The double gyroid (DG) structure is a 3-dimensionally periodic bicontinuous structure with Ia3 d symmetry that consists of two interpenetrating networks of the minority component and a matrix of the majority component (Figure 1). This ordered structure with Ia3 d symmetry was first identified in strontium lipids in 1967. 1 The first report of the DG structure in block copolymers was made in 1994. 2 Since then the DG has intrigued the polymer science community due to its structural complexity compared to the more common ordered structures of layers and hexagonally-packed cylinders found in linear block copolymers. 2 Moreover, the percolating domains of the DG morphology contribute to enhancing mechanical properties such as modulus, toughness, and creep resistance of materials relative to the other morphologies. [3][4][5] The bicontinuous domains of DG can also improve ionic conductivities when the interpenetrating domains contain ionic functionalities. [6][7][8][9] The triply periodic structure of DG can even control the optical properties and its application into metamaterials. [10][11][12][13] Followed by the first assignment of DG in the linear polystyrene-polyisoprene diblock copolymers (PS-b-PI), 2 comprehensive investigations of polystyrene-polyisoprene block copolymers have advanced the understanding of the structural characteristics of DG in polymeric systems. [14][15][16][17][18] Note that the DG structures in star block copolymers and other block copolymer systems were originally reported to be double-diamond morphologies with Pn3m symmetry and were later correctly identified as exhibiting the DG. 18 Subsequently, various block chemistries including polystyrene-b-poly(2-vinyl pyridine), 19,20 polyethylene-bpolyethylethylene, 21,22 and many others 23,24 also identified the DG in diblock copolymers. Building upon the design rules to generate DG structures from linear diblock copolymers, self-consistent field theory (SCFT) was utilized to predict the equilibrium morphologies within the χN-f phase diagram, where χ, N, and f are the Flory-Huggins interaction parameter, degree of polymerization, and the volume fraction of one block component, respectively. The SCFT revealed that the DG structure is thermodynamically stable within a relatively narrow volume fraction range of ~3.7 vol% at intermediate segregation strength of χN = 20 and ~1.5 vol% at strong segregation of χN = 100 (Figure 2a). 25 Note that the DG was originally found to be unstable at high-χN values due to the increased sharpness at the interface and packing frustration, while experiments and recent SCFT calculations find that the DG exists at high χN. [25][26][27][28] The narrow composition range of DG in conventional linear diblock copolymers is due to the triply periodic minimal surface and chain packing within the DG. Specifically, polymer chains should minimize the interfacial area and uniformly fill the domains defined for each block species. 29,30 Linear diblock copolymers that satisfy both conditions are often limited to only a few volume fraction range according to the self-consistent field theory (SCFT) calculation. Experimentally, PS-b-PI diblock copolymers selfassemble into the DG structures at f PI = 0.36 -0.39 and 0.65 -0.68 (Figure 2b). 17,31 The DG structures have been widely observed in more complicated polymer systems than simple diblock copolymers, such as AB/A diblock copolymer/homopolymer blends, 2,32,33 AB/A`B` binary diblock copolymer blends, 21,34 tapered block copolymers, 35 and ABA triblock copolymers. 4,[36][37][38] In addition, ABC triblock copolymers exhibit a core-shell type DG (Ia3 d symmetry) [39][40][41][42] and alternating gyroid (I4 1 32 symmetry) [43][44][45] structures depending on the block species. DG structures exist at a wider volume fraction range in diblock/homopolymer, due to the additional chain ends and increased chain length dispersity that alleviates packing frustration. [46][47][48] For example, polystyrene-b-polyisoprene diblock copolymers (10.1 kg/mol PS and 17.3 kg/mol PI) with homopolystyrene (760 g/mol hPS) or homopolyisoprene (650 g/mol hPI) blends demonstrated that the volume fraction window for the DG structure increases up to ~12 vol% after a long time of annealing (Figure 2c). 2 Alternatively, introducing a more flexible linkage between blocks of diblock copolymers expands the accessible DG window (Figure 2d). 49 A comprehensive phase map of PI-b-PS-b-PEO identified core-shell DG structures with PS encapsulating PEO to form both gyroid structures when the PI matrix volume fraction was 0.45 -0.50. [40][41][42] Note that these block copolymers have been synthesized using predominantly anionic polymerization methods to control the composition range required to produce the DG with a molecular weight dispersity index of 1.01 -1.11. 20,24,50 While the polydispersity index can significantly impact the self-assembly of block copolymers, [51][52][53][54] dissipative particle dynamics simulations showed that increasing polydispersity index of ABA linear block copolymers results in the destabilization of ordered DG structure into disordered bicontinuous morphology. 55 Experimental studies on ABA triblock copolymers identified the disordered bicontinuous morphologies at a composition window of ~10 vol%. 56,57 In contrast, the layered morphologies of ABC triblock copolymers can transform into core-shell DG structures upon increasing the polydispersity of one block species. 58 In summary, the DG morphology has been found in a wide variety of systems containing AB, ABA, and ABC block copolymers and the composition window within which the DG morphology resides is narrowest for diblock copolymers and broadens for more complex systems.
