensional information related to the device performance, and reference standards containing known LER values are needed for fundamental research on lithography-related technology. We propose a novel framework for developing LER reference standards that contain the necessary characteristics. The proposed framework has three main features: (i) a self-affine fractal roughness model for the development of next-generation devices; (ii) a reliable evaluation method based on LER reference metrology using SI-traceable atomic force microscopy (AFM); and (iii) a versatile evaluation method using scanning electron microscopy-based LER metrology verified by the previous AFM-based LER reference metrology. In an experiment to verify the proposed framework, we fabricated an LER reference standard by using electron-beam lithography. The results show that the proposed framework can be used to develop LER reference standards for next-generation functional micro-/nanostructures.

On the contact between elasto-plastic media with self-affine fractal roughness

Modeling the elasto-plastic contact between rough interfaces may require high computational effort as real surfaces present broad roughness spectra. In this work, we propose an efficient multi-asperity model where each asperity follows Jackson and Green’s equations and both coupling and coalescence of contact spots are considered. In agreement with previous studies, the contact area A is found to rise linearly with the applied load. Moreover, under the assumption of yield stress independent of the asperity size, no differences are found with respect to the elastic contact solution for nanometric root mean square (rms) roughness amplitudes hrms. On the contrary, for micrometric hrms, the slope of the area-load relation is observed to increase when reducing the yield strength σY. We have also investigated the effect of increasing the rms roughness gradient of the surface hrms′ by adding fine-scale wavelengths to the roughness spectrum. Due to plastic deformations, the contact area A is found to be independent of the high-frequency cut-off of the roughness spectrum as it converges when increasing the number of roughness scales.

A closer look into slickensides: Deformation on and under fault surfaces

Accurate descriptions of natural fault surfaces and associated fault rocks are important for understanding fault zone processes and properties. Slickensides--grooved polished surfaces that record displacement and wear along faults-- develop measurable roughness and characteristic microstructures during fault slip. We quantify the roughness of natural slickensides from three different fault surfaces by calculating the surfaces power spectra and height distributions and analyze the microstructures formed below the slickensides. Slickenside surfaces exhibit anisotropic self-affine roughness with corresponding mean Hurst exponents in directions parallel-- 0.53 ± 0.07-- and perpendicular --0.6 ± 0.1-- to slip, consistent with reports from other fault surfaces. Additionally, surfaces exhibit non-Gaussian height distributions, with their skewness and kurtosis roughness parameters having noticeable dependence on the scale of observation. Below the surface, microstructural analyses reveal that S–C–C′ fabrics develop adjacent to a C-plane-parallel principal slip zone characterized by a sharp decrease in clast size and a thin (≤100 μm) nanoparticulate-rich principal slip surface (PSS). These microstructures are present in most analyzed samples suggesting they commonly form during slickenside development regardless of lithology or tectonic setting. Our results suggests that 1) PSS likely arise by progressive localization along weaker oriented fabrics 2) deformation along PSS's is energetic enough to comminute the rocks into nanometric grains, and 3) fault geometry can be further characterized by studying the height distributions of fault surfaces, which are likely to impact stress distributions and frictional responses along faults.

A novel analytic model for sealing performance of static metallic joint considering the yield hardening effect

The static metallic joints’ sealing performance is deeply impacted by the plastic deformation and the interfacial separation of the contact surfaces with self-affine roughness. The yield hardening effect, unavoidable at the contact interface, is of vital importance to the plastic deformation and the distribution of the interfacial separation. However, most of the previous research ignores the effect of yield hardening, assuming that the contact surfaces are elastic-perfectly plastic. To address the problem, a novel analytic model for investigating the sealing performance under the effect of yield hardening has been developed in this paper. The upper boundary of contact stress in the conventional Persson model has been enlarged from a constant nominal yield stress to a maximum yield stress that varies with contact strain, making the model adaptable to actual scenarios with yield hardening effects. Utilizing the measured data of contact surfaces as input, the corresponding leakage rates are calculated. Besides, the contact stress distribution as well as the real contact area at the interface is also discussed. The sealing experiments are carried out accordingly, verifying that the proposed model owns the ability to predict the leakage rate under the effect of yield hardening.

