Ivantsov Solution Research Articles

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.

Simulation of dendritic growth of Al-4 wt.% Cu alloy from an undercooled melt

Abstract Three-dimensional dendritic growth of an Al-4 wt.% Cu alloy during solidification in the presence of forced flow was simulated by using a three-dimensional cellular automaton model. The three-dimensional Lattice–Boltzmann method was used in computing the flow dynamics. The effects of forced flow on the growth Péclet number, dendrite tip selection parameter, and secondary arm spacing were investigated. The simulated relationships of growth Péclet number and dendrite tip selection parameter with the forced flow are in good agreement with the Oseen–Ivantsov solution. The results indicate that with the increase in forced flow velocity, the secondary arm spacing first increases and then decreases.

Analytic solution for the morphology of a soil‐mantled valley undergoing steady headward growth: Validation using case studies in southeastern Arizona

The geomorphic literature contains many analytic solutions for the topographic evolution of gently sloping soil‐mantled hillslopes responding to base level changes. Most of these solutions are limited to vertical base level changes and/or to simplified geometries, however. In this paper we present an analytic solution for the morphology of a valley and its adjacent hillslopes undergoing steady headward growth. The mathematics of this problem were first solved by Ivantsov (1947) in the context of heat flow near a parabolic solidification boundary. Here we test whether the Ivantsov solution provides an accurate first‐order prediction of the morphology of valley heads and their adjacent hillslopes by comparing the model predictions to survey data from two study sites in southeastern Arizona. The model predicts that elevation contours of valley heads are parabolas and that topographic transects normal to contour lines are error functions. High‐resolution Digital Elevation Models (DEMs) were constructed for the two study sites using Real‐Time Kinematic Global Positioning System (RTK‐GPS) measurements and a Terrestrial Laser Scanner (TLS). Our analyses show that the model reproduces the first‐order morphology of headward‐growing valleys and their adjacent hillslopes. We also show that by analyzing hillslope profiles at different distances from the valley head, the model framework can be used to infer likely changes in the valley head migration rate through time.

Adaptive phase field simulation of dendritic crystal growth in a forced flow: 2D vs 3D morphologies

Two- and three-dimensional (3D) adaptive phase field simulations of dendritic crystal growth in a forced flow are presented. The simulations are based on an adaptive finite volume mesh for a better resolution on the dendrite morphology. It also allows the simulation in a large domain without much additional computing cost, so that the boundary effect can be neglected for the comparison with classic solutions. With the efficient simulations, the effect of forced convection on the growth behavior at high undercooling is discussed, and the results agree well with Oseen–Ivantsov solution and the reported results. For the case of low undercooling, the simulated tip radius and speed are also consistent with the experimental ones. As compared with 2D morphologies, side branches are easily induced in 3D dendrites, and the dramatic difference can be explained through the simulated flow structures and temperature fields. The effect of the flow on the side branching for different undercoolings is also illustrated.

Solidification velocity of undercooled Ni–Cu alloys

The solidification velocity of Ni–Cu alloys was measured as a function of bulk undercooling using high-speed thermal imaging of electromagnetically levitated samples. Two departures from power law growth (approximating plateaus) in the velocity versus undercooling data were observed: the first occurred at intermediate undercoolings and is attributed to copper solute, while the second occurred at high undercoolings and is hypothesized to be an effect of oxygen. The Ivantsov solution with marginal stability arguments (IMS model) is a widely used model that relates dendrite growth velocity to total undercooling for dilute alloy systems. However, the model does not predict a plateau at intermediate undercoolings for alloys with a large equilibrium partition coefficient, k E. Satisfactory agreement between the model and experimental results can be obtained by using a value of k E that is smaller than the alloy’s actual value, but this is physically unreasonable and causes disagreement with currently accepted kinetic models.

