Order parameters such as the surface gradient in thin film growth can be unbounded as the size of an underlying system increases. Such unbounded order parameters can be modeled by a variational problem in which the effective free energy consists of a negative logarithmic function of the order parameter and a usual regularizing term. In this talk, we will first describe such a model for unbounded order parameters and compare it with a usual Ginzberg-Landau type model for domain walls. We will also give heuristic arguments to show why low energy configurations should have large value of the order parameter. Rigorous results will then be presented, and proved using the direct method in the calculus of variations. We will conclude the talk by a discussion on the related dynamics.