Fractal Surface Research Articles

A revised contact stiffness model of rough curved surfaces based on the length scale

Based on the continuity of length scale for asperities, a complete model of normal stiffness between curved fractal surfaces considering friction factor is proposed. Through the coupling of critical base diameter and contact area, deformation modes of all asperities are determined, and contact stiffness of the whole rough surface is derived by double integral. Accordingly, effects of contact load, surface topography, friction factor, size parameters and material properties on the contact stiffness are investigated. Results show that contact stiffness depends sensitively on the fractal dimension, and exhibits nonmonotonic characteristics as fractal roughness varies across scales. Finally, predicted values of normal stiffness are compared with existing contact models as well as experimental data.

Physicochemical characterisation of kafirins extracted from sorghum grain and dried distillers grain with solubles related to their biomaterial functionality

Kafirin, the hydrophobic prolamin storage protein in sorghum grain is enriched when the grain is used for bioethanol production to give dried distillers grain with solubles (DGGS) as a by-product. There is great interest in DDGS kafirin as a new source for biomaterials. There is however a lack of fundamental understanding of how the physicochemical properties of DDGS kafirin having been exposed to the high temperature conditions during ethanol production, compare to kafirin made directly from the grain. An understanding of these properties is required to catalyse the utilisation of DDGS kafirin for biomaterial applications. The aim of this study was to extract kafirin directly from sorghum grain and from DDGS derived from the same grain and, then perform a comparative investigation of the physicochemical properties of these kafirins in terms of: polypeptide profile by sodium-dodecyl sulphate polyacrylamide gel electrophoresis; secondary structure by Fourier transform infra-red spectroscopy and x-ray diffraction, self-assembly behaviour by small-angle x-ray scattering, surface morphology by scanning electron microscopy and surface chemical properties by energy dispersive x-ray spectroscopy. DDGS kafirin was found to have very similar polypeptide profile as grain kafirin but contained altered secondary structure with increased levels of β-sheets. The structure morphology showed surface fractals and surface elemental composition suggesting enhanced reactivity with possibility to endow interfacial wettability. These properties of DDGS kafirin may provide it with unique functionality and thus open up opportunities for it to be used as a novel food grade biomaterial.

Understanding the granule, growth ring, blocklets, crystalline and molecular structure of normal and waxy wheat A- and B- starch granules

Understanding the structural characteristics of starch can help in determining its functional properties and choosing appropriate applications. This work aimed to comprehensively characterize and understand the morphological, crystalline, and molecular structure of the normal and waxy wheat A- and B- starch granules. X-ray diffraction analysis revealed that both normal and waxy wheat A- and B- starch granules have a typical A-type crystalline structure. Additionally, scanning electron microscopy, small-angle X-ray scattering, and Fourier transform infrared spectroscopy analyses showed differences in shape, the thickness of semi-crystalline growth rings, and short-range molecular order of the normal and waxy A- and B- starch granules. The wheat A-starch granules contained lower proportions of A-chains (DP 6–12) and B1- chains (DP 13–24) than those of the wheat B- starch granules. Also, the results revealed that B1-chains (DP 13–24) play an important role in the formation of surface/mass fractal characteristics of starch granules. Furthermore, the atomic force microscopy analysis showed that the normal and waxy wheat A-starch granules are mass fractal, while the normal and waxy wheat B- starch granules are surface fractal.

Adsorption on Fractal Surfaces: A Non Linear Modeling Approach of a Fractional Behavior

This article deals with the random sequential adsorption (RSA) of 2D disks of the same size on fractal surfaces with a Hausdorff dimension 1<d<2. According to the literature and confirmed by numerical simulations in the paper, the high coverage regime exhibits fractional dynamics, i.e., dynamics in t−1/d where d is the fractal dimension of the surface. The main contribution this paper is that it proposes to capture this behavior with a particular class of nonlinear model: a driftless control affine model.

