Microscopic Solvability Theory Research Articles

Dendritic tip selection during solidification of alloys: Insights from phase-field simulations

Abstract The effect of undercooling ΔT and the interface energy anisotropy parameter ε 4 on the shape of the equiaxed dendritic tip has been investigated by using a quantitative phase-field model for solidification of binary alloys. It was found that the tip radius ρ increases and the tip shape amplitude coefficient A 4 decreases with the increase of the fitting range for all cases. The dendrite tip shape selection parameter σ* decreases and then stabilizes with the increase of the fitting range, and σ* increases with the increase of ε 4. The relationship between σ* and ε 4 follows a power-law function σ ∗ ∝ ε 4 α , and α is independent of ΔT but dependent on the fitting range. Numerical results demonstrate that the predicted σ* is consistent with the curve of microscopic solvability theory (MST) for ε 4 < 0.02, and σ* obtained from our phase-field simulations is sensitive to the undercooling when ε 4 is fixed.

Free dendritic growth model for binary alloy based on microscopic solvability theory and nonisothermal nature caused by anisotropy and curved interface

Abstract Considering both the effect of interfacial anisotropy and the effect of the non-planar interface, an extended free dendritic growth model for binary alloys was proposed. It can be achieved by introducing the microscopic solvability theory to replace the marginal stability theory and by adopting a more improved interface response function. The thermal diffusion equation is also re-solved strictly with the new interfacial response function as boundary condition. Both the morphological stability model and the interface response function take into account the interfacial nonisothermal nature caused by interfacial anisotropy and non-planar interface. Therefore, the present model is truly self-consistent. Model comparisons with the related models and the available experiment data for the Cu70Ni30 alloy were also made. An agreement between the model predictions with the experiment data was obtained.

Free dendritic growth model based on nonisothermal interface and microscopic solvability theory

Abstract Considering both the effect of nonisothermal nature of the solid/liquid interface and the microscopic solvability theory (MicST), a further improved version of free dendritic growth model for pure materials was proposed. Model comparison indicates that there is a higher temperature at the tip of dendrite predicted by the present model compared with the corresponding model with the isothermal solid/liquid interface assumption. This is attributed to the sidewise thermal diffusion, i.e. the gradient of temperature along the nonisothermal interface. Furthermore, it is indicated that the distinction between the stability criteria from MicST and marginal stability theory (MarST) is more significant with the increase of bath undercoolings. Model test also indicates that the present model can give an agreement with the available experimental data. It is finally concluded that the nonisothermal nature of the solid/liquid interface and the stability criterion from MicST should be taken into account in modeling free dendritic growth.

Research of three-dimensional dendritic growth using phase-field method based on GPU

The interface thickness of phase-field model limits its simulation scale. In order to shorten the computation time, the large undercooling degree higher than that under the conventional casting process condition is often employed in the three-dimensional dendritic growth simulation, and many qualitative simulations are mainly aimed at a single dendrite and multi-dendrites morphologies for pure substances or binary alloys. The problems of low computational efficiency, small-scale simulation and limited to qualitative research exist on a single CPU computation using phase-field method. To simulate microstructure evolution process during actual casting solidification more effectively, a high performance computing method based on CUDA+GPU architecture is explored in this paper, and larger-scale computation is implemented by the concurrent execution of multiple threads to improve computational efficiency. The three-dimensional dendritic growth of pure SCN is quantitatively simulated on a single GPU, and the computational efficiency based on a single GPU and a single CPU is also compared under the same condition. The simulation results show that a speedup of 98.52 is achieved when the grid size is 5123 on the GPU, with computational efficiency being greatly improved. Meanwhile, the calculated values of the dendritic tip velocities and the tip radius on the GPU are identical with the values of the microscopic solvability theory and the references’ simulation results, which validates the GPU parallel algorithm.

Laser-Driven Accelerated Growth of Dendritic Patterns in Liquids

We report a scheme for very significantly accelerated growth of dendritic patterns in organic, inorganic, polymeric, and biological liquids, using laser power as low as a few hundred microwatts in the presence of an efficient absorber such as carbon nanotubes (CNTs). The CNTs act as a heat source that drives dendritic growth; their anisotropy ensures a rich diversity of branched patterns. We have studied the time evolution of the accelerated growth patterns; growth patterns are seen on millisecond time scales. We rationalize such unprecedented speed of dendritic growth using a diffusion equation for the temperature field with an additional source term. The predictions of the well-established microscopic solvability theory (MST) are seen to hold on time scales in excess of 100 ms even with the introduction of an additional source term to account for our laser beam. On the other hand, on time scales shorter than 100 ms, MST is seen to break down. Our method opens new vistas for studies on the dynamics of de...

