Theory of Anisotropic Fluctuations in Ordered Block Copolymer Phases An-Chang Shi* and Jaan Noolandi Xerox Research Centre of Canada, 2660 Speakman Drive, Mississauga, Ontario, Canada L5K 2L1 Rashmi C. Desai Department of Physics, University of Toronto, Toronto, Ontario Canada M5S 1A7 Received March 18, 1996; Revised Manuscript Received June 12, 1996 X ABSTRACT: A general theoretical framework for the study of anisotropic composition fluctuations about an ordered block copolymer phase is developed. The approach is based on the idea that, in order to study the effects of fluctuations around an ordered broken symmetry phase, the theory must be formulated as a self-consistent expansion around the mean-field solution of this ordered state. A random phase approximation treatment of the theory leads to anisotropic correlation functions for the system. It is shown that the calculation of the polymer correlation functions in an ordered phase is equivalent to the calculation of the energy bands and eigenfunctions for an electron in a periodic potential. This general method is applied to the lamellar phase of block copolymers. The calculated anisotropic scattering intensity captures the main features observed experimentally, including the secondary peaks due to fluctuations with hexagonal symmetry. The origin of the anisotropic fluctuations can be traced to the formation of “energy” bands, similar to electronic states in solids. I. Introduction Block copolymers are fascinating materials with unique structural and mechanical properties. 1,2 Due to their amphiphilic nature, AB diblock copolymers self- assemble into a variety of ordered microphases. 1-3 At high temperatures, A and B blocks mix homogeneously to form a disordered phase. As the temperature is decreased (or the Flory-Huggins  parameter is in- creased), the blocks undergo a spatial segregation. However, a macroscopic phase separation cannot occur because the AB blocks are chemically connected at the junctions. The phase separation is, therefore, neces- sarily on a mesoscopic scale, forming A- and B-rich domains separated by internal interfaces. The competi- tion between the spontaneous curvature of the internal interfaces and the entropic stretching (or packing) of the A and B blocks dictates the symmetry of the equilibrium phases. The symmetry of the ordered phases is con- trolled by the degree of segregation and the chemical composition of the copolymers. The segregation of the blocks is quantified by the product Z, where Z ) Z A + Z B is the degree of polymerization of the copolymers, and Z A and Z B are the degrees of polymerization of the A and B blocks, respectively. The chemical composition of the copolymers is quantified by the ratio f ) Z A /Z.A decrease in Z drives the melt from an ordered phase to the disordered phase. Changes in f affect the shape and packing symmetry of the ordered structure. Be- sides the well-known lamellar, cylindrical, and spherical phases, 1,3,4 more complicated structures such as the bicontinuous cubic phase and the hexagonally modu- lated and perforated layered phases have been recently identified. 3,5,6 Theoretically, the study of diblock copolymer melts has been based on a simplified, standard model, in which the diblock copolymer chains are assumed to be flexible chains obeying Gaussian statistics. 7 The poly- mer conformations are specified by the position of the monomer, R(t). Each monomer is further assumed to have a statistical length b R . The hard-core repulsive interactions between the molecules are accounted for by an incompressibility constraint where the average monomer concentration is required to be uniform. The remaining interactions are modeled by a local enthalpy term φ A (r)φ B (r), where φ A (r) and φ B (r) are the volume fractions of the A and B monomers at position r. This simple model captures the three most important fea- tures of the system: chain conformation entropy, in- compressibility, and immiscibility between unlike mono- mers. The thermodynamics of a diblock copolymer melt can, therefore, be described by the partition function of the model. The partition function may be cast in several equivalent forms. On particularly simple and conve- nient formulation is to write the partition function of the melt as a functional integral over the monomer concentrations φ R (r) and two auxiliary fields ω R (r)(R) A, B). 8,9 The functional integral is carried out with the incompressibility constraint and with an integrand of the form exp{-F({φ}, {ω})}. The free energy functional is the sum of three terms: an enthalpic contribution, an entropic contribution, and a coupling term, where V is the volume of the system, and Q c is the partition function of a single diblock copolymer chain in the external fields ω R (r). 9 It is important to note that Q c ({ω}) is a functional of ω R (r). Specifically, the single- chain partition function Q c is determined by where the propagators Q R (r, t|r′) are X Abstract published in Advance ACS Abstracts, August 1, 1996. F ) ∫ dr [φ A (r)φ B (r) - ∑ R ω R (r)φ R (r)] - V Z ln Q c ({ω R }) (1) Q c ) 1 V ∫ dr 1 dr 2 dr 3 Q A (r 1 , Z A |r 2 )Q B (r 2 , Z B |r 3 ) (2) 6487 Macromolecules 1996, 29, 6487-6504 S0024-9297(96)00411-1 CCC: $12.00 © 1996 American Chemical Society