Diffusion in Lamellae, Cylinders, and Double Gyroid Block Copolymer

Aug 30, 2018 - Thus, among randomly oriented standard minority phase structures with no grain boundary effects, lamellae is preferable for transport...
0 downloads 0 Views 829KB Size
Download PDF
Letter Cite This: ACS Macro Lett. 2018, 7, 1092−1098

pubs.acs.org/macroletters

Diffusion in Lamellae, Cylinders, and Double Gyroid Block Copolymer Nanostructures Kuan-Hsuan Shen, Jonathan R. Brown, and Lisa M. Hall* William G. Lowrie Department of Chemical and Biomolecular Engineering, The Ohio State University, 151 West Woodruff Avenue, Columbus, Ohio 43210, United States

ACS Macro Lett. Downloaded from pubs.acs.org by UNIV OF SOUTH DAKOTA on 08/31/18. For personal use only.

S Supporting Information *

ABSTRACT: We study transport of penetrants through nanoscale morphologies motivated by common block copolymer morphologies, using confined random walks and coarse-grained simulations. Diffusion through randomly oriented grains is 1/3 for cylinder and 2/3 for lamellar morphologies versus an unconstrained (homopolymer) system, as previously understood. Diffusion in the double gyroid structure depends on the volume fraction and is 0.47−0.55 through the minority phase at 30−50 vol % and 0.73−0.80 through the majority at 50−70 vol %. Thus, among randomly oriented standard minority phase structures with no grain boundary effects, lamellae is preferable for transport.

B

through one of the microphases is significantly slower than that through the other and the pathways are straight, the overall diffusion depends on the fraction of orientations of the morphology that allow diffusion. For instance, cylinders allow conduction in one direction and lamellae in two; therefore, the overall diffusion is reduced to 1/3 and 2/3, respectively, of the diffusion in a bulk system. To help explain relative conduction through different morphologies, the Balsara group defined a morphology factor f = σ/(σcϕc)=D/Dc, where σ and D are the conductivity and diffusion constant in BCPs, respectively; σc and Dc are those in homopolymers of the conducting phase, and ϕc is the volume fraction of the conducting phase.8,9 Note that this relationship between conductivity and diffusion constant ratios is true only if the degree of correlation in ion motion (fractional deviation from the Nernst−Einstein equation) is the same in the BCP and homopolymer, which is expected at constant local chemistry and concentration of ions. Some studies have instead reported the tortuosity τ = σcϕc/σ,35−37 the reciprocal of the morphology factor. Multiple experiments have validated the prediction of f = 1/3 for cylinders and f = 2/3 for lamellae, especially at high polymer molecular weight or high salt concentration.10−13,33,34 However, there remains significant debate regarding diffusion through BCPs with the gyroid morphology or other network phases. Some have hypothesized that diffusion through the gyroid phase could be as efficient as in the unconfined case since ion transport is possible in all three dimensions, regardless of grain orientation;8,14,20,22,38 nevertheless, these works did not consider the internal tortuosity of

lock copolymers (BCPs) are widely used in transport applications, as their chemically distinct polymer components locally microphase separate into domains with different material properties.1−4 For battery electrolyte applications, ions dissolve in and diffuse through one microphase while the other provides mechanical robustness, potentially allowing for both ion conduction and the ability to block lithium dendrite growth at the same time.5−15 The BCP systems of interest are locally ordered with well-known microphases such as hexagonally packed cylinders, the double gyroid (which we refer to as gyroid here for brevity) or other bicontinuous network phases, and lamellae. However, on a longer length scale such as the width of a typical membrane, multiple grains with these structures are present, and these grains are often considered to be randomly oriented with respect to each other. Thus, conduction depends on both the local tortuosity of the microphase structure and the overall connectivity of pathways across different grains.16,17 In some cases, grains can be aligned to increase the overall conductivity, though this adds significant complexity to the material’s processing.18,19 The gyroid or other bicontinuous network morphologies have attracted interest, as they are inherently interconnected in three dimensions and potentially can provide both good transport and mechanical properties.20−26 The double gyroid phase is also of interest in different kinds of materials such as ceramics,27,28 lipids,29 photonic crystals,30,31 and lyotropic liquid crystals.32 Prior experimental and theoretical work has developed a consistent understanding of how topology and grain orientations contribute to the overall ion transport in BCPs with cylinder and lamellar morphologies. Sax and Ottino used effective medium theory to predict diffusion through randomly oriented grains by averaging the diffusion coefficients in three principal directions within each grain.33,34 When the diffusion © XXXX American Chemical Society

