Auswahl der wissenschaftlichen Literatur zum Thema „Phase-Field Models (PFM)“

The phase-field model (PFM) is gaining increasing attention in the application of multiphase flows due to its advantages, in which the phase interface is treated as a narrow layer and phase parameters change smoothly and continually at this thin layer. Thus, the construction or tracking of the phase interface can be avoided, and the bulk phase and phase interface can be simulated integrally. PFM provides a useful alternative that does not suffer from problems with either the mass conservation or the accurate computation of surface tension. In this paper, the state of the art of PFM in the numerical modeling and simulation of multiphase flows is comprehensively reviewed. Starting with a brief description of historical developments in the PFM, we continue to take a tour into the basic concepts, fundamental theory, and mathematical models. Then, the commonly used numerical schemes and algorithms for solving the governing systems of PFM in the application of multiphase flows are presented. The various applications and representative results, especially in non-match density scenarios of multiphase flows, are reviewed. The primary challenges and research focus of PFM are analyzed and summarized as well. This review is expected to provide a valuable reference for PFM in the application of multiphase flows.

In cases with complicated crack topologies, the computational modeling of failure processes in materials owing to fracture based on sharp crack discontinuities fails. Diffusive crack modeling based on the insertion of a crack phase-field can overcome this. The phase-field model (PFM) portrays the fracture geometry in a diffusive manner, with no abrupt discontinuities. Unlike discrete fracture descriptions, phase-field descriptions do not need numerical monitoring of discontinuities in the displacement field. This considerably decreases the complexity of implementation. These qualities enable PFM to describe fracture propagation more successfully than numerical approaches based on the discrete crack model, especially for complicated crack patterns. These models have also demonstrated the ability to forecast fracture initiation and propagation in two and three dimensions without the need for any ad hoc criteria. The phase-field model, among numerous options, is promising in the computer modeling of fracture in solids due to its ability to cope with complicated crack patterns such as branching, merging, and even fragmentation. A brief history of the application of the phase-field model in predicting solid fracture has been attempted. An effort has been made to keep the conversation focused on recent research findings on the subject. Finally, some key findings and recommendations for future research areas in this field are discussed.

Phase field method (PFM) was employed to investigate the crystal growth of Mg-Al alloy, on the basis of binary alloy model, the fluid field equation was coupled into the phase-field models, and the marker and cell (MAC) method was used in the numerical calculation of micro structural pattern. In the cast process, quantitative comparison of different anisotropy values that predicted the dendrite evolution were discussed in detail, and when the fluid flow rate reaches a high value, we can see the remelting of dendrite arms.

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)’.

Additive Manufacturing (AM) is a process that fabricates products by manufacturing materials according to a three-dimensional model. It has recently gained attention due to its environmental advantages, including reduced energy consumption and high material utilization rates. However, controlling defects such as melting issues and residual stress, which can occur during metal additive manufacturing, poses a challenge. The trial-and-error verification of these defects is both time-consuming and costly.Consequently, efforts have been made to develop phenomenological models that understand the influence of process variables on defects, and mechanical/electrical/thermal properties of geometrically complex products. This paper introduces modeling techniques that can simulate the powder additive manufacturing process. The focus is on representative metal additive manufacturing processes such as Powder Bed Fusion (PBF), Direct Energy Deposition (DED), and Binder Jetting (BJ) method.To calculate thermal-stress history and the resulting deformations, modeling techniques based on Finite Element Method (FEM) are generally utilized. For simulating the movements and packing behavior of powders during powder classification, modeling techniques based on Discrete Element Method (DEM) are employed. Additionally, to simulate sintering and microstructural changes, techniques such as Monte Carlo (MC), Molecular Dynamics (MD), and Phase Field Modeling (PFM) are predominantly used.

