Exact solutions for the Cahn-Hilliard equation in terms of Weierstrass-elliptic and Jacobi-elliptic functions.

Despite the historical position of the F-expansion method as a method for acquiring exact solutions to nonlinear partial differential equations (PDEs), this study highlights its superiority over alternative auxiliary equation methods. The efficacy of this method is demonstrated through its application to solve the convective-diffusive Cahn-Hilliard (cdCH) equation, describing the dynamic of the separation phase for ternary iron alloys (Fe-Cr-Mo) and (Fe-X-Cu). Significantly, this research introduces an extensive collection of exact solutions by the auxiliary equation, comprising fifty-two distinct types. Six of these are associated with Weierstrass-elliptic function solutions, while the remaining solutions are expressed in Jacobi-elliptic functions. I think it is important to emphasize that, exercising caution regarding the statement of the term 'new,' the solutions presented in this context are not entirely unprecedented. The paper examines numerous examples to substantiate this perspective. Furthermore, the study broadens its scope to include soliton-like and trigonometric-function solutions as special cases. This underscores that the antecedently obtained outcomes through the recently specific cases encompassed within the more comprehensive scope of the present findings.

Patterning and folding of intestinal villi by active mesenchymal dewetting.

Tissue folds are structural motifs critical to organ function. In the intestine, bending of a flat epithelium into a periodic pattern of folds gives rise to villi, finger-like protrusions that enable nutrient absorption. However, the molecular and mechanical processes driving villus morphogenesis remain unclear. Here, we identify an active mechanical mechanism that simultaneously patterns and folds the intestinal epithelium to initiate villus formation. At the cellular level, we find that PDGFRA+ subepithelial mesenchymal cells generate myosin II-dependent forces sufficient to produce patterned curvature in neighboring tissue interfaces. This symmetry-breaking process requires altered cell and extracellular matrix interactions that are enabled by matrix metalloproteinase-mediated tissue fluidization. Computational models, together with in vitro and in vivo experiments, revealed that these cellular features manifest at the tissue level as differences in interfacial tensions that promote mesenchymal aggregation and interface bending through a process analogous to the active dewetting of a thin liquid film.

Modelling lipid rafts formation through chemo-mechanical interplay triggered by receptor-ligand binding.

Cell membranes, mediator of many biological mechanisms from adhesion and metabolism up to mutation and infection, are highly dynamic and heterogeneous environments exhibiting a strong coupling between biochemical events and structural re-organisation. This involves conformational changes induced, at lower scales, by lipid order transitions and by the micro-mechanical interplay of lipids with transmembrane proteins and molecular diffusion. Particular attention is focused on lipid rafts, ordered lipid microdomains rich of signalling proteins, that co-localise to enhance substance trafficking and activate different intracellular biochemical pathways. In this framework, the theoretical modelling of the dynamic clustering of lipid rafts implies a full multiphysics coupling between the kinetics of phase changes and the mechanical work performed by transmembrane proteins on lipids, involving the bilayer elasticity. This mechanism produces complex interspecific dynamics in which membrane stresses and chemical potentials do compete by determining different morphological arrangements, alteration in diffusive walkways and coalescence phenomena, with a consequent influence on both signalling potential and intracellular processes. Therefore, after identifying the leading chemo-mechanical interactions, the present work investigates from a modelling perspective the spatio-temporal evolution of raft domains to theoretically explain co-localisation and synergy between proteins' activation and raft formation, by coupling diffusive and mechanical phenomena to observe different morphological patterns and clustering of ordered lipids. This could help to gain new insights into the remodelling of cell membranes and could potentially suggest mechanically based strategies to control their selectivity, by orienting intracellular functions and mechanotransduction.

Connecting Structural Characteristics and Material Properties in Phase-Separating Polymer Solutions: Phase-Field Modeling and Physics-Informed Neural Networks.