Received: July 20, 2018 Accepted: August 27, 2018

1092

DOI: 10.1021/acsmacrolett.8b00506 ACS Macro Lett. 2018, 7, 1092−1098

Letter

ACS Macro Letters

phase”. In our simplest model, the random walk (RW) model, we predict diffusion based on numerical results of random walks confined by nonpenetrable surfaces to account for only the tortuosity of the overall microphase structure. We also study the influence of the interface on diffusion by performing Langevin dynamics simulations on particles confined by surfaces of different roughness that represent BCPs at different segregation strength χN (results are shown in the Supporting Information). Finally, we perform molecular dynamics (MD) simulations using a coarse-grained bead−spring model with added selective penetrants, allowing us to study the effect of molecular weight on the transport properties. For each model, we study morphologies of cylinders (C), lamellae (L), inner gyroid (Ginner, where the minority component is the conducting part), and outer gyroid (Gouter, where the majority component is the conducting part) with the span of fA = 0.2− 0.3 for C, 0.3−0.5 for Ginner, 0.35−0.65 for L, and 0.5−0.7 for Gouter (based roughly on the regions where these structures are preferred).57 In all models, we consider diffusion through a single grain, and predict overall diffusion based on a randomly oriented structure with no grain boundary effects. We note that the fact that experimental groups can find close to the predicted 1/3 and 2/3 factors for randomly oriented cylinders and lamellae, even though these structures have grain boundaries, suggests that, under the right conditions, grain boundary effects are small (microphases connect at grain boundaries to locally reduce the unfavorable interfacial area). Such effects should not be significantly different for gyroid versus cylinders and lamellae. Details of models and dynamic analyses are discussed in the Supporting Information. Over long enough time scales, a diffusing particle’s motion follows a random walk, and random walk simulations have been used to study transport in various heterogeneous media.59−63 Here, we consider particles placed in the conducting domain that move as random walks constrained by the surfaces of the domain (if any random walk step would to cross the surface, it instead is reflected). We calculate the diffusion constant (D) of the constrained particles relative to that of particles with no constraints (DNC), as shown in Figure 1. As expected, D/DNC is around 33% and 67% for cylinders and lamellae, as diffusion occurs only in one and two dimensions in these respective systems. Interestingly, D/DNC

the gyroid structure. Multiple experimental groups have worked to measure the tortuosity of gyroid14,20,34,39 or other bicontinuous structures.37,38,40−43 There has been significant disagreement, even regarding whether conductivity through gyroid14,20,34,39 or other bicontinuous networks38,40 is higher or lower than that through unaligned lamellae. Meanwhile, there is some clustering of results for tortuosity of gyroid or other networks between 1.5 and 3, but it is unclear whether the discrepancies are due to differences in structures considered, experimental error, or nonidealities in transport through different systems.36,44−48 Theory has provided insight into the ideal tortuosity of various regular network structures, especially certain minimal surface structures. Anderson and Wennerstroem calculated the constrained diffusion on a minimal surface of cubic symmetry (considering diffusion to be confined to the 2D surface), showing it to be lower than unconstrained diffusion by a factor of 2/3, and suggested this factor would be similar for all such surfaces.49 Relatedly, Chen et al. numerically solved the diffusion equations in 3D for several minimal-surface structures such as Schwartz primitive, diamond, and Schoen gyroid at a volume fraction of 0.5 (the Schoen gyroid or single gyroid is the shape of one network of the double gyroid; however, placing two of these would translate to a volume fraction of 1.0, which is obviously not a reasonable minority fraction for double gyroid); they showed the tortuosities of these structures are slightly different but all are close to 1.5.50 Meanwhile, Torquato and co-workers investigated these minimal surfaces regarding the thermal conductivity of one phase relative to electrical conductivity of the other and regarding fluid flow, but did not focus on diffusion.51−53 There have been some theoretical arguments regarding transport in the double gyroid structure based on various approximate methods. Considering the length of a pathway along the gyroid struts, Hamersky et al. estimated that τ ≈ 2, implying an intermediate conductivity of gyroid between that of randomly oriented cylinders and lamellae.39 Another group claimed that 1D diffusion through the gyroid phase yields a tortuosity of 1.5.54−56 It is not clear why these estimates differ, but the overall expectation based on these results is that diffusion through the minority phase of the gyroid would be similar to or less than that through randomly oriented lamellae. However, the exact tortuosity for 3D diffusion through the double gyroid structure is not available, and these works did not address transport through the majority phase of the gyroid or the effect of volume fraction. We aim to compare diffusion through cylinder, gyroid, and lamellar structures and show how the tortuosity of the gyroid structure impacts overall diffusion in both ideal structures and simulated diblock copolymers. The morphology of AB diblock copolymer systems is understood to be primarily governed by three variables: Flory parameter (χ, which describes how chemically incompatible two components are), composition (fA, denoting the volume fraction of A monomers), and molecular weight (N).57,58 These factors vary across studies, making experimental works hard to compare directly with each other. We compare three models of varying complexity, in which the structure can be explicitly controlled, to help clarify how morphology and fA (and to a lesser extent, χ and N) impact diffusion in BCP electrolytes. Note that we calculate diffusion constants, which are proportional to conductivity only if correlated ion motion is negligible, but for simplicity, we refer to the phase that allows transport as the “conducting

Figure 1. Diffusion constant of particles moving by random walks confined in various morphologies normalized by that with no constraints (D/DNC) as a function of volume fraction of the conducting domain ( fA). Images of structures show the conducting domain as red and the nonconducting one as blue. 1093

