Interfacial energy anisotropy governs nucleation, growth and microstructure selection in materials, yet it has long been regarded as an intrinsic and immutable material constant fixed by crystallographic symmetry and thermodynamic state variables such as composition and temperature. Whether this interfacial energy landscape can be actively reconstructed under full thermodynamic equilibrium remains a fundamental unresolved question. Here we demonstrate that a static magnetic field acts as a symmetry-breaking thermodynamic contribution capable of reconstructing the solid/liquid interfacial energy landscape. Using equilibrium droplet shape measurements and three-dimensional X-ray tomography, we show that magnetic fields reconstruct the interfacial energy landscape, inducing a robust reduction of crystallographic symmetry. In both weakly and strongly anisotropic Al-Cu interfaces, the intrinsic symmetry collapses into a dominant twofold form, accompanied by pronounced changes in equilibrium interface shape. A magnetically coupled phase-field crystal framework identifies a directional magnetic contribution to interfacial stiffness that underlies this symmetry reconstruction. The reconstructed energy landscape deterministically controls morphological selection during phase transition. Our findings elevate magnetic fields from external processing parameters to symmetry-breaking thermodynamic variables, establishing a general paradigm for engineering crystal morphologies by reconstructing interfacial energy landscapes.

Interfacial energy anisotropy plays a central role in nucleation, crystal growth, and microstructure evolution in materials. It is generally considered to be governed primarily by crystallographic symmetry and thermodynamic state variables such as composition and temperature. Whether external fields can modify solid/liquid interfacial energy anisotropy under near-equilibrium conditions remains an open question. Here, equilibrium droplet shape measurements and three-dimensional X-ray tomography are combined with a magnetically coupled phase-field crystal (MPFC) model to investigate the influence of static magnetic fields on solid/liquid interfacial anisotropy in Al-Cu alloys. The experimental results show that both equilibrium droplet shapes and dendritic growth patterns evolve systematically under applied magnetic fields, accompanied by pronounced changes in anisotropic interface contours. In both weakly and strongly anisotropic Al-Cu interfaces, the interfacial shape exhibits enhanced twofold symmetry characteristics under applied magnetic fields, together with significant changes in equilibrium interface shape. The MPFC model further suggests that the magnetic fields may introduce orientation-dependent modifications to the interfacial stiffness, which are qualitatively consistent with the experimentally observed morphology evolution. These results provide evidence that static magnetic fields can influence the interfacial energy anisotropy and correlate with morphology selection during non-directional solidification. The present work provides a framework for exploring how externally applied magnetic fields may influence crystal growth behavior through modifications of effective interfacial energy anisotropy.

Due to its analytical flexibility and thermodynamic consistency, the phase field methodology is widely used in the analysis of equilibrium states and transformation between phases. The present review is devoted to a class of hyperbolic phase field models, which applies to slow and fast phase transformations. Focusing on the example of solidification of metastable liquid, an analysis is presented for the important procedure of reducing the diffuse interface to the sharp interface. An asymptotic analysis is discussed for application to solidifying binary mixture with diffuse phase interface under arbitrary concentration of species and isothermal and isobaric conditions. The analysis reveals that the hyperbolic phase field model can be mapped onto the known hyperbolic Stefan problem within the sharp interface limit. This result, together with the common tangent construction, allows us to analyze (i) nonequilibrium effects in the form of solute trapping and (ii) the complete transition from the diffusion-limited to the diffusionless (chemically partitionless) solidification at finite interface velocity. A comparison with other theoretical models is summarized and a discussion, which is attributed to experimental results, is given.

