Article Cite This: Macromolecules XXXX, XXX, XXX−XXX
Ion Distribution in Microphase-Separated Copolymers with Periodic Dielectric Permittivity Weiwei Chu,† Jian Qin,†,‡,§ and Juan J. de Pablo*,†,‡ †
Institute for Molecular Engineering, University of Chicago, Chicago, Illinois 60637, United States Argonne National Laboratory, Argonne, Illinois 70439, United States § Department of Chemical Engineering, Stanford University, Stanford, California 94305, United States ‡
ABSTRACT: We examine the distribution of ionic species in charged, microphase-separated diblock copolymers by relying on coarse-grained simulations of the underlying materials. The model adopted here has been particularly useful in understanding the behavior of neutral block polymer systems. In this work, it is extended to describe charged molecules. A simulation methodology is proposed in which dielectric inhomogeneities within the phase-separated systems are taken into account by solving on-the-fly Poisson’s equation for spatially varying dielectric permittivity and calculating the corresponding electric fields. The Green’s function appropriate for a periodic and ionic system is explicitly calculated and used in simulations to arrive at phase diagrams as a function of salt concentration and copolymer composition. The systems considered here are representative of lithium salt-doped diblock copolymers, which could be of potential use for solid-electrolyte batteries.
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informed coarse grained (TICG) simulations approaches.11,12 Recent work from our own group has incorporated charges into such models, thereby permitting simulation of phase transitions and critical behavior in charged block copolymers.13 More recent efforts have built on the SCMF and TICG formalisms to examine coacervation and microphase segregation in polyelectrolytes.14 In those studies, however, the dielectric constant of the system was assumed to be uniform. In this work, we propose a methodology that bridges molecular-level simulations with a continuum description of dielectric inhomogeneity, thereby establishing a self-consistent framework with which to couple the effects of fluctuations and dielectric inhomogeneities. This framework is implemented within the context of lithium salt-doped diblock copolymers. These materials have shown promise as separation membranes for lithium ion batteries.15 Typically, the block copolymers contain a glassy block that confers mechanical strength, such as polystyrene (PS), and one lithium ion associating block that confers ionic conductivity, such as poly(ethylene oxide) (PEO). The dielectric permittivity of PEO at room temperature is more than twice that of polystyrene.16 In microphase-separated materials, a spatial modulation of the dielectric constant arises, which severely affects the localization of ions, largely as a consequence of the Born solvation energy.4,17 The Born solvation energy measures the energetic cost of transferring an ion from one dielectric medium to the other, and it decreases with increasing dielectric constant. In recent work it has been identified as a key contributor to the shift in the order disorder transition temperature of salt-doped diblock copolymers.18
INTRODUCTION A wide range of biological aggregates and synthetic materials exhibit a spatially heterogeneous dielectric response. Examples include protein fibrils in aqueous solution,1 ion channels across lipid bilayer membranes,2 polymer−ceramic composites,3 and block polymers with spatially varying dielectric permittivity.4 In these examples, the dielectric inhomogeneities affect a diverse range of properties, ranging from the ion distribution across the interface between hydrophobic and hydrophilic domains to the ionic conductivity across the channel, the capacitance and dielectric response of a composite, or the transport of ions in solid electrolytes. Nearly all systems that exhibit a structural or morphological inhomogeneity at mesoscopic length scales (between 5 nm and 1 μm) exhibit a heterogeneous dielectric response. Inhomogeneities in dielectric permittivity are often ignored, but as shown in this work, they can have important consequences. At atomistic length scales, properly parametrized force fields are used to describe the effects of polarization explicitly.5 Such approaches, however, generally assume a uniform dielectric constant for the evaluation of long-range interactions (e.g., in the evaluation of Ewald sums6,7). It is unclear what the consequences of that assumption are. At continuum length scales, the effects of polarization (or charge accumulation at interfaces) may be modeled by introducing a spatially varying dielectric constant.8 In that limit, electrostatic interactions can be handled by solving Poisson’s equation for an inhomogeneous dielectric profile. In past work, however, Poisson’s equation has been used in the context of mean-field treatments,4,9 and as such, important fluctuation effects have not been taken into account. A promising alternative to incorporate the effects of fluctuations on the behavior of ordered block polymers is to rely on single-chain in mean field (SCMF)10 or theoretically © XXXX American Chemical Society
