Diffuse Interface Approach Research Articles

Deciphering the influence of multi-component blends and their electronic band structure on the performance of All-Solid-State Batteries

The emergence of all-solid-state batteries (ASSBs) introduces a paradigm shift in energy storage technology, offering enhanced safety compared to conventional liquid-based metal-ion batteries. Significant effort is directed toward optimizing the solid-electrolyte blend composition to enhance the battery’s electrochemical performance. Despite some promising results, a lack of guidelines persists, particularly for optimizing multicomponent solid electrolytes given their large parameter window. This study aims to address this challenge by implementing a unified diffuse-interface approach to model and simulate the solid electrolyte morphologies and their corresponding electrochemical performance when incorporated in a battery. The electrolyte microstructures are simulated using the Cahn–Hilliard formulation while a diffuse-interface framework formulated in terms of electrochemical potential is utilized for exploring Li-ion transport across the battery. It is found that, while the variegated microstructures arising from various solid electrolyte blend compositions influence the power density of the battery, the electronic band structure of the blend phases is an important consideration. The proposed model is versatile and can be adapted for various battery technologies beyond ASSBs. This expands its potential impact and could lead to innovations in energy storage technology.

A computational algorithm for optimal design of a bioartificial organ scaffold architecture.

We develop a computational algorithm based on a diffuse interface approach to study the design of bioartificial organ scaffold architectures. These scaffolds, composed of poroelastic hydrogels housing transplanted cells, are linked to the patient's blood circulation via an anastomosis graft. Before entering the scaffold, the blood flow passes through a filter, and the resulting filtered blood plasma transports oxygen and nutrients to sustain the viability of transplanted cells over the long term. A key issue in maintaining cell viability is the design of ultrafiltrate channels within the hydrogel scaffold to facilitate advection-enhanced oxygen supply ensuring oxygen levels remain above a critical threshold to prevent hypoxia. In this manuscript, we develop a computational algorithm to analyze the plasma flow and oxygen concentration within hydrogels featuring various channel geometries. Our objective is to identify the optimal hydrogel channel architecture that sustains oxygen concentration throughout the scaffold above the critical hypoxic threshold. The computational algorithm we introduce here employs a diffuse interface approach to solve a multi-physics problem. The corresponding model couples the time-dependent Stokes equations, governing blood plasma flow through the channel network, with the time-dependent Biot equations, characterizing Darcy velocity, pressure, and displacement within the poroelastic hydrogel containing the transplanted cells. Subsequently, the calculated plasma velocity is utilized to determine oxygen concentration within the scaffold using a diffuse interface advection-reaction-diffusion model. Our investigation yields a scaffold architecture featuring a hexagonal network geometry that meets the desired oxygen concentration criteria. Unlike classical sharp interface approaches, the diffuse interface approach we employ is particularly adept at addressing problems with intricate interface geometries, such as those encountered in bioartificial organ scaffold design. This study is significant because recent developments in hydrogel fabrication make it now possible to control hydrogel rheology and utilize computational results to generate optimized scaffold architectures.

Upscaling and Effective Behavior for Two-Phase Porous-Medium Flow Using a Diffuse Interface Model

We investigate two-phase flow in porous media and derive a two-scale model, which incorporates pore-scale phase distribution and surface tension into the effective behavior at the larger Darcy scale. The free-boundary problem at the pore scale is modeled using a diffuse interface approach in the form of a coupled Allen–Cahn Navier–Stokes system with an additional momentum flux due to surface tension forces. Using periodic homogenization and formal asymptotic expansions, a two-scale model with cell problems for phase evolution and velocity contributions is derived. We investigate the computed effective parameters and their relation to the saturation for different fluid distributions, in comparison to commonly used relative permeability saturation curves. The two-scale model yields non-monotone relations for relative permeability and saturation. The strong dependence on local fluid distribution and effects captured by the cell problems highlights the importance of incorporating pore-scale information into the macro-scale equations.

