Surface Evolution Model Research Articles

Inverse methods to gradient etch three-dimensional features with prescribed topographies using abrasive jet micro-machining: Part I – Modeling

Abrasive jet micro-machining (AJM) can be used to machine a wide range of features on the micro-scale. Given a set of process parameters, surface evolution models capable of predicting the shape of straight, constant-depth channels in a wide variety of materials are well-established. This paper presents a novel framework for solving the unaddressed more challenging and industrially relevant inverse problem of determining the erosive source traverse velocity and path required to machine a particular user-specified feature topography. It is shown that the problem can be expressed as a Fredholm integral equation of the first kind where, for a given desired topography, the source erosive efficacy is a difference kernel, and the unknown function to be determined is the dependency of the instantaneous source velocity on its location. For Gaussian sources typical of AJM, an analytical solution based on the Gaussian transform is presented that can be used for a wide range of desired topographies. For non-Gaussian sources, an approximate approach based on a point-source solution is suggested, and its accuracy is discussed. Once the velocity function to create the desired shallow topography has been determined, it can be used in existing surface evolution models to predict the shape at any depth. The methodologies are demonstrated to be applicable for the AJM of constant depth micro-channels with user-specified cross-sectional shape, gradient etched micro-channels with specified texture along their length, and pockets with texture in two perpendicular directions. The accompanying paper validates the methodology by comparison with measured three-dimensional profiles arising from the predicted velocity functions. The study is the first to consider the inverse problem for AJM, and to demonstrate the ability to etch and predict user specified three-dimensional topographies.

Abrasive jet machining of channels on highly-curved glass and PMMA surfaces

The paper presents a modification to the existing surface evolution models for predicting channel profiles machined using AJM in brittle and ductile materials. Considering the variation in both the local nozzle standoff distance and the divergence angle of each particle trajectory in the jet plume enabled the modified model to address all the shortcomings of existing models for the prediction of channel profiles machined on inclined and highly-curved surfaces. It was observed that the jet footprint differed on the flat and curved surfaces, becoming wider as the rod diameter decreased. Computational fluid dynamics modeling showed that this was not due solely to the expected conical divergence in the jet, but also due to differences in the erosion caused by secondary impacts of rebounding particles. The model predictions compared well with measured surface profiles machined in flat glass and PMMA targets inclined at 30°, and with axial grooves made in 3 and 5mm diameter glass rods and a 4.72mm PMMA rod (maximum error of approximately 8%). With an appropriate characterization of the footprint, the model is in theory applicable to virtually any target geometry and incident angle.

Texture Evolution during Zinc Electrodeposition

This talk investigates the impact of additives and ionic liquids on the stability of zinc interfaces by measuring the temporal evolution of the surface roughness. Experiments are designed to quantify the morphological evolution during charge and discharge cycles in battery electrolytes. We use of in situ electrochemical atomic force microscopy and in situ ultra-small angle x-ray scattering (EC USAXS) to track surface morphology during zinc electrodeposition from two electrolytes: alkaline potassium hydroxide and an ionic liquid, 1-butyl-3-methyl-imidazolium trifluoromethanesulfonate (BMIM OTF). Images are used to quantify the temporal evolution of lateral correlations and surface roughness as a function of deposition thickness. Zinc growth within BMIM OTF displays a surface morphology that has not been described for zinc before; the surface organizes into domains of parallel platelets with the prism facets perpendicular to the surface. Because of this textured morphology, the surface roughness stops growing under some conditions. This cessation of roughening is counter to current models of surface evolution. The coarsening of the grains and the evolution of the roughness are quantitatively described using scaling exponents. The goal of our project is to link early time nucleation events with the onset of large-scale instabilities such as dendrites. Auspices: Portions of this work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

Abrasive waterjet micro-machining of channels in metals: Model to predict high aspect-ratio channel profiles for submerged and unsubmerged machining

Recent advances in the development of miniaturized nozzles have made possible the use of abrasive waterjets to perform controlled-depth micro-milling. Haghbin et al. (2015) discussed the effects of the surrounding fluid in the micro-machining of shallow channels in 316L stainless steel and 6061-T6 aluminum using a prototype nozzle having an orifice diameter of 127μm and a 254μm mixing tube diameter. This paper uses those results to develop a new surface evolution model that predicts the size and shape of relatively deep micro-channels resulting from unsubmerged and submerged abrasive water jet micro-machining (AWJM). For both unsubmerged and submerged AWJM, and for both materials, the erosive efficacy distribution changed suddenly after the initial formation of the channel. The initially wide distribution was due to backflow of the abrasive slurry along the channel walls, which did not occur once the channel was formed and most of the flow was directed along the channel length. A novel approach in which two different erosive efficacy expressions are sequentially used in a surface evolution equation is presented and shown to accurately predict the evolving surface topography for micro-channels up to aspect ratios of 3.

