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La nucleació és un procés molt comú pel qual la separació de fases es dóna en sistemes líquids multicomponents. Els models de camps de fase són habitualment emprats per estudiar la separació de fases, però són insuficients per estudiar els processos de nucleació, degut a què aquestes transicions són activades tèrmicament i requereixen superar una barrera d’energia. Per superar aquesta dificultat, hem combinat els models de camps de fase amb el “string method” (mètode de la corda), un esquema numèric capaç de trobar camins de transició i estats crítics, per estudiar la nucleació en sistemes binaris i ternaris. El nostre model és capaç de capturar la nucleació homogènia i heterogènia tant en sistemes binaris com ternaris. Validem la nostra aproximació comparant-la amb la teoria de nucleació clàssica, vàlida en el límit de interfases abruptes, i examinen i discutim les discrepàncies entre aquestes dues aproximacions. En el futur aquest model pot ser emprat per estudiar configuracions més complexes, algunes de les quals són interessants des d’un punt de vista de la biologia perquè simulen la separació de proteïnes en el fluid intracel·lular., La nucleación es un proceso muy común por el cual la separación de fases se da en sistemas líquidos multicomponentes. Los modelos de campos de fase son habitualmente empleados para estudiar la separación de fases, pero son insuficientes para estudiar los procesos de nucleación, debido a que estas transiciones son activadas térmicamente y requieren superar una barrera de energía. Para superar esta dificultad, hemos combinado los modelos de campos de fase con el “string method” (método de la cuerda), un esquema numérico capaz de hallar caminos de transición y estados críticos, para estudiar la nucleación en sistemas binarios y ternarios. Nuestro modelo es capaz de capturar la nucleación homogénea y heterogénea tanto en sistemas binarios como ternarios. Validamos nuestra aproximación comparándola con la teoría de nucleación clásica, válida en el límite de interfases abruptas, y examinamos y discutimos las discrepancias entre estas dos aproximaciones. En el futuro este modelo puede ser empleado para estudiar configuraciones más complejas, algunas de las cuales son interesantes desde un punto de vista de la biología porque simulan la separación de proteínas en el fluido intracelular., Nucleation is a very common process by which phase separation takes place in multicomponent liquid mixtures. Phase field models are usually applied to study phase separation, but they are insufficient to study nucleation events, since these transitions are thermally activated and require overcoming an energy barrier. To overcome this difficulty, we combined phase field models with the string method, a numerical scheme capable of finding transition pathways and critical states, to study nucleation in binary and ternary systems. Our model is able to capture homogeneous and heterogeneous nucleation both in binary and ternary systems. We validate our approach by comparing it with classical nucleation theory, valid in a sharp interface limit, and further examine and discuss the discrepancies between these two approaches. In the future this model can be applied to study more complex configurations, some of which are interesting from a biological point of view because they simulate protein separation in the intracellular fluid.

The thin interface limit of the phase-field model is extended to include transport via melt convection. A double-sided model (equal diffusivity in liquid and solid phases) is considered for the present analysis. For the coupling between phase-field and Navier-Stokes equations, two commonly used schemes are investigated using a matched asymptotic analysis: (i) variable viscosity and (ii) drag force model. While for the variable viscosity model, the existence of a thin interface limit can be shown up to the second order in the expansion parameter, difficulties arise in satisfying no-slip boundary condition at this order for the drag force model. Nevertheless, detailed numerical simulations in two dimensions show practically no difference in dendritic growth profiles in the presence of forced melt flow obtained for the two coupling schemes. This suggests that both approaches can be used for the purpose of numerical simulations. Simulation results are also compared to analytic theory, showing excellent agreement for weak flow. Deviations at higher fluid velocities are discussed in terms of the underlying theoretical assumptions. © 2020 The Author(s) Published by the Royal Society. All rights reserved.

Kinetics of crystal growth in undercooled melts is analyzed by methods of theoretical modeling. Special attention is paid to rapid growth regimes occurring at deep undercoolings at which non-linearity in crystal velocity appears. A traveling wave solution of the phase field model (PFM) derived from the fast transitions theory is used for a quantitative description of the crystal growth kinetics. The "velocity - undercooling" relationship predicted by the traveling wave solution is compared with the data of molecular dynamics simulation (MDS) which were obtained for the crystal-liquid interfaces growing in the 100-direction in the Ni50Al50 alloy melt. © Published under licence by IOP Publishing Ltd.

