The shape of dendritic tips: a test of theory with computations and experiments

Scaling laws for two-dimensional dendritic crystal growth in a narrow channel

We investigate analytically and computationally the dynamics of two-dimensional needle crystal growth from the melt in a narrow channel. Our analytical theory predicts that, in the low supersaturation limit, the growth velocity V decreases in time t as a power law V∼t^{-2/3}, which we validate by phase-field and dendritic-needle-network simulations. Simulations further reveal that, above a critical channel width Λ≈5l_{D}, where l_{D} is the diffusion length, needle crystals grow with a constant V<V_{s}, where V_{s} is the free-growth needle crystal velocity, and approaches V_{s} in the limit Λ≫l_{D}.

A Stable Mode of Dendritic Growth in Cases of Conductive and Convective Heat and Mass Transfer

In this paper, we develop a theory of stable dendritic growth in undercooled melts in the presence of conductive and convective heat and mass transfer boundary conditions at the solid/liquid interface of a dendrite. To simplify the matter and construct the analytical theory, conductive and convective mechanisms are considered separately. Namely, the laws for total undercooling and selection criterion defining the stable growth mode (dendrite tip velocity and diameter) are derived for conductive and convective boundary conditions. To describe the case of simultaneous occurrence of these heat and mass transfer mechanisms, we sew together conductive and convective laws using power stitching functions. The generalised selection theory is compared with experimental data for Al24Ge76 and Ti45Al55 undercooled melts.

Analysis of the boundary integral equation for the growth of a parabolic/paraboloidal dendrite with convection

The growth of a parabolic/paraboloidal dendrite streamlined by viscous and potential flows in an undercooled one-component melt is analyzed using the boundary integral equation. The total melt undercooling is found as a function of the Péclet, Reynolds, and Prandtl numbers in two- and three-dimensional cases. The solution obtained coincides with the modified Ivantsov solution known from previous theories of crystal growth. Varying Péclet and Reynolds numbers we show that the melt undercooling practically coincides in cases of viscous and potential flows for a small Prandtl number, which is typical for metals. In cases of water solutions and non-metallic alloys, the Prandtl number is not small enough and the melt undercooling is substantially different for viscous and potential flows. In other words, a simpler potential flow hydrodynamic model can be used instead of a more complicated viscous flow model when studying the solidification of undercooled metals with convection.

Induced side-branching in smooth and faceted dendrites: theory and phase-field simulations

The present work is devoted to the phenomenon of induced side branching stemming from the disruption of free dendrite growth. We postulate that the secondary branching instability can be triggered by the departure of the morphology of the dendrite from its steady state shape. Thence, the instability results from the thermodynamic trade-off between non monotonic variations of interface temperature, surface energy, kinetic anisotropy and interface velocity within the Gibbs-Thomson equation. For the purposes of illustration, the toy model of capillary anisotropy modulation is prospected both analytically and numerically by means of phase-field simulations. It is evidenced that side branching can befall both smooth and faceted dendrites, at a normal angle from the front tip which is specific to the nature of the capillary anisotropy shift applied. This article is part of the theme issue 'Transport phenomena in complex systems (part 2)'.

Microstructure and morphology of Si crystals grown in pure Si and Al-Si melts

Microstructure of Al-40 wt%Si samples solidified in electromagnetic levitation furnace is studied at high melt undercooling. Primary Si with feathery and dendritic structures is observed. As this takes place, single Si crystals either contain secondary dendrite arms or represent faceted structures. Our experiments show that at a certain undercooling, there exists the microstructural transition zone of faceted to non-faceted growth. Also, we analyze the shape of dendritic crystals solidifying from liquid Si as well as from hypereutectic Al-Si melts at high growth undercoolings. The shapes of dendrite tips grown at undercoolings >100 K along the surface of levitated Al-40 wt%Si droplets are compared with pure Si dendrite tips from the literature. The dendrite tips are digitized and superimposed with theoretical shape function recently derived by stitching the Ivantsov and Brener solutions. We show that experimental and theoretical dendrite tips are in good agreement for Si and Al-Si samples.

A review on the theory of stable dendritic growth

This review article summarizes the main outcomes following from recently developed theories of stable dendritic growth in undercooled one-component and binary melts. The nonlinear heat and mass transfer mechanisms that control the crystal growth process are connected with hydrodynamic flows (forced and natural convection), as well as with the non-local diffusion transport of dissolved impurities in the undercooled liquid phase. The main conclusions following from stability analysis, solvability and selection theories are presented. The sharp interface model and stability criteria for various crystallization conditions and crystalline symmetries met in actual practice are formulated and discussed. The review is also focused on the determination of the main process parameters-the tip velocity and diameter of dendritic crystals as functions of the melt undercooling, which define the structural states and transitions in materials science (e.g. monocrystalline-polycrystalline structures). Selection criteria of stable dendritic growth mode for conductive and convective heat and mass fluxes at the crystal surface are stitched together into a single criterion valid for an arbitrary undercooling. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

Transport phenomena in complex systems (part 1)

The issue, in two parts, is devoted to theoretical, computational and experimental studies of transport phenomena in various complex systems (in porous and composite media; systems with physical and chemical reactions and phase and structural transformations; in biological tissues and materials). Various types of these phenomena (heat and mass transfer; hydrodynamic and rheological effects; electromagnetic field propagation) are considered. Anomalous, relaxation and nonlinear transport, as well as transport induced by the impact of external fields and noise, is the focus of this issue. Modern methods of computational modelling, statistical physics and hydrodynamics, nonlinear dynamics and experimental methods are presented and discussed. Special attention is paid to transport phenomena in biological systems (such as haemodynamics in stenosed and thrombosed blood vessels magneto-induced heat generation and propagation in biological tissues, and anomalous transport in living cells) and to the development of a scientific background for progressive methods in cancer, heart attack and insult therapy (magnetic hyperthermia for cancer therapy, magnetically induced circulation flow in thrombosed blood vessels and non-contact determination of the local rate of blood flow in coronary arteries). The present issue includes works on the phenomenological study of transport processes, the derivation of a macroscopic governing equation on the basis of the analysis of a complicated internal reaction and the microscopic determination of macroscopic characteristics of the studied systems. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

Dendrite tips as elliptical paraboloids

This review article summarizes current theories of the steady-state growth mode of dendrites in the form of elliptical paraboloids. The shape of dendrite tips is analyzed, temperature and solute concentration distributions are described in its vicinity, and a solution of the hydrodynamic problem of a viscous incompressible fluid flowing against a dendrite tip is developed. A significant difference in analytical solutions describing a dendrite tip as an elliptic paraboloid as compared to an axisymmetric morphology is shown. The system of nonlinear equations for determining the stationary velocity of dendrite growth and the radii of curvature of the dendrite tip along the major and minor axis of the ellipse, respectively, is derived. The developed theory is compared with experimental data on the growth of ice crystals consisting of D₂O or H₂O.