Weak segregation theory and non-conventional morphologies in the ternary ABC triblock copolymers

The permeability of diamond and other microphase-separated morphologies in copolymer films

A method is proposed for the theoretical assessment of potential membrane properties in materials based on microphase-separated block copolymer films subjected to the etching of one of the components. The concepts of topological permeability and connectivity contrast introduced by the authors are used to compare the percolation characteristics of simple cubic, diamond, and diamond-like morphologies compatible with the slit geometry and specified by a given distribution of the order parameter. The diamond-like morphology, which has the most promising transport characteristics, can be formed in a thin film of diblock copolymer AB placed on a chemically structured substrate. For this case, the quantitative analysis of permeability for certain parameters of the model system is carried out using the local distribution of A and B units calculated with the self-consistent field theory of microphase separation. The results obtained may be useful for the design of isoporous membranes based on block copolymers.

Design of Linear Block Copolymers and ABC Star Terpolymers That Produce Two Length Scales at Phase Separation

Quasicrystals (materials with long-range order but without the usual spatial periodicity of crystals) were discovered in several soft matter systems in the last 20 years. The stability of quasicrystals has been attributed to the presence of two prominent length scales in a specific ratio, which is 1.93 for the 12-fold quasicrystals most commonly found in soft matter. We propose design criteria for block copolymers such that quasicrystal-friendly length scales emerge at the point of phase separation from a melt, basing our calculations on the Random Phase Approximation. We consider two block copolymer families: linear chains containing two different monomer types in blocks of different lengths, and ABC star terpolymers. In all examples, we are able to identify parameter windows with the two length scales having a ratio of 1.93. The models that we consider that are simplest for polymer synthesis are, first, a monodisperse ALBASB melt and, second, a model based on random reactions from a mixture of AL, AS, and B chains: both feature the length scale ratio of 1.93 and should be relatively easy to synthesize.

Liquid-Crystal Ordering and Microphase Separation in the Lamellar Phase of Rod-Coil-Rod Triblock Copolymers. Molecular Theory and Computer Simulations

A molecular model of the orientationally ordered lamellar phase exhibited by asymmetric rod-coil-rod triblock copolymers has been developed using the density-functional approach and generalizing the molecular-statistical theory of rod-coil diblock copolymers. An approximate expression for the free energy of the lamellar phase has been obtained in terms of the direct correlation functions of the system, the Flory-Huggins parameter and the Maier-Saupe orientational interaction potential between rods. A detailed derivation of several rod-rod and rod-coil density-density correlation functions required to evaluate the free energy is presented. The orientational and translational order parameters of rod and coil segments depending on the temperature and triblock asymmetry have been calculated numerically by direct minimization of the free energy. Different structure and ordering of the lamellar phase at high and low values of the triblock asymmetry is revealed and analyzed in detail. Asymmetric rod-coil-rod triblock copolymers have been simulated using the method of dissipative particle dynamics in the broad range of the Flory-Huggins parameter and for several values of the triblock asymmetry. It has been found that the lamellar phase appears to be the most stable one at strong segregation. The density distribution of the coil segments and the segments of the two different rods have been determined for different values of the segregation strength. The simulations confirm the existence of a weakly ordered lamellar phase predicted by the density-functional theory, in which the short rods separate from the long ones and are characterized by weak positional ordering.

Different Mechanisms of Translational Symmetry Breaking in Liquid-Crystal Coil–Rod–Coil Triblock Copolymers

A molecular-statistical theory of coil-rod–coil triblock copolymers with orientationally ordered rod-like fragments has been developed using the density functional approach. An explicit expression for the free energy has been obtained in terms of the direct correlation functions of the reference disordered phase, the Flory–Huggins parameter and the potential of anisotropic interaction between rigid rods. The theory has been used to derive several phase diagrams and to calculate numerically orientational and translational order parameter profiles for different polymer architecture as a function of the Flory–Huggins parameter, which specifies the short-range repulsion and as functions of temperature. In triblock copolymers, the nematic–lamellar transition is accompanied by the translational symmetry breaking, which can be caused by two different microscopic mechanisms. The first mechanism resembles a low dimensional crystallization and is typical for conventional smectic liquid crystals. The second mechanism is related to the repulsion between rod and coil segments and is typical for block copolymers. Both mechanisms are analyzed in detail as well as the effects of temperature, coil fraction and the triblock asymmetry on the transition into the lamellar phase.

