Surface Diffusion Equation Research Articles

A shallow physics-informed neural network for solving partial differential equations on static and evolving surfaces

In this paper, we introduce a shallow (one-hidden-layer) physics-informed neural network (PINN) for solving partial differential equations on static and evolving surfaces. For the static surface case, with the aid of a level set function, the surface normal and mean curvature used in the surface differential expressions can be computed easily. So, instead of imposing the normal extension constraints used in literature, we write the surface differential operators in the form of traditional Cartesian differential operators and use them in the loss function directly. We demonstrate a series of performance study for the present methodology by solving Laplace–Beltrami equations and surface diffusion equations on complex static surfaces. With just a moderate number of neurons used in the hidden layer, we are able to attain satisfactory prediction results. We then extend the present methodology to solve the advection–diffusion equation on an evolving surface with a given velocity. To track the deforming surface, we additionally introduce a network, in which a prescribed hidden layer is employed to enforce the topological structure of the surface and learn the homeomorphism between the surface and the prescribed topology. The proposed network structure is designed to track the surface and solve the equation simultaneously. Again, the numerical results show comparable accuracy as the static cases. As an application, we simulate surfactant transportation on a droplet surface under shear flow and obtain some physically plausible results.

Phase-field model of bilipid membrane electroporation.

This paper proposes a new model of membrane electropermeabilisation that combines the water content of the membrane and the transmembrane voltage. Interestingly, thanks to a well defined free-energy of the membrane, we somehow generalise the seminal approach of Chizmadzhev, Weaver and Krassowska, getting rid of the geometrical cylindrical assumption upon which most of the current electroporation models are based. Our approach is physically relevant and we recover a surface diffusion equation of the lipid phase proposed by Leguèbe et al. in a previous phenomenological model. We also perform a fine analysis of the involved nonlocal operators in two simple configurations (a spherical membrane and a flat periodic membrane) that enables us to compare the time constants of the phenomenon in spherical and flat membranes. An accurate splitting scheme combined with Fast Fourier Transforms is developed for efficient computations of the model. Our numerical results enable us to make a link between the molecular dynamics simulations of membrane permeabilisation and the experimental observations on vesicles and cells.

A symmetrized parametric finite element method for simulating solid-state dewetting problems

We propose an accurate, area/mass-conservative and energy-dissipative parametric finite element method for solving the sharp-interface continuum model of solid-state dewetting with arbitrary anisotropic surface energy. The model is governed by the anisotropic surface diffusion equation with suitable boundary conditions at the contact points, and could be used to describe the kinetic evolution of the film/vapor interface with contact line migration. The sharp-interface continuum model is firstly reformulated into a conservative form by introducing a symmetric positive definite matrix Zk(n) that depends on the Cahn–Hoffman ξ-vector and a stabilizing function k(n). A new symmetrized variational formulation with arbitrary anisotropic surface energy is built and then its area/mass conservation and energy dissipation properties are proved. By using piecewise linear elements in space, we propose the spatial semi-discretized scheme, and using backward Euler in time with an appropriate approximation of the unit normal vector, we further derive the fully discretized parametric finite element approximation. We show, theoretically, that the semi-discretized and fully discretized schemes are both area/mass-conservative and energy-dissipative. Finally, numerical experiments are reported to show the accuracy and efficiency of the numerical method as well as the anisotropic effects on the morphological evolution processes for solid-state dewetting of thin films.

Global dynamics of solutions for a sixth-order parabolic equation describing continuum evolution of film-free surface

This paper is concerned with a sixth-order diffusion equation, which describes continuum evolution of film-free surface. By using the regularity estimates for the semigroups, iteration technique and the classical existence theorem of global attractors we verified the existence of global attractor for this surface diffusion equation in the spaces H3(Ω) and fractional-order spaces Hk(Ω), where 0 ≤ k < ∞.

On a semi-implicit numerical method for the surface diffusion equation for triangulated surfaces

On a semi-implicit numerical method for the surface diffusion equation for triangulated surfaces

Theoretical description of molecular permeation via surface diffusion through graphene nanopores.

We establish a theoretical model to describe the surface molecular permeation through two-dimensional graphene nanopores based on the surface diffusion equation and Fick's law. The model is established by considering molecular adsorption and desorption from the surface adsorption layer and the molecular diffusion and concentration gradient on the graphene surface. By comparing with the surface flux obtained from molecular dynamics simulations, it is shown that the model can predict well the overall permeation flux especially for strongly adsorbed molecules (i.e. CO2 and H2S) on graphene surfaces. Although good agreement between the theoretical and simulated density distribution is hard to achieve owing to the large uncertainty in the calculation of surface diffusion coefficients based on the Einstein equation, the model itself is very competent to describe the surface molecular permeation both from the aspects of the overall permeation flux and detailed density distribution. This model is believed to supplement the theoretical description of molecular permeation through graphene nanopores and provide a good reference for the description of mass transport through two-dimensional porous materials.