Electrically enhanced free dendrite growth in polar and non-polar systems

We describe the diffusion-limited growth of dendritic needle crystals from the vapor phase, for which the application of an electrical potential to the crystal can profoundly affect the growth morphology, leading to an electrically induced morphological instability. With an improved theoretical treatment of this phenomenon, as well as new experimental data, we find the following below a threshold potential: (1) attachment kinetics is often more important than surface tension for selecting the Ivantsov solution in growth from the vapor phase, even in the absence of significant faceting; (2) the electrically modified growth velocity contains both polarization and electrostatic contributions, in contrast to previous treatments; (3) the threshold electrical potential is roughly independent of supersaturation, and (4) non-polar systems also exhibit the same electrically induced instability. Above threshold the underlying assumptions of solvability theory break down, and a new stabilization mechanism is required to explain the high-velocity shape-preserving needle growth observed in this regime. Guided by new electric needle data, we propose that the growth rates are being stabilized primarily by structural deformation driven by surface tension. We also describe the use of chemical additives to alter the needle growth axis and greatly increase the growth velocity.

Effect of surface free energy anisotropy on dendrite tip shape

In previous work, approximate solutions were found for paraboloids having perturbations with four-fold axial symmetry in order to model dendritic growth in cubic materials. These solutions provide self-consistent corrections through second order in a shape parameter ε to the Peclet number vs supercooling relation of the Ivantsov solution. The parameter ε is proportional to the amplitude of the four-fold correction to the dendrite shape, as measured from the Ivantsov paraboloid of revolution. The equilibrium shape for anisotropic surface free energy to second order in the anisotropy is calculated. The value of ε is determined by comparing the dendrite tip shape with the portion of the equilibrium shape near the growth direction, [001], for anisotropic surface free energy of the form γ= γ 0[1+4 ε 4( n 1 4+ n 2 4+ n 3 4)], where the n i are components of the unit normal of the crystal surface. This comparison results in ε=−2 ε 4−24 ε 4 2+ O( ε 4 3), independent of the Peclet number. From the experimental value of ε 4, it is found that ε≈−0.012±0.004, in good agreement with the measured value ε≈−0.008 of LaCombe et al. ( Phys. Rev. E, 1995, 52, 2778)

Analytic solution for a non-axisymmetric isothermal dendrite

The Ivantsov solution for an isothermal paraboloid of revolution growing into a pure, supercooled melt provides a relation between the bulk supercooling and a dimensionless product (the Peclet number P) of the growth velocity and tip radius of a dendrite. Horvay and Cahn generalized this axisymmetric analytical solution to a paraboloid with elliptical cross-section. They found that as the deviation of the dendrite cross-section from a circle increases, the two-fold symmetry of the interface shape causes a systematic deviation from the supercooling/Peclet number relation of the Ivantsov solution. To model dendritic growth in cubic materials, we find approximate solutions for paraboloids having perturbations with four-fold axial asymmetry. These solutions are valid through second order in the perturbation amplitude, and provide self-consistent corrections through this order to the supercooling/Peclet number relation of the Ivantsov solution. Glicksman and colleagues have measured the shape and the supercooling/Peclet number relation for growth of succinonitrile dendrites in microgravity. For a Peclet number of P≈0.004 and the experimentally observed shape, we calculate a correction corresponding to a 9% increase in the supercooling, in general agreement with the experimental results.

Numerical Simulation of Isothermal Dendritic Growth by Phase-field Model.

The phase-field model is applied to simulate microstructural evolution during isothermal dendritic growth of the δ-phase in undercooled Fe-C liquid. The parameters of phase-field equation are determined by a thin interface limit condition. Realistic dendritic structures at various temperatures and compositions of the alloy can be obtained using the phase-field model with the thin interface limit condition. The calculated results show various solidification features consistent with our experience. The calculated tip radius, the velocity and the concentration at the interface are compared with the predicted values from the conventional dendritic growth theory using Ivantsov solution and marginal stability criterion.

The tip temperature of a freely growing dendrite: the influence of various modifications to the Ivantsov solution

The space shuttle dendritic growth data for high-purity succinonitrile (SCN) are reinterpreted here in terms of the model proposed earlier by the present author. Unlike other recent growth models that have been used to rationalize the SCN growth data, all of which include two adjustable parameters, the present model includes only one unknown parameter, λ t. It has been shown that λ t = 1/Nu, the reciprocal of the “local” value of the heat transfer Nusselt number at the dendrite tip. Thus λ t may also be interpreted as a measure of the heat flux generated by the dendritic sidebranches. Estimates for λ t based on the space shuttle dendritic growth data are provided. A comparison of this model with the solutions to the heat transfer problem for simple geometric shapes, such as a slowly growing sphere (λ t = 1), the Fisher solution for a cylindrical body with a hemispherical tip (λ t = 1/2), and the Ivantsov paraboloid, λ t = (φ t/2p t), suggest that λ t can be bounded between the values of 1/2 ≤ λ t ≤ φ t/2p t.

