Self-affine Surfaces Research Articles

Coastlines violate the Schramm–Loewner Evolution

Mandelbrot’s empirical observation that the coast of Britain is fractal has been confirmed by many authors, but it can be described by the Schramm–Loewner Evolution? Since the self-affine surface of our planet has a positive Hurst exponent, one would not expect a priori any critical behavior. Here, we investigate numerically the roughness and fractal dimension of the isoheight lines of real and artificial landscapes. Using a novel algorithm to take into account overhangs, we find that the roughness exponent of isoheight lines is consistent with unity regardless of the Hurst exponent of the rough surface. Moreover, the effective fractal dimension of the iso-height lines decays linearly with the Hurst exponent of the surface. We perform several tests to verify if the complete and accessible perimeters would follow the Schramm–Loewner Evolution and find that the left passage probability test is clearly violated, implying that coastlines violate SLE.

Impact of fractal dimension and lateral correlation length on surface plasmon resonance activity in sputtered silver layers

Fractal and optical characteristics of self-affine surfaces of silver(Ag) thin films deposited through direct current (dc) magnetron sputtering as a function of thickness are investigated and explored here. The surface morphology of Ag thin films is characterized by field emission electron microscopy, and atomic force microscopy technique. The cube counting algorithm is used to extract the fractal dimension of Ag thin film. The surface roughness (interface width) shows monotonic increases with film thickness, while the other parameters, such as lateral correlation length, roughness exponent, and fractal dimension exhibit linear variation with thickness. Our findings reveal distinctive scaling behaviors, with scaling exponents α, β, and 1/z indicating unique growth characteristics. The interface width w increases as a power law of thickness t, w(t)∝tβ, with β=0.39± 0.007, and the lateral correlation length ξ grows as ξt∝t1/z with 1/z=0.14± 0.002. The roughness exponent extracted from height-height correlation analysis is α=0.61–0.41. The self-affine nature of the Ag thin films is further confirmed by the autocorrelation function. X-ray photoelectron spectroscopy (XPS) is used to the confirm the growth of Ag thin film. Additionally, we have studied the role of fractal dimensions and lateral correlation length (ξ) on the surface plasmon resonance (SPR) of Ag thin film. Our results indicate a red-shifting behavior of SPR with increasing interface width (w), lateral correlation length (ξ), and fractal dimensions. This study suggests the significance of not only the roughness exponent and fractal dimension but also the local surface slope in SPR activity in Ag thin films.

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.

Numerical Evaluation of Planetary Radar Backscatter Models for Self-Affine Fractal Surfaces

Numerous analytical radar-scattering laws have been published through the past decades to interpret planetary radar observations, such as Hagfors’ law, which has been commonly used for the Moon, and the cosine law, which is commonly used in the shape modeling of asteroids. Many of the laws have not been numerically validated in terms of their interpretation and limitations. This paper evaluates radar-scattering laws for self-affine fractal surfaces using a numerical approach. Traditionally, the autocorrelation function and, more recently, the Hurst exponent, which describes the self-affinity, have been used to quantify the height correlation. Here, hundreds of three-dimensional synthetic surfaces parameterized using a root-mean-square (rms) height and a Hurst exponent were generated, and their backscattering coefficient functions were computed to evaluate their consistency with selected analytical models. The numerical results were also compared to empirical models for roughness and radar-scattering measurements of Hawaii lava flows and found consistent. The Gaussian law performed best at predicting the rms slope regardless of the Hurst exponent. Consistent with the literature, it was found to be the most reliable radar-scattering law for the inverse modeling of the rms slopes and the Fresnel reflection coefficient from the quasi-specular backscattering peak, when homogeneous statistical properties and a ray-optics approach can be assumed. The contribution of multiple scattering in the backscattered power increases as a function of rms slope up to about 20% of the backscattered power at normal incidence when the rms slope angle is 46°.

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.