The formed morphology during phase separation is crucial for determining the properties of the resulting product, e.g., a functional membrane. However, an accurate morphology prediction is challenging due to the inherent complexity of molecular interactions. In this study, the phase separation of a two-dimensional model polymer solution is investigated. The spinodal decomposition during the formation of polymer-rich domains is described by the Cahn-Hilliard equation incorporating the Flory-Huggins free energy description between the polymer and solvent. We circumvent the heavy burden of precise morphology prediction through two aspects. First, we systematically analyze the degree of impact of the parameters (initial polymer volume fraction, polymer mobility, degree of polymerization, surface tension parameter, and Flory-Huggins interaction parameter) in a phase-separating system on morphological evolution characterized by geometrical fingerprints to determine the most influential factor. The sensitivity analysis provides an estimate for the error tolerance of each parameter in determining the transition time, the spinodal decomposition length, and the domain growth rate. Secondly, we devise a set of physics-informed neural networks (PINN) comprising two coupled feedforward neural networks to represent the phase-field equations and inversely discover the value of the embedded parameter for a given morphological evolution. Among the five parameters considered, the polymer-solvent affinity is key in determining the phase transition time and the growth law of the polymer-rich domains. We demonstrate that the unknown parameter can be accurately determined by renormalizing the PINN-predicted parameter by the change of characteristic domain size in time. Our results suggest that certain degrees of error are tolerable and do not significantly affect the morphology properties during the domain growth. Moreover, reliable inverse prediction of the unknown parameter can be pursued by merely two separate snapshots during morphological evolution. The latter largely reduces the computational load in the standard data-driven predictive methods, and the approach may prove beneficial to the inverse design for specific needs.

The Influence of Physical Properties on the Membrane Morphology Formation during the Nonisothermal Thermally Induced Phase Separation Process.

The physical properties of a polymer solution that are composition- and/or temperature-dependent are among the most influential parameters to impact the dynamics and thermodynamics of the phase separation process and, as a result, the morphology formation. In this study, the impact of composition- and temperature-dependent density, heat capacity, and heat conductivity on the membrane structure formation during the thermally induced phase separation process of a high-viscosity polymer solution was investigated via coupling the Cahn-Hilliard equation for phase separation with the Fourier heat transfer equation. The variations of each physical property were also investigated in terms of different boundary conditions and initial solvent volume fractions. It was determined that the physical properties of the polymer solution have a noteworthy impact on the membrane morphology in terms of shorter phase separation time and droplet size. In addition, the influence of enthalpy of demixing in this case is critical because each physical property showed a nonhomogeneous pattern owing to the heat generation during phase separation, which in turn influenced the membrane morphology. Accordingly, it was determined that investigating spinodal decomposition without including heat transfer and the impact of physical properties on the morphology formation would lead to an inadequate understanding of the process, specifically in high-viscosity polymer solutions.

Controlling the structure of spin-coated multilayer ethylcellulose/hydroxypropylcellulose films for drug release.

Porous phase-separated ethylcellulose/hydroxypropylcellulose (EC/HPC) films are used to control drug transport out of pharmaceutical pellets. Water-soluble HPC leaches out and forms a porous structure that controls the drug transport. Industrially, the pellets are coated using a fluidized bed spraying device, and a layered film exhibiting varying porosity and structure after leaching is obtained. A detailed understanding of the formation of the multilayered, phase-separated structure during production is lacking. Here, we have investigated multilayered EC/HPC films produced by sequential spin-coating, which was used to mimic the industrial process. The effects of EC/HPC ratio and spin speed on the multilayer film formation and structure were investigated using advanced microscopy techniques and image analysis. Cahn-Hilliard simulations were performed to analyze the mixing behavior. A gradient with larger structures close to the substrate surface and smaller structures close to the air surface was formed due to coarsening of the layers already coated during successive deposition cycles. The porosity of the multilayer film was found to vary with both EC/HPC ratio and spin speed. Simulation of the mixing behavior and in situ characterization of the structure evolution showed that the origin of the discontinuities and multilayer structure can be explained by the non-mixing of the layers.

An operator splitting scheme for numerical simulation of spinodal decomposition and microstructure evolution of binary alloys.

