Dual composition block copolymers made from conjugated diene and monovinyl aromatic monomers in batch organolithium initiated polymerization show advantageous performance in the production of crosslinked microcellular rubber compounds and pressure sensitive hot melt adhesives. The dual composition block copolymers are partially coupled with a coupling agent linking inner monovinyl aromatic blocks. Their un-coupled low molecular weight fraction has greater monovinyl aromatic repeating unit content than their coupled high molecular weight fraction. Crosslinked microcellular rubber articles made from the dual composition block copolymers exhibit lower density, smaller and more homogeneous cell size, higher softness and higher resiliency than prior art block copolymers. Rubber compounding of formulations comprising the dual composition block copolymers proceed at slightly lower torque and slightly lower temperature than with prior art block copolymers. Pressure sensitive hot melt adhesives formulated with the dual composition block copolymers are very well suited for labels, exhibiting higher tack and higher softening temperature than those made from prior art block copolymers.