Gustav Amberg

Intracerebral microdialysis: I. Experimental studies of diffusion kinetics

Intracerebral microdialysis is a brain perfusion technique in which a tubular, semipermeable membrane perfused with a physiological solution is implanted into a selected brain region. Molecules in the extracellular space diffuse into the perfusate and may be recovered and their concentration determined. Hence, the level of substances such as neurotransmitters may be monitored, and the response to different treatments may be studied. The technique also allows for administration of substances locally to the region of the brain surrounding the perfused tubular membrane. Basic principles of the microdialysis technique are described, and the results from methodological experiments are examined. It is concluded that there is a direct linear relation between the concentration of a molecule in the medium surrounding the dialysis membrane and the concentration measured in the collected perfusate. Relative changes of molecular concentration in brain extracellular space may be calculated even when the molecular diffusion rate is unknown. In addition, a method is presented for calculating the real concentration of a substance in the extracellular space from its concentration in the perfusate. Applied in striatum of rat brain using microdialysis in vivo, the average extracellular concentration of the following substances is estimated to be: substance P, 0.9 nM; dopamine, 1 microM; and dihydroxyphenylacetic acid, 0.05 mM.

Intracerebral microdialysis: II. mathematical studies of diffusion kinetics

The kinetics of intracerebral microdialysis are studied mathematically. In the microdialysis technique, a tubular membrane that is permeable to diffusion is implanted in the brain and perfused with an artificial cerebrospinal fluid. Molecular diffusion causes substances in the brain to enter the flowing perfusate. The perfusate is collected outside the brain and the content of various substances is determined. The mathematical problem of diffusion through the porous brain tissue into the flowing perfusate is formulated. Solutions for the concentration distributions in the brain and in the perfusate are derived. It is found that the factor limiting the transport is the diffusion through the brain and not through the membrane. A theoretical expression for the recovery ratio is also obtained. This ratio may be used to infer the extracellular concentration in the brain from the concentration in the collected perfusate.

Nonlinear analysis of buoyant convection in binary solidification with application to channel formation

We consider the problem of nonlinear thermal-solutal convection in the mushy zone accompanying unstable directional solidification of binary systems. Attention is focused on possible nonlinear mechanisms of chimney formation leading to the occurrence of freckles in solid castings, and in particular the coupling between the convection and the resulting porosity of the mush. We make analytical progress by considering the case of small growth Peclet number, δ, small departures from the eutectic point, and infinite Lewis number. Our linear stability results indicate a small O (δ) shift in the critical Darcy-Rayleigh number, in accord with previous analyses. We find that nonlinear two-dimensional rolls may be either sub- or supercritical, depending upon a single parameter combining the magnitude of the dependence of mush permeability on solids fraction and the variations in solids fraction owing to melting or freezing. A critical value of this combined parameter is given for the transition from supercritical to subcritical rolls. Three-dimensional hexagons are found to be transcritical, with branches corresponding to upflow and lower porosity in either the centres or boundaries of the cells. These general results are discussed in relation to experimental observations and are found to be in general qualitative agreement with them.

Hydrodynamical instabilities of thermocapillary flow in a half-zone

The stability of the flow in a half-zone configuration is analysed with the aid of direct numerical simulation. The work is concentrated on the small Prandtl numbers relevant for typical semiconductor melts. The axisymmetric thermocapillary flow is found to be unstable to a steady non-axisymmetric state with azimuthal wavenumber 2, for a zone with aspect ratio 1. The critical Reynolds number for this bifurcation is 1960. This three dimensional steady solution loses stability to an oscillatory state at a Reynolds number of 6250. For small Prandtl numbers, both bifurcations are seen to be quite insensitive to changes in the Prandtl number, and are thus hydrodynamic in nature. An analogy to the instability of thin vortex rings is made. This analogy suggests a physical mechanism behind the instability and also gives an explanation of how the azimuthal wavenumber of the bifurcated solution is selected. The implications of this for the floating-zone crystal growth process are discussed.

Phase-field simulations of non-isothermal binary alloy solidification

A phase-field method for two-dimensional simulations of binary alloy solidification is studied. Phase-field equations that involve both temperature and solute redistribution are formulated. The equations are solved using the finite element method with triangular elements on unstructured meshes, which are adapted to the solution. Dendritic growth into a supersaturated melt is simulated for two temperature regimes: (a) the temperature is prescribed on the boundary of the computational domain; and (b) the heat is extracted through the domain boundary at a constant rate. In the former regime the solute redistribution is compared with the one given by an isothermal model. In the latter case the influence of the size of the computational domain and of the heat extraction rate on dendritic structure is investigated. It is shown that at high cooling rates the supersaturation is replaced by thermal undercooling as the driving force for growth.

The phase-field approach and solute drag modeling of the transition to massive γ → α transformation in binary Fe-C alloys

The transition between diffusion controlled and massive transformation gamma --> alpha in Fe-C alloys is investigated by means of phase-field simulations and thermodynamic functions assessed by the ...

Phase-field simulation of dendritic growth in a shear flow

Abstract We study how the dendritic evolution of an initially small nucleus is affected by a mean external flow. The nucleus is considered to be attached to a solid wall, and it grows away from the wall into the melt. The melt is assumed to be flowing due to an applied shear stress far away from the wall. The fluid flow alters the local heat transfer at the solidification front, and thus the shape of the dentrite. Due to the flow the nucleus evolves to an asymmetric dentrite that tilts. Another effect of the flow is that the sidebranch-growth gets promoted and inhibited on the upstream and downstream side, respectively. We use an adaptive grid and finite element applied to the phase-field method in 2D. The adaptivity results in a high local resolution at the solidification front and a much coarser mesh away from the front.

Finite element simulations using symbolic computing

In our work on modelling of phase change, fluid flow and heat transfer in materials processes, it is crucial to have complete control of the mathematical models, and the numerical simulation of the models. We have developed a toolbox in Maple which can be used to generate complete finite element codes in 1, 2 or 3 dimensions, from a symbolic specification of the mathematical problem. It has been used to create codes used in our research on welding, crystal growth, flow of viscoelastic liquids and more. The possibility of defining a complex problem in a compact readable form, without compromising the flexibility and accessibility of the model and the numerical methods, has changed the way we use simulations in research radically.

The splash of a solid sphere impacting on a liquid surface: Numerical simulation of the influence of wetting

The impact of a solid sphere on a liquid surface has challenged researchers for centuries and remains of interest today. Recently, Duez [Nat. Phys. 3, 180 (2007)] published experimental results of ...

Dendritic growth of randomly oriented nuclei in a shear flow

A numerical study of the effect of an external mean flow on dendritic growth has been performed. All simulations are 2D phase-field computations using an adaptive finite element method. The dendritic growth starts from a small nucleus that is attached to a solid insulated wall. A flow of melt past the wall is maintained by prescribing a shear stress far from the wall. The orientation of the preferred growth directions of the nucleus is varied. In the simulations, a non-dimensional undercooling of 0.1 is used. The Prandtl number of the melt is taken to be 23, which corresponds approximately to that of SCN. Different initial orientations of the preferred growth directions of the nucleus gives different vertical growth velocities. The results depend also on the flow strength (the flow Peclet number) and the degree of anisotropy. The maximal vertical growth velocity tends to be achieved for crystals with a preferred growth direction that is moderately inclined in the upstream direction.