Solute trapping in rapid solidification of a binary dilute system: a phase-field study

Towards eliminating friction and wear in plain bearings operating without lubrication

Plain bearings, renowned for their versatility and simplicity, are extensively utilized in engineering design across various industries involving moving parts. Lubrication is vital to the functioning of these bearings, yet their usage is inhibited under dynamic load conditions, or at elevated or reduced temperatures due to this dependency on lubrication. This study introduces an innovative method to significantly mitigate friction and wear in plain bearings operating without lubrication. The plain bearings were constructed from steel-bronze pairs, where the steel shafts were alloyed with bismuth oxide via short-pulse laser treatment. MnO2 was utilized as a carrier to incorporate the bismuth oxide into the surface layers of the steel. Insights from transmission electron microscopy and X-ray photoelectron spectroscopy revealed a highly non-equilibrium state of matter, unattainable through conventional engineering methods. The tribological performance of the modified steel disks was assessed via a block-on-ring sliding test, demonstrating superior wear and friction performance without lubrication, as well as an ultra-low coefficient of friction. Remarkably, the modified friction pairs remained functional after 200 km of linear sliding at a load of 250 N (12.5 MPa) and a sliding speed of 9 m/s. To substantiate the technique's viability, we tested the performance of an internal combustion engine turbocharger fitted with a modified steel shaft. The turbocharger's performance validated the long-term effectiveness of the steel-bronze coupling operating without lubrication at 75,000 rpm. The simplicity and resilience of this technique for modifying steel-bronze pairs offer a ground-breaking and promising approach for a wide range of applications.

Microstructural Pattern Formation during Far-from-Equilibrium Alloy Solidification

We introduce a new phase-field formulation of rapid alloy solidification that quantitatively incorporates nonequilibrium effects at the solid-liquid interface over a very wide range of interface velocities. Simulations identify a new dynamical instability of dendrite tip growth driven by solute trapping at velocities approaching the absolute stability limit. They also reproduce the formation of the widely observed banded microstructures, revealing how this instability triggers transitions between dendritic and microsegregation-free solidification. Predicted band spacings agree quantitatively with observations in rapidly solidified Al-Cu thin films.

The hodograph equation for slow and fast anisotropic interface propagation

Using the model of fast phase transitions and previously reported equation of the Gibbs-Thomson-type, we develop an equation for the anisotropic interface motion of the Herring-Gibbs-Thomson-type. The derived equation takes the form of a hodograph equation and in its particular case describes motion by mean interface curvature, the relationship 'velocity-Gibbs free energy', Klein-Gordon and Born-Infeld equations related to the anisotropic propagation of various interfaces. Comparison of the present model predictions with the molecular-dynamics simulation data on nickel crystal growth (obtained by Jeffrey J. Hoyt et al. and published in Acta Mater. 47 (1999) 3181) confirms the validity of the derived hodograph equation as applicable to the slow and fast modes of interface propagation. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

Kinetics of solid-liquid interface motion in molecular dynamics and phase-field models: crystallization of chromium and silicon

The results of molecular dynamics (MD) simulations of the crystallization process in one-component materials and solid solution alloys reveal a complex temperature dependence of the velocity of the crystal-liquid interface featuring an increase up to a maximum at 10-30% undercooling below the equilibrium melting temperature followed by a gradual decrease of the velocity at deeper levels of undercooling. At the qualitative level, such non-monotonous behaviour of the crystallization front velocity is consistent with the diffusion-controlled crystallization process described by the Wilson-Frenkel model, where the almost linear increase of the interface velocity in the vicinity of melting temperature is defined by the growth of the thermodynamic driving force for the phase transformation, while the decrease in atomic mobility with further increase of the undercooling drives the velocity through the maximum and into a gradual decrease at lower temperatures. At the quantitative level, however, the diffusional model fails to describe the results of MD simulations in the whole range of temperatures with a single set of parameters for some of the model materials. The limited ability of the existing theoretical models to adequately describe the MD results is illustrated in the present work for two materials, chromium and silicon. It is also demonstrated that the MD results can be well described by the solution following from the hodograph equation, previously found from the kinetic phase-field model (kinetic PFM) in the sharp interface limit. The ability of the hodograph equation to describe the predictions of MD simulation in the whole range of temperatures is related to the introduction of slow (phase field) and fast (gradient flow) variables into the original kinetic PFM from which the hodograph equation is obtained. The slow phase-field variable is responsible for the description of data at small undercoolings and the fast gradient flow variable accounts for local non-equilibrium effects at high undercoolings. The introduction of these two types of variables makes the solution of the hodograph equation sufficiently flexible for a reliable description of all nonlinearities of the kinetic curves predicted in MD simulations of Cr and Si. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