Onset of turbulence in channel flows with scale-invariant roughness

Using 3D direct numerical simulations of the Navier--Stokes equations, we study the effect of a self-affine wall roughness on the onset of turbulence in channel flow. We quantify the dependence of the turbulent intensity (proportional to the mean-squared velocity fluctuations) on the Reynolds number $\mathrm{Re}$ for different roughness amplitudes $A$. We find that for sufficiently high amplitudes, $A>{A}_{b}$, the transition changes its nature from being subcritical (as is known at $A=0$) to supercritical, i.e., the boundary roughness renders the flow unstable for $\mathrm{Re}>{\mathrm{Re}}_{l}$, where the critical ${\mathrm{Re}}_{l}$ decays nontrivially with increasing $A$. The dependence of the friction factor on $\mathrm{Re}$ is found to follow a generalized Forchheimer law, which interpolates between the laminar and inertial asymptotes. The transition between these two asymptotes occurs at a second critical ${\mathrm{Re}}_{\text{c}}$ which is comparable in magnitude to ${\mathrm{Re}}_{l}$. This implies that transitional flow is an integral part of flow in open fractures when $\mathrm{Re}$ is sufficiently high, and should be accounted for in effective modeling approaches.

Magnetoimpedance of a ferromagnetic thin film in the presence of isotropic self-affine surface roughness cross correlations

Magnetoimpedance of a ferromagnetic thin film in the presence of isotropic self-affine surface roughness cross correlations

Influence of humidity on the binding of stone fragments via capillary bridges

We study the humidity dependency of the adhesion (or pull-off) force between granite fragments and a silica glass plate. The particles bind to the glass plate via capillary bridges. The granite particles are produced by cracking a granite stone in a mortar and have self-affine fractal surface roughness. Theory shows that the surface roughness results in an interaction force between stone fragments and the glass plate which is independent of the size of the particles, in contrast to the linear size dependency expected for particles with smooth surfaces. We measure the adhesion force by depositing the granite particle powder, with particle sizes ranging from mm to μm (or less), on the glass plate. By turning the glass plate upside-down all particles with a gravitational force larger than the adhesion force will fall off the glass plate. By studying the size (and hence the mass) of biggest still attached particles we obtained the adhesion force, which is found to be in good agreement with the theory prediction.

Fractal geometry of contacting patches in rough elastic contacts

Many naturally formed and processed surfaces are rough over a broad range of length scales. Surface roughness reduces the area of contact between solids, with ramifications for phenomena that depend on the geometry of the interface and the amount of direct contact, including friction and adhesion. In this work, we employ large-scale boundary-element simulations for nonadhesive, elastic solids to study the size dependence of contact patch mean pressure and geometry for patches formed between solids with self-affine fractal surface roughness across seven decades in patch area. Contact patches with diameters smaller than a crossover length scale of order the minimum wavelength of roughness are generally compact with simple geometries and bear pressures well described by Hertz theory. The patch pressure in contact patches larger than the crossover scale rises logarithmically before saturating at a finite value. Furthermore, the largest contact patches formed during our simulations are ramified and populated with regions out of contact, or bubbles, which reduce patch area and increase patch perimeter. As a result, we show that the mean contact diameter of the largest patches saturates, indicating that the patch contact area is proportional to the total patch perimeter. We quantify the effects of bubbles on patch area and perimeters as a function of Hurst exponent and contrast our findings with results of comparable bearing-area model calculations. The slow evolution of the mean patch pressure with patch size in our large-scale calculations explains the common observation that the global mean contact pressure depends on the structure of the roughness, the contact area, and even on system size.

Comprehensive topography characterization of polycrystalline diamond coatings

The surface topography of diamond coatings strongly affects surface properties such as adhesion, friction, wear, and biocompatibility. However, the understanding of multi-scale topography, and its effect on properties, has been hindered by conventional measurement methods, which capture only a single length scale. Here, four different polycrystalline diamond coatings are characterized using transmission electron microscopy to assess the roughness down to the sub-nanometer scale. Then these measurements are combined, using the power spectral density (PSD), with conventional methods (stylus profilometry and atomic force microscopy) to characterize all scales of topography. The results demonstrate the critical importance of measuring topography across all length scales, especially because their PSDs cross over one another, such that a surface that is rougher at a larger scale may be smoother at a smaller scale and vice versa. Furthermore, these measurements reveal the connection between multi-scale topography and grain size, with characteristic scaling behavior at and slightly below the mean grain size, and self-affine fractal-like roughness at other length scales. At small (subgrain) scales, unpolished surfaces exhibit a common form of residual roughness that is self-affine in nature but difficult to detect with conventional methods. This approach of capturing topography from the atomic- to the macro-scale is termed comprehensive topography characterization, and all of the topography data from these surfaces has been made available for further analysis by experimentalists and theoreticians. Scientifically, this investigation has identified four characteristic regions of topography scaling in polycrystalline diamond materials.