A high-order, localized-artificial-diffusivity method for Eulerian simulation of multi-material elastic-plastic deformation with strain hardening

A high-order method for Eulerian simulation of material undergoing large elastic–plastic deformation is developed. Thermodynamically consistent hyperelastic constitutive relations are assumed, facilitating the treatment of solids, liquids, and gases in a unified manner. The method enables the simulation of multi-material interactions using a diffuse interface approach. Numerical capturing of material interfaces, shock waves, contact surfaces, and elastic-plastic strain discontinuities using high-order compact-difference schemes is assisted by Localized Artificial Diffusivity (LAD). In the new setting involving elastic–plastic deformation, the previously established terms for the artificial properties are verified to effectively regularize normal shocks. Additional LAD terms are introduced to the elastic and plastic kinematic equations to regularize shear shocks and other strain discontinuities, improving solution stability. Other important features of the method that improve robustness include the numerical treatment of compatibility terms in the kinematic equations, and the treatment of rotation. Particular emphasis is focused toward new advancements of the methods for plastic-deformation integration and the associated strain hardening of the material, including rate-dependent plasticity. The method is demonstrated on a variety of test problems, including 1-D impacts, a variant of the Shu-Osher problem, a Taylor impact, and a Richtmyer-Meshkov instability between two elastic–plastic solids with strain hardening.

A Semi-implicit Finite Volume Scheme for Incompressible Two-Phase Flows

This paper presents a mass and momentum conservative semi-implicit finite volume (FV) scheme for complex non-hydrostatic free surface flows, interacting with moving solid obstacles. A simplified incompressible Baer-Nunziato type model is considered for two-phase flows containing a liquid phase, a solid phase, and the surrounding void. According to the so-called diffuse interface approach, the different phases and consequently the void are described by means of a scalar volume fraction function for each phase. In our numerical scheme, the dynamics of the liquid phase and the motion of the solid are decoupled. The solid is assumed to be a moving rigid body, whose motion is prescribed. Only after the advection of the solid volume fraction, the dynamics of the liquid phase is considered. As usual in semi-implicit schemes, we employ staggered Cartesian control volumes and treat the nonlinear convective terms explicitly, while the pressure terms are treated implicitly. The non-conservative products arising in the transport equation for the solid volume fraction are treated by a path-conservative approach. The resulting semi-implicit FV discretization of the mass and momentum equations leads to a mildly nonlinear system for the pressure which can be efficiently solved with a nested Newton-type technique. The time step size is only limited by the velocities of the two phases contained in the domain, and not by the gravity wave speed nor by the stiff algebraic relaxation source term, which requires an implicit discretization. The resulting semi-implicit algorithm is first validated on a set of classical incompressible Navier-Stokes test problems and later also adds a fixed and moving solid phase.

Multi-scale modelling of boiling heat transfer: Exploring the applicability of an enhanced volume of fluid method in sub-micron scales

The advancement of technology has led to a significant increase in thermal loads, thus presenting new challenges in heat dissipation. Traditional single-phase cooling systems are often inadequate to meet these demands. As a result, phase-change technologies utilizing boiling and condensation, which can achieve high heat transfer coefficients, have garnered considerable attention. To delve into the complex physics of boiling heat transfer, researchers are increasingly turning to numerical simulation methods such as the Volume of Fluid (VOF) and the Diffuse Interface (DI) approaches. The VOF method, widely employed for macro-scale simulations ranging from micrometers to millimeters, effectively tracks bubble growth and detachment. Conversely, the DI method represents the interface as a continuous phase field and is primarily used for mesoscale simulations spanning from nanometers to micrometers. While the DI method excels in resolving mesoscale interfacial phenomena, it is computationally expensive for larger domains. Considering the strengths and weaknesses of both the VOF and DI methods, there is a growing interest in developing a multi-scale modeling approach that amalgamates their benefits. To pursue this objective, initial efforts are being made to evaluate the scaling capability of VOF towards lower spatial and temporal limits. Hence, an enhanced and customized VOF methodology has been developed within the OpenFOAM toolbox. This methodology is employed to investigate various bubble growth scenarios, progressively exploring its applicability at lower temporal and spatial scales to identify the lower limits of its application. By taking this first step towards combining the strengths of both the VOF and DI methods through a multi-scale modeling approach, the presented paper paves the way for enhancing the accuracy and efficiency of modelling approaches for boiling heat transfer while tackling a challenge associated with varying spatial and temporal scales. This endeavor not only pushes the boundaries of computational fluid dynamics but also holds promise for addressing real-world thermal management issues in diverse technological applications.

