Gyroid Phase Research Articles

Towards an understanding of phase transitions between inverse bicontinuous cubic lyotropic liquid crystalline phases

The inverse bicontinuous cubic phases that form in some lipid–water mixtures are both important structural elements in cells and emerging vehicles for nanotechnological applications. We model the relative phase behaviour of the three known inverse bicontinuous cubic lyotropic phases as the sum of the curvature elastic and chain packing energy of the membrane, under the assumption that the membranes form interfaces of constant mean curvature. The model correctly predicts a number of apparently universal, qualitative features of the relative phase behaviour of the gyroid (QGII), double diamond (QDII) and primitive (QPII) inverse bicontinuous cubic phases. These are: the phase sequence QGII → QDII → QPII with increasing water composition; the absence in certain cases of QPII from the phase diagram; the destabilisation of QPII with an increase in the temperature and the negative slope of phase boundaries with respect to temperature. Unexpectedly the model predicts the potential existence of a re-entrant QGII at high water dilutions that swells indefinitely. This has yet to be reported, which may reflect the difficulty of stabilising and then detecting such swollen, fluid interfacial structures. However, in the model the presence of the highly swollen QGII phase causes the adjacent phase to exhibit an almost vertical phase boundary, a phenomenon which should be more readily detectable.

Fast and accurate SCFT calculations for periodic block-copolymer morphologies using the spectral method with Anderson mixing

We study the numerical efficiency of solving the self-consistent field theory (SCFT) for periodic block-copolymer morphologies by combining the spectral method with Anderson mixing. Using AB diblock-copolymer melts as an example, we demonstrate that this approach can be orders of magnitude faster than competing methods, permitting precise calculations with relatively little computational cost. Moreover, our results raise significant doubts that the gyroid (G) phase extends to infinite chi N . With the increased precision, we are also able to resolve subtle free-energy differences, allowing us to investigate the layer stacking in the perforated-lamellar (PL) phase and the lattice arrangement of the close-packed spherical ( S (cp) phase. Furthermore, our study sheds light on the existence of the newly discovered Fddd ( O(70) morphology, showing that conformational asymmetry has a significant effect on its stability.

Block Copolymers with a Twist

Chiral block copolymers (BCPs*) comprising chiral entities were designed to fabricate helical architectures (i.e., twisted morphologies) from self-assembly. A new helical phase (H*) with P622 symmetry was discovered in the self-assembly of poly(styrene)-b-poly(l-lactide) (PS-PLLA) BCPs*. Hexagonally packed, interdigitated PLLA helical microdomains in a PS matrix were directly visualized by electron tomography. The phase diagram of the PS-PLLA BCPs* was also established. Phase transitions from the H* phase to the stable cylinder and gyroid phases were found after long-time annealing, suggesting that the H* is a long-lived metastable phase. In contrast to racemic poly(styrene)-b-poly(d,l-lactide) BCPs, chiral interaction significantly enhances the incompatibility between achiral PS and chiral PLLA blocks in the PS-PLLA BCPs* and can be estimated through the determination of the interaction parameter.

Strongly segregated cubic microdomain morphology consistent with the double gyroid phase in high molecular weight diblock copolymers of polystyrene and poly(dimethylsiloxane)

AbstractWe report the observation of a cubic phase consistent with the double gyroid structure in strongly segregated diblock copolymers of PS‐b‐PDMS over a volume fraction (φPDMS) range of ∼0.39 to 0.45. The samples have respective molecular weights of 127 kg/mol and 73 kg/mol and degree of segregation Nχ equal to 187 and 106, respectively, at annealing temperature of 130 °C. It is important to highlight that two out of the total four samples investigated, exhibited hexagonally close packed cylindrical domains of PDMS and alternating lamellae at φPDMS = 0.39 and 0.45, respectively, indicating the possible narrow range of the DG morphology for the specific diblock copolymers. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 2419–2427, 2009

Ordered Three- and Five-ply Nanocomposites from ABC Block Terpolymer Microphase Separation with Niobia and Aluminosilicate Sols