A linear instability analysis of the planar solid–liquid interface propagating into a binary liquid is revisited for a steady-state mode of crystallization. The model statement includes convective and conductive transport of heat and mass in bulk phases together with these transfer contributions at the solid–liquid interface. Following our analysis [D. V. Alexandrov and P. K. Galenko, “The Mullins-Sekerka theory: 60 years of morphological stability,” J. Appl. Phys. 136 (2024) 055103], it is shown that the directional solidification with the convective and conductive transport also becomes possible only if the finite distance h of the solidification front from the cooling unit (cold boundary) exists and is taken into account in the formal analysis. If the cooling unit is removed from the interface to the spatially infinite distance (as accepted in many previous works), the directional solidification stops. The obtained dispersion relation for the system with the conductive and convective transport takes into account the existence of perturbations appearing from the cooling unit, solid–liquid interface, and bulk liquid. Therefore, the range of instability essentially depends on the distance h and friction velocity of the flow, which characterizes the liquid convection and convective contributions of heat and mass fluxes. Special cases of bounded and unbounded solidification domains that may affect the front instability are investigated. It is shown that within the bounded domain perturbations from convective flow or temperature fluctuations affect the solid–liquid interface more strongly than in the unbounded domain. This effect leads to a broader range of wavenumbers that provide the front instability.

Motivated by important applications in materials physics, we study the shape of dendritic crystals with sixfold crystalline symmetry grown in aqueous solutions of various substances and pure water. Based on recently developed Geometrically Morphological Theory [Philos. Trans. R. Soc. A 378 (2020) 20190243; Phys. Lett. A 501 (2024) 129375] we demonstrate that the shape of the main stem and internal/external envelope in the secondary branches of dendrite at the steady-state growth mode is described by the scaling law z(x) proportional to -|x|(n) with the values of exponent n = 3.0 and n = 1.0 for ice dendrites growing in solutions of Secale cereale, glucose, sucrose and pure water. In addition, by changing the crystalline symmetry from sixto four-fold, the scaling law exponent and the crystallographic growth direction change from n = 1.0 to n = 1.164 and < 110> to < 100 >, respectively. The evolutionary route of how dendrites reach steady-state growth velocity is also described by supporting and finding parameters from laboratory experiments.

The phase field models address the equilibrium and evolution of density fields in elemental systems or mixtures of atoms of different types. Suggested in present work multiscale approach presents a successful link between macroscopic heat transfer model of selective laser melting (sintering) and temperature sensitive phase field crystal model (PFC). The PFC is used in the present work for the study of the effect of thermal fluctuations on the crystalline structure and formation of defects during solidification. The defects density has a maximum as the laser beam speed increases, and its formation dynamics depends on the balance between noise-induced defects at high temperatures and annealing of defects during prolonged exposure at lower temperatures. Obtained cooling curves can be interpolated by a function dependent on the beam speed, and then introduced as the source condition to the microscopic two-mode PFC model with thermal fluctuations.

Motivated by important applications in materials physics, we study the shape of dendritic crystals based on recently developed morphological theory [Phys. Lett. A 501 (2024) 129375]. The generalized shape function describing dendrite's tip, primary stem and external envelope is tested against the phase-field simulations. Our computations confirm good agreement with the theory describing the shape of dendritic crystals using a unified nonlinear shape function.

One of the classes of the kinetic phase-field model in the form of the two-mode hyperbolic phase-field crystal model (modified PFC model) is used for the study of the noise effect of the crystalline structure. Special attention is paid to the origin of the defect’s microstructure in the crystalline honeycomb lattice due to induced colored noise. It shows that the noise–time correlation coefficient τζ, comparable to the diffusion time, enhances the grain boundary mobilities. Instead, a small spatial correlation coefficient, λζ, close to the first lattice parameter of the honeycomb crystal, stabilizes the structure. The finite non-zero value of the relaxation time τ for the atomic flux significantly slows the local relaxation of the fluctuated field and leads to the grains’ fragmentation and formation of the disordered phases. The obtained results are applicable to the hexagonal atomic structures and, in particular, to honeycomb crystals, such as boron nitride, in which the lattice defects might be simulated through the induced colored noise.