Effect of shock impedance of mesoscale inclusions on the shock-to-detonation transition in liquid nitromethane

Two-dimensional, meso-resolved numerical simulations are performed to investigate the effect of shock impedance of mesoscale inclusions on the shock-to-detonation transition (SDT) in liquid nitromethane (NM). The shock-induced initiation behaviors resulting from the cases with NM mixed with randomly distributed, 100-μm-sized air-filled cavities, polymethyl methacrylate (PMMA), silica, aluminum (Al), and beryllium (Be) particles with various shock impedances are examined. In this paper, hundreds of inclusions are explicitly resolved in the simulation using a diffuse-interface approach to treat two immiscible fluids. Without using any empirically calibrated, phenomenological models, the reaction rate in the simulations only depends on the temperature of liquid NM. The sensitizing effect of different inclusion materials can be rank-ordered from the weakest to the strongest as PMMA → silica → air → Al → Be in the hot-spot-driven regime of SDT. Air-filled cavities have a more significant sensitizing effect than silica particles, which is in agreement with the experimental finding. For different solid-phase inclusions, hot spots are formed by Mach reflection upon the interaction between the incident shock wave and the particle. The sensitizing effect increases roughly with the shock impedance of the inclusion material. More details of the hot-spot formation process for each solid-phase inclusion material are revealed via zoom-in simulations of a shock passing over a single particle.

Geometric control by active mechanics of epithelial gap closure.

Epithelial wound healing is one of the most important biological processes occurring during the lifetime of an organism. It is a self-repair mechanism closing wounds or gaps within tissues to restore their functional integrity. In this work we derive a new diffuse interface approach for modelling the gap closure by means of a variational principle in the framework of non-equilibrium thermodynamics. We investigate the interplay between the crawling with lamellipodia protrusions and the supracellular tension exerted by the actomyosin cable on the closure dynamics. These active features are modeled as Korteweg forces into a generalised chemical potential. From an asymptotic analysis, we derive a pressure jump across the gap edge in the sharp interface limit. Moreover, the chemical potential diffuses as a Mullins-Sekerka system, and its interfacial value is given by a Gibbs-Thompson relation for its local potential driven by the curvature-dependent purse-string tension. The finite element simulations show an excellent quantitative agreement between the closure dynamics and the morphology of the edge with respect to existing biological experiments. The resulting force patterns are also in good qualitative agreement with existing traction force microscopy measurements. Our results shed light on the geometrical control of the gap closure dynamics resulting from the active forces that are chemically activated around the gap edge.

A diffuse interface approach for vector-valued PDEs on surfaces

A diffuse interface approach for vector-valued PDEs on surfaces

Modeling the morphological-dependent performance of metal-ion battery: A promising yet overlooked strategy

All-solid-state batteries (ASSBs) have emerged as a promising alternative to traditional lithium-ion batteries (LIBs) due to their potential to mitigate the safety concerns which are associated with the use of liquid electrolytes. However, the practical implementation of ASSBs is hindered by the low mobility of Li-ions, and the inefficient ion migration within the solid-state electrolytes, which can be enhanced altogether by selecting a suitable material blend as solid-electrolyte with an optimum blend ratio. This enhancement provides us with a wide range of blend microstructures that can significantly improve the Li-ion mobility across the electrolyte, leading to enhanced battery performance. However, the vast space of material blends poses a challenge for experimental screening. So, to tackle this, computational models have been utilized to identify the most suitable blend for ASSBs. This research aims to enhance the electrochemical characteristics, and the Li-ion conductivity across the solid-electrolyte for an ASSB using numerical methods to zero in on the suitable blend ratio that can be used as a solid-electrolyte. The study presents a detailed analysis of a novel diffuse-interface approach, to model, and assess the impact of phase-separating blends on the electrochemical performance of the ASSBs.