Fatigue delamination damage is one of the most important fatigue failure modes for laminated composite structures. However, there are still many challenging problems in the development of the theoretical framework, mathematical/physical models, and numerical simulation of fatigue delamination. What is more, it is essential to establish a systematic classification of these methods and models. This article reviews the experimental phenomena of delamination onset and propagation under fatigue loading. The authors reviewed the commonly used phenomenological models for laminated composite structures. The research methods, general modeling formulas, and development prospects of phenomenological models were presented in detail. Based on the analysis of finite element models (FEMs) for laminated composite structures, several simulation methods for fatigue delamination damage models (FDDMs) were carefully classified. Then, the whole procedure, range of applications, capability assessment, and advantages and limitations of the models, which were based on four types of theoretical frameworks, were also discussed in detail. The theoretical frameworks include the strength theory model (SM), fracture mechanics model (FM), damage mechanics model (DM), and hybrid model (HM). To the best of the authors’ knowledge, the FDDM based on the modified Paris law within the framework of hybrid fracture and damage mechanics is the most effective method so far. However, it is difficult for the traditional FDDM to solve the problem of the spatial delamination of complex structures. In addition, the balance between the cost of acquiring the model and the computational efficiency of the model is also critical. Therefore, several potential research directions, such as the extended finite element method (XFEM), isogeometric analysis (IGA), phase-field model (PFM), artificial intelligence algorithm, and higher-order deformation theory (HODT), have been presented in the conclusions. Through validation by investigators, these research directions have the ability to overcome the challenging technical issues in the fatigue delamination prediction of laminated composite structures.

During the solidification process of the alloy, the temperature lies in the range between the solid-phase line and the liquidus. Dendrite growth exhibits high sensitivity to even slight fluctuations in temperature, thereby significantly influencing the tip growth rate. The increase in temperature can result in a reduction in the rate of tip growth, whereas a decrease in temperature can lead to an augmentation of the tip growth rate. In cases where there is a significant rise in temperature, dendrites may undergo fracture and subsequent remelting. Within the phenomenon of ultrasonic cavitation, the release of internal energy caused by the rupture of cavitation bubbles induces a substantial elevation in temperature, thereby causing both dendrite remelting and fracture phenomena. This serves as the main mechanism behind microstructure refinement induced by ultrasonic cavitation. Although dendrite remelting and fracture exert significant influences on the solidification process of alloys, most studies primarily focus on microscopic characterization experiments, which fail to unveil the transient evolution law governing dendrite remelting and fracture processes. Numerical simulation offers an effective approach to address this gap. The existing numerical models primarily focus on predicting the dendrite growth process, while research on remelting and fracture phenomena remains relatively limited. Therefore, a dendrite remelting model was established by incorporating the phase field method (PFM) and finite element difference method (FDM) into the temperature-induced modeling, enabling a comprehensive investigation of the entire process evolution encompassing dendrite growth and subsequent remelting.