3D bicontinuous diamond-like morphologies in thin films of soluted block copolymers

As shown theoretically earlier via both weak segregation and self-consistent field theories, ordering of confined molten di- and tri-block copolymer morphologies in the presence of a proper 1D patterned substrate could induce the formation of 3D bicontinuous (in particular, diamond-like) morphologies (DLMs). The purpose of the present paper is to study, unlike the previous studies, how the stable DLMs are formed not in a melt but in a solution of symmetric diblock copolymers with a nonselective solvent that wets the thin film on the patterned substrate. It is shown, via a straightforward self-consistent field calculation of the total solution free energy for various competing phases, that the DLM could be formed in the solutions (with the solvent volume fraction of 0.5), which provides much faster thermodynamic equilibration of the solution as compared to the melt. The last circumstance can ease the production of stable DLMs in thin films of copolymers. The phase diagram describing the stable phases on the plane "the pattern period-the film thickness" is built.

Identification of Some New Triply Periodic Mesophases from Molten Block Copolymers

Using field-theoretic simulations based on a self-consistent field theory (SCFT) with or without finite compressibility, nanoscale mesophase formation in molten linear AB and ABC block copolymers is investigated in search of candidates for new and useful nanomaterials. At selected compositions and segregation strengths, the copolymers are shown to evolve into some new nanostructures with either unusual crystal symmetry or a peculiar morphology. There exists a holey layered morphology with Im3 symmetry, which lacks one mirror reflection compared with Im3m symmetry. Also, a peculiar cubic bicontinuous morphology, whose channels are connected with tetrapod units, is found to have Pn3m symmetry. It is shown that there is another network morphology with tripod connections, which reveals P432 symmetry. The optimized free energies of these new mesophases and their relative stability are discussed in comparison with those of double gyroids and double diamonds.

Thermodynamics of 3D diamond-like epitaxial (film) morphologies on 1D modulated substrate: Weak crystallization theory

Being inspired by recent theoretical findings in block copolymer ordering, we present, within the simplest version of the Landau weak crystallization theory, a consistent treatment of the thin film thermodynamics. It is first shown that a proper design of the period of a 1D modulated substrate results in the formation of a 3D morphology whose symmetry is close to a cubic diamond one and differs from the latter due to a symmetry violation caused by the block copolymer-substrate interaction. The corresponding phase portraits are built. With a decrease in the temperature, the diamondlike morphology becomes less stable until it turns (via the 1st order phase transition) into the lamellar phase. On the contrary, with an increase in the substrate selectivity, the lamellae become less stable and, finally, turn into the diamondlike morphology.

Emergence of ordered network mesophases in kinetic pathways of order-order transition for linear ABC triblock terpolymers

Applying the string method to the self-consistent field theory (SCFT) of ABC linear triblock copolymers, we developed a new strategy to design kinetic pathways for the formation of stable or metastable network mesophases in order-order transition (OOT) processes. The design principle regarding the kinetic pathways between distinct mesophases is based on the matching relationships of both domain spacing and dominant Fourier components of the density distributions. The results suggest that complex ordered network mesophases, such as alternating diamond (D^A) and alternating plumber's nightmare (P^A) could be obtained in kinetic pathways between simple phases covering lamellae, cylinders and spheres. By virtue of the minimal free energy pathway (MEP) obtained, we could acquire the epitaxial relationship and phase transition mechanism. Furthermore, we managed to regulate the MEP by changing the block composition to adjust packing frustration. Two new metastable networks, core-shell five-pronged and six-pronged morphologies, were found in the kinetic pathways, further demonstrating the regulating mechanism. The results will contribute to a better understanding of the kinetic relationship between simple phases and complex networks, thus providing a platform for soft materials design via the OOT route and guiding experimental procedures to fabricate ordered network mesophases.