Self-similar grooving solutions to the Mullins’ equation

In 1957, Mullins proposed surface diffusion motion as a model for thermal grooving. By adopting a small slope approximation, he reduced the model to the Mullins’ linear surface diffusion equation, ( M E ) y t + B y x x x x = 0 , \begin{equation} \nonumber ({\mathrm {ME}})\quad \quad y_t + B y_{xxxx}=0, \end{equation} known also more simply as the Mullins’ equation. Mullins sought self-similar solutions to (ME) for planar initial conditions, prescribing boundary conditions at the thermal groove, as well as far field decay. He found explicit series solutions which are routinely used in analyzing thermal grooving to this day. While (ME) and the small slope approximation are physically reasonable, Mullins’ choice of boundary conditions is not always appropriate. Here we present an in depth study of self-similar solutions to the Mullins’ equation for general self-similar boundary conditions, explicitly identifying four linearly independent solutions defined on R ∖ { 0 } \mathbb {R}\setminus \{0\} ; among these four solutions, two exhibit unbounded growth and two exhibit asymptotic decay, far from the origin. We indicate how the full set of solutions can be used in analyzing the effective boundary conditions from experimental profiles and in evaluating the governing physical parameters.

The Surface Diffusion Flow with Elasticity in Three Dimensions

We establish the short-time existence of a smooth solution to the surface diffusion equation with an elastic term and without an additional curvature regularization in three space dimensions. We also prove the asymptotic stability of strictly stable stationary sets.

Solution for 4th-order nonlinear axisymmetric surface diffusion by inverse method

We present a method for constructing similarity solutions of a fourth-order nonlinear partial differential equation for axisymmetric surface diffusion by extending an inverse method previously used for the second-order one-dimensional nonlinear diffusion equation. After imposing a solution profile, both a feasible surface tension, and a compatible mobility function are deduced simultaneously. Although the profile is not one-to-one, an optimization algorithm is implemented to construct a mobility function that is a function of surface orientation, with no practical difference in mobility between different arms of the many-to-one profile. It is shown that the solution of the linear model well approximates the solution of the nonlinear model, in which the surface tension and mobility are close to constant for a wide range of surface angles, even when nonlinear geometric terms are included.

Asymmetry in crystal facet dynamics of homoepitaxy by a continuum model

In the absence of external material deposition, crystal surfaces usually relax to become flat by decreasing their free energy. We study analytically an asymmetry in the relaxation of macroscopic plateaus, facets, of a periodic surface corrugation in 1+1 dimensions via a continuum model below the roughening transition temperature. The model invokes a continuum evolution law expressed by a highly degenerate parabolic partial differential equation (PDE) for surface diffusion, which is related to the nonlinear gradient flow of a convex, singular surface free energy with a certain exponential mobility in homoepitaxy. This evolution law is motivated both by an atomistic broken-bond model and a mesoscale model for crystal steps. By constructing an explicit solution to this PDE, we demonstrate the lack of symmetry in the evolution of top and bottom facets in periodic surface profiles. Our explicit, analytical solution is compared to numerical simulations of the continuum law via a regularized surface free energy.

Stability of axisymmetric CMC surfaces as steady states for the evolution by surface diffusion

The surface diffusion equation is one of the geometric evolution laws and was first derived by Mullins [6] to model the development of surface grooves at the grain boundaries of a heated polycrystal. It is known that the surface diffusion equation is obtained as the H-1-gradient flow of the area functional for the evolving surfaces. Owing to this variational structure, we expect that the constant mean curvature surfaces (CMC surfaces) are steady states for this equation. In this paper, the criteria of the stability of the axisymmetric CMC surfaces will be studied with the help of the linearized stability analysis of the surface diffusion equation.

The Surface Diffusion Flow with Elasticity in the Plane

In this paper we prove short-time existence of a smooth solution in the plane to the surface diffusion equation with an elastic term and without an additional curvature regularization. We also prove the asymptotic stability of strictly stable stationary sets.

Existence and Uniqueness Results for a Kinetic Model in Bulk-Surface Surfactant Dynamics

In this paper we study a new multidimensional mixed-kinetic adsorption model which consists of a nonlinear evolution system of two parabolic partial differential equations: a convective diffusion equation for the bulk surfactant concentration in a bounded domain and a surface diffusion equation for its surface concentration on a compact Lipschitz manifold. The two equations are coupled with a nonlinear relationship, consistent with the Langmuir--Hinshelwood model, which describes the adsorption-desorption transport of surfactant molecules between the bulk phase and one part of its boundary. We provide results on the unique weak solvability of the system and non-negativity of its solution. We use the truncation method combined with the fixed point approach and the theory of time-dependent partial differential equations on manifolds.