The Tip Temperatures of Freely Growing High-Purity Succinonitrile Dendrites

The tip temperatures of “freely” growing, high-purity succinonitrile dendrites have been estimated here using the Ivantsov solution and space shuttle dendritic growth data. The estimated tip temperatures are found to be significantly higher than the currently accepted equilibrium melting point of succinonitrile (58.083°C or 331.24 K), for a large majority of these experiments. Because a solid cannot grow above its melting point, it is suggested that this discrepancy is due to dendritic side-branching effects that should be included, in a systematic manner, in the analysis of the growth data. The earth-based dendritic growth data reveal the same discrepancies.

Solute Redistribution Model of Dendritic Solidification Considering Diffusion in Both the Liquid and Solid Phases.

To estimate micro-segregation with dendritic solidification, a common method using an integral profile method has been developed. We considered three cases; (1) no diffusion in the solid and finite diffusion in the liquid, (2) finite diffusion in the solid and complete diffusion in the liquid, and (3) finite diffusion in both the solid and liquid. Expressing the solute distribution in the liquid and/or in the solid as quadratic polynomials, and assuming the linear growth case or the parabolic growth case, ordinary differential equations have been obtained. We introduced Ivantsov's solution as the initial condition of the diffusion problem to expand the model to rapid solidification conditions. Using the Runge–Kutta method, the solute distribution profile in both the liquid and solid over time was estimated. After calculation, good agreement with rigorous solutions for each case was gained. For easy way of programming and short computating time, this method has practical merits.

Dendritic crystal growth for weak undercooling

Through an asymptotic expansion for small undercooling (i.e., P\'eclet number $P\ensuremath{\ll}1$), different regions of a two-dimensional evolving dendrite in a one-sided model are identified, when the dendrite shape is initially parabolic in the far field. There is an ${o(P}^{\ensuremath{-}1})$ tip region (region I) where the temperature field is a Laplacian with an appropriate matching condition to the far field. For deviations to an Ivantsov field that are initially limited to this region, it is shown to be consistent to assume that the Ivantsov solution is asymptotically valid in the far-field asymptotic regions when $t\ensuremath{\ll}{P}^{\ensuremath{-}1}.$ Within this interval of time, we consider the dynamics in region I through an integro-differential equation derived from a conformal mapping formulation, where the upper-half \ensuremath{\zeta} plane is mapped to the exterior of the dendrite. It is shown that a linear analysis of an initially localized disturbance fails to remain localized later in time. Instead, it results in a partially wavy interface where the waviness has a sharp leading edge. This feature is shown to be related to the Stokes phenomenon associated with the inner equation around an initial complex singularity in the lower-half \ensuremath{\zeta} plane, as it advects into the upper-half \ensuremath{\zeta} plane. The specific linear growth rate is shown to be consistent with prior results of Barber, Barbieri, and Langer [Phys. Rev. A 36, 3340 (1987)], though not with Caroli et al. [J. Phys. (Paris) 48, 1423 (1987)]. The results do not depend crucially on the nature of initial disturbance or the singularity in the lower-half complex \ensuremath{\zeta} plane that is used to represent such a disturbance. However, nonlinear effects are shown to cause important changes in the prediction for both qualitative and quantitative aspects. This understanding is advanced by considering the dynamics of singularities in the lower-half complex \ensuremath{\zeta} plane. We also present many features of the zero-surface-energy dynamics including tip splitting, sidebranching, as well as cusp formation. We also present a scenario for dendrite sidebranch coarsening in terms of the motion of complex singularities.

