In recent years the statistical mechanics of non-spherical molecules, such as polypeptide chains and protein molecules, has garnered considerable attention as the phase behavior of these systems has important scientific and health implications. More recently, it has been realized that surface binding may have a considerable impact on this behavior. With this in mind, we examine here the role of surface interactions on the phase behavior of Y-shaped molecules (a simplified model of immunoglobulin) using grand-canonical Monte Carlo simulation. In particular, we investigate the critical behavior of a system of such molecules on a hexagonal lattice using histogram reweighting, multicanonical sampling, and finite-size scaling. After obtaining the critical properties of the three-dimensional bulk system, we investigate the impact of a patterned solid surface on these properties as a function of patterning geometry and surface interaction strength. Our results suggest avenues for tailoring phase behavior by selectively controlling surface characteristics. (C) 2015 Elsevier B.V. All rights reserved.