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Comparisons are made among Molecular Dynamics (MD), Classical Density Functional Theory (c-DFT), and Poisson-Boltzmann (PB) modeling of the electric double layer (EDL) for the nonprimitive three component model (3CM) in which the two ion species and solvent molecules are all of finite size. Unlike previous comparisons between c-DFT and Monte Carlo (MC), the present 3CM incorporates Lennard-Jones interactions rather than hard-sphere and hard-wall repulsions. c-DFT and MD results are compared over normalized surface charges ranging from 0.2 to 1.75 and bulk ion concentrations from 10 mM to 1 M. Agreement between the two, assessed by electric surface potential and ion density profiles, is found to be quite good. Wall potentials predicted by PB begin to depart significantly from c-DFT and MD for charge densities exceeding 0.3. Successive layers are observed to charge in a sequential manner such that the solvent becomes fully excluded from each layer before the onset of the next layer. Ultimately, this layer filling phenomenon results in fluid structures, Debye lengths, and electric surface potentials vastly different from the classical PB predictions.

We present optimization algorithms for the design of complex hierarchical systems, motivated by applications to the design of nanoporous materials. Nanoporous materials have a broad range of engineering applications, including gas storage and filtration, electrical energy storage in batteries and capacitors, and catalysis. The design of such materials involves modeling of the material over many length scales, leading to a hierarchy of mathematical models. Our algorithms are also hierarchical in structure with the goal of exploiting the model hierarchy to obtain solutions more rapidly. We discuss the choice of optimization models, initialization schemes, the hierarchical optimization algorithm, software design, and computational results.

Measured electrochemical impedance spectra of porous electrodes comprised of redox-active ruthenium oxide and inert niobium hydroxide are compared with the results of structurally consistent mathematical models describing coupled processes of electron transport in the solid matrix, ion transport in the electrolyte, proton transport within the ruthenium oxide particles, and redox reaction on particle surfaces. Addition of moderate amounts of niobium to crystalline ruthenium oxide is found to improve the frequency response due to enhanced intraparticle proton transport. However, excessive niobium reduces ion and electron transport through the electrode thickness, reducing the available capacitance. Thus, an optimum composition is needed to achieve the best balance in transport properties. Near this optimum, the intraparticle proton transport undergoes a transition from a constant phase element (CPE) response for Ru-rich materials to a classical Warburg diffusion response for Nb-rich compositions. The CPE regime is analyzed in detail to identify fractal-like structures as well as alternative radial distributions of intraparticle proton diffusivity consistent with measured response. The models involving variations in radial diffusivity appear most probable and have nearly exponential decreases in radial diffusivity with distance from particle surfaces similar to a Debye distribution of charge carriers in an electric double layer.

A simple solution-based method has been developed to vary the composition of redox active ruthenium oxide with highly proton-conducting niobium hydroxide to create stable, high capacitance electrodes at elevated temperatures. This method presents a dramatic departure from most other ruthenium oxide systems, which are prepared through annealing of hydrous ruthenium oxide. Typically RuO 2 processed at high temperature only exhibits high electrical conductivity and suffers from poor proton conduction, giving low overall capacitances. Here, the optimized Ru/Nb oxide composition can be used to achieve high power densities, high capacitances, and stabilized electrodes while significantly reducing ruthenium content. Extensive materials characterization including high-resolution cross-sectional TEM, elemental mapping, XRD, electrochemical impedance spectroscopy, and proton NMR were used to evaluate the structure of the material system. The electrochemically inert niobium oxide serves as a network former enhancing accessibility to redox active ruthenium oxide. The dispersion of RuO 2 in the NbO(OH) x matrix results in reduced RuO 2 particle size, as observed via TEM and XRD, while also increasing the proton concentration in the material. Interconnected RuO 2 particles provide electrically conducting pathways, even at low Ru contents, where percolation networks remain intact. Ruthenium is more efficiently utilized in the Ru/Nb composites and ruthenium content can be significantly reduced without decreasing capacitive performance. In addition, the composite electrodes, with the fine mixing of Ru and Nb, give higher power performance than for RuO 2 alone.

Numerical methods are employed to examine the resolution and optimization of a relatively new technique for charged species separation that is based on flow along nano-scale channels having an electric double-layer thickness comparable to the channel size. In such channels, the electric field inherent to the double-layer produces transverse species distributions that depend on the species charge. Flow along the channel thus yields mean axial species speeds that also depend on the species charge, enabling species separation and identification. Building on earlier work describing retention and plate heights, here we characterize this new type of field-flow fractionation via the classic metric of resolution. Sample results are presented and discussed for a wide range of conditions for both pressure-driven and electroosmotic flows. Optimum design and operating conditions are also examined. We find that resolution is maximized for optimum values of the zeta potential and Debye layer thickness and that these optima depend strongly on the species charge or range of charges of interest. Under optimum conditions, acceptable resolution can be obtained over a wide range of species charges for pressure-driven flows. This separable range of charges is much smaller for electroosmotic flows. Finally, sample calculations are presented showing that all species in the range of charge between -8 to 10 can be separated simultaneously with resolutions above unity, and this is possible in less than 6 s under optimum conditions that are readily achievable.

Analytical and numerical solutions are presented for steady evaporating flow in open microchannels having a rectangular cross section and a uniform depth. The flow, driven by the axial gradient of capillary pressure, generally consists of an entry region where the meniscus is attached to the top corners of the channel followed by a jump-like transition to a corner-flow region in which the meniscus progressively recedes into the bottom corners of the channel. Illustrative numerical solutions are used to guide the derivation of an easily applied analytical approximation for the maximum sustainable heat flux or capillary limit.

