Electrostatically Tuned Microdomain Morphology and Phase

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Electrostatically Tuned Microdomain Morphology and PhaseDependent Ion Transport Anisotropy in Single-Ion Conducting Block Copolyelectrolytes Chenxi Zhai,† Huanhuan Zhou,† Teng Gao,‡ Lingling Zhao,‡ and Shangchao Lin*,† †

Department of Mechanical Engineering, Materials Science and Engineering Program, FAMU-FSU College of Engineering, Florida State University, Tallahassee, Florida 32310, United States ‡ Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of Energy & Environment, Southeast University, Nanjing, Jiangsu 210096, China S Supporting Information *

ABSTRACT: Block copolyelectrolytes are solid-state singleion conductors which phase separate into ubiquitous microdomains to enable both high ion transference number and structural integrity. Ion transport in these charged block copolymers highly depends on the nanoscale microdomain morphology; however, the influence of electrostatic interactions on morphology and ion diffusion pathways in block copolyelectrolytes remains an obscure feature. In this paper, we systematically predict the phase diagram and morphology of diblock copolyelectrolytes using a modified dissipative particle dynamics simulation framework, considering both explicit electrostatic interactions and ion diffusion dynamics. Various experimentally controllable conditions are considered here, including block volume fraction, Flory−Huggins parameter, block charge fraction or ion concentration, and dielectric constant. Boundaries for microphase transitions are identified based on the computed structure factors, mimicking small-angle X-ray scattering patterns. Furthermore, we develop a novel “diffusivity tensor” approach to predict the degree of anisotropy in ion diffusivity along the principal microdomain orientations, which leads to highthroughput mapping of phase-dependent ion transport properties. Inclusion of ions leads to a significant leftward and upward shift of the phase diagram due to ion-induced excluded volume, increased entropy of mixing, and reduced interfacial tension between dissimilar blocks. Interestingly, we discover that the inverse topology gyroid and cylindrical phases are ideal candidates for solid-state electrolytes in metal-ion batteries. These inverse phases exhibit an optimal combination of high ion conductivity, well-percolated diffusion pathways, and mechanical robustness. Finally, we find that higher dielectric constants can lead to higher ion diffusivity by reducing electrostatic cohesions between the charged block and counterions to facilitate ion diffusion across block microdomain interfaces. This work significantly expands the design space for emerging block copolyelectrolytes and motivates future efforts to explore inverse phases to avoid engineering hurdles of aligning microdomains or removing grain boundaries.

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INTRODUCTION

transport strongly depends on the alignment of specific anisotropic nanostructures. Efforts to control the alignment of these nanostructures to avoid ionic diffusion energy barriers across microdomains or grain boundaries are extensive, with significant interests in using substrate templating,6−8 mechanical shearing,9 kinetic control,10 and solvent engineering.11 As solid-state electrolytes, conventionally charge-neutral BCPs, such as polystyrene-b-poly(ethylene oxide) (PS-bPEO), lead to better controls of the ion transport pathways compared to homopolymer electrolytes such as poly(ethylene glycol) (PEG). However, both polymers need to function with

Block copolymers (BCPs) are macromolecular amphiphiles composed of chemically distinct homopolymer blocks linked end-to-end covalently. Their self-assembly is driven by demixing (or microphase separation) of the chemically dissimilar polymer segments into different microdomains.1,2 The self-assembly of BCPs can enable “bottom-up” approaches to provide access to nanostructures with a desired range of characteristic lengths, both at scalable scales and with reduced costs.1 BCPs are one of the leading candidate materials for use as solid-state electrolytes (for ion transport) and ion-conductive membranes (for ion exchanging) in electrochemical energy storage systems, such as Li-ion batteries and fuel cells.3 They self-assemble into nanostructures that enable both ionic transport and maintain structural integrity,4,5 while ionic © XXXX American Chemical Society

Received: March 1, 2018 Revised: May 24, 2018

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DOI: 10.1021/acs.macromol.8b00451 Macromolecules XXXX, XXX, XXX−XXX

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RESULTS AND DISCUSSION DPD-Simulated Phase Diagrams of Diblock Copolyelectrolytes. First, we carefully validated the predictability of the DPD simulation framework (see Computational Methods) by comparing the DPD-simulated phase diagram of conventional neutral diblock copolymers with that predicted using SCFT (see Figure 1). On the basis of the computed structure

a dissociated salt, in which one of the mobile ions is the charge carrier (such as the Li+ cations) and the other does not contribute to charge transport at all. The occurrence of salt concentration gradients causes a concentration polarization,12,13 which leads to lower ion transference number for the actual charge carrier and, hence, poor cell performances.14 Therefore, single-ion conducting block copolyelectrolytes, which are synthesized by doping one block of the conventional BCP with charged chemical groups, can serve as localized ions, while keeping the mobility of the actual charge carrying ions high.15−17 Examples of these charged BCP systems include proton (H +) or Li +-doped polystyrenesulfonate-b-poly(ethylene oxide) (PSS-b-PEO), polystyrenesulfonate(trifluoromethylsulfonyl)imide-b-poly(ethylene oxide) (PSTFSI-b-PEO), and poly(oligo(oxyethylene) methacrylate)-b-polystyrene (POEM-b-PS).18 The ionic conductivity in block copolyelectrolytes highly depends on the morphology of self-assembled microdomains at the molecular level. Despite the wealth of the BCP literature, the influence of electrostatic interactions on morphology and associated ionic conductivity remains an obscure feature of charged BCPs.4,5,19 Therefore, there are urgent needs to develop predictive computational tools to allow fundamental understanding and de novo design of charged BCP systems. Current theoretical understandings on charged BCP melts, whether obtained from mesoscopic dissipative particle dynamics (DPD) simulations20,21 (which treat a polymer as a bead− spring system with each bead representing a monomer), Monte Carlo (MC) simulations,22,23 or the self-consistent field theory (SCFT),24,25 have not addressed the inherent difficulties of capturing the following elements simultaneously: (i) discretized local charge distributions and electrostatic interactions, (ii) quantifying the dynamic ionic transport process, and (iii) direct correlation between the ionic diffusivity and the microdomain anisotropy.26−28 To address the above challenges, we predict the morphology of A−nB-type diblock copolyelectrolyte (with negatively charged monomers in block A) single-ion conductors using a modified DPD simulation framework, considering both explicit electrostatic interactions and dynamics of counterions (monovalent cations M+ to mimic Li+ or Na+ ions). Boundaries at microphase transitions are identified based on the computed structure factor patterns. The entire phase diagrams are mapped out under various experimentally controllable conditions, including (i) the volume fraction of block A ( fA), (ii) Flory−Huggins parameter (χ) for the repulsive enthalpy between blocks A and B, (iii) charge fraction (ϕ) defined by the partial charge on each A monomer, and (iv) dielectric constant (εr). To quantify the ionic diffusivity under these phases, we develop a novel diffusivity tensor approach that can capture anisotropic diffusivities along principal microdomain orientations for high-throughput data analysis. Although the synergetic interplay between electrostatic interactions and BCP microdomains leads to highly anisotropic ionic transport, we discover that the block-A-rich (also ion-rich) inverse topology gyroid (G′) and cylindrical (C′) phases are the optimal candidates for single-ion conductors with high-flux ion conductivity, well-percolated isotropic diffusion pathways, and mechanical robustness. This work significantly expands the design space for emerging charged BCP systems and will motivate future efforts to explore inverse phases without the engineering hurdle of aligning microdomains or removing grain boundaries in order to enhance ion transport.

