  Here we derive and experimentally test a free energy functional that captures
the adhesion of DNA-coated emulsion droplets. Generalizing previous approaches,
the theory combines important energetic and entropic effects of microscopic DNA
mechanics and droplet elasticity. It simultaneously predicts adhesion size,
morphology, and binder concentration as a function of experimental control
parameters. Notably, droplets transition from undeformed binding to flat
droplet interfaces at a characteristic DNA coverage. These equilibrium
predictions agree quantitatively with experiments on droplet-substrate and
droplet-droplet binding, revealing a weak effective binding strength of
$3.7\pm0.3\text{k}_{\text{B}}\text{T}$ owing to entropic costs. Our results
open the path to rich design strategies for making colloidal architectures.
