Nondimensional Particle Research Articles

Showerhead-Enhanced Inertial Particle Deposition in Parallel-Plate Reactors

ABSTRACT Particle deposition is modeled in a parallel-plate reactor geometry characteristic of a wide range of semiconductor process tools: uniform downward flow exiting a perforated-plate showerhead separated by a small gap from a circular wafer resting on a parallel susceptor. Because the area available to flow is constricted inside the showerhead, high gas velocities within showerhead holes can accelerate particles and lead to inertia-enhanced particle deposition on the wafer below. Numerical models are presented for determining the extent of inertia-enhanced deposition in a parallel-plate geometry for particles originating upstream of the showerhead. The problem is treated in two steps, for which particle and fluid transport are determined: 1) within a showerhead-hole, and 2) between the showerhead and susceptor. Analytic expressions are presented for the gas velocity in the showerhead holes. Particle transport analysis leads to an analytic expression for the dimensionless particle velocity at the exit of the showerhead as a function of the dimensionless showerhead thickness, the showerhead porosity, and the particle Stokes number. The fluid flow field in the inter-plate region is calculated by both an analytic 1-D creeping-flow approximation and by a numerical 2-D finite element method; for Reynolds numbers less than about four, the two methods are found to be in good agreement. To calculate the net effect of inertia-enhanced deposition, the analysis of particle acceleration in the showerhead is coupled with that for inter-plate transport. The final result is the particle collection efficiency (fraction of particles entering the reactor that deposit on the wafer) as a function of the dimensionless parameters characterizing: showerhead porosity, showerhead thickness, inter-plate Reynolds number, dimensionless particle drift velocity (a measure of the strength of applied external forces), and particle Stokes number. Conditions are found under which particle acceleration in the showerhead greatly enhances particle deposition on wafers. Numerical predictions are presented for the critical Stokes number, which is the nondimensional particle size at which inertial effects lead to substantial particle deposition even in the absence of external forces. Specific guidelines for reducing inertia-enhanced particle deposition include: increase the number of and/or enlarge the size of the showerhead holes, make the showerhead very thin, keep the inter-plate gap large, apply an external force that resists deposition, use low mass flow rates, avoid low pressure, or use a high molecular weight gas. For Reynolds numbers less than four, particle deposition efficiencies calculated with the analytic inter-plate flow field approximation are found to be in good agreement with numerical 2-D simulations.

Motions of Small Rigid Spheres in Simulated Random Velocity Field

Motions of particles in a simulated three‐dimensional random Gaussian flow field are numerically studied. The equation of motion of a solid particle (the generalized BBO equation), in addition to the Stokes drag, the Basset force, and the virtual mass effects, includes the three‐dimensional lift force. An ensemble of 300 particle trajectories are generated for evaluating various statistics. Turbulent diffusivity of small particles is calculated and its variations with nondimensional particle size, particle Reynolds number, density ratio, and particle response time are studied. Particle velocity autocorrelations and particle velocity‐fluid velocity cross‐correlations, as well as fluctuation energy transport from the fluid phase to the particulate phase, are evaluated. A simplified model for evaluating the particle response statistics is also described. The simulation results are compared with the predictions of the simplified model and the available experimental data and good agreement is observed. Effects...

Turbulent eddy deposition of particles on smooth and rough surfaces

An analytical model for turbulent eddy impaction of particles is presented which is uniformly valid for smooth and rough surfaces. Turbulent Schmidt number expressions for particle diffusion in the limits of residence times much smaller than and of the same order as particle relaxation and turbulent energy time scales are adapted for determining the particle approach velocity and diffusion in the buffer layer. The model correctly predicts the experimentally observed fall-off in the deposition velocity for non-dimensional particle relaxation time τ p +, greater than about 25. The model predictions also compare favorably with limited deposition data for rough surfaces.

Measurements of the mean force on a particle near a boundary in turbulent flow

The transport of particles through gaseous systems is controlled by three factors: their arrival to the surface; whether or not they bounce upon impact; and when (if ever) they are resuspended from the surface. One of the parameters required in determining whether or not a particle is suspended is the lift force acting on the particle. We demonstrate that the fluid lift forces acting on particles as small as 1 μm in diameter can be modelled by particles of several mm in diameter. However, the forces involved in modelling such small particles are around 10−8 N, which is several orders of magnitude smaller than reported in published measurements of fluid lift forces. A system to determine such lift forces has been developed and is described. Measurements of the mean force acting on particles on both rough and smooth surfaces are presented.The data recorded here for the mean fluid lift force on a sphere on a smooth surface are in good agreement with the relationship \[ F^{+} = (20.90\pm 1.57)(a^{+})^{2.31\pm 0.02}, \] where F+ is the non-dimensional force and a+ the non-dimensional particle radius scaled on fluid-boundary-layer parameters. It was observed that surface roughness can change the force by up to a factor of six.