Figure 1. DPD-simulated phase diagram of neutral diblock copolymers with representative simulation snapshots. Phase diagrams predicted using the SCFT by Schick et al. are shown in solid lines to make a comparison.29 The DPD-simulated phase diagram here is corrected by the low-molecular-weight effect studied by Matsen31 (see section S2) to allow a fair comparison with SCFT results for infinitely long polymer chains (N → ∞). The sketch of a single neutral diblock copolymer in the DPD simulation is shown on the top. For block-Apoor phases, only the A monomers (in red) are shown in the snapshots, while for block-A-rich inverse phases, only the B monomers (in blue) are shown in the snapshots for clarity.

factor profiles (see Figure S1 and section S1 of the Supporting Information), which resemble experimental small-angle X-ray scattering (SAXS) patterns, we identified all the microphases of neutral diblock copolymers with a constant length of N = 16 (the number of beads/monomers per copolymer in DPD simulations). Specifically, we mapped out the order−disorder transition (ODT) and order−order transition (OOT) boundaries as a function of the block-A ratio ( fA) and Flory− Huggins parameter (χAB) (see Figure 1). The DPD-simulated symmetric phase diagram of conventional neutral diblock copolymers agrees very well with that predicted using SCFT (usually applicable for high-molecular-weight, long-chain BCPs with N → ∞),29 after correcting the low-molecular-weight factor of the BCPs simulated in DPD using linear extrapolations between SCFT-predicted results at different N B


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Figure 2. DPD-simulated phase diagram of An−B-type diblock copolyelectrolyte system with representative simulation snapshots. Specifically, it is simulated at a charge fraction ϕ = 1:4 and dielectric constant εr = 1.1 for an aqueous environment in the DPD simulations here. The dashed lines serve as a guide to the eye and map out the boundaries for major microphase transitions. The sketch of a single diblock copolyelectrolyte chain with counterions is shown on the top. For block-A-poor phases, only the A monomers (in red) and the counterions (in green) are shown in the snapshots, while for block-A-rich inverse phases, only the B monomers (in blue) are shown in the snapshots for clarity.

lectrolyte. Therefore, we investigated cases under ϕ = 1:2, 1:4, and 1:8 that correspond to qA = −0.5, −0.25, and −0.125 e, respectively, where each monovalent counterion bead carries a constant charge of +1. We also investigated cases under εr = 1.1, which corresponds to a dipolar aqueous solution, and εr = 1.1/4 = 0.275, which corresponds to a nonpolar organic solvent with less charge screening, to model different background solvent environments. Please note that the background solvent effect is only modeled through the εr value in an implicit fashion in the DPD simulations here, considering that the amount of solvents is minimal in typical BCP melts. We systematically explored the structural, phase, and ion diffusion properties of diblock copolyelectrolytes under ϕ = 1:4 and εr = 1.1 as a representative case. The DPD-simulated phase diagram of a representative diblock copolyelectrolyte system under ϕ = 1:4 and εr = 1.1 is shown in Figure 2 with DPD simulation snapshots of the equilibrated microdomains. We applied the same computational method to identify the phase diagram of this charged BCP system as that for its neutral counterpart. The charged BCP system exhibits all the microphases present in its neutral counterpart, and it also follows the same phase transition order (D → S → C → G → L → G′ → C′ → S′ → D) when increasing the fA value. However, the phase diagram of the charged BCP system exhibits both upward and leftward shifts compared to its neutral counterpart. Specifically, the location of the ODT at fA = 0.5 is at χN ≈ 15

values (see section S2 and Figure S2).30 All the block-A-poor and block-A-rich (inverse topology phases when fA > 0.5) phases are identified in Figure 1, including the disordered phase (D) and seven ordered phases: lamellar (L), spherical (S), cylindrical (C, hexagonally packed), gyroid (G, bicontinuous), and their block-A-rich inverse counterparts (S′, C′, and G′). A modified DPD simulation framework, considering explicit electrostatic interactions between smeared charges and explicit ion diffusions, is developed here (see Computational Methods). After introducing A monomers with uniformly distributed partial charge qA (in units of elementary charge, e) for an A−nBtype diblock copolyelectrolyte system (see the top of Figure 2), the total charge of each block A in a polymer would be NAqA = −n (where NA = NfA is the number of beads or monomer units in block A) which will be balanced by the added n M+ counterions to achieve electroneutrality. For the smeared charges on block A, we define the charge fraction as ϕ = n/NA = −qA, reflecting not only the number ratio between M+ counterions (n) and A monomers (NA) but also the partial charge qA on each A monomer. In addition, polar-solventinduced electrostatic screening will be reflected by the dielectric constant εr used in the DPD simulations. Electrostatic interactions under different charge fractions ϕ (or the partial charge qA on each A monomer) and dielectric constants εr can be introduced in DPD simulations to fine-tune the phase diagram and microdomain morphology of a diblock copolyeC


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under εr = 0.275, reflecting net enhanced electrostatic attractions (or cohesions) among A monomers in the blockA-rich (also counterion-rich) phases, which should be due to the reduced electrostatic screening. The introduced electrostatic interactions (on top of the Flory−Huggins repulsive energy, χ, between blocks A and B) and explicit counterions are the dominant controlling factors behind the shifted phase diagrams of the charged BCP systems compared to their neutral counterpart. First, counterions will bind closely to A monomers due to strong electrostatic attractions, which will contribute to additional excluded volumes in a well-mixed (A block + counterion) single phase. This will increase the effective volume fraction of block A (fA) and therefore shift the OOT boundaries leftward. Second, electrostatically induced mixing of counterions with A monomers would increase the entropy of mixing, which would hinder microphase separation due to the reduced free energy of mixing and therefore lift the ODT boundary toward higher χN values.32 Finally, the presence of counterions might reduce the effective interfacial tension that drives phase separation between blocks A and B, which would also reduce the free energy of mixing and lift the ODT boundary.24 The above three factors are also reflected by the fact that the phase diagram can be fine-tuned by ϕ and εr values (see Figure 3). Specifically, at higher charge fractions (e.g., ϕ = 1:2) where the amount ratio between counterions and A monomers is high, the counterions would contribute to a greater amount of excluded volumes and therefore shift the OOT boundaries further leftward compared to the lower charge fraction cases (e.g., ϕ = 1:8). When reducing εr from 1.1 to 0.275 at ϕ = 1:4, there is a leftward shift of the OOT boundaries only at lower fA values and a downward shift of the ODT boundary only at higher fA values. Reducing the εr value would enhance both electrostatic cohesions and repulsions but will not affect the entropy of mixing counterions and A monomers under fixed fA and ϕ values. At higher εr = 1.1, the mixing entropic effect overcomes the net electrostatic interactions (slightly attractive after summing up attractions and repulsions), which hinders phase separation at all fA values (see Figure 2). But at lower εr = 0.275, the enhanced net attractive electrostatic interactions start to overcome the mixing entropic effect toward higher fA values, which favors phase separation (see Figure 3c). Interestingly, the DPD-simulated phase diagrams of diblock copolyelectrolyte systems agree qualitatively but lie somewhere between the two phase diagrams predicted by de la Cruz et al.25 and Neimark et al.24 using SCFTs integrated with the Poisson− Boltzmann equation and the liquid state theory (LS-SCFT), respectively. Since de la Cruz et al. have not considered the S and S′ phases in their SCFT study to map out the actual ODT boundary,25 we regard the transition between the C/C′ and D phases as the ODT boundary in their study. Specifically, they observed remarkable suppression of phase separation between C′ and D phases at higher fA values, which has been experimentally verified.33 Such a phenomenon results from the increased entropy of mixing counterions with A monomers at larger fA values, when there are more counterions present to balance the charges of A monomers, which hinders phase separation at low χN values. However, a large difference is observed in the ODT at lower fA values, where an obvious leftward and downward (favors phase separation even at very low χN values) shift of the ODT boundary is predicted by de la Cruz et al.,25 while our DPD simulation results show less

for the neutral BCP system, while it shifts up to χN ≈ 30 for the charged BCP system. Furthermore, each ordered phase is observed at smaller fA values under constant χN for the charged BCP system compared to its neutral counterpart. Such a leftward shift of the OOT results in an asymmetric phase diagram of the charged BCP system around fA = 0.5 compared to its symmetric neutral counterpart. When reducing the charge fraction ϕ from 1:2 to 1:8 in the DPD simulations, we find that the phase diagrams of the charge BCPs converge to that of the neutral case (see both Figure 2 and Figure 3a,b). While the locations of the ODT at fA = 0.5