We report the first use of a non-frustrated block terpolymer for the synthesis of highly ordered oxide nanocomposites containing multiple plies. The morphological behavior of 15 ISO-oxide nanocomposites was investigated spanning a large range of compositions along the ƒ(I)=ƒ(S) isopleth using aluminosilicate and niobia sols. Morphologies were determined by TEM and SAXS measurements. Four morphologies were identified, including core-shell hexagonal, core-shell double gyroid, three-domain lamellae, and core-shell inverse-hexagonal, in order of increasing O+oxide vol fraction. All of the resulting nanocomposites had three- or five-ply morphologies containing domains that were continuous in one, two, or three dimensions. The five-ply core-shell double gyroid phase was only found to be stable when the O+oxide domain was a minority. Removal of the polymer enabled simple and direct synthesis of mesoporous oxide materials while retaining the ordered network structure. We believe that advances in the synthesis of multi-ply nanocomposites will lead to advanced materials and devices containing multiple plies of functional materials.

Gyroid Phase in Nuclear Pasta

Nuclear matter is considered to be inhomogeneous at subnuclear densities that are realized in supernova cores and neutron star crusts, and the structures of nuclear matter change from spheres to cylinders, slabs, cylindrical holes, and spherical holes as the density increases. In this Letter, we discuss other possible structures, that is, gyroid and double-diamond morphologies, which are periodic bicontinuous structures discovered in a block copolymer. Utilizing the compressible liquid drop model, we show that there is a chance of gyroid appearance near the transition point from a cylinder to a slab and the volume fraction at this point is also similar for nuclear and polymer systems. Although the five shapes listed initially have been long thought to be the only major constituents of so-called nuclear pasta at subnuclear densities, our findings imply that this belief needs to be reconsidered.

Self-Assembly of Dendronized Polymers

The gradual rigidification of a single dendronized chain upon increasing the size and density of the dendron units attached to it is studied using a Monte Carlo simulation. The dependence of the backbone flexibility on the size and density of the dendrons is used as an input to study the self-assembly of dendronized polymers using a real space, self-consistent field theoretic method. These calculations predict different phases in melts and solutions, ranging from lamellar to gyroid phases, depending upon various physical (temperature, concentration) and architectural (relative volume fraction of the backbone and the dendron units) parameters.

Relationship between Morphological Change and Crystalline Phase Transitions of Polyethylene−Poly(ethylene oxide) Diblock Copolymers. 3. Dependence of Morphological Transition Phenomena on the PE/PEO Segmental Lengths and Its Possible Origins

By measurement of the small-angle and wide-angle X-ray scatterings and infrared and Raman spectra and thermal data, microphase separation phenomena have been investigated for a series of polyethylene-poly(ethylene oxide) diblock copolymer (PE-b-PEO) in both the heating and cooling processes and compared with the structural changes occurring inside the PE and PEO domains. The complicated morphological changes between lamella, perforated lamella, gyroid, cylinder, and sphere phases were detected for the copolymer with relatively short PE segments. The orthorhombic crystalline structure of PE was kept unchanged in the lamella-to-gyroid transition. When the PE orthorhombic phase transformed to the pseudohexagonal or rotator phase, the gyroid morphology changed to the cylinder. On the other hand, the diblock copolymer with relatively long PE segment was found to show only the lamellar morphology, in which the order-disorder structural transition between the orthorhombic and pseudohexagonal phases occurred in the PE crystal region. As a possibility, the large difference in morphological change between the copolymers with short and long PE segments has been ascribed to the difference in thermal mobility of PE segments, which is controlled by the conformation of chains and their packing mode, i.e., an extended chain or a folded chain. The extended chains may move thermally and actively along the interfacial boundary in addition to the librational motion around the chain axis, resulting in a variety of morphological changes, whereas the thermal motion of the folded chains may be suppressed because of the geometrical constraint and does not cause such a large-scale morphological change from the lamellar structure. This concept, a thermal activity and geometrical constraint, is considered to be quite important in the interpretation of complicated morphological changes observed for many crystalline-amorphous and crystalline-crystalline diblock copolymers when viewed from the molecular level.

Spherical/gyroid phase diagram of the diblock copolymer in the median selective solvent

The effect of the median selective solution on the lamellar, spherical and gyroid structures is studied. The self-consistent field equations of the diblock copolymer solution are solved by using the reciprocal space method. It is shown that the spherical and gyroid phases have the lowest free energy in the certain range of the solution concentration. Furthermore, the phase diagram of the ordered structures in the diblock copolymer solution with the median selective solvent is calculated, which is consistent with the experimental results.

