The buoyant motion of a bubble rising through a continuous liquid phase can be retarded by the adsorption onto the bubble surface of surfactant dissolved in the liquid phase. The reason for this retardation is that adsorbed surfactant is swept to the trailing pole of the bubble where it accumulates and lowers the surface tension relative to the front end. The difference in tension creates a Marangoni force which opposes the surface flow, rigidifies the interface and increases the drag coefficient. Surfactant molecules adsorb onto the bubble surface by diffusing from the bulk to the sublayer of liquid adjoining the surface, and kinetically adsorbing from the sublayer onto the surface. The surface surfactant distribution which defines the Marangoni force is determined by the rate of kinetic adsorption and bulk diffusion relative to the rate of surface convection. In the limit in which the rate of either kinetic or diffusive transport of surfactant to the bubble surface is slow relative to surface convection and surface diffusion is also slow, surfactant collects in a stagnant cap at the back end of the bubble while the front end is stress free and mobile. The size of the cap and correspondingly the drag coefficient increases with the bulk concentration of surfactant until the cap covers the entire surface and the drag coefficient is that of a bubble with a completely tangentially immobile surface. Previous theoretical research on the stagnant cap regime has not studied in detail the competing roles of bulk diffusion and kinetic adsorption in determining the size of the stagnant cap angle, and there have been only a few studies which have attempted to quantitatively correlate simulations with measurements.This paper provides a more complete theoretical study of and a validating set of experiments on the stagnant cap regime. We solve numerically for the cap angle and drag coefficient as a function of the bulk concentration of surfactant for a spherical bubble rising steadily with inertia in a Newtonian fluid, including both bulk diffusion and kinetic adsorption. For the case of diffusion-limited transport (infinite adsorption kinetics), we show clearly that very small bulk concentrations can immobilize the entire surface, and we calculate the critical concentrations which immobilize the surface as a function of the surfactant parameters. We demonstrate that the effect of kinetics is to reduce the cap angle (hence reduce the drag coefficient) for a given bulk concentration of surfactant. We also present experimental results on the drag of a bubble rising in a glycerol–water mixture, as a function of the dissolved concentration of a polyethoxylated non-ionic surfactant whose bulk diffusion coefficient and a lower bound on the kinetic rate constants have been obtained separately by measuring the reduction in dynamic tension as surfactant adsorbs onto a clean interface. For low concentrations of surfactant, the experiments measure drag coefficients which are intermediate between the drag coefficient of a bubble whose surf...

A rigorous method is presented for measuring chemical reaction rates for CMP of copper with a slurry containing an oxidizer and complexing agent. A model reaction network is first proposed that includes copper oxidation/reduction at the wafer surface, liquidphase copper complexation, and copper oxide film formation. Each reaction is studied in a non-polish environment via experiments specially configured to isolate chemical kinetics from transport effects. Oxidation and reduction are characterized in a rotating-disk electrochemical cell, complexation in a stopped-flow reactor, and oxide formation using Auger electron spectroscopy. Rate constants and reaction orders are determined from the data by applying Tafel analysis for oxidation, the Koutecky-Levich approach for reduction, stirred-tank reactor kinetics for complexation, and boundary layer theory for oxide formation. Oxide composition is found to vary with oxidizer level, offering a concentration-dependent resistance to mass transfer. Results are discussed in the context of an integrated CMP model including contact, kinetics, and transport.

The chemical and mechanical properties of wafer materials have a direct impact on the performance of the CMP process. This report presents a systematic study to characterize surface films on copper wafers in the presence of simple peroxide solutions as well as representative bulk copper and copper barrier CMP slurries offered by the Dow Chemical Company. Measurements are made of chemical composition, thickness, modulus, and removal rate of films formed under both static etch and CMP conditions. Sample aging-a major obstacle in past efforts of this type-is minimized by temporarily setting up the equipment for wafer film preparation in close proximity to the surface analytical instruments. It is found that Dow's commercial bulk and barrier Cu CMP slurries form permeable surface films less than 4 nm deep consisting of reaction products with copper. Compared to native copper, the films have compressive and shear moduli that are 5X and 2X lower respectively-enabling mechanical removal by sliding contact with pad asperities at a lower threshold contact pressure. The results provide insight into the mechanism of surface film formation during CMP and lay groundwork for modeling the mechanical and chemical steps in the wafer planarization process.

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