Linear (AB) n alternating multiblock copolymers have the potential for designing ordered nanostructures as they can self-assemble into various nanostructures. 59 For example, SCFT calculations predict that the equilibrium morphologies of (AB) n multiblock copolymers are identical to the conventional diblock copolymer morphologies with layers, DG, cylinders, and spheres. 31 The number of repeating units (n) impacts various characteristics of (AB) n alternating multiblock copolymers. First, the microphase separation of multiblocks requires ~50% higher segregation strength (χN crit ~ 15.1 at n > 20) than the diblock copolymers (χN crit ~ 10.5 at n = 1). 60 Also, the domain spacings of ordered structures of (AB) n alternating multiblock copolymers decrease with an increasing n, particularly at n < 10. 61,62 Therefore, theory predicts that (AB) n multiblock copolymers at sufficiently high segregation strength will produce ordered structures, as well as the desired DG structures. Nevertheless, experimental studies showing ordered structures in linear (AB) n multiblock copolymers are far fewer as compared to studies of AB diblock copolymers, presumably due to the (1) synthetic challenges to produce alternating and monodisperse blocks and (2) insufficient block segregation strength upon increasing the number of repeating blocks. Producing (AB) n alternating multiblocks using anionic polymerization requires 2n polymerization steps and the polydispersity of each block length impedes the ordering of microphase separated structures. In linear (PS-b-PI) n multiblock copolymers, an increase of n resulted in the decrease of the grain size of the layers, i.e. less ordering with increasing n. 63,64 For these reasons relative to diblock copolymers, accessing ordered morphologies including the DG in linear (AB) n multiblock copolymers has been quite limited.
Recent studies of precise ion-containing copolymers highlight the phase behavior and formation of DG structures in alternating multiblock architectures. Figure 3a presents the step-growth synthesis of polyester sulfonate multiblock copolymers (PESxM) using a sulfosuccinate diester with a counterion and an alkyl diol of fixed length of x-carbons. These PESxM polymers are linear (AB) n multiblock copolymers with strictly alternating polar ionic and non-polar blocks. The polydispersity index for the length of AB repeating subunit (N) is exactly 1.00, because the polar and non-polar block lengths are fixed by the monomers. Note that we define N as the number of backbone atoms in the AB repeat unit, N = x+ 6. In these (AB) n multiblock copolymers, the number of AB repeating units (n) becomes trivial with increasing n as the chain-end effects become negligible. In PES23Li the hydrocarbon blocks crystallize below the melting temperature (T m ) ~ 123 °C and the short polar blocks self-assemble into a layered (LAY) morphology as indicated by the differential scanning calorimetry (DSC) traces (Figure 3b). Upon heating above the T m , X-ray scattering with reflections peaks at √6, √8, √14, √16, √20, √22, √24, and √26 indicate that PES23Li (n = 11) forms a well-defined DG morphology at 150 °C (Figure 3C). 7 The phase transitions found in PES23Li are altered by the choice of counterion. Figure 4 illustrates the ionic aggregate and hydrocarbon chain structures of PES23M as a function of temperature and cation species (M = Li + , Na + , Cs + , and NBu 4 ). PES23NBu 4 is an intermediate product that includes bulky quaternary ammonium cations prior to the cation exchange with Li, Na, and Cs. For all PES23M polymers, the hydrocarbon blocks form hexagonally packed crystals below the T m . For PES23NBu 4 with a bulky quaternary ammonium cation and weak electrostatic interaction, the melting transition is relatively low, T m ~32 °C, and the polymer transitions directly from layered to disordered ionic aggregates upon heating. In contrast, PESxM polymers with metal cations show melting transition temperatures higher than 100 °C and produce DG structures above the T m . The order-to-order transition temperatures (T OOT ) as measured by DSC increase with the increase of cation size (Li < Na < Cs). PES23Li and PES23Na show GYR -HEX transitions upon heating, while the HEX morphology is inaccessible in PES23Cs below 210 °C. When polar ionic diblocks strongly interact via Coulombic cohesion, OOT phase boundaries shift to lower values of χN. 65 Since χ ~ 1/T, this implies that the T OOT increases with increased Coulombic interaction between ionically charged blocks. For PES23M, the GYR to HEX T OOT increases from Li to Na to Cs, indicating stronger electrostatic cohesion with increasing cation size.