On the dendritic tip stability parameter for aluminium alloy solidification

The dendritic tip stability parameter σ∗ is estimated for two aluminium alloys, namely Al–4 wt.%Cu and Al–2 wt.%Si, based on recently measured values of their crystal-melt surface energy anisotropy strength and linear scaling of the microscopic solvability theory. The influence of the stability parameter on columnar dendritic growth models of solidification, and consequently on columnar to equiaxed transition (CET), is determined, with the help of modified Hunt CET maps and meso-scale front tracking simulations.

Sharp interface Cartesian grid method III: Solidification of pure materials and binary solutions

A numerical technique is presented for computing dendritic growth of crystals from pure melts and binary solutions. The governing equations are solved on a fixed Cartesian mesh and the immersed phase boundary is treated as a sharp solid–fluid interface. The interface is tracked using a level-set field. A finite-difference scheme is presented that incorporates the immersed phase boundary with only a small change to a standard Cartesian grid Poisson solver. The scheme is simple to implement in three-dimensions. The results from our calculations show excellent agreement with two-dimensional microscopic solvability theory for pure material solidification. It is shown that the method predicts dendrite tip characteristics in excellent agreement with the theory. The sharp interface treatment allows discontinuous material property variation at the solid–liquid interface. This facilitates sharp-interface simulations of dendritic solidification of binary solutions.

A FINITE-VOLUME SHARP INTERFACE SCHEME FOR DENDRITIC GROWTH SIMULATIONS: COMPARISON WITH MICROSCOPIC SOLVABILITY THEORY

We present and validate a numerical technique for computing dendritic growth of crystals from pure melts. The solidification process is computed in the diffusion-driven limit. The governing equations are solved on a fixed Cartesian mesh and a mixed Eulerian-Lagrangian framework is used to treat the immersed phase boundary as a sharp solid-fluid interface. A conservative finite-volume discretization is employed which allows the boundary conditions to be applied exactly at the moving surface. The results from our calculations are compared with two-dimensional microscopic solvability theory. It is shown that the method predicts dendrite tip characteristics in good agreement with the theory. The sharp interface treatment allows discontinuous material property variation at the solid-liquid interface. Calculations with such discontinuities are also shown to produce results in agreement with solvability and with other sharp interface simulations.

Microstructure evolution in equiaxed dendritic growth

Microstructure evolution in equiaxed dendritic solidification is investigated through the study of free dendritic growth in a supercooled melt. A detailed measurement of microstructural features (such as side-branch spacings, envelope shape, projection area, and contour length) of freely growing succinonitrile dendrites is performed using images from the microgravity experiment of Glicksman and co-workers. The measurements show that the microstructure evolution of an equiaxed dendrite is divided into two regimes: an initial linear regime and a subsequent non-linear coarsening regime. It is found that unique scaling relations exist between the measured geometry parameters and the primary tip radius or speed for both regimes. The underlying mechanisms involved in dendritic structure evolution are discussed. In addition, using the phase-field method, we perform numerical experiments to investigate the effects of melt convection on equiaxed dendritic growth. The dendrite tip operating state (i.e. the tip velocity and radius) is quantitatively evaluated as a function of the flow velocity and dendrite orientations and compared with Microscopic Solvability Theory. Other structural features (such as the side-branches) of an equiaxed dendrite in the presence of flow are also examined in order to show how convection influences microstructure evolution in equiaxed dendritic growth.

Adaptive Mesh Refinement Computation of Solidification Microstructures Using Dynamic Data Structures

We study the evolution of solidification microstructures using a phase-field model computed on an adaptive, finite element grid. We discuss the details of our algorithm and show that it greatly reduces the computational cost of solving the phase-field model at low undercooling. In particular, we show that the computational complexity of solving any phase-boundary problem scales with the interface arclength when using an adapting mesh. Moreover, the use of dynamic data structures allows us to simulate system sizes corresponding to experimental conditions, which would otherwise require lattices greater than 217×217elements. We examine the convergence properties of our algorithm. We also present two-dimensional, time-dependent calculations of dendritic evolution, with and without surface tension anisotropy. We benchmark our results for dendritic growth with microscopic solvability theory, finding them to be in good agreement with theory for high undercoolings. At low undercooling, however, we obtain higher values of velocity than solvability theory at low undercooling, where transients dominate, in accord with a heuristic criterion which we derive.

Surface tension and dynamics of fingering patterns

We study the minimal class of exact solutions of the Saffman-Taylor problem with zero surface tension, which contains the physical fixed points of the regularized (non-zero surface tension) problem. New fixed points are found and the basin of attraction of the Saffman-Taylor finger is determined within that class. Specific features of the physics of finger competition are identified and quantitatively defined, which are absent in the zero surface tension case. This has dramatic consequences for the long-time asymptotics, revealing a fundamental role of surface tension in the dynamics of the problem. A multifinger extension of microscopic solvability theory is proposed to elucidate the interplay between finger widths, screening and surface tension.