Order‐Order Transition of C → sdG → sL → S in ABC Triblock Copolymer Thin Film Induced by Solvent Vapor

The morphology transition of polystyrene-block-poly(butadiene)-block-poly(2-vinylpyridine) (SBV) triblock thin film induced in benzene vapor showing weak selectivity for PS is investigated. The order-order transitions (OOT) in the sequence of core-shell cylinders (C), sphere in 'diblock gyroid' (sdG), sphere in lamella (sL) and sphere (S) are observed. The projection along (111) direction in Gyroid phase (sdG(111)) is found to epitaxially grow from C(001) in the film. Instead of sdG(111), sdG(110)(0.1875) develops to the phase of sL. Consequently, the film experiences the transition sequence of sdG(111) → sdG(211) → sdG(110)(0.25) → sdG(110)(0.1875) between C and sL. The mechanism is analyzed from the total surface area of the blocks.

Order-to-Order Transitions of Block Copolymer in Film Geometry

The various mesophasic transitions for an asymmetric polystyrene-block-polyisoprene (PS-b-PI) in film geometry, like the order-to-order transitions, were investigated by in situ grazing incidence small-angle X-ray scattering (GISAXS). Compared with the sequential phase transitions in bulk, lamella (LAM) to hexagonally perforated layer (HPL) to gyroid (GYR) to hexagonally packed cylinder (HEX) to disorder (DIS) by thermal energy, the only difference in geometry leads to the enhancement of LAM phase, the parallel shifts of transition temperatures for HPL and GYR phases, and a little increase of order-to-disorder temperature (TODT) as a consequence of weak interfacial interactions. However, a short-range of HEX phase in bulk disappears. These results may be correlated to the suppressed compositional fluctuation in film geometry due to the interactions of polyisoprene block with the substrate and air.

Reinvestigation of the Block Copolymer Modulated Lamellar Structure

We report extended crystallographic studies on shear-aligned block copolymer systems within the metastable modulated lamellae (ML) state. With studies limited to the “classical” orientations parallel and perpendicular to shear plane, the apparent modulated state would likely have been assigned simple lamellar. Surprisingly, upon rotating the sample to intermediate angles additional scattering reflections appear, which reveal the apparent ML phase much beyond what was expected. The modulated structure is a slightly distorted fcc structure. With the sample sheared at relatively low temperature, presumably below the stable gyroid phase, we find a very well resolved ML texture corresponding to a simple twin structure of the distorted fcc structure. When shear-aligned within the hexagonal cylinder phase, and quenced to the gyroid phase or slightly below, we find ML alignment into a two-dimensional powder texture.

Relationship between Morphological Change and Crystalline Phase Transitions of Polyethylene−Poly(ethylene Oxide) Diblock Copolymers, Revealed by the Temperature-dependent Synchrotron WAXD/SAXS and Infrared/Raman Spectral Measurements

The phase transition behaviors of low-molecular-weight polyethylene-poly(ethylene oxide) (PE-b-PEO) diblock copolymers with the monomeric units of PE/PEO = 17/40 and 39/86 have been successfully investigated through the temperature-dependent measurements of wide-angle X-ray diffraction (WAXD), small-angle X-ray scattering (SAXS), infrared and Raman spectra, as well as thermal analysis. These diblock copolymers had been believed to show only order-to-disorder transition of lamellar morphology in a wide temperature region, but it has been found here for the first time that this copolymer clearly exhibits the three stages of transitions among lamella, gyroid, cylinder, and spherical phases in the heating and cooling processes. The WAXD and IR/Raman spectral measurements allowed us to relate these morphological changes to the microscopic changes in the aggregation states of PEO and PE segments. In the low-temperature region the PEO segments form the monoclinic crystal of (7/2) helical chain conformation and the PE segments of planar-zigzag form take the orthorhombic crystalline phase. These crystalline lamellae of PEO and PE segments are alternately stacked with the long period of 165 Angstroms. In a higher temperature region, where the PEO crystalline parts are on the way of melting but the PE parts are still in the orthorhombic phase, the gyroid morphology is detected in the SAXS data. By heating further, the gyroid morphology changes to the hexagonally packed cylindrical morphology, where the orthorhombic phase of PE segments is gradually disordered because of thermally activated molecular motion and finally transforms to the pseudohexagonal or rotator phase. Once the PE segments are perfectly melted, the higher-order structure changes from the cylinder to the spherical morphology. These morphological transitions might relate to the thermally activated motions of two short chain segments of the diblock copolymer, although the details of the transition mechanism are unclear at the present stage.