DOI: 10.1021/acsmacrolett.8b00506 ACS Macro Lett. 2018, 7, 1092−1098

Letter

ACS Macro Letters for both gyroid cases increases with fA; over the studied fA range, D/DNC ranges from 47% to 55% (tortuosity of 2.13− 1.82) and from 73% to 80% (tortuosity of 1.37−1.25) for Ginner and Gouter, respectively. Thus, we show a large effect of volume fraction, though there has been a lack of discussion that volume fraction impacts tortuosity in prior work. The tortuosity of Ginner is always higher than 1.5, in contrast to the hypothesis that the gyroid would show superior transport versus randomly oriented lamellae. However, Gouter is significantly lower, which helps explain why higher conductivity has sometimes been observed in gyroid/network phases than in lamellae in BCP or related systems (as in these works the majority phase is the conducting phase).20,38,40 In addition to the long-range morphology, ion transport is thought to be significantly affected by the interfaces between the microphase separated domains. For instance, Balsara and co-workers compared conductivity in salt-doped BCPs over a wide range of molecular weights and suggested that conduction is reduced near interfaces.5,6,10 To include effects of interfacial behavior and chain dynamics together with microphase morphology, we use coarse-grained MD simulations. Several prior studies have used coarse-grained block copolymers with selective penetrants (which are not charged but are driven by their interactions with the polymer beads to preferentially dissolve in one microphase) as an approximate model for ion containing BCP systems.64,65 It has been found that the molecular weight dependence of diffusion in BCP lamellae, which contrasts that in homopolymers, can be captured by such a simple model.64 Here, we apply an attractive Kremer-grest type of bead−spring model with added monomer-sized penetrants, identical to that used in our prior work, and simulate systems in different morphologies.65,66 We use ε = 1 as the unit of energy and interaction strength for like−like interactions, which are of Lennard−Jones (LJ) form and are cut off at 2.5σ (σ is the unit of length and monomer diameter). The unlike LJ interaction parameters (εij) are chosen to yield a given χij based on the simple calculation in refs 65−67 involving an integral over the interaction times an approximate radial distribution function for the homogeneous fluid, which gives χij ≈ 14.07(1 − εij/ε). This is a rough χ estimate that allows a clear comparison to prior work. We note that χ could be more accurately determined by using a more exact distribution function or advanced methods such as thermodynamic integration.68−71 We create BCP systems with short polymer chains (N = 40) at χN = 90 and 140 (εAB = 0.84 and 0.75) and systems with long polymer chains (N = 100) at χN = 140 (εAB = 0.9). We also consider homopolymer systems with chain lengths N = 8−640. To match previous work on the mapping for selective penetrant solvation,64,65,72 the interaction strengths of penetrant C with A and B are selected so that χBC − χAC = 8 (εAC = 1.28 and εBC = 0.72), representing the strong tendency of ions to dissolve in the higher dielectric constant microphase in salt-doped BCP electrolytes. The ratio of A monomers to penetrants is fixed at [A]:[C] = 16:1. Systems are initialized in the desired morphology by a similar approach as in refs 65−67 (see section S2 of the Supporting Information for details). Because the gyroid phase has a preferred spacing in all three dimensions, standard MD simulations (with periodic boundary conditions and constant number of polymers) cannot ensure the proper domain spacing is found, even if a barostat is applied and the box size is allowed to fluctuate.73−77 We predict the domain spacing for our gyroid systems based on the lamellar systems’ spacing for

the results presented here. We also studied the effect of box size on the diffusion constant to ensure the results do not change significantly over a range of box sizes (the range is likely large in comparison to the uncertainty in our predicted domain spacing, as there is a clear understanding of the relationship between lamellar and gyroid domain spacings).57,78−81 Further data showing box size effects are shown in Figure S3. We compare penetrant diffusion constant D in the BCP morphologies versus that in amorphous homopolymers of A monomers (DHP) to quantify the morphology effect in the same way as in experiments (morphology factor is equivalent to D/DHP). Experimental observations suggest that ion transport in homopolymers is coupled with chain dynamics especially for low molecular weight polymers, leading to decreasing conductivity as N increases.82,83 We also observe decreasing penetrant diffusion with increasing N in our MD simulations of homopolymers, as shown by the dashed lines in Figure 2a and discussed further in the Supporting Information.

Figure 2. Penetrant diffusion constant (a) parallel and (b) perpendicular to the interfaces vs fraction of A monomers (fA) for cylinders and lamellae. The dashed line indicates that in homopolymers (DHP) with chain length corresponding to the length of the A block. Points give the average across 10 time blocks of 1.5 × 105 τ, and error bars show the standard error.

The DHP versus N curve initially decreases significantly at low N (likely due to the lower density of short chain systems) but approximately plateaus at N > 100. Specifically, the curve is well fit with the equation DHP = D0 + K/N, analogous to that suggested in prior work,82,83 with D0 = 0.0276σ2/τ and K = 0.0849σ2/τ, where τ is the unit of time. Some experimental works normalize conductivity in BCPs with different N or fA to that in homopolymers of fixed N because their systems are in the regime that ion transport is decoupled with chain 1094

DOI: 10.1021/acsmacrolett.8b00506 ACS Macro Lett. 2018, 7, 1092−1098

Letter

ACS Macro Letters dynamics.11−14 But since our systems are in the low molecular weight regime where the effect of N is significant, we compare the diffusion in each BCP system to DHP of an analogous homopolymer system that has identical chain length as the BCP’s conducting block (A block). The diffusion constants of penetrants in the directions parallel (D||) and perpendicular (D⊥) to the domain interfaces in cylinders and lamellae for various BCP systems are presented in Figure 2. It is found that the parallel diffusion of penetrants decreases as fA or N increases (as the A block length increases), apparently due to the impact of polymers’ dynamics on penetrants’ dynamics. D|| in the BCP is always lower than the corresponding DHP, and it increases with χN. This is generally consistent with previous experimental observations showing increasing χN increases conductivity.5,6,12,13 On the other hand, D⊥ decreases as χN increases because ions are less likely to cross interfaces when there is a stronger repulsion between two blocks. Interestingly, it is easier for penetrants to diffuse across cylinders (especially smaller cylinders) than lamellae. However, the penetrant diffusion across interfaces in these systems only impacts overall diffusion slightly, as there is an order of magnitude difference between D|| and D⊥, which is also observed experimentally in BCP electrolytes.16,18,19 We calculate isotropic penetrant diffusion constants D and normalize them versus the appropriate DHP; the result is given in Figure 3 (see Figure S5 for the original D values). D/DHP is