This article compares the operator splitting scheme to linearly stabilized splitting and semi-implicit Euler's schemes for the numerical solution of the Cahn-Hilliard equation. For the purpose of validation, the spinodal decomposition phenomena have been simulated. The efficacy of the three schemes has been demonstrated through numerical experiments. The computed results show that the schemes are conditionally stable. It has been observed that the operator splitting scheme is computationally more efficient.

Phase field modeling and computation of vesicle growth or shrinkage.

We present a phase field model for vesicle growth or shrinkage induced by an osmotic pressure due to a chemical potential gradient. The model consists of an Allen-Cahn equation describing the evolution of the phase field parameter that describes the shape of the vesicle and a Cahn-Hilliard-type equation describing the evolution of the ionic fluid. We establish conditions for vesicle growth or shrinkage via a common tangent construction using free energy curves. During the membrane deformation, the model ensures total mass conservation of the ionic fluid, and we weakly enforce a surface area constraint of the vesicle. We develop a stable numerical scheme and an efficient nonlinear multigrid solver to evolve the phase and concentration fields, and we use this to evolve the fields to near equilibrium for 2D vesicles. Convergence tests confirm an [Formula: see text] accuracy for our scheme and near-optimal convergence for our multigrid solver. Numerical results reveal that the diffuse interface model captures the main features of cell shape dynamics: for a growing vesicle, there exist circle-like equilibrium shapes if the concentration difference across the membrane and the initial osmotic pressure are large enough; while for a shrinking vesicle, there exists a rich collection of finger-like equilibrium morphologies.

Non-reciprocity induces resonances in a two-field Cahn-Hilliard model.

We consider a non-reciprocally coupled two-field Cahn-Hilliard system that has been shown to allow for oscillatory behaviour and suppression of coarsening. After introducing the model, we first review the linear stability of steady uniform states and show that all instability thresholds are identical to the ones for a corresponding two-species reaction-diffusion system. Next, we consider a specific interaction of linear modes-a 'Hopf-Turing' resonance-and derive the corresponding amplitude equations using a weakly nonlinear approach. We discuss the weakly nonlinear results and finally compare them with fully nonlinear simulations for a specific conserved amended FitzHugh-Nagumo system. We conclude with a discussion of the limitations of the employed weakly nonlinear approach. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

A dual resolution phase-field solver for wetting of viscoelastic droplets.

We present a new and efficient phase-field solver for viscoelastic fluids with moving contact line based on a dual-resolution strategy. The interface between two immiscible fluids is tracked by using the Cahn-Hilliard phase-field model, and the viscoelasticity incorporated into the phase-field framework. The main challenge of this approach is to have enough resolution at the interface to approach the sharp-interface methods. The method presented here addresses this problem by solving the phase field variable on a mesh twice as fine as that used for the velocities, pressure, and polymer-stress constitutive equations. The method is based on second-order finite differences for the discretization of the fully coupled Navier-Stokes, polymeric constitutive, and Cahn-Hilliard equations, and it is implemented in a 2D pencil-like domain decomposition to benefit from existing highly scalable parallel algorithms. An FFT-based solver is used for the Helmholtz and Poisson equations with different global sizes. A splitting method is used to impose the dynamic contact angle boundary conditions in the case of large density and viscosity ratios. The implementation is validated against experimental data and previous numerical studies in 2D and 3D. The results indicate that the dual-resolution approach produces nearly identical results while saving computational time for both Newtonian and viscoelastic flows in 3D.

An Explicit Adaptive Finite Difference Method for the Cahn-Hilliard Equation.

In this study, we propose an explicit adaptive finite difference method (FDM) for the Cahn-Hilliard (CH) equation which describes the process of phase separation. The CH equation has been successfully utilized to model and simulate diverse field applications such as complex interfacial fluid flows and materials science. To numerically solve the CH equation fast and efficiently, we use the FDM and time-adaptive narrow-band domain. For the adaptive grid, we define a narrow-band domain including the interfacial transition layer of the phase field based on an undivided finite difference and solve the numerical scheme on the narrow-band domain. The proposed numerical scheme is based on an alternating direction explicit (ADE) method. To make the scheme conservative, we apply a mass correction algorithm after each temporal iteration step. To demonstrate the superior performance of the proposed adaptive FDM for the CH equation, we present two- and three-dimensional numerical experiments and compare them with those of other previous methods.

