Simulation of Particle Dissolution Using the Phase Field Approach

Simulation of Particle Dissolution using the Phase Field Approach

D. Sleziona¹, D. R. Ely², M. Thommes¹

¹Technical University Dortmund, Emil-Figge-Str. 68, 44227 Dortmund, Germany

²Ivy Tech Community College, 3101 S Creasy Ln, Lafayette IN 47905, USA

PURPOSE

The aim of this work is to describe the dissolution behavior of single crystals with a numerical model. Already existing analytical approaches for the description of crystal dissolution like Fick’s law of diffusion¹, the Nernst-Brunner equation^2,3, the Hixson-Crowell cubic root law⁴ and the Higuchi⁵ equation are suitable. However they do not take into account the anisotropy or the particle shape. The motivation of this work is to model the anisotropic dissolution behaviour of single crystal particles by using a finite volume method. Dissolution is understood as the transformation of a solid (crystalline state) in a liquid environment in a solution. This process is divided into two parts. The phase transition and diffusion. The first part can be modelled numerically with the so called phase field simulation⁶. In this numerical tool a partial differential equation is used to replace boundary conditions at the interface⁶. This leads to an evolving ancillary field. In this field the state of the materials is described by a non-conserved order parameter, which is a function of position and time⁶. It models the phase as a continuous variable ranging from disordered (liquid) to perfectly ordered (solid)^6,7. Also the second part, diffusion, can be calculated with a finite volume method. This approach offers the advantage that no diffusion layer has to be implemented. It is formed independently during the simulation.

RESULTS

An isotropic, circular, two-dimensional Xylitol particle dissolving in water (25 °C) was simulated as a model system. The time evolution of the phase (phase transition) of the binary system could be described by the Allen-Cahn-equation^6,8, as follows:

Equation 1 gives the rate change of the phase (∂Φ/∂t) in terms of the mobility of the phase (M_Φ), the free energy density (f) and the gradient energy coefficient of the phase (ε_Φ). The corresponding concentration field (diffusion) was modelled using the approach of Cahn and Hilliard^8,9 given by the equation,

As seen in equation 2, the rate change of the concentration field (∂c/∂t) is expressed as a function of the concentration mobility (M_c), the free energy density (f) and the gradient energy coefficient of the concentration (ε_c). Equations 1 and 2 form a system of two coupled partial differential equations linked via the free energy density f(Φ,c,T), which is a function of phase, concentration, and temperature. The system was solved numerically to obtain the phase transition and diffusion (concentration field) as a function of time as shown in Figure 1.

Figure 1: dissolution of a 100 µm xylitol particle in water (25 °C)

A simultaneously evolving phase and concentration field could be observed. Because of the isotropic properties of the simulated, two dimensional particle a validation of the numerical results based on the previous mentioned Nernst-Brunner equation is possible. Due to the fact that the Nernst-Brunner law describes the dissolution behaviour of a three dimensional, isotropic spherical particle, a new validation equation for the circular two dimensional, isotropic, Xylitol crystal was derived.

This equation 3 represents the dissolution process of an infinite long, three dimensional cylinder which dissolves only over its lateral surface. There the particle radius (r) is a function of the starting radius (r₀), diffusion coefficient (D), saturation concentration (c_S), concentration (c), diffusion layer thickness (d ), particle density (ρ ) and time. Geometrically this reflects the two dimensional dissolution of the simulated circular particle. Figure 2 shows the course of the simulated remaining particle radius over the simulation time, which follows continuously the derived validation equation and ends up in a similar total dissolution time.

Figure 2: particle radius of a 100 µm, circular, isotropic, two dimensional xylitol particle in water (25 °C)

So a successful coupling of the Allen-Cahn and Cahn-Hilliard equation can be presented, which leads to a simulation where the change in phase transition and diffusion can be observed over time.

CONCLUSION

A finite volume simulation was developed for the investigation of the dissolution behaviour of crystal particles. Phase transition and diffusion could be coupled by linking the Allen-Cahn- and Cahn-Hilliard-equation over the free energy density function f(Φ,c,T). The dissolution of a circular, isotropic, two dimensional Xylitol particle in water at 25°C was simulated and validated with a derived two dimensional Nernst-Brunner equation. The extension of this simulation into the third dimension and incorporation of anisotropy into this model is part of the ongoing work.

REFERENCES

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