Numerical methods are employed to examine the work, electric power input, and efficiency of electrokinetic pumps at a condition corresponding to maximum pump work. These analyses employ the full Poisson-Boltzmann equations and account for both convective and conductive electric currents, including surface conductance. We find that efficiencies at this condition of maximum work depend on three dimensionless parameters, the normalized zeta potential, normalized Debye layer thickness, and a fluid property termed the Levine number indicating the nominal ratio of convective to conductive electric currents. Efficiencies at maximum work exhibit a maximum for an optimum Debye layer thickness when the zeta potential and Levine number are fixed. This maximum efficiency increases with the square of the zeta potential when the zeta potential is small, but reaches a plateau as the zeta potential becomes large. The maximum efficiency in this latter regime is thus independent of the zeta potential and depends only on the Levine number. Simple analytical expressions describing this maximum efficiency in terms of the Levine number are provided. Geometries of a circular tube and planar channel are examined.

Analytical expressions are derived for the mean velocity of a liquid flowing in an open rectangular microchannel. Solutions are decomposed into additive components driven by pressure gradients and by shear stresses on the liquid/vapor interface. Speeds are computed numerically for meniscus contact angles ranging from 0° to 90°, arbitrary channel aspect ratios, and for wetting regimes ranging from liquid-full to nearly-dry corner flows. These numerical results are used to guide the development of several analytical expressions that apply in asymptotic limits of fluid depth and contact angle. The resulting asymptotes are then blended analytically to obtain relatively simple, accurate, and comprehensive expressions for the mean velocity.

Although acoustic agitation is known to en- hance the development rate of LIGA resists, the operative physical mechanism is not well understood. Analytical and numerical models presented here suggest that the ob- served enhancement results from steady acoustic stream- ing and associated polymer fragment transport induced by the high frequency sound waves. The computed steady flow within the feature is torroidal with downflow along the feature walls and upflow in the center. The numerically calculated enhancement of fragment transport is shown to be well approximated by a closed-form expression based on a pair of asymptotic formulas for weak and strong agitation. These transport models are combined with a model of PMMA dissolution kinetics and used to predict development rates for a range of LIGA feature sizes. The results are in good agreement with experimental data from

Analytical and numerical methods are employed to investigate the role of PMMA fragment transport in resist development for the LIGA process. We demonstrate that the overall development time can be expressed as the sum of the kinetic-limited development time and a time for fragment transport. The kinetic-limited time depends only on the resist thickness, dose profile and development temperature and is independent of feature size. The transport time grows as the square of the resist thickness and falls inversely with the Sherwood number. A new analytical model describing the Sherwood number for forced convective transport in deep cavities is also developed. This model, applicable to both resist development and electroforming, is compared with numerical simulations and with data previously reported. Based on this model, we find that forced convective transport can significantly reduce development times only for features having an aspect ratio less than about five. Acoustic agitation is also discussed, and sample calculations of the development time are presented for both forced convective transport and acoustic agitation over a wide range of feature sizes.

Analytical and numerical methods are employed to determine the electric potential, fluid velocity, and late-time solute distribution for electroosmotic flow in a tube and channel at zeta potentials that are not necessarily small. The electric potential and fluid velocity are in general obtained by numerical means. In addition, new analytical solutions are presented for the velocity in a tube and channel in the extremes of large and small Debye layer thickness. The electroosmotic fluid velocity is used to analyze late-time transport of a neutral nonreacting solute. Zero- and first-order solutions describing axial variation of the solute concentration are determined analytically. The resulting expressions contain eigenvalues representing the dispersion and skewness of the axial concentration profiles. These eigenvalues and the functions describing transverse variation of the concentration field are determined numerically using a shooting technique. Results are presented for both tube and channel geometries over a wide range of the normalized Debye layer thickness and zeta potential. Simple analytical approximations to the eigenvalues are also provided for the limiting cases of large and small values of the Debye layer thickness.

Integral methods are used to derive an analytical expression describing fluid condensation pressures in slit pores bounded by parallel plane walls. To obtain this result, the governing equations of density functional theory (DFT) are integrated across the pore width assuming that fluid densities within adsorbed layers are spatially uniform. The thickness, density, and free energy of these layers are expressed as composite functions constructed from asymptotic limits applicable to small and large pores. By equating the total free energy of the adsorbed layers to that of a liquid-full pore, we arrive at a closed-form expression for the condensation pressure in terms of the pore size, surface tension, and Lennard-Jones parameters of the adsorbent and adsorbate molecules. The resulting equation reduces to the Kelvin equation in the large-pore limit. It further reproduces the condensation pressures computed by means of the full DFT equations for all pore sizes in which phase transitions are abrupt. Finally, in...

A combined analytical and numerical method is employed to optimize process conditions for composites fiber coating by chemical vapor infiltration. For a first-order deposition reaction, the optimum pressure yielding the maximum deposition rate at a preform center is obtained in closed form and is found to depend only on the activation energy of the deposition reaction, the characteristic pore size, and properties of the reactant and product gases. It does not depend on the preform specific surface area, effective diffusivity or preform thickness, nor on the gas-phase yield of the deposition reaction. Further, this optimum pressure is unaltered by the additional constraint of a prescribed deposition uniformity. Optimum temperatures are obtained using an analytical expression for the optimum value along with numerical solutions to the governing transport equations. These solutions account for both diffusive and advective transport, as well as both ordinary and Knudsen diffusion. Sample calculations are presented for coating preform fibers with boron nitride.

