Fractal Rough Surface Research Articles

Chip-to-Chip Copper Interconnects With Rough Surfaces: Analytical Models for Parameter Extraction and Performance Evaluation

In multigigahertz integrated circuit design, the extra delay and power losses due to surface roughness in copper (Cu) interconnects is more evident than ever before and needs utmost consideration. This paper reports novel closed-form expressions for parameter extraction in chip-to-chip Cu interconnects at several gigahertz frequency ranges considering experimentally measured fractal surface roughness. Backplane (BP) and printed circuit board (PCB) links are considered here to study the effect of surface roughness in chip-to-chip Cu interconnects. Our proposed model is verified with experimental characterization performed on fabricated interconnects structures as well as simulation data obtained using full-wave electromagnetic field solvers, i.e., HFSS and Q3D extractor, that show excellent accuracy. Our analysis, for the first time, quantitatively proves the penalty incurred due to surface roughness in chip-to-chip interconnects using experimental measured realistic surface profiles. We observe that surface roughness strongly affects the interconnect performance metrics, i.e., delay, energy-delay product, and bandwidth density of BP and PCB chip-to-chip interconnects at higher frequency regimes. At 100 GHz and 50 cm length, BP interconnects with all four rough surfaces exhibit $2.6\times $ and $2.8\times $ higher delay and energy budget, respectively, when compared to interconnects with smooth surfaces. Similarly, PCB interconnects in the worst case exhibit $2.5\times $ and $2.6\times $ higher delay and energy budget, respectively, due to surface roughness. We also present computational overhead for simulating BP and PCB interconnects with rough surfaces. The simulation time, physical memory, matrix size, and number of tetrahedrons increase by several times due to surface roughness. Finally, we present eye diagrams at 10- and 15-Gb/s data rates that show 53.4% and 40% penalty on eye height and signal-to-noise ratio, respectively.

Wavelet Method Identifying the Characteristic Length Scale Parameter of Fractal Rough Surface

Wavelet Method Identifying the Characteristic Length Scale Parameter of Fractal Rough Surface

Research on fractal model of normal contact stiffness for mechanical joint considering asperity interaction

The normal contact stiffness of mechanical joint has immediate impact on dynamic characteristics and machining precision of machine tools. A fractal contact stiffness model considering the effect of asperity interaction is proposed in this paper. On the basis of elastic theory, the deformation of a given asperity is deduced. The relationship between contact stiffness, normal load and contact area is derived from Hertz contact theory and fractal theory. The effect of asperity interaction on contact stiffness is studied by comparing with fractal model. The fractal smoothness is defined from relation between fractal parameters and surface roughness. It is concluded that with the increase of the fractal dimension D and the decrease of fractal roughness parameter G, the contact stiffness of the joint increases, and the effect of asperity interaction becomes greater. The fractal smoothness also affects the normal contact stiffness and the effect of surface smoothness can be offset by the asperity interaction. Experiment shows that the proposed model is more in line with the actual load-stiffness relationship. The model reveals the factors that affect stiffness of the joint, which can provide a theoretical basis for improving the rigidity and precision of integrated machineries.

Interfacial contact stiffness of fractal rough surfaces

In this work we describe a theoretical model that predicts the interfacial contact stiffness of fractal rough surfaces by considering the effects of elastic and plastic deformations of the fractal asperities. We also develop an original test rig that simulates dovetail joints for turbo machinery blades, which can fine tune the normal contact load existing between the contacting surfaces of the blade root. The interfacial contact stiffness is obtained through an inverse identification method in which finite element simulations are fitted to the experimental results. Excellent agreement is observed between the contact stiffness predicted by the theoretical model and by the analogous experimental results. We demonstrate that the contact stiffness is a power law function of the normal contact load with an exponent α within the whole range of fractal dimension D(1 < D < 2). We also show that for 1 < D < 1.5 the Pohrt-Popov behavior (α = 1/(3 − D)) is valid, however for 1.5 < D < 2, the exponent α is different and equal to 2(D − 1)/D. The diversity between the model developed in the work and the Pohrt-Popov one is explained in detail.