Figure 3. DPD-simulated phase diagrams of An−B-type diblock copolyelectrolyte systems at ϕ = 1:2 (a), 1:8 (b), and 1:4 (c), while εr = 1.1 for (a) and (b) and εr = 1.1/4 = 0.275 for (c).

remain at χN ≈ 30 for the charged BCPs under a range of ϕ from 1:8 to 1:2, a higher ϕ value leads to more significant leftward shift of the phase diagram. In addition, increasing the ϕ value to 1:2 will also lead to more G and G′ phases at the ODT boundary to replace the original L phase, suggesting a small level of disordering and microdomain interfacial fluctuations due to a reduced interfacial tension between blocks A and B, as mediated by the counterions. On the other hand, when reducing the dielectric constant εr from 1.1 to 0.275 (see Figure 3c) in the DPD simulations under constant ϕ = 1:4, we find a similar trend (upward and leftward shifts) in the phase diagram as that when increasing ϕ. Interestingly, more inverse phases (G′ and S′) emerge at lower χN values D


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0.4−0.45,34 which is consistent with the DPD-simulated phase diagram at ϕ = 1:2 where both phases are observed at fA > 0.4 (see Figure 3a). It is important to note that the phase behavior of the above proton-containing PSS-b-PMB system is expected to differ a little from the DPD-simulated single-ion containing BCP system because protons are harder to dissociate from the polymer backbone than counterions. DPD-Simulated Microdomain Morphology of Diblock Copolyelectrolytes. DPD particle radial distribution functions (RDFs) under the L and D phases are shown in Figure 4

degree of leftward shift and an upward shift similar to ODT at larger fA values. The observed common leftward shifts of the ODT and OOT boundaries are partially due to the fact that both DPD and LS-SCFT consider the excluded volume contributed by the counterions. Furthermore, the significantly enhanced phase separation at very low χN values predicted by de la Cruz et al. is primarily due to the sole electrostatic cohesions they introduced between counterions and A monomers, which is expected to drive phase separation, but inconsistent with some experimental findings.19,25,32 In our DPD simulations, in addition to the electrostatic cohesions, electrostatic repulsions among counterions and among A monomers are also considered, which mediates the cohesiondriven phase separation at very low χN values observed by de la Cruz et al.25 On the other hand, Neimark et al.24 observed both a rightward and an upward shift in the SCFT-predicted phase diagram of diblock copolyelectrolytes, which is very different from that predicted using LS-SCFT.25 While our DPD simulation also exhibits the upward shift due to the increased entropy of mixing, the observed rightward shift (our DPD simulation shows a leftward shift) by Neimark et al. is partly due to the negligence of the excluded volume effect contributed by the counterions in their SCFT model. Interestingly, similar to our DPD results, they also did not observe a significant phase separation (ODT at very low χN values) at lower fA values as that predicted using LS-SCFT with solely electrostatic cohesions. Finally, our DPD-simulated phase diagram is also consistent with recent MC simulation studies, although the counterions were treated implicitly in these MC studies.22,23 Effective χ parameters were also introduced between charged A monomers in these MC studies to implicitly treat electrostatic interactions.22,23 In summary, we believe that our DPD simulation, when compared to the above two SCFT models,24,25 captures both excluded volume of counterions and the attractive/repulsive electrostatic interactions, which compares more closely with experimentally determined phase diagrams.17,34−37 For example, Balsara et al. extensively studied single-ion conducting PSTFSI-b-PEO doped with Li+ and Mg2+ experimentally, where they found that when introducing ions, the L phase would form under much lower A-block (in this case, PSTFSI) ratios compared to the conventional charge-neutral BCPs.38 Specifically, for Li+-doped systems, they observed the L phase at f PSTFSI = 0.24−0.36 with the charge fraction ϕ ≈ 0.2.17,36,37 In our DPD simulations, we also observed the L phase under the similar fA range under ϕ = 1:4 and εr = 0.275 (see Figure 3c), confirming the experimentally observed leftward shifts of the OOT boundaries between L and other ordered phases (C, C′, G, and G′) upon introducing counterions. Furthermore, our DPD-simulated phase diagram of diblock copolyelectrolyte is consistent with the experimental phase diagram of protoncontaining poly(styrenesulfonate)-b-polymethylbutylene (PSSb-PMB) block copolymers at proton fractions ϕ (similar to charge fraction) up to 0.53.34,35 Specifically, the L phase is observed experimentally at fA = 0.25−0.3 and ϕ ≈ 0.4,35 which is consistent with the DPD-simulated phase diagram at ϕ = 1:2 where the L phase is observed at fA = 0.25−0.45 (see Figure 3a). The L phase is also observed experimentally at fA = 0.45− 0.5 and ϕ < 0.25,34 which is consistent with the DPD-simulated phase diagram at ϕ = 1:4 and 1:8 where the L phase is observed at fA = 0.35−0.55 (see Figures 2 and 3b). Finally, the G′ and C′ phases were observed experimentally at fA = 0.4−0.5 and ϕ =

Figure 4. (a) RDFs (among the A and B monomers and the counterions) for a representative L phase of the DPD-simulated diblock copolyelectrolytes. (b) RDFs for a representative D phase of the DPD-simulated diblock copolyelectrolytes.

to characterize the microstructures of diblock copolyelectrolytes. A key finding from these RDFs is that the sum of electrostatic cohesions and repulsions between the charged species (block A and counterions) is only slightly attractive, as reflected by the similar RDF peak heights within blocks A−A and B−B as well as between block A and counterions. The introduced electrostatic repulsions in our DPD model would balance out the electrostatic cohesions under very low fA values to suppress the extremely strong phase separations as predicted by de la Cruz et al. using LS-SCFT.25 As shown in Figure 4a, the first major RDF peak between A and B monomers is located at ∼6Rc (where Rc is the characteristic length in DPD, see section S3 in the Supporting Information), which reflects the thickness of the lamellar microdomain. The first major RDF peaks between A and B monomers themselves, as well as the minor RDF peak between A and B monomers, are all located at ∼1Rc, reflecting the covalent bond length in the polymer. Furthermore, due to electrostatic cohesions between A monomers and counterions, the RDF peak between them is located closely at ∼1Rc, similar to those between A monomers and counterions themselves (see Figure 4a). However, the RDF peak between B monomers and counterions is located at ∼6Rc, reflecting the fact that counterions are confined within block A to make a well-mixed single phase. On the other hand, the observed long-range ordering at ∼6Rc for the L phase is missing for the D phase (see Figure 4b), where only an RDF peak at ∼1Rc is observed for all pairs of particles, as expected for such a homogeneous and isotropic morphology. Similar long-range ordering features are observed from the RDF profiles of the G/G′ and C/C′ phases (see Figures S3 and S4), where the locations of the primary peaks can vary a lot, reflecting the different length scales of the E


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Figure 5. (a) Illustration of the principal axis z normal to the lamellar microdomains, along which the number density profiles are computed. (b−f) Computed number density profiles for the lamellar (L) phases of the diblock copolyelectrolytes under various simulation parameters. Each subplot corresponds to (b) ϕ = 1:2, fA = 7/16, χN = 40, and εr = 1.1; (c) ϕ = 1:8, fA = 7/16, χN = 40, and εr = 1.1; (d) ϕ = 1:4, fA = 6/16, χ = 1.1; (e) ϕ = 1:4, fA = 7/16, χN = 120, and εr = 1.1; and (f) ϕ = 1:4, fA = 7/16, χN = 40, and εr = 0.275.

Figure 6. DPD simulation snapshots showing the morphologies of the (a) anisotropic L, (b) perfectly isotropic D, and (c) fairly isotropic C′ phases of diblock copolyelectrolytes, which result in (d) anisotropic, (e) perfectly isotropic, and (f) fairly isotropic ion diffusivity contour plots computed using the ion diffusivity tensor D. Diffusivities normal (D⊥) and parallel (D∥) to the lamellar and the inverse cylindrical microdomains are computed from the eigenvalues of matrix D.