Dendritic growth in microgravity and forced convection

Abstract A unified treatment of natural and forced convection, based on solutions of Navier-Stokes and energy equations, is used to analyze the experimental data on dendritic crystal growth from undercooled succinonitrile melts. The melt velocities that have been induced in the experiments reported for, μ g , 1 g and forced convection differ by 10 5 , the order being 10 −5 , 10 −2 , 1 cm/s, respectively. Transport theory predicts quite well the growth Peclet number, P= VR /2α, as a function of the Stefan, St, Prandtl, Pr, and Grashof, Gr, or Reynolds, Re, numbers in thermal or forced convection. For μ g , the theoretical simulations and the experiments suggest a relation P − P IV = aP IV + bGr 1/4 , where P IV is the growth Peclet number given by Ivantsov's conduction theory with equal melt and solid densities, ϱ 1 = ϱ s , and a , b are constants. The microgravity (μ g ) data show systematic deviations above and below Ivantsov's conduction theory for an isothermal dendrite at low and high undercooling, respectively. We show here theoretically that convection in the melt, at low undercooling is partly responsible for the higher rates of growth. Conversely, advection, (ϱ 1 ≠ ϱ s ), solid phase conduction and the Gibbs-Thompson curvature at large undercoolings could account for most of the deviations below Ivantsov's solution in the μ g experiments. At 1 g , the convection theory of Ananth and Gill for an isothermal dendrite agrees well with the experiments over the entire range of undercoolings studied. The critical undercooling required to produce by convection a 1% change in P decreases with the level of gravity and is given approximately by the slowly varying function ΔT c ∼ (g/g 0 ) 0.15 . Thus a large change in g/g 0 produces a small change in ΔT c , and the critical undercooling needed to render convection negligible is rather insensitive to the strength of the gravitational field. Existing theories predict that the pattern selection parameter, σ*, is independent of undercooling. However, the thermal (10 −5 –10 −2 cm/s) and forced (10 −1 −1 cm/s) convection data show that σ* does depend on both the undercooling and the intensity of convection at larger values of Re and depends only on ΔT at small Re. In the forced convection experiments, Re is large, and σ* clearly increases with Reynolds number at fixed values of the undercooling. In the thermal convection experiments, σ* increases with decreasing ΔT and is independent of the intensity (up to 0.01 cm/s) of convection because Re is small. This dependence of σ* on ΔT and Re is not well understood and needs further study because of theoretical interest.

Thermal diffusion dominated dendritic growth — an analysis of the wall proximity effect

It is demonstrated that using a simple correction to the original Ivantsov solution to account for wall proximity effects is sufficient to describe the Peclet number microgravity data of Glicksman et al. [M.E. Glicksman, M.B. Koss and E.A. Winsa, Phys. Rev. Lett. 73 (1994) 573; M.E. Glicksman, M.B. Koss, L.T. Bushnell, J.C. LaCombe and E.A. Winsa, ISLJ International 35 (1995) 1216; MRS Fall Meeting, Symp. P, Boston MA, 1995, in press] at low supercooling. The analytical correction provides for the enhanced diffusive heat transfer when the thermal diffusion length becomes comparable to the physical chamber dimension. The wall proximity effect is also responsible for the existence of a lower supercooling limit below which the dendrite cannot grow in a steady-state manner. It is concluded that Glicksman's USMP-2 microgravity data is thermal diffusion dominated and thus entirely appropriate for comparison with dendritic growth theories.

Dendritic Growth of Succinonitrile in Terrestrial and Microgravity Conditions as a Test of Theory.

Dendritic growth is the common mode of solidification encountered when metals and alloys freeze under low thermal gradients. The growth of dendrites in pure melts is generally acknowledged to be controlled by the transport of latent heat from the moving crystal-melt interface into its supercooled melt. The Ivantsov formulation solves the equation of heat flow from a paraboloidal dendrite tip for the case of diffusive heat transport. However, this formulation is incomplete, and the physics of an additional selection rule, coupled to the Ivantsov solution, is necessary to predict the dendrite tip velocity and radius of curvature as a unique function of the supercooling. Unfortunately, the experimental evidence is not definitive because dendritic growth can be complicated by buoyancy-induced convection, which is normally unavoidable under terrestrial conditions. Recent experiments performed in the microgravity environment of the space shuttle Columbia (STS-62) show quantitatively that convection alters the tip velocities and radii of curvature of dendrites in both terrestrial and microgravity conditions. In addition, these data can be used to evaluate both how well the Ivantsov diffusion solution and the selection rule (the product of the dendrite tip velocity and the tip radius of curvature squared is a constant) match the dendritic growth data under microgravity conditions.