Oleoylethanolamide-Based Lyotropic Liquid Crystals as Vehicles for Delivery of Amino Acids in Aqueous Environment

We have investigated the phase behavior of self-assembled lyotropic liquid crystals (LC) formed by ternary mixtures of oleoylethanolamide (OEA), water and arginine. OEA, a natural analog of the endogenous cannabinoid anandamide involved in the peripheral regulation of feeding, was selected as a main component due to its capacity to induce efficient decreases in food intake and gains in body mass. Arginine was selected as representative hydrophilic amino acid and added to the OEA-water mixture at different concentrations. The phase diagrams were determined by combining cross-polarized optical microscopy and small angle x-ray scattering. First, the phase diagram for the OEA-water system was determined. It was shown that these two compounds give rise to reverse Ia3d double gyroid and reverse Pn3m double diamond cubic phases existing in bulk over a large window of temperature and composition, and that for water content beyond 25% Pn3m coexisted with excess water. Successively, the influence of arginine as guest molecule in the water channels of the reverse LC was investigated. For the sake of comparison, results for the OEA-water-arginine system were compared with analog series of OEA-water-glucose. The results showed that, at a fixed water content and temperature, the phase behavior of the liquid crystalline phases is strongly dependent on arginine concentration. In more detail, arginine could be encapsulated in the bulk OEA-water LC up to 2.0% wt, whereas transitions from Ia3d to Pn3m cubic phase were observed with increasing arginine concentration. Interestingly, upon an increase of water concentration beyond 20–25%, Pn3m phase started to coexist with excess water releasing the arginine in external water solution. Quantitative measurements of arginine content inside the LC water channels and in the excess external water solution revealed a complete release of the amino acid, demonstrating that the investigated lyotropic liquid crystalline systems can be used as ideal vehicles for the delivery of functional hydrophilic active molecules in aqueous environment.

A coarse-grained simulation study of mesophase formation in a series of rod–coil multiblock copolymers

The mesophase behaviour of a rod-coil multiblock copolymer is assessed by means of a new soft-core simulation model, which is suitable for the simulation of combinations of isotropic and anisotropic particles. The simulations demonstrate the presence of isotropic melt, micelles, lamellar, nematic and gyroid phases, with ordered phases able to grow spontaneously from the isotropic melt. The influence of increasing the length of the rigid rod-component of the polymer is studied, with mesophase stability enhanced by increases in rod length. Increased nematic phase stability is demonstrated also as the length of rod component is increased. For longer rods (smaller fraction of coils) the models show evidence of metastable chevron-like structures that initially form on cooling from the polymer melt. These are eventually lost in favour of true lamellar ordering over long annealing runs. The structure of molecules within the phases formed is also assessed. Chains in the lamellar phase are shown to produce both bridging and loop behaviour, with the latter preferred slightly. Simulations at a low occupied volume fraction, corresponding to self-assembly in solution, demonstrate the formation of a structurally ordered nanowire.

Shear rheology of amphiphilic cubic liquid crystals from large-scale kinetic lattice–Boltzmann simulations

We investigate the rheological characteristics of ternary amphiphilic gyroid, diamond and primitive cubic phases under applied Couette flow simulated using a kinetic lattice–Boltzmann model and periodic Lees–Edwards boundary conditions. The simulated rheological response of the cubic phases is compared to experimental observations in lyotropic liquid crystals. We relate the variations in the non-Newtonian response and deformation under strain in these cubic phases to their triply bicontinuous cubic morphologies as well as to the differences in the interaction parameters between the three species present in the amphiphilic system. The large system sizes allow simulation of multiple domains which elucidate the correlation between the evolution of the defect texture and the change in the stress field of the cubic phase under applied Couette flow.

