The development of a methodology for the simulation of structure forming processes is highly desirable. The smoothed particle hydrodynamics (SPH) approach provides a respective framework for modeling the self-structuring of complex geometries. In this paper, we describe a diffusion-controlled phase separation process based on the Cahn-Hilliard approach using the SPH method. As a novelty for SPH method, we derive an approximation for a fourth-order derivative and validate it. Since boundary conditions strongly affect the solution of the phase separation model, we apply boundary conditions at free surfaces and solid walls. The results are in good agreement with the universal power law of coarsening and physical theory. It is possible to combine the presented model with existing SPH models.
Phase inversion and phase separation play an important role in the preparation of precipitation membranes. Dozens of experiments and simulations have been performed to extend empirical correlations and the understanding of the major aspects of these processes.
Phase separation models based on the Cahn-Hilliard equation [
After the fundamental work of Cahn and Hilliard, several researchers [
For the SPH approach, two models for phase separation are developed. Okuzono [
Thieulot et al. [
In this paper, we develop a phase separation model that describes the diffusion-controlled stage using the Cahn-Hilliard equation. It can easily be extended to describe all three stages of coarsening by combination with existing SPH models. As a prerequisite, we introduce an approach to calculate the fourth-order derivative in SPH.
Most models use periodic boundary conditions. But the effects of boundary conditions on the solution of the Cahn-Hilliard equation are not negligible. Therefore, we discuss different kinds of boundary conditions. Since an extension to multicomponent mixtures is straightforward, we consider only a binary fluid mixture in diffusion-controlled stage. Analog to [
In Section
The model describes phase separation on a mesoscale. Therefore, we make one fundamental assumption. We assume completed nucleation on microscopic scale for the description with a continuum method. The initial state of the system will contain uniformly distributed nuclei in space.
First, we review a phase separation model based on the Cahn-Hilliard equation. Next, we introduce the Smoothed Particle Hydrodynamics method (SPH) and an approximation for the fourth-order derivative. Then, we apply the SPH method to the phase separation model. For simplicity, we neglect viscous and inertia effects (i.e., the momentum balance). That is valid in case of equal molar volume, density, and negligible excess values. Finally, since boundary conditions have an important influence on the solution, we focus on free surfaces and solid walls.
Phase decomposition is the result of phase changes as it occurs in, for example, immersion precipitation, condensation, and evaporation. A change of temperature or concentration of a mixture can cause phase decomposition. After exceeding the binodal of the mixture, a fluid mixture switches from stable (miscible) to metastable or unstable (immiscible) state, respectively. In an immiscible system, nucleation occurs spontaneously after exceeding the spinodal. The nuclei grow or disappear depending on their chemical potentials. After infinite time, the system reaches its equilibrium state with homogeneous chemical potential in space.
A phase separation model describes the dynamics of an immiscible fluid mixture based on fundamental thermodynamics. The thermodynamics describes the total free energy of the system. It is the same energy on microscopic, mesoscopic, and macroscopic scale.
In the late 1950s, Cahn and Hilliard [
The free energy results from Legendre transformation of the internal energy
Two parts represent the energy
The Smoothed Particle Hydrodynamics method (SPH) is a Lagrangian, particle-based, and meshfree simulation method. Originally developed for astrophysical problems [
In SPH, one calculates a quantity
In discrete formulation of (
For example, we could calculate the density with (
Brookshaw [
Due to the sensitivity of the second-order and higher order derivatives of the kernel function to particle disorder, it is not possible to do stable simulations using the fourth-order derivative of the kernel function. Therefore, we approximate the fourth-order derivative of a quantity
We determine the error of (
Even though the approximation, (
We consider a binary, isothermal, incompressible, and equimolar fluid mixture. If we neglect viscous and inertia effects, the governing equations are
Equation (
In a large system, when boundary effects are negligible, it is valid to use periodic boundary conditions to describe phase separation. But there remain systems where boundaries have an effect on the phase separation process.