Comparing the RDF profiles of DPD-simulated charged BCPs and those of their neutral counterparts (see Figures S6 and S7), the addition of counterions does not significantly alter the

microdomains formed in each phase. RDF profiles of the S/S′ phases exhibit similar peak distribution as the D phase due to their isotropic and homogeneous nature (see Figure S5). F


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Figure 7. Predicted contour maps of orientationally averaged ion diffusivities D̅ (left column) and degrees of anisotropy η (right column) at various ϕ and εr values. These phase-dependent maps match the DPD-simulated phase diagram of diblock copolyelectrolytes in Figures 2 and 3. Each subplot corresponds to (a) ϕ = 1:2 and εr = 1.1, (b) ϕ = 1:4 and εr = 1.1, (c) ϕ = 1:8 and εr = 1.1, (d) ϕ = 1:4 and εr = 0.275.

observed for both blocks. Finally, comparing Figures 5b and 5f, as εr increases, wider density peaks for block A are observed, similar to those observed when increasing ϕ, due to the increased net electrostatic attractions between counterions and A monomers. DPD-Simulated Orientational Anisotropy in Ion Transport. We take advantage of the explicit-ion DPD model to investigate the dynamic anisotropic ion diffusivities within the microdomains as a function of the diverse morphologies of diblock copolyelectrolytes (see Figure 6a−c for the anisotropic L phase, the perfectly isotropic D phase, and the fairly isotropic C′ phase). Using the computed ion velocity self/auto- and cross-correlation functions (see Computational Methods and Figure S8a,b), we determined a unique ion diffusivity tensor D for each phase as a function of fA and χN. The eigenvalues of this matrix D are the principal ion diffusivities along the principal axes determined by the eigenvectors of matrix D. The computed principal diffusivities quantify the contribution from each characteristic microdomain orientation. For example, for the anisotropic L phase, the principal ion diffusivities are D⊥ < D∥ (see Figure 6d for the corresponding ellipsoid-shaped 3D contour plot showing orientational anisotropy), which are normal and parallel to the lamellar microdomains, respectively. As expected, the principal ion diffusivities for the isotropic D phase are almost identical to each other (see Figure 6e for the corresponding spherical 3D contour plot showing orientational isotropy). Similar, for the C′ phase, we would expect that D⊥ < D∥, which

distributions between A and B monomers and among the monomers themselves. The number density profiles for the DPD-simulated diblock copolyelectrolytes, along the principal axis normal to the lamellar microdomains (see Figure 5a), reflect microphase separation (interface formation) between blocks A and B and signify the typical alternating layer-by-layer lamellar morphology. In addition, the density peak locations of the counterions match very well with those of block A due to the electrostatic cohesions between them, in consistent with the mixed singlephase of counterions confined in block A and the resulting RDF profiles (see Figure 4a). The width of each density peak is ∼6Rc reflecting the lamella thickness (see Figure 5), also in consistent with the RDF profiles in Figure 4a. DPD simulation parameters, such as ϕ, εr, fA, and χN, would affect the resulting number density profiles of the L phase. Comparing Figures 5b and 5c, the increase in ϕ would increase the width of the density peaks for block A due to the simultaneously increased density of counterions. Since the counterions can infiltrate into block B due to larger diffusivity and higher density, block A will also infiltrate into block B together with the electrostatically coupled counterions to form larger density peak width. Comparing Figures 5b and 5d, as fA increases, the density peaks of block A are higher than those of block B, as a direct result of increased block A composition. Comparing Figures 5b and 5e, as χN increases, a sharper interface between blocks A and B is formed due to a higher block repulsive energy and interfacial tension, in addition to the wider density peaks G


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Figure 8. Phase-dependent (a) Ea, (b) D0, and (c) α values in the modified Arrhenius equation, predicted from the maximum (left column), minimum (middle column), and orientationally averaged (right column) ion diffusivities at different temperatures. These DPD simulations are conducted under ϕ = 1:2 and εr = 1.1, which matches the predicted phase diagram in Figure 2.

Figure 3 when increasing ϕ. Furthermore, as χN decreases, η also decreases when getting closer to the ODT boundary and entering the D phase where η = 1. Finally, as εr decreases (see Figure 7d), much lower D̅ values are observed under all fA and χN values due to stronger electrostatic cohesions between block A and counterions, which greatly hinder ion diffusion into block B microdomains. In addition, as εr decreases, higher degrees of anisotropy in ion diffusivity are present for the C phase due to the sustained phase separations, as a result of strong electrostatic cohesions, even at lower counterion concentrations (or lower fA values). In summary, the computed degrees of anisotropy nicely reflect our DPD-simulated phase diagrams in Figures 2 and 3, which theoretically validate our phase diagram predictions. On the basis of Figure 7, we can conclude that the degrees of anisotropy in ion diffusivity strongly correlate with the anisotropy of the self-assembled polymer microdomains. From the most anisotropic L phase, as reducing χN or changing the fA values, both internal micropore formation across different lamella layers and the increased curvature of the lamella layers lead to the formation of gyroid phase. These two factors reduce the anisotropies of the polymer microdomain and the resulting ion diffusivity within them. On the other hand, the anisotropic C′ phase exhibits a low degree of anisotropy in ion diffusivity. This is due to the fact that the ioncontaining, A-block-rich microdomains serve as a bulk matrix for ion diffusion, which is weakly affected by the anisotropic cylindrical block-B microdomains. This interesting finding suggests that the degree of anisotropy in ion diffusivity is only correlated to the anisotropy of the ion-containing microdomains (which is formed by block A here). Evidently, ion conductivity is influenced by many factors, including ion diffusivity, ion concentration, electrostatic

are normal and parallel to the cylindrical microdomains, respectively. Using the three principal diffusivities, we can compute the orientationally averaged diffusivities D̅ (arithmetic mean over the three principal diffusivities) as well as the degrees of orientational anisotropy in ion diffusivities η, defined as the ratio between the maximum and minimum principal diffusivities (e.g., η = D∥/D⊥ for the L and C′ phases). The phase-dependent maximum and minimum principal diffusivities are shown in Figure S9. One would expect that η = 1 for perfectly isotropic phases (D, S, G, S′, and G′) and η > 1 for anisotropic phases (C, L, and C′). As shown in Figure 7 (left column), the computed average ion diffusivity D̅ exhibit strong dependence on fA and χN values, where higher D̅ is observed at higher fA and lower χN values, while lower D̅ is observed at lower fA and higher χN values. This trend reflects the underlying mechanisms that (i) higher ion concentration leads to higher ion diffusivity, consistent with the Kohlrausch’s (or Debye−Hückel−Onsager, DHO) limiting law on ion conductivity,39,40 and (ii) enhanced phase separation will hinder ion diffusivity due to the higher energy barrier for counterions, confined within block A, to diffuse into and across block B. As expected, the highest degree of anisotropy in ion diffusivity, η, is observed for the L phase, as shown in Figure 7 (right panel) for all the ϕ and εr values considered. The degree of anisotropy decreases when moving away from the L phase to other less anisotropic ordered phases, except for the highly anisotropic C phase. The absence of high η values for the C phase when ϕ = 1:2 (see Figure 7a, right panel) reflects the minimal C phase region and the more disordered microdomains when increasing ϕ. The computed anisotropic ion diffusion regions (with higher η values) also exhibit more leftward shifts when increasing ϕ (from Figures 7c to 7a), similar to the observed leftward shifts in OOT boundaries in H


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fact that phases exhibiting larger free energy barriers for ion diffusion should also behave more like a solid, whose transition rates depend more weakly on T with smaller α values. Such temperature dependence is well reflected by the synergy between the microdomain ODT and OOT in BCPs and the solid−liquid phase transition of materials in general.