Received: November 27, 2017 Revised: January 25, 2018
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DOI: 10.1021/acs.macromol.7b02508 Macromolecules XXXX, XXX, XXX−XXX
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as illustrated in Figure 1. The order of the distribution can be adjusted31 by relying on mapping schemes of different order. In
Recent experimental studies have shown that the morphology of microphase-separated copolymers affects salt distribution and ionic conductivity.19−21 More specifically, addition of salts influences phase behavior22−25 and the characteristics of the underlying phases (e.g., domain spacing26). In the past, such effects have been summarized by resorting to an effective “Flory−Huggins” parameter, which can be accessed in both experiments and theory.4,27,28 In this work, we examine the consequences of introducing a locally varying dielectric constant. More specifically, we use theory and simulations to address the following two questions: (1) how is ion distribution affected by the physical (dielectric permittivity) and chemical (chemical components) properties of the underlying polymers and (2) how does the phase diagram topology of polymers change with addition of salts. By relying on a coarse-grained model of ionic polymeric systems, we are able to explore the relevant parameter space systematically and present results for properties that are not easily accessible in experiments, such as the localization of salts.29 The effect of dielectric inhomogeneity is incorporated by solving Poisson’s equation explicitly. We find that ions are localized in the domains with higher dielectric permittivity, thereby lowering the Born solvation energy. The phase diagram of neat diblock copolymers is also altered considerably by the addition of ions; within the range of ion concentration explored here, the corresponding values of (χN)ODT become smaller, and the phase diagrams become asymmetric.
Figure 1. Schematics of dielectric inhomogeneity in lamellar morphology of salt-doped diblock copolymers. In the TICG model, the bead number density fields and charge density fields are constructed by distributing the mass or charge of the beads onto the underlying lattice sites in the neighborhood (labeled by a, b, c, and d). Within the PM1 scheme, eight such lattices sites are involved for each bead. The spatially varying dielectric permittivity is the numberaverage of the dielectric permittivities of two blocks.
this work, we use the PM1 scheme, which maps each bead onto eight neighboring sites by linear interpolation; additional details can be found in refs 30 and 31. To model salt-doped diblock copolymers, we use four types of beads, A, B, C, or D, which represent PEO-like, PS-like, cations, and anions, respectively. The χ parameter is assigned to bead pairs A/B and A/C, and is set to vanish for all other bead pairs, to reduce the number of free parameters. A positive value for χAB is used to model the thermodynamic incompatibility between PEO- and PS-like blocks. A negative value for χAC is used to model the complexation between lithium ions and PEO-like blocks. The details have been reported previously.13 We note that, given the coarse-grained nature of our model, the reference to PEO- and PS-like domains should not be taken literally, unlike the case of atomistic simulations. Coulombic Interaction. The Born solvation energy Hs measures the energetic cost of transferring ions into a medium 2 of dielectric permittivity ϵ(r) and is given by Hs = ∑C,D i qi / 8πϵ(r)ai, where qi is the charge on bead i and ai are ion radii. The summation is performed over all charged beads (C or D) in the system. For a homogeneous medium with constant dieletric permittivity, the Born solvation energy is uniform. It does not affect ion partitioning, and has been adapted to investigate the ordering transition in the weak segregation regime.13 For a heterogeneous