Rarefaction cloaking: Influence of the fractal rough surface in gas slider bearings

For ultra-thin gas lubrication, the surface-to-volume ratio increases dramatically when the flow geometry is scaled down to the micro/nano-meter scale, where surface roughness, albeit small, may play an important role in gas slider bearings. However, the effect of surface roughness on the pressure and load capacity (force) in gas slider bearings has been overlooked. In this paper, on the basis of the generalized Reynolds equation, we investigate the behavior of a gas slider bearing, where the roughness of the slider surface is characterized by the Weierstrass-Mandelbrot fractal function, and the mass flow rates of Couette and Poiseuille flows are obtained by deterministic solutions to the linearized Bhatnager-Gross-Krook equation. Our results show that the surface roughness reduces the local mass flow rate as compared to the smooth channel, but the amount of reduction varies for Couette and Poiseuille flows of different Knudsen numbers. As a consequence, the pressure rise and load capacity in the rough bearing become larger than the ones in the smooth bearing in the slip and early transition flow regimes, e.g., a 6% roughness could lead to an increase of 20% more bearing load capacity. However, this situation is reversed in the free-molecular flow regime, as the ratio of the mass flow rates between Couette and Poiseuille flows is smaller than that in the smooth channel. Interestingly, between the two extremes, we have found a novel “rarefaction cloaking” effect, where the load capacity of a rough bearing equals to that of a smooth bearing at a certain range of Knudsen numbers, as if the roughness does not exist.

Surface-Roughness Control of a Detail Treated by Electric Discharge Machining

The possibility of using the fractal dimension as a criterion for assessing the sustainability of surface-roughness formation after electric-discharge machining (EDM) was justified scientifically. The nature of the changes in surface roughness after EDM of AK34 and VT20 alloys and the fractal dimension were studied experimentally as functions of the main process parameters (pulse rate and porosity). The fractal dimension and surface roughness after EDM were closely correlated, which allowed the roughness of the machined surface to be controlled in real time.

Characterization of Three - Dimensional Rough Fractal Surfaces from Backscattered Radar Data

This paper discusses the scattering of electromagnetic (EM) waves from three – dimensional (3D) rough fractal surfaces using the Kirchhoff approximation. In particular, it introduces a novel method to characterize 3D rough fractal surfaces from spectral information of the backscattered EM wave in Synthetic Aperture Radar (SAR) applications. It represents an important extension of a previous recent paper by the same research group from 2D fractal surfaces to 3D fractal surfaces (the latter representing real – life SAR radar scenes). More specifically, in the present simulation scenarios it is assumed that the radar emits a burst of radar pulses of increasing carrier frequency [therefore a ‘stepped – frequency’ (SF) SAR radar]. By calculating the backscattered EM wave from 3D fractal surfaces as a function of the above radar frequencies (therefore, a ‘spectral method’), it is found here that the slope between the main backscattered lobe and its adjacent side lobes increases with increasing surface fractal dimension (i.e. with increasing surface roughness). In this way a characterization of 3D fractal rough surfaces from backscattered SAR radar data is achieved, as explained in detail in this paper.DOI: http://dx.doi.org/10.5755/j01.eie.23.4.18721

Contact stiffness of multiscale surfaces by truncation analysis

In this paper, we study the contact stiffness of a fractal rough surface compressed by a rigid flat plane. A numerical model based on the analysis of flat punch indentation is proposed for simulated hierarchical surfaces, which are generated using statistical and fractal descriptors collected by surface profilometry. The contact stiffness of surfaces under increasing normal load is determined on the basis of the total truncated area at varying heights. The results are compared with experimental data from nanoindentation on four types of treated rough surfaces, showing good agreement with experimental observations below a certain truncation depth. Furthermore, the limits of the model's validity are discussed by focusing on surface geometries and deformation of contacting asperities. With this proposed truncation method, we present a parametric analysis to establish a correlation between contact stiffness and surface roughness descriptors. The contact stiffness shows a unified power-law scaling with respect to the applied load over a wide range for simulated surfaces with distinct sets of roughness descriptors. The exponent of the power-law relationship is found to correlate positively to the fractal dimension while its amplitude is inversely correlated to the surface roughness amplitude. This study provides an easily implemented and computationally efficient method to connect mechanical behaviour with multi-scale surface structure, which can be utilized in design and optimization of engineering applications involving rough contacts.