interactions between ions under high ion concentrations, temperature, and solvent effects. Conventional DHO theory applies to dilute ionic solutions, where the ion conductivity increases as their concentration increases, as reflected from the strong dependence of the computed average ion diffusivity D̅ on fA in Figure 7. Chandra et al. further extended the DHO theory to highly concentrated ionic solutions and experimentally verified their theory.41 In their theory, the molar ionic conductivity would decrease and reach a plateau, in contrast to the continuously decayed molar ionic conductivity predicted by conventional DHO theory. We expect that such extended DHO theory for highly concentrated ionic solutions would be also captured by our DPD-simulated ion diffusivities, as reflected from the ion diffusivity in the D phase (see Figure 7). Assuming that the ionic conductivity is linearly dependent on their diffusivity, we use the Nernst−Einstein equation, σ = ∑i

cizi 2e 2 D, kBT i

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CONCLUSIONS In summary, we developed a modified DPD simulation framework to overcome the inherent difficulties of capturing, simultaneously, (i) electrostatic interactions, (ii) dynamic ionic transport process, and (iii) correlation between ionic diffusivity and microdomain structure. We first validated the predictability of our DPD simulation framework by comparing DPDsimulated phase diagram of conventional charge-neutral diblock copolymers with that predicted using the SCFT. A modified DPD simulation framework with explicit electrostatic interactions was then developed to systematically explore the structural, phase separation, and ion diffusion properties of diblock copolyelectrolytes. We found that when ϕ increases or when εr decreases, there is a significant leftward and upward shift of the phase diagram. The microdomain morphology of diblock copolyelectrolytes was successfully verified by computed structure factors (mimicking SAXS patterns), RDFs, and number density profiles. High consistency was found between experimental morphologies of single-ion-conducting diblock copolyelectrolytes and our DPD-predicted phase diagrams, suggesting that this DPD framework is very useful for the rational design of emerging charged BCP systems. Using a novel diffusivity tensor approach, we predicted the phase-dependent ion diffusivity along the principal microdomain orientations and found that the degree of anisotropy in ion diffusivity strongly correlates with that of the polymer microdomains. Ion conductivities depend strongly on ion concentrations and temperatures but weakly on phase diagram and other experimentally controllable parameters. Such a weak dependence is expected to become stronger for higher ion concentrations when the DHO theory dominates. The computed phase-dependent free energy barriers for ion diffusion verify the DPD-predicted morphology and nicely reflect the unique ion transport features within each type of microdomain. For the practical application of block copolyelectrolytes as solid-state electrolytes in Li-ion batteries, the perfect microphase candidate should have an optimal combination of both high ion conductivity and low degree of anisotropy in ion transport. Contrary to the conventional choice of the L phase with the highest degree of anisotropy in ion transport, we found that better candidates would be the block-A-rich inverse phases (G′, C′, and S′) with higher fA values and ion concentrations. Finally, we can also conclude that higher dielectric constants εr can lead to higher ion diffusivity by reducing the electrostatic cohesion between block A and the counterions and therefore facilitate ion diffusion into block B microdomains and across the A−B microdomain interfaces. Furthermore, mechanical robustness of these block copolyelectrolytes should also be considered, when inhibiting dendrite growth at the solid−electrolyte interphase (SEI) becomes critical. Although the DPD model used here does not allow the crystallization of blocks and can only provide qualitative trends, we would like to discuss the implication of block crystallinity here to help one design mechanically robust block copolyelectrolytes. For most of the existing block copolyelectrolytes, such as PSTFSI-b-PEO, block A (PSTFSI) is usually amorphous due

to predict the phase-dependent ion conductiv-

ity, as shown in Figure S10. Experiments have shown that ion conductivity in single-ion conducting block copolyelectrolytes increases as counterion concentration (proportional to the vol % of the charged block) increases.17,37,42,43 Note that such increase will last until the vol % of the ion-conducting neutral block reduces to the level that again hinders ion diffusion, since experiments were carried out under the L phase where the ionconducting block impacts more on ion diffusion. We observed the same trend here, where ion conductivities exhibit a very strong dependence on ion concentrations (or fA values) and weakly depend on the block repulsive energies (χN), charge fractions (ϕ), and dielectric constants (εr). This is due to the fact that the ion concentration term in the Nernst−Einstein equation dominates over the phase-dependent ion diffusivity term. We further predicted the phase-dependent free energy barriers, Ea, for ion diffusion in diblock copolyelectrolytes based on the principal and averaged ion diffusivities obtained from DPD simulations conducted at different temperatures T (see Figure 8). To fit the temperature-dependent ion diffusivities, we utilized the modified Arrhenius equation based on the transition-state theory: D = D0Tαe−Ea/kBT, where D0 is the intrinsic ion diffusivity, α is the power factor on T, and kB is the Boltzmann constant. This more generalized equation origins from the conventional Arrhenius equation (when α = 0) and the Eyring equation (when α = 1), while it accounts for the unique ion diffusion characteristics for each type of microdomains, as reflected by the D0, α, and Ea values obtained from nonlinear fitting (see Figure S11). As shown in Figure 8a, the predicted Ea values for ion diffusions normal to the lamellar microdomains (corresponds to the minimum diffusivity) are significantly larger than those parallel (corresponds to the maximum diffusivity) to the lamellar microdomains, consistent with the predicted phase diagram (Figure 2) and anisotropic ion diffusivities (Figure 7a). The Ea values corresponding to the averaged ion diffusivities exhibit a general trend that they are higher in ordered phases and under higher ion concentrations (or fA values). Furthermore, the predicted D0 values increase as the ion concentration increases, as a result of the DHO theory (see Figure 8b). Significantly reduced D0 values are observed for the minimum ion diffusivities across block B in the ordered block-A-poor phases. Interestingly, the predicted α values exhibit the exactly opposite trend as the predicted Ea values (see Figure 8c), consistent with the phase state of the selfassembled BCP microdomains. This can be explained by the I


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electrostatic screening is considered here by adjusting the dielectric constant, as discussed below using eq 1. All simulations were carried out using the massively parallelized LAMMPS package51 with modifications for electrostatic interactions based on the general DPD model (see section S3 in the Supporting Information). The simulated systems have a cubic dimension (32 × 32 × 32 Rc3, where Rc = 0.814 nm is the physical length scale in the DPD model here; see section S3) with 3D periodic boundary conditions. Each system consisted of a total of ∼100 000 DPD particles, representing AB copolymers for neutral BCP systems or An−B copolymers and associated counterions for charged BCP systems. Each copolymer chain is composed of a total of N = 16 covalently bonded DPD particles. The number of DPD particles (or A monomers) in block A, NA, is defined as NA = 16fA. For the initial system configuration, we grew polymer chains randomly with a fixed incremental length a (where a = Rc = 1 is the equilibrium bond distance between particles in the chain) in the simulation box up to the maximum polymer volume fraction, considering the number/volume fraction of counterions to be added. We then filled the box with counterions until the number density of DPD particles reaches ρRc3 = 3. For charged BCP systems with different charge fractions ϕ = 1:2, 1:4, and 1:8, partial charges of −0.5, −0.25, and −0.125 are assigned to each A block particle, respectively, while a constant partial charge of +1 is assigned to each counterion particle. The numbers of counterions added into the system under different ϕ values are adjusted for electroneutrality. After creating the initial random system configuration, we equilibrated each system for 8 × 106 steps (equivalent to a physical time scale of ∼90 μs) under the NVT ensemble in DPD. After equilibration, the internal normal stresses (pressures) are relaxed to ∼22 kBT/Rc3 along all three dimensions, which is very close to the pressure needed under the NPT ensemble to render a desired particle density of 3.52,53 The soft-core conservative potential used in DPD also avoids the accumulation of large internal stresses,54,55 and therefore the use of NVT ensemble here is expected to have a small impact on the simulated BCP microdomain morphologies. In the DPD model, length is in units of Rc, time in τ, and energy in kBT, which are all set to 1 here based on the Lennard-Jones (LJ) units in LAMMPS. A constant temperature of T = 1 (close to room temperature) is used here for integrating the DPD motion of equation with a fixed time step of Δt = 0.04 τ, where τ = 0.28 ns is the physical time scale in the DPD model here (see section S3).56 Details on the DPD formulizm and the parametrization procedure can be found in our previous work.56 The soft-core repulsive potential parameter, aij, in the DPD model dictates the interparticle interactions, in addition to electrostatic interactions.20 We took aAA = aBB = aCC (“C” denotes counterions) = 25 for interactions between the same particle type for which χ = 0. The interaction parameters between counterions and A or B particles are also fixed at aAC = aBC = 25 to focus only on the electrostatic interactions between the counterions and the A and B particles. To allow phase separations, interactions between A and B blocks should be more repulsive, and therefore a series of aAB values, ranging from 27 to 54 (correspond to χN values from 10 to 140), were used to render different microphases. A cutoff distance of Rc = 1 is used for the soft-core repulsive interactions.20 Each DPD particle represents ∼0.18 nm3 in volume and a polymer segment of Rc = 0.814 nm in length, representing ∼3 EO units