As one of the representative patterns in nature and laboratory experiments, dendritic structures control the properties of a broad range of advanced materials. Dendrites arise during different phase and structural transformation processes. Generally, the formation of dendritic structures are stipulated by transport processes in bulk phases, together with thermodynamic properties and kinetic phenomena at the phase interfaces. The formation of a dendritic microstructure under the influence of external fields (electromagnetic and gravitational) is considered in this review. These fields involve the liquid and gaseous phases in a forced convective flow, causing the transfer of energy and matter in addition to the usual conductive (diffusion) transport. The formulated model takes into account rapid solidification from an undercooled liquid phase as well as intermediate and low growth velocities of dendritic crystals in pure one-component systems extended to binary mixtures and alloys. The areas of undercooling are identified, in which the influence of convection caused by the electromagnetic and/or gravitational field is most noticeable. The solidification regimes (from the diffusion-limited mode to the thermally and kinetically controlled mode) are reviewed in connection with the different liquid flow velocities that dictate various boundary conditions (conductive and convective) on the surface of growing crystals. A comparison of model predictions with experimental data and computational results provides the grounds for a discussion about the applicability of the formulated model to interpreting known and unexpected phenomena in the formation of a crystalline structure. By changing the power of the considered fields or reducing them almost to zero (for instance, in microgravity), it is possible to control the dispersion of a dendritic microstructure, as well as separate accompanying phases (eutectic, peritectic, monotectic, intermetallic phases, etc.) during the solidification of materials and, in the general case, during phase transformations. (c) 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Classic kinetic theories for crystal growth and melting, such as diffusion- or collision-limited theories, encounter challenges in quantitative describing the interface velocity at large values of the driving force. One possible solution of such a crisis of classic theories is seen in using the kinetic phase field model (KPFM) formulated for small and large driving forces on solidification, melting and evaporation. The predictions of the nonisothermal phase field model are discussed in the present work to describe the nonlinear behavior of the nickel crystal growth obtained in molecular dynamic (MD) simulation.

The solidification of Inconel 718 alloy (IN718) from undercooled liquid is studied. The solidification kinetics is evaluated in melted and undercooled droplets processed using the electromagnetic levitation (EML) technique by the temperature–time profiles and solid/liquid (S/L) interface movement during recalescence. The kinetics is monitored in real time by special pyrometrical measurements and high-speed digital camera. It is shown that the growth velocity of γ-phase (the primary phase in IN718), the final crystalline microstructure (dendritic and grained), and the mechanical properties (microhardness) are strongly dependent on the initial undercooling ΔT at which the samples started to solidify with the originating γ-phase. Particularly, with the increase in undercooling, the secondary dendrite arm spacing decreases from 28 μm to 5 μm. At small and intermediate ranges of undercooling, the solidified droplets have a dendritic crystalline microstructure. At higher undercooling values reached in the experiment, ΔT>160 K (namely, for samples solidified with ΔT=170 K and ΔT=263 K), fine crystalline grains are observed instead of the dendritic structure of solidified drops. Such change in the crystalline morphology is qualitatively consistent with the behavior of crystal growth kinetics which exhibits the change from the power law to linear law at ΔT≈160 K in the velocity–undercooling relationship (measured by the advancement of the recalescence front in solidifying droplets). Study of the local mechanical properties shows that the microhardness increases with the increase in the γ″-phase within interdendritic spacing. The obtained data are the basis for testing the theoretical and computational of multicomponent alloy samples.

Over 60 years of studying morphological stability under fundamental ideas of William Wilson Mullins and Robert Floyd Sekerka [J. Appl. Phys. 34, 323 (1963) and J. Appl. Phys. 35, 444 (1964)] it has become possible to explain the origin and selection of surface structures from planar to cellular, dendritic, and fractal patterns. The Mullins–Sekerka (MS) morphological instability theory provides a condition for stability or reconstruction of interfaces, which separates the phases during phase transformation. The MS-theory has come a long way in the conceptual understanding of the incipience of morphological instability and the formation of structures, although today, certain aspects of this theory continue to be discussed at the fundamental and quantitative level of its interpretation. In the sixtieth anniversary of this theory, we re-examine the MS-analysis under boundary conditions satisfying the smooth existence of temperature and its gradients in directional crystallization of a binary melt. These boundary conditions are dependent on the finite distance from the solidification front for providing directional solidification that quantitatively affects the amplification rate of perturbations in the solid–liquid front morphology.