Calculation of surface diffusivity and residence time by molecular dynamics with application to nanoscale selective-area growth

The surface diffusivity and residence time were calculated by molecular dynamics simulations in order to solve the surface diffusion equations for selective-area growth. The calculations for CdTe/CdS material system were performed in substrates with Cd termination and S termination. The surface diffusivity and residence time were obtained at different temperatures (600K, 800K, 1000K, 1200K, and 1400K). The thermal activation energies were extracted from Arrhenius equation for each substrate termination. Thereafter, values obtained by molecular dynamics were used in a surface diffusion model to calculate the surface concentration profile of adatoms. Alternating the surface termination has the potential to achieve nanoscale selective-area growth without the need of a dielectric film as a mask.

SOME PROPERTIES OF SOLUTIONS FOR A SIXTH-ORDER PARABOLIC EQUATION IN ONE SPATIAL DIMENSION

In this paper, we consider the existence and uniqueness of global weak solution for a sixth-order classical surface-diffusion equation in one spatial dimension. Moreover, the regularity and blow-up of solutions are also studied.

Initial stage of Free Pressureless Spark-Plasma Sintering of vanadium carbide: Determination of surface diffusion parameters

Inter-particle neck growth kinetics of vanadium carbide powder during initial stages of conventional and spark-plasma sintering has been investigated. Vanadium carbide (V8C7) micron-size powder has been subjected to both Free Pressureless Spark Plasma Sintering (FPSPS) and conventional sintering dilatometry. While the latter resulted not to be effective in the consolidation process, FPSPS procedure proved to be successful in terms of the initial stages of densification and inter-particle neck evolution, even without applying an external load. FPSPS was conducted with and without holding time, in order to obtain experimental data on the inter-particle contact growth for different temperature regimes. The neck sizes derived from scanning electron microscopy observations were utilized in the inverse regression of the surface diffusion equation and led to the calculation of the value of the surface diffusion activation energy for the vanadium carbide powder.

Comparative Study of Membrane Humidifier and Enthalpy Wheel Humidifier for Large Power Fuel Cell System

The performance of membrane humidifier (MH) and enthalpy wheel humidifier (EWH) for a large power proton exchange membrane fuel cell (PEMFC) system is compared using simulations and experiments. The MH model is based on one dimensional diffusion equation and the EWH model is based on the porous media surface diffusion equation. Simulation results agree well with experimental data. According to the results, the effect of intake air temperature has a lower effect on the MH performance than it does on the EWH performance while the air mass flow has a much higher effect on the MH performance than it does on the EWH performance. MH performs better than EWH at a low flow rate but worse at a high flow rate. Vapor transfer mechanism in the humidifiers is also studied.

Plasma-controlled metal catalyst saturation and the initial stage of carbon nanostructure array growth

The kinetics of the nucleation and growth of carbon nanotube and nanocone arrays on Ni catalyst nanoparticles on a silicon surface exposed to a low-temperature plasma are investigated numerically, using a complex model that includes surface diffusion and ion motion equations. It is found that the degree of ionization of the carbon flux strongly affects the kinetics of nanotube and nanocone nucleation on partially saturated catalyst patterns. The use of highly ionized carbon flux allows formation of a nanotube array with a very narrow height distribution of half-width 7 nm. Similar results are obtained for carbon nanocone arrays, with an even narrower height distribution, using a highly ionized carbon flux. As the deposition time increases, nanostructure arrays develop without widening the height distribution when the flux ionization degree is high, in contrast to the fairly broad nanostructure height distributions obtained when the degree of ionization is low.

Curvature driven diffusion, Rayleigh-Plateau instability, and Gregory-Laflamme instability

It can be expected that the respective endpoints of the Gregory-Laflamme black brane instability and the Rayleigh-Plateau membrane instability are related because the bifurcation diagrams of the black hole--black string system and the liquid drop--liquid bridge system display many similarities. In this paper, I investigate the nonlinear dynamics of the Rayleigh-Plateau instability in a range of dimensions, including the critical dimension at which the phase structure changes. I show that near the critical dimension and above, depending on a parameter in initial conditions an unstable cylinder will either pinch off or converge to an equilibrium state. The equilibrium state is apparently nonuniform but has a constant mean curvature everywhere. The results suggest that in the gravity side, near the critical dimension and above, the final state of an unstable black string (which is not too long) is a nonuniform black string. The equation of motion adopted to describe the dynamics is the surface diffusion equation, which was originally proposed to describe a grooving process of heated metal surfaces. An interesting correspondence between the diffusion dynamics and black hole (thermo)dynamics is discussed.

A new stable finite volume method for predicting thermal performance of a whole building

Discretised governing equations involving only temperatures and heat fluxes at both surfaces of a solid wall layer were obtained by combining a new stable finite volume scheme for the two inner nodes of the wall layer with the surface diffusion equations (discretised by third order equations). The finite volume scheme for the inner nodes of the layer is proved to be stable with its truncation error being O ( Δ x 4 , Δ x 2 Δ t 2 ) . A special analytical solution for a solid wall was used to evaluate different schemes for the inner nodes, showing that the new proposed scheme performs better than all other schemes for time steps of 3600 and 600 s. Finally, this scheme was used to simulate a whole house and the predicted zonal air temperature, and surface temperatures agreed well with measured values.