to ion doping, while block B (PEO) also tends to be amorphous to allow faster ion diffusion, which leads to mechanical vulnerability. The anisotropic nature of the L phase selected in these studies is the reason behind the need to use amorphous block B as ion conducting segments, since crystalline block B obtained below the glass transition temperature will block ion diffusion. On the other hand, in conventional charge-neutral BCPs, such as the poly(styrene-bn-butyl methacrylate) (PS-b-PBMA),44−46 block B (PS) is usually a mechanically robust crystalline segment to serve as structural holders, while block A (PBMA) is usually a soft amorphous segment to allow ion diffusion. With the above in mind, to harness the benefits of the crystalline segment (block B) as structural holders to enhance mechanical robustness, lower fA values are required, in addition to the critical need of well-percolated block A microdomains, such that ions do not need to diffuse through the crystalline block B. Such rational design can be indeed achieved by the G′ and C′ phases, not the S′ phase which possesses too high fA values and isolated crystalline block B. Therefore, we believe that the inverse G′ and C′ phases hold great potentials with good balance among several key design criteria for solid-state electrolytes, including ion conductivity (requiring higher fA), degree of anisotropy (requiring well-percolated amorphous block A), and mechanical robustness (requiring lower fA and well-percolated crystalline block B). This theoretical work significantly expands the design space for emerging charged BCP systems and will motivate future efforts to explore inverse phases without the engineering hurdle of aligning microdomains or removing grain boundaries in order to enhance ion transport.

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COMPUTATIONAL METHODS We carried out mesoscale simulations based on a modified DPD framework to explicitly model electrostatic interactions and ion transport in charged diblock copolymer systems. Such DPD framework poses a good balance between all-atomistic molecular dynamics simulations (limited by small time and length scales to predict equilibrium BCP morphologies) and SCFT calculations (suitable for thermodynamically equilibrated states without dynamical information). In our DPD simulations, we treat each solvated counterion as a DPD particle. Each A−B diblock copolymer is modeled by two covalently connected linear chains of different DPD particles, representing blocks A and B. The composition of the BCP system is controlled by the volume fraction of block A, fA = 1 − f B = vA/ (vA + vB), where v is the molar volume of each block. Immiscibility between the blocks A and B is measured by the Flory−Huggins interaction (repulsive) parameter, χ. Under specific fA values and temperatures, the χ value serves as the criterion for ODT, beyond which the copolymer segments phase separate thermodynamically into various types of microdomains.2 We modeled the solvent effects implicitly here, similar to earlier DPD studies on BCP systems,20 due to several reasons below: (i) the solvents are typically removed in BCP systems through evaporation to achieve solid state and reduce the moisture level (in the case of water); (ii) although the solvent effects could be modeled implicitly using the effective Flory− Huggins parameter between A and B blocks to reflect their hydrophobic/hydrophilic nature,47−50 the focus of this work is on electrostatic interactions; and (iii) the solvent-induced J


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between the velocity component of ion i along directions α and β, νiα(0) and νiβ(t), respectively. Note that conventional velocity autocorrelation functions (VACFs) only compute the normal diffusivities (diagonal terms of the tensor D): Dxx, Dyy, and Dzz. For isotropic diffusion in a homogeneous phase, the diagonal terms are expected to be identical: Dxx = Dyy = Dzz, while all the off-diagonal terms in D are zero. On the other hand, for general anisotropic diffusion in a heterogeneous phase, similar to what we are investigating here, the off-diagonal terms in D are nonzero. The principal diffusivity values of a general diffusivity tensor D are equal to the three eigenvalues of D, which are denoted as λ1, λ2, and λ3, while the three orthogonal eigenvectors of D denote the directions of the principal diffusivities. The principal diffusivity values can be projected into 3D as the three principal axes, a, b, and c, of an ellipsoid, such that a = λ1, b = λ2, and c = λ3 (see Figure 6c). Under the special condition of isotropic diffusivity, the ellipsoid becomes a sphere with a radius R = a = b = c = λ1 = λ2 = λ3 = 1/Dxx = 1/Dyy = 1/Dzz (see Figure 6d). The accuracy of the diffusivity tensor calculation is verified by comparing Dxx, Dyy, and Dzz computed using VACFs with those computed by the conventional mean-square displacement (MSD) calculations, as shown in Figure S12.

in a common diblock copolymer PEO-b-PS.56 Each counterion DPD particle represents a solvated counterion, which is larger than the stand-alone naked counterion in size. To address the local charge concentration effect at the mesoscale, we incorporated a new electrostatic interaction module into the current LAMMPS software package for DPD simulations. Since the DPD particles are modeled as soft-core repulsive particles, the charge will spread over a finite volume based on the Slater smearing charge distribution to avoid overlaps between oppositely charged particles,57,58 which ultimately lead to the modified Coulomb’s law: Uijelec =

⎡ ⎛ 2rij ⎞⎤ rij ⎞ ⎛ ⎢1 − ⎜1 + ⎟ exp⎜ − ⎟⎥ ⎝ 4πε0εrrij ⎣ L⎠ ⎝ L ⎠⎦ qiqj

(1)

where ε0 is the vacuum permittivity, εr is the dielectric constant of the background medium, qi and qj are partial charges of a pair of DPD particles, rij is the distance between a pair of DPD particles, and L = 0.32 is the decay length of electrostatic interactions according to Sindelka et al.58 Because of the coarsegraining nature of the DPD method, the dielectric constant of the background medium is reflected by εr from a mean-field aspect. Using the modified Coulomb’s law, one can incorporate the solvent conditions implicitly into the current simulation framework. Different dielectric media, such as water and other organic solvents, in which the BCPs are dissolved, could be modeled by the εr value.59 Long-range electrostatic interactions are treated using the particle−particle particle-mesh (PPPM) solver method60 with a cutoff distance of 2Rc = 2 and a convergence tolerance value of 0.1% on force calculations. To mimic aqueous polar environment, we set the dielectric constant εr = 1.1 in DPD units.57 To mimic nonpolar environment with organic solvents, a lower dielectric constant εr = 1.1/4 = 0.275 is used. The degree of ion transport in the diblock copolyelectrolyte system is quantified using the DPD-simulated ion trajectories and the associated ion self-diffusions. To quantify the ion selfdiffusion process, the ion diffusivity tensor, D, for mobile counterions in different microphases is determined from the velocity correlation functions (VCFs). A tensor form of the ion diffusivity is deployed here to characterize the orientational anisotropy in ion diffusivity in 3D, such that we can decompose the conventional, orientationally averaged diffusivity into contributions from each principal microdomain orientation (e.g., the cylindrical axis for the C phase or the two axes in parallel to the lamellae in the L phase). Specifically, we will compute the diffusivity tensor D, using VCFs obtained from simulated ion trajectories: ⎡ Dxx Dxy Dxz ⎤ ⎢ ⎥ D = ⎢ Dyx Dyy Dyz ⎥ ⎢ ⎥ ⎢⎣ Dzx Dzy Dzz ⎥⎦

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The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b00451. Structure factor calculations; correcting DPD-simulated phase diagrams by the low-molecular-weight effect; general description of the DPD simulation and physical time/length scales; and Figures S1−S12 (PDF)

■

1 N

N

⎡

∑ ⎢⎣∫ i=1

0

∞

⎤ ⟨ναi (0)νβi (t )⟩ dt ⎥ ⎦

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (S.L.). ORCID

Lingling Zhao: 0000-0002-1532-7247 Shangchao Lin: 0000-0002-6810-1380 Author Contributions

C.Z. and H.Z. contributed equally to this work. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge startup funding from the Energy and Materials Initiative from the Florida State University. Acknowledgment is made to the donors of the American Chemical Society (ACS) Petroleum Research Fund (PRF) under the Doctoral New Investigator (DNI) Award # 56499-DNI7. The authors would also acknowledge computational resources provided by the XSEDE program under Grant TGDMR160044.

