Symmetric Blends of Complementary Diblock Copolymers: Multiorder Parameter Approach and Monte Carlo Simulations June Huh, Henk Angerman, and Gerrit ten Brinke* Laboratory of Polymer Chemistry and Materials Science Center, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands Received November 17, 1995; Revised Manuscript Received June 4, 1996 X ABSTRACT: Symmetric diblock copolymer blends AfB1-f/A1-fBf (0 e f e 0.5) are theoretically discussed in terms of a multiorder parameter approach and numerically investigated by Monte Carlo simulations. Theoretically, our main result is that below f = 0.3, but still in the microphase separation region given by 0.21 e f e 0.5, the concentration profiles of the long and short A-blocks as well as the long and short B-blocks are out of phase. Monte Carlo simulations were used to investigate the nature of the phase transition, micro versus macro, as a function of f. Using the canonical ensemble, the microphase separation temperature (MIST) was determined. The macrophase separation temperature (MAST) was studied with the semi-grand-canonical ensemble combined with the histogram extrapolation technique. The phase diagram differs considerably from the theoretical predictions due to the stretching/polarization of the molecules already far above the transition temperature, thus stabilizing the macroscopically homogeneous state. The out-of-phase behavior between the long and short blocks near the critical value f = 0.21, separating the micro- and macrophase separation regimes, was confirmed by the simulations. 1. Introduction Technologically important block copolymer systems are often characterized by a distribution in chain length as well as block length. Thermoplastic elastomers consist of multiblock copolymers with alternating soft and hard blocks, where the number of blocks, the block length, and the chain length all vary from polymer to polymer. 1 Theories to deal more accurately with these types of systems have been developed over the past five years. A single order parameter approach was intro- duced by Fredrickson et al., 2 who demonstrated the presence of micro- or macro-phase separation depending on the chemical correlations within the polymers. Cor- related random copolymers were subsequently studied in more detail in refs 3 and 4. In the last reference, a very general theory developed by Dobrynin and Erukhi- movich 5 was applied. This theory is a refinement of the original Leibler description 6 obtained by introducing separate order parameters for each type of molecule present. These order parameters are also called de- tailed densities. As usual, the Landau free energy is developed in a power series of all order parameters (in principle, infinitely many), and this free energy must be minimized with respect to all these order parameters. At first, this seems a very complicated task, which, however, can be solved by observing that at the spinodal the system becomes critical with respect to one particu- lar linear combination of order parameters. This linear combination is therefore called the strongly fluctuating field, all others being weakly fluctuating fields. The procedure is now to expand the free energy up to fourth order in the strongly fluctuating fields and to such order in the weakly fluctuating fields that the terms in which they occur are at least comparable to this fourth-order contribution. In polydisperse polymer systems charac- terized by infinitely many order parameters, it is rather difficult to discuss in detail the precise nature of the strongly and weakly fluctuating fields. It is obvious of interest to indicate the additional information that becomes available by this detailed description. For this reason and for simulation technical reasons that will become clear further on, we selected to study a particu- larly simple system consisting of a symmetric binary melt of diblock copolymers A f B 1-f /A 1-f B f . Already a decade ago, Erukhimovich 7,8 considered this symmetric blend and showed that microphase separa- tion can only occur for (1 - 1/ 3) e 2f e 1, where for symmetry reasons we restrict ourselves to f e 0.5. For smaller values of f, the two components differ so much that at first only a separation into spatially uniform macroscopic phases can occur. Subsequently, these macrophases can then microphase separate. The criti- cal point where the phase transition changes its nature from macroscopic to microscopic is known as Lifshitz point, 9 so that this point represents the connecting point between three distinctive states: the spatially uniform ordered phase, the spatially modulated ordered phase, and the disordered phase. Multicritical behavior of this kind has been explored for polymeric systems theoreti- cally as well as experimentally. Theoretically, Broseta and Fredrickson 10 demonstrated isotropic Lifshitz be- havior, i.e. the characteristic wave vector q* changes continuously from 0 to a finite value, for blends of a block copolymer and a homopolymer by using the random phase approximation. Within the mean-field approach, i.e. neglecting fluctuation effects, they found that for large copolymer content, the vertex function in the RPA equation has its minimum at q* * 0, whereas at a small copolymer content the vertex function is a monotonically increasing function. For the same sys- tem, Bates and co-workers 11 observed experimentally that the critical exponent at the isotropic Lifshitz point was consistent with that of mean-field Lifshitz behavior. In recent publications, 12,13 general binary diblock copolymer blends, A f B 1-f /A g B 1-g , were considered at all possible compositions. In ref 13, a detailed phase diagram was calculated using a grand-canonical self- consistent field theory developed in ref 14. In the present paper, we will consider symmetric binary diblock copolymer blends A f B 1-f /A 1-f B f at equal volume fraction 0.5 of both types of block copolymers. In the first section, a theoretical analysis in terms of the four order parameters, describing the concentration profiles of the A- and B-units of block copolymer 1 (A f B 1-f ) and X Abstract published in Advance ACS Abstracts, August 15, 1996. 6328 Macromolecules 1996, 29, 6328-6337 S0024-9297(95)01715-3 CCC: $12.00 © 1996 American Chemical Society