Growth of a needle crystal from an undercooled alloy melt

The growth of a free dendrite from a supercooled alloy melt is studied with the phase-field model in the isothermal approximation; the surface tension is assumed to be asymmetric with fourfold anisotropy. For fixed supercooling and when the anisotropy parameter \ensuremath{\gamma} is not too high, the computed tip radius \ensuremath{\rho} and velocity ${v}_{\mathrm{tip}}$ obey the scaling law ${\ensuremath{\rho}}^{2}{v}_{\mathrm{tip}}\ensuremath{\propto}{\ensuremath{\gamma}}^{\ensuremath{-}7/4},$ predicted by the microscopic solvability theory. The solute diffusion across the solid-liquid interface reflects the nonequilibrium conditions of the growth process: the concentration jump deviates from the static value and decreases with increasing interface velocity. We also explore the effect of different diffusivities in the solid and liquid phases. Previous studies based on the equilibrium formulation of the free boundary equations predict a monotonic increase of ${\ensuremath{\rho}}^{2}{v}_{\mathrm{tip}}$ with the solute diffusivity of the solid phase; due to nonequilibrium effects this result is not recovered in the present investigation.

Phase-field simulation of three-dimensional dendrites: is microscopic solvability theory correct?

Dendritic growth theory and experiment are critically examined in the light of recent three-dimensional phase-field simulations of the crystallization of a pure undercooled melt. These simulations make use of a new thin-interface limit of phase-field equations that makes it possible to perform efficient computations in three dimensions with negligible interface kinetics. These computations are applied to perform a quantitative test of microscopic solvability theory, which is reviewed here, and to make contact with low undercooling experiments by exploiting the scaling property of the dendrite growth problem. A good overall agreement is found between the steady-state operating state of the dendrite tip selected dynamically in the simulations and the one predicted by solvability theory over a wide range of crystalline anisotropy. Quantitative differences between solvability predictions and simulations are present at larger anisotropy but reflect the limitation of the axisymmetric approximation presently used to carry out these predictions rather than a conceptual breakdown of this theory. Phase-field simulation results yield an improved agreement with experiment for succinonitrile. A poor agreement however is still obtained for pivalic acid. Possible origins of this disagreement are discussed.

Evaluation of Dendritic Growth Theories with Experimental Data

A comparison of dendritic growth kinetics of various materials with the transport theories suggests that dendritic growth kinetics of materials might be dependent on microstructure of crystals. Also advective flow created during solidification seems to be taken into account when the growth kinetics of dendrites is determined, which has not been considered before. A plot of stability parameters, σ * , of different anisotropic materials as a function of anisotropy of interfacial energy, e m , indicates that relationship σ * ∼ e m 7/4 does not hold, which is the main prediction of microscopic solvability theory. This fact suggests that stability parameter seems to be determined not only by anisotropy of given material but also other factors which is not yet found.

Optimum stability conjecture for the role of interface kinetics in selection of the dendrite operating state

We develop a model that allows interface kinetics to be incorporated in a simple way in the determination of the dendrite operating state. The model is based on an optimum stability conjecture, according to which the dendrite tip radius is related to the wavelength of the fastest growing Fourier component in a linear stability analysis that includes consideration of interface kinetics. The basic idea is similar to that of the marginal stability hypothesis but leads to a value of the selection parameter σ = 2κ d 0/( Vϱ 2) that decreases as the Peclet number Pe = Vϱ/(2κ) is increased ( V = dendrite growth velocity, ϱ = dendrite tip radius, κ = thermal diffusivity and d 0 = capillary length). This results in 1/σ being a linear function of 1/ϱ. The Ivantsov equation for the dimensionless supercooling S in terms of Pe is modified to account for capillarity and kinetics by using the dendrite tip temperature instead of the melting temperature in the original Ivantsov equation. This yields a complete parametric solution to the problem, and the resulting “scaling laws” are exhibited for the case of two spatial dimensions. We also reexamine the data of Rubinstein and Glicksman for dendritic growth in pivalic acid for which σ varies by about 100% over the measured range of supercooling. The value of 1/σ is nearly linear in 1/ϱ, in agreement with the optimum stability conjecture. Comparison with the present theory results in a kinetic coefficient of μ = 0.218cm/s·K. The resulting calculated values of V and ϱ are found to be in better agreement with experiment than for constant σ, but still show considerable discrepancy at small supercoolings, probably due to convection. This stability theory is based on a planar geometry rather than on a parabolic geometry, the modification of the Ivantsov relation is crude, and the theory completely neglects treatment of anisotropy of either capillarity or interface kinetics, which are known to be important from microscopic solvability theory. Nevertheless, we identify a dimensionless group m = μ/ T Mγ(κ L V) which must be small for interface kinetics to be important ( T M = absolute melting temperature, γ = crystal-melt surface free energy, and L V = latent heat per unit volume).