(2)

where each diffusivity term in the tensor is computed by the VCF: Dαβ =

ASSOCIATED CONTENT

S Supporting Information *

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REFERENCES

(1) Hu, H. Q.; Gopinadhan, M.; Osuji, C. O. Directed self-assembly of block copolymers: a tutorial review of strategies for enabling nanotechnology with soft matter. Soft Matter 2014, 10 (22), 3867− 3889. (2) Mai, Y. Y.; Eisenberg, A. Self-assembly of block copolymers. Chem. Soc. Rev. 2012, 41 (18), 5969−5985.

(3)

where α and β are the indices of the original Cartesian coordinate of the simulation box, i denotes the ith ion, N is the total number of ions in the system, and ⟨νiα(0)νiβ(t)⟩ is the VCF K


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Model with Short-Range Interactions. Macromolecules 2013, 46 (14), 5724−5730. (23) Knychała, P.; Banaszak, M.; Balsara, N. Effect of composition on the phase behavior of ion-containing block copolymers studied by a minimal lattice model. Macromolecules 2014, 47 (7), 2529−2535. (24) Yang, S. A.; Vishnyakov, A.; Neimark, A. V. Self-assembly in block polyelectrolytes. J. Chem. Phys. 2011, 134 (5), 054104. (25) Sing, C. E.; Zwanikken, J. W.; Olvera de la Cruz, M. Electrostatic control of block copolymer morphology. Nat. Mater. 2014, 13 (7), 694−698. (26) Böker, A.; Elbs, H.; Hänsel, H.; Knoll, A.; Ludwigs, S.; Zettl, H.; Urban, V.; Abetz, V.; Müller, A. H. E.; Krausch, G. Microscopic Mechanisms of Electric-Field-Induced Alignment of Block Copolymer Microdomains. Phys. Rev. Lett. 2002, 89 (13), 135502. (27) Morkved, T. L.; Lu, M.; Urbas, A. M.; Ehrichs, E. E.; Jaeger, H. M.; Mansky, P.; Russell, T. P. Local Control of Microdomain Orientation in Diblock Copolymer Thin Films with Electric Fields. Science 1996, 273 (5277), 931−933. (28) Olszowka, V.; Hund, M.; Kuntermann, V.; Scherdel, S.; Tsarkova, L.; Böker, A. Electric Field Alignment of a Block Copolymer Nanopattern: Direct Observation of the Microscopic Mechanism. ACS Nano 2009, 3 (5), 1091−1096. (29) Matsen, M. W.; Schick, M. Stable and unstable phases of a diblock copolymer melt. Phys. Rev. Lett. 1994, 72 (16), 2660. (30) Matsen, M. W. Effect of Architecture on the Phase Behavior of AB-Type Block Copolymer Melts. Macromolecules 2012, 45 (4), 2161−2165. (31) Matsen, M. Self-consistent field theory for melts of lowmolecular-weight diblock copolymer. Macromolecules 2012, 45 (20), 8502−8509. (32) Nakamura, I.; Balsara, N. P.; Wang, Z.-G. Thermodynamics of ion-containing polymer blends and block copolymers. Phys. Rev. Lett. 2011, 107 (19), 198301. (33) Huang, J.; Tong, Z.-Z.; Zhou, B.; Xu, J.-T.; Fan, Z.-Q. Saltinduced microphase separation in poly (ε-caprolactone)-b-poly (ethylene oxide) block copolymer. Polymer 2013, 54 (12), 3098− 3106. (34) Park, M. J.; Balsara, N. P. Phase behavior of symmetric sulfonated block copolymers. Macromolecules 2008, 41 (10), 3678− 3687. (35) Wang, X.; Yakovlev, S.; Beers, K. M.; Park, M. J.; Mullin, S. A.; Downing, K. H.; Balsara, N. P. On the origin of slow changes in ionic conductivity of model block copolymer electrolyte membranes in contact with humid air. Macromolecules 2010, 43 (12), 5306−5314. (36) Thelen, J. L.; Chen, X. C.; Inceoglu, S.; Balsara, N. P. Influence of Miscibility on Poly (ethylene oxide) Crystallization from Disordered Melts of Block Copolymers with Lithium and Magnesium Counterions. Macromolecules 2017, 50 (12), 4827−4839. (37) Rojas, A. A.; Inceoglu, S.; Mackay, N. G.; Thelen, J. L.; Devaux, D.; Stone, G. M.; Balsara, N. P. Effect of lithium-ion concentration on morphology and ion transport in single-ion-conducting block copolymer electrolytes. Macromolecules 2015, 48 (18), 6589−6595. (38) Bates, F. S.; Fredrickson, G. H. Block Copolymers-Designer Soft Materials. Phys. Today 1999, 52 (2), 32−38. (39) Onsager, L. Theories of Concentrated Electrolytes. Chem. Rev. 1933, 13 (1), 73−89. (40) Debye, P.; Hückel, E. The theory of electrolytes. I. Lowering of freezing point and related phenomena. Phys. Z. 1923, 24 (9), 185− 206. (41) Chandra, A.; Biswas, R.; Bagchi, B. Molecular Origin of the Debye− Huckel− Onsager Limiting Law of Ion Conductance and Its Extension to High Concentrations: Mode Coupling Theory Approach to Electrolyte Friction. J. Am. Chem. Soc. 1999, 121 (16), 4082−4083. (42) Porcarelli, L.; Shaplov, A. S.; Salsamendi, M.; Nair, J. R.; Vygodskii, Y. S.; Mecerreyes, D.; Gerbaldi, C. Single-ion block copoly (ionic liquid) s as electrolytes for all-solid state lithium batteries. ACS Appl. Mater. Interfaces 2016, 8 (16), 10350−10359. (43) Thelen, J. L.; Inceoglu, S.; Venkatesan, N. R.; Mackay, N. G.; Balsara, N. P. Relationship between ion dissociation, melt morphology,