Automated matching of two-time X-ray photon correlation maps from phase-separating proteins with Cahn-Hilliard-type simulations using auto-encoder networks.

Machine learning methods are used for an automated classification of experimental two-time X-ray photon correlation maps from an arrested liquid-liquid phase separation of a protein solution. The correlation maps are matched with correlation maps generated with Cahn-Hilliard-type simulations of liquid-liquid phase separations according to two simulation parameters and in the last step interpreted in the framework of the simulation. The matching routine employs an auto-encoder network and a differential evolution based algorithm. The method presented here is a first step towards handling large amounts of dynamic data measured at high-brilliance synchrotron and X-ray free-electron laser sources, facilitating fast comparison with phase field models of phase separation.

T cell therapy against cancer: A predictive diffuse-interface mathematical model informed by pre-clinical studies.

T cell therapy has become a new therapeutic opportunity against solid cancers. Predicting T cell behaviour and efficacy would help therapy optimization and clinical implementation. In this work, we model responsiveness of mouse prostate adenocarcinoma to T cell-based therapies. The mathematical model is based on a Cahn-Hilliard diffuse interface description of the tumour, coupled with Keller-Segel type equations describing immune components dynamics. The model is fed by pre-clinical magnetic resonance imaging data describing anatomical features of prostate adenocarcinoma developed in the context of the Transgenic Adenocarcinoma of the Mouse Prostate model. We perform computational simulations based on the finite element method to describe tumor growth dynamics in relation to local T cells concentrations. We report that when we include in the model the possibility to activate tumor-associated vessels and by that increase the number of T cells within the tumor mass, the model predicts higher therapeutic effects (tumor regression) shortly after therapy administration. The simulated results are found in agreement with reported experimental data. Thus, this diffuse-interface mathematical model well predicts T cell behavior in vivo and represents a proof-of-concept for the role such predictive strategies may play in optimization of immunotherapy against cancer.

Analysis and Optimal Velocity Control of a Stochastic Convective Cahn-Hilliard Equation.

A Cahn-Hilliard equation with stochastic multiplicative noise and a random convection term is considered. The model describes isothermal phase-separation occurring in a moving fluid, and accounts for the randomness appearing at the microscopic level both in the phase-separation itself and in the flow-inducing process. The call for a random component in the convection term stems naturally from applications, as the fluid's stirring procedure is usually caused by mechanical or magnetic devices. Well-posedness of the state system is addressed, and optimisation of a standard tracking type cost with respect to the velocity control is then studied. Existence of optimal controls is proved, and the Gâteaux-Fréchet differentiability of the control-to-state map is shown. Lastly, the corresponding adjoint backward problem is analysed, and the first-order necessary conditions for optimality are derived in terms of a variational inequality involving the intrinsic adjoint variables.

Enhancement of damaged-image prediction through Cahn-Hilliard image inpainting.

We assess the benefit of including an image inpainting filter before passing damaged images into a classification neural network. We employ an appropriately modified Cahn-Hilliard equation as an image inpainting filter which is solved numerically with a finite-volume scheme exhibiting reduced computational cost and the properties of energy stability and boundedness. The benchmark dataset employed is Modified National Institute of Standards and Technology (MNIST) dataset, which consists of binary images of handwritten digits and is a standard dataset to validate image-processing methodologies. We train a neural network based on dense layers with MNIST, and subsequently we contaminate the test set with damages of different types and intensities. We then compare the prediction accuracy of the neural network with and without applying the Cahn-Hilliard filter to the damaged images test. Our results quantify the significant improvement of damaged-image prediction by applying the Cahn-Hilliard filter, which for specific damages can increase up to 50% and is advantageous for low to moderate damage.

Asymptotic Behaviour of Time Stepping Methods for Phase Field Models.