Gravitational stresses limit maximum temperatures for processing semiconductor wafers. Allowable rates of heating and cooling are also limited by the combined affects of thermal and gravitational stresses, particularly in batch processing of stacked wafers. With increasing wafer diameter these limitations become more severe, requiring such measures as improved wafer support to maintain a desired processing temperature or increased wafer spacing to maintain desired rates of heating and cooling. These issues are addressed here through the derivation of scaling rules and engineering formulas relating allowable operating conditions to wafer dimensions. Gravitational stresses and displacements are first determined analytically for wafers supported by rings or at discrete points. The maximum operating temperature is then computed by comparing the calculated gravitational stresses with the temperature-dependent wafer strength. The difference between wafer strength and gravitational stress is used to determine the allowable thermal stress and allowable temperature variation across the wafer. Finally, an analytical model of radial heat transfer in a batch furnace is used to compute the maximum permissible rates of heating and cooling as a function of the instantaneous wafer temperature. Example calculations illustrate the dependence of maximum operating temperatures and allowable ramp rates on wafer dimensions, support geometry, and wafer spacing.

To better understand and to help optimize the electroforming portion of the LIGA process, we have developed one and two-dimensional numerical models describing electrode-position of metal into high aspect-ratio molds. The one-dimensional model addresses dissociation, diffusion, electromigration, and deposition of multiple ion species. The two-dimensional model is limited to a single species, but includes transport induced by forced flow of electrolyte outside the mold and by buoyancy associated with metal ion depletion within the mold. To guide model development and to validate these models, we have also conducted a series of laboratory experiments using a sulfamate bath to deposit nickel in cylindrical molds having aspect ratios up to twenty-five. The experimental results indicate that current densities well in excess of the diffusion-limited currents may still yield metal deposits of acceptable morphology. However, the numerical models demonstrate that such large ion fluxes cannot be sustained by convection within the mold resulting from flow across the mold top. Instead, calculations suggest that the observed enhancement of transport probably results from natural convection within the molds, and that buoyancy-driven flows may be critical to metal ion transport even in micron-scale features having very large aspect ratios. Taking advantage of this enhanced ion transportmore » may allow order-of-magnitude reductions in electroforming times for LIGA microdevice fabrication. 42 refs., 14 figs., 1 tab.« less

The fluid density distribution within microscopic pores is determined by solving integral equations relating the local chemical potential to the van der Waals attractions and hard sphere repulsions of surrounding material. To avoid resolving the density distribution on submolecular scales, the governing equations are averaged over zones of molecular size using analytic functions to represent local density variations within each zone. These local density profiles range from singularities to uniform distributions depending on the local variation of the potential field. Sample calculations indicate that this integral approach yields results in very good agreement with those based on traditional density functional theory, while reducing computing times by three or four orders of magnitude for one-dimensional geometries.

A second-order perturbation solution describing the radial transport of a reactive species and concurrent deposition on wafer surfaces is derived for use in optimizing chemical vapor deposition process conditions. The result is applicable to various deposition reactions and accounts for both diffusive and advective transport, as well as both ordinary and Knudsen diffusion. Based on the first-order approximation, the deposition rate is maximized subject to a constraint on the radial uniformity of the deposition rate. For a fixed reactant mole fraction, the optimum pressure and optimum temperature are obtained using the method of Lagrange multipliers. This yields a weak one-sided maximum; deposition rates fall as pressures are reduced but remain nearly constant at all pressures above the optimum value. The deposition rate is also maximized subject to dual constraints on the uniformity and particle nucleation rate. In this case, the optimum pressure, optimum temperature, and optimum reactant fraction are similarly obtained, and the resulting maximum deposition rate is well defined.

A one-dimensional isobaric reactor model is used to simulate hydrogen isotope exchange processes taking place during flow through a powdered palladium bed. This simple model is designed to serve primarily as a platform for the initial development of detailed chemical mechanisms that can then be refined with the aid of more complex reactor descriptions. The one-dimensional model is based on the Sandia in-house code TPLUG, which solves a transient set of governing equations including an overall mass balance for the gas phase, material balances for all of the gas-phase and surface species, and an ideal gas equation of state. An energy equation can also be solved if thermodynamic properties for all of the species involved are known. The code is coupled with the Chemkin package to facilitate the incorporation of arbitrary multistep reaction mechanisms into the simulations. This capability is used here to test and optimize a basic mechanism describing the surface chemistry at or near the interface between the gas phase and a palladium particle. The mechanism includes reversible dissociative adsorptions of the three gas-phase species on the particle surface as well as atomic migrations between the surface and the bulk. The migration steps are more general than those used previously in that they do not require simultaneous movement of two atoms in opposite directions; this makes possible the creation and destruction of bulk vacancies and thus allows the model to account for variations in the bulk stoichiometry with isotopic composition. The optimization code APPSPACK is used to adjust the mass-action rate constants so as to achieve the best possible fit to a given set of experimental data, subject to a set of rigorous thermodynamic constraints. When data for nearly isothermal and isobaric deuterium-to-hydrogen (D {yields} H) and hydrogen-to-deuterium (H {yields} D) exchanges are fitted simultaneously, results for the former are excellent, while those for the latter show pronounced deviations at long times. These discrepancies can be overcome by postulating the presence of a surface poison such as carbon monoxide, but this explanation is highly speculative. When the method is applied to D {yields} H exchanges intentionally poisoned by known amounts of CO, the fitting results are noticeably degraded from those for the nominally CO-free system but are still tolerable. When TPLUG is used to simulate a blowdown-type experiment, which is characterized by large and rapid changes in both pressure and temperature, discrepancies are even more apparent. Thus, it can be concluded that the best use of TPLUG is not in simulating realistic exchange scenarios, but in extracting preliminary estimates for the kinetic parameters from experiments in which variations in temperature and pressure are intentionally minimized.