A nanoscale view of the origin of boiling and its dynamics

In this work, we present a dynamical theory of boiling based on fluctuating hydrodynamics and the diffuse interface approach. The model is able to describe boiling from the stochastic nucleation up to the macroscopic bubble dynamics. It covers, with a modest computational cost, the mesoscale area from nano to micrometers, where most of the controversial observations related to the phenomenon originate. In particular, the role of wettability in the macroscopic observables of boiling is elucidated. In addition, by comparing the ideal case of boiling on ultra-smooth surfaces with a chemically heterogeneous wall, our results will definitively shed light on the puzzling low onset temperatures measured in experiments. Sporadic nanometric spots of hydrophobic wettability will be shown to be enough to trigger the nucleation at low superheat, significantly reducing the temperature of boiling onset, in line with experimental results. The proposed mesoscale approach constitutes the missing link between macroscopic approaches and molecular dynamics simulations and will open a breakthrough pathway toward accurate understanding and prediction.

Numerical treatment of reactive diffusion using the discontinuous Galerkin method

AbstractThis work presents a new finite element variational formulation for the numerical treatment of diffusional phase transformations using the discontinuous Galerkin method (DGM). Steep concentration and property gradients near phase boundaries require particular focus on a sound numerical treatment. There are different ways to tackle this problem ranging from (i) the well-known phase field method (PFM) (Biner et al. in Programming phase-field modeling, Springer, Berlin, 2017, Emmerich in The diffuse interface approach in materials science: thermodynamic concepts and applications of phase-field models, Springer, Berlin, 2003), where the interface is described continuously to (ii) methods that allow sharp transitions at phase boundaries, such as reactive diffusion models (Svoboda and Fischer in Comput Mater Sci 127:136–140, 2017, 78:39–46, 2013, Svoboda et al. in Comput Mater Sci 95:309–315, 2014). Phase transformation problems with continuous property changes can be implemented using the continuous Galerkin method (GM). Sharp interface models, however, lead to stability problems with the GM. A method that is able to treat the features of sharp interface models is the discontinuous Galerkin method. This method is well understood for regular diffusion problems (Cockburn in ZAMM J Appl Math Mech 83(11):731–754, 2003). As will be shown, it is also particularly well suited to model phase transformations. We discuss the thermodynamic background by review of a multi-phase, binary system. A new DGM formulation for the phase transformation problem with sharp interfaces is then introduced. Finally, the derived method is used in a 2D microstructural evolution simulation that features a binary, three-phase system that also takes the vacancy mechanism of solid body diffusion into account.

Analysis of a diffuse interface method for the Stokes-Darcy coupled problem

We consider the interaction between a free flowing fluid and a porous medium flow, where the free flowing fluid is described using the time dependent Stokes equations, and the porous medium flow is described using Darcy’s law in the primal formulation. To solve this problem numerically, we use a diffuse interface approach, where the weak form of the coupled problem is written on an extended domain which contains both Stokes and Darcy regions. This is achieved using a phase-field function which equals one in the Stokes region and zero in the Darcy region, and smoothly transitions between these two values on a diffuse region of width (ϵ) around the Stokes-Darcy interface. We prove convergence of the diffuse interface formulation to the standard, sharp interface formulation, and derive rates of convergence. This is performed by deriving a priori error estimates for discretizations of the diffuse interface method, and by analyzing the modeling error of the diffuse interface approach at the continuous level. The convergence rates are also shown computationally in a numerical example.

Numerical solution of coupled Cahn–Hilliard Navier–Stokes equations for two‐phase flows having variable density and viscosity

This work is concerned with the implementation of a decoupled diffuse interface approach for the numerical solution of two‐phase flows with a moving interface. The underlying two‐phase flow model consists of a mass‐preserving Cahn–Hilliard (CH) equation coupled with the incompressible Navier–Stokes equations (NSEs) through surface tension. Due to the higher order nonlinearity and stiff nature of the CH equation, its numerical solution is very challenging. We have used the decoupled pressure projection method for the numerical simulation of the governing equations. The spatial variables are discretized using the finite difference scheme on the staggered grids, while the explicit Euler method is used for time discretization. Four examples include the coalescence of inline rising bubbles, the obliques coalescence of two rising bubbles, and the deformation of single and two different‐sized bubbles in a shear flow field is numerically simulated. It is observed that the scheme is capable of tracking the interface accurately and respecting mass conservation and energy dissipation. Moreover, the scheme is efficient, easily implementable, energy stable, and fully decoupled.