Deuterated Bacterial Cellulose Dissolution in Ionic Liquids

Understanding the dissolution mechanism of deuterated bacterial cellulose (DBC) is important to engineer advanced material applications such as in quantifying and visualizing biomolecules at the cellulose interface for diagnostics. Small-angle neutron scattering (SANS) is applied to evaluate the distribution and volume fraction of dissolved DBC chains in 1-ethyl-3-methylimidazolium acetate (EMIM-Ac) ionic liquid (IL-h) solvent in three different ways: (i) DBC in IL-h, (ii) DBC in a mixture of N,N-dimethylformamide (DMF) with IL-h (IL-h/DMF), and (iii) modified DBC by dissolution in IL-h with dichloromethane (DCM), (DCM-DBC). EMIM-Ac is a highly viscous solvent, and the incorporation of DMF reduces its viscosity. DCM incorporation into EMIM-Ac leads to partial acetylation of the cellulose chains. The DBC dissolves differently in all the modified solvents studied. The DBC and DCM-DBC dissolution in IL-h shows the presence of surface fractals (power law relation of intensity to a scattering vector, q, of q-3.4) indicating compact aggregated DBC structures. The DBC structure is more open in the DMF/IL-h solvent, which is reflected in the SANS curve mass fractal analysis with a power law of q-2.5. At intermediate values of the scattering vector, a q-1 power law is observed, indicative of rigid segments of dissolved DBC chains. Analysis of the intensity in this range provides insights as to the dissolution mechanism. The observed higher intensity measured in the solutions of DBC and DCM-DBC in IL-h can be attributed to the tight binding adsorption of the acetate ions on the DBC surface. Moreover, the unique aspect of this experiment, using deuterated cellulose in a mixture of deuterated DMF with protiated EMINM-Ac, provides direct proof for formation of a shell layer of IL-h surrounding the DBC surface. The results obtained shed light on the dissolution mechanism of cellulose in EMIM-Ac, highlighting its potential application in engineering biosensors and bio-diagnostics. (Less)

Fractal approach in expansive clay-based materials with special focus on compacted GMZ bentonite in nuclear waste disposal: a systematic review.

Knowledge of the behavior of highly compacted expansive clays, as an engineered barrier, in disposal of high-level nuclear waste (HLW) systems to prevent the pollution due to migration of radionuclide is extremely essential. The prominent properties of globally and widely used bentonites have been extensively studied during past two decades. In China, GaoMiaoZi (GMZ) bentonite is the first choice as a buffer or backfill material for deep geological repositories. This review article presents the recent progresses of knowledge on water retention properties, hydromechanical behavior, and fractal characteristics of GMZ bentonite-based materials, by reviewing 217 internationally published research articles. Firstly, the current literature regarding hydrogeochemical and mechanical characteristics of GMZ bentonite influenced by various saline solutions are critically summarized and reviewed. Then, the role of osmotic suction π alongside the application of surface fractal dimension Ds is presented from the standpoint of fractal theory. Finally, the strength characteristics of GMZ bentonites using fractal approach have been discussed. Furthermore, this study sheds light on gaps, opportunities, and further research for understanding and analyzing the long-term hydromechanical characteristics of the designed backfill material, from the standpoint of surface fractality of bentonites, and implications of sustainable buffer materials in the field of geoenvironmental engineering.

Lacunarity exponent and Moran index: A complementary methodology to analyze AFM images and its application to chitosan films

In this work, we developed new scripts to calculate the lacunarity exponent and Moran’s index of Atomic Force Microscopy (AFM) images. The lacunarity exponent was estimated by combining the Otsu binarization and gliding-box algorithm, and Moran index was introduced to evaluate the surfaces’ spatial autocorrelation. Developed scripts were first validated using numerical simulation of self-similar fractal and self-affine isotropic surfaces. Then, we successfully synthesized chitosan films with different glycerol concentrations and used the lacunarity and Moran’s index for a thorough characterization. The validation of the proposed scripts using simulated Sierpinski Carpets and 3D artificial surfaces showed promising potential for analyzing AFM images. Finally, the methodology application to AFM images of chitosan films suggested that lacunarity analysis and Moran index determination could complement thin films’ quality processing control.

Organic Non-Wettable Superhydrophobic Fullerite Films.

A long-standing quest in material science has been the development of non-wettable superhydrophobic films based on a single organic material, without the requirement of fluorination or silane treatment. Here, such films and coatings, which are developed using colloidal gels of fullerite C60 and C70 nanocrystals, are described. It is illustrated that despite the high surface energy of these van der Waals molecular crystals their gelation can create films having self-affine fractal surfaces with multiscale roughness. Water droplets on resulting surfaces of fullerite films bead like a pearl resting in a Fakir state with contact angle exceeding 150°. The films are extremely water repellent and non-wettable; when submerged in water they stay dry up to 3 h even at a water depth of two feet and exhibit the plastron effect. A series of experiments are presented to provide comprehensive inspection of water droplet dynamics on these films. These include rolling, bouncing, squeezing, freezing, melting, evaporating; along with acidic and alkaline tests. Non-wettable films of such materials are unique as fullerites get photosensitized instantaneously generating extremely high yields (≈100%) of singlet oxygen (1 O2 ) that can destroy viruses and bacteria; thereby enabling their use in rheology, water purification, and medicinal devices.