Channel widths and spacing in latticelike hierarchical transport networks are optimized to achieve maximum extraction of gas or electrical charge from nanoporous energy-storage materials during charge and discharge cycles of specified duration. To address a range of physics, the effective transport diffusivity is taken to vary as a power, $m$, of channel width. Optimal channel widths and spacing in all levels of the hierarchy are found to increase in a power-law manner with normalized system size, facilitating the derivation of closed-form approximations for the optimal dimensions. Characteristic response times and ratios of channel width to spacing are both shown to vary by the factor $2/m$ between successive levels of any optimal hierarchy. This leads to fractal-like self-similar geometry, but only for $m=2$. For this case of quadratic dependence of diffusivity on channel width, the introduction of transport channels permits increases in system size on the order of ${10}^{4}$, ${10}^{8}$, and ${10}^{10}$, without any reduction in extraction efficiency, for hierarchies having 1, 2 and, 8 levels, respectively. However, we also find that for a given system size there is an optimum number of hierarchical levels that maximizes extraction efficiency.

Porous materials having multiple scales of porosity afford the opportunity to combine the high surface area and functionality of nanopores with the superior charge/discharge characteristics of wider transport channels. However, the relative volume fractions assigned to nanopores and transport channels must be thoughtfully balanced because the introduction of transport channels reduces the volume available for nanopore functionality. In the present paper, the optimal balance between nanopore capacity and system response time is achieved by adjusting the aperture and spacing of a family of transport channels that provide access to adjacent nanopores during recharge/discharge cycles of materials intended for storage of gas or electric charge. A diffusive transport model is used to describe alternative processes of viscous gas flow, Knudsen gas flow, and ion diffusion or electromigration. The coupled transport equations for the nanopores and transport channels are linearized and solved analytically for a periodic variation in external gas pressure, ion concentration, or electric potential using a separation-of-variables approach in the complex domain. Optimization of these solutions yields closed-form expressions for channel apertures and spacing that provide maximum discharge of gas or electric charge for a fixed system volume and a desired discharge time.

Analytical solutions are derived for the quasisteady shape and speed of a single wormhole resulting from the coupled processes of Darcian fluid motion and chemical dissolution in a soluble permeable material. For an initially unsaturated medium, two-dimensional solutions are obtained by addressing an inverted free-boundary problem in which the spatial coordinates are treated as dependent variables on the plane of a complex potential. For initially saturated materials, solutions are obtained by analogy to Ivantsov's problem of dendrite growth.

Numerical methods are employed to examine the transport of charged species in pressure-driven and electroosmotic flow along nanoscale channels having an electric double-layer thickness comparable to the channel size. In such channels, the electric field inherent to the double layer produces transverse species distributions that depend on species charge. Flow along the channel thus yields mean axial species speeds that also depend on the species charge, enabling species separation and identification. Here we characterize field-flow separations of this type via the retention and plate height. For pressure-driven flows, we demonstrate that mean species speeds along the channel are uniquely associated with a single species charge, allowing species separation based on charge alone. In contrast, electroosmotic flows generally yield identical speeds for several values of the charge, and these speeds generally depend on both the species charge and electrophoretic mobility. Coefficients of dispersion for charged species in both planar and cylindrical geometries are presented as part of this analysis.

Molecular density profiles and charge distributions determined by density functional theory (DFT) are used in conjunction with the continuum Navier-Stokes equations to compute electro-osmotic flows in nanoscale channels. The ion species of the electrolyte are represented as centrally charged hard spheres, and the solvent is treated as a dense fluid of neutral hard spheres having a uniform dielectric constant. The model explicitly accounts for Lennard-Jones interactions among fluid and wall molecules, hard sphere repulsions, and short range electrical interactions, as well as long range Coulombic interactions. Only the last of these interactions is included in classical Poisson-Boltzmann (PB) modeling of the electric field. Although the proposed DFT approach is quite general, the sample calculations presented here are limited to symmetric monovalent electrolytes. For a prescribed surface charge, this DFT model predicts larger counterion concentrations near charged channel walls, relative to classical PB modeling, and hence smaller concentrations in the channel center. This shifting of counterions toward the walls reduces the effective thickness of the Debye layer and reduces electro-osmotic velocities as compared to classical PB modeling. Zeta potentials and fluid speeds computed by the DFT model are as much as two or three times smaller than corresponding PB results. This disparity generally increases with increasing electrolyte concentration, increasing surface charge density and decreasing channel width. The DFT results are found to be comparable to those obtained by molecular dynamics simulation, but require considerably less computing time.

The oxidation of zirconium alloys is one of the most studied processes in the nuclear industry. The purpose of this report is to provide in a concise form a review of the oxidation process of zirconium alloys in the moderate temperature regime. In the initial ''pre-transition'' phase, the surface oxide is dense and protective. After the oxide layer has grown to a thickness of 2 to 3 {micro}m's, the oxidation process enters the ''post-transition'' phase where the density of the layer decreases and becomes less protective. A compilation of relevant data suggests a single expression can be used to describe the post-transition oxidation rate of most zirconium alloys during exposure to oxygen, air, or water vapor. That expression is: Oxidation Rate = 13.9 g/(cm{sup 2}-s-atm{sup -1/6}) exp(-1.47 eV/kT) x P{sup 1/6} (atm{sup 1/6}).

