The lattice cluster theory of corrections to Flory-Huggins theory is applied to binary compressible blends (at a pressure of one atmosphere) that are formed by polymers having structured monomers. Calculations are performed in the high molecular weight limit for the dependence of the small angle neutron scattering effective interaction parameter CHI(eff) on composition PHI(1), monomer structure, microscopic interaction energies, and temperature. The limiting high molecular weight CHI(eff)(PHI(1)) curves have an overall general parabolic behavior with center, curvature, and magnitude that vary significantly with monomer structures and with interaction energies. The latter variation is stronger and occurs even at constant Flory-Huggins interaction parameter CHI(12)FH where incompressible blend models cannot describe the strong dependence on the self-polymer-polymer-interactions obtained here. A quasi-athermal limit, in which CHI(eff)(PHI(1)) is nearly temperature independent, appears for high molecular weights only when CHI(PHI)(1) is nearly temperature independent, appears for high molecular weights only when CHI(12)FH is vanishingly small. Phase diagrams are studied by evaluating the cloud points for coherent scattering from binary blends. Blends with negative CHI(12)FH have only a LCST, but ones with positive CHI(12)FH may have closed loop phase diagram or both LCST and UCST. However, one of the latter two critical points may be unobservable due to an intervening glass transition or because of thermal degradation.