Power spectral density analysis to investigate morphology of Ni films

Ni thin films grown by thermal evaporation and sputtering under different deposition conditions are characterised by structural and morphological properties using X-ray diffraction (XRD) and atomic force microscopy (AFM) techniques. XRD results suggested the growth of polycrystalline face-centred cubic Ni phase for all the samples. Morphological characteristics of the films were compared by analysing AFM data for root mean square roughness, height–height correlation function and power spectral density (PSD) measurements. Applying fractal and k-correlation fitting models to the PSD data, different morphological parameters are quantified. The study suggested that Ni films grown at higher substrate temperature (∼150 °C) by thermal evaporation and at low Ar pressure (∼0.2 Pa) by sputtering techniques yielded films of small surface roughness with a Brownian fractal self-affine surface.

A story of two transitions: From adhesive to abrasive wear and from ductile to brittle regime.

Atomistic simulations performed with a family of model potential with tunable hardness have proven to be a great tool for advancing the understanding of wear processes at the asperity level. They have been instrumental in finding a critical length scale, which governs the ductile to brittle transition in adhesive wear, and further helped in the understanding of the relation between tangential work and wear rate or how self-affine surfaces emerge in three-body wear. However, so far, the studies were mostly limited to adhesive wear processes where the two surfaces in contact are composed of the same material. Here, we propose to study the transition from adhesive to abrasive wear by introducing a contrast of hardness between the contacting surfaces. Two wear processes emerge: one by gradual accretion of the third body by detachment of chips from both surfaces and the other being a more erratic mixed process involving large deformation of the third body and removal of large pieces from the soft surface. The critical length scale was found to be a good predictor of the ductile to brittle transition between both processes. Furthermore, the wear coefficients and wear ratios of soft and hard surfaces were found to be consistent with experimental observations. The wear particle is composed of many concentric layers, an onion-like structure, resulting from the gradual accretion of matter from both surfaces. The distribution of sizes of these layers was studied, and it appears that the cumulative distribution of hard surface's chip sizes follows a power law.

Spectral wear modelling of rubber friction on a hard substrate with large surface roughness

Soft-hard matter friction is a long-standing tribology problem that remains unclarified, requiring engineers to empirically predict the wear life. To clarify this issue, this study examines the transient running-in regime of rubber friction on a hard rough substrate and models the temporal wear progression using the spectrum curves of surface roughness for both materials. Performing a series of friction tests and three-dimensional surface-height measurements, the time-dependent behaviours of the power spectral densities (PSDs) are divided into two phases, namely the initial non-steady and long-term steady phases. The detailed spectral analyses of worn rubber surfaces in the initial phase lead to a blended PSD function between self-affine and K -correlation surface models, consisting of one variable (the Hurst exponent) that is saturated by the substrate self-affinity. Supported by the Greenwood–Williamson theory concerning rough contact mechanics, the volumetric estimate with the blended PSD function is used to assess the volume rate of wear debris in the steady phase, which is validated experimentally. These findings not only improve the wear predictions of soft materials from the initial measurements of worn surfaces but also help clarify the constrained multiscale mechanism of wear.

Drying behaviour of nanofluid sessile droplets on self-affine vis-à-vis corrugated nanorough surfaces.

In recent years, evaporative self-assembly of sessile droplets has gained considerable attention owing to its wide applicability in many areas. While the phenomenon is well studied for smooth and isotropically rough (self-affine) surfaces, investigations comparing the outcomes on self-affine vis-à-vis corrugated surfaces remains to be done. In this experimental work, we compare the wetting and evaporation dynamics of nano-colloidal microlitre droplets on self-affine and corrugated nanorough surfaces having identical roughnesses and interface properties. The coupled influence of particle size, concentration, and surface structuring has been explored. Differences in wettability and evaporation dynamics are observed, which are explained via the interaction between wetting fluid and anisotropic surface roughness. Our findings exhibit different temporal behaviour of contact radius and angle in the evaporation process of the droplets. Further, the corrugated surface exhibits anisotropic wettability with a monotonic change in droplet shape as evaporation proceeds, finally giving rise to irregular dried patterns. The scaled rim width and crack spacing of the particulate deposits are examined. Our results can inspire fabrication of surfaces that can facilitate direction-dependent droplet motion for specific applications.