Stability of the Frank-Kasper σ-phase in BABC linear tetrablock terpolymers

The phase behavior of B1AB2C tetrablock terpolymer melts is systematically studied using the self-consistent field theory, focusing on the emergence and stability of the complex Frank-Kasper σ-phase. Our study starts with an investigation of the stability region of the σ phase for a generic model of B1AB2C terpolymers, in which the C-blocks form spherical domains immersed in the A/B matrix. Then, we examine the stability of the σ phase for a model system with a specific set of parameters mimicking poly(styrene-b-isoprene-b-styrene-b-ethylene oxide) (SISO) block copolymers which were examined in recent experiments. Our results reveal that the formation of the σ phase is mainly governed by two factors. The first factor is the conformational asymmetry between the A/B-blocks and the C-block, similar to that in conformationally asymmetric AB-type block copolymers. The second factor is the specific chain architecture of B1AB2C. The tetrablock architecture with a specific set of interaction parameters and compositions leads to the formation of large core-shell spherical domains, which amplifies the effect of interfacial energy and thereby stabilizes the σ phase.

Perpendicular lamellar-in-lamellar and other planar morphologies in A-b-(B-b-A)2-b-C and (B-b-A)2-b-C ternary multiblock copolymer melts

Ordered planar morphologies in A-b-(B-b-A)2-b-C and (B-b-A)2-b-C terpolymer melts are studied within the framework of the self-consistent field theory for volume fractions of components A, B, and C in the ratio 1:1:2 and the Flory-Huggins interaction parameters satisfying χ(AB) = 2χ(AC). The stable phases turn out to be the disordered, hexagonal, parallel lamellar-in-lamellar L∥ (including the simple lamellar) as well as non-shifted and shifted (L⊥ and SL⊥) perpendicular lamellar-in-lamellar morphologies. Depending on the value of the ratio r = Θ(AB)/Θ(BC), where Θ is a characteristic temperature of the units involved, different sequences of phase transitions are shown to occur. The hexagonal phase is characteristic for r ≅ 1. The L⊥ and SL⊥ morphologies occur at weak and intermediate segregations whereas the L∥ morphology appears for stronger degrees of segregation. For (B-b-A)2-b-C a reduction in r favors the shifted SL⊥ phase over the non-shifted L⊥ one, whereas for A-b-(B-b-A)2-b-C we find re-entrant phase transitions SL⊥ - L⊥. The physics determining the particular phase behavior is discussed.

Morphologies of ABC triblock terpolymer melts containing poly(cyclohexadiene): effects of conformational asymmetry

We have synthesized linear ABC triblock terpolymers containing poly(1,3-cyclohexadiene), PCHD, as an end block and characterized their morphologies in the melt. Specifically, we have studied terpolymers containing polystyrene (PS), polybutadiene (PB), and polyisoprene (PI) as the other blocks. Systematically varying the ratio of 1,2- /1,4-microstructures of poly(1,3-cyclohexadiene), we have studied the effects of conformational asymmetry among the three blocks on the morphologies using transmission electron microscopy (TEM), small-angle X-ray scattering (SAXS), and self-consistent field theory (SCFT) performed with PolySwift++. Our work reveals that the triblock terpolymer melts containing a high percentage of 1,2-microstructures in the PCHD block are disordered at 110 °C for all the samples, independent of sequence and volume fraction of the blocks. In contrast, the triblock terpolymer melts containing a high percentage of 1,4-microstructure form regular morphologies known from the literature. The accuracy of the SCFT calculations depends on calculating the χ parameters that quantify the repulsive interactions between different monomers. Simulations using χ values obtained from solubility parameters and group contribution methods are unable to reproduce the morphologies as seen in the experiments. However, SCFT calculations accounting for the enhancement of the χ parameter with an increase in the conformational asymmetry lead to an excellent agreement between theory and experiments. These results highlight the importance of conformational asymmetry in tuning the χ parameter and, in turn, morphologies in block copolymers.

Self-assembly of (A-comb-C)-b-(B-comb-C) diblock copolymer-based comb copolymers

The phase behavior of (A-comb-C)-b-(B-comb-C) diblock copolymer melts is investigated using the strong segregation theory approach. Three different regimes are distinguished. In regime 1 both disordered comb blocks are microphase separated from each other, in regime 2 the side chains C are microphase separated from the disordered A-b-B diblock backbones, and, finally, in regime 3 all species A, B, and C are microphase separated. In the first regime the behavior is similar to that of a simple diblock copolymer melt with a renormalized Flory-Huggins interaction parameter. In regime 2 the region of stability of the different phases is significantly changed compared to simple diblocks due to the comb architecture. The fully microphase separated case, regime 3, is characterized by hierarchical structure formation. We restrict the analysis to systems where self-assembly results in the formation of alternating C layers and internally microphase separated AB layers. The latter consist of alternating A and B layers or disks of the minority component. In the former case, the A and B layers are generally perpendicular to the C layers. The parallel orientation is only possible for small grafting densities.

