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When a cubic lattice packaged by a boundary layer is subjected to a mechanical and temperature load, the force and length change of the bonds are equivalently evaluated by the average stress and strain of the unit cell. Provided a displacement gradient variation at a certain stress state, the variation of stress related to the strain variation provides the effective stiffness of the material at the corresponding configuration. It is discovered that the effective elasticity and thermal expansion coefficient can be tailored by the prestress through the boundary layer, which generates a configurational stress. Because the bonds of a cubic lattice depend on the material types, we consider the harmonic potential of springs for cellular lattices and Hertz's contact potential of balls for granular lattices, respectively. The cubic symmetry of effective elasticity is demonstrated for the three types of cubic lattices. By taking the orientational average, isotropic elastic constants can be obtained for randomly oriented lattices. As the bond length changes with the prestress of the boundary layer and controls the thermoelastic behavior, a novel design method of lattice-based materials confined in a spherical shell is demonstrated to achieve zero thermal expansion and a positive temperature derivative of elasticity.

Highly solvent swollen poly(N‐isopropylacrylamide‐co‐acrylic acid) microgels are synthesized without exogenous crosslinker, making them extremely soft and deformable. These ultralow crosslinked microgels (ULC) are incubated under controlled osmotic pressure to provide a slow (and presumably thermodynamically controlled) approach to higher packing densities. It is found that ULC microgels show stable colloidal packing over a very wide range of osmotic pressures and thus packing densities. Surprising observation of co‐existence between hexagonal and square lattices is also made over the lower range of studied osmotic pressures, with microgels apparently changing shape from spheres to cubes in defects or grain boundaries. It is proposed that the unusual packing behavior observed for ULC microgels is due to the extreme softness of these particles, where deswelling causes deformation and shrinking of the particles that result in unique packing states governed by contributions to the entropy at the colloidal system, single particle and ionic levels. These observations further suggest that more detailed experimental and theoretical studies of ultra‐soft microgels are required to obtain a complete understanding of their behavior in packed and confined geometries. [ABSTRACT FROM AUTHOR]

We study the local optimality of simple cubic, body-centred-cubic and face-centred-cubic lattices among Bravais lattices of fixed density for some finite energy per point. Following the work of Ennola (Math Proc Camb 60:855–875, 1964), we prove that these lattices are critical points of all the energies, we write the second derivatives in a simple way and we investigate the local optimality of these lattices for the theta function and the Lennard–Jones-type energies. In particular, we prove the local minimality of the FCC lattice (resp. BCC lattice) for large enough (resp. small enough) values of its scaling parameter and we also prove the fact that the simple cubic lattice is a saddle point of the energy. Furthermore, we prove the local minimality of the FCC and the BCC lattices at high density (with an optimal explicit bound) and its local maximality at low density in the Lennard–Jones-type potential case. We then show the local minimality of FCC and BCC lattices among all the Bravais lattices (without a density constraint). The largest possible open interval of density's values where the simple cubic lattice is a local minimizer is also computed. [ABSTRACT FROM AUTHOR]