Pyrough: A tool to build 3D samples with rough surfaces for atomistic and finite-element simulations

Natural samples are characterized by surface roughness which is intrinsically multi-scale as depicted by the well known concept of fractal dimension. Nevertheless, surface asperities are barely taken into account in simulations and modeling where flat surfaces and sharp corners or edges are generally preferred for the sake of simplicity. In this context, we propose here a versatile Python program called Pyrough that aims at building virtual samples characterized by configurable surface roughness for numerical applications such as atomistic and finite-element simulations. The program is open source and relies on the classical roughness theory that integrates the concept of self-affine surface. Several basic shapes including basic blocks, spheres, grains and wires with self-affine surface asperities are implemented and the object-oriented structure of the program simplifies the implementation of more complex objects. Virtual sample design is improved using Pyrough, which enables more realistic simulations to be made. Several application examples including e.g., the design of wavy grain boundaries or nanoindentation testing using a roughened indenter tip are presented. Program summaryProgram Title: PyroughCPC Library link to program files:https://doi.org/10.17632/x7jdtrbf4s.1Developer's repository link:https://github.com/jamodeo12/PyroughLicensing provisions: GNU General Public License 3Programming language: PythonExternal routines/libraries: Gmsh, Meshio, Wulffpack, ASE, Atomsk, cv2Nature of problem: 3D virtual samples used for atomistic or finite-element simulations generally rely on simplified geometries and surfaces for the sake of design simplicity. However, the influence of surface roughness play a crucial role in various fields of applications (e.g., mechanics, catalysis, lubrication) and must be taken into account.Solution method: Pyrough allows for the design of 3D virtual objects with rough surfaces by means of the classical roughness theory. The user can easily tune the morphology of surfaces and shape 3D objects. Output samples can be used in finite-element or atomistic simulations according to the user's needs.Additional comments including restrictions and unusual features: The program documentation is available at https://jamodeo12.github.io/Pyrough/

Accessibility of the surface fractal dimension during film growth.

Fractal properties on self-affine surfaces of films growing under nonequilibrium conditions are important in understanding the corresponding universality class. However, measurement of the surface fractal dimension has been intensively investigated and is still very problematic. In this work, we report the behavior of the effective fractal dimension in the context of film growth involving lattice models believed to belong to the Kardar-Parisi-Zhang (KPZ) universality class. Our results, which are presented for growth in a d-dimensional substrate (d=1,2) and use the three-point sinuosity (TPS) method, show universal scaling of the measure M, which is defined in terms of discretization of the Laplacian operator applied to the height of the film surface, M=t^{δ}g[Θ], where t is the time, g[Θ] is a scale function, δ=2β,Θ≡τt^{-1/z},β, and z are the KPZ growth and dynamical exponents, respectively, and τ is a spatial scale length used to compute M. Importantly, we show that the effective fractal dimensions are consistent with the expected KPZ dimensions for d=1,2, if Θ≲0.3, which include a thin film regime for the extraction of the fractal dimension. This establishes the scale limits in which the TPS method can be used to accurately extract effective fractal dimensions that are consistent with those expected for the corresponding universality class. As a consequence, for the steady state, which is inaccessible to experimentalists studying film growth, the TPS method provided effective fractal dimension consistent with the KPZ ones for almost all possible τ, i.e., 1≲τ<L/2, where L is the lateral size of the substrate on which the deposit is grown. In the growth of thin films, the true fractal dimension can be observed in a narrow range of τ, the upper limit of which is of the same order of magnitude as the correlation length of the surface, indicating the limits of self-affinity of a surface in an experimentally accessible regime. This upper limit was comparatively lower for the Higuchi method or the height-difference correlation function. Scaling corrections for the measure M and the height-difference correlation function are studied analytically and compared for the Edwards-Wilkinson class at d=1, yielding similar accuracy for both methods. Importantly, we extend our discussion to a model representing diffusion-dominated growth of films and find that the TPS method achieves the corresponding fractal dimension only at steady state and in a narrow range of the scale length, compared to that found for the KPZ class.

Contact angle hysteresis on random self-affine rough surfaces in Wenzel's wetting regime: Numerical study.