Traveling waves of the solidification and melting of cubic crystal lattices

Using the phase field crystal model (PFC model), an analysis of slow and fast dynamics of solid-liquid interfaces in solidification and melting processes is presented. Dynamical regimes for cubic lattices invading metastable liquids (solidification) and liquids propagating into metastable crystals (melting) are described in terms of the evolving amplitudes of the density field. Dynamical equations are obtained for body-centered cubic (bcc) and face-centered cubic (fcc) crystal lattices in one- and two-mode approximations. A universal form of the amplitude equations is obtained for the three-dimensional dynamics for different crystal lattices and crystallographic directions. Dynamics of the amplitude's propagation for different lattices and PFC mode's approximations is qualitatively compared. The traveling-wave velocity is quantitatively compared with data of molecular dynamics simulation previously obtained by Mendelev et al. [Modell. Simul. Mater. Sci. Eng. 18, 074002 (2010)MSMEEU0965-039310.1088/0965-0393/18/7/074002] for solidification and melting of the aluminum fcc lattice.

On Directional Dendritic Growth and Primary Spacing—A Review

The primary spacing is intrinsically linked with the mechanical behavior of directionally solidified materials. Because of this relationship, a significant amount of solidification work is reported in the literature, which relates the primary spacing to the process variables. This review provides a comprehensive chronological narrative on the development of the directional dendritic growth problem over the past 85 years. A key focus within this review is detailing the relationship between key solidification parameters, the operating point of the dendrite tip, and the primary spacing. This review critiques the current state of directional dendritic growth and primary spacing modelling, briefly discusses dendritic growth computational and experimental research, and suggests areas for future investigation.

Fast traveling waves in the phase-field theory: effective mobility approach versus kinetic energy approach

A phase-field model for small and large driving forces on solidification and melting of a pure substance or alloys is formulated. Derivations of the phase-field model are based on the effective mobility approach and on the kinetic energy approach to analyze fast phase transformation from metastable liquid to solid phase. A hodograph equation (an acceleration-velocity dependent equation of the Gibbs-Thomson type) which predicts the non-linear behavior in the velocity of the crystal-liquid interface is found at the large driving force on transformation and analyzed for different thermodynamic potentials. Traveling wave solutions of this equation are found for double-well and double-obstacle potentials. The velocity-dependent traveling waves as a function of driving force on transformation exhibit non-linearity of the solutions. Namely, in the relationship 'velocity-driving force' exists a maximum at a fixed undercooling which is very well known in the solidification of glass-forming metals and alloys. The predicted solidification velocity is quantitatively compared with the molecular dynamics simulation data obtained by Tang and Harrowell (2013 Nat. Mater. 12 507-11) for the solidification of congruently melting Cu-Zr binary alloy. The comparison confirms a crucial role of local non-equilibrium such as relaxation of gradient flow in the quantitative description of fast phase transformations.

Thermodynamics of rapid solidification and crystal growth kinetics in glass-forming alloys

Thermodynamic driving forces and growth rates in rapid solidification are analysed. Taking into account the relaxation time of the solute diffusion flux in the model equations, the present theory uses, in a first case, the deviation from local chemical equilibrium, and ergodicity breaking. The second case of ergodicity breaking may exist in crystal growth kinetics of rapidly solidifying glass-forming metals and alloys. In this case, a theoretical analysis of dendritic solidification is given for congruently melting alloys in which chemical segregation does not occur. Within this theory, a deviation from thermodynamic equilibrium is introduced for high undercoolings via gradient flow relaxation of the phase field. A comparison of the present derivations with previously verified theoretical predictions and experimental data is given. This article is part of the theme issue 'Heterogeneous materials: metastable and non- ergodic internal structures'.

Kinetic transition in the order-disorder transformation at a solid/liquid interface

Phase-field analysis for the kinetic transition in an ordered crystal structure growing from an undercooled liquid is carried out. The results are interpreted on the basis of analytical and numerical solutions of equations describing the dynamics of the phase field, the long-range order parameter as well as the atomic diffusion within the crystal/liquid interface and in the bulk crystal. As an example, the growth of a binary A₅₀B₅₀ crystal is described, and critical undercoolings at characteristic changes of growth velocity and the long-range order parameter are defined. For rapidly growing crystals, analogies and qualitative differences are found in comparison with known non-equilibrium effects, particularly solute trapping and disorder trapping. The results and model predictions are compared qualitatively with results of the theory of kinetic phase transitions (Chernov 1968 Sov. Phys. JETP26, 1182-1190) and with experimental data obtained for rapid dendritic solidification of congruently melting alloy with order-disorder transition (Hartmann et al. 2009 Europhys. Lett.87, 40007 (doi:10.1209/0295-5075/87/40007)).This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

