# (720e) Simulation of Particle Dissolution Using the Phase Field Approach

- Conference: AIChE Annual Meeting
- Year: 2017
- Proceeding: 2017 Annual Meeting
- Group: Pharmaceutical Discovery, Development and Manufacturing Forum
- Session:
- Time: Thursday, November 2, 2017 - 1:58pm-2:20pm

Simulation of Particle Dissolution using the Phase

Field Approach

D. Sleziona^{1}, D. R. Ely^{2}, M. Thommes^{1}

^{}

^{1}Technical University Dortmund,

Emil-Figge-Str. 68, 44227 Dortmund, Germany

^{2}Ivy 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 diffusion^{1}, the Nernst-Brunner equation^{2,3}, the Hixson-Crowell cubic root law^{4} and

the Higuchi^{5} 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

simulation^{6}. In this numerical tool a partial differential equation

is used to replace boundary conditions at the interface^{6}. 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 time^{6}. 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*). The

_{Φ}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

(*r _{0}*), 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**

1. Fick, A. (1855) Ueber Diffusion, Annalen der Physik

170, 59-86.

2.

Nernst, W. (1904) Theory of

reaction velocity in heterogenous systems, Zeit. physikal. Chem 47, 52-55.

3.

Brunner, E.

(1903) Reaktionsgeschwindigkeit in heterogenen Systemen,

Georg-Augusts-Universitat, Gottingen.

4.

Hixson, A. and Crowell, J. (1931) Dependence of Reaction

Velocity upon surface and Agitation, Industrial & Engineering Chemistry 23,

923-931.

5.

Higuchi, T. (1961) Rate of release of medicaments from

ointment bases containing drugs in suspension, Journal of Pharmaceutical

Sciences 50, 874-875.

6.

Boettinger, W. J., Warren, J. A., Beckermann, C.,Karma, A.

(2002) Phase-field simulation of solidification, Annual review of materials

research 32, 163-194.

7.

Cui, Z., Gao, F., Cui, Z.,Qu, J. (2011) Developing a second

nearest-neighbor modified embedded atom method interatomic potential for

lithium, Modelling and simulation in materials science and engineering 20,

015014.

8.

Allen, S. M. and Cahn, J. W. (1979) A microscopic theory for

antiphase boundary motion and its application to antiphase domain coarsening,

Acta Metallurgica 27, 1085-1095.

9.

Cahn, J. W. and Hilliard, J. E. (1958) Free energy of a

nonuniform system. I. Interfacial free energy, The Journal of chemical physics

28, 258-267.