Surface microtexture and wettability analysis of quasi two-dimensional (Ti, Al)N thin films using fractal geometry

In this work, image and fractal analysis of Aluminium doped Titanium Nitride (Ti, Al)N thin films deposited by direct current (DC)/radio frequency (RF) magnetron sputtering is reported. Atomic force microscopy (AFM) data were studied to understand the impact of the Al doping concentration on the quasi two-dimensional surface morphology as well as surface texture parameters, evaluated by fractal and advanced fractal analysis. Conventional analysis shows a decrease in roughness of film surfaces along with peaks, valleys, and high surface uniformity. Furrows and contour lines generated from the topographical profile and grains were analyzed to gain a deep insight into the microtexture of the film surfaces. Power spectrum density approaches confirmed the self-affine behavior of the films with a semi-regular behavior of fractal dimension. The surface entropy did not show any significant difference indicating a globally uniform surface. Succolarity study showed a different behavior from that of furrow analysis. The effect on wettability of TiN films due to Al doping is studied based on the modified Wenzel equation and curvature pressure for fractal surfaces The hydrophobic behavior of (Ti, Al)N thin films exhibits a dependence on the fractal dimension, Hurst coefficient and fractal succolarity of the film surfaces. This study offers a deep understanding about the applicability of (Ti, Al)N thin films.

Surface Evolution and Fractal Dimensions of CH3NH3PbI3-XClX Thin Films and its Photovoltaic Performance

We fabricated the CH 3 NH 3 PbI 3-X Cl X perovskite crystalline films by solution processed method in air and nitrogen ambience to investigate the impact of the growth conditions on surface morphology and interface properties of the films. The surface morphology of CH 3 NH 3 PbI 3-X Cl X thin films are captured by scanning electron microscopy (SEM) and fractal analysis is performed on SEM micrographs. The value of fractal dimension/hurst exponent is found to be larger/smaller in N 2 environment as comparison to open air environment. The greater value of fractal dimension is corresponding to larger surface jaggedness which plays a crucial role for improving the interface between the perovskite and hole transportation layer. Furthermore, this analytical prediction is confirmed by photovoltaic devices fabricated on these films and observed a 60% enhancement in power conversion efficiency (PCE) of N 2 environment based thin films as compare to open air processed films. This may be due to the formation of compact and voids free perovskite self-affine fractal surfaces under nitrogen environment as compare to open air environment. Our findings represent a step forward to the realization of the peculiar morphological behaviour of the mixed halide perovskite fabricated in different environmental conditions.

Erosion potential mapping using analytical hierarchy process (AHP) and fractal dimension

Assessing landform vulnerability to soil erosion is crucial for improved sustainable land use planning and management. In the Loess Plateau of the Northern Shaanxi Province of China, soil erosion has been reported as a major threat to sustainable land management and impacts on driving the socio-economic benefits that can be accrued from the landforms. Several studies especially on Erosion Potential Mapping (EPM) in the region have been conducted but the role of the fractal dimension (FD) of the terrain features has been limited. In this study, the paper assessed the role of fractal terrain features on the overall EPM. The Analytical Hierarchical Process (AHP) was adopted using 6 criteria, FD of the terrain, Land Use Land Cover, Slope, Elevation, Geomorphology and Flow Accumulation. These were developed in a Geographic Information System (GIS) framework. Eight Scales (8) were evaluated in order to select the best Scale with the lowest Consistency Ratio (CR) and the Minimum Relative Error (MRE). The results from this study shows that fractal features of terrain when integrated with the rest of the criteria produced a reliable EPM for the study area. The absence of the FD also gives unrealistic results for the EPM. The EPM with FD distribution recorded 29.4% for low erosion potential whereas EPM without FD recorded 46.7%. A larger portion of the Shaanxi province (70%) is found to be at a higher risk of erosion. Therefore, it is hoped that the findings from this research would further boost the integration of fractals into EPM in China and similar regions across the World. The study further recommends that sustainable soil management measures are put in place to reduce the erosion risk in the province to protect the natural ecological habitat.

