Multifunctional Composites: Optimizing Microstructures for Simultaneous Transport of Heat and Electricity

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 discover that the optimal three-dimensional structures are bicontinuous triply periodic minimal surfaces.

Figure 1

Schematic illustrating the allowable region in which all composites, with specified phase properties and microstructural constraints, must lie for the case of two different effective properties.

Figure 2

Cross-property bounds and simulation datum for the effective electrical ${\sigma }_e$ and thermal ${\lambda }_e$ conductivities for ill-ordered phases (${\sigma }_1/{\sigma }_2<1$ and ${\lambda }_1/{\lambda }_2>1$) as specified by (4). The datum on the upper bound corresponds to the bicontinuous structures described in the text. Elsewhere [18] we have shown that the two points corresponding to the intersections of the upper and lower bounds are three-dimensional generalizations of the two-dimensional single-scale Vigdergauz structures [19].

Figure 3

A bicontinuous optimal structure at ${\varphi }_1={\varphi }_2=1/2$ corresponding to maximization of the sum of the effective electrical and thermal conductivities (${\sigma }_e+{\lambda }_e$) for ill-ordered phases as specified by (4). Left panel: A $2\times 2\times 2$ unit cell of the composite. Right panel: Corresponding morphology of phase 2.

Figure 4

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