Molecular dynamics simulations were used to study the interlayer structure and dynamics of polystyrene (PS) and polystyrene-polyisoprene (PS-PI) block copolymers intercalated in organically modified layered silicates. In the case of PS the polymer chains displace the aliphatic surfactant chains and reside adjacent to the silicate layers. The electrostatic interactions between the aromatic rings on the PS chains and the silicate surface drive the intercalation of the polymer into the host galleries. PI, which lacks such electrostatic interactions, is immiscible (does not intercalate) with the host. There appears to be a minimum number of PS mers for intercalation of PS-PI copolymers to take place. The intercalated copolymer appears to structure inside the host galleries with the PS mers adjacent to the silicate layers and the corresponding PI away from the surface and towards the middle of the gallery. Using the mean square displacements we find that PS is the least mobile species in the galleries with the surfactant chains been the most mobile of all.