To delineate to what extent these results are due to the topological effect of domain interfaces versus the polymer dynamics which also change with χN, we use Langevin dynamics simulations of particles in implicit solvent (without A monomers present) confined by a microstructure representative of the nonconductive domain at various χN (see model details in section S3 of the Supporting Information). The diffusion constant in each morphology versus that in a system with no constraints is available in Figure S6, which shows the same overall trends as Figure 3, except with somewhat smaller normalized diffusion constants, especially at lower χN. We believe the fixed nature of the interfaces in our simple method serves to overemphasize the effect of interfaces. Nonetheless, this model reproduces that (1) particle diffusion is slowed down as χN decreases (as interfaces become more rough), as expected, and (2) the larger the fA, the more efficient particle diffusion is, because particles are less likely to interact with interfaces when there is more free volume in the system. Thus, these effects can be explained by the (nonideal) structures rather than other features of the BCP systems. While the bicontinuous nature of the gyroid phase has been thought to make it ideal for transport, we have shown in ideal and simulated systems that the minority phase tortuosity is higher than that of randomly oriented lamellae. Our precise calculation of the gyroid tortuosity shows that (1) the tortuosity of the gyroid structure significantly depends on the volume fraction of the conducting domain, which may help explain why different outcomes have been observed in comparing the conductivity in the gyroid phase with that of homopolymers experimentally, and (2) the tortuosity factors of Ginner and Gouter are drastically different, and the majority component of gyroid is less tortuous than randomly oriented lamellae. We discussed the topological effects alone; although grain boundary effects (which we expect can be small under appropriate conditions) and other features such as the different mechanical properties or defects of these morphologies will be important factors in the design of BCP electrolytes, we hope to provide a theoretical benchmark for experimentalists to better predict the relative trade-off using one morphology over another.

■

ASSOCIATED CONTENT

* Supporting Information S

Figure 3. Normalized diffusion constants (D/DHP for MD model and D/DNC for RW model) in different morphologies vs fA for the noted segregation strengths (χN) and polymer chain lengths (N). Snapshot of each structure shows A monomers, B monomers, and penetrants in red, blue, and white, respectively, with some of the monomers cut away to show the structure.

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.8b00506. Details on RW, MD, and Langevin dynamics methods; additional data for structural and dynamic properties; and the discussion on the incommensurability effect for

not dependent on N (and is not dependent on fA for cylinders and lamellae) as the effect of chain dynamics is removed by the normalization. The trend in the morphology factor from the MD model follows the predictions from the RW model across all morphologies, including tortuosity dependence on fA for the gyroid phase. The MD morphology factors are generally slightly lower than the ideal factors from the RW model, and this deviation is slightly more significant at lower χN, apparently due to the effect of interfaces. The cylinder results are an exception, with very slightly higher morphology factors than ideal, due to the non-negligible diffusion across interfaces (D⊥) in these systems.

■

the gyroid phase (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Jonathan R. Brown: 0000-0002-4859-9118 Lisa M. Hall: 0000-0002-3430-3494 Notes

