Fractal Surface Research Articles

Flow Characteristics Considering the Temperature Viscosity of Oils in the Gap based on Micro Geometric Roughness Fractal Surface

Due to the limitation of manufacturing conditions in China, the valve core will always exhibit random, disordered and multi-scale micro geometric characteristics in recent years. Based on the fractal features of the micro-scale surfaces of the cores and the chambers of a multi-way valve, this research establishes3-danisotropic models for the surface shapes of the valve cores and chambers using the Weierstrass-Mandelbrot (W-M) function. Then, the boundary conditions of the Navier slip model and computational fluid dynamics (CFD) are used to analyse the flow characteristics of the hydraulic oil in the gap and the wall slip. The research results demonstrate that the surface roughness of the solid walls exerts a significant influence on the flow characteristics of the hydraulic oil as it flows in the gaps: the greater the surface roughness, the higher the probability of occurrence of slip between the surfaces of the solid walls. With varying surface roughness of the solid walls, the greater the slip coefficient b is, the larger the slip velocity on the walls, and the greater the flow velocity of the hydraulic oil in the whole gap; meanwhile, this phenomenon occurs when the temperature difference between the upper and lower walls increases as well, and the flow velocity of the hydraulic oil increases significantly.

Theory for local EIS at rough electrode under diffusion controlled charge transfer: Onset of whiskers and dendrites

Theory is developed for the local electrochemical impedance spectroscopy (LEIS) at rough electrode under reversible charge transfer. Theory is used for analyzing the onset of non-uniform local growth or deposition on rough electrode which may cause unstable growth, viz. whiskers and dendrites, on lithium electrode. Model is also used to simulate temporal frequency dependent local admittance density and local growth velocity over a sinusoidal and a random fractal surfaces. For sinusoidal checkers surface, peak positions of electrode show preferential “whisker” growth for low roughness while for moderate and high roughness, anomalous “dendritic” growth may occur besides peaks. At low temporal frequency, LEIS indicates higher growth rate at cliff positions and depletion at the peak positions. The multiscale (fractal) roughness is modeled using Weierstrass function and shows a complex distribution of growth velocity over the surface. These surfaces show the curvature dependent localization of growth sites. The complex random growth pattern over fractal surface is described in terms of probability distribution function of growth velocities. Distribution function of growth velocity is analyzed for different roughness parameters, viz. width of roughness, finest scale of roughness and fractal dimensions. Results show that high growth rate sites increases with increase in these roughness features. Finally, LEIS theory for rough electrode provides an alternative approach to understand the problem of unstable growth in various electrochemical systems.

Critical heat flux on heterogeneous fractal surfaces with micro-pin-fins in pool boiling – Part II: Model establishment and analysis

The heterogeneous fractal surfaces can achieve higher critical heat flux (CHF) than that with the uniformly distributed surfaces, which is reported in Part Ⅰ. In order to explain the phenomenon and understand the trigger mechanism of CHF, a theoretical model of CHF for the heterogeneous structure surfaces is proposed based on the bubble coalescence caused by the hydrodynamic instability and the wall dryout induced by the liquid supply impediment. The continuous expansion and coalescence of the dry spots lead to the formation of the vapor blanket covering the whole surface, which triggers the occurrence of critical phenomenon. The CHF is obtained according to the arrangement of the vapor columns on the heating surface. Both the distribution of the heterogeneous structure and the subcooling of liquid affect the bubble behaviors and liquid supply, which result in an optimal structure distribution to reach the highest CHF under different subcoolings. Compared with the uniform distribution, the width of smooth region surrounding the bubble increases with the increase of heat flux and bubble diameter on the fractal structure, which can effectively provide liquid replenishment at high heat fluxes and achieve higher CHF. The CHFs calculated from this model agree well with the experimental results, and the vapor column arrangement is consistent with the bubble distribution observed from the experimental phenomenon. Moreover, the model can also be used to calculate the CHF of the uniformly distributed heterogeneous surfaces and the homogeneous microstructure surfaces with good precision, which verifies the correctness and extensive applicability of the model.

