| Self-organization of soft condensed matters has been one of the hot topics in the field of condensed matter physics. Full understanding of their structural formations, along with their dynamics features during phase transitions will make contributions to design novel functional materials. Soft matters include polymers, colloids, surfactant, biomacromolecules etc. As a typical soft matter, block copolymers have taken considerable attentions for decades due to their fruitful nano-scale ordered structures and potential applications. It has been well known that there are five equilibrium ordered phases in block copolymer melts, alternating lamellae, hexagonally-packed cylinders, center-body cubic spheres, double gyroid and Fddd (O70).Self-consistent field theory (SCFT) has been one of the most important and successful theories in polymer science. The SCFT has its origin in the work of Edwards in 1960s. Later, this theory was adapted explicitly to treat block copolymers by Helfand and Noolandi etc. During last decades, establishment of various numerical methods for solving SCFT equations, which include "spectral method" proposed by Matsen and Schick in 1994, "real-space method" proposed by Drolet and Fredrickson in 1999 and "pseudo-spectral method" proposed by Rasmussen etc. in 2001, have greatly improved the applications of SCFT in the field of polymer science. These methods have their advantages and disadvantages. Take "real-space method" for instance, although this method is good at searching new ordered structures in block copolymers, difficulties are encountered at reaching long-range ordered structures and determination of space group of complex ordered phases, especially those of complex networks, like gyroid and Fddd. For Matsen-Schick spectral method, all spatially varying functions are expanded in terms of basis functions with priori symmetry and SCFT equations are cast in Fourier space. Consequently, the spectral method is only good at computation of phase diagrams. Recently, Guo et al proposed a generic method for solving SCFT equations, in which all spatially varying functions are expanded in terms of cosine functions, provided that most of ordered phases observed experimentally are centrosymmetric. There are, however, some non-centrosymmetric structures in block copolymer melts, especially in star-shaped terpolymers, where the generic method for SCFT is no longer valid. Under this situation, we aim at proposing a generic approach to solution of self-consistent field theory (SCFT) equations for block copolymers, which combines the capabilities of searching new ordered phases (including centrosymmetric and non-centrosymmetric structures) and determination of space groups of obtained structures together. In this method, all spatially varying functions are expanded in terms of Fourier series (including cosine and sine functions) which are essential determined by computational box parameters. Then, SCFT equations can be cast in terms of expansion coefficients. This method can be looked as an expansion of Matsen-Schick spectral method. The advantage of the approach consists in fact that structural symmetries of resulting ordered phases can be easily deduced from expansion coefficients of nonzero values, which will be clearly demonstrated for complex phases in block copolymers.To evaluate the capabilities of generic Fourier-space method for SCFT, we firstly applied this method to AB diblock copolymer melts. As expected, we successfully obtained the ordered phases including alternating lamellae, hexagonally-packed cylinders, body-centered cubic spheres and double gyroid. Then, the equilibrium ordered phases in ABC linear triblock copolymers melts are computed using the generic Fourier method proposed. Some typical phases, including alternating type, core-shell type and the decorated type structures, are obtained successfully. For some non-centrosymmetric structures in ABC star-shaped terpolymers, we extend the Fourier method to the equilibrium phases in ABC star-shaped terpolymers. Some two-dimensional cylindrical structures are computed, among which are [6.6.6], [8.8.4], [8.6.4;8.6.6], [10.6.4;10.6.6], [12.6.4;8.6.4], [12.6.4]. Note that [6.6.6] and [10.6.4; 10.6.6] are two typical structures of non-centrosymmetric space group. With the above verifications, a conclusion is reached that the generic Fourier method for SCFT is capable of searching new ordered structures with any space groups.Structural determinations of ordered phases in experiments are made via small-angle X-ray scattering (SAXS) method. We have mentioned that the Fourier method proposed aims at combination of capabilities of searching new structures and of determination of space-groups of obtained ordered phases. In order to make a close comparison with SAXS results in experiments, derivations of the scattering intensities of ordered phases in terms of Fourier coefficients in our method is made. Then, the scattering intensities of some typical ordered morphologies in AB diblock copolymers and ABC triblock copolymers are computed and compared with those in experiments. A good consistence between the theoretical scattering functions and those in experiments proves the capability of determination of space groups of ordered structures in block copolymers.ABC star-shaped terpolymers have attracted attentions for years due to their fruitful ordered morphologies. The most distinction of phase behaviors of star terpolymers from that of linear triblock copolymers lies in fact that junction points in ABC star terpolymers be arranged along one-dimensional lines resulting from topological constraint, while in linear ABC triblock copolymers two-dimensional plane could be allowed for the connecting points between neighboring blocks. The spatial arrangement of junction point in star-shaped terpolymers leads to two-dimensional cylinder-type morphologies under assumption that three polymer chains are totally incompatible and long enough. With the generic Fourier method for SCFT, the equilibrium phases of ABC star-shaped terpolymers have been studies. Two broad types are investigated in detail:one with symmetric interactions,χABN=χBCN=χACN=30.0, and one with asymmetric interactions,χABN=χBCN=25.0,χACN=37.0, corresponding to ISP star-shaped terpolymers. The triangle phase diagrams are obtained. For the former, six ordered morphologies are obtained, including [6.6.6], [8.6.4;8.6.6], [8.8.4], [10.6.4; 10.6.6], [12.6.4;8.6.4] and [12.6.4], which is in consistence with those simulations by Monte Carlo and DPD. For asymmetric interactions, two series of star terpolymers are studied, A1.0B1.0Cx, A1.0B1.8Cx. After comparison with experimental results for ISP, a qualitative agreement is reached, while there are few quantitative agreements. These discrepancies maybe due to the inconsistence of molecular parameters used in SCFT with those in real star terpolymers, which are hardly determined experimentally for now, like Flory-Huggins interaction parameter, statistical segment length, etc.ABC linear triblock copolymers reveal fruitful phase behaviors in comparison with AB diblock copolymers, due to their vast parameter space. Thanks to efforts of experimental and theoretical workers on block copolymers, some knowledge has been collected on equilibrium phase behaviors of ABC linear triblock copolymers. Only few samples of ABC linear triblock copolymers, however, have been studied in detail, including ISO and SBM (SEBM). The phase diagrams that have been computed theoretically are for these two types of triblock copolymers, which provide the information of equilibrium phases under each molecular parameter. For obtaining another insight into phase behaviors of ABC linear triblock copolymers, we studied in detail the effects of interaction parameters upon phases with fixed volume fractions of each segment. For simplicity, our focus are located on symmetric samples,χABN=χBCN,fA=fC. We have studied three samples, fA=fC=0.201, fA=fC=0.25 and fA=fC=0.18. For the first one, four different ordered phases have been obtained:alternating sphere, alternating cylinders, alternating gyroid, alternating diamond and double gyroid, while there are two phases including alternating gyroid and lamellae for the second sample. For the third, lamellae and a decorated phase, spheres on lamellae are computed with various interaction parameters. The physical mechanism underlying these phase transitions with varying interaction parameters is discussed.As a conclusion, we proposed a generic Fourier-space method for solving SCFT equations. This method combines the capabilities of searching new structures in block copolymers and determination of space groups of ordered phases. With this method, equilibrium phase behaviors of ABC star-shaped triblock copolymers and ABC linear triblock copolymers have been investigated, where phase diagrams are obtained.
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