The authors declare no competing financial interest. 1095

DOI: 10.1021/acsmacrolett.8b00506 ACS Macro Lett. 2018, 7, 1092−1098

Letter

ACS Macro Letters

■

(16) Timachova, K.; Villaluenga, I.; Cirrincione, L.; Gobet, M.; Bhattacharya, R.; Jiang, X.; Newman, J.; Madsen, L. A.; Greenbaum, S. G.; Balsara, N. P. Anisotropic Ion Diffusion and Electrochemically Driven Transport in Nanostructured Block Copolymer Electrolytes. J. Phys. Chem. B 2018, 122 (4), 1537−1544. (17) Arges, C. G.; Kambe, Y.; Dolejsi, M.; Wu, G.-P.; Segal-Pertz, T.; Ren, J.; Cao, C.; Craig, G. S. W.; Nealey, P. F. Interconnected Ionic Domains Enhance Conductivity in Microphase Separated Block Copolymer Electrolytes. J. Mater. Chem. A 2017, 5 (11), 5619−5629. (18) Majewski, P. W.; Gopinadhan, M.; Jang, W.-S.; Lutkenhaus, J. L.; Osuji, C. O. Anisotropic Ionic Conductivity in Block Copolymer Membranes by Magnetic Field Alignment. J. Am. Chem. Soc. 2010, 132 (49), 17516−17522. (19) Majewski, P. W.; Gopinadhan, M.; Osuji, C. O. Understanding Anisotropic Transport in Self-Assembled Membranes and Maximizing Ionic Conductivity by Microstructure Alignment. Soft Matter 2013, 9 (29), 7106. (20) Cho, B. K.; Jain, A.; Gruner, S. M.; Wiesner, U. Mesophase Structure-Mechanical and Ionic Transport Correlations in Extended Amphiphilic Dendrons. Science 2004, 305 (5690), 1598−1601. (21) Meuler, A. J.; Hillmyer, M. A.; Bates, F. S. Ordered Network Mesostructures in Block Polymer Materials. Macromolecules 2009, 42 (19), 7221−7250. (22) Orilall, M. C.; Wiesner, U. Block Copolymer Based Composition and Morphology Control in Nanostructured Hybrid Materials for Energy Conversion and Storage: Solar Cells, Batteries, and Fuel Cells. Chem. Soc. Rev. 2011, 40 (2), 520−535. (23) Ichikawa, T.; Yoshio, M.; Hamasaki, A.; Kagimoto, J.; Ohno, H.; Kato, T. 3D Interconnected Ionic Nano-Channels Formed in Polymer Films: Self-Organization and Polymerization of Thermotropic Bicontinuous Cubic Liquid Crystals. J. Am. Chem. Soc. 2011, 133 (7), 2163−2169. (24) Martínez-Veracoechea, F. J.; Escobedo, F. A. Lattice Monte Carlo Simulations of the Gyroid Phase in Monodisperse and Bidisperse Block Copolymer Systems. Macromolecules 2005, 38 (20), 8522−8531. (25) Martínez-Veracoechea, F. J.; Escobedo, F. A. Simulation of the Gyroid Phase in Off-Lattice Models of Pure Diblock Copolymer Melts. J. Chem. Phys. 2006, 125 (10), 104907. (26) Prasad, I.; Jinnai, H.; Ho, R. M.; Thomas, E. L.; Grason, G. M. Anatomy of Triply-Periodic Network Assemblies: Characterizing Skeletal and Inter-Domain Surface Geometry of Block Copolymer Gyroids. Soft Matter 2018, 14 (18), 3612−3623. (27) Chan, V. Z.; Hoffman, J.; Lee, V. Y.; Iatrou, H.; Avgeropoulos, A.; Hadjichristidis, N.; Miller, R. D.; Thomas, E. L. Ordered Bicontinuous Nanoporous and Nanorelief Ceramic Films from Self Assembling Polymer Precursors Downloaded From. Science 1999, 286 (5445), 1716−1719. (28) Crossland, E. J. W.; Kamperman, M.; Nedelcu, M.; Ducati, C.; Wiesner, U.; Smilgies, D.-M.; Toombes, G. E. S.; Hillmyer, M. A.; Ludwigs, S.; Steiner, U.; Snaith, H. J. A Bicontinuous Double Gyroid Hybrid Solar Cell. Nano Lett. 2009, 9 (8), 2807−2812. (29) Leal, C.; Bouxsein, N. F.; Ewert, K. K.; Safinya, C. R. Highly Efficient Gene Silencing Activity of SiRNA Embedded in a Nanostructured Gyroid Cubic Lipid Matrix. J. Am. Chem. Soc. 2010, 132 (47), 16841−16847. (30) Urbas, A. M.; Maldovan, M.; DeRege, P.; Thomas, E. L. Bicontinuous Cubic Block Copolymer Photonic Crystals. Adv. Mater. 2002, 14 (24), 1850−1853. (31) Saranathan, V.; Osuji, C. O.; Mochrie, S. G. J.; Noh, H.; Narayanan, S.; Sandy, A.; Dufresne, E. R.; Prum, R. O. Structure, Function, and Self-Assembly of Single Network Gyroid (I4132) Photonic Crystals in Butterfly Wing Scales. Proc. Natl. Acad. Sci. U. S. A. 2010, 107 (26), 11676−11681. (32) Sorenson, G. P.; Coppage, K. L.; Mahanthappa, M. K. Unusually Stable Aqueous Lyotropic Gyroid Phases from Gemini Dicarboxylate Surfactants. J. Am. Chem. Soc. 2011, 133 (38), 14928− 14931.

ACKNOWLEDGMENTS We thank Nitash Balsara and Gregory Grason for helpful discussions. We also acknowledge the Ohio Supercomputer Center for compute resources. This material is based upon work supported by the National Science Foundation under Grant No. 1454343 (L.M.H. and K.-H.S.). Additionally, this material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award No. DE-SC0014209 (confined random walk code of J.R.B.).

