Theory Of Dendritic Growth Research Articles

On the theory of non-stationary dendritic growth in a channel

On the theory of non-stationary dendritic growth in a channel

Exact analytical solution of the equations for a quasiequilibrium two-phase domain: permeability and interdendritic spacing

This study is concerned with the theoretical description of a quasi-stationary process of directional crystallization of binary melts and solutions in the presence of a quasi-equilibrium two-phase region. The quasi-equilibrium process is ensured by the fact that the system supercooling is almost completely compensated by heat released during the phase transformation. Quasi-stationarity of the process determining constancy of the crystallization rate is ensured by given temperature gradients in the solid and liquid phases. The system of heat and mass transfer equations and boundary conditions to them under these assumptions is dependent on a single spatial variable in the reference frame moving with the crystallization rate relative to a laboratory coordinate system. Exact analytical solutions to the formulated problem in parametric form are obtained. The parameter of the solution is the solid phase fraction in a two-phase region. The distributions of temperature and impurity concentration in the solid, liquid and two-phase regions of the crystallizing system, the rate of solidification, and the spatial coordinate in the two-phase region depending on the solid phase fraction in it are found. An algebraic equation for the solid phase fraction at the interface between the solid material and the two-phase region is derived. Exact analytical solutions show that the impurity concentration in the two-phase layer increases as the solid phase fraction increases. Moreover, the solid phase fraction at the interface solid phase — two phase region and its thickness increase as the temperature gradient in the solid phase and the solidification rate increase. The developed theory allows us to determine analytically the permeability of the two-phase region and a characteristic interdendritic spacing in it. Analytical solutions show that the relative permeability in the two-phase region increases from a certain value at the interface with the solid phase to unity at the interface with the liquid phase. The selection theory of stable dendritic growth allows us to determine analytically a characteristic interdendritic distance in the two-phase layer that decreases as the temperature gradient in the solid phase increases. An increase of impurity in the molten phase gives a decrease in the interdendritic spacing within a two-phase region.

Liquid state property and rapid peritectic solidification of refractory Mo-33.3 at.% Zr alloy under electrostatic levitation condition

The thermophysical properties and rapid solidification mechanism of liquid Mo-33.3 at.% Zr peritectic alloy were investigated by electrostatic levitation technique, which attained a maximum undercooling of 387 K (0.16 TL). At its liquidus temperature, the density, surface tension, viscosity and solute diffusion coefficient of this refractory alloy were determined as 8.14 g/cm3, 1.60 N/m, 11.32 mPa·s and 1.41 × 10−9 m2/s, respectively. The primary (Mo) dendrite growth started from multi-point nucleation, while its growth velocity agreed well with the prediction of LKT/BCT dendrite growth theory and reached an upper value of 43 mm/s. The subsequent peritectic transition was characterized by a power-law kinetics relation between the nominal growth velocity of peritectic Mo2Zr phase and peritectic undercooling, displaying a maximum velocity of 46 mm/s. The increase of liquid undercooling facilitated the completion of peritectic transition and refined the microstructure of residual primary (Mo) phase, thus enhancing the Vickers hardness of this alloy gradually up to 1190.9 HV at the maximum undercooling.

Study on undercooling of dendrite tip during electron beam treatment

The surface of 5CrNiMo steel was treated using electron beam, and the sample's cross-section was separated into three layers: re-melting layer, heat affected layer, and substrate. The re-melting layer's microstructure is primarily made up of ultrafine dendrites. The undercooling of dendritic tip during rapid solidification was calculated using BCT dendrite growth theory. According to the calculations, the overall undercooling of dendritic tip is 344.8 K, of which kinetic undercooling is 259.5 K. The results presented above demonstrate our study can provide methodological reference for the calculation of undercooling during electron beam treatment.