Numerical analysis of interaction of a high-voltage solar array withionospheric plasma

Three-dimensional analysis of the solar array that exposes the surfaces at a relatively high potential and flies in the ionospheric plasma was conducted. For the standard ionospheric conditions, the drain power was calculated at about 0.3% of the solar generated power, which is rather insignificant. On the other hand, the ion forces were found to be enhanced by the large electric potential difference between the solar array and the plasma and must affect significantly the motion of spacecraft with the neutral particle drag. Nomenclature c = unit cord vector = s x n D = characteristic length of solar array d = thickness of space charge sheath d = unit vector of ion drag = - u e - charge of electron F - vector of total ion force / = normalized charge density k = Boltzmann constant lie - ion-electron mean free path / = unit vector of ion lift = dxp M =ion molecular mass N,n = dimensional and nondimensional particle densities n =unit normal vector on conductive surface of solar array P, Q = collision and reflection momenta

Quantitative criteria for emulsion phase characterization and for the transition between particulate and bubbling fluidization

A quantitative characterization for powder mixtures (Geldart Type A and C powders) in the emulsion phase was developed. The plastic deformation coefficient Y, the dissipation energy for powder packing rearrangement E p of the emulsion phase in fluidized beds and a nondimensional particle interaction term Ko were derived and experimentally obtained. A quantitative criterion for the transition between particulate and bubbling fluidization was developed taking the particle interactions into account and compared with experimental results.

Dynamic Stability of a Beam Loaded by a Sequence of Moving Mass Particles

The dynamic stability for the lateral response of a finite Bernoulli-Euler beam loaded by a continuous sequence of identical, equally spaced, mass particles attached to and traveling at a constant speed across the beam is investigated. This paper points out that, in general, multiple regions of unstable response will occur. However, for certain particle spacings and foundation moduli a single region of unstable response occurs. In general, the critical speed ratio Beta cr (or the nondimensional particle speed at which instability of the beam occurs) is increased by decreasing Alpha T, the nondimensional particle mass; increasing rho, the nondimensional particle spacing; increasing gamma, the nondimensional foundation stiffness; increasing P*, the nondimensional axial tensile force on the beam; increasing xi sub n, the damping coefficient.

Fluid Dynamics of Particle (Platelet) Deposition for Filtering Walls: Relationship to Atherosclerosis

Particle deposition onto filtering surfaces is expected to occur when the drag force of filtration overcomes the fluid mechanic wall repulsive force. A nondimensional particle deposition parameter has been formulated and experimentally validated utilizing red blood cells. The parameter is applied to platelet deposition in the vasculature and shows that platelets are expected to contact arterial surfaces only in low shear, separated regions. This may be the link between the filtration and thrombosis theories of atherosclerosis. Platelet radius and plasma viscosity are predicted to be important parameters of atherogenesis.

Separating Characteristics of Disc-type Centrifuges for Solid-Liquid System

As there had been only a few experimental and theoretical studies made on a disc-type centrifuge in the past, experiments in separating characteristics of centrifuges for solid-liquid system were carried out, using low concentrations of fly-ash particles throughly dispersed in liquid. The particle-size distributions of spherical particles in the feed and in the clarified liquid employed in the test were measured through a microscope and the performance of the centrifuge was evaluated, as follows:1. In order to describ the perfomance of a centrifuge based on the experimental results, two nondimensional parameters were introduced. One was a fractional penetration, as is commonly used in an aerosol collection, and the other, a non-dimensional particle diameter. In the latter, the particle diameter was determined by the ratio of a particle diameter to a critical particle diameter calculated from the factors of mechanism of separation, such as physical properties of suspension, geometrical dimensions of a disc and operating conditions.2. The separating characteristics seemed to be determined by two parameters, and the characteristic curves are expressed as: Pf=f(Dpr)3. As is obviously shown by the characteristic curve, the critical particle diameter proposed by one of the authors might be considered a standard for the operation.4. The necessaary and sufficient condition for scaling-up is given by:This condition proved to be more generally applicable than the one presented by C.M. Ambler.