This report summarizes the results of a five-year effort to understand the mechanisms and develop models that predict the corrosion of refractories in oxygen-fuel glass-melting furnaces. Thermodynamic data for the Si-O-(Na or K) and Al-O-(Na or K) systems are reported, allowing equilibrium calculations to be performed to evaluate corrosion of silica- and alumina-based refractories under typical furnace operating conditions. A detailed analysis of processes contributing to corrosion is also presented. Using this analysis, a model of the corrosion process was developed and used to predict corrosion rates in an actual industrial glass furnace. The rate-limiting process is most likely the transport of NaOH(gas) through the mass-transport boundary layer from the furnace atmosphere to the crown surface. Corrosion rates predicted on this basis are in better agreement with observation than those produced by any other mechanism, although the absolute values are highly sensitive to the crown temperature and the NaOH(gas) concentration at equilibrium and at the edge of the boundary layer. Finally, the project explored the development of excimer laser induced fragmentation (ELIF) fluorescence spectroscopy for the detection of gas-phase alkali hydroxides (e.g., NaOH) that are predicted to be the key species causing accelerated corrosion in these furnaces. The development ofmore » ELIF and the construction of field-portable instrumentation for glass furnace applications are reported and the method is shown to be effective in industrial settings.« less

A new generation of irreversible, chemically reacting getters specifically targeted toward assuring the integrity of the local environment within microsystem packages were developed and evaluated. These reactive getters incorporate volatile species into a polymer through covalent bonds, thus producing a non-volatile product. These reactive getters will be combined with getters that rely on absorption media (e.g. zeolites and high surface area carbon fibers) to scavenge non-reactive species, like solvents. Our getter systems will rely on device packaging to limit exchange between the microsystem and the global environment. Thus, the internal getters need only provide local environmental control within the microsystem package. A series of experiments were conducted to determine uptake rates and capacities absorption and reactive-based getters. Diffusion rates through the binder used to hold the getter particles together were also investigated. Getters were evaluated in environments with a saturated headspace and with a limited amount of the volatile species of interest. One- and two-dimensional numerical models and analysis techniques have been developed and used to predict the transport of contaminant species within a representative microsystem package consisting of an open gas-filled volume adjacent to a polymer layer containing embedded particles of getter. The two-dimensional model features explicit representation ofmore » the individual getter particles while the one-dimensional treatment assumes a homogeneous distribution of getter material within the getterlpolymer layer. Example calculations illustrate the dependence of getter performance on reaction rates, polymer diffusivity, and getter particle volume fraction. In addition, the model is used to deduce surface reaction rates, solid phase diffusivities, and maximum-loading densities by least-squares fitting of model predictions to measured histories of gas-phase contaminant concentration and getter weight gain.« less

Analytical methods are employed to investigate band broadening in a microchannel turn and adjoining straight channel segments for species transport by electrophoresis or electroosmotic flow. On the basis of closed-form solutions, we find that turn-induced broadening is negligible relative to total broadening when the radius of the turn exceeds some minimum. This minimum radius is about six-tenths of the product of the channel width and the Peclet number. We also find that the minimum radius is significantly reduced when a straight channel segment adjoins the turn in a folded configuration. Such straight segments noticeably reduce the minimum radius even for segment lengths comparable to the turn radius. The application of these results to folded and spiral channels is discussed, and sample calculations for practical conditions are presented. New pleated and coiled geometries for the compact layout of separation channels are also presented.

Numerical simulations are used to analyze injection of sample bands from an open channel into a packed separation column. Injection procedures commonly used for open channel systems are shown to produce poorly-formed dilute sample bands when the electroosmotic mobility within the packed separation column is considerably less than that of the adjacent open channels. Alternative procedures are suggested.

Numerical methods are employed to optimize the geometry of two-dimensional microchannel turns such that the turn-induced spreading of a solute band is minimized. An inverted numerical method is first developed to compute the electric potential and local species motion in turns of arbitrary geometry. The turn geometry is then optimized by means of a nonlinear least-squares minimization algorithm using the spatial variance of the species distribution leaving the turn as the object function. This approach yields the turn geometry producing the minimum possible dispersion, subject only to prescribed constraints. The resulting low-dispersion turns provide an induced variance 2-3 orders of magnitude below that of a comparable conventional turns. Sample results are presented for 180 and 90 degrees turns, and the use of these turns to form wyes and tees is discussed. A sample 45 degrees wye is presented. The use of low-dispersion turns in folding separation columns is also discussed, and sample calculations are presented for folding a column 100 microm in width and up to 900 mm in length onto a region of only 10 by 10 mm. These low-dispersion geometries are applicable to electroosmosis, electrophoresis, and some pressure-driven flows.

Analytical and numerical methods are employed to investigate species transport by electrophoretic or electroosmotic motion in the curved geometry of a two-dimensional turn. Closed-form analytical solutions describing the turn-induced diffusive and dispersive spreading of a species band are presented for both the low and high Peclet number limits. We find that the spreading due to dispersion is proportional to the product of the turn included angle and the Peclet number at low Peclet numbers. It is proportional to the square of the included angle and independent of the Peclet number when the Peclet number is large. A composite solution applicable to all Peclet numbers is constructed from these limiting behaviors. Numerical solutions for species transport in a turn are also presented over a wide range of the included angle and the mean turn radius. On the basis of comparisons between the analytical and numerical results, we find that the analytical solutions provide very good estimates of both dispersive and diffusive spreading provided that the mean turn radius exceeds the channel width. These new solutions also agree well with data from a previous study. Optimum conditions minimizing total spreading in a turn are presented and discussed.