Growth shape of 3He needle crystals.

We present experiments on the growth shape of $^{3}\mathrm{He}$ needle crystals. These crystals show a surprisingly weak sidebranching. Growth velocities v and the tip radii \ensuremath{\rho} have been measured, and v${\mathrm{\ensuremath{\rho}}}^{2}$ is found to be roughly constant. We also measured the surface tension anisotropy. We compare our results with the microscopic solvability theory, which needs to be modified due to the existence of a thermal (Kapitza) resistance ${\mathit{R}}_{\mathit{K}}$ at the interface. By studying the linear stability of a spherical seed, we show that ${\mathit{R}}_{\mathit{K}}$ can be taken into account by modifying the ratio of the bulk heat conductivities. The selected values v and \ensuremath{\rho} at the tip are found, in satisfactory agreement with the modified theory. Some qualitative arguments are also presented which may explain the weakness of the sidebranching instability.

Comment on "Surface-tension-anisotropy measurements of succinonitrile and pivalic acid: Comparison with microscopic solvability theory"

Comment on "Surface-tension-anisotropy measurements of succinonitrile and pivalic acid: Comparison with microscopic solvability theory"

Selection of a length scale in unconstrained dendritic growth with convection in the melt

For the first time the free growth of fully developed succinonitrile, SCN, dendrites is studied experimentally with carefully controlled, well defined, forced convection velocities, U ∞, in the ultrapure melt up to 1 cm/s, which is about 40 times larger than the velocity due to natural (thermal) convection, U N, and is 300 times larger than the growth velocity of the dendrite, ν. Therefore thermal convection and advection have a negligible effect on our experimental data. The selection parameter, σ ∗ = 2αd 0⧸ν R 2 , increases by over 50% as the ratio, U ∞⧸ν, of the forced convection velocity, U ∞, to the growth velocity, ν, increases from 3 to 300. This result is opposite to the prediction of microscopic solvability theory for a pure material. Our result also is opposite to that reported for binary experiments which support solvability theory and indicate that σ ∗ decreases as U ∞/ν increases up to values of about 19.

Computation of dendrites using a phase field model

A phase field model is used to numerically simulate the solidification of a pure material. We employ it to compute growth into an undercooled liquid for a one-dimensional spherically symmetric geometry and a planar two-dimensional rectangular region. The phase field model equation are solved using finite difference techniques on a uniform mesh. For the growth of a sphere, the solutions from the phase field equations for sufficiently small interface widths are in good agreement with a numerical solution to the classical sharp interface model obtained using a Green's function approach. In two dimensions, we simulate dendritic growth of nickel with four-fold anisotropy and investigate the effect of the level of anisotropy on the growth of a dendrite. The quantitative behavior of the phase field model is evaluated for varying interface thickness and spatial and temporal resolution. We find quantitatively that the results depend on the interface thickness and with the simple numerical scheme employed it is not practical to do computations with an interface that is sufficiently thin for the numerical solution to accurately represent a sharp interface model. However, even with a relatively thick interface the results from the phase field model show many of the features of dendritic growth and they are in surprisingly good quantitative agreement with the Ivantsov solution and microscopic solvability theory.

Surface-tension-anisotropy measurements of succinonitrile and pivalic acid: Comparison with microscopic solvability theory.

New determinations of the surface-tension-anisotropy parameter ${\mathrm{\ensuremath{\varepsilon}}}_{4}$ of succinonitrile (SCN), pivalic acid (PVA), and a PVA--1%-ethanol mixture are reported. Effects of temperature gradients and crystal orientation are analyzed in detail. The experiments utilized numerical interpolation techniques previously employed by Dougherty and Gollub to enhance the digital image resolution. The values found for ${\mathrm{\ensuremath{\varepsilon}}}_{4}$ in SCN, PVA, and PVA--1%-ethanol are 0.0055\ifmmode\pm\else\textpm\fi{}0.0015, 0.025\ifmmode\pm\else\textpm\fi{}0.002, and 0.026\ifmmode\pm\else\textpm\fi{}0.002, respectively. From these values, the selection parameter ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{theor}}^{\mathrm{*}}$ predicted by microscopic solvability theory for the three-dimensional axisymmetric case was computed and compared to the ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{expt}}^{\mathrm{*}}$ values determined directly from previous dendritic-growth experiments. We find that ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{theor}}^{\mathrm{*}}$/${\mathrm{\ensuremath{\sigma}}}_{\mathrm{expt}}^{\mathrm{*}}$ is 0.56\ifmmode\pm\else\textpm\fi{}0.20 for SCN, and 2.14\ifmmode\pm\else\textpm\fi{}0.50 for PVA. Possible sources for these discrepancies are discussed.