■

REFERENCES

(1) Hamley, I. W. Ordering in Thin Films of Block Copolymers: Fundamentals to Potential Applications. Prog. Polym. Sci. 2009, 34 (11), 1161−1210. (2) Phillip, W. A.; O’Neill, B.; Rodwogin, M.; Hillmyer, M. A.; Cussler, E. L. Self-Assembled Block Copolymer Thin Films as Water Filtration Membranes. ACS Appl. Mater. Interfaces 2010, 2 (3), 847− 853. (3) Elabd, Y. A.; Hickner, M. A. Block Copolymers for Fuel Cells. Macromolecules 2011, 44 (1), 1−11. (4) Ozcam, A. E.; Petzetakis, N.; Silverman, S.; Jha, A. K.; Balsara, N. P. Relationship between Segregation Strength and Permeability of Ethanol/Water Mixtures through Block Copolymer Membranes. Macromolecules 2013, 46 (24), 9652−9658. (5) Singh, M.; Odusanya, O.; Wilmes, G. M.; Eitouni, H. B.; Gomez, E. D.; Patel, A. J.; Chen, V. L.; Park, M. J.; Fragouli, P.; Iatrou, H.; Hadjichristidis, N.; Cookson, D.; Balsara, N. P. Effect of Molecular Weight on the Mechanical and Electrical Properties of Block Copolymer Electrolytes. Macromolecules 2007, 40 (13), 4578−4585. (6) Gomez, E. D.; Panday, A.; Feng, E. H.; Chen, V.; Stone, G. M.; Minor, A. M.; Kisielowski, C.; Downing, K. H.; Borodin, O.; Smith, G. D.; Balsara, N. P. Effect of Ion Distribution on Conductivity of Block Copolymer Electrolytes. Nano Lett. 2009, 9 (3), 1212−1216. (7) Morris, M. A.; An, H.; Lutkenhaus, J. L.; Epps, T. H. Harnessing the Power of Plastics: Nanostructured Polymer Systems in LithiumIon Batteries. ACS Energy Lett. 2017, 2 (8), 1919−1936. (8) Hallinan, D. T.; Balsara, N. P. Polymer Electrolytes. Annu. Rev. Mater. Res. 2013, 43 (1), 503−525. (9) Chintapalli, M.; Chen, X. C.; Thelen, J. L.; Teran, A. A.; Wang, X.; Garetz, B. A.; Balsara, N. P. Effect of Grain Size on the Ionic Conductivity of a Block Copolymer Electrolyte. Macromolecules 2014, 47 (15), 5424−5431. (10) Yuan, R.; Teran, A. A.; Gurevitch, I.; Mullin, S. A.; Wanakule, N. S.; Balsara, N. P. Ionic Conductivity of Low Molecular Weight Block Copolymer Electrolytes. Macromolecules 2013, 46 (3), 914− 921. (11) Chintapalli, M.; Le, T. N. P.; Venkatesan, N. R.; Mackay, N. G.; Rojas, A. A.; Thelen, J. L.; Chen, X. C.; Devaux, D.; Balsara, N. P. Structure and Ionic Conductivity of Polystyrene-Block-Poly(Ethylene Oxide) Electrolytes in the High Salt Concentration Limit. Macromolecules 2016, 49 (5), 1770−1780. (12) Mullin, S. A.; Stone, G. M.; Panday, A.; Balsara, N. P. Salt Diffusion Coefficients in Block Copolymer Electrolytes. J. Electrochem. Soc. 2011, 158 (6), A619. (13) Panday, A.; Mullin, S.; Gomez, E. D.; Wanakule, N.; Chen, V. L.; Hexemer, A.; Pople, J.; Balsara, N. P. Effect of Molecular Weight and Salt Concentration on Conductivity of Block Copolymer Electrolytes. Macromolecules 2009, 42 (13), 4632−4637. (14) Wanakule, N. S.; Panday, A.; Mullin, S. A.; Gann, E.; Hexemer, A.; Balsara, N. P. Ionic Conductivity of Block Copolymer Electrolytes in the Vicinity of Order−Disorder and Order−Order Transitions. Macromolecules 2009, 42 (15), 5642−5651. (15) Young, W.-S.; Kuan, W.-F.; Epps, T. H. Block Copolymer Electrolytes for Rechargeable Lithium Batteries. J. Polym. Sci., Part B: Polym. Phys. 2014, 52 (1), 1−16. 1096