Effect of physicochemical properties of coal gasification fine ash on its wettability

Coal gasification, recognized as one of the most effective coal utilization technologies, will produce a certain amount of fine ash during the high-temperature reaction process. The wettability of gasification fine ash is a critical parameter to characterize the degree of high-temperature reaction and determine the separation efficiency of purification process. In the present work, the effect of physicochemical properties on the wetting behavior of different gasification fine ash is studied. The difference in wettability between particle size and types can be explained by the unique properties of particles (i.e. pore structure, mineral composition, and chemical structure). The results show that with the decrease in particle size, the surface morphology transforms from an irregular shape with a rough surface to a spherical shape with a smooth surface. The pore structure, characterized by fractal surface dimension Ds, presents positive correlations with the contact angle for a single type of fine ash. The mineral composition can just qualitatively assess the hydrophilicity of particles. Only the ratio of hydrophilic chemical structure can be used as a generic parameter to describe the wetting performance. Meanwhile, the wettability of hydrophilic particles can be enhanced by increasing moisture content, but there is no significant effect of moisture content on the wettability of hydrophobic particles.

Asperity-based modification on theory of contact mechanics and rubber friction for self-affine fractal surfaces

Modeling the real contact area plays a key role in every tribological process, such as friction, adhesion, and wear. Contact between two solids does not necessarily occur everywhere within the apparent contact area. Considering the multiscale nature of roughness, Persson proposed a theory of contact mechanics for a soft and smooth solid in contact with a rigid rough surface. In this theory, he assumed that the vertical displacement on the soft surface could be approximated by the height profile of the substrate surface. Although this assumption gives an accurate pressure distribution at the interface for complete contact, when no gap exists between two surfaces, it results in an overestimation of elastic energy stored in the material for partial contact, which typically occurs in many practical applications. This issue was later addressed by Persson by including a correction factor obtained from the comparison of the theoretical results with molecular dynamics simulation. This paper proposes a different approach to correct the overestimation of vertical displacement in Persson’s contact theory for rough surfaces with self-affine fractal properties. The results are compared with the correction factor proposed by Persson. The main advantage of the proposed method is that it uses physical parameters such as the surface roughness characteristics, material properties, sliding velocity, and normal load to correct the model. This method is also implemented in the theory of rubber friction. The results of the corrected friction model are compared with experiments. The results confirm that the modified model predicts the friction coefficient as a function of sliding velocity more accurately than the original model.

Elastic Rough Surface Contact and the Root Mean Square Slope of Measured Surfaces over Multiple Scales

This study investigates the predictions of the real contact area for perfectly elastic rough surfaces using a boundary element method (BEM). Sample surface measurements were used in the BEM to predict the real contact area as a function of load. The surfaces were normalized by the root-mean-square (RMS) slope to evaluate if contact area measurements would collapse onto one master curve. If so, this would confirm that the contact areas of manufactured, real measured surfaces are directly proportional to the root mean square slope and the applied load, which is predicted by fractal diffusion-based rough surface contact theory. The data predicts a complex response that deviates from this behavior. The variation in the RMS slope and the spectrum of the system related to the features in contact are further evaluated to illuminate why this property is seen in some types of surfaces and not others.

Metamaterial Loaded Circularly Polarized Fractal Antenna for 2.4 GHz Frequency Applications

A metamaterial loaded circularly polarized fractal boundary patch antenna is investigated for applications at the ISM band. The suggested patch comprises of Minkowski fractal over metamaterial inspired reactive impedance substrate. Initially, asymmetrical Minkowski fractal curves are deployed at the edges of a square patch for generating the circular polarization (CP). Later, using metamaterial reactive impedance substrate as antenna loading, CP bandwidth is enhanced. The characteristics of the Minkowski fractals are adjusted to obtain the pure CP radiation. The metamaterial loaded Koch fractal antenna is also studied for CP emission. Minkowski antenna with fractal reactive impedance surface as metamaterial with iteration 1 is experimentally examined. The achieved 10-dB return loss and 3-dB axial ratio bandwidths are 40.7% and 7.47%, respectively.

