**Optimal Multifunctional Materials and Minimal Surfaces**

Composite materials are ideally suited to achieve multifunctionality since the best features of different materials can be combined to form a new material that has a broad spectrum of desired properties. Nature’s ultimate multifunctional composites are biological materials. There are presently no simple examples that rigorously demonstrate the effect of competing property demands on composite microstructures. To illustrate the fascinating types of microstructures that can arise in multifunctional optimization, we maximize the simultaneous transport of heat and electricity in three-dimensional, two-phase composites using rigorous optimization techniques. Interestingly, we have discovered that the optimal three-dimensional structures are bicontinuous triply periodic minimal surfaces (see the figures below).

A bicontinuous optimal structure (for which volume fractions of both phases are identical) corresponding to maximization of the sum of the effective electrical and thermal conductivities for ill-ordered phases. Left panel: A 2 x 2 x 2 unit cell of the composite. Right panel: Corresponding morphology of phase 2 which has low thermal conductivity but high electrical conductivity.

Significantly, the aforementioned optimal bicontinuous composites can be made using sol-gel processing techniques. Interestingly, cell membranes resembling periodic minimal surfaces have been observed in cytoplasmic organelles, such as mitochondria and chloroplasts, in which a variety of different transport processes occur. Our work tantalizingly suggests that it may be fruitful to explore the origins of such structures under a new light, namely, whether the optimization of competing functionalities within organelles can explain their resulting structures. Finally, note that our methodology is quite general and can be employed to discover the novel optimal microstructures that are expected to emerge when any combination of functionalities (e.g., mechanical, optical, chemical, electrical, thermal, and flow properties) compete against one another. Such analyses will lead to insights into the genesis of optimal microstructures and will be pursued in future work.

Unit cells of two different minimal surfaces with a resolution of 64 x 64 x 64. Left panel: Schwartz simple cubic surface. Right panel: Schwartz diamond surface.

This work is described in the paper by

S. Torquato, S. Hyun, and A. Donev, *Multifunctional Composites: Optimizing Microstructures for Simultaneous Transport of Heat and Electricity*, **Physical Review Letters**, **89**, 266601 (2002).

We further established the multifunctionality of the aforementioned bicontinuous two-phase systems by showing that they are also extremal when a competition is set up between the effective bulk modulus and the electrical (or thermal) conductivity of the composite. The implications of our findings for materials science and biology, which provides the ultimate multifunctional materials, are discussed.

This work is described in the paper by

S. Torquato and A. Donev, *Minimal Surfaces and Multifunctionality*, **Proceedings of the Royal Society of London A**, **460**, 1849 (2004).

Do triply periodic minimal surfaces optimize any other properties? We computed the fluid permeabilities of the Schwatrz P and D surfaces and other triply periodic bicontinuous at a porosity = 1/2 using the immersed-boundary finite-volume method. The other triply periodic porous media that we studied include the Schoen gyroid (G) minimal surface, two different pore-channel models, and an array of spherical obstacles arranged on the sites of a simple cubic lattice. We find that the Schwartz P porous medium has the largest fluid permeability among all of the six triply periodic porous media considered in this paper. The fluid permeabilities are shown to be inversely proportional to the corresponding specific surfaces for these structures. This leads to the conjecture that the maximal fluid permeability for a triply periodic porous medium with a simply connected pore space at a porosity = 1/2 is achieved by the structure that globally minimizes the specific surface.

This work is described in the paper by

Y. Jung and S. Torquato, *Fluid Permeabilities of Triply Periodic Minimal Surfaces*, **Physical Review E**,**92**, 255505 (2005).

Questions concerning this work should be directed to Professor Torquato.