A comparison of interatomic potentials for modeling tungsten–hydrogen–helium plasma–surface interactions

We compare the hydrogen and helium clustering characteristics of three interatomic potential energy models intended for simulation of plasma-facing materials for fusion applications. Our simulations compare a Finnis–Sinclair potential and two different Tersoff-style bond order potentials created by Juslin et al. (2005) and Li et al. (2011), respectively, with respect to both helium and hydrogen clustering behavior in tungsten. We find significant differences between the Juslin and Li potentials in terms of both hydrogen and helium clustering behavior as well as the spatial distribution of hydrogen below the surface. These simulations are an important test on the road to more accurate models of gas clustering and surface evolution of tungsten divertors in ITER and other plasma devices.

Automated recognition of quasi‐planar ignimbrite sheets as paleosurfaces via robust segmentation of digital elevation models: an example from the Central Andes

ABSTRACTQuasi‐planar morphological surfaces may become dissected or degraded with time, but still retain original features related to their geologic‐geomorphic origin. To decipher the information hidden in the relief, recognition of such features is required, possibly in an automated manner. In our study, using Shuttle Radar Topography Mission (SRTM) digital elevation model (DEM), an existing algorithm has been adapted to recognize quasi‐planar features fulfilling specified criteria. The method has been applied to a study area of the Central Andes with Miocene to Quaternary volcanic edifices, tilted ignimbrite surfaces, and basin‐filling sediments. The result is a surface segmentation, whereas non‐planar features (gullies, tectonic faults, etc.) are sorted out. The main types of geomorphic features that can be distinguished and interpreted are as follows. (1) The west‐dipping western margin of the Altiplano is differentiated into segments of the lower sedimentary cover that of increased erosion by tectonic steepening at intermediate levels, and an upper plane with limited erosion. (2) In the central part of the Western Cordillera, the Oxaya ignimbrite block shows a ‘striped’ bulging pattern that results from a smoothly changing surface dip. This pattern is due to continuous folding/warping of the ignimbrite block possibly related to gravitational movements. (3) To the west, large, uniform planes correspond to flat, smooth, tectonically undisturbed surfaces of young sedimentary cover of the Central Basin. (4) The evolution of Taapaca volcanoes with sector collapse events and cone‐building phases is shown by several segments with overlapping clastic aprons. (5) To the east, on the western margin of the Altiplano, young intermontane basins filled by Upper Miocene sediments show progressively increasing dip toward basin margins, reflected by a circular pattern of the segmentation planes. We show that the segmentation models provide meaningful images and additional information for geomorphometric analysis that can be interpreted in terms of geological and surface evolution models. Copyright © 2014 John Wiley & Sons, Ltd.

An optimization method for ion etching yield modeling combined with factual etching data

The profile surface simulation is an important method to study the ion etching mechanism. In profile surface simulation, the result of surface evolution is primarily determined by the surface evolution model and the etching yield optimization model as well. However, the currently available surface evolution model is not accurate enough. What's more, most of the data used in etching yield optimization model are based on simulation, while no factual data are used to optimize the parameters of ion etching yield model. In order to solve these problems, the accuracy of current evolution model is improved, the optimal objects of etching yield model are redefined, and the factual etching data are introduced to optimize the etching yield model for the first time. In this paper, parallel method is also adopted to speed up the optimization process, whose optimized parameters are then applied to the etching simulation process that is based on cellular automata. The experimental results show that our proposed approach does improve the accuracy of simulation and greatly shorten the optimization process.

A comparison of empirical and physically based glacier surface melt models for long-term simulations of glacier response

AbstractWe investigate the performance of five glacier melt models over a multi-decadal period in order to assess their ability to model future glacier response. The models range from a simple degree-day model, based solely on air temperature, to more-sophisticated models, including the full shortwave radiation balance. In addition to the empirical models, the performance of a physically based energy-balance (EB) model is examined. The melt models are coupled to an accumulation and a surface evolution model and applied in a distributed manner to Rhonegletscher, Switzerland, over the period 1929–2012 at hourly resolution. For calibration, seasonal mass-balance measurements (2006–12) are used. Decadal ice volume changes for six periods in the years 1929–2012 serve for model validation. Over the period 2006–12, there are almost no differences in performance between the models, except for EB, which is less consistent with observations, likely due to lack of meteorological in situ data. However, simulations over the long term (1929–2012) reveal that models which include a separate term for shortwave radiation agree best with the observed ice volume changes, indicating that their melt relationships are robust in time and thus suitable for long-term modelling, in contrast to more empirical approaches that are oversensitive to temperature fluctuations.

