Dendrite tips as elliptical paraboloids

Abstract

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.

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

“…The boundary integral method [9,11,15,16] allows to describe the steady-state shape of a solid/liquid interface in a binary undercooled melt by a single integro-differential equation in a dimensionless form, where ζ (x, t) is the interface function, and x and t represent the spatial and time variables as…”

“…The temperature I T ζ and solute concentration I C ζ boundary integrals, respectively, are given in the general form for the 3D case [9,11,15,16] as…”

“…In the present article, tip shape and radius of Si dendrites grown from pure Si melts as documented in the literature are used for comparison with the Al-Si dendrites. Parabolic dendritic Si tips are compared with the recently developed theory [9][10][11] on the dendrite tip shapes and the region behind the tip to a distance of several tip radii. The theory is based on analytical results by Ivantsov [12,13], and represents the generalized shape function smoothly transforms between these limiting cases within the transition region.…”

The boundary integral equation for curved solid/liquid interfaces propagating into a binary liquid with convection

“…Let us especially note that the convective boundary integral method developed below has potential applications to the selection theory of stable dendritic growth [13][14][15][16], morphological stability analysis of curvilinear fronts [17][18][19], numerical calculations of pattern formation [10,[20][21][22], and comparison with the phase-field simulations.…”

Convective boundary integral equation: the case of a non-axisymmetric dendrite with a forced viscous flow

“…It is worth noting that one of the important applications of the boundary integral theory is the study of various shapes of growing patterns in an undercooled liquid. Studies carried out recently in [12][13][14] have made it possible to investigate in detail the shape of dendritic crystal vertices.…”