What are real examples where fractals were used in Matter Modeling?

Question

A fractal is, accordingly Oxford English Dictionary:

A curve or geometric figure, each part of which has the same statistical character as the whole. Fractals are useful in modeling structures (such as eroded coastlines or snowflakes) in which similar patterns recur at progressively smaller scales, and in describing partly random or chaotic phenomena such as crystal growth, fluid turbulence, and galaxy formation.

One of the must famous fractals is the Mandelbrot set:

My question is: Are there real examples where fractals were used to solve problems in Matter Modeling?

Answer 1

The review that @Anyon cited focuses on the use of fractal geometry to classify and model the structure of disordered materials, e.g. structures synthesized by the sol-gel method.

The computational work is nicely summarized in the following figure taken from the paper:

Here "Reaction-Limited", "Ballistic" and "Diffusion-Limited" correspond to 3 different types of simulations, and each model specifies different growth kinetics. In these simulations, particles are moving in random walks and growth occurs when certain conditions are met.

"Monomer-Cluster" simulations start with a seed at a particular site and the growth events happen when a monomer lands on a site neighboring the seed, increasing the seed size. On the other hand, in "Cluster-Cluster" simulations, the seeds are allowed to move around and interact with each other, resulting in extended structures.

The D values on the bottom-left corner of each simulation correspond to the fractal dimension. In a 3D embedding space, this dimension relates the mass ($M$) of an object to its size ($R$) \begin{equation} M \sim R^{D}. \end{equation}

In a 2D embedding space, the surface area ($S$) and size of an object are related through the surface fractal dimension ($D_S$) \begin{equation} S \sim R^{D_S}. \end{equation}

Moreover, when scattering techniques are used on fractal objects, the intensity ($I$) of the incidental beam is related to the wave vector ($K$), through: \begin{equation} I \sim K^{-2D+D_S}. \end{equation}

These relationships, along with the simulations mentioned previously, allow to classify structures and identify the growth factors in kinetic growth systems.

Answer 2

It has been recognized that fractal features are exploited in numerous natural materials. The properties conferred by fractal geometry (for instance, efficient utilization of material and filling of space) have in addition inspired some material modelers to incorporate the scale-invariant properties of fractals into their designs. An understanding of fractal principles can therefore be implemented by material modelers in two ways: a priori, to design features into materials conferred by the fractal nature; or a posteriori, to understand or describe the properties of materials.

I post just two examples found quickly with an online search for the keywords "fractal material properties simulation".

The first example [Ref. 1] rather stunningly exploits the scale-free geometric property of fractals to design a stretchable electrode through appropriate fractal slicing of the starting polymeric material.

A second example [Ref. 2] uses fractal principles to describe the hierarchical structure of palm fibers, and uses a model derived from this description to relate material properties to structure at different scales.

References

1. Yigil Cho, Joong-Ho Shin, Avelino Costa, Tae Ann Kim, Valentin Kunin, Ju Li, Su Yeon Lee, Shu Yang, Heung Nam Han, In-Suk Choi, and David J. Srolovitz. Engineering the shape and structure of materials by fractal cut. PNAS December 9, 2014 111 (49) 17390-17395; first published November 24, 2014 https://doi.org/10.1073/pnas.1417276111

2. Wang, Y., Zhang, T., Jing, L., Deng, P., Zhao, S., and Guan, D. (2020). "Exploring natural palm fiber’s mechanical performance using multi-scale fractal structure simulation," BioRes. 15(3), 5787-5800.