FRACTAL DERIVATIVE MODEL FOR TSUNAMI TRAVELING

Tsunami traveling in an unsmooth boundary is studied, where the boundary is considered as a fractal surface, and mass conservation and moment conservation in a fractal space are established, and a fractal derivative-based tsunami model is established. Other discontinuous boundaries can be treated in a similar way.

A Comparison of Multiscale Surface Curvature Characterization Methods for Tribological Surfaces

Surface curvature affects adhesion forces, deformations of surfaces in contact, leakage of mechanical seals, friction, wear, paintability, and electrical conductivity. However, potential benefits of surface curvature have not yet been fully utilized. One problem is the lack of comparison data helping to make an informed decision on the selection of curvature characterization method. In this paper, five multiscale curvature characterization methods, namely Nowicki, Bigerelle-Nowicki, Gleason-Heron, Kalin, and Bartkowiak are compared. The comparison was conducted on large image databases of computer-generated fractal surfaces, sine waves and real engineering surfaces. Specifically, the methods were evaluated for their ability to differentiate between surfaces that: (i) exhibit increasing curvature complexity, (ii) have varying curvatures at a single scale, and (iii) represent minute multiscale curvature changes encountered in real engineering applications. The results obtained indicate that the Bigerelle-Nowicki method exhibits the best overall performance.

Interactive Effects of Rarefaction and Surface Roughness on Aerodynamic Lubrication of Microbearings

The aerodynamic lubrication performance of gas microbearing has a particularly critical impact on the stability of the bearing-rotor system in micromachines. Based on the Duwensee’s slip correction model and the fractal geometry theory, the interactive effects of gas rarefaction and surface roughness on the static and dynamic characteristics were investigated under various operation conditions and structure parameters. The modified Reynolds equation, which governs the gas film pressure distribution in rough bearing, is solved by employing the partial derivative method. The results show that high values of the eccentricity ratio and bearing number tend to increase the principal stiffness coefficients significantly, and the fractal roughness surface considerably affects the ultra-thin film damping characteristics compared to smooth surface bearing.

Correlation between surface topography, optical band gaps and crystalline properties of engineered AZO and CAZO thin films

In this work, the stereometric 3-D surface topography as well as the optical properties of the Al doped ZnO (AZO), and Al/Cu co-doped ZnO (CAZO) thin films were investigated. These films were deposited with different thicknesses of 50 and 150 nm. The effect of dopants (Al, Cu) and sputtering parameters on the microstructures, crystalline structures, and fractal features were probed by Atomic Force Microscopy (AFM), X-ray diffraction, and Rutherford back scattering. AFM analysis of 3-D surface texture provided a deeper insight into their characteristics and implementation in graphical models and computer simulation. Studying surface roughness of samples at nanometer scale revealed a fractal structure which confirmed the relationship between the value of the fractal dimension and surface roughness parameters. Moreover, the optical properties of AZO and CAZO thin films and the relationship between their optical band gaps and their varied thicknesses were evaluated by Ultraviolet–visible spectrophotometry.

Critical heat flux on heterogeneous fractal surfaces with micro-pin-fins in pool boiling Part I: The effects of distribution and subcooling

Heterogeneous surfaces can offer superior heat transfer performance by using characteristics of different structures compared with the homogeneous surfaces. The micro-pin-finned structure facilitating bubble nucleation and the smooth area as a good liquid supply pathway constitute a heterogeneous surface for further enhancing boiling heat transfer. Inspired by the fractal geometry’s enhancement for the heat and mass transfer in nature, the created surface is designed into a fractal pattern. In this paper, five fractal structures F0–F4 with the same micro-pin-finned area ratio of 49% were designed. The smooth surface S and homogeneous micro-pin–fin surface PF30-60 were also used as comparisons. Pool boiling experiments using FC-72 were conducted under different subcoolings of 0 K, 15 K, 25 K and bubble behaviors were observed with a high-speed camera. The results indicated that under saturated conditions, the critical heat flux (CHF) of fractal surfaces increases first, and then decreases with the increase of fractal dimension. Besides, with the increase of subcooling, the fractal dimension with the highest CHF surface also increases. These results are explained from the liquid supply and the dry spots on the heating surface based on the bubble behaviors. There exists an optimal fractal dimension that can effectively impede the bubble coalescence and promote the liquid supply to achieve the highest CHF. The subcooling has also influence on CHF by affecting the bubble size.