dielectric system, however, the Born solvation energy will drive ions preferentially into domains with higher dielectric permittivity. In the salt-doped PS−PEO system, the dielectric permittivity of the PEO-like block is nearly twice that of the PS-like block, and the Born solvation has a strong effect on the distribution of ions. The Coulombic energy Hc includes interactions from all pairs of ionic beads and is given by Hc = (1/2)∫ dr q(r)ϕ(r), where q(r) is the charge density field and ϕ(r) is the electrostatic potential. The charge density field can be constructed in the same way as the bead number density field, using the same grid that was depicted in Figure 1. The electrostatic potential can be obtained by solving Poisson’s equation:
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METHODOLOGY We model dense ionic polymeric liquids by incorporating charge interactions into the framework of a theoretically informed coarse-grained (TICG) model. The TICG framework has been successfully applied to study the phase behavior of block copolymers in the bulk, in thin films, and in solutions.30 In this model, polymers are treated as bead−spring chains, and solvents and ions are treated as simple beads. For neutral systems, the Hamiltonian of the system includes two terms: one term Hb describes bonded, intramolecular interactions, and the other, Hnb, quantifies nonbonded intermolecular interactions. For charged systems, contributions from the Born solvation Hs and pairwise Coulomb interaction Hc must also be included. We have recently employed a simplified version of this model to investigate the ordering transition in salt-doped symmetric diblock copolymers.13 Neutral Interaction. The bond energy Hb/kBT is modeled as a harmonic potential energy, and is written in the form (3/ N 2 2b2)∑M m=1∑i=2(Rm,i − Rm,i−1) , where M is the number of polymers, N is the number of beads on each polymer, and Rm,i are bead position vectors. The prefactor 3/(2b2) is the effective spring constant for Gaussian chains, and b is the statistical segment length representing chain flexibility. The nonbonded energy Hnb is evaluated by relying on coarsegrained bead density fields ϕα(r), which measure the number densities of type α beads at position r. The energy is expressed in terms of ϕα(r) as Hnb/kBT = ∑αβχαβ∫ dr ϕα(r)ϕβ(r) + (κ/ 2)∫ dr(∑αϕα(r) − 1)2. Here χαβ are Flory−Huggins parameters that quantify the thermodynamic incompatibility between beads of type α and β, and κ denotes the compressibility of the material. The fields ϕα(r) are defined by (1/ρ)∑αi δ(Ri − r), where ρ is the average bead number density and i runs over all beads in the system of type α. The Dirac delta function is regularized by assigning the bead mass onto nearby lattice sites,
∇·ϵ(r)∇ϕ(r) = −q(r) B
(1) DOI: 10.1021/acs.macromol.7b02508 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules Here ϵ(r) is the spatially varying dielectric permittivity, which may be expressed as a density-weighted average by using a proper mixing rule based on the instantaneous density profile. For simplicity, we use a linear mixing rule, i.e., ϵ(r) = ϕA(r)ϵA + ϕB(r)ϵB. Higher-order interpolation schemes such as a 1/3 mixing rule8 may also be employed. In order to reduce the computational costs associated with solution of Poisson’s equation, the following Green’s function is introduced. Note that the Green’s function is uniquely specified by the dielectric permittivity profile. It is given implicitly by ϕ(r) = ∫ dr′ G(r,r′)q(r′), where G(r,r′) is the electrostatic potential at point r induced by a point charge located at position r′. For a homogeneous dielectric medium, the Green’s function is determined by the relative position vector, and we have G(r,r′) = G(|r − r′|). For a heterogeneous dielectric medium, the translational symmetry is broken, and the Green’s function depends on both arguments. Because we use periodic boundary conditions, the dielectric profile is also a periodic function of position r. We therefore need to solve the Green’s function subject to the constraint of periodic boundary conditions. The proper basis functions for that purpose are the Fourier waves, and we adopt the following convention for the discrete Fourier transform (DFT). The DFT ûj = û(qj) of function ui = u(ri) is defined by ûj = ∑iuie−iqj·ri = ∑iuie−i2π(i1j1/n1+i2j2/n2+i3j3/n3), where qj = 2π(j1/L1, j2/ L2, j3/L3) and rj = (i1L1/n1, i2L2/n2, i3L3/n3). According to such a convention, û and u have the same physical dimension. The 1 corresponding inverse-DFT is given by ui = n n n ∑i uĵ eiq j·ri .