Tin whiskers are hair-like single crystals that spontaneously grow from tin-coated surfaces. Whiskers are commonly found in electronic components, where tin coatings are used in, e.g., soldering applications, and to protect components from corrosion. Whiskers are known to cause short-circuits leading to failure of electronic components. The exact mechanisms responsible for whisker formation and growth are not fully understood. It is, however, believed that whiskers grow as a way to relax stress in the tin coating and that compressive stress gradients drive the growth. The stress in the coating is mainly caused by the formation and growth of the intermetallic compound Cu6Sn5 in the interface region between the copper substrate and the tin layer.This thesis presents numerical simulations and experimental investigations related to the growth and formation of tin whiskers on tin coated copper substrates. The aim of the experimental work has been to verify the existence of a compressive stress gradient and to characterise the microstructure around tin whiskers in 2D as well as 3D. This was realised using different x-ray diffraction methods, namely scanning Laue microdiffraction, differential aperture x-ray microscopy (DAXM), and scanning 3D x-ray diffraction (3DXRD). Laue microdiffraction was used to study the evolution of the microstructure around two whiskers over a period ranging from 4 to 21 months of ageing. The hydrostatic stress field in the tin coating was estimated by assuming plane stress conditions. It was found that the stress field was highly inhomogeneous.It was possible to identify ridges of high compressive stress leading to the whisker. These ridges, which have not been observed previously, are potentially driving diffusion of tin from specific regions of the coating towards the root of the whisker.DAXM and 3DXRD were used to study the microstructure around a whisker in three dimensions. Using DAXM, through-depth variations of the deviatoric strain field

This work concerns spontaneous fracture of growing brittle precipitates in an elastic plastic matrix. The mass of the precipitate is increasing as more transformed matrix material is added to it. Under stress-free conditions, the new phase occupies a larger volume than the original matrix material. Just outside the expanding precipitate, the matrix undergoes stretching beyond the elastic limit. The influence of the elastic plastic material behaviour is studied. A phase field model that keeps track of the phase composition is used. Both cases with a crack and without a crack are included. The growth histories from microscopic to macroscopic precipitate sizes are followed. Growth of the precipitate is very slow and quasi-static mechanical equilibrium is assumed at all time. The result is compared with observations of hydride blisters that are formed on surfaces of zirconium alloys. The numerical model is qualified against a derived exact solution for a cylindrical precipitate without a crack. The numerical result predicts a position of the growing crack that is confirmed by the observations. Also, the predicted length of the crack is in fair agreement with the experimental observations. The depth of the blister is slightly larger than what is found at the experiments. Also, it is found that the incorporated transformed phase rejects the compression, which creates an increasing tensile stress in the inner part of the precipitate. © 2018 Wiley Publishing Ltd.

The formation of self-organized domain patterns during polarization reversal in highly non-equilibrium switching conditions was studied by us earlier in the single crystals of lithium niobate and lithium tantalate. In this paper, we used the phase-field simulation to verify the analogy between self-organized growth of domains and new phase during the first order phase transition. The crystal symmetry C3v was taken into account. The similarity of simulated and experimentally observed shapes of isolated domains was achieved. © Published under licence by IOP Publishing Ltd.

We report a direct measurement of the friction coefficient ξc of two fluctuating contact lines formed on a fiber surface when a long glass fiber intersects the two water-air interfaces of a thin soap film. The glass fiber of diameter d in the range of 0.4–4 μm and length 100–300 μm is glued onto the front end of a rectangular cantilever used for atomic force microscopy. As a sensitive mechanical resonator, the hanging fiber probe can accurately measure a minute change of its viscous damping caused by the soap film. By measuring the broadening of the resonant peak of the hanging fiber probe with varying viscosity η of the soap film and different surface treatments of the glass fiber, we confirm that the contact line dissipation obeys a universal scaling law, ξc=απdη, where the coefficient α=1.1±0.3 is insensitive to the change of liquid-solid contact angle. The experimental result is in good agreement with the numerical result based on the phase field model under the generalized Navier boundary conditions.