A "best fit" approach for synergistic surface parameters to guide the design of candidate implant surfaces.

Human bone resorption surfaces can provide a template for endosseous implant surface design. We characterized the topography of such sites using four synergistic parameters (fractal dimension, lacunarity, porosity, and surface roughness) and compared the generated values with those obtained from two groups of candidate titanium implant surfaces. For the first group (n = 5/group): grit-blasted acid etched (BAE), BAE with either discrete calcium phosphate crystal deposition or nanotube formation, machined titanium with nanotubes, or a nanofiber surface; each measured synergistic parameter was statistically compared with that of the resorbed bone surface and scored for inclusion in a "best fit" analysis. The analysis informed changes that could be made to a candidate implant surface to render it a closer "best fit" to that of the resorbed bone surface. In a second group of either titanium or titanium alloy implants their micro-topography, created by dual acid etching, was the same for each material substrate; but their nanotopographic complexity was changed by varying the degree of calcium phosphate crystalline deposits. These implants were also used in vivo where bone anchorage was tested using a tensile disruption test; and the "best fit" of synergistic parameters coincided with the best biological outcome for both titanium and titanium alloy implants. In conclusion, the four chosen synergistic parameters can be used to guide the sub-micron surface design of candidate implants, and our "best fit" approach is capable of identifying the surfaces with the best biological outcomes. © 2019 Wiley Periodicals, Inc. J Biomed Mater Res Part B: Appl Biomater 107B: 2165-2177, 2019.

New types of fractal interpolation surfaces

In this paper we present new methods to generate fractal interpolation surfaces which generalize the existing fractal interpolation surfaces. We ensure that attractors of nonlinear iterated function systems constructed by Geraghty contractions are graphs of some continuous functions which interpolate the given data. In particular, we give two explicit illustrative examples to demonstrate the effectiveness of obtained results.

Thermal diffusion and flow property of CO2/CH4 in organic nanopores with fractal rough surface

The kerogen is rich in complex pore networks with a random rough surface, which is a factor controlling the thermal diffusion and flow property of gases. In this work, we construct organic-rich nanopore with fractal surfaces by inserting and deleting carbon atoms. The adsorption ability, thermal diffusion property, and flow velocity of CO2 /CH4 in the nanopore are analyzed using with molecular simulations. The results showed that the adsorption capacity of CO2 is nearly twice that of CH4, which is decided by adsorption enthalpy, whereas the maximum thermal diffusion ability of CO2 is only 23.7% that of CH4. With external pressure gradients imposed on the system, the flow speed of CO2 was lower than that of CH4 for nanopores with different roughness. These findings provide a theoretical basis for the feasibility of CO2 exploitation of shale gas.

Friction Dynamics on Rough Agar Gel Surfaces

Gels exhibit complex friction behavior. This study aims to evaluate the friction forces between two fractal agar gel substrates under sinusoidal motion to show the effect of rough surfaces on friction dynamics. In a previous study, we observed an asymmetric friction profile during reciprocating motion and an ultra-low friction state on flat agar gel surfaces. On the other hand, these distinct friction profiles were not observed on rough agar gel surfaces. We determined that this distinction was caused by the contact state between fractal agar gel surfaces; no thick water film was formed on the fractal surfaces because the rough structure provided channels to drain water from the interface. These physical insights are useful not only for developing biofunctional materials but also for understanding surface phenomena on biosurfaces including tongues and small intestinal walls.

