  Open structures can display a number of unusual properties, including a
negative Poisson's ratio, negative thermal expansion, and holographic
elasticity, and have many interesting applications in engineering. However, it
is a grand challenge to self-assemble open structures at the colloidal scale,
where short-range interactions and low coordination number can leave them
mechanically unstable. In this paper we discuss the self-assembly of open
structures using triblock Janus particles, which have two large attractive
patches that can form multiple bonds, separated by a band with purely
hard-sphere repulsion. Such surface patterning leads to open structures that
are stabilized by orientational entropy (in an order-by-disorder effect) and
selected over close-packed structures by vibrational entropy. For different
patch sizes the particles can form into either tetrahedral or octahedral
structural motifs which then compose open lattices, including the pyrochlore,
the hexagonal tetrastack and the perovskite lattices. Using an analytic theory,
we examine the phase diagrams of these possible open and close-packed
structures for triblock Janus particles and characterize the mechanical
properties of these structures. Our theory leads to rational designs of
particles for the self-assembly of three-dimensional colloidal structures that
are possible using current experimental techniques.