In the general case, it can be shown that the Green’s function is both symmetric and real-valued. To see this, we denote the inverse of the Green’s operator G by H. The elements of H are given by Hi,j = (qi·qj)ϵ̂i−j. Since the dielectric permittivity ϵ is real, its Fourier coefficients satisfy ϵ̂−i = ϵ̂*i . It then follows that (1) Hj,i = H*i,j (Hermitian) and (2) H−i,−j = H*i,j . Since H is Hermitian, both of these two properties can be carried onto G. Property 1 ensures that G is symmetric, G(r′,r) = G(r,r′), and property 2 ensures that G(r,r′) is real. We now evaluate the total energy. Let the net charge at the ith site be Qi. The total electrostatic energy can be calculated by E = 1/2∑i,jQiQjG(ri,rj). The energy scale is set by the number of grid points. Suppose the total absolute charge in the system is Q. Then the site charge Qi is of order Q/Ng, and the interaction between two lattice sites scales with Q2/Ng2ϕ ≃ Q2/Ng. In addition, we also include the Born solvation energy term, EBorn = qi2/(8πϵ(r))ai, in the Hamiltonian. Unless otherwise stated, our simulation box is 4 × 4 × 4Re3, where Re is the average end-to-end distance of the polymer molecules. The box is divided into 12 × 12 × 12 lattice sites. The number of polymer beads in one chain is N = 16. The invariant degree of interdigitation is N̅ = 256. The Flory− Huggins parameter χACN = −40 is used for PEO and lithium ions,13 and χAB is varied in a systematic manner to examine the response of the material. Following past reports,30 the compressibility κN is set to 50. The dielectric permittivity of PS is 4, and that of PEO is twice that of PS. The number of polymer chains is 1024. The ratio between ion beads and polymer beads is 1/16. The radius of the ions is half that of the grid size. These parameters are chosen according to past reports that applied successfully the TICG model to neutral block copolymers. The model is solved by resorting to Monte Carlo simulations using a variety of Monte Carlo moves. Specifically, we use 20% single-bead moves, 40% of chainflipping moves, and 40% of chain-reptation moves. We update the dielectric permittivity profile and solve the Green’s function for Poisson’s equation at intervals of 3000 MC steps. The energy profile is shown in Figure 2.
1 2 3
To solve the Green’s function G(r,r′) in Fourier space, we begin with the conventional definition ∇·ϵ(r) ∇ ϕ(r) = −δ(r − r′). In the Fourier representation, this equation can be written Ng as ∑ ((q + q ) ·q )ϵ̂ ϕ ̂ei(q k + q j)r = ∑ eiq i·(r − r ′), where Ng k ,j
k
j
j
k j
v
i
≡ n1n2n3 is the number of lattice sites, and v ≡ L1L2L3/(n1n2n3) is the site volume. Multiplying both sides by e −iq i ·r , and summing over lattice sites labeled by r, gives Ng ∑ ((q + q ) ·q )ϵ̂ ϕδ̂ = e−iq i·r ′ or, alternatively,
k j i,k+j Ng −iq ·r ′ ∑j (q i·q j)ϵ̂i − jϕĵ = v e i . −1 ̂ k ,j
k
j
j
v
This equation can be written in
matrix form as G ·ϕ = ê(r′), with (G−1)ij = (qi·qj)ϵ̂i − j, and ê(r′)i = (n1n2n3)e−iqi·r′/v. The Fourier component ϕ̂ is given by G·ê(r′). The electrostatic potential at r generated by a point charge at r′ is therefore given by G(r,r′) = (1/Ng)∑iϕ̂ ieiqi·r = (1/ v)∑i,jeiqi·rGije−iqj·r′, the inverse Fourier transform of G that converts parameters (qi, qj) to (r, −r′). Direct calculation of G(r,r′) requires Ng4 floating point operations. By implementing a fast Fourier transform (FFT), the order of the calculations can be reduced to Ng2 log(Ng). The wave indices cover the range from −Ni/2 to Ni/2 for each dimensional index i. When constructing the Green’s matrix, whenever the wave index difference falls out this bound, we shift them back by adding or subtracting an index Ni. Furthermore, the q = 0 components are excluded, since they are related to the electrostatic response for the total number of charges in the system, which obviously vanish for a charge-neutral system. In a medium with a homogeneous dielectric profile, the Fourier component of dielectric permittivity ϵk only has one homogeneous component, and G is diagonal. We may explicitly write G(r;r′) = G(r − r′) = (1/v)∑ieiqi·(r−r′)/ϵ0qi2 = (1/4π2ϵ0) v∑ieiqi·(r−r′)/[(i1/L1)2 + (i2/L2)2 + (i3/L3)2]. This is identical to the results obtained previously.32
Figure 2. Variation of energies with the number of MC simulation steps. The dielectric profile is updated every 3000 steps, following each of which the solvation energy displays a jump.