A fractal model for predicting thermal contact conductance considering elasto-plastic deformation and base thermal resistances

A prediction model of thermal contact conductance is developed. Engineering rough surfaces are characterized by three-dimensional fractal Weierstrass-Mandelbrot fractal function. Three deformation modes, including fully plastic deformation, elasto-plastic deformation and elastic deformation, are considered to analyze the contact mechanism. Fractal surface and three deformation modes are incorporated into the calculation of thermal contact conductance. A comprehensive thermal contact conductance computation model considering both base thermal resistance and constricted thermal resistance is established. The results show that thermal contact conductance increases with the increase of normal contact pressure; the relative contribution of constricted resistance component to base resistance component tends to increase with the increase of normal contact pressure; fractal dimension and fractal roughness both have significant influences on thermal contact conductance.

The structure of deterministic mass and surface fractals: theory and methods of analyzing small-angle scattering data.

Small-angle scattering (SAS) of X-rays, neutrons or light from ensembles of randomly oriented and placed deterministic fractal structures is studied theoretically. In the standard analysis, a very few parameters can be determined from SAS data: the fractal dimension, and the lower and upper limits of the fractal range. The self-similarity of deterministic structures allows one to obtain additional characteristics of their spatial structures. In the present work, we consider models that can describe accurately SAS from such structures. The developed models of deterministic fractals offer many advantages in describing fractal systems, including the possibility to extract additional structural information, an analytic description of SAS intensity, and effective computational algorithms. The generalized Cantor fractal and few of its variants are used as basic examples to illustrate the above concepts and to model physical samples with mass, surface, and multi-fractal structures. The differences between the deterministic and random fractal structures in analyzing SAS data are emphasized. Several limitations are identified in order to motivate future investigations of deterministic fractal structures.

A New Probe: AFM Measurements for Random Disorder Systems**Supported by TUBITAK under Grant No 115F315.

We study the quenched random disorder (QRD) effects created by aerosil dispersion in the octylcyanobiphenyl (8CB) liquid crystal (LC) using atomic force microscopy technique. Gelation process in the 8CB+aerosil gels yields a QRD network which also changes the surface topography. By increasing the aerosil concentration, the original smooth pattern of LC sample surfaces is suppressed by the emergence of a fractal aerosil surface effect and these surfaces become more porous, rougher and they have more and larger crevices. The dispersed aerosil also serves as pinning centers for the liquid crystal molecules. It is observed that via the diffusion-limited-aggregation process, aerosil nano-particles yield a fractal-like surface pattern for the less disordered samples. As the aerosil dispersion increases, the surface can be described by more aggregated regions, which also introduces more roughness. Using this fact, we show that there is a net correlation between the short-range ordered x-ray peak widths (the results of previous x-ray diffraction experiments) and the calculated surface roughness. In other words, we show that these QRD gels can also be characterized by their surface roughness values.

A fractal rate model for adsorption kinetics at solid/solution interface

Langmuir?s linear rate equation has limited applications in the adsorption kinetics at solid/solution interface. Considering the fractal properties of adsorption surfaces, a fractal derivative model is proposed, its initial slope agrees well with Azizian-Fallah?s modified rate equation.

Rheological properties and small-angle X-ray scattering studies of phosphate dust obtained from baghouse collectors