Analysis of convection heat transfer on multiscale rough superhydrophobic and liquid infused surfaces

Multiscale rough superhydrophobic or slippery liquid infused porous surfaces have gained much interest in recent years for their improved transport phenomena properties. While there have been several studies on drag reduction and condensation on non-wetting surfaces, convection heat transfer that is important in many thermal and thermochemical applications has not been addressed systematically. This article utilizes a fractal description of rough surface topographies to develop analytical models for the Nusselt number and the thermal hydraulic factor for fluid flow and heat transfer inside a cylinder with non-wetting surfaces. For air-infused superhydrophobic surfaces, the model considers the dynamic stability of the air/fluid interface in the asperities. Using the analytical formulations and the stability criteria, systematic studies are presented on the effects of the fractal surface parameters, cylinder radius and Reynolds number on the convective heat transfer characteristics, from which surface texture design maps are developed for maximizing the convection heat transfer. It is shown that multiscale non-wetting surfaces are most effective in the range of lower Reynolds number and small cylinder radius for achieving the best convective heat transfer and thermal hydraulic performance. Applying the models to actual non-wetting surface topographies fabricated using electrodeposition and chemical etching, it is shown that contrary to prevailing notion, superhydrophobicity, characterized by the highest contact angles, does not always lead to the maximum convective heat transfer performance, and that under certain fluid flow conditions, hydrophobic surfaces may offer a greater thermal performance.

Construction and box dimension of recurrent fractal interpolation surfaces

In this paper, we present a general framework to construct recurrent fractal interpolation surfaces (RFISs) on rectangular grids. Then we introduce bilinear RFISs, which are easy to be generated while there are no restrictions on interpolation points and vertical scaling factors. We also obtain the box dimension of bilinear RFISs under certain constraints, where the main assumption is that vertical scaling factors have uniform sums under a compatible partition.

Feature Analysis of Fractal Surface Roughness Based on Three-dimensional W-M Function

The three-dimensional W-M fractal function was used to simulate the rough surface, and the different fractal parameters correspond to the different surface morphology. In order to explore the influence rule of fractal dimension D and scale coefficient G on the surface profile, the fractal dimension D value was 2.5, and the scale factor G value was 1E-14. The mean height, standard deviation and spectrum characteristics of the surface contour were analyzed. The results show that under the action of a single fractal parameter, the surface roughness Ra and the surface contour height standard deviation std all decrease with the increase of D, and increase with the increase of G, and the change trend is basically the same. When D is kept constant and only G is a variable, the surface height value of frequency components remain unchanged, the amplitude of each frequency decreases with the decrease of G; G remains unchanged but only D is a variable, with the increase of D, the frequency amplitude changes slowly, and the amplitude decreases.

Stress-dependent electrical impedance behaviours at fractal rough interfaces

This work investigates interfacial electro-mechanical properties, including electrical contact resistance, interfacial capacitance and characteristic frequency of contacts formed with various surface structures. Fractal rough surfaces were generated and characterised by fractal dimension and root-mean-square (RMS) roughness. The rough surface with a thin oxide layer was compressed by the rigid flat to form a capacitor. Electrical impedances of this contact capacitor were simulated using the finite element method across a wide range of frequencies. A power-law relationship was found between the electrical contact resistance and applied compression load. An analytical model is proposed to capture the interfacial capacitance behaviour with increasing contact loads, revealing a transition of predominated modes for the capacitance. Higher fractal dimension yields smaller overall capacitance in the gap dominant and transition zones. The dependence of the characteristic frequency on compression was found to follow a power-law function at the low load range. It is found that the exponent and magnitude of obtained power-law relations show strong correlations to the fractal dimension and RMS roughness, respectively. Results of this work provide insights into developing a potential impedance measurement protocol to determine the thickness of the oxide layer on conductive fractal rough surfaces.

Analyzing the surface dynamics of titanium thin films using fractal and multifractal geometry

Titanium thin films are prepared by DC magnetron sputtering on the glass substrate with varying sputtering power. The nanoscale surface morphology and nanotexture are captured via atomic force microscopy (AFM). The value of average roughness and interface width show the increasing trend with sputtering power. The autocorrelation and height-height correlation function are applied to characterize the formation of the self-affine and mounded surfaces. The roughness (or Hurst) exponent and fractal dimension are estimated by height-height correlation function while lateral correlation length is measured by autocorrelation. It is found that the surface complexity is affected with various sputtering power. The value of Hurst exponent of the topography greater than 0.5 indicates a persistent behavior and such system shows memory effect. The fractal lacunarity, succolarity and surface entropy is used for investigating the surface microtexture and percolation. The 2D Multifractal Detrended Fluctuation Analysis (2D MFDFA) method is used to study the multifractal nature of thin-film surfaces. Computed Δα and Δf show an increase with sputtering power indicating a surface with many characteristic scale patterns. Furthermore, the thin film deposited with the highest sputtering power has greater topographic uniformity, lower surface percolation and greater homogeneity of the surface microtexture.