The volume fraction of the polar block in these (AB) n alternating multiblock copolymers is readily controlled by the selection of the alkyl chain length of x-carbons. This kinetic competition between the self-assembly of the polar blocks and the crystallization of the hydrocarbon blocks is also evident by comparing PES48Na and PES10Na upon cooling below T m . For PES48Na, the 48-carbon alkyl block crystallizes to form layered ionic aggregates during the in situ X-ray experiment, while the 10-carbon alkyl block of PES10Na persists in the HEX morphology without crystallization. Since f p of PES48Na and PES10Na are 0.16 and 0.45, respectively, PES48Na exhibits a stronger driving force to crystallization than PES10Na. The hysteresis displayed in Figure 5 for PESxNa multiblock copolymers suggests that the accessible temperature window for DG morphologies can be extended to lower temperatures, perhaps even room temperature, by impeding the crystallization of the hydrocarbon backbones. For example, substituting the linear diol with a non-crystallizable diol may produce the DG at room temperature. In the PESxNa alternating multiblock copolymers the DG structures are observed over an unexpectedly wide composition range. One explanation might be the strong electrostatic cohesion between the polar ionic blocks, which lead to asymmetric phase diagrams. In block copolymers with charged moieties tethered to the backbone, the composition of the charged block disproportionally impacts the microphase-separated morphologies. [67][68][69] The physical properties of PESxNa, such as the non-Gaussian chain statistics of the short alternating blocks and conformational asymmetry of blocks, may further skew the phase diagram. While PESxM systems have charges covalently tethered to the backbone, extensive research on diblock copolymers (e.g., PS-b-PEO) with added salt have found that the salt content significantly impacts the equilibrium morphologies. [70][71][72][73][74][75] Specifically, a substantial shift of the phase boundaries of salt-doped block copolymers is often explained by factors including ion solvation energy and ion-ion correlations. Clearly, the thermodynamics of equilibrium morphologies in ion-containing block copolymer systems are not yet fully understood. To the best of our knowledge, theoretical studies are lacking to describe the wide range of DG structures and the phase behavior of these ion-containing (AB) n multiblock copolymers.
Figure 6 shows the isothermal lattice parameters (a) of the three ordered morphologies observed in PESxNa polymers along with their scaling relationships to the number of backbone atoms (N = x + 6).
The lattice parameters are small (< 10 nm) and exceptionally well-controlled by selecting the length of the aliphatic diol monomers. The effect of n on the lattice parameter appears to be negligible for the PESxNa polymers, because the value of n spans from ~10 for PES23Na to ~37 for PES12Na. The SCFT for (AB) n multiblock copolymers predicts the domain spacings of layers when n = 10 is only slightly larger (< 2%) than that of n = 37. The scaling relationship of a ~ N 0.92 for LAY is attributed to the crystallization of hydrocarbon, and therefore the distance between the layered ionic aggregates is proportional to the number of carbons in the hydrocarbon block. The relationships for DG (a ~ N 0.67 ) and HEX (a ~ N 0.52 ) morphologies with amorphous hydrocarbon chains coincide with the scaling relationship of strongly and weakly segregated neutral diblock copolymers, respectively. By comparison, experimental results for diblock copolymers observed exponents of ~0.8 -1.0. [76][77][78][79] The a-N relationships and morphology map of PESxNa reveal that the nanoscale ordered structures and their length scales in PESxNa multiblock copolymers can be finely tuned at the length scale of 2 -8 nm. 15 25 35 45 55 2 3 4 5 6 7 8 a (nm) N a GYR ~ N 0.67 a LAY ~ N 0.92
Further development of (AB) n multiblock copolymer thermodynamics requires knowledge of block interactions. The Flory-Huggins interaction parameter (χ) is obtained from the disordered morphology of PES12Li using the random phase approximation for (AB) n multiblock copolymers (Figure 8a). 62 The quality of the fit of the random phase approximation theory to the experimental scattering data is high even though the alternating block lengths in PES12Li are short. In Figure 8b, the temperature dependence of χ = 77.4/T + 2.95 (T in Kelvin) with a reference volume of 0.118 nm 3 indicates a high enthalpic contribution from ionic interactions and a high entropic contribution from short block lengths. The value of χ at 25 °C is 3.21, identifying the PESxLi ion-containing multiblock copolymers as ultrahigh-χ and low-N block copolymers. This is consistent with the formation of ordered nanostructures with sub-3 nm domain spacings.
