Commensurability effect in diblock copolymer lamellar phase under d-dimensional nanoconfinement

We theoretically consider the commensurability problem of AB diblock lamellar phase confined between parallel plates, in cylinder, and in sphere calculating the free energy of confined lamellar phase which is generalized in terms of dimensionality of confinement (d) and conformational asymmetry (ε). We find that the first-order layer-addition transition of lamellar layers parallel to the confining surface (L_∥) becomes suppressed as the dimensionality of confinement increases. For lamellae confined in curved space, the conformational asymmetry alters the location of layer-addition transition point and the stability of L_∥ against nonconcentric layers. When the surface-preferential block becomes flexible, the radius of cylindrically or spherically confined space at the layer-addition transition, where the number of A-B layers of L_∥ changes from l layers to l+1 layers, increases if l is odd and decreases otherwise due to the tendency of less flexible block filling innermost layer. The curved space also weakens the stability L_∥ competing with nonconcentric layers when the surface-preferential block becomes less flexible. The phase maps in the parameter space of conformational asymmetry and degree of confinement are constructed for different cases of the confinement dimensionality and the surface fields, demonstrating the effects of various system variables on the confined lamellar structures.