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.

Péclet number and crystal growth in a channel.

We present experimental results on the growth of a needle-shaped crystal from an undercooled liquid in a rectangular capillary. We have studied the crossover from the Saffman-Taylor solution at small P\'eclet number to the Ivantsov solution at large P\'eclet number. Surface tension anisotropy is studied by comparing succinonitrile and pivalic acid for different orientations of the crystalline axis. Finally, we show evidence that, at least above a certain value of the P\'eclet number, the growth at constant velocity is not steady state.

Dendritic solidification of krypton and xenon

Rare gases form simple liquids and thus they can be used as model substances to study the dendritic solidification of metals and the developments of spatial structures at conditions far from equilibrium. Krypton and xenon form an ideal couple to study scaling laws, because the physical properties of heavy rare gases scale according to their simple interaction forces. The growth of rare gas dendrites into a volume of about 100 cm 3 supercooled melt was investigated using the capillary injection technique. The supercooling of the melt, ΔT = T m - T ∞, was held constant to within ±0.0002 K during an experiment. T m designates the triple point temperature of the rare gas and T ∞ the temperature of the melt far away from the dendrite. Tip growth velocity v tip , tip radius R and secondary side arm spacing S tip of krypton and xenon dendrites have been measured in the supercooling range 0.005 < ΔT < 0.3 K. The tip regions showed a fourfold symmetry with wings reaching almost to the tip. At low supercooling ( ΔT < 0.02 K) the dendrite tips become rotationally symmetric, suggesting that anisotropy of surface free energy in kinetics is not important in this regime. The dependences of v tip [ cm/ s], R[ cm] and S tip [ cm] on ΔT [ K] were determined under steady state conditions and could be fitted for krypton by v tip = 3.12 × 10 -2 ΔT 1.73, R = 6.70 × 10 -4 ΔT -0.68 and S tip = 1.47 × 10 -3 ΔT -0.80, and for xenon by v tip = 1.89 × 10 -2 ΔT 1.74, R = 7.48 × 10 -4 ΔT -0.68 and S tip = 1.78 × 10 -3 ΔT 0.79. Thus the Péclet number, p = v tip R/(2 α), where α is the thermal diffusivity in the melt, is proportional to Δ T. This is in agreement with the so-called Ivantsov solution. The measured growth velocities of Kr and Xe are indistinguishable if measured in dimensionless units. The data of Kr and Xe do not confirm a crucial result of stability theories and of more recent numerical analyses of the diffusion problem. In these theoretical treatments it is assumed that the kinetics at the solid-liquid interface do not at all influence the solidification process, and it is found that v tip R 2 = constant, i.e. the volume solidification rate does not depend on supercooling. The rare gas data show that the volume solidification rate increases with increasing supercooling.

Asymptotic behavior of steady dendrite growth at far field.

The asymptotic behavior of the solution to steady dendrite growth with isotropic surface tension is examined. The results show that (1) at far field (z\ensuremath{\rightarrow}\ensuremath{\infty}), the deviation of the nonisothermal dendrite from an Ivantsov , where the eigenvalue ${\ensuremath{\alpha}}_{1}$ is within the interval (0,(1/2) and can be determined by the undercooling of the melt and (2) the asymptotic character of the dendrite at far field cannot be fully determined locally. It is also influenced by the behavior of the solution at the tip region. These results demonstrate that the conventional far-field boundary condition in which the difference between the dendrite and the Ivantsov paraboloid is vanishingly small is not correct. Moreover, the results also imply that for any finite surface tension linearization around the Ivantsov solution corresponding to zero surface tension may not be uniformly valid in the whole region (0\ensuremath{\le}z\ensuremath{\infty}). As a consequence, the conclusions drawn from the linearized system, such as the nonexistence of the steady solution and the ``solvability condition theory,'' are questionable.