A revised Majumdar and Bushan model of elastoplastic contact between rough surfaces

A revision of the elastoplastic contact model of fractal rough surfaces, initiated by Majumdar and Bushan, is further developed in this paper. According to the revised contact model, the critical contact areas of a single asperity are scale dependent. With an increase in contact load and contact area, a transition from elastic, elastoplastic to full plastic deformation takes place in this order. The relation between size distribution function for all level asperities and size distribution function for each level asperity is given. The total contact load, real contact area, contact load of each level asperity and contact area of each level asperity are obtained. The revised model shows that mechanical property of rough surface depends on asperity levels. The minimum level asperities and sequential five levels asperities are major contribution to rough surface. When asperity levels nmin and nmin+5 are less than elastic critical level nec, the rough surface appears to have elastic property under contact load. When asperity level nmin is greater than the second elastoplastic critical level npc, the rough surface appears to be carrying inelastic properties because the ratio of elastic deformation to inelastic deformation is very small.

Characterization of 2D and 3D Rough Fractal Surfaces from Backscattered Radar Data

Abstract The scattering of electromagnetic (EM) waves, emitted by a monostatic radar, from two - dimensional (2D) rough fractal surfaces is examined by using the Kirchhoff approximation. We examine the way that the level of roughness of the fractal surface affects the backscattered EM wave captured by a radar as a function of frequency (therefore, a ‘spectral method’) and whether the roughness of the surface can be estimated from these radar measurements. The backscattering coefficient is calculated for a number of radar frequencies and for different values of the surface fractal dimension. It is found that the values of the slopes between the main lobe and the first sidelobes of the backscattering coefficient as a function of the wave number (frequency) of the incident EM waves increase with the surface fractal dimension. Therefore, we conclude that the magnitude of the above slopes provides a reliable method for the classification of the rough fractal surfaces. Furthermore, concerning three - dimensional (3D) fractal surfaces, investigations similar to the above are already performed by the authors and will be presented during the Conference. The above are also investigated in the presence of electronic noise in the radar receiver (effect of SNR values in the above proposed technique).

Structural Analysis on Reduced Graphene Oxide Prepared from Old Coconut Shell by Synchrotron X-Ray Scattering

This work is aimed to identify the distribution of size and structure of the reduced graphene oxide (rGO) particles before and after the mechanical exfoliation process from coconut shell. The burnt sample was heated at 400°C in ambient air, followed by sonication and centrifugation method for various concentration. The characterization were performed by using wide angle X-ray scattering (WAXS) and synchrotron small angle X-ray scattering (SAXS). Based on SAXS result, the structure of rGO showed that in low range of momentum transfer (Q), the particle size was too big to be measured using SAXS. In the high Q, the graph shown the apparent absence of change in curvature which indicates that the small particles are typically seen on high Q range, having aggregate with a new shape and big particles. The average fitting slope is -3.3, implying that the particles structure tends to be fractal (rough) surface, where the particles form aggregates to become bigger size and irregular structure. Further, the WAXS characterization shows that the exfoliation step has not affected the rGO phase formed.

Dynamics of Deformable Fractal Surface in Contact with Harmonically Excited Rigid Flat

Dynamics of contact between a deformable fractal rough surface and a rigid flat is studied under harmonic excitation to the flat surface. Fractal surface is generated from the modified Weierstrass-Mandelbrot function and is imported to ANSYS to construct the finite element model of the same. A parameter called ‘nonlinearity exponent', is obtained from the force-displacement relationship of the rough surface and is used to find out the dynamic properties of the contacting interface using single spring-mass-damper model. The effect of variation in surface roughness and material properties on the system response is analyzed. The system exhibits superharmonic responses for different values of the nonlinearity exponent. The phase plot and time-displacement plots for the system are also furnished.