We present a numerical study of the advancing and receding apparent contact angles for a liquid meniscus in contact with random self-affine rough surfaces in Wenzel's wetting regime. Within the framework of the Wilhelmy plate geometry, we use the full capillary model to obtain these global angles for a wide range of local equilibrium contact angles and for different parameters that determine the self-affine solid surfaces: Hurst exponent, wave vector domain, and root-mean-square roughness. We find that the advancing and receding contact angles are single-valued functions that depend only on the roughness factor determined by the set of values of the parameters of the self-affine solid surface. Moreover, the cosines of these angles are found to depend linearly on the surface roughness factor. The relations between the advancing, the receding, and Wenzel's equilibrium contact angles are investigated. It is shown that for materials with self-affine surface structure, the hysteresis force is the same for different liquids and it depends only on the surface roughness factor. A comparison with existing numerical and experimental results is carried out.

Evolution of the real contact area of self-affine non-Gaussian surfaces

The impact of non-Gaussian height distribution on the real contact area evolution of elastic, frictionless and non-adhesive contact is studied. The Weibull probability function is used to model the height distribution, as it can capture surfaces of practical relevance. The set of variables needed to parametrise the problem is discussed, including the shape parameter of the height distribution, the topography’s wavelength ratio and the Hurst exponent. As the topographies become more non-Gaussian, a significant deviation from the Gaussian case is observed. Moreover, the spectral properties show a distinct effect on different non-Gaussian surfaces and the dependency on Nayak’s parameter is not inherited. A power-law evolution of the real contact area is found to fit the numerical results obtained with the boundary element method precisely. Finally, two semi-analytical asperity-based models are compared with numerical results, using the statistics of artificially generated topographies. The qualitative behaviour predicted with the numerical simulations is captured.

Normal Contact Analysis Between Two Self-affine Fractal Surfaces at the Nanoscale by Molecular Dynamics Simulations

Normal Contact Analysis Between Two Self-affine Fractal Surfaces at the Nanoscale by Molecular Dynamics Simulations

Modeling contact deformation of bare and coated rough metal bodies

The effect of the presence of a passivation layer on a metal rough surface during contact loading is investigated by means of dislocation dynamics simulations. The metal body is modeled as an FCC single crystal with a self-affine rough surface that is either bare, or covered by a thin coating, impenetrable to dislocations. This analysis permits to isolate the effect of surface roughening driven by dislocation motion: when the surface is bare the dislocations can glide out, leaving crystallographic steps at the surface that modify the local roughness; when the surface is passivated, dislocations are stopped by the interface.

Thermal-hydraulic performance of flat-plate microchannel with fractal tree-like structure and self-affine rough wall

ABSTRACTInspired by the natural bifurcating structures, tree-like microchannels have been applied for microelectronics cooling. In order to understand the thermal-hydraulic performance of a flat-plat tree-like microchannel, successive branching ratios of tree-like structure are optimized based on minimization of flow resistance. It is shown that the optimal successive diameter ratio of symmetrical and dichotomous structures under volume constraint follows Murray’s law, while the optimal successive length ratio under the constraint of fixed channel area follows the power law 2−2/3. A mathematical model of convection in disc-shaped heat sink composed of a tree-like microchannel with self-affine rough surface is developed by the fractal geometry and finite element method. The flat-plate tree-like micro-channel with optimal successive diameter and length ratio shows enhanced thermal-hydraulic performance. The Nusselt number of the flat-plat tree-like micro-channel increases with the inlet Reynolds number and the self-affine fractal dimension of the rough wall. The present optimization method and mathematical model for the flat-plate tree-like microchannel shed light on the design of flat-plate micro-channel heat sinks and flow channel in fuel cell among other potential cooling applications.