Nonconventional morphologies in two-length scale block copolymer systems beyond the weak segregation theory

The order-disorder and order-order transitions (ODT and OOT) in the linear multiblock copolymers with two-length scale architecture A(fmN)(B(N2)A(N2))(n)B((1-f)mN) are studied under intermediate cooling below the ODT critical point where a nonconventional sequence of the OOTs was predicted previously [Smirnova et al., J. Chem. Phys. 124, 054907 (2006)] within the weak segregation theory (WST). To describe the ordered morphologies appearing in block copolymers (BCs) under cooling, we use the pseudospectral version of the self-consistent field theory (SCFT) with some modifications providing a good convergence speed and a high precision of the solution due to using the Ng iterations [J. Chem. Phys. 61, 2680 (1974)] and a reasonable choice of the predefined symmetries of the computation cell as well as initial guess for the iterations. The WST predicted sequence of the phase transitions is found to hold if the tails of the BCs under consideration are symmetric enough (mid R:0.5-fmid R:</=0.05); the quantitative agreement between the WST and SCFT phase diagrams is reasonable in a narrow (both in f and chi=chiN) region close to the critical point, though. For mid R:0.5-fmid R:>0.05, a large region of the face-centered cubic phase stability is found (up to our knowledge, first within the SCFT framework) inside of the body-centered cubic phase stability region. Occurrence of the two-dimensional and three-dimensional phases with the micelles formed, unlike the conventional diblock copolymers, by the longer (rather than shorter) tails, and its relationship to the BC architecture is first described in detail. The calculated spectra of the ordered phases show that nonmonotonous temperature dependence of the secondary peak scattering intensities accompanied by their vanishing and reappearance is rather a rule than exception.

Discovering ordered phases of block copolymers: new results from a generic Fourier-space approach

A generic Fourier-space approach to solve the self-consistent field theory of block copolymers is developed. This approach is based on the fact that, for any computational box with periodic boundary conditions, all spatially varying functions are spanned by the Fourier series determined by the size and shape of the box. The method reproduces all known diblock copolymer phases. The application of this method to a model "frustrated" triblock copolymer leads to a phase diagram with a number of new phases. Furthermore, the capability of the method to reproduce experimentally observed structures is demonstrated using the knitting pattern of triblock copolymers.

Microphase separation of double-grafted copolymers (centipedes) with gradient, random, and regular sequence of the branch points

We propose a theory of microphase separation of the double-grafted copolymer melts with different types of branch point distribution. It is shown that copolymers with gradient sequence of the branch points form a microstructure easier (at smaller values of the Flory-Huggins parameters) than equivalent copolymers with random or regular sequences. Two characteristic types of the microphase separation, "diblock-" and "comblike" are predicted. Due to the existence of the two characteristic length scales (the length of the backbone and the length of the side chains), the two-scale switching is possible.

Microphase separation in multiblock copolymer melts: Nonconventional morphologies and two-length-scale switching

The phase behavior of AfmN(BN2AN2)B(1-fmN) multiblock copolymer melts is studied within the weak segregation theory. The interplay between ordering on different length scales is shown to cause dramatic changes both in the ordered phase symmetry and periodicity upon small variation of the architectural parameters of the macromolecules. Phase diagrams are presented in the (f,chiN) plane (chi is the Flory-Huggins parameter) for various values of the architecture parameters n and m. Near the critical surface, i.e., for (f-0.5)2<<1, such nonconventional cubic phases as the face-centered cubic (FCC), simple cubic (SC), (double) gyroid, and the so-called BCC(2) (single gyroid) are found to be stable. The lamellar morphology is shown to be replaced by BCC2, FCC, or SC (depending on the structural parameters) as the most stable low-temperature phase.