# Discrete model of diffusion-limited dissolution (DLD)

One of the most well-known and widely applied models for pattern-forming physical processes is the diffusion-limited aggregation (DLA) model (Witten and Sander 1981). The myriad growth processes it models share the same underlying mathematics, i.e. the growth rate at the interface is determined by the gradient of some field $u$, e.g. concentration for colloidal aggregation, temperature for solidification, and pressure for viscous fingering in Hele-Shaw flow. The field satisfies the diffusion equation, which in steady state becomes the Laplace equation, $\nabla^2 u =0$. A diffusion-limited aggregation cluster in 2D. Color indicates the time when a particle is attached to the cluster. Manganese dendrites in rocks. The dendritic form of the pattern can be well captured by the DLA model. Image credit: Rossman, Cal Tech.

This allows us to borrow techniques known in electrostatics for analysis. In electrostatics, the gradient of the electric potential increases with curvature. Analogously,  protruding parts of a DLA cluster have higher probability of aggregation; the instability leads to branching, creating complex fractal structures. In the presence of surface tension, e.g. in Hele-Shaw flow, the surface tension at the interface makes the resulting structures appear more rounded.

Here we explore a diffusion-limited dissolution (DLD) model (Krug and Meakin 1991, Meakin and Deutch 1986) by flipping the sign of growth. This provides a simple model for dissolution, erosion, or melting. DLD has not been investigated as thoroughly, since its stable dynamics does not lead to fractals. However, there remains a trove of interesting phenomena and unanswered questions to explore.

# DLD Model Description

### Basic DLD steps: A circular initial cluster. A random walker that can annihilate a particle on the cluster does random walk until it comes into contact with the cluster.

• Create initial cluster on lattice;
• introduce off-lattice particle from far away, one at a time, and let it diffuse freely;
• random walker reaches the cluster, annihilates contacted particle;
• repeat until only one particle left.

### Conformal Mapping and First Passage:

• Remapping far-away walker: we use conformal map from unit circle to straight line $f(z) = \frac{az + b}{cz +d},~ad\ne bc$ and a first-passage problem calculation, to remap far-away walkers back to bounding circle.
• Precise contact probability (see schematic below):
1. Use conformal map from unit circle to an eclipsed circle around the random walker. The shape is constructed by drawing an arc with radius equal the distance to the second nearest neighbor, the use an arc of the nearest neighbor to complete the boundary. The conformal map is $f(z) = \frac{A(z-1)^\alpha + B(z+1)^\alpha}{C(z-1)^\alpha + D(z+1)^\alpha}$ where $A=-B = r\sin\theta (e^{i \theta}+1)$, $C= e^{i\theta}+1$, $D = e^{-i\theta}+1$, $\alpha = \frac{\pi - \theta -\phi}{\pi}$.
2. For the first-passage PDF, solve Laplace equation with point charge and absorbing boundary at the real axis. Using image method, the PDF is obtained: $P(x) = \frac{y_0}{\pi(x^2+y_0^2)}$. Using a simple circle around the random walker leads to a point contact with the nearest neighbor. The contact probability is infinitesimal. Here, contact probability is finite because it is integrated over the arc on the nearest neighbor. However, if random walker goes to the larger arc, more steps are required to reach the cluster.

# Simulations

Some example simulations are shown here.

# Ongoing work

Rycroft and Bazant (my PhD advisor, and my advisor’s PhD advisor, respectively) developed a continuum model of advection-diffusion-limited dissolution (Rycroft and Bazant 2016). The evolution of the shape of an object in a flow is represented by time-dependent conformal map. By simplifying the problem using asymptotic expansion in small Péclet number regime, the boundary can be tracked by a system of ODEs. One of the goals of our current research is to connect the continuum model (without flow) to the discrete model. In particular, the discrete model can help establish statistical properties of the dissolving front and final collapse point when the system is stochastic.

In our model, toggling between dissolution and aggregation is easy, hence the model is well-suited to study Diffusion-limited aggregation, as well as a hybrid dissolution and aggregation model. Example of a cluster produced by Diffusion-limited aggregation and dissolution. Notice that some branches of the cluster are disconnected.

# References

J. Krug and P. Meakin. Phys. Rev. Lett., 66:703–706, Feb 1991.

P. Meakin and J. M. Deutch. J. Chem. Phys., 85(4):2320–2325, 1986.

C. H. Rycroft and M. Z. Bazant. Proc. R. Soc. A, 472(2185), 2016.

T. A. Witten and L. M. Sander. Phys. Rev. Lett., 47:1400–1403, Nov 1981. ## Y Luna Lin View All →

I’m a grad student studying Applied Mathematics at SEAS, Harvard University. My interest lies in using mathematical models and computation to explore problems and phenomena in the natural world. Together with my Ph.D. advisor, Prof. Chris Rycroft and my collaborators, we explore topics such as numerical methods for fluid-solid interaction problems, simulations of diffusion-limited dissolutions, modeling bacteria growth and pattern formation in biofilm.

When I’m not doing math or coding, I enjoy being outdoors and playing music with Tobi.