(3) 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. (4) Soo, P. P.; Huang, B. Y.; Jang, Y. I.; Chiang, Y. M.; Sadoway, D. R.; Mayes, A. M. Rubbery block copolymer electrolytes for solid-state rechargeable lithium batteries. J. Electrochem. Soc. 1999, 146 (1), 32− 37. (5) Bouchet, R.; Maria, S.; Meziane, R.; Aboulaich, A.; Lienafa, L.; Bonnet, J. P.; Phan, T. N. T.; Bertin, D.; Gigmes, D.; Devaux, D.; Denoyel, R.; Armand, M. Single-ion BAB triblock copolymers as highly efficient electrolytes for lithium-metal batteries. Nat. Mater. 2013, 12 (5), 452−457. (6) Kim, S. O.; Solak, H. H.; Stoykovich, M. P.; Ferrier, N. J.; de Pablo, J. J.; Nealey, P. F. Epitaxial self-assembly of block copolymers on lithographically defined nanopatterned substrates. Nature 2003, 424 (6947), 411−414. (7) Nie, Z. H.; Kumacheva, E. Patterning surfaces with functional polymers. Nat. Mater. 2008, 7 (4), 277−290. (8) Tavakkoli, K. G. A.; Gotrik, K. W.; Hannon, A. F.; AlexanderKatz, A.; Ross, C. A.; Berggren, K. K. Templating Three-Dimensional Self-Assembled Structures in Bilayer Block Copolymer Films. Science 2012, 336 (6086), 1294−1298. (9) Angelescu, D. E.; Waller, J. H.; Adamson, D. H.; Deshpande, P.; Chou, S. Y.; Register, R. A.; Chaikin, P. M. Macroscopic orientation of block copolymer cylinders in single-layer films by shearing. Adv. Mater. 2004, 16 (19), 1736−1740. (10) Cui, H. G.; Chen, Z. Y.; Zhong, S.; Wooley, K. L.; Pochan, D. J. Block copolymer assembly via kinetic control. Science 2007, 317 (5838), 647−650. (11) Kim, S. H.; Misner, M. J.; Xu, T.; Kimura, M.; Russell, T. P. Highly oriented and ordered arrays from block copolymers via solvent evaporation. Adv. Mater. 2004, 16 (3), 226−231. (12) Doyle, M.; Fuller, T. F.; Newman, J. The importance of the lithium ion transference number in lithium/polymer cells. Electrochim. Acta 1994, 39 (13), 2073−2081. (13) Rey, I.; Lassègues, J. C.; Baudry, P.; Majastre, H. Study of a lithium battery by confocal Raman microspectrometry. Electrochim. Acta 1998, 43 (10−11), 1539−1544. (14) Sun, X.-G.; Kerr, J. B. Synthesis and Characterization of Network Single Ion Conductors Based on Comb-Branched Polyepoxide Ethers and Lithium Bis(allylmalonato)borate. Macromolecules 2006, 39 (1), 362−372. (15) Ryu, S.-W.; Trapa, P. E.; Olugebefola, S. C.; Gonzalez-Leon, J. A.; Sadoway, D. R.; Mayes, A. M. Effect of Counter Ion Placement on Conductivity in Single-Ion Conducting Block Copolymer Electrolytes. J. Electrochem. Soc. 2005, 152 (1), A158−A163. (16) Feng, S.; Shi, D.; Liu, F.; Zheng, L.; Nie, J.; Feng, W.; Huang, X.; Armand, M.; Zhou, Z. Single lithium-ion conducting polymer electrolytes based on poly[(4-styrenesulfonyl)(trifluoromethanesulfonyl)imide] anions. Electrochim. Acta 2013, 93, 254−263. (17) Inceoglu, S.; Rojas, A. A.; Devaux, D.; Chen, X. C.; Stone, G. M.; Balsara, N. P. Morphology−Conductivity Relationship of SingleIon-Conducting Block Copolymer Electrolytes for Lithium Batteries. ACS Macro Lett. 2014, 3 (6), 510−514. (18) Hallinan, D. T.; Balsara, N. P. Polymer Electrolytes. Annu. Rev. Mater. Res. 2013, 43, 503−525. (19) Wanakule, N. S.; Virgili, J. M.; Teran, A. A.; Wang, Z. G.; Balsara, N. P. Thermodynamic Properties of Block Copolymer Electrolytes Containing Imidazolium and Lithium Salts. Macromolecules 2010, 43 (19), 8282−8289. (20) Groot, R. D.; Madden, T. J. Dynamic simulation of diblock copolymer microphase separation. J. Chem. Phys. 1998, 108 (20), 8713−8724. (21) Gavrilov, A. A.; Kudryavtsev, Y. V.; Chertovich, A. V. Phase diagrams of block copolymer melts by dissipative particle dynamics simulations. J. Chem. Phys. 2013, 139 (22), 224901. (22) Knychała, P.; Dzięcielski, M.; Banaszak, M.; Balsara, N. P. Phase Behavior of Ionic Block Copolymers Studied by a Minimal Lattice L


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Macromolecules and electrochemical performance of lithium and magnesium single-ion conducting block copolymers. Macromolecules 2016, 49 (23), 9139− 9147. (44) Weidisch, R.; Michler, G.; Fischer, H.; Arnold, M.; Hofmann, S.; Stamm, M. Mechanical properties of weakly segregated block copolymers: 1. Synergism on tensile properties of poly (styrene-b-nbutylmethacrylate) diblock copolymers. Polymer 1999, 40 (5), 1191− 1199. (45) Weidisch, R.; Michler, G.; Arnold, M. Mechanical properties of weakly segregated block copolymers 2. The influence of phase behaviour on tensile properties of poly (styrene-b-butylmethacrylate) diblock copolymers. Polymer 2000, 41 (6), 2231−2240. (46) Weidisch, R.; Stamm, M.; Michler, G.; Fischer, H.; Jérôme, R. Mechanical properties of weakly segregated block copolymers. 3. Influence of strain rate and temperature on tensile properties of poly (styrene-b-butyl methacrylate) diblock copolymers with different morphologies. Macromolecules 1999, 32 (3), 742−750. (47) Léonforte, F.; Mü ller, M. Morphology Modulation of Multicomponent Polymer Brushes in Selective Solvent by Patterned Surfaces. Macromolecules 2015, 48 (1), 213−228. (48) Wang, J.; Müller, M. Microphase separation of diblock copolymer brushes in selective solvents: Single-chain-in-mean-field simulations and integral geometry analysis. Macromolecules 2009, 42 (6), 2251−2264. (49) Hannon, A.; Bai, W.; Alexander-Katz, A.; Ross, C. Simulation methods for solvent vapor annealing of block copolymer thin films. Soft Matter 2015, 11 (19), 3794−3805. (50) Chang, J.-B.; Son, J. G.; Hannon, A. F.; Alexander-Katz, A.; Ross, C. A.; Berggren, K. K. Aligned sub-10-nm block copolymer patterns templated by post arrays. ACS Nano 2012, 6 (3), 2071−2077. (51) Plimpton, S. Fast Parallel Algorithms for Short-Range Molecular-Dynamics. J. Comput. Phys. 1995, 117 (1), 1−19. (52) Trofimov, S.; Nies, E.; Michels, M. Constant-pressure simulations with dissipative particle dynamics. J. Chem. Phys. 2005, 123 (14), 144102. (53) Jakobsen, A. F. Constant-pressure and constant-surface tension simulations in dissipative particle dynamics. J. Chem. Phys. 2005, 122 (12), 124901. (54) Dequidt, A.; Solano Canchaya, J. G. Bayesian parametrization of coarse-grain dissipative dynamics models. J. Chem. Phys. 2015, 143 (8), 084122. (55) Solano Canchaya, J. G.; Dequidt, A.; Goujon, F.; Malfreyt, P. Development of DPD coarse-grained models: From bulk to interfacial properties. J. Chem. Phys. 2016, 145 (5), 054107. (56) Lin, S.; Ryu, S.; Tokareva, O.; Gronau, G.; Jacobsen, M. M.; Huang, W. W.; Rizzo, D. J.; Li, D.; Staii, C.; Pugno, N. M.; Wong, J. Y.; Kaplan, D. L.; Buehler, M. J. Predictive modelling-based design and experiments for synthesis and spinning of bioinspired silk fibres. Nat. Commun. 2015, 6, 6892. (57) Groot, R. D. Electrostatic interactions in dissipative particle dynamics-simulation of polyelectrolytes and anionic surfactants. J. Chem. Phys. 2003, 118 (24), 11265−11277. (58) Sindelka, K.; Limpouchova, Z.; Lisal, M.; Prochazka, K. Dissipative Particle Dynamics Study of Electrostatic Self-Assembly in Aqueous Mixtures of Copolymers Containing One Neutral WaterSoluble Block and One Either Positively or Negatively Charged Polyelectrolyte Block. Macromolecules 2014, 47 (17), 6121−6134. (59) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Chemistry, 3rd ed.; Marcel Dekker: New York, 1997; p xix, 650 pp. (60) Deserno, M.; Holm, C. How to mesh up Ewald sums. II. An accurate error estimate for the particle−particle−particle-mesh algorithm. J. Chem. Phys. 1998, 109 (18), 7694−7701.

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