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RESULTS Our simulations are initiated from random configurations; the corresponding dielectric profile is determined from the bead density fields, and then Green’s function is solved when the dielectric profile is updated. The energy of a configuration is calculated using the electrostatic potential ϕ(r). The computaC
DOI: 10.1021/acs.macromol.7b02508 Macromolecules XXXX, XXX, XXX−XXX
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importance of the Born solvation energy term for structured dielectric media.4 In Figure 4 results are shown for the density profile of phases with cylindrical symmetries. Results are presented along two
tional cost associated with calculation of the energy scales with the number of grid points Ng. Figure 2 shows how different energy components evolve with the number of MC steps. Every time that the dielectric profile is updated, the solvation energy drops substantially, serving to reflect the convergence of the simulated dielectric profile. During the course of a simulation, the bead density fields, and the corresponding ion distributions, converge relatively slowly. For sufficiently long simulations, the dielectric density profile and the bead number densities become self-consistent. Figure 3 shows the morphology and density profile of lamellar-forming symmetric diblock copolymers. ϕ(z) repre-
Figure 4. Density profiles for cylindrical morphologies. Results are shown for both hexagonal and inverted hexagonal phases along characteristic directions α[0, 1] and β[1/2, √3/2]. (a) NPEO:NPS = 4:12. PEO (green) forms a cylinder and salts (red for cation and blue for anion) seggregate into the middle of the PEO domains. (b) NPEO:NPS = 11:5. PS (orange) forms cylinder. Figure 3. (a) Density profiles of all components corresponding to different dielectric permittivity ratios, namely ϵPEO/ϵPS = 1.0 (red), 1.2 (blue), and 1.8 (green). The green region corresponds to the PEO-like domains, and the red region denotes the PS-like domains. The highest three lines in the PEO-like domain represent the fraction of PEO-like material for different dielectric permittivity ratios. Correspondingly, the highest three lines in the PS-like domain represent the fraction of PS-like material. The three lower-valued lines represent the salt fraction as a function of position. (b) Ion concentration profile as a function of strength of dielectric heterogeneity, shown in an expanded scale. The inset shows the equilibrium lamellar morphology for a saltdoped symmetric diblock copolymer. Lithium ion (red) and anions (blue) are concentrated in the PEO-like domain (green).