A model for solute evolution in laser powder bed fused Al-Si alloys and its application to improve thermal conductivity

A model capable of describing the solute evolution in laser powder bed fused (LPBF) Al-Si alloys has been proposed. The proposed model incorporates the use of three-dimensional Rosenthal equation and a rapid dendrite growth theory. Due to the analytical nature of the equation and the theory, the avoidance of modeling detailed microstructure of cells/dendrites as well as the eutectic, and the neglect of grain nucleation/elimination process, the proposed model is efficient at calculating, taking merely ∼10 min for portraying the spatial distribution of dissolved solute in an entire melt pool. The model was validated by scanning transmission electron microscopic-energy disperse spectrum (STEM-EDS) analysis. The validated model was subsequently applied in designing fabrication strategy of LPBF Al-Si alloys, in hope of improving the alloys’ thermal conductivity through purposefully adjusting processing parameters. This objective was successfully achieved, based on the fact that among imperfections induced by LPBF, dissolved solute atoms are the dominant influencing factor for thermal conductivity, and that the model can reveal the correlations between the overall amount of dissolved solute in the solid solution matrix and laser/feedstock powder parameters.

Lattice Boltzmann model for predicting frosting process on surfaces considering wettability.

The process of frosting is a multiscale problem, which leads to challenges of proposing accurate numerical methods. In this study, a lattice Boltzmann model for predicting frost formation and growth on surfaces of various wettabilities is proposed based on the heterogeneous nucleation and dendrite growth theories. Three lattice Boltzmann equations are used to calculate the velocity, humidity, and temperature distributions. Furthermore, the heterogeneous nucleation theory and dendrite growth theory are used to construct the equations that govern ice production during the frosting process, so that the surface wettability can be considered. After experimental validation, the model was used in the analysis of frosting behaviors on plates and in microchannels with different wettabilities. The effects of the intrinsic contact angles and roughness on the frost layer properties were evaluated. This study will likely facilitate a better understanding of frosting on the mesoscopic level.

Gradient-controlled freeze casting of preceramic polymers

Solidification is the foundation upon which freeze casting is built, and this work seeks to understand the solidification process in freeze casting, especially with respect to the growth of dendrites. To this end, two solidification parameters, freezing front velocity and temperature gradient, were independently controlled using a gradient-controlled freeze casting setup to investigate the effects of each parameter. Changes in dendritic pore size and morphology with solidification parameters were investigated by scanning electron microscopy and mercury intrusion porosimetry. The results agree with dendrite growth theory. The theory of constitutional supercooling is used to describe the morphological changes of dendritic pores to cellular pores by the control of freezing front velocity, temperature gradient, and preceramic polymer concentrations.

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.

Mathematical modeling of dendrite growth in an Al–Ge alloy with convective flow

A theory of stable dendrite growth in an undercooled binary melt is developed for the case of intense convection. Conductive heat and mass transfer boundary conditions are replaced by convective conditions, where the flux of heat (or solute) is proportional to the temperature or concentration difference between the surface of the dendrite and far from it. The marginal mode of perturbation wavelengths is calculated using the linear morphological stability analysis. Combining this analysis with the solvability theory, we have derived a selection criterion that represents the first condition to define a combination of dendrite tip velocity and tip diameter. The second condition—the undercooling balance—is derived for intense convection. The theory under consideration determines the dendrite tip velocity and tip diameter for low undercooling. This convective theory is combined with the classical theory of dendritic growth (conductive boundary conditions), which is valid for moderate and high undercooling. Thus, the entire range of melt undercooling is covered. Our results are in good agreement with experiments on Al–Ge crystallization.

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)'.

Selection constants in the theory of stable dendritic growth

A stationary growth of the dendritic tip having the tip shape of an elliptical paraboloid in a binary liquid is considered. A method for constructing a selection criterion that allows one to obtain the growth rate and two radii of the dendritic tip as functions of undercooling is proposed. Selection constants are found theoretically for the n -fold crystalline symmetry. The theory is in good agreement with experimental data obtained on water and water solution previously described in literature.

A Stable Dendritic Growth with Forced Convection: A Test of Theory Using Enthalpy-Based Modeling Methods

The theory of stable dendritic growth within a forced convective flow field is tested against the enthalpy method for a single-component nickel melt. The growth rate of dendritic tips and their tip diameter are plotted as functions of the melt undercooling using the theoretical model (stability criterion and undercooling balance condition) and computer simulations. The theory and computations are in good agreement for a broad range of fluid velocities. In addition, the dendrite tip diameter decreases, and its tip velocity increases with increasing fluid velocity.

