We present some discrete models with surface scaling of the interaction energy. We assume that at least nearest and next-to-nearest neighbor interactions are taken into account. Our purpose is to show that interactions beyond nearest neighbors have the role of penalizing changes of orientation and, to some extent, they may replace the positive-determinant constraint that is usually required when only nearest neighbors are accounted for. From a joint work in collaboration with R. Alicandro and M. Palombaro.