Analysis of Lava Tubes’ Roughness and Radar Near-Nadir Regime Backscattering Properties

Lava tubes are terrestrial tunnel-like natural subsurface caves. Mounting evidence suggests their presence on the Moon and Mars. Planetary radar sounders are nadir-looking instruments operating in the high-frequency (HF)/very-HF (VHF) part of the spectrum with subsurface penetration capabilities. Recently, several studies either proposed future mission concepts for lava tubes’ detection or attempted to locate them on the Moon and Mars with the available radar-sounding data. Lava tubes are typically modeled as quasi-cylindrical structures but their actual geometry and their influence on the radar backscattering in near-nadir regime have never been investigated in the literature. These are crucial information for understanding the feasibility of detecting lava tubes by current and future planetary radar sounding systems. Accordingly, in this article: 1) we assess whether lava tubes are self-affine fractal surfaces at horizontal scales relevant to radio and microwave scattering and 2) we evaluate the effect of lava tube topography on the radar backscattering response in the near-nadir regime. Our experimental results, which are inferred from 3-D terrestrial laser scanning (TLS) data of planetary lava tube analogs, show that lava tubes: 1) are self-affine fractals at horizontal scales relevant to radar sounding and 2) they are electromagnetically rough surfaces, especially in the VHF band. We provide quantitative values on the lava tube fractal parameters and radar roughness losses along with a discussion on both: 1) the implication of our results on current radar sounding systems’ ability to detect lava tubes and 2) the planning of future missions devoted to lava tube detection and characterization.

Elastic contact of random surfaces with fractal and Hurst effects

Most of the recent research on random surface contact mechanics has been on self-affine surfaces. In such models, the fractal dimension (which represents the ‘roughness’) and the Hurst parameter (which represents the ‘spatial memory’) are linearly dependent. In this study, we investigate the non-adhesive, frictionless contact between elastic solids with non-self-affine manifolds. In particular, we use Cauchy and Dagum covariance functions, which can decouple the fractal and Hurst effects, to describe the height distribution of the random surfaces. A numerical model based on the Boussinesq point load fundamental solutions is employed along with the discrete convolution FFT method to perform the contact analysis. We investigate the true contact area evolution under increasing load for surfaces with a wide range of fractal and Hurst parameters. It is observed that the contact area evolution at light loads is almost independent of the Hurst parameter and non-monotonically dependent on the fractal dimension. By contrast, previous studies predicted the contact evolution to be weakly dependent on the Hurst parameter and the fractal dimension. The curvature of the plots of the slope of the contact area evolution is found to depend on the fractal dimension, contrary to previous studies, which predicted either convexity or concavity.

The osmocapillary effect on a rough gel surface

A gel surface is generally biphasic at equilibrium. Asperities can be submerged under solvent due to osmocapillary phase separation and be flattened due to elastocapillary deformation. The surface behavior of a gel is governed by the interaction between osmocapillary phase separation and elastocapillary deformation. We develop a linear osmocapillary model that numerically predicts the phase separation and deformation on a gel surface. We use the model to simulate the deformation of a sinusoidal surface and a computer-generated self-affine rough surface. The simulation shows that the behavior of a gel surface is governed by the osmocapillary length and elastocapillary length relative to the characteristic length of the surface roughness. As the relative magnitude of these length scales change, the surface can behave like a rigid material with little deformation, a liquid with a completely flattened surface, a dry elastomer where deformation is not accompanied by osmocapillary phase separation, or a wet gel where a large portion of the surface is covered by the solvent. Contrary to the existing understandings, we show that a gel surface can be mostly covered by the liquid solvent, although the gel is solid, and the resulting surface energy can be significantly lower than the prediction from existing models neglecting osmocapillary phase separation.

Friction behaviors of rice husk silica-reinforced elastomer composites in contact with rough self-affine surfaces

This study presents rice husk silica-reinforced elastomer composites fabricated by wet compounding, exhibiting 10% higher dynamic friction coefficients than the reference composites made by conventional dry compounding. Depending on the fabrication method, the filler micro-dispersion was characterized to correlate the measured friction coefficients with the viscoelastic properties with respect to sliding velocity and substrate lubrication. The effect of viscoelastic properties of the elastomer composites on the adhesion and hysteresis friction coefficients was investigated based on the elastomer friction theory. It was found that the constructed friction master curves showed a reasonable correlation with the theory. This study demonstrates that the rice husk silica can be used as a reinforcing filler for sustainable tires. More importantly, the well-dispersed filler microstructures can ensure the higher friction coefficients of the elastomer composites on rough surfaces, improving the tire performances (e.g., wet traction, cornering, and acceleration) of the ever-growing electric vehicles and future mobility.