The development of exposed PMMA resist for the LIGA process is very difficult and time-consuming when the resist thickness is large and feature aspect ratios exceed about four. This is due mainly to limitations on the development rate imposed by the diffusive transport of PMMA fragments away form the dissolution surface. Development rates under these conditions can be enhanced significantly by high- frequency acoustic agitation. To study this enhancement, analytical solutions describing the periodic flow field are used to evaluate the time-mean Reynolds stresses that drive streaming fluid motion. The resulting steady flow and transport rates within a feature are then computed by solving the Navier Stokes and species transport equations. For typical acoustic frequencies and power levels of 1 MHz and 6 w/cm2 the streaming flow within the feature is torroidal, with inflow along the feature walls at maximum speeds approaching 1 mm/min, coupled with a slower outflow along the feature center. The computed increase in transport, relative to diffusion, is typically on the order of 4 to 6 for feature aspect ratios ranging from 3 to 10 and for polymer fragment diffusivity on the order of 10-11 m2/s, provided that the feature width is greater than about 10 microns. In smaller features, the streaming speed may be suppressed by overlapping viscous boundary layers on opposing feature walls. Higher frequencies help reduce the boundary layer thickness but may lead to less efficient multi-cellular flow patterns when the acoustic wavelength is less than the feature depth.© (2000) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Analytical and numerical methods are employed to investigate species transport by electrophoretic or electroosmotic motion in the curved geometry of a two-dimensional turn. Closed-form analytical solutions describing the turn-induced diffusive and dispersive spreading of a species band are presented for both the low and high Peclet number limits. We find that the spreading due to dispersion is proportional to the product of the turn included angle and the Peclet number at low Peclet numbers. It is proportional to the square of the included angle and independent of the Peclet number when the Peclet number is large. A composite solution applicable to all Peclet numbers is constructed from these limiting behaviors. Numerical solutions for species transport in a turn are also presented over a wide range of the included angle and the mean turn radius. Based on comparisons between the analytical and numerical results, we find that the analytical solutions provide very good estimates of both dispersive and diffusive spreading provided that the mean turn radius exceeds the channel width. These new solutions also agree well with data from a previous study.

Steady electroosmotic flow of uniform liquids in uniform media is irrotational provided the electric double layers adjacent to surfaces are negligibly thin, the surfaces are non-conducting and impermeable, and the total pressure imposed at inlets and outlets is uniform. Because many microfluidic devices employing electroosmosis approximately satisfy these requirements, this ideal electrosmosis is a limiting case with considerable practical significance. In ideal electroosmosis, fluid motion follows current lines. Flow-fields have no Reynolds number dependence and are everywhere proportional to the electric field. Both fields may be obtained by a single solution of the Laplace equation. In this paper, we discuss these features of ideal electroosmotic flows and present particle-image derived velocity fields that confirm ideal flow conditions in glass microchannel networks.© (1999) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Electroosmotic flow is fluid motion driven by an electric field acting on the net fluid charge produced by charge separation at a fluid-solid interface. Under many conditions of practical interest, the resulting fluid velocity is proportional to the local electric field, and the constant of proportionality is everywhere the same. Here the authors show that the main conditions necessary for this similitude are a steady electric field, uniform fluid and electric properties, an electric Debye layer that is thin compared to any physical dimension, and fluid velocities on all inlet and outlet boundaries that satisfy the Helmholtz-Smoluchowski relation normally applicable to fluid-solid boundaries. Under these conditions, the velocity field can be determined directly from the Laplace equation governing the electric potential, without solving either the continuity or momentum equations. Three important consequences of these conditions are that the fluid motion is everywhere irrotational, that fluid velocities in two-dimensional channels bounded by parallel planes are independent of the channel depth, and that such flows exhibit no dependence on the Reynolds number.

Adsorption and condensation are critical to many applications of porous materials including filtration, separation, and the storage of gases. Integral methods are used to derive an analytical expression describing fluid condensation pressures in slit pores bounded by parallel plane walls. To obtain this result, the governing equations of Density Functional Theory (DFT) are integrated across the pore width assuming that fluid densities within adsorbed layers are spatially uniform. The thickness, density, and energy of these layers are expressed as composite functions constructed from asymptotic limits applicable to small and large pores. By equating the total energy of the adsorbed layers to that of a liquid-full pore, the authors arrive at a closed-form expression for the condensation pressure in terms of the pore size, surface tension, and Lennard-Jones parameters of the adsorbent and adsorbate molecules. The resulting equation reduces to the Kelvin equation in the large-pore limit. It further reproduces the condensation pressures computed by means of the full DFT equations for all pore sizes in which phase transitions are abrupt. Finally, in the limit of extremely small pores, for which phase transitions may be smooth and continuous, this simple analytical expression provides a good approximation to the apparent condensation pressure indicated by the steepest portion of the adsorption isotherm computed via DFT.

LIGA is a micromachining technology that uses high energy x-rays from a synchrotron to create patterns with small lateral dimensions in a deep, non-conducting polymeric resist. Typical dimensions for LIGA parts are microns to tens of microns in lateral size, and hundreds of microns to millimeters in depth. Once the resist is patterned, metal is electrodeposited in the features to create metal microparts, or to create a metal mold for subsequent replication. The acronym LIGA comes from the German words for lithography, electroforming, and molding, and the technology has been under worldwide development for more than a decade. over the last five years, a full-service capability to produce metal microparts using the LIGA process has been established at Sandia national Laboratories, California. This report describes the accomplishments made during the past two years in infrastructure establishment funded by a Laboratory Directed Research and Development (LDRD) project entitled ''LIGA Micromachining.'' Specific topics include photoresist processing for LIGA mask making, x-ray scanning equipment, plating bath instrumentation, plating uniformity, and software architecture.