We present a Lagrangian phase-field method to study the low Reynolds number dynamics of vesicles embedded in a viscous fluid. In contrast to previous approaches, where the field variables are the phase-field and the fluid velocity, here we exploit the fact that the phasefield tracks a material interface to reformulate the problem in terms of the Lagrangian motion of a background medium, containing both the biomembrane and the fluid. We discretize the equations in space with maximum-entropy approximants, carefully shown to perform well in phase-field models of biomembranes in a companion paper. The proposed formulation is variational, lending itself to implicit time-stepping algorithms based on minimization of a time-incremental energy, which are automatically nonlinearly stable. The proposed method deals with two of the major challenges in the numerical treatment of coupled fluid/phase-field models of biomembranes, namely the adaptivity of the grid to resolve the sharp features of the phase-field, and the stiffness of the equations, leading to very small time-steps. In our method, local refinement follows the features of the phasefield as both are advected by the Lagrangian motion, and large time-steps can be robustly chosen in the variational time-stepping algorithm, which also lends itself to time adaptivity. The method is presented in the axisymmetric setting, but it can be directly extended to 3D., Peer Reviewed, Postprint (published version)

We present an adaptive meshfree method to approximate phase-field models of biomembranes. In such models, the Helfrich curvature elastic energy, the surface area, and the enclosed volume of a vesicle are written as functionals of a continuous phase-field, which describes the interface in a smeared manner. Such functionals involve up to second-order spatial derivatives of the phase-field, leading to fourth-order Euler–Lagrange partial differential equations (PDE). The solutions develop sharp internal layers in the vicinity of the putative interface, and are nearly constant elsewhere. Thanks to the smoothness of the local maximum-entropy (max-ent) meshfree basis functions, we approximate numerically this high-order phase-field model with a direct Ritz–Galerkin method. The flexibility of the meshfree method allows us to easily adapt the grid to resolve the sharp features of the solutions. Thus, the proposed approach is more efficient than common tensor product methods (e.g. finite differences or spectral methods), and simpler than unstructured Cº finite element methods, applicable by reformulating the model as a system of second-order PDE. The proposed method, implemented here under the assumption of axisymmetry, allows us to show numerical evidence of convergence of the phase-field solutions to the sharp interface limit as the regularization parameter approaches zero. In a companion paper, we present a Lagrangian method based on the approximants analyzed here to study the dynamics of vesicles embedded in a viscous fluid., Peer Reviewed, Postprint (published version)

A recently developed ternary phase-field model is applied to study the precipitation behavior in Ni-Al-Cr alloys. The effect of diffusivity and initial compositional/ordering profiles on the growth process of precipitates are examined. Specifically, the composition evolution with time in the precipitates is simulated and compared with experimental observations by Sudbrack et al. The simulation work reveals that the CR- trapping phenomenon can only be reproduced with no significant partitioning of Cr at the nucleation stage. By looking into the composition dependent diffusivity and thermodynamic driving force for both Al and Cr, it is found that the key difference resides in the much stronger composition dependence of the diffusivity of Al., In-house contract. To be submitted for publication in Acta Materialia. Prepared in collaboration with UBS, Inc., Dayton, OH.

This effort has focused on verification and refinement of the mechanism transitions at intermediate temperatures in the disk alloy Rene 104, with the observation of microtwinning, continuous faulting and dislocation by-pass at successively higher temperatures above 650 degrees C. Evidence for the twin initiation process has also been obtained via TEM studies of specimens interrupted after small strain levels. A model for the novel microtwinning regime. Modeling at the ab initio, atomistic and phase field levels is providing insight into the activation parameters associated with the observed deformation mechanisms. Single crystals of Rene 104 have successfully been grown. A novel phase field model of directional coarsening (rafting) during high temperature, low stress creep of blade alloys has been developed. This model accounts for the local stress fields associated with matrix dislocations as well as the lattice misfit, and demonstrates promising qualitative agreement with experiment. Finally, deformation mechanisms in high temperature creep of several "generation 4" single crystal blade alloys under rafting conditions have been studied and a model for creep under these conditions developed. The goal of the program is to develop improved models that will (a) incorporate more realistic representation of the relevant microstructures and micromechanisms, (b) enable modeling for a range of relevant service conditions (c) address time-dependent deformation in both disk and blade alloys, and (d) provide this information to the component design process, building upon the paths to the design process created in the DARPA Accelerated Insertion of Materials (AIM) program., Prepared in cooperation with the Department of Materials Science and Engineering, The Ohio State University, Columbus, OH. This effort comprises a coordinated team of researchers from the Ohio State University (under Grant #FA9550-05-1-0135), University of Michigan (under Grant #FA9550-05-1-0100) and Johns Hopkins University (under Grant #FA9550-05-1-0102). The original document contains color images.