Adaptive time stepping methods for metastable dynamics of the Allen-Cahn and Cahn-Hilliard equations are investigated in the spatially continuous, semi-discrete setting. We analyse the performance of a number of first and second order methods, formally predicting step sizes required to satisfy specified local truncation error σ in the limit of small length scale parameter ϵ → 0 during meta-stable dynamics. The formal predictions are made under stability assumptions that include the preservation of the asymptotic structure of the diffuse interface, a concept we call profile fidelity. In this setting, definite statements about the relative behaviour of time stepping methods can be made. Some methods, including all so-called energy stable methods but also some fully implicit methods, require asymptotically more time steps than others. The formal analysis is confirmed in computational studies. We observe that some provably energy stable methods popular in the literature perform worse than some more standard schemes. We show further that when Backward Euler is applied to meta-stable Allen-Cahn dynamics, the energy decay and profile fidelity properties for these discretizations are preserved for much larger time steps than previous analysis would suggest. The results are established asymptotically for general interfaces, with a rigorous proof for radial interfaces. It is shown analytically and computationally that for most reaction terms, Eyre type time stepping performs asymptotically worse due to loss of profile fidelity.

Parallel splitting solvers for the isogeometric analysis of the Cahn-Hilliard equation.

Modeling tumor growth in biological systems is a challenging problem with important consequences for diagnosis and treatment of various forms of cancer. This growth process requires large simulation complexity due to evolving biological and chemical processes in living tissue and interactions of cellular and vascular constituents in living organisms. Herein, we describe with a phase-field model, namely the Cahn-Hilliard equation the intricate interactions between the tumors and their host tissue. The spatial discretization uses highly-continuous isogeometric elements. For fast simulation of the time-dependent Cahn-Hilliard equation, we employ an alternating directions implicit methodology. Thus, we reduce the original problems to Kronecker products of 1 D matrices that can be factorized in a linear computational cost. The implementation enables parallel multi-core simulations and shows good scalability on shared-memory multi-core machines. Combined with the high accuracy of isogeometric elements, our method shows high efficiency in solving the Cahn-Hilliard equation on tensor-product meshes.

A computational study of lateral phase separation in biological membranes.

Conservative and non-conservative phase-field models are considered for the numerical simulation of lateral phase separation and coarsening in biological membranes. An unfitted finite element method is proposed to allow for a flexible treatment of complex shapes in the absence of an explicit surface parametrization. For a set of biologically relevant shapes and parameter values, the paper compares the dynamic coarsening produced by conservative and non-conservative numerical models, its dependence on certain geometric characteristics and convergence to the final equilibrium.

Mesoscale Computational Modeling of Protein-Membrane Interactions Based on Continuum Mean-Field Theory.

Quantitative computational modeling of protein-membrane interactions is of great importance as it aids in the interpretation of experimental results and enables design and exploration of new experimental systems. This review describes one such computational approach conceived specifically to treat electrostatically driven interactions between a lipid membrane and a protein (or protein domains) adsorbing onto the membrane. The methodology is based on self-consistent minimization of the governing free energy functional which is expressed in the mean-field approximation and has contributions from electrostatic interactions as well as from mixing entropy of lipids in the membrane and ions in the solution. The method enables calculation of the free energy of the binding process and quantification of the steady-state lipid distribution around the adsorbing protein. The extension of the method to include membrane deformation degrees of freedom further allows calculation of the equilibrium bilayer shape upon the protein binding.

A multilevel Monte Carlo finite element method for the stochastic Cahn-Hilliard-Cook equation.

In this paper, we employ the multilevel Monte Carlo finite element method to solve the stochastic Cahn-Hilliard-Cook equation. The Ciarlet-Raviart mixed finite element method is applied to solve the fourth-order equation. In order to estimate the mild solution, we use finite elements for space discretization and the semi-implicit Euler-Maruyama method in time. For the stochastic scheme, we use the multilevel method to decrease the computational cost (compared to the Monte Carlo method). We implement the method to solve three specific numerical examples (both two- and three dimensional) and study the effect of different noise measures.