A note on the pull-off force for a pattern of contacts distributed over a halfspace

We consider a series of flat contact spots distributed over a half-space, for which the pull-off force is proportional to the square root of the total contact area over the elastic compliance. By using an electro-mechanical analogy to compute the compliance using the well-known Greenwood–Holm equation, we show how the pull-off decays for fractal patterns of contact spots with simple scaling laws, tending to zero in a fractal limit, as the contact area goes to zero. Moreover, a qualitative assessment is made for contact of fractal rough surfaces, and it is shown that pull-off in this case is dominated by the value of the contact area reached during the loading process, which depends on the applied load, suggesting pressure-sensitive adhesion.

Characterization of Rough Fractal Surfaces from Backscattered Radar Data

In this paper the scattering of electromagnetic (EM) waves, emitted by a monostatic radar, from rough fractal surfaces is examined by using the Kirchhoff approximation. Of particular interest here is the way that the level of roughness of the fractal surface affects the backscattered EM wave captured by a synthetic aperture radar (SAR) and whether the roughness of the surface can be estimated from these SAR radar measurements. More specifically, the scattering coefficient of the backscattered signal is calculated for a number of radar frequencies and for different values of the surface fractal dimension. It is found here that the slopes between the main lobe and the first sidelobes emerging in the backscattering coefficient as a function of the wavenumber of the incident EM waves increase with the surface fractal dimension. Therefore, we conclude in this paper that the magnitude of the above slopes provides a reliable method for the classification of the rough fractal surfaces. Applications of the proposed method can be found, for example, in the characterization of the sea state from measured SAR radar data.DOI: http://dx.doi.org/10.5755/j01.eie.22.6.17226

Dynamic contact interactions of fractal surfaces

Roughness parameters and material properties have significant influence on the static and dynamic properties of a rough surface. In the present paper, fractal surface is generated using the modified two-variable Weierstrass-Mandelbrot function in MATLAB and the same is imported to ANSYS to construct the finite element model of the rough surface. The force-deflection relationship between the deformable rough fractal surface and a contacting rigid flat is studied by finite element analysis. For the dynamic analysis, the contacting system is represented by a single degree of freedom spring mass-damper-system. The static force-normal displacement relationship obtained from FE analysis is used to determine the dynamic characteristics of the rough surface for free, as well as for forced damped vibration using numerical methods. The influence of fractal surface parameters and the material properties on the dynamics of the rough surface is also analyzed. The system exhibits softening property for linear elastic surface and the softening nature increases with rougher topography. The softening nature of the system increases with increase in tangent modulus value. Above a certain value of yield strength the nature of the frequency response curve is observed to change its nature from softening to hardening.

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.

Numerical Analysis of Normal Contact Stiffness of Rough Surfaces

A numerical model was proposed to investigate the contact behaviour of a solid with a rough surface squeezed against a rigid flat plane. We considered simulated hierarchical surface structures as well as scanned surface data obtained by the profilometry of isotropically roughened specimens. The simulated and treated surfaces were characterised using statistical and fractal parameters. The evolution of contact stiffness under increasing normal compression was analysed through the total truncated area at varying heights, in order to relate contact mechanics to different surface parameters employed for surface characterisation. For a relatively small surface interference, the predicted stress-dependent normal contact stiffness of both scanned and simulated surfaces is in good agreement with experimental observation from nanoindentation tests, revealing a power-law function of the normal load, with the exponent of this relationship closely depending on the fractal dimension of rough surfaces. The numerical results show that the amplitude of a fractal rough surface mainly contributes to the magnitude of the contact stiffness at a given normal load.

Multi-parameter identification and shape reconstruction for unbounded fractal rough surfaces with tapered wave incidence

In this paper, we study the inverse scattering of tapered waves from sound soft surfaces or horizontally polarized electromagnetic waves from perfectly conducting surfaces. By the related Fréchet derivative with respect to a given surface of the solution of direct scattering problems, an efficient algorithm is proposed to reconstruct the parameters of unbounded rough surfaces from a set of scattered field measurements. Tapered wave is introduced to realize the asymptotic truncation for unbounded fractal rough surface. In order to improve the results of reconstruction, multi-angle and multi-frequency incidence strategy has been used. The numerical results show that the method yields satisfactory reconstructions for some rough surface profiles.