The effect of applied mechanical strains on the ferroelectric and dielectric properties of a model single crystal is investigated using a phase field model based on the time-dependent Ginzburg-Landau equation, which takes both multiple-dipole-dipole-clectric and -elastic interactions into account. The evolution of the ferroelectric domain structure is simulated at different temperatures and applied strains. The results show that the paraelectric/ferroelectric phase transition temperature linearly increases with the applied mechanical strain under mechanical clamping conditions. Analogous to the classical Ehrenfest equation, a thermodynamics equation is derived to describe the relationship between the transition temperature and the applied strain. The change in the domain structure with temperature under applied inequiaxial strains is different from that under applied equiaxial strains. The simulations also illustrate the changes in the coercive field, the remanent polarization and the nonlinear dielectric constant with the applied strain. (c) 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

The effect of applied mechanical strains on the ferroelectric and dielectric properties of a model single crystal is investigated using a phase field model based on the time-dependent Ginzburg-Landau equation, which takes both multiple-dipole-dipole-clectric and -elastic interactions into account. The evolution of the ferroelectric domain structure is simulated at different temperatures and applied strains. The results show that the paraelectric/ferroelectric phase transition temperature linearly increases with the applied mechanical strain under mechanical clamping conditions. Analogous to the classical Ehrenfest equation, a thermodynamics equation is derived to describe the relationship between the transition temperature and the applied strain. The change in the domain structure with temperature under applied inequiaxial strains is different from that under applied equiaxial strains. The simulations also illustrate the changes in the coercive field, the remanent polarization and the nonlinear dielectric constant with the applied strain. (c) 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

The effect of applied mechanical strains on the ferroelectric and dielectric properties of a model single crystal is investigated using a phase field model based on the time-dependent Ginzburg-Landau equation, which takes both multiple-dipole-dipole-clectric and -elastic interactions into account. The evolution of the ferroelectric domain structure is simulated at different temperatures and applied strains. The results show that the paraelectric/ferroelectric phase transition temperature linearly increases with the applied mechanical strain under mechanical clamping conditions. Analogous to the classical Ehrenfest equation, a thermodynamics equation is derived to describe the relationship between the transition temperature and the applied strain. The change in the domain structure with temperature under applied inequiaxial strains is different from that under applied equiaxial strains. The simulations also illustrate the changes in the coercive field, the remanent polarization and the nonlinear dielectric constant with the applied strain. (c) 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Phase boundaries play an important role in setting the properties of multicomponent materials at both the meso- and nano-scales. In this review, we provide an overview of the modelling methods utilized in state-of-the-art research and engineering applications. We review the current physical understanding of how phase boundaries evolve, focusing on multicomponent systems. The recent advances in numerical modelling, fueled by powerful computers, have provided accurate and robust results that allow problems that are beyond the reach of analytic methods to be addressed. While the approaches used in engineering-oriented applications employ simplified microstructures, it is found that such models are quite useful in many problems. The ability to simulate realistic microstructures will further increase the power of materials modelling. We also highlight the differences between the sharp interface and diffuse interface approaches for modelling microstructural evolution. In addition, we identify future research topics in this area., QC 20100525

We present numerical simulations directed at the description of smectic-B germs growing into the supercooled nematic phase for two different liquid crystalline substances. The simulations are done by means of a phase-field model appropriate to study strong anisotropy and also faceted interfaces. The most important ingredient is the angle-dependent surface energy, but kinetic effects are also relevant. The simulations reproduce qualitatively a rich variety of morphologies observed in the experiments for different value of undercooling, extending from the faceted equilibrium shape to fully developed dendrites., Peer Reviewed, Postprint (published version)