To better understand and to help optimize the electroforming portion of the LIGA process, we have developed one and two- dimensional numerical models describing electrodeposition of metal into high aspect-ratio molds. The one-dimensional model addresses dissociation, diffusion, electromigration, and deposition of multiple ion species. The two-dimensional model is limited to a single species, but includes transport induced by forced flow of electrolyte outside the mold and by buoyancy associated with metal ion depletion within the mold. To guide model development and to validate these models, we have also conducted a series of laboratory experiments using a sulfamate bath to deposit nickel in cylindrical molds having aspect ratios up to twenty-five. The experimental results indicate that current densities well in excess of diffusion-limited currents may still yield acceptable morphologies in the deposited metal. However, the numerical models demonstrate that such large ion fluxes cannot be sustained by convection within the mold resulting from flow across the mold top. Instead, calculations suggest that the observed hundred-fold enhancement of transport probably results from natural convection within the molds and that buoyancy-driven flows may be critical to metal ion transport even in micron-scale features having very large aspect ratios. Taking advantage of this enhanced ion transport may allow order-of-magnitude reductions in electroforming times for LIGA microdevice fabrication.

To better understand and to help optimize the electroforming portion of the LIGA process, we have developed one and two-dimensional numerical models describing electrode-position of metal into high aspect-ratio molds. The one-dimensional model addresses dissociation, diffusion, electromigration, and deposition of multiple ion species. The two-dimensional model is limited to a single species, but includes transport induced by forced flow of electrolyte outside the mold and by buoyancy associated with metal ion depletion within the mold. To guide model development and to validate these models, we have also conducted a series of laboratory experiments using a sulfamate bath to deposit nickel in cylindrical molds having aspect ratios up to twenty-five. The experimental results indicate that current densities well in excess of the diffusion-limited currents may still yield metal deposits of acceptable morphology. However, the numerical models demonstrate that such large ion fluxes cannot be sustained by convection within the mold resulting from flow across the mold top. Instead, calculations suggest that the observed enhancement of transport probably results from natural convection within the molds, and that buoyancy-driven flows may be critical to metal ion transport even in micron-scale features having very large aspect ratios. Taking advantage of this enhanced ion transport may allow order-of-magnitude reductions in electroforming times for LIGA microdevice fabrication. 42 refs., 14 figs., 1 tab.

A combined analytical and numerical method is employed to optimize process conditions for composites fiber coating by chemical vapor infiltration (CVI). For a first-order deposition reaction, the optimum pressure yielding the maximum deposition rate at a preform center is obtained in closed form and is found to depend only on the activation energy of the deposition reaction, the characteristic pore size, and properties of the reactant and product gases. It does not depend on the preform specific surface area, effective diffusivity or preform thickness, nor on the gas-phase yield of the deposition reaction. Further, this optimum pressure is unaltered by the additional constraint of a prescribed deposition uniformity. Optimum temperatures are obtained using an analytical expression for the optimum value along with numerical solutions to the governing transport equations. These solutions account for both diffusive and advective transport, as well as both ordinary and Knudsen diffusion. Sample calculations are presented for coating preform fibers with boron nitride.

The fluid density distribution within microscopic pores is determined by solving integral equations relating to the local chemical potential to the Van der Waals attractions and hard sphere repulsions of surrounding material. To avoid resolving the density distribution on sub-molecular scales, the governing equations are averaged over zones of molecular size using analytic functions to represent local density variations within each zone. These local density profiles range form singularities to uniform distributions depending on the local variation of the potential field. Sample calculations indicate that this integral approach yields results in very good agreement with those based on traditional density functional theory (DFT), while reducing computing times by factors of 10{sup 3} to 10{sup 4} for one- dimensional geometries.

Analytical and numerical methods are used to investigate ion transport and storage within the pore network of a porous bed of nanoporous particles. Under simplifying assumptions that electromigration dominates ion transport, that the pore capacitance is constant, and that the electrolyte exhibits ideal behavior, transport within this hierarchical pore network is modeled using a coupled pair of one-dimensional equations describing net charge motion into the bed by way of the interparticle void and subsequently into smaller pores within particles. These diffusion-like equations are solved analytically via Laplace transforms, and the results are used to determine the optimum interparticle porosity yielding the maximum energy deliverable in a specified time. Both step and periodic boundary conditions are considered, and closed-form expressions for the optimum porosity are derived for the periodic case. We find that the optimum porosity is remarkably similar for the periodic and step boundary conditions when the normalized bed thickness is very large or very small. We also find that the optimum porosity increases at least linearly with bed thickness when the thickness is small but is independent of both the thickness and prescribed time for delivery when the bed thickness is large. Sample results are presented and discussed.

Filling deep trench-like features by electrodeposition is often limited by ion transport from the electrolyte bath to the plating surface at the feature bottom. This transport may be substantially enhanced by buoyancy-driven convection induced by metal-ion depletion adjacent to the plating surface. Numerical solutions of the Navier-Stokes and species transport equations are used to determine the magnitude of transport enhancement, expressed as a Sherwood number, for Rayleigh numbers ranging from 10 3 to 10 8 and for feature aspect ratios of depth to width ranging from 1 to 16 both for open trenches and for fully enclosed rectangular domains. To facilitate extrapolation of these numerical results, an exact analytical solution is derived for aspect ratios much greater than unity. This is used in conjunction with the known asymptotic behavior for large Rayleigh numbers to construct a composite formula relating the Sherwood number to the Rayleigh number and aspect ratio. The results indicate that buoyancy-driven convection may provide significant transport enhancement during electrodeposition into features having depths greater than about 100 μm and that enhancement exceeding a factor of ten may occur in LIGA features having depths of 1 mm or more. It is also shown that a moderate inclination of the substrate helps to suppress the formation of multiple vertically stacked convective cells that would otherwise reduce the overall transport.