DOI: 10.1021/acsmacrolett.8b00506 ACS Macro Lett. 2018, 7, 1092−1098

Letter

ACS Macro Letters (33) Sax, J.; Ottino, J. M. Modeling of Transport of Small Molecules in Polymer Blends: Application of Effective Medium Theory. Polym. Eng. Sci. 1983, 23 (3), 165−176. (34) Kinning, D. J.; Thomas, E. L.; Ottino, J. M. Effect of Morphology on the Transport of Gases in Block Copolymers. Macromolecules 1987, 20 (5), 1129−1133. (35) Irwin, M. T.; Hickey, R. J.; Xie, S.; So, S.; Bates, F. S.; Lodge, T. P. Structure−Conductivity Relationships in Ordered and Disordered Salt-Doped Diblock Copolymer/Homopolymer Blends. Macromolecules 2016, 49 (18), 6928−6939. (36) Chopade, S. A.; Au, J. G.; Li, Z.; Schmidt, P. W.; Hillmyer, M. A.; Lodge, T. P. Robust Polymer Electrolyte Membranes with High Ambient-Temperature Lithium-Ion Conductivity via PolymerizationInduced Microphase Separation. ACS Appl. Mater. Interfaces 2017, 9 (17), 14561−14565. (37) Kim, O.; Kim, S. Y.; Lee, J.; Park, M. J. Building Less Tortuous Ion-Conduction Pathways Using Block Copolymer Electrolytes with a Well-Defined Cubic Symmetry. Chem. Mater. 2016, 28 (1), 318−325. (38) Villaluenga, I.; Chen, X. C.; Devaux, D.; Hallinan, D. T.; Balsara, N. P. Nanoparticle-Driven Assembly of Highly Conducting Hybrid Block Copolymer Electrolytes. Macromolecules 2015, 48 (2), 358−364. (39) Hamersky, M. W.; Hillmyer, M. A.; Tirrell, M.; Bates, F. S.; Lodge, T. P.; Von Meerwall, E. D. Block Copolymer Self-Diffusion in the Gyroid and Cylinder Morphologies. Macromolecules 1998, 31 (16), 5363−5370. (40) Choi, J.-H.; Ye, Y.; Elabd, Y. A.; Winey, K. I. Network Structure and Strong Microphase Separation for High Ion Conductivity in Polymerized Ionic Liquid Block Copolymers. Macromolecules 2013, 46 (13), 5290−5300. (41) Kehrwald, D.; Shearing, P. R.; Brandon, N. P.; Sinha, P. K.; Harris, S. J. Local Tortuosity Inhomogeneities in a Lithium Battery Composite Electrode. J. Electrochem. Soc. 2011, 158 (12), A1393. (42) Cecen, A.; Wargo, E. A.; Hanna, A. C.; Turner, D. M.; Kalidindi, S. R.; Kumbur, E. C. 3-D Microstructure Analysis of Fuel Cell Materials: Spatial Distributions of Tortuosity, Void Size and Diffusivity. J. Electrochem. Soc. 2012, 159 (3), B299−B307. (43) Sahimi, M.; Jue, V. L. Diffusion of Large Molecules in Porous Media. Phys. Rev. Lett. 1989, 62 (6), 629−632. (44) Zhou, N.; Bates, F. S.; Lodge, T. P. Mesoporous Membrane Templated by a Polymeric Bicontinuous Microemulsion. Nano Lett. 2006, 6 (10), 2354−2357. (45) Seo, M.; Hillmyer, M. A. Reticulated Nanoporous Polymers by Controlled Polymerization-Induced Microphase Separation. Science 2012, 336 (6087), 1422−1425. (46) Phillip, W. A.; Amendt, M.; O’Neill, B.; Chen, L.; Hillmyer, M. A.; Cussler, E. L. Diffusion and Flow Across Nanoporous Polydicyclopentadiene-Based Membranes. ACS Appl. Mater. Interfaces 2009, 1 (2), 472−480. (47) Chen, L.; Phillip, W. A.; Cussler, E. L.; Hillmyer, M. A. Robust Nanoporous Membranes Templated by a Doubly Reactive Block Copolymer. J. Am. Chem. Soc. 2007, 129 (45), 13786−13787. (48) Milhaupt, J. M.; Lodge, T. P. Homopolymer and SmallMolecule Tracer Diffusion in a Gyroid Matrix. J. Polym. Sci., Part B: Polym. Phys. 2001, 39 (8), 843−859. (49) Anderson, D. M.; Wennerstrom, H. Self-Dlffusion in Bicontinuous Cubic Phases, L3 Phases, and Microemulsions. J. Phys. Chem. 1990, 94, 8683−8694. (50) Chen, H.-Y.; Kwon, Y.; Thornton, K. Multifunctionality of Three-Dimensional Self-Assembled Composite Structure. Scr. Mater. 2009, 61 (1), 52−55. (51) Torquato, S.; Hyun, S.; Donev, A. Multifunctional Composites: Optimizing Microstructures for Simultaneous Transport of Heat and Electricity. Phys. Rev. Lett. 2002, 89 (26), 266601. (52) Torquato, S.; Hyun, S.; Donev, A. Optimal Design of Manufacturable Three-Dimensional Composites with Multifunctional Characteristics. J. Appl. Phys. 2003, 94 (9), 5748−5755.