Electrochemical applications play a key role for the topic of “green hydrogen” for the de-carbonization of the energy and mobility sectors. Electrochemical systems and processes, including fuel cells and electrolysers, have witnessed several benefits over conventional combustion-based technologies currently being widely used in power plants and vehicles. The ceramic high-temperature technologies by means of SOC exhibits high efficiencies, with a thermoelectric conversion efficiency as high as 60% and a total efficiency of up to 90% in fuel cell operation and even higher in electrolysis mode. The SOC technology, therefore, sees a promising future in the production of green hydrogen and electricity. Providing high operating temperature, over 600 oC, the SOC system shows the capability to operate with diverse types of gas mixtures, for example, hydrogen, ammonia, and carbon-containing mixtures such as methane (CH4), carbon monoxide (CO) in fuel cell operation (SOFC) and steam and/or carbon dioxide (CO2) in electrolysis operation (SOEC). The design of the SOC stack enables a reversible operation (rSOC) between fuel cell and electrolysis modes. It indicates the SOC system can perform with high efficiencies in both operating modes, which also widens the scope of possible applications. Challenges remain when it comes to commercialization of the SOC technology, in both the investment costs (CAPEX) and operating costs (OPEX) aspects. From the technological and scientific point of view, the physical transport phenomena in SOCs need to be understood which can be done by the help of experimental and numerical investigations. Cheaper and long-lasting material alternatives may be found for the cell, stack and system development afterwards. Detailed experimental investigations usually require a lot of effort in time and data analysis. To promote the scientific and technological studies on the SOC technology, numerical investigations by using multiscale and multiphysical models are carried out in this work. The models include an in-house designed/written phase field model (PFM) 1, and an open-source based computational fluid dynamics (CFD) model, openFuelCell2 2 (based on OpenFOAM). The former accounts for the microstructure evolution of the Ni/YSZ composition. The latter addresses the multiphysical transport processes in different phases, i.e., ionic transfer in YSZ, electronic transfer in Ni, and the gas diffusion in the gas phase. The figure below shows the computational domain and different phases in the numerical simulations. The evolution of Ni/YSZ composition can be predicted by the PFM. It is supposed to reproduce the Ni agglomeration that has been observed in SOFC long-term experiments 3. The simulation result shown at the left-most is obtained by performing the PFM simulation (with 96 x 96 x 96 voxels) representing the microstructure change for a certain time duration. The computational domain (192 x 192 x 192 voxels) consists of three phases, namely, YSZ, Ni, and gas, as shown in the middle and on the right side. It is refined in each direction to better capture the triple phase regions (lower right) shared by the three phases, which refers to the active sites that enable the electrochemical reaction to be conducted. By applying different governing equations on these phases, the CFD model, openFuelCell2, can describe the transport phenomena numerically. Hence, the performance degradation of a SOFC due to Ni agglomeration can be captured by carrying out the simulations for different time durations. The effective properties may be derived as well so that they can be used in numerical simulations with larger scales. Acknowledgement The authors would like to thank their colleagues at Forschungszentrum Jülich GmbH for their great support and the Helmholtz Society, the German Federal Ministry of Education and Research as well as the Ministry of Culture and Science of the Federal State of North Rhine-Westphalia for financing these activities as part of the Living Lab Energy Campus. References Q. Li, L. Liang, K. Gerdes, and L.-Q. Chen, Appl. Phys. Lett., 101, 033909 (2012). S. Zhang, S. Hess, H. Marschall, U. Reimer, S. B. Beale, and W. Lehnert, Computer Physics Communications, to be submitted (2023). C. E. Frey, Q. Fang, D. Sebold, L. Blum, and N. H. Menzler, J. Electrochem. Soc., 165, F357 (2018). Figure 1

To facilitate the analysis of pattern formation and the related phase transitions in Bose–Einstein condensates, we present an explicit approximate mapping from the nonlocal Gross–Pitaevskii equation with cubic nonlinearity to a phase field crystal (PFC) model. This approximation is valid close to the superfluid–supersolid phase transition boundary. The simplified PFC model permits the exploration of bifurcations and phase transitions via numerical path continuation employing standard software. While revealing the detailed structure of the bifurcations present in the system, we demonstrate the existence of localized states in the PFC approximation. Finally, we discuss how higher-order nonlinearities change the structure of the bifurcation diagram representing the transitions found in the system.

In this paper, we review the Fourier-spectral method for some phase-field models: Allen–Cahn (AC), Cahn–Hilliard (CH), Swift–Hohenberg (SH), phase-field crystal (PFC), and molecular beam epitaxy (MBE) growth. These equations are very important parabolic partial differential equations and are applicable to many interesting scientific problems. The AC equation is a reaction-diffusion equation modeling anti-phase domain coarsening dynamics. The CH equation models phase segregation of binary mixtures. The SH equation is a popular model for generating patterns in spatially extended dissipative systems. A classical PFC model is originally derived to investigate the dynamics of atomic-scale crystal growth. An isotropic symmetry MBE growth model is originally devised as a method for directly growing high purity epitaxial thin film of molecular beams evaporating on a heated substrate. The Fourier-spectral method is highly accurate and simple to implement. We present a detailed description of the method and explain its connection to MATLAB usage so that the interested readers can use the Fourier-spectral method for their research needs without difficulties. Several standard computational tests are done to demonstrate the performance of the method. Furthermore, we provide the MATLAB codes implementation in the Appendix A.