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Many foraging microorganisms rely upon cellular transport networks to deliver nutrients, fluid and organelles between different parts of the organism. Networked organisms ranging from filamentous fungi to slime molds demonstrate a remarkable ability to mix or disperse molecules and organelles in their transport media. Here we introduce mathematical tools to analyze the structure of energy efficient transport networks that maximize mixing and sending signals originating from and arriving at each node. We define two types of entropies on flows to quantify mixing and develop numerical algorithms to optimize the combination of entropy and energy on networks, given constraints on the amount of available material. We present an in-depth exploration of optimal single source-sink networks on finite triangular grids, a fundamental setting for optimal transport networks in the plane. Using numerical simulations and rigorous proofs, we show that, if the constraint on conductances is strict, the optimal networks are paths of every possible length. If the constraint is relaxed, our algorithm produces loopy networks that fan out at the source and pour back into a single path that flows to the sink. Taken together, our results expand the class of optimal transportation networks that can be compared with real biological data, and highlight how real network morphologies may be shaped by tradeoffs between transport efficiency and the need to mix the transported matter.