The generalized shape function for the dendritic tip, primary stem and external envelope is derived and tested against experimental data for succinonitrile and succinonitrile-acetone dendrites. In addition, scaling dependencies for the dendritic shape were obtained in terms of the dendritic primary stem and external envelope. These original scaling relations are in good agreement with previous measurements in pure succinonitrile provided by Li and Beckermann. The dendrite shape theory for rapid solidification of a binary melt based on the hyperbolic mass transfer equation is also discussed.

A linear morphological stability analysis of a planar solid-liquid phase interface describing the solidification processes of a binary melt with convection is carried out. The developed theory includes conductive and convective heat and mass transfer mechanisms near the phase interface and generalizes previously known theories of morphological stability. The amplification rate as a function of wavenumber of perturbations and neutral stability curve that divides the stability/instability parametric domains are obtained. It is shown that these domains are highly dependent of convection intensity, which represents a stabilizing factor for solidification processes. A criterion of concentration supercooling in the steady-state solidification conditions with convection is found. The obtained dispersion relation and neutral stability curve define various crystallization scenarios such as (i) morphological instability and concentration supercooling appearing to the formation of a two-phase mushy layer, (ii) morphological stability and concentration supercooling leading to the existence of a slurry layer, (iii) morphological stability without concentration supercooling when the planar solidification front is stable, and (iv) morphological instability without concentration supercooling forming the mesoscopically rough phase interface.

Convection affect primary crystalline structure, particularly, the dendrite crystal velocity and dendrite tip radius. The present work aims on the influence of the convective flow on the primary dendrite spacing between neighbouring crystals within the dendrite ensemble. Solidification of a binary alloy is considered within the model of stagnant boundary layer, under imposed thermal gradient which influence on the crystal microstructure and chemical microsegregation. Chemical composition in the solidifying liquid and crystalline solid is derived from the solution of the solute diffusion transport equation taking into account the convective flow. The model results are consistent with the Scheil-Brody-Flemings model, experimental data and computational results.

The boundary integral equation defining the interface function for a curved solid/liquid phase transition boundary is analytically solved in steady-state growth conditions. This solution describes dendrite tips evolving in undercooled melts with a constant crystallization velocity, which is the sum of the steady-state and translational velocities. The dendrite tips in the form of a parabola, paraboloid, and elliptic paraboloid are considered. Taking this solution into account, we obtain the modified boundary integral equation describing the evolution of the patterns and dendrites in undercooled binary melts. Our analysis shows that dendritic tips always evolve in a steady-state manner when considering a kinetically controlled crystallization scenario. The steady-state growth velocity as a factor that is dependent on the melt undercooling, solute concentration, atomic kinetics, and other system parameters is derived. This expression can be used for determining the selection constant of the stable dendrite growth mode in the case of kinetically controlled crystallization.

This study is concerned with the question of what is the shape of a dendritic tip grown from an undercooled melt in the presence of external impacts? To answer this question we extend the recent theory (Alexandrov and Galenko in Philos Trans R Soc A 378:20190243, 2020) to the case of external processes influencing the crystal growth phenomenon. The tip shape function is derived and tested against experimental data and numerical simulations when forced convection and dissolved impurities play a decisive role. It is shown that the tip shape function taking external impacts into account is in good agreement with the theory, experiments and computations. Using our well tested formula for the dendrite tip shape we show that the mechanisms of heat and mass transfer in inclined fluid currents can be essentially different. Namely, heat and mass fluxes at the crystal surface can be described by Fick’s or Newton’s laws or even by a more general mixed-type heat and mass transfer formula.