The ground state of an induced electrorheological (ER) solid is a body-centered tetragonal (bct) lattice with conventional Bravais vectors [Formula: see text], [Formula: see text], and [Formula: see text] where a is the radius of dielectric spheres and [Formula: see text] is the field direction. The reciprocal lattice vectors are [Formula: see text], [Formula: see text], and [Formula: see text]. Three order parameters are defined as [Formula: see text], (j = 1, 2, 3), where N is the total number of particles. Among them, ρ3 characterizes the formation of chains in the z direction, while ρ1 and ρ2 characterize the structure in the x-y plane. Monte Carlo simulations of canonical ensemble have shown that three different phases and two phase transitions exist in the ER fluid. At a fixed temperature, there are two critical electric fields Ec2 < Ec1. When the applied electric field E < Ec2, ρj (j = 1, 2, 3) are all vanishing and the ER fluid is a liquid. When Ec2 < E < Ec1, ρ1 = ρ2 = 0, but ρ3 is not vanishing, indicating that the ER fluid begins to form chains between two electrodes, but the distribution of these chains is random. This state is similar to nematic liquid crystal. The second phase transition occurs as E exceeds Ec1. Then ρ1, ρ2, and ρ3 are all non-vanishing, indicating that the chains aggregate together to form thick columns which have the bct lattice structure. These two phase transitions can also be realized at a fixed field by lowering the temperature.
