Making Materials Modelled Under Murray’s Law

Murray’s law is quite a simple statement relating the radii of the many branching stems that come from a single stem:

r_{p}^{3}=r_{{d_{1}}}^{3}+r_{{d_{2}}}^{3}+r_{{d_{3}}}^{3}+...+r_{{d_{n}}}^{3}
Thank you to Wikipedia for the image of this equation

It is often associated as being true for the splitting of veins, arteries and capillaries in the circulatory system or perhaps the tree-like nature of alveoli in the respiratory system. As well as holding true for animal physiology it also exists in plants such as the water transport system, the xylem. In general any biological system where diffusion is required needs an optimising of the surface area to volume ratio, and when there is a rank of important tissue to supply the splitting structure is very effective.

The hope is, that by mimicking the ever decreasing size of the pores in a material, such materials would become sufficiently better at various processes such as acting as catalysts or chemical sensing. Of course creating a network of multi scale interconnected pores requires quite a bit of effort. One process is to use mini polymer spheres as a starting block and growing the porous structure off them. At the end, however, the polymer template has to be removed which normally involves washing with hydrofluoric acid (the really nasty one) at 500°C in a high concentration oxygen atmosphere. Not many materials can survive this process and so the choice of materials to build the “Murray material” is limited. The other option is a self assembling material based on nanocrystals but although aggregation is achievable the structured order is still beyond our reach. The idea suggested in this paper is that of self assembling materials with units of various sizes (called macro-meso-micropores) being created through the theoretical framework of Murray’s law. Created layer by layer these nanomaterials were shown to improve reaction rates by five times (in laboratory conditions) as photocatalysts. It is suggested that this design based on Murray’s law can be applied to many porous material production methods to hopefully increase efficiency and reduce the use of raw materials in the future.

Paper links: Bio-inspired Murray materials for mass transfer and activity

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