Theory of Microphase Separation in Graft and Star Copolymers

Phase stability criteria and static structure factors have been calculated for simple AB graftcopolymers, for star copolymers with equal numbers of A and B arms, and for n-arm star diblock copolymers.The A–B interactions are characterized by the usual x parameter. The fraction of A monomer in the graftcopolymer is denoted as f and the fractional position along the A chain backbone at which the B graft ischemically linked is denoted as r. When r = 0 or 1 the graft copolymer degenerates to a simple diblock copolymer.Leibler previously calculated that the critical value, (xN)_c, at which an AB diblock copolymer containing Nmonomer units undergoes microphase separation is 10.5. This critical value occurs at f = 0.5 and is the onlycomposition for which the transition is second order. According to the present theory, a graft copolymer (0< τ < 1) does not have a critical point for any f; i.e., all transitions are first order. For a given τ, the spinodalvalues, (xN)_s always reach a minimum value at f = 0.5; for τ= f = 0.5, (xN)_s = 13.5. However, star copolymerswith equal numbers (n) of A and B arms each containing N/2 monomers (f = 0.5) have a critical point at(x-N)c = 10.5 for all values of n. Like the graft copolymers, the n-arm star diblock copolymers (each arm isa diblock copolymer of composition f containing N monomer units) do not have a critical point. At f = 0.5,(xN)s equals 8.86, 7.07, 5.32, and 4.33 for n = 2, 4, 10, and 30, respectively. At a spinodal point the staticstructure factor S(q) diverges at a finite wave vector q. Near a critical point q/2π determines the periodicityof the lowest symmetry-ordered structure (mesophase) and is expressed in units of the copolymer’s radiusof gyration R.