Abstract Baghouse dust collectors are using in the drying unit of Beni Idir situated in Beni-Idir Khouribga city, Morocco, to retrieve phosphates particles from dust air-drying before its expulsion through the smokestacks. The phosphate dust samples used in this study were taken from the filtration chamber of the baghouse dust collectors. The first sample (S1) is untreated calcium phosphate dust, the second (S2) is the calcium phosphate dust from the outside of filter media while the third one (S3) is the calcium phosphate dust from the inside of filter media which causes clogging depth. In this paper, the rheology and the small-angle X-ray scattering (SAXS) of the three samples were investigated to elucidate the changes in terms of local structure, the viscosity, and the shear stress parameters. The rheological behavior of the dust samples was investigated for a solid mass concentration ranging from 50 to 60%, the three samples (S1) (S2) and (S3) had a solid mass concentration of C1=60%, C2=55% and C3=50% and a shear rate in the range from 1 to 1000 s−1. The results indicated that during the filtration process, the pseudo-plastic behavior of the dust phosphate changed to that of Bingham. Comparing the results of the sample’s viscosity, we found that the viscosity decreased during the process filtration. The SAXS results suggested that the dust phosphate samples possess a fractal surface structure of enormous dust particles with a rough surface interface. This new study highlights the rheological behavior of grain phosphates that could be extrapolated to other mining powder as grains material or in solution. It is important to understand the rheological characteristics of materials, their flow, and the subsequent deformation of matter as a result of the flow.

Analyzing the Surface Structure of the Binding Domain on DNA and RNA Binding Proteins

The study of nucleic acid-binding protein (NBP) has important significance for us to understand critical intracellular activities, such as the transmission of cellular genetic information, cell metabolism, substance transport, and signal transduction. DNA-binding proteins (DBPs) and RNA-binding proteins (RBPs) interact through their diverse binding domains and different types of nucleic acid molecules. In this paper, we used a novel method that combines the CX algorithm and the fractal surfaces algorithm. This method gets the molecular volume and solvent surface area in the local area of the NBPs binding domain residues. Then, based on the algorithm results, the requisite domain residues are divided into three types: peak, flat, and valley. At the same time, we analyzed the solvent accessibility and secondary structural characteristics of the DBPs and RBPs binding domains. Finally, we found that there was an important difference in the distribution of peak residues and valley residues in the two types of NBPs binding domains. Similarly, there were significant differences in the solvent accessibility and secondary structural distribution of the two types of NBPs binding domains. To verify the existence of differences, we constructed SVM classifier to make a distinction between DBPs and RBPs using a 10-fold cross-validation method. Lastly, the SVM classification model achieves AUC of 78%. In summary, we have proposed a new perspective for the study of NBPs binding domains. This method not only calculates the geometric characteristics of the molecule, but also analyzes the protein properties associated with the structure, which will assist in the study of NBPs binding domains.

Interactions between nanoparticles and fractal surfaces

This study evaluated attachment of a 30-nm nanoparticle to and detachment from fractal surfaces by calculating Derjaguin-Landau-Verwey-Overbeek (DLVO) interaction energies in three-dimensional space using the surface element integration technique. The fractal surfaces were generated using the Weierstass-Mandelbrot function with varying values of fractal dimension D (2.3 ≤ D ≤ 2.7) and fractal roughness G (0.000136 ≤ G ≤ 0.136). Results show that maximum energy barrier is reduced at peak areas of a fractal surface, and hence attachment in primary minima is favored. Some nanoparticles attached in primary minima at the peak areas can be detached by decreasing ionic strength (IS) due to monotonic decrease of interaction energy with increasing separation distance at low ISs. While the attachment in primary minima at valley areas is irreversible to IS reduction, the attachment is inhibited due to enhanced maximum energy barrier at these areas. A nonmonotonic variation of attachment efficiency in primary minimum (AEPM) with IS is present at high fractal dimension (D ≥ 2.4) or low fractal roughness (G < 0.00136), whereas the AEPM decreases monotonically with decreasing IS at low fractal dimension (D < 2.4) or high fractal roughness (G ≥ 0.00136). The AEPM decreases monotonically with increasing D or decreasing G at ISs from 1 mM to 200 mM. The decrease of AEPM with D or G is much slower at 10 mM compared to other ISs. These theoretical findings can explain various experimental observations in the literature, and can have important utility to development of water filtration techniques in engineered systems and to assessment of environmental risks of nanoparticles.

Fractal surface maghemite nanoparticles prepared by co-precipitation: the influence of Iron concentration and base nature

Fractal surface maghemite nanoparticles prepared by co-precipitation: the influence of Iron concentration and base nature