Fabrication of microelectronic and LIGA microdevices often requires transport of chemical species into or out of liquid-filled microcavities. Two common examples are the chemical development of patterned photoresists to produce recessed features and the filling of such features by electrodeposition. During chemical development, fragments of the resist must be transported from the bottom of narrow features into the development bath. Similarly, filling of a patterned resist by electrodeposition requires transport of metal ions from the electroplating bath into the features. This plating process is commonly used to form microelectronic interconnects having submicrometer linewidths. Although the depths of these features may be several times greater than their widths, they rarely exceed a depth of 1 mm. Over these short length scales, diffusion provides very effective transport of chemical species. Transport rates are, however, smaller by orders of magnitude in the LIGA process used to produce detailed metal parts having depth dimensions of a millimeter or more. The acronym LIGA is derived from the German words for lithography, electroforming, and molding. 1 In LIGA, a high-energy X-ray source is used to expose a thick photoresist, typically polymethylmethacrylate ~PMMA!, through a patterned absorber mask. The exposed material is then removed by chemical dissolution in a development bath. This development process yields a nonconducting mold having a conducting substrate beneath deep cavities that are subsequently filled by electrodeposition. The resulting metal parts may be the final product or may be used as injection or embossing molds for mass production. Development and deposition rates in recessed features depend on both species transport and surface reaction kinetics. Transport is rarely an issue in conventional electroplating on flat surfaces where surface ion concentrations can be maintained by pumping bath fluids or by moving the substrate relative to the bath. However, in plating or development of patterned photoresists, even a very strong external flow is not effective in providing increased transport into recessed features having aspect ratios greater than one or two. 2-4 This is because the convective cell that circulates the fluid in the top of each feature penetrates only about one feature width. Additional counter-rotating convective cells are formed deeper within high aspect ratio features, but the circulation speeds decrease by nearly two

A general method-of-lines numerical approach for modeling hydraulically-driven fractures is developed and tested. The methodology employs several novel features: a straining coordinate system that elongates as the fracture grows, an evolutionary equation to describe growth of the fracture length, direct treatment of the fluid/elastic-solid coupling, and a control volume equation which governs fluid motion near the tip and thus circumvents local degeneracy of the differential equations. Spatial discretization of the governing equations leads to a nonsparse system of implicit, coupled ordinary differential equations that is solved for time derivatives that are then integrated with a Runge-Kutta algorithm. The numerical solutions agree very well with known similarity solutions for laminar and for turbulent flow. New solutions for nonsimilar flows are also presented and these converge to proper limits as the fracture becomes very long. Acceptable accuracy, in all cases, is obtained using a very few numerical grid points and with modest execution times.

Gas-driven fracture propagation is important in a variety of blasting technologies. In most applications, the driving gases are very hot and so heat transfer from the gas to the fracture walls strongly influences the speed and the ultimate extent of fracture propagation. In this paper, similarity solutions are presented for the problem of a planar gas-driven fracture propagating into a brittle elastic solid. Local values of the fluid pressure, temperature and speed are computed from the one-dimensional mass, momentum and energy equations governing the turbulent flow of an ideal gas. Crack opening displacements and the stress intensity at the crack tip are calculated from the gas pressure distribution, in accordance with the quasi-steady integral relations from linear/elastic fracture mechanics. Four thermal models are considered in the analysis: isothermal, isentropic, adiabatic with frictional heating, and the more general case with heat transfer to a cold fracture wall. When heat transfer is very weak, the general solution approaches the adiabatic result; when heat transfer is very strong, the general result approaches the isothermal limit. The applicable range of each model is described. Example calculations are presented for a number of applications, including: underground nuclear testing, tailored-pulse well shooting, and near surface blasting.

A possible explanation for the closely associated magmas of contrasting compositions erupted from many mature volcanic centers can be found in the large differences of density produced by relatively small compositional variations in liquids that evolve by crystallization or melting at the walls of shallow magma chambers. A mechanism of liquid fractionation in which differentiated liquids segragate gravitationally to form compositionally graded columns of magma may surmount the long-standing problem of explaining large volumes of highly evolved liquids that reach advanced degrees of differentiation in times that are too short to be consistent with conventional models of crystal fractionation based on crystal settling. In those types of magmas that decrease in density as they differentiate, a fractionated liquid next to a wall may form a buoyant compositional boundary layer that flows up the wall and accumulates as a separate zone in the upper levels of the reservoir. Magmas that increase in density as they differentiate will have the opposite behavior; they descend along the wall and pond on the floor. Both types of systems can be modeled using simple aqueous solutions and techniques similar to those developed by Chen and Turner (1980). The insights gained through experiments of this kind suggest a number of processes that may be responsible for common types of volcanic behavior and patterns of differentiation in shallow plutons.

The processes of liquid fractionation described in part I of this report are evaluated quantitatively to determine the ranges of physical conditions in which they are likely to be effective under realistic geological conditions. Similitude analyses and numerical modeling are used to identify the conditions consistent with counterflow of a rising compositional boundary layer and a descending thermally driven flow in the interior of a cooling intrusion. Estimates are derived for the rates of accumulation of differentiated liquids in the upper levels of stratified reservoirs; these are more than adequate to account for rates inferred from field observations. Parametric studies illustrate the influence of various factors (e.g. Prandtl, Lewis, Reyleigh, and Grashof numbers, the ratio of buoyancy forces, turbulence, and viscosity contrasts between the wall and interior) for values spanning many orders of magnitude. In particular, it is shown that flow in the buoyant boundary layer is relatively unaffected by turbulence of the interior and that viscosity variations can be accounted for by an appropriately chosen effective or mean viscosity. Results are used to assess the fidelity of laboratory models outlined in Part I, and specific examples are given to illustrate the distribution of velocity, temperature, and composition at various levels within a calc-alkaline magma chamber.