Travelling-wave amplitudes as solutions of the phase-field crystal equation

The dynamics of the diffuse interface between liquid and solid states is analysed. The diffuse interface is considered as an envelope of atomic density amplitudes as predicted by the phase-field crystal model (Elder et al. 2004 Phys. Rev. E70, 051605 (doi:10.1103/PhysRevE.70.051605); Elder et al. 2007 Phys. Rev. B75, 064107 (doi:10.1103/PhysRevB.75.064107)). The propagation of crystalline amplitudes into metastable liquid is described by the hyperbolic equation of an extended Allen-Cahn type (Galenko & Jou 2005 Phys. Rev. E71, 046125 (doi:10.1103/PhysRevE.71.046125)) for which the complete set of analytical travelling-wave solutions is obtained by the [Formula: see text] method (Malfliet & Hereman 1996 Phys. Scr.15, 563-568 (doi:10.1088/0031-8949/54/6/003); Wazwaz 2004 Appl. Math. Comput.154, 713-723 (doi:10.1016/S0096-3003(03)00745-8)). The general [Formula: see text] solution of travelling waves is based on the function of hyperbolic tangent. Together with its set of particular solutions, the general [Formula: see text] solution is analysed within an example of specific task about the crystal front invading metastable liquid (Galenko et al. 2015 Phys. D308, 1-10 (doi:10.1016/j.physd.2015.06.002)). The influence of the driving force on the phase-field profile, amplitude velocity and correlation length is investigated for various relaxation times of the gradient flow.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

From atomistic interfaces to dendritic patterns

Transport processes around phase interfaces, together with thermodynamic properties and kinetic phenomena, control the formation of dendritic patterns. Using the thermodynamic and kinetic data of phase interfaces obtained on the atomic scale, one can analyse the formation of a single dendrite and the growth of a dendritic ensemble. This is the result of recent progress in theoretical methods and computational algorithms calculated using powerful computer clusters. Great benefits can be attained from the development of micro-, meso- and macro-levels of analysis when investigating the dynamics of interfaces, interpreting experimental data and designing the macrostructure of samples. The review and research articles in this theme issue cover the spectrum of scales (from nano- to macro-length scales) in order to exhibit recently developing trends in the theoretical analysis and computational modelling of dendrite pattern formation. Atomistic modelling, the flow effect on interface dynamics, the transition from diffusion-limited to thermally controlled growth existing at a considerable driving force, two-phase (mushy) layer formation, the growth of eutectic dendrites, the formation of a secondary dendritic network due to coalescence, computational methods, including boundary integral and phase-field methods, and experimental tests for theoretical models-all these themes are highlighted in the present issue.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

Coarse-graining for fast dynamics of order parameters in the phase-field model

In standard descriptions, the master equation can be obtained by coarse-graining with the application of the hypothesis of full local thermalization that is equivalent to the local thermodynamic equilibrium. By contrast, fast transformations proceed in the absence of local equilibrium and the master equation must be obtained with the absence of thermalization. In the present work, a non-Markovian master equation leading, in specific cases of relaxation to local thermodynamic equilibrium, to hyperbolic evolution equations for a binary alloy, is derived for a system with two order parameters. One of them is a conserved order parameter related to the atomistic composition, and the other one is a non-conserved order parameter, which is related to phase field. A microscopic basis for phenomenological phase-field models of fast phase transitions, when the transition is so fast that there is not sufficient time to achieve local thermalization between two successive elementary processes in the system, is provided. In a particular case, when the relaxation to local thermalization proceeds by the exponential law, the obtained coarse-grained equations are related to the hyperbolic phase-field model. The solution of the model equations is obtained to demonstrate non-equilibrium phenomenon of solute trapping which appears in rapid growth of dendritic crystals.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

Microscopic treatment of solute trapping and drag

The long wavelength limit of a recent microscopic phase-field crystal (PFC) theory of a binary alloy mixture is used to derive an analytical approximation for the segregation coefficient as a function of the interface velocity, and relate it to the two-point correlation function of the liquid and the thermodynamic properties of solid and liquid phases. Our results offer the first analytical derivation of solute segregation from a microscopic model, and support recent molecular dynamics and numerical PFC simulations. Our results also provide an independent framework, motivated from classical density functional theory, from which to elucidate the fundamental nature of solute drag, which is still highly contested in the literature.

Phase-field-crystal study of solute trapping

In this study we have incorporated two time scales into the phase-field-crystal model of a binary alloy to explore different solute trapping properties as a function of crystal-melt interface velocity. With only diffusive dynamics, we demonstrate that the segregation coefficient, K as a function of velocity for a binary alloy is consistent with the model of Kaplan and Aziz where K approaches unity in the limit of infinite velocity. However, with the introduction of wavelike dynamics in both the density and concentration fields, the trapping follows the kinetics proposed by Sobolev [Phys. Lett. A 199, 383 (1995)], where complete trapping occurs at a finite velocity.