Physics of heat flow in the tails of needle crystals (original) (raw)
Selected mode for rapidly growing needle-like dendrite controlled by heat and mass transport
Acta Materialia, 2017
The boundary integral method is developed for fast anisotropic interfaces. A general integro-differential equation for curved interfaces controlled by heat and mass transport is derived and applied to the problem of rapid dendritic growth. A selection criterion for the steady-state mode of growing parabolic interfaces is obtained and, in common solution with the undercooling balance, it is compared with experimental data on rapid dendritic solidification of deeply supercooled liquid droplets. In this comparison, transitions from solute diffusion-limited to thermo-solutal regime and, finally, to pure thermally controlled regime of the anisotropic dendrite are discussed and revealed. Limiting cases of known selection criteria for anisotropic dendrites growing at small and high growth P eclet numbers are provided.
Steady-state dendritic growth at non-zero capillarity
Scripta Metallurgica, 1984
Recently there has been renewed interest in the dendritic growth exhibited by substances solidifying from a supercooled melt(I,2). The growth process gives rise to a complex morphology of the resulting solid, making solidification a prime example of non-equilibrium pattern formation. Under the assumption that the interface motion is completely controlled by heat diffusion, one can derive a set of equations which should be able to predict the long-time evolution of the growing crystal. To date, most attention has focused on the rate at which the dendritic tip grows and the spatial distribution of the branches generated along the initial needle crystal(3). Schematically, we can represent (I) as the operator equation },M 1 [oJ] + M 2 [~o] --O (3)
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2018
Motivated by important applications in materials science and geophysics, we consider the steady-state growth of anisotropic needle-like dendrites in undercooled binary mixtures with a forced convective flow. We analyse the stable mode of dendritic evolution in the case of small anisotropies of growth kinetics and surface energy for arbitrary Péclet numbers and n -fold symmetry of dendritic crystals. On the basis of solvability and stability theories, we formulate a selection criterion giving a stable combination between dendrite tip diameter and tip velocity. A set of nonlinear equations consisting of the solvability criterion and undercooling balance is solved analytically for the tip velocity V and tip diameter ρ of dendrites with n -fold symmetry in the absence of convective flow. The case of convective heat and mass transfer mechanisms in a binary mixture occurring as a result of intensive flows in the liquid phase is detailed. A selection criterion that describes such solidific...
Physical Review E, 2000
Viscous fingering morphologies during the displacement of a high viscosity fluid by a low viscosity immiscible fluid in a radial fourfold anisotropic Hele-Shaw cell are examined. By using the kerosene-glycerin system for which the /T ratio (being the relative viscosity and T the interfacial tension between the fluids͒ is about ten times higher than that for the commonly used air-glycerin system, we have been able to access the hitherto unexplored N ca տ1 regime ͑capillary number N ca ϭU/T, U being the advancing fingertip velocity͒. Within the anisotropy-dominated regime, and when flow rates are significantly high ͑capillary number well beyond N ca ϭ1), a new phase is seen to evolve wherein the dendrites grow simultaneously along the channels and along the directions making an angle of 45°with the channels, both being kinetically driven. This new phase resembles the one observed in a miscible fluid system at all flow rates of the displacing fluid.
Physical Review Letters, 1985
We develop a systematic boundary-layertype formalism for diffusion-controlled dendritic growth, which yields an expression for the shape of steady-state needle solutions valid at large un- dercoolings. Both physical and analytical considerations suggest the general existence of a continu- ous family of steady-state needlelike solutions of the heat-flow equations. Simple modifications of the boundary-layer model of Ben-Jacob et al. exhibit this behavior.
2020
The dendritic form is one of the most common forms of crystals growing from supercooled melts and supersaturated solutions. In recent decades, an analytical theory has been developed that describes a stable dendrite growth mode under the conditions of a conductive heat and mass transfer process. However, in experiments, the growth of dendritic crystals is often observed under the conditions of convective fluid flow. In the present work, the theory of the growth of dendritic crystals is developed taking into account the convective mechanism of heat and mass transfer at the crystal-melt interface. A stable mode of dendritic growth in the case of intense convective flows near the steady-state growing dendritic tip is analyzed. The selection theory determining a stable growth mode in the vicinity of parabolic solutions as well as the undercooling balance condition are used to find the dendrite tip velocity and its tip diameter as functions of the melt undercooling. It is shown that the theoretical predictions in the case of convective boundary conditions are in agreement with experimental data for small undercoolings. In addition, the convective and conductive heat and mass transfer mechanisms near the growing dendritic surfaces are compared. Our calculations show that the convective boundary conditions essentially influence the stable mode of dendritic growth.
Physical Review E, 1999
The evolution of viscous fingering morphology is examined for the case of a system of miscible fluids in an anisotropic radial Hele-Shaw cell. It is shown that dendritic morphologies similar to the kinetic and surfacetension morphology types coexist for this case. The critical role of the means of introducing anisotropy in the Hele-Shaw cell is established, and an explanation of the pattern behavior is offered on the basis of shape discontinuities of the individual elements of the lattice used to induce anisotropy. The ramifications of such an explanation are experimentally verified by demonstrating a clear difference in the morphology evolution in two halves of a single Hele-Shaw cell, one half of which contains square lattice elements, and the other half of which contains circular lattice elements. ͓S1063-651X͑99͒14202-8͔
Phase-field simulations of dendritic crystal growth in a forced flow
2001
Convective effects on free dendritic crystal growth into a supercooled melt in two dimensions are investigated using the phase-field method. The phase-field model incorporates both melt convection and thermal noise. A multigrid method is used to solve the conservation equations for flow. To fully resolve the diffuse interface region and the interactions of dendritic growth with flow, both the phase-field and flow equations are solved on a highly refined grid where up to 2.1 million control volumes are employed. A multiple time-step algorithm is developed that uses a large time step for the flow-field calculations while reserving a fine time step for the phase-field evolution. The operating state ͑velocity and shape͒ of a dendrite tip in a uniform axial flow is found to be in quantitative agreement with the prediction of the Oseen-Ivantsov transport theory if a tip radius based on a parabolic fit is used. Furthermore, using this parabolic tip radius, the ratio of the selection parameters without and with flow is shown to be close to unity, which is in agreement with linearized solvability theory for the ranges of the parameters considered. Dendritic sidebranching in a forced flow is also quantitatively studied. Compared to a dendrite growing at the same supercooling in a diffusive environment, convection is found to increase the amplitude and frequency of the sidebranches. The phase-field results for the scaled sidebranch amplitude and wavelength variations with distance from the tip are compared to linear Wentzel-Kramers-Brillouin theory. It is also shown that the asymmetric sidebranch growth on the upstream and downstream sides of a dendrite arm growing at an angle with respect to the flow can be explained by the differences in the mean shapes of the two sides of the arm.
A Stable Mode of Dendritic Growth in Cases of Conductive and Convective Heat and Mass Transfer
Crystals
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.
Heat diffusion anisotropy in dendritic growth
Journal of Crystal Growth, 1998
An anisotropic heat diffusion coefficient is introduced in order to study some interfacial growth phenomena. This anisotropy has been incorporated in a phase field model which has been studied numerically to reproduce some fundamental solidification situations (needle crystal growth) as well as the dynamics of a nematic-smectic-B interface. As a general result, we find that dendrites grow faster in the lower heat diffusion direction. Simulation results are compared with experiments with remarkable qualitative agreement.