Crystal Growth Simulator

The core of the simulation is the random walk of each particle, which models diffusion. The particle's position is updated step-by-step with no preferred direction.

Here, $\vec{r}$ is the particle's position, $\delta$ is the step length (often 1 pixel), and $\theta$ is a random angle chosen uniformly between $0$ and $2\pi$ at each step. This creates the "drunken walk" you see on screen.

The fractal dimension $D_f$ is estimated using the radius-of-gyration method. As the cluster grows, we measure how its mass $M$ scales with its characteristic size $R_g$.

Here, $M$ is the number of particles in the cluster, and $R_g$ is the root-mean-square distance of particles from the cluster's center of mass. A plot of $\log(M)$ vs $\log(R_g)$ gives a slope of $D_f$. This is the number the simulator estimates and displays.

Electrodeposition & Battery Technology: When charging a lithium-ion battery, lithium metal can deposit on the electrodes in dendritic, DLA-like structures. These branches can short-circuit the battery, making DLA simulations crucial for understanding and preventing this dangerous failure mode.

Mineral & Snowflake Formation: The intricate patterns of ice crystals growing from water vapor, or minerals precipitating from a solution, are often governed by diffusion-limited processes. The simulator's branchy shapes mimic how real snowflakes and certain mineral deposits form in nature.

Porous Material & Filter Design: The structure of materials like activated carbon filters or bone tissue can resemble DLA clusters. Simulating their growth helps engineers design materials with optimal surface area and porosity for filtration or medical implants.

Dielectric Breakdown & Lightning: The path of a lightning bolt or an electrical spark through an insulating material follows a fractal pattern similar to DLA. The model helps physicists study how these unpredictable, branching discharges propagate.

When you start using this simulator, there are a few points that are easy to misunderstand, so let's cover them first. You might tend to think that increasing the number of particles makes the simulation closer to a real crystal, but it's not that simple. While more particles do create a finer structure, if the lattice size for calculation is insufficient, the cluster will hit the screen edges and its growth will become distorted. For instance, generating 100,000 particles on a 1000x1000 lattice will make edge effects non-negligible. The trick is to first increase the lattice size and then increase the particle count.

Next, the interpretation of the "Sticking Probability" parameter. While lowering it is described as "increasing bounces," remember this is an abstracted model that "probabilistically decides whether a particle sticks." In real-world phenomena, like electrodeposition, this corresponds to factors like the energy barrier at the interface or the influence of additives. So, understand that setting it to "0.1" doesn't mean the real-world probability is 10%; it's a parameter for observing relative trends.

Finally, how to read the fractal dimension value. You might see the value fluctuate between 1.5 and 1.8 during simulation and wonder, "Is the calculation wrong?" This is normal. In the early growth stages where few data points (cluster radius and mass) are used for estimation, the value is unstable. It converges toward the theoretical value of approximately 1.71 as the particle count increases. Therefore, refer to the value after growth has progressed somewhat (e.g., after the particle count exceeds 5000).