Robert H. Nilson

Enhanced Transport by Acoustic Streaming in Deep Trench-Like Cavities

.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

CONDENSATION PRESSURES IN SMALL PORES : AN ANALYTICAL MODEL BASED ON DENSITY FUNCTIONAL THEORY

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...

Natural Convection in Trenches of High Aspect Ratio

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.

Comparison of Molecular Dynamics with Classical Density Functional and Poisson–Boltzmann Theories of the Electric Double Layer in Nanochannels

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.

Influence of atomistic physics on electro-osmotic flow: An analysis based on density functional theory

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.

Numerical analysis of hydraulically-driven fractures☆

Abstract 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.

Irrotationality of uniform electro-osmosis

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.

Transport limitations in electrodeposition 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 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.

Optimum Interparticle Porosity for Charge Storage in a Packed Bed of Nanoporous Particles

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.

Modeling the Electrochemical Impedance Spectra of Electroactive Pseudocapacitor Materials

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.