Synthetic aperture radar image of fractal rough surface

The synthetic aperture radar imaging of fractal rough surface is studied. The natural surface can be very accurately described in terms of fractal geometry. Such a two-dimensional fractional Brownian motion (FBM) stochastic process provides a very sound description of natural surface. The samples of band-limited FBM process are realized by using physical Weierstrass-Mandelbrot function. The parameters of fractal rough surface are discussed and how to choose the value is analyzed. The roughness is mostly determined by the Hurst coefficient or the fractal dimension. In the actual simulation, a fractal rough surface can be seen as the superposition of finite sinusoidal tones, and any scattering measurement is limited to a finite set of scales. In this paper, the surface is described with two-scale model, i. e., locally approximated by plane facets with dimension smaller than that of resolution cells, but much larger than wavelength. Because this paper focuses on the texture of the synthetic aperture radar (SAR) image and the overall image texture is related to the macroscopic scale, the microscopic roughness superimposed on the facets is neglected. For the macroscopic scale scattering problem, a facet Kirchhoff approach is proposed. The fractal rough surface consists of many triangle facets, and the scatter field of each facet can be obtained by the facet Kirchhoff approach. The principle of dimension selection is studied. The dimension of facet must follow the principle that the surface profile is not damaged. At the same time, the facet dimension should be as large as possible in order to increase the efficiency of imaging. After establishing the fractal geometry model and obtaining the field from each facet, the SAR image can be realized through Rang-Doppler method in the stripmap mode. The results show that in the SAR image, the effects of fractal parameters on the rough surface can be obviously observed. The peaks and ravines of rough surface are obviously observed at low fractal dimension or high Hurst coefficient. However, when the fractal dimension gets higher or Hurst coefficient gets higher, the peaks and ravines disappear because the surface becomes rougher and diffuse scattering is enhanced. The effect of fractal parameter on the SAR image can be specifically expressed with entropy and angle second moment. With the increase of fractal dimension D, the texture of SAR image behaves more randomly and disorderly. So the entropy of SAR image becomes larger and the angle second moment of SAR image becomes smaller. The texture of SAR image is also related to the squint angle and frequency of incidence wave. The relative roughness will become larger when the squint angle and frequency of incidence wave become larger. The research on a complete fractal surface SAR imaging system consists of establishing the environmental model, developing the electromagnetic scattering model, and using the SAR imaging technique. The achievements show the characteristics of fractal rough surface SAR image, which have a theoretical support for natural environment remote sensing and the environment parameters inversion.

Estimation of fractal dimension and surface roughness based on material characteristics and cutting conditions in the end milling of carbon steels

Cutting parameters and material properties have important effects on the quality of milled surface, which can be characterized by fractal dimension and surface roughness. The relationships between two surface parameters (surface roughness and fractal dimension) and material hardness, elongation, spindle speed and feed rate were investigated, respectively, in this study. Four carbon steels, that is, AISI 1020, Gr 50, 1045 and 1566, were milled with five spindle speeds and four feed rates on a computer numerical control machine. The surface topographies were measured with a three-dimensional profiler. The surface profiles were obtained by re-sampling the data points on the surface topography in the measurement direction. The surface roughness and fractal dimension were calculated from the two-dimensional profiles, where the fractal dimension was obtained by the root-mean-square method. The results showed that for specific spindle speed and feed rate, the roughness of the milled surface decreased with the workpiece hardness, whereas the elongation and fractal dimension increased with the hardness. Based on the material hardness and elongation, spindle speed and feed rate, empirical formulae were established to quantitatively estimate the surface roughness and fractal dimension. Moreover, the spindle speed and feed rate can be easily calculated from the empirical formulae to achieve a surface with the desired surface roughness and fractal dimension. The empirical formulae have been demonstrated with the experiments and were shown to be applicable in estimating the surface roughness and fractal dimension for all carbon steels in end milling. The results are instructive for the fractal dimension estimation of the machined surfaces of carbon steel, which has not been previously studied.