Browsing by Author "Braun, Richard J."
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Item A 2+1 Dimensional Insoluble Surfactant Model for a Vertical Draining Free Film(Department of Mathematical Sciences, 2002-05-09) Naire, S.; Braun, Richard J.; Snow, S.A.A 2 + 1 dimensional mathematical model is constructed to study the evolution of a vertically-oriented thin, free liquid film draining under gravity when there is an insoluble surfactant, with finite surface viscosity, on its free surface. Lubrication theory for this free film results in four coupled nonlinear partial differential equations (PDEs) describing the free surface shape, the surface velocities and the surfacant transport, at leading order. Numerical experiments are performed to understand the stability of the system to perturbations across the film. In the limit of large surface viscosities, the evolution of the free surface is that of a rigid film. In addition, these large surface viscosities act as stabilizing factors due to their energy dissipating effect. An instability is seen for the mobile case; this is caused by a competition between gravity and the Marangoni effect. The behavior observed from this model qualitatively matches some structures observed in draining film experimentsItem A1-L1 Phase Boundaries and Anisotropy via Multiple-Order-Parameter Theory for an FCC Alloy(Department of Mathematical Sciences, 2003) Tanoglu, G.B.; Braun, Richard J.; Cahn, J.W.; McFadden, G.B.The dependence of thermodynamic properties of planar interphase boundaries (IPBs) and antiphase boundaries (APBs) in a binary alloy on an FCC lattice is studied as a function of their orientation. Using a recently-developed diffuse interface model based on three non-conserved order parameters and the concentration, and a free energy density that gives a realistic phase diagram with one disordered phase (A1) and two ordered phases (L12 and L10) such as occurs in the Cu-Au system, we are able to find IPBs and APBs between any pair of phases and domains, and for all orientations. The model includes bulk and gradient terms in a free energy functional, and assumes that there is no mismatch in the lattice parameters for the disordered and ordered phases. We catalog the appropriate boundary conditions for all IPBs and APBs. We then focus on the IPB between the disordered A1 phase and the L10 ordered phase. For this IPB we compute the numerical solution of the boundary value problem to find the interfacial energy, γ, as a function of orientation, temperature, and chemical potential (or composition). We determine the equilibrium shape for a precipitate of one phase within the other using the Cahn-Hoffman ‘ξ-vector’ formalism. We find that the profile of the interface is determined only by one conserved and one non-conserved order parameter, which leads to a surface energy which, as a function of orientation, is “transversely isotropic” with respect to the tetragonal axis of the L10 phase. We verify the model’s consistency with the Gibbs adsorption equation.Item Bounded Film Evolution with Nonlinear Surface Properties(Department of Mathematical Sciences, 2001-09-12) Debisschop, C.A.; Braun, Richard J.; Snow, S.A.We study the evolution of a Newtonian free surface of a thin film above a solid wall. We consider the case in which the horizontal solid is covered by a non-wetting fluid and an insoluble monolayer of surfactant is present on the fluid-air interface. We pose a model that incorporates a variety of interfacial effects: van der Waals forces, variable surface tension and surface viscosity. The surface tension and surface viscosity depend nonlinearly on the surfactant concentration. Using lubrication theory we obtain a leading order description of the shape and velocity of the fluid-air interface, and the surfactant concentration, in the form of coupled nonlinear partial differential equations. A linear stability analysis reveals that the wavenumber that characterizes the marginal state is independent of the presence of the surfactant and the nonlinearity of the surface properties. We solve the 1+1-dimensional system numerically to obtain the spatio-temporal evolution of the free surface in the nonlinear regime, and observe the progression to rupture.Item Hydrodynamics of Bounded Vertical Film with Nonlinear Surface Properties(Department of Mathematical Sciences, 2001) Heidari, A.H.; Braun, Richard J.; Hirsa, A. H.; Snow, S.A.; Naire, S.The drainage of a thin liquid film with an insoluble monolayer down a vertical wall is studied. Lubrication theory is used to develop a model where the film is pinned at the top with a given thickness and the film drains into a bath at the bottom. A nonlinear equation of state is used for the surface tension and the surface viscosity is a nonlinear function of the surfactant concentration; these are appropriate for some aqueous systems. The three partial differential equations are solved via discretization in space and then solving the resulting differential algebraic system. Results are described for a wide range of parameters, and the conditions under which the free surface is immobilized are discussed.Item A Model for Anisotropic Epitaxial Lateral Overgrowth(Department of Mathematical Sciences, 2001) Khenner, M.; Braun, Richard J.; Mauk, M.G.The model for anisotropic crystal growth on a substrate covered by a mask material with a periodic series of parallel long trenches where the substrate is exposed to the vapor phase if developed. The model assumes that surface diffusion and deposition flux are the main mechanisms of the growth, and that the three key surface quantities (energy, mobility and adatom diffusivity) are anisotropic with either four- or six-fold symmetry. A geometrical approach to the motion of crystal surface in two dimensions is adopted and nonlinear evolution equations are solved by a finite-difference method. The model allows the direct computation of the crystal surface shape and the study of effects due to finite mask thickmness. As in experiments, lateral overgrowth of crystal onto the mask if found, as well as comparable crystal shapes; the anisotropy of the surface mobility is found to play the dominant role in the shape selection. The amount of the overgrowth and the shapes can be effectively controlled by orienting the fast and slow growth directions with respect to the substrate.Item A Model for Isotropic Crystal Growth from Vapor on a Patterned Substrate(Department of Mathematical Sciences, 2001) Khenner, M.; Braun, Richard J.; Mauk, M.G.We developed a consistent mathematical model for isotropic crystal growth on a substrate covered by the mask material with periodic series of parallel long trenches where the substrate is exposed to the vapor phase. Surface diffusion and the flux of particles from vapor are assumed to be the main mechanisms of growth. A geometrical approach to the motion of crystal surface in two dimensions is adopted and nonlinear evolution equations are solved by finite-difference method. The model allows the direct computation of the crystal surface shape, as well as the study of the effects due to mask regions of effectively nonzero thickness. As in experiments, lateral overgrowth of crystal onto the mask and enhanced growth in the region near the contact of the crystal and the mask is found, as well as the comparable crystal shapes. The growth rates inveritcal and lateral directions are investigated.Item Modelling drainage of the precorneal tear film after a blink(Department of Mathematical Sciences, 2002) Braun, Richard J.; Fitt, A.D.We study the drainage of the precorneal tear lm in humans. A uid dynamic model for the drainage of the aqueous layer is developed that includes the effects of evaporation and gravity. The model may be reduced to a single nonlinear partial differential equation for the thickness of the aqueous layer. The equation is solved numerically and accurate times for film rupture are obtained for physically realistic parameters. The results indicate that though gravity and evaporation are not the most dominant effects in some parts of the film, they can nevertheless materially affect the film drainage process and should therefore be included in models for tear film drainage.