Ultrafast growth kinetics of titanium dendrites investigated by electrostatic levitation experiments and molecular dynamics simulations

An ultrafast crystal growth velocity of 122 m/s for β–Ti dendrites was achieved at a liquid undercooling of 352 K (0.18Tm) by electrostatic levitation technique. In contrast, most refractory metals attained slower dendrite growth velocities but displayed similar power laws with liquid undercooling. A combined analysis by current dendrite growth theory with molecular dynamics simulation indicates that liquid thermal diffusion rather than solid-liquid (S/L) interface kinetics is still the dominant factor to control dendrite growth. Besides, the atomic-scale thickness of the S/L interface layer may determine the dendrite growth velocity difference for various refractory metals.

Strongly Out-of-Equilibrium Columnar Solidification During Laser Powder-Bed Fusion in Additive Manufacturing

Laser-based additive manufacturing offers a promising route for 3D printing of metallic parts. We evidence experimentally a particular columnar solidification microstructure in a Laser Powder-Bed Fusion processed Inconel 718 nickel-based alloy, that we interpret using phase-field simulations and classical dendritic growth theories. Owing to the large temperature gradient and cooling rate, solidification takes places through dendritic arrays wherein the characteristic length scales, i.e tip radius, diffusion length and primary spacing, are of the same order. This leads to a weak mutual interaction between dendrite tips, and a drastic reduction of side-branching. The resulting irregular cellular-like solidification pattern then remains stable on time scales comparable to the complete melt pool solidification, as observed in the as-built material.

Dynamics of Nanoscale Dendrite Formation in Solution Growth Revealed Through in Situ Liquid Cell Electron Microscopy.

Formation mechanisms of dendrite structures have been extensively explored theoretically, and many theoretical predictions have been validated for micro- or macroscale dendrites. However, it is challenging to determine whether classical dendrite growth theories are applicable at the nanoscale due to the lack of detailed information on the nanodendrite growth dynamics. Here, we study iron oxide nanodendrite formation using liquid cell transmission electron microscopy (TEM). We observe "seaweed"-like iron oxide nanodendrites growing predominantly in two dimensions on the membrane of a liquid cell. By tracking the trajectories of their morphology development with high spatial and temporal resolution, it is possible to explore the relationship between the tip curvature and growth rate, tip splitting mechanisms, and the effects of precursor diffusion and depletion on the morphology evolution. We show that the growth of iron oxide nanodendrites is remarkably consistent with the existing theoretical predictions on dendritic morphology evolution during growth, despite occurring at the nanoscale.

Crystal Nucleation and Growth in Undercooled Melts of Pure Zr, Binary Zr-Based and Ternary Zr-Ni-Cu Glass-Forming Alloys

Glass formation of a liquid undercooled below its melting temperature requires the complete avoidance of crystal nucleation and subsequent crystal growth. Even though they are not part of the glass formation process, a detailed knowledge of both processes involved in crystallization is mandatory to determine the glass-forming ability of metals and metallic alloys. In the present work, methods of containerless processing of drops by electrostatic and electromagnetic levitation are applied to undercool metallic melts prior to solidification. Heterogeneous nucleation on crucible walls is completely avoided giving access to large undercoolings. A freely suspended drop offers the additional benefit of showing the rapid crystallization process of an undercooled melt in situ by proper diagnostic means. As a reference, crystal nucleation and dendrite growth in the undercooled melt of pure Zr are experimentally investigated. Equivalently, binary Zr-Cu, Zr-Ni and Zr-Pd and ternary Zr-Ni-Cu alloys are studied, whose glass-forming abilities differ. The experimental results are analyzed within classical nucleation theory and models of dendrite growth. The findings give detailed knowledge about the nucleation-undercooling statistics and the growth kinetics over a large range of undercooling.