The hydraulic efficiency of single fractures: correcting the cubic law parameterization for self-affine surface roughness and fracture closure

Abstract. Quantifying the hydraulic properties of single fractures is a fundamental requirement to understand fluid flow in fractured reservoirs. For an ideal planar fracture, the effective flow is proportional to the cube of the fracture aperture. In contrast, real fractures are rarely planar, and correcting the cubic law in terms of fracture roughness has therefore been a subject of numerous studies in the past. Several empirical relationships between hydraulic and mechanical aperture have been proposed based on statistical variations of the aperture field. However, often, they exhibit non-unique solutions, attributed to the geometrical variety of naturally occurring fractures. In this study, a non-dimensional fracture roughness quantification scheme is acquired, opposing effective surface area against relative fracture closure. This is used to capture deviations from the cubic law as a function of quantified fracture roughness, here termed hydraulic efficiencies. For that, we combine existing methods to generate synthetic 3-D fracture voxel models. Each fracture consists of two random, 25 cm2 wide self-affine surfaces with prescribed roughness amplitude, scaling exponent, and correlation length, which are separated by varying distances to form fracture configurations that are broadly spread in the newly formed two-parameter space (mean apertures in submillimeter range). First, we performed a percolation analysis on 600 000 synthetic fractures to narrow down the parameter space on which to conduct fluid flow simulations. This revealed that the fractional amount of contact and the percolation probability solely depend on the relative fracture closure. Next, Stokes flow calculations are performed, using a 3-D finite differences code on 6400 fracture models to compute directional permeabilities. The deviations from the cubic law prediction and their statistical variability for equal roughness configurations were quantified. The resulting 2-D solution fields reveal decreasing cubic law accordance down to 1 % for extreme roughness configurations. We show that the non-uniqueness of the results significantly reduces if the correlation length of the aperture field is much smaller than the spatial extent of the fracture. An equation was provided that predicts the average behavior of hydraulic efficiencies and respective fracture permeabilities as a function of their statistical properties. A model to capture fluctuations around that average behavior with respect to their correlation lengths has been proposed. Numerical inaccuracies were quantified with a resolution test, revealing an error of 7 %. By this, we propose a revised parameterization for the permeability of rough single fractures, which takes numerical inaccuracies of the flow calculations into account. We show that this approach is more accurate compared to existing formulations. It can be employed to estimate the permeability of fractures if a measure of fracture roughness is available, and it can readily be incorporated in discrete fracture network modeling approaches.

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

Most natural and man-made surfaces appear to be rough on many length scales. There is presently no unifying theory of the origin of roughness or the self-affine nature of surface topography. One likely contributor to the formation of roughness is deformation, which underlies many processes that shape surfaces such as machining, fracture, and wear. Using molecular dynamics, we simulate the biaxial compression of single-crystal Au, the high-entropy alloy Ni36.67Co30Fe16.67Ti16.67, and amorphous Cu50Zr50 and show that even surfaces of homogeneous materials develop a self-affine structure. By characterizing subsurface deformation, we connect the self-affinity of the surface to the spatial correlation of deformation events occurring within the bulk and present scaling relations for the evolution of roughness with strain. These results open routes toward interpreting and engineering roughness profiles.

Hydraulic resistance in open-channel flows over self-affine rough beds

ABSTRACTKnowledge of hydraulic resistance of single-valued self-affine fractal surfaces remains very limited. To advance this area, a set of experiments have been conducted in two separate open-channel flumes to investigate the effects of the spectral structure of bed roughness on the drag at the bed. Three self-affine fractal roughness patterns, based on a simple but realistic three-range spectral model, have been investigated with spectral scaling exponents of −1, −5/3 and −3, respectively. The different widths of the flumes and a range of flow depths also afforded an opportunity to consider effects of the flow aspect ratio and relative submergence. The results show that with all else equal the friction factor increases as the spectral exponent decreases. In addition, the relationship between the spectral exponent and effective slope of the roughness is demonstrated, for the first time. Aspect ratio effects on the friction factor within the studied range were found to be negligible.

Wave scattering from two-dimensional self-affine Dirichlet and Neumann surfaces and its application to the retrieval of self-affine parameters

Wave scattering from two-dimensional self-affine Dirichlet and Neumann surfaces is studied for the purpose of using the intensity scattered from them to obtain the Hurst exponent and topothesy that characterize the self-affine roughness. By the use of the Kirchhoff approximation a closed form mathematical expression for the angular dependence of the mean differential reflection coefficient is derived under the assumption that the surface is illuminated by a plane incident wave. It is shown that this quantity can be expressed in terms of the isotropic, bivariate ($\alpha$-stable) L\'evy distribution of a stability parameter that is two times the Hurst exponent of the underlying surface. Features of the expression for the mean differential reflection coefficient are discussed, and its predictions compare favorably over large regions of parameter space to results obtained from rigorous computer simulations based on equations of scattering theory. It is demonstrated how the Hurst exponent and the topothesy of the self-affine surface can be inferred from scattering data it produces. Finally several possible scattering configurations are discussed that allow for an efficient extraction of these self-affine parameters.