Surface evolution models for abrasive slurry jet micro-machining of channels and holes in glass

Abrasive slurry jet micro-machining (ASJM) uses a jet of abrasive slurry to erode features with relatively high resolution without the need for a patterned mask. The present study investigated the ability of a surface evolution model to predict the profiles of micro-channels and holes machined in borosilicate glass with a newly developed ASJM system. The system could produce micro-channels with depth and width variations along their length of less than 3%, and a channel-to-channel repeatability within 5%. The fundamental erosion rate of the borosilicate glass was measured as a function of impact angle using a slurry of water mixed with a low concentration of 10 and 25µm nominal diameter aluminum oxide particles. This erosion rate-impact angle relationship was used in an existing model developed previously for the abrasive air jet micro-machining of brittle materials. The results demonstrated that, despite the differences in abrasive flow patterns between air and slurry based systems, the surface evolution model accurately predicted the profiles of micro-channels with a maximum error of 7% for aspect ratios (depth/width) of up to 5. The predicted profiles of holes were also in reasonable agreement with a maximum error of 14% for aspect ratios close to 1.

A quasi-variational inequality problem arising in the modeling of growing sandpiles

Existence of a solution to the quasi-variational inequality problem arising in a model for sand surface evolution has been an open problem for a long time. Another long-standing open problem concerns determining the dual variable, the flux of sand pouring down the evolving sand surface, which is also of practical interest in a variety of applications of this model. Previously, these problems were solved for the special case in which the inequality is simply variational. Here, we introduce a regularized mixed formulation involving both the primal (sand surface) and dual (sand flux) variables. We derive, analyse and compare two methods for the approximation, and numerical solution, of this mixed problem. We prove subsequence convergence of both approximations, as the mesh discretization parameters tend to zero; and hence prove existence of a solution to this mixed model and the associated regularized quasi-variational inequality problem. One of these numerical approximations, in which the flux is approximated by the divergence-conforming lowest order Raviart-Thomas element, leads to an efficient algorithm to compute not only the evolving pile surface, but also the flux of pouring sand. Results of our numerical experiments confirm the validity of the regularization employed. 0 (Ω) and the nonnegative density of the distributed source f ∈ L 2 (0 ,T ; L 2 (Ω)) are given. We consider the growing sandpile y = w(x ,t ) and set an open boundary condition w|∂Ω =0 . Denoting by q(x ,t ) the horizontal projection of the flux of material pouring down the evolving pile surface, we can write the mass balance equation ∂w ∂t + ∇ .q = f. (1.1)

A Monte Carlo simulation model for surface evolution by plasma etching

A Monte Carlo simulation model, i.e. sputtering etch model, was established with Monte Carlo method to simulate plasma etching induced surface morphology evolution. The surface morphology images and fractal exponents were obtained, with α=0.5, β=0.27, z=1.76. Germanium samples were etched in SF6 plasma and the post-etch surface was analyzed in order to compare with the model. The surface morphology images and fractal exponents by simulation describe the experimental results very well.

Prediction of machined surface evolution in the abrasive jet micro-machining of metals

Surface evolution models have been used in the past to accurately predict the cross-sectional profile of micro-channels resulting from the abrasive jet micro-machining (AJM) of glass and polymeric substrates. In the present paper, the models are suitably modified and applied for the first time to the AJM of metallic substrates. The dependence of erosion rate on abrasive jet inclination angle was measured for aluminum 6061-T6, Ti–6Al–4V alloy, and 316L stainless steel using 50μm Al2O3 abrasive powder launched at an average velocity of 106m/s. For all three systems the peak erosion rate was found to occur when the jet was inclined between 20° and 35° relative to the surface. The AJM etch rate was found to be much lower than that found in glass and polymers, and it was found that a significant amount of particle embedding occurred in the 316L stainless steel. When the erosion data were used in an AJM surface evolution model, the resulting predicted cross-sectional profiles of unmasked and masked channels were in reasonable agreement with the measured profiles up to an aspect ratio (channel depth/width) of 1.25. The results demonstrate that surface evolution during the abrasive jet micro-machining of metals can be predicted using existing models, originally developed for glass and polymers.