Simulation and analysis of [formula omitted]-Ni cellular growth during laser powder deposition of Ni-based superalloys

Cellular or dendritic microstructures that result as a function of additive manufacturing solidification conditions in a Ni-based melt pool are simulated in the present work using three-dimensional phase-field simulations. A macroscopic thermal model is used to obtain the temperature gradient G and the solidification velocity V which are provided as inputs to the phase-field model. We extract the cell spacings, cell core compositions, and cell tip as well as mushy zone temperatures from the simulated microstructures as a function of V. Cell spacings are compared with different scaling laws that correlate to the solidification conditions and approximated by G-mV-n. Cell core compositions are compared with the analytical solutions of a dendrite growth theory and found to be in good agreement. Through analysis of the mushy zone, we extract a characteristic bridging plane, where the primary γ phase coalesces across the intercellular liquid channels at a γ fraction between 0.6 and 0.7. The temperature and the γ fraction in this plane are found to decrease with increasing V. The simulated microstructural features are significant as they can be used as inputs for the simulation of subsequent heat treatment processes.

Measurements and modelling of dendritic growth velocities of pure Fe with thermoelectric magnetohydrodynamics convection

Dendritic growth velocities of pure Fe under static magnetic fields of intensity ranging from B=0T to B=6T were measured using a high speed camera. The data measured at undercoolings up to ΔT=190K show a depression followed by a recovery of the growth velocities as the magnetic field intensity increased from a low range, B=1–3T to a high range, B=4–6T. These magnetic field effects are similar to those previously observed for pure Ni and can be attributed to competing thermoelectric magnetohydrodynamic (TEMHD) convection patterns in the local liquid. The experimental measurements for the two metals were modelled using a three-dimensional dendritic growth theory taking into account convection to estimate the effective flow velocities in the tip growth direction. The calculated effective flow velocities identify two undercooling dependences and a distinct type of magnetic field intensity dependence in common for the two metals. In comparison, the calculated effective flow velocities for pure Fe are generally smaller in magnitude. This difference between the two metals can be related to their differences in material-dependent properties as is revealed by a simple model proposed for a transverse TEMHD flow.

Dendrite growth velocity in the undercooled melt of glass forming Ni50Zr50 compound

Glass-forming Ni50Zr50 intermetallic compound is containerless undercooled and solidified using electrostatic levitation. Large undercoolings up to ∆T = 300 K are achieved. The dendrite growth velocity of the congruently melting alloy is measured as a function of undercooling using a high-speed camera technique. The experimental data is analysed within a sharp interface theory. It is found that the driving force of crystallisation is controlling the growth kinetics at ∆T < 250 K but at larger undercoolings the growth kinetics is progressively controlled by atomic diffusion. This leads to a slowing down of the growth velocity. The maximum velocity and the temperature at which the maximum occurs (Tmax) are inferred from the dendrite growth velocity – undercooling relation. The relation of the temperature Tmax and the glass temperature fits into a general classification scheme for glass-forming systems. The kinetic and thermal undercooling terms are calculated within dendrite growth theory as a function of the total undercooling. At ∆T > 126 K, the kinetic undercooling dominates and increases rapidly with the undercooling ∆T. The maximum prefactor of the kinetic undercooling is plotted vs. the reciprocal temperature. Its temperature dependence is discussed.

Microstructure selection in thin-sample directional solidification of an Al-Cu alloy: In situ X-ray imaging and phase-field simulations

We study microstructure selection during directional solidification of a thin metallic sample. We combine in situ X-ray radiography of a dilute Al-Cu alloy solidification experiments with three-dimensional phase-field simulations. We explore a range of temperature gradient G and growth velocity V and build a microstructure selection map for this alloy. We investigate the selection of the primary dendritic spacing Λ and tip radius ρ. While ρ shows a good agreement between experimental measurements and dendrite growth theory, with ρ∼V−1/2, Λ is observed to increase with V (∂Λ/∂V>0), in apparent disagreement with classical scaling laws for primary dendritic spacing, which predict that ∂Λ/∂V<0. We show through simulations that this trend inversion for Λ(V) is due to liquid convection in our experiments, despite the thin sample configuration. We use a classical diffusion boundary-layer approximation to semi-quantitatively incorporate the effect of liquid convection into phase-field simulations. This approximation is implemented by assuming complete solute mixing outside a purely diffusive zone of constant thickness that surrounds the solid-liquid interface. This simple method enables us to quantitatively match experimental measurements of the planar morphological instability threshold and primary spacings over an order of magnitude in V. We explain the observed inversion of ∂Λ/∂V by a combination of slow transient dynamics of microstructural homogenization and the influence of the sample thickness.