Theory of generalized Gerischer impedance for quasi-reversible charge transfer at rough and finite fractal electrodes

A theoretical model for the generalized Gerischer impedance at an irregular interface under quasi-reversible heterogeneous charge transfer reaction has been developed. The dynamic impedance response is predicted to depend on three phenomenological components, viz. diffusion, charge transfer reaction and bulk homogeneous kinetics as well as roughness. The theory has been developed for both deterministic and stochastic electrode roughness. Statistical properties of the stochastic electrode roughness are characterized through their power spectral density. The impedance response is influenced by the finite self-affine fractal roughness which is analyzed in detail. Three governing phenomenological length scales are diffusion length, charge transfer length and reaction layer thickness. The anomalous impedance response of the finite fractal electrode is due to the synergistic effect of the phenomenological lengths and finite-fractal morphological parameters, namely, fractal dimension (DH), lower cut-off length (ℓ) and topothesy length (ℓτ). The impedance response has three distinct characteristic frequency regimes: the low frequency bulk kinetics controlled regime, roughness dependent intermediate frequency region and heterogeneous charge transfer controlled high frequency regime. A contrasting diffusive behavior is predicted, wherein the system switches from sub-diffusive to super-diffusive response with increase in electrode roughness. This theory unravels the influence of both bulk and interfacial kinetics on the impedance response of a rough electrode.

A Generalized Johnson Parameter for Pull-Off Decay in the Adhesion of Rough Surfaces

There is no simple theory at present to predict accurately the decay of pull-off in the adhesion of randomly rough surfaces. The asperity model of Fuller and Tabor has shown significant error in recent numerical investigations by Pastewka and Robbins of self-affine random roughness from micrometer to atomic scale which corresponds to low values of Tabor parameter. For sinusoidal contact, the Johnson parameter, originally introduced for the JKR regime (from Johnson-Kendall-Roberts) is the dominant parameter ruling the pull-off at intermediate Tabor values. Hence, we define a generalized Johnson parameter as the ratio between the adhesive energy to the elastic strain energy to flatten the surface in the case of multiscale roughness and find that it correlates very well with the data of Pastewka and Robbins spanning almost five orders of magnitude of reduction from theoretical strength, improving significantly with respect to other possible single parameter criteria. For the most important case in practice, that of low fractal dimensions, this suggests the product of amplitude and slope of the largest wavelength components of roughness dominate pull-off decay, and not small scales features like slopes and curvatures, as suggested by Pastewka and Robbins.

Self-affine subglacial roughness: consequences for radar scattering and basal water discrimination in northern Greenland

Abstract. Subglacial roughness can be determined at a variety of length scales from radio-echo sounding (RES) data either via statistical analysis of topography or inferred from basal radar scattering. Past studies have demonstrated that subglacial terrain exhibits self-affine (power law) roughness scaling behaviour, but existing radar scattering models do not take this into account. Here, using RES data from northern Greenland, we introduce a self-affine statistical framework that enables a consistent integration of topographic-scale roughness with the electromagnetic theory of radar scattering. We demonstrate that the degree of radar scattering, quantified using the waveform abruptness (pulse peakiness), is topographically controlled by the Hurst (roughness power law) exponent. Notably, specular bed reflections are associated with a lower Hurst exponent, with diffuse scattering associated with a higher Hurst exponent. Abrupt waveforms (specular reflections) have previously been used as a RES diagnostic for basal water, and to test this assumption we compare our radar scattering map with a recent prediction for the basal thermal state. We demonstrate that the majority of thawed regions (above pressure melting point) exhibit a diffuse scattering signature, which is in contradiction to the prior approach. Self-affine statistics provide a generalised model for subglacial terrain and can improve our understanding of the relationship between basal properties and ice-sheet dynamics.

Leakage of Metallic Seals: Role of Plastic Deformations

I study the influence of plastic deformations on the leakage of metallic seals. The solids are assumed to deform elastically as long as the stress is below the penetration hardness of the material, and yield plastically when the local stress reaches the penetration hardness. The metal surfaces are assumed to exhibit self-affine fractal surface roughness. We use the Persson contact mechanics theory and the Bruggeman effective medium theory to estimate the leak rate of the plastically deformed interface.