Free surfaces influence the phase separation of a fluid mixture. There is not a contribution of the boundary to the free energy of the fluid mixture, because of the negligible phase.
A contact angle between two fluids and a solid phase results from solid wall boundary conditions. This was observed in experiments. Now, we consider both kinds of boundaries.
In SPH, we use imaginary particles to model free surfaces by mirroring the particles normal to the boundary. This is important, because of particle deficiency to conserve mass.
Near the solid wall, the influence of the wall free energy on the free energy of the fluid mixture is not negligible. We take this into account with an extension of the free energy:
First, we validate the previously developed model by comparing the time-dependent interfacial area of the system with the universal power law for diffusion-controlled phase separation. Then, we present results of the phase separation with free surface and solid wall boundary conditions and discuss these results.
In the early 1960s, Lifshitz and Slyozov found a universal power law for diffusion-controlled coarsening [
We validate our model using a binary mixture (component 1 as
Figure
Time series of the fluid mixture at
The fluid mixture decomposes into two phases. At
Next, we track the interface size
Figure
(a) Interface size
Simulation results and the power law, within resolution error, are in good agreement with regard to the gradient. At times
The effect of particle disorder is investigated by repeating the previous simulation with particles randomly displaced in space by
In the previous simulation, we used periodic boundary conditions. Now, we investigate the effect of free surfaces and solid walls on phase separation.
First, we investigate free surface boundary conditions. The simulation parameters are the previous ones except for the computational domain. In this and the following simulations, we choose the particle number equal to 50 × 50 to reduce the computation time. We only use free surface boundary conditions at all boundaries. As described in Section
Consider the equilibrium of this system. The system is in equilibrium when the interface size between both phases is minimal. The shape of the equilibrium depends on the initial composition of the fluid mixture. There exist two possible shapes of the equilibrium. One shape is, in two-dimensional case, a circle of one phase. The other shape is a layering of both phases. Now, we analyze the initial composition of the fluid mixture for both shapes.
Consider a circle inside of a square. If layering of both phases is the equilibrium shape, the circumference
Figures
Time series of composition of the fluid mixture at
For quantitative comparison of these results, we also calculated the same system using finite differences on a regular grid with 2500 grid points. Figures
Next, we qualitatively investigate solid wall boundary conditions. As described in Section
The computational domain and the simulation parameters, except for
Equilibrium state of composition of the fluid mixture with various contact angles. Initial state (a),
We vary the specified contact angle between
As a last point, we qualitatively consider phase separation of a binary fluid mixture in a box. The computational domain and parameters are the previous ones. We apply solid wall boundary conditions to all boundaries. The contact angle is
Figure
Time series of composition of the fluid mixture at
We developed a SPH approach of a phase separation model. The model describes diffusion-controlled, equimolar coarsening dynamics based on the Cahn-Hilliard equation. The model is extensible to multicomponent systems and nonequimolar systems, as shown in the literature. Since a diffusion equation can be combined with existing SPH models, see, for example, [
We calculated the chemical potential of a fluid mixture from free energy and its second derivative. The gradient of the chemical potential is proportional to the diffusion flux; see transport equation (
The novelty of the present SPH model is a direct Cahn-Hilliard approach using a fourth-order derivative of a quantity
We compared the time-dependent interface size of the computational domain with the universal power law for diffusion-controlled phase separation, and we found good agreement with the power law.
In the last part, we investigated free surface and solid wall boundary conditions. We validated the boundary condition with the equilibrium state of the system. For solid wall boundary conditions, we introduced a phenomenological approach. This approach is based on a wall free energy correction term. We applied the solid wall boundary conditions to model static contact angles. Finally, we applied solid wall boundary conditions to present model.
The extension to three or more component systems is straightforward, as seen in the literature [
The authors declare that there is no conflict of interests regarding the publication of this paper.
The work is supported by the German Science Foundation (DFG) SFB 716 project and within the funding program Open Access Publishing.