For comparison, the value of χ at 25 °C for polystyrene-b-poly(methyl methacrylate) is 0.043. 80,81 Although the χ value was inaccessible for PESxNa, the presence of ordered morphologies at T > T m for PES12Na and PES10Na suggests χ values even higher than PESxLi. The ultrahigh-χ and ordered morphologies with sub-3 nm domain spacings suggest a new direction for template-assisted nanofabrication technologies using (AB) n multiblock copolymers. The experimentally observed morphology transitions of PESxLi polymers are offset from the SCFT predictions: GYR-HEX transitions for PES23Li and PES18Li, and LAY-DIS transition for PES12Li are shifted to a lower f p relative to the theoretical boundaries. The discrepancies between the phase behavior of PESxLi and the SCFT phase boundaries can be attributed to the short block lengths and the electrostatic interactions between the polar blocks. Theoretically, mean-field approximations including the selfconsistent field theory (SCFT) used to describe the uncharged polymer systems are insufficient for predicting phase behavior in electrostatically charged systems because of the field fluctuations. [82][83][84] Theoretical models have introduced the fluctuation effects to consider the electrostatic interactions in the phase-separated polymer systems. [82][83][84] More recently, a polarizable field-theoretic model explored the electrostatically stabilized microphase separation in a blend of oppositely charged polymer systems, which showed a drastic shift of phase boundaries. 84 A hybrid self-consistent field theory and liquid state integral equation theory (SCFT-LS) demonstrates that the ionically charged blocks of diblock copolymers significantly skew the phase boundaries toward a lower volume fraction of charged blocks. 65 The extent of boundary shift increases as a function of charge fraction and Coulombic interaction strengths. Similarly, dissipative particle dynamics (DPD) simulations show this shift of phase boundaries with an increasing charge fraction. 85 Further development of these theoretical approaches incorporating polarized field effects are needed to improve the understanding of phase behavior in precise ion-containing (AB) n multiblock copolymers. Copyright 2021 American Chemical Society.
To provide a deeper understanding of morphology characteristics of precise ion-containing multiblock copolymers in confined geometries, PESxLi thin films were prepared and examined with in-situ grazing-incidence X-ray scattering (Figure 10). 86 At 40 °C, distinct in-plane scattering peaks at q y ~ 1.5 Å - 1 indicate that the crystalline hydrocarbon blocks pack with an interchain distance of 0.4 nm and the chain axis vertically aligned relative to the substrate (Figure 10a). The out-of-plane scattering peaks along the q z -axis indicate well-defined ionic layers parallel to the substrate, where the layer spacing is 3.1 nm (Figure 10d). Upon heating above T m , the layered morphology spontaneously transitions into highly-oriented DG with an epitaxial transition from the (100) plane of LAY to the (211) plane of DG parallel to the substrate (Figure 10e). 87 The domain spacing calculated from the primary (211) peak of the DG structure is ~2.5 nm.
Further heating to 180 °C allows an epitaxial transition of DG into HEX morphology, where the cylinders are parallel to the substrate (Figure 10f). 19 The transitions between the DG and HEX are reversible, and the DG to LAY transition is kinetically trapped due to slow crystallization consistent with bulk behavior. The
So far, this perspective has highlighted the DG morphology in one class of (AB) n multiblock copolymers composed of alternating polyester sulfonate with a metal counterion and hydrocarbon blocks.
After inspecting results from other classes of (AB) n polymers, we will propose criteria for achieving the DG morphology. First, the removal of ionic groups from PES23M results in the absence of ordered morphologies above T m , and we attribute this to a significant reduction in χ. 7 Therefore, the blocks of (AB) n multiblock copolymers should be highly incompatible to achieve ordered morphologies at sub 10-nm length scales.
Figure 11 shows a polyethylene oxide (PEO) based multiblock copolymer synthesized by the A 2 + B 2 melt polycondensation of a hydroxy-terminated oligo(ethylene glycol) monomer and a 5sulfoisophthalate salt. 88 In these ionomers, PEO lengths of m ~ 9 and 13 with Li + , Na + , and Cs + exhibit disordered ionic aggregate morphologies with an amorphous PEO backbone. When the length of PEO is long enough at m ~ 25 and 75, the PEO blocks crystallize and the ionic aggregates form layers. Upon heating above the T m , these PEO-based multiblock copolymers do not form ordered morphologies such as DG and HEX. The absence of ordered morphologies in these PEO-based ionomers can be attributed to the absence of precision in block lengths and the lack of chain flexibility in the polar block. First, the PEO block lengths are polydisperse as compared to the precise hydrocarbon lengths of the PESxM polymers. It is well-established that randomly distributed spacer lengths give rise to poorly defined ionic aggregate morphologies compared to precise spacer lengths. [89][90][91] Second, the polar blocks with rigid phenyl ring in the backbone are less flexible and impede the chain packing necessary to form DG or HEX morphologies.