(53) Jung, Y.; Torquato, S. Fluid Permeabilities of Triply Periodic Minimal Surfaces. Phys. Rev. E - Stat. Nonlinear, Soft Matter Phys. 2005, 72 (5), 056319. (54) Li, L.; Schulte, L.; Clausen, L. D.; Hansen, K. M.; Jonsson, G. E.; Ndoni, S. Gyroid Nanoporous Membranes with Tunable Permeability. ACS Nano 2011, 5 (10), 7754−7766. (55) Li, L.; Szewczykowski, P.; Clausen, L. D.; Hansen, K. M.; Jonsson, G. E.; Ndoni, S. Ultrafiltration by Gyroid Nanoporous Polymer Membranes. J. Membr. Sci. 2011, 384 (1−2), 126−135. (56) Guo, F.; Schulte, L.; Vigild, M. E.; Ndoni, S. Load-Release of Small and Macromolecules from Elastomers with Reversible Gyroid Mesoporosity. Soft Matter 2012, 8 (45), 11499−11507. (57) Matsen, M. W.; Schick, M. Stable and Unstable Phases of a Diblock Copolymer Melt. Phys. Rev. Lett. 1994, 72 (16), 2660−2663. (58) Fredrickson, G. H.; Bates, F. S. DYNAMICS OF BLOCK COPOLYMERS: Theory and Experiment. Annu. Rev. Mater. Sci. 1996, 26, 501−550. (59) Torquato, S. Effective Electrical Conductivity of Two-phase Disordered Composite Media. J. Appl. Phys. 1985, 58 (10), 3790− 3797. (60) Torquato, S. Random Heterogeneous Media: Microstructure and Improved Bounds on Effective Properties. Appl. Mech. Rev. 1991, 44 (2), 37. (61) Kim, I. C.; Torquato, S. Effective Conductivity of Suspensions of Hard Spheres by Brownian Motion Simulation. J. Appl. Phys. 1991, 69 (4), 2280−2289. (62) Kim, I. C.; Torquato, S. Diffusion of Finite-sized Brownian Particles in Porous Media. J. Chem. Phys. 1992, 96 (2), 1498−1503. (63) Schwartz, L. M.; Banavar, J. R. Transport Properties of Disordered Continuum Systems. Phys. Rev. B: Condens. Matter Mater. Phys. 1989, 39 (16), 11965−11970. (64) Ganesan, V.; Pyramitsyn, V.; Bertoni, C.; Shah, M. Mechanisms Underlying Ion Transport in Lamellar Block Copolymer Membranes. ACS Macro Lett. 2012, 1 (4), 513−518. (65) Seo, Y.; Brown, J. R.; Hall, L. M. Diffusion of Selective Penetrants in Interfacially Modified Block Copolymers from Molecular Dynamics Simulations. ACS Macro Lett. 2017, 6 (4), 375−380. (66) Grest, G. S.; Lacasse, M. D.; Kremer, K.; Gupta, A. M. Efficient Continuum Model for Simulating Polymer Blends and Copolymers. J. Chem. Phys. 1996, 105 (23), 10583−10594. (67) Seo, Y.; Brown, J. R.; Hall, L. M. Effect of Tapering on Morphology and Interfacial Behavior of Diblock Copolymers from Molecular Dynamics Simulations. Macromolecules 2015, 48, 4974− 4982. (68) Glaser, J.; Qin, J.; Medapuram, P.; Morse, D. C. Collective and Single-Chain Correlations in Disordered Melts of Symmetric Diblock Copolymers: Quantitative Comparison of Simulations and Theory. Macromolecules 2014, 47 (2), 851−869. (69) Medapuram, P.; Glaser, J.; Morse, D. C. Universal Phenomenology of Symmetric Diblock Copolymers near the OrderDisorder Transition. Macromolecules 2015, 48 (3), 819−839. (70) Zhang, W.; Gomez, E. D.; Milner, S. T. Predicting FloryHuggins χ from Simulations. Phys. Rev. Lett. 2017, 119 (1), 017801. (71) Seo, Y. Structure and Dynamic Properties of Interfacially Modified Block Copolymers from Molecular Dynamics Simulations, Ph.D. Thesis, The Ohio State University, 2017. (72) Wang, Z.-G. Effects of Ion Solvation on the Miscibility of Binary Polymer Blends. J. Phys. Chem. B 2008, 112 (50), 16205− 16213. (73) Cochran, E. W.; Garcia-Cervera, C. J.; Fredrickson, G. H. Stability of the Gyroid Phase in Diblock Copolymers at Strong Segregation. Macromolecules 2006, 39 (7), 2449−2451. (74) Wang, Q.; Yan, Q.; Nealey, P. F.; De Pablo, J. J. Monte Carlo Simulations of Diblock Copolymer Thin Films Confined between Two Homogeneous Surfaces. J. Chem. Phys. 2000, 112 (1), 450−464. (75) Arora, A.; Morse, D. C.; Bates, F. S.; Dorfman, K. D. Commensurability and Finite Size Effects in Lattice Simulations of Diblock Copolymers. Soft Matter 2015, 11 (24), 4862−4867. 1097

DOI: 10.1021/acsmacrolett.8b00506 ACS Macro Lett. 2018, 7, 1092−1098

Letter

ACS Macro Letters (76) Š kvor, J.; Posel, Z. Simulation Aspects of Lamellar Morphology: Incommensurability Effect. Macromol. Theory Simul. 2015, 24 (2), 141−151. (77) Beardsley, T. M.; Matsen, M. W. Monte Carlo Phase Diagram for Diblock Copolymer Melts. Eur. Phys. J. E: Soft Matter Biol. Phys. 2010, 32 (3), 255−264. (78) Rancon, Y.; Charvolin, J. Fluctuations and Phase Transformations in a Lyotropic Liquid Crystal. J. Phys. Chem. 1988, 92 (22), 6339−6344. (79) Clerc, M.; Levelut, A. M.; Sadoc, J. F. Transitions between Mesophases Involving Cubic Phases in the Surfactant-Water Systems. Epitaxial Relations and Their Consequences in a Geometrical Framework. J. Phys. II 1991, 1 (10), 1263−1276. (80) Schulz, M. F.; Bates, F. S.; Almdal, K.; Mortensen, K. Epitaxial Relationship for Hexagonal-to-Cubic Phase Transition in a Book Copolymer Mixture. Phys. Rev. Lett. 1994, 73 (1), 86−89. (81) Hajduk, D. A.; Harper, P. E.; Gruner, S. M.; Honeker, C. C.; Kim, G.; Fetters, L. J.; Kim, G. The Gyroid: A New Equilibrium Morphology in Weakly Segregated Diblock Copolymers. Macromolecules 1994, 27 (15), 4063−4075. (82) Shi, J.; Vincent, C. A. The Effect of Molecular Weight on Cation Mobility in Polymer Electrolytes. Solid State Ionics 1993, 60 (1−3), 11−17. (83) Teran, A. A.; Tang, M. H.; Mullin, S. A.; Balsara, N. P. Effect of Molecular Weight on Conductivity of Polymer Electrolytes. Solid State Ionics 2011, 203 (1), 18−21.

1098

DOI: 10.1021/acsmacrolett.8b00506 ACS Macro Lett. 2018, 7, 1092−1098