characteristic directions for both PEO- and PS-core cylinders. In both cases, salts are concentrated in the PEO-like domains. However, when the PEO domain represents the minority block, the ion concentration is much higher (the overall ion concentrations are identical). Figure 5 compares several copolymer phase diagrams obtained without and with salt. One can see that the width of the ordered regime is expanded by the addition of salt. The phase diagram is also shifted slightly toward the right side (high PS concentration) of the composition axis. This shift results from the colocalization of the ions and the PEO domain, which effectively increases the volume fraction of the PEO-like domains. The results also indicate that the ordering transition occurs at a lower value of χN in salt-doped systems than in neutral diblocks. This observation is consistent with the view of Wang et al., who proposed that the effective χ parameter increases with the addition of salt by using an argument based on a mean-field theory. Figure 5c shows salt distribution in different morphologies. Salts are all concentrated in PEO domain irrespective of the morphologies. By comparing the first and last spherical morphology, we noticed that, in the latter, salt concentration is high in PEO as the minor domain; the spherical morphology is more ordered and the interface is sharper in this case. This indicates that the self-assemble of ion pairs enhances phase separation. We also note that when the salt concentration is increased from 1/21 to 1/5 for PEO:PS = 5:11 block copolymers, an order-to-order transition, from cylinder to lamellar morphology, is observed. Similar experimental results have been obtained previously.33
sents the bead density along the z direction. The lines represent the density for all species. The three lines with the maximum value in the green, PEO-like domain correspond to the density of PEO beads. The three lines with a maximum in the orange, PS-like domains represent the density of PS beads. The three lines with a lower density value represent the salt density. Each color represents a different dielectric permittivity ratio. To highlight the effect of dielectric inhomogeneity, the interaction between PEO and cations was turned off. Thus, the only driving force for ionic segregation is the spatially heterogeneous dielectric profile. One can appreciate in the figure that the ions are localized in the PEO-like domains (green). This heterogeneous distribution is further quantified by the density profile of different species, as seen in Figure 3b, where results are shown for three different ratios of dielectric permittivity between the blocks, namely PS and PEO. As the dielectric heterogeneity increases, the ions become increasingly localized in the PEO-like domains, in agreement with experimental results24 and with the theoretical work that emphasized the D
DOI: 10.1021/acs.macromol.7b02508 Macromolecules XXXX, XXX, XXX−XXX
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toward the PS-rich side. All these observations can be rationalized by the favorable solvation of ions in domains with higher dielectric permittivity.4 Detailed density profiles serve to confirm that ions are preferentially dissolved in PEOlike domains, regardless of whether they are minority or majority.24,34 Compared to recent work that coupled SCFT with liquid state theory9 (Figure 2), our model solves Poisson’s equation directlyrather than relying on a Poisson−Boltzmann approach, it does not bind ions to the PEO-like block; it captures composition fluctuations, and it produces the 3D morphology of salt-doped systems. However, the qualitative observations about the variation in phase diagram topology are similar. In both cases, the diagram is shifted to the regime with lower χN values and becomes asymmetric, with a broader ordered composition range on the PS-rich side. The effects observed in our work are not as dramatic as those reported previously.9 The most striking observation there was that (χN)ODT can become negative; i.e., the ordering transition is possible even when the two blocks are fully compatible. We found a much milder variation, as shown in Figure 5. Still, the difference between the undoped and doped cases is appreciable. The fact that the diagram becomes asymmetric in the doped system suggests clearly that a composition-independent effective χ parameter alone can not rationalize the effects of added salts. The strength of our model lies in its efficiency to rapidly explore the morphological behavior and produce phase diagrams. Its main limitation is the use of soft potentials among polymer beads and ions, an approximation needed for the sake of simulation efficiency. Thus, we are not able to accurately assess the details of ion aggregation, which may be captured by models that incorporate liquid-state theory.9 These local details are particularly important for the study of ion transport, which is best addressed by relying on atomistic simulations35 or by coarse-grained approaches backed by such simulations.36
Figure 5. Part of phase diagrams for pure (a) and salt-doped (b) block copolymers. Results are plotted on the Flory−Huggins (χABN)−PS monomer fraction (ϕPS) projection. The ODT is χABN = 18 for pure polymers and χABN = 16 for polymers doped with 6.2% of ionic sites. The phase diagram is symmetric for pure diblock copolymers and becomes slightly asymmetric for salt-doped block copolymers. The region where ordered phases are observed is shifted toward the highPS concentration side. In our simulation, we do not identify an explicit boundary between the gyroid and perforated lamellar phases, and a dashed line is drawn at the approximate location where the transition is expected to occur. (c) 3D representations extracted from simulations showing different morphologies and the corresponding distribution of salts. The color bar indicates the salt fraction, with blue being the highest and red the lowest. Symbols denote: S, ○ = sphere; C, ▽ = cylinder; G, ◇ = gyroid; PL, + = perforated lamellar; L, □ = lamellar.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected] (J.J.d.P.). ORCID
Jian Qin: 0000-0001-6271-068X Juan J. de Pablo: 0000-0002-3526-516X Notes
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The authors declare no competing financial interest.