Control of nonlinear bulk deformation and large shear strain on first-order phase transformation kinetics

Phase transformations play a key role in numerous coupled natural processes, and they are important for many industrial applications. However, the kinetics of phase transformations in coupled chemo-mechanical systems undergoing large mechanical deformations still needs to be better quantified. Here, we study the phase transformation kinetics of a two-phase binary mixture using the diffuse interface approach. We couple a Cahn–Hilliard type model with a mechanical model for a compressible viscous flow. The bulk compressibility is a nonlinear function of the pressure, and the shear viscosity is a nonlinear function of the concentration. The mechanical coupling is achieved by employing a pressure-dependent mechanical mixing term in the equation for the Gibbs energy. We derive a dimensionless system of equations which we solve numerically with a pseudo-transient method using conservative finite differences for discretization. We perform numerical simulations in 1D and 2D model setups considering far-field simple shear and pure shear. For a chemo-mechanically coupled system, we show that the velocity of the phase boundary is a linear function of the degree of metastability and, hence, confirm the hypothesis of “normal growth.” A stronger mechanical coupling and a larger volumetric effect of the chemical reaction result in lower phase boundary velocities. The 2D results show a significant impact of the mechanical coupling and the far-field deformation on the orientation and kinetics of the phase transformations. Under far-field simple shear and pure shear in 2D, the phase transformations generate string-like patterns. The orientation of these patterns is controlled by the applied far-field deformation and orientations differ by 45 degrees between simple shear and pure shear.

Phase-field modeling of pitting and mechanically-assisted corrosion of Mg alloys for biomedical applications.

A phase-field model is developed to simulate the corrosion of Mg alloys in body fluids. The model incorporates both Mg dissolution and the transport of Mg ions in solution, naturally predicting the transition from activation-controlled to diffusion-controlled bio-corrosion. In addition to uniform corrosion, the presented framework captures pitting corrosion and accounts for the synergistic effect of aggressive environments and mechanical loading in accelerating corrosion kinetics. The model applies to arbitrary 2D and 3D geometries with no special treatment for the evolution of the corrosion front, which is described using a diffuse interface approach. Experiments are conducted to validate the model and a good agreement is attained against in vitro measurements on Mg wires. The potential of the model to capture mechano-chemical effects during corrosion is demonstrated in case studies considering Mg wires in tension and bioabsorbable coronary Mg stents subjected to mechanical loading. The proposed methodology can be used to assess the in vitro and in vivo service life of Mg-based biomedical devices and optimize the design taking into account the effect of mechanical deformation on the corrosion rate. The model has the potential to advocate further development of Mg alloys as a biodegradable implant material for biomedical applications. STATEMENT OF SIGNIFICANCE: A physically-based model is developed to simulate the corrosion of bioabsorbable metals in environments that resemble biological fluids. The model captures pitting corrosion and incorporates the role of mechanical fields in enhancing the corrosion of bioabsorbable metals. Model predictions are validated against dedicated in vitro corrosion experiments on Mg wires. The potential of the model to capture mechano-chemical effects is demonstrated in representative examples. The simulations show that the presence of mechanical fields leads to the formation of cracks accelerating the failure of Mg wires, whereas pitting severely compromises the structural integrity of coronary Mg stents. This work extends phase-field modeling to bioengineering and provides a mechanistic tool for assessing the service life of bioabsorbable metallic biomedical devices.

A Diffuse Interface Approach to Drop Impact Undergoing Solidification under a Horizontal Electric Field

A Diffuse Interface Approach to Drop Impact Undergoing Solidification under a Horizontal Electric Field

Diffuse-Interface Blended Method for Imposing Physical Boundaries in Two-Fluid Flows.