In addition, the segregation strength of these PEO-based ionomers will be weaker than polyethylene-based PESxM multiblock copolymers, pushing them toward the disordered state. The lack of precision in block length, a rigid polar block, and a smaller χ combine to impede the self-assembly of these PEO-based sodium sulfonated polyesters into ordered nanostructures. Figure 12a shows the precise acid-containing polymers synthesized via acyclic diene metathesis (ADMET) polymerization of diene monomers containing a symmetric pendant functionality. All-atom molecular dynamics simulations at T > T m show that the precise polymer containing COO -Li + on every 21st carbon (p21AA-Li, f p = 0.15) exhibits disordered, stringy, and percolated ionic aggregates, Figure 12b. 92 Similar stringy ionic aggregates are formed above T m when the periodicity between the ionic groups is shorter (p9AA-Li and p15AA-Li), 92,93 indicating that the non-periodic packing of ionic aggregates is preferred relative to ordered morphologies in these ADMET polymers with pendant acid or ionic groups.
One exception to this generalization is the precise ADMET polymer containing geminal phosphonic acids on exactly every 21st carbon (p21gPA, f p = 0.25) that exhibits low symmetry diffraction peaks assigned to spherical aggregates on a face-centered cubic (FCC) lattice symmetry (Figure 12c). 94,95 In comparison with the PESxM polymers that contain ester linkages and short polar blocks, the ADMET polymers have allcarbon polymer backbones and the pendant groups on just one carbon. These features of the acid-and ioncontaining ADMET polymers provide fewer chain conformations to accommodate microphase separation of the functional groups and prevent self-assembly into ordered morphologies, including the DG morphology. Given the chemical similarities between these ADMET polymers and the ionic lipids that originally displayed the DG morphology, 1 the absence of the DG in these polymers is unexpected. The physicochemical properties of the ionic groups, chain flexibility, and block incompatibility of the PESxM polymers provide insight into designing DG-forming multiblock copolymers. Below we propose four criteria for (AB) n multiblock copolymers to produce the DG and other ordered morphologies at sub-10 nm length scales.
The A and B blocks are highly incompatible, for example, pairing ionic and non-ionic blocks.
Both the A and B blocks are comprised of flexible chains to accommodate the surface curvature of the double gyroid structure and other ordered morphologies.
The lengths of the A and B blocks are precise so that the polydispersity index of the AB unit is exactly 1.00.
The volume fraction of the minority block is 0.27 -0.41 to form double gyroid structures.
Lower and higher volume fraction can form hexagonally packed cylinders or layers, respectively.
This Perspective summarizes the recent developments of precise ion-containing multiblock copolymers that self-assemble into double-gyroid structures at an unusually wide composition range of > 14 vol%.
Step-growth polymerization methods could be developed to synthesize an even wider variety of precise ion-containing multiblock copolymers, and greatly expand the investigation of self-assembly in (AB) n copolymers. Current self-consistent field theories fail to capture the phase behavior of these precise ion-containing polymers, and this is primarily attributed to the presence of charges that induce electrostatic interactions and significant density fluctuations. A combination of experimental, theoretical, and simulation studies are required to fully establish a foundation for designing ordered nanostructures including DG with precise ion-containing multiblock copolymers.
Numerous applications can be envisioned for new (AB) n multiblock copolymer synthesized by stepgrowth polymerization. First, precise multiblock copolymers could be used for nanopatterning templates to achieve sub-3 nm domain spacings by having ultrahigh-χ and low-N. Key next steps include exploring multiblock copolymer kinetics, directed self-assembly of ultrahigh-χ polymers, and sequential infiltration synthesis within sub-3 nm domains to improve etching contrast. In addition, porous materials templated from the precise multiblock copolymers could be explored as filtration membranes. Another set of potential applications involves selective ion transport, including single-ion conductors having covalently bonded ionic functionalities to the polymer backbone as in the PESxM polymers. The ordered morphologies in thin films of these ion-containing polymers create well-aligned ion transport channels, which can be further optimized for high ionic conductivity by selecting optimal block chemistries from step-growth polymerization and by the selective solvation with additives to swell the polar domains and dissociate the ion pairs. Precise ion-containing multiblock copolymers have great potential to expand understanding of block copolymer physics and address a variety of technological challenges.