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SUMMARY Our key contribution has been to couple coarse-grained molecular simulations of an ordered, charged copolymer electrolyte with the explicit solution of electrostatics in a heterogeneous dielectric medium. The dielectric profile is determined by solving electrostatics and the Poisson’s equation from particle-based, simulated molecular configurations. The electrostatic potential in turn influences the evolution of morphology, which is used to update the dielectric profile. Upon convergence, the iteration of dielectric profile updates and morphology evolution leads to self-consistent solutions. Using this approach, we studied diblock copolymer phase diagrams change with the addition of dissolved salts. From the results presented in Figure 5, three observations can be made. First, the value of (χN)ODT decreases; second, the composition range for ordered phases widens; third, the diagram shifts
ACKNOWLEDGMENTS This work was supported by the Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Division of Materials Science and Engineering. The development of publicdomain codes for coupling particle-based simulations to continuum solutions of Poisson's equation in the COPSS software suite was supported by DOE, BES, Materials Science and Engineering Division, through the Midwest Center for Computational Materials (MiCCoM).
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REFERENCES
(1) Chi, E. Y.; Krishnan, S.; Randolph, T. W.; Carpenter, J. F. Physical Stability of Proteins in Aqueous Solution: Mechanism and Driving Forces in Nonnative Protein Aggregation. Pharm. Res. 2003, 20, 1325−1336.
E
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Article
Macromolecules (2) Hille, B. Ion Channels of Excitable Membranes; Sinauer Associates: 1984. (3) Cooke, T. R. Inorganic fibres: A Literature Review. J. Am. Ceram. Soc. 1991, 74, 2959−2978. (4) Nakamura, I.; Wang, Z.-G. Salt-doped block copolymers: ion distribution, domain spacing and effective χ parameter. Soft Matter 2012, 8, 9356−9367. (5) Borodin, O. Polarizable Force Field Development and Molecular Dynamics Simulations of Ionic Liquids. J. Phys. Chem. B 2009, 113, 11463−11478. (6) Deserno, M.; Holm, C. How to mesh up Ewald sums. I. A theoretical and numerical comparison of various particle mesh routines. J. Chem. Phys. 1998, 109, 7678−7693. (7) 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, 7694−7701. (8) Landau, L. D.; Pitaevskii, L. P.; Lifshitz, E. M. Electrodynamics of Continuous Media; Butterworth-Heinemann: 1982. (9) Sing, C. E.; Zwanikken, J. W.; Olvera de la Cruz, M. Electrostatic control of block copolymer morphology. Nat. Mater. 2014, 13, 694− 698. (10) Daoulas, K. C.; Müller, M. Single chain in mean field simulations: Quasi-instantaneous field approximation and quantitative comparison with Monte Carlo simulations. J. Chem. Phys. 2006, 125, 184904. (11) Detcheverry, F. A.; Kang, H.; Daoulas, K. C.; Müller, M.; Nealey, P. F.; de Pablo, J. J. Monte Carlo simulations of a coarse grain model for block copolymers and nanocomposites. Macromolecules 2008, 41, 4989−5001. (12) Detcheverry, F. A.; Pike, D. Q.; Nealey, P. F.; Müller, M.; de Pablo, J. J. Monte Carlo simulation of coarse grain polymeric systems. Phys. Rev. Lett. 2009, 102, 197801. (13) Qin, J.; de Pablo, J. J. Ordering transition in salt-doped diblock copolymers. Macromolecules 2016, 49, 3630. (14) Lytle, T. K.; Radhakrishna, M.; Sing, C. E. High Charge Density Coacervate Assembly via Hybrid Monte Carlo Single Chain in Mean Field Theory. Macromolecules 2016, 49, 9693. (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, 1−16. (16) Kumar, M.; Sekhon, S. Role of plasticizer’s dielectric constant on conductivity modification of PEO-NH 4 F polymer electrolytes. Eur. Polym. J. 2002, 38, 1297−1304. (17) Nakamura, I.; Balsara, N. P.; Wang, Z.