Synthesis and self-assembly in bulk of six-arm star-block copolymers with a C60 core

This study focuses on the synthesis and the structural characterization of various branched star-block copolymers (polyisoprene-block-polystyrene)6C60 with a fullerene C60 as a core. Well defined 6-arm stars (PI-b-PS)6C60 with a low polydispersity and a precise control of the number of branches were prepared by grafting PI-b-PS diblock copolymers through the polystyrene block onto the C60 core. The self-assembled structures formed in bulk were studied by Transmission Electron Microscopy (TEM) and Small Angle X-ray Scattering (SAXS) for both symmetric and asymmetric polystyrene-block-polyisoprene (PS-b-PI) diblock arms in the strong-segregation regime (65≤χN≤115). Various microstructures including lamellae, hexagonal packings of PS and PI cylinders as well as a gyroid phase were obtained by varying the volume fraction of polystyrene (fPS) of the branches, leading to the formation of ordered, periodic and localized nanoscale dispersions of the C60 in a polymer matrix including planes, threads and a 3D bicontinuous network of C60.

Universal sequence of ordered structures obtained from mesoscopic description of self-assembly

Mesoscopic theory for soft-matter systems that combines density functional and statistical field theory is derived by a systematic coarse-graining procedure. For particles interacting with spherically symmetric potentials of arbitrary form, the grand-thermodynamic potential consists of two terms. The first term is associated with microscopic length-scale fluctuations, and has the form of the standard density functional. The second term is associated with mesoscopic length-scale fluctuations, and has the form known from the statistical field theory. For the correlation function between density fluctuations in mesoscopic regions, a pair of equations similar to the Ornstein-Zernicke equation with a new closure is obtained. In the special case of weak ordering on the mesoscopic length scale, the theory takes a form similar to either the Landau-Ginzburg-Wilson (LGW) or the Landau-Brazovskii (LB) field theory, depending on the form of the interaction potential. Microscopic expressions for the phenomenological parameters that appear in the Landau-type theories are obtained. Within the framework of this theory, we obtain a universal sequence of phases: disordered, bcc, hexagonal, lamellar, inverted hexagonal, inverted bcc, disordered, for increasing density well below the close-packing density. The sequence of phases agrees with experimental observations and with simulations of many self-assembling systems. In addition to the above phases, more complex phases may appear depending on the interaction potentials. For a particular form of the short-range attraction long-range repulsion potential, we find the bicontinuous gyroid phase ( Ia3d symmetry) that may be related to a network-forming cluster of colloids in a mixture of colloids and nonadsorbing polymers.

Morphological Changes in Block Copolymer Melts Due to a Variation of Intramolecular Branching. Dissipative Particles Dynamics Study

We performed dissipative particles dynamics (DPD) simulations for the melts of various star and comb diblock copolymers with equal molecular weight (L = 41 beads) but with various types of internal branching. To assist formation of native morphologies, an anisotropic NPT ensemble was used. For the linear diblock polymer, we reevaluated the phase diagram in a high segregation regime and confirmed existence of the gyroid phase. The equilibrium morphologies and molecular conformations of branched molecules are compared with those found in the linear diblock copolymers at the same composition fraction. The analysis of the morphologies is performed in terms of metrics, spacial arrangement, and conformations of the molecules. Both weak (no phase change) and strong (phase changes) branching effects are observed. Promotion of a less ordered phase occurs in the case when highly constrained beads are unable to fill in required space regions (defined by the phase pitch) homogeneously. As a result, more ordered phase (observed for the linear diblock) cannot be formed. In some cases, an opposite effect is observed, e.g., promotion of more ordered lamellar phase (in place of the hexagonal one for the linear diblock) by suitable symmetry of the comb architecture.

Possible Formation of Multicontinuous Structures by Rodlike Particles

It is shown by extending the Landau theory of freezing that a tendency to form multicontinuous complex structures is possessed by simple systems consisting of rigid rodlike particles. The excellent stability of the Gyroid phase, which is most widely observed in nature, is naturally explained. An exotic structure found in thermotropic liquid crystals can be further regarded as an example.