Multiphase flows are commonly found in chemical engineering processes such as distillation columns, bubble columns, fluidized beds and heat exchangers. The physical boundaries of domains in numerical simulations of multiphase flows are generally defined by a conformal unstructured mesh which, depending on the complexity of the physical system, results in time-consuming mesh generation which frequently requires user-intervention. Furthermore, the resulting conformal unstructured mesh could potentially contain a large number of skewed elements, which is undesirable for numerical stability and accuracy. The diffuse-interface approach allows for the use of a simple structured meshes to be used while still capturing the desired physical (e.g., solid-fluid) boundaries. In this work, a novel diffuse-interface method for the imposition of physical boundaries is developed for the incompressible two-fluid multiphase flow model. This model is appropriate for dispersed multiphase flows which are pervasive in chemical engineering processes, in that this flow regime results in high levels of mass and energy transfer between phases. A diffuse interface is used to define the physical boundaries and boundary conditions are imposed by blending the conservation equations from the two-fluid model with that of the nondeformable solid. The results from the diffuse-interface method are compared with results from a conformal unstructured mesh for different interface functions and widths. For small interface widths, the accuracy of the flow profile is unaffected by the choice of interface function and the phase fraction distribution and flow behavior are within 3% compared to those from a conformal mesh. As the interface width increases, the diffuse-interface solution deviates from the conformal mesh solution in both the localized gas fraction and the overall gas hold-up, resulting in a difference up to 30%. In the case of flow past a cylinder, where the solid interacts with the flow, the presence of the diffuse interface extends the thickness of the solid boundary and results in a deviation from the conformal mesh solution as time increases.

Analysis of a phase‐field finite element implementation for precipitation

AbstractPrecipitation hardening is an essential mechanism in materials design of age‐hardenable aluminium alloys. The occurrence and distribution of nano‐sized particles in such alloys can lead to superior material properties. During thermo‐mechanical processing, these particles evolve dynamically as function of temperature and applied load. Therefore, sophisticated modelling frameworks are required to study the underlying phenomena of this microstructural evolution in depth. Phase‐field method based on the diffuse interface approach has been successfully employed in literature to study particle nucleation and growth, as well as equilibrium particle shapes. Although phase‐field models provide reliable results due to the flexible adaption of the free energy, the method is computationally expensive, requiring efficient solution schemes. The finite‐element discretization in deal.II can overcome scalability disadvantages and can outperform standard finite‐difference codes. In this work, we used adaptive mesh refinement and adaptive time‐stepping and investigate how AMR and the use of the same stiffness matrix for a certain amount of time steps affect the performance of the phase‐field model. Particle growth simulations are performed to outline the major benefits of the finite element phase‐field model. The numerical strategy is shown to be effective regardless of the initial particle shape by considering different particle morphologies. The results illustrate a significant increase in simulation performance with the applied numerical techniques.

PERFORMANCE OF SHARP-VERSUS-DIFFUSE INTERFACE-BASED LEVEL SET METHOD ON A STAGGERED-VERSUS-CO-LOCATED GRID FOR CMFD

The present work is on comparison of computational performance for various level set methods (LSMs): diffuse interface level set method on staggered grid (DI-LSM<sub>stag</sub>), sharp-interface level set method on staggered grid (SI-LSM<sub>stag</sub>), diffuse interface level set method on co-located grid (DI-LSM<sub>col</sub>), and sharp interface level set method on co-located grid (SI-LSM<sub>col</sub>). Even though the implementations of the diffuse and sharp interface (DI and SI) approaches on staggered grid are straightforward, an additional pressure-interfacial force balance needs to be ensured on the co-located grid. This is established here with balanced force method (BFM) for the DI-LSM and ghost fluid method (GFM) for the SI-LSM. Computational performances of these LSMs are presented for a variety of computational multi-fluid dynamics (CMFD) problems: static drop, dam break, rising bubble, falling droplet, and droplet coalescence. Greater accuracy is found with SI-LSMs for the static drop, dam break, and rising bubble, whereas for the other problems, both SI-LSM and DI-LSM result in almost the same accuracy. Smaller computational time is taken by the SI-LSM for rising bubble and falling droplet, and by DI-LSM for the dam break and droplet coalescence. Comparing between grid systems, co-located grid resulted in greater accuracy for all the problems except falling droplet, for which both grid systems resulted in similar accuracy, whereas, a smaller computational time is taken by the co-located grid for rising bubble and falling droplet, and by the staggered grid for dam break and droplet coalescence. Overall, SI-LSM on the co-located grid shows better results with a slight increase in computational time as compared to the other LSMs, and is a suitable alternative to the staggered grid.