-G. Thermodynamics of ion-containing polymer blends and block copolymers. Phys. Rev. Lett. 2011, 107, 198301. (18) Nakamura, I.; Balsara, N. P.; Wang, Z.-G. First-order disordereto-lamellar phase transition in Lithium salt doped block copolymers. ACS Macro Lett. 2013, 2, 478−481. (19) Cho, B.-K.; Jain, A.; Gruner, S. M.; Wiesner, U. Mesophase Structure-Mechanical and Ionic Transport Correlations in Extended Amphiphilic Dendrons. Science 2004, 305, 1598−1601. (20) Young, W.-S.; Epps, T. H. Ionic Conductivities of Block Copolymer Electrolytes with Various Conducting Pathways: Sample Preparation and Processing Considerations. Macromolecules 2012, 45, 4689−4697. (21) 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, 1−16. (22) 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, 8282−8289. (23) 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, 1212−1216. (24) 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, 5642−5651. (25) Huang, J.; Tong, Z.-Z.; Zhou, B.; Xu, J.-T.; Fan, Z.-Q. Phase behavior of LiClO4-doped poly(Iţ̂ -caprolactone)-b-poly(ethylene oxide) hybrids in the presence of competitive interactions. Polymer 2014, 55, 1070−1077. (26) Gunkel, I.; Thurn-Albrecht, T. Thermodynamic and structural changes in ion-containing symmetricdiblock copolymers: a small-angle X-Ray scattering study. Macromolecules 2012, 45, 283−291. (27) Teran, A. A.; Balsara, N. P. Thermodynamics of Block Copolymers with and without Salt. J. Phys. Chem. B 2014, 118, 4−17. (28) Sing, C. E.; Olvera de la Cruz, M. Polyelectrolyte Blends and Nontrivial Behavior in Effective Flory-Huggins Parameters. ACS Macro Lett. 2014, 3, 698−702. (29) Gilbert, J. B.; Luo, M.; Shelton, C. K.; Rubner, M. F.; Cohen, R. E.; Epps, T. H. Determination of Lithium-Ion Distributions in Nanostructured Block Polymer Electrolyte Thin Films by X-ray Photoelectron Spectroscopy Depth Profiling. ACS Nano 2015, 9, 512−520. (30) Müller, M. Concurrent coupling between a particle simulation and a continuum description. Eur. Phys. J.: Spec. Top. 2009, 177, 149− 164. (31) Pike, D. Q.; Detcheverry, F. A.; Müller, M.; de Pablo, J. J. Theoretically informed coarse grain simulations of polymeric systems. J. Chem. Phys. 2009, 131, 084903. (32) Marshall, S. L. A periodic Green function for calculation of coulombic lattcie potentials. J. Phys.: Condens. Matter 2000, 12, 4575− 4601. (33) Young, W.-S.; Epps, T. H. Salt Doping in PEO-Containing Block Copolymers: Counterion and Concentration Effects. Macromolecules 2009, 42, 2672−2678. (34) Gilbert, J. B.; Luo, M.; Shelton, C. K.; Rubner, M. F.; Cohen, R. E.; Epps, T. H. Determination of Lithium-Ion Distributions in Nanostructured Block Polymer Electrolyte Thin Films by X-ray Photoelectron Spectroscopy Depth Profiling. ACS Nano 2015, 9, 512−520. (35) Webb, M. A.; Jung, Y.; Pesko, D. M.; Savoie, B. M.; Yamamoto, U.; Coates, G. W.; Balsara, N. P.; Wang, Z.-G.; Miller, T. F. Systematic Computational and Experimental Investigation of Lithium-Ion Transport Mechanisms in Polyester-Based Polymer Electrolytes. ACS Cent. Sci. 2015, 1, 198−205. (36) Webb, M. A.; Savoie, B. M.; Wang, Z.-G.; Miller, T. F. Chemically Specific Dynamic Bond Percolation Model for Ion Transport in Polymer Electrolytes. Macromolecules 2015, 48, 7346− 7358.
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DOI: 10.1021/acs