Modeling the growth of stylolites in sedimentary rocks

Stylolites are ubiquitous pressure solution seams found in sedimentary rocks. Their morphology is shown to follow two self‐affine regimes. Analyzing the scaling properties of their height over their average direction shows that (1) at small scale, they are self‐affine surfaces with a Hurst exponent around 1, and (2) at large scale, they follow another self‐affine scaling with Hurst exponent around 0.5. In the present paper, we show theoretically the influence of the main principal stress and the local geometry of the stylolitic interface on the dissolution reaction rate. We compute how it is affected by the deviation between the principal stress axis and the local interface between the rock and the soft material in the stylolite. The free energy entering in the dissolution reaction kinetics is expressed from the surface energy term and via integration from the stress perturbations due to these local misalignments. The resulting model shows the interface evolution at different stress conditions. In the stylolitic case, i.e., when the main principal stress is normal to the interface, two different stabilizing terms dominate at small and large scales which are linked respectively to the surface energy and to the elastic interactions. Integrating the presence of small‐scale heterogeneities related to the rock properties of the grains in the model leads to the formulation of a Langevin equation predicting the dynamic evolution of the surface. This equation leads to saturated surfaces obeying the two observed scaling laws. Analytical and numerical analysis of this surface evolution model shows that the crossover length separating both scaling regimes depends directly on the applied far‐field stress magnitude. This method gives the basis for the development of a paleostress magnitude marker. We apply the computation of this marker, i.e., the morphological analysis, on a stylolite found in the Dogger limestone layer located in the neighborhood of the ANDRA Underground Research Laboratory at Bure (eastern France). The results are consistent with the two scaling regimes expected, and the practical determination of the major principal paleostress, from the estimation of a crossover length, is illustrated on this example.

The Fate of Terrestrial Organic Carbon in the Marine Environment

Understanding the fate of terrestrial organic carbon (Corg) delivered to oceans by rivers is critical for constraining models of biogeochemical cycling and Earth surface evolution. Corg fate is dependent on both intrinsic characteristics (molecular structure, matrix) and the environmental conditions to which fluvial Corg is subjected. Three distinct patterns are evident on continental margins supplied by rivers: (a) high-energy, mobile muds with enhanced oxygen exposure and efficient metabolite exchange have very low preservation of both terrestrial and marine Corg (e.g., Amazon subaqueous delta); (b) low-energy facies with extreme accumulation have high Corg preservation (e.g., Ganges-Brahmaputra); and (c) small, mountainous river systems that sustain average accumulation rates but deliver a large fraction of low-reactivity, fossil Corg in episodic events have the highest preservation efficiencies. The global patterns of terrestrial Corg preservation reflect broadly different roles for passive and active margin systems in the sedimentary Corg cycle.

Cryogenic abrasive jet machining of polydimethylsiloxane at different temperatures

The temperature dependence of the solid particle erosion of polydimethylsiloxane (PDMS) using aluminum oxide particles was investigated between the temperatures of −178 and 17°C for a variety of angles of attack using a novel cryogenic abrasive jet machining apparatus. It was found that the most efficient machining of PDMS (volume removed per kinetic energy of erodent) occurred at approximately −178°C, at angles of attack between 30° and 60° from the surface. A previously developed surface evolution model was used to predict the size and shape of unmasked channels at various temperatures. A good agreement between the predicted and measured channel profiles was obtained when the average blasting temperature was between approximately −127 and −178°C. At −82°C, the fit was poorer, probably because of an increase in particle embedding. Although it was demonstrated that PDMS could be machined at temperatures above its glass transition, the erosion rate increased by a factor of more than 10 when the machining temperature was below this point.

Laser shadowgraphy measurements of abrasive particle spatial, size and velocity distributions through micro-masks used in abrasive jet micro-machining

In abrasive jet micromachining (AJM), a jet of particles is passed through narrow mask openings in order to define the features to be micro-machined. The size and shape of the micro-machined features depends on the distribution of the particle velocity and mass flux through the mask openings. In this work, a high speed laser shadowgraphy technique was used to demonstrate experimentally, for the first time, the significant effect that the mask opening size and powder shape and size have on the resulting distribution of particle mass flux and velocity through the mask opening. In particular, it was found that the velocity through the mask was approximately constant, but different in magnitude than the velocity in the free jet incident to the mask. The measured mass flux distributions were in excellent agreement with a previously developed analytical model, thus directly confirming its validity. Additional measurements also showed that an existing numerical model could be used to predict the velocity distribution in free jets of spherical particles, and, if a modification to the particle drag coefficient is made, in free jets of angular particles. The direct experimental verification of these models allows for their use in surface evolution models that can predict the evolving shape of features micro-machined using AJM.

Thermal analysis of cryogenically assisted abrasive jet micromachining of PDMS

In cryogenic abrasive jet micromachining (CAJM), it is important to sufficiently cool the machined surface. However, over-cooling or cooling a larger volume than necessary, reduces the cost-effectiveness of the process. With a view to process optimization, a finite element thermal analysis has been made of the CAJM of channels and holes in polydimethylsiloxane (PDMS). For the machining of holes, using a previously developed experimental CAJM setup, it was found that the thermal front moved nearly 12 times faster than the material removal front, implying that the target PDMS was cooled sufficiently prior to micromachining. The time required to cool a chip removed by a single particle was calculated to be 0.7 ms while the time required to machine it was 7.7 ms indicating the possibility of optimizing the cooling process by reducing the heat flux. A finite element analysis (FEA) was used to simulate a moving heat sink to investigate the general cooling behavior during the machining of unmasked and masked channels and to assess the influence of the jet scanning speed. The temperatures predicted by the FEA agreed reasonably well with those that were measured using embedded thermocouples. Under steady-state conditions, the temperature distribution within the moving jet footprint was highly asymmetric, being much colder toward the trailing edge and the target temperature at a given position decreased with decreasing scan speed. A coupled thermal/structural finite element analysis was used to examine the influence of thermally induced strains on the final room-temperature shape of micromachined features. It was found that accounting for these thermal strains in the surface evolution model significantly improved the agreement of predicted with measured profiles.

An analytical model of the effect of particle size distribution on the surface profile evolution in abrasive jet micromachining

An analytical model to estimate the spatial distribution of erosive efficacy across the mask opening in the abrasive jet micromachining (AJM) of substrates is presented. A closed form analytical expression is derived which allows the erosive efficacy in the vicinity of the mask edge to be estimated as a function of the measured abrasive particle size distribution and the width of the mask opening. This analytical expression was used in a previously developed analytical surface evolution model to predict the time dependent eroding surface profiles of micro-holes and micro-channels of various sizes in glass and polymethylmethacrylate (PMMA), using aluminum oxide abrasive powders of different sizes. Use of the measured powder size distributions in the analytical models resulted in excellent agreement between the measured and model predicted channel profiles. The results of the study demonstrate that the particle size distribution and mask opening width can greatly affect the shape and depth of micro-channel profiles. A major improvement over previously developed models is ease-of-application since the erosive efficacy is given by an analytical expression rather than by the use of a computer simulation or a semi-empirical approach.

Effect of elastic anisotropy on surface pattern evolution of epitaxial thin films

Stress-induced surface instability and evolution of epitaxial thin films leads to formation of a variety of self-assembled surface patterns with feature sizes at micro- and nanoscales. The anisotropy in both the surface and bulk properties of the film and substrate has profound effects on the nonlinear dynamics of surface evolution and pattern formation. Experimentally it has been demonstrated that the effect of anisotropy strongly depends on the crystal orientation of the substrate surface on which the film grows epitaxially. In this paper we develop a nonlinear model for surface evolution of epitaxial thin films on generally anisotropic crystal substrates. Specifically, the effect of bulk elastic anisotropy of the substrate on epitaxial surface pattern evolution is investigated for cubic germanium (Ge) and SiGe films on silicon (Si) substrates with four different surface orientations: Si(0 0 1), Si(1 1 1), Si(1 1 0), and Si(1 1 3). Both linear analysis and nonlinear numerical simulations suggest that, with surface anisotropy neglected, ordered surface patterns form under the influence of the elastic anisotropy, and these surface patterns clearly reflect the symmetry of the underlying crystal structures of the substrate. It is concluded that consideration of anisotropic elasticity reveals a much richer dynamics of surface pattern evolution as compared to isotropic models.

Morphologies and kinetics of a dewetting ultrathin solid film

The surface evolution model based on a geometric partial differential equation is used to numerically study the kinetics of dewetting and the dynamic morphologies for the localized pinhole defect in the surface of an ultrathin solid film with the strongly anisotropic surface energy. Depending on the parameters such as the initial depth and width of the pinhole, the strength of the attractive substrate potential and the strength of the surface-energy anisotropy, the pinhole may either extend to the substrate and thus rupture the film, or evolve to the quasiequilibrium shape while the rest of the film surface undergoes a phase separation into a hill-and-valley structure followed by coarsening. Emergence of the quasiequilibrium shape and the termination of a dewetting are associated with the faceting of the pinhole tip. Overhanging (nongraph) morphologies are possible for deep, narrow (slitlike) pinholes.