Theory of two-dimensional self-assembly of Janus colloids: crystallization and orientational ordering

We study the rich crystalline phase behavior of amphiphilic spherical Janus colloids using a new formulation of self-consistent phonon theory that includes coupled translational and rotational entropic and enthalpic contributions to the free energy. In contrast to homogeneous spheres, broken rotational symmetry can result in more exotic crystals that possess distinct orientational patterns, and also plastic crystals. Ground states are identified based on the compatibility between the patch geometry of particles (e.g., patch coverage, number, shape) and lattice symmetry. We derive the explicit coupled self-consistent equations for translational and rotational localization parameters for effectively 2-dimensional dense monolayers of Janus crystals. The equations are numerically solved for a given crystal symmetry, thermodynamic state, and patch orientational order, and the thermodynamic stability of different phases is determined. For hexagonal packing, we predict with increasing temperature or decreasing attraction strength the possibility of a phase sequence of maximally bonded zigzag stripe, trimer, and rotationally disordered plastic crystal phases (or a phase sequence of trimer, dimer, and plastic crystal), which depends sensitively on particle chemical composition (Janus balance) and pressure. The role of rotational entropy in stabilizing the intermediate trimer (or dimer) phase at intermediate temperatures and high pressures is discussed in detail. Evolution of the center-of-mass vibrational and rotational amplitudes with thermodynamic state and Janus balance is also determined.