  This paper is the second in a two-part exposition on {\it surface-directed
spinodal decomposition} (SDSD), i.e., the interplay of kinetics of wetting and
phase separation at a surface which is wetted by one of the components of a
binary mixture. In our first paper [P. Das, P.K. Jaiswal and S. Puri, Phys.
Rev. E {\bf 102}, 012803 (2020)], we studied SDSD on chemically heterogeneous
and physically flat substrates. In this paper, we study SDSD on a chemically
homogeneous but morphologically patterned substrate. Such substrates arise in a
vast variety of technological applications. Our goal is to provide a
theoretical understanding of SDSD in this context. We present detailed
numerical results for domain growth both inside and above the grooves in the
substrate. The morphological evolution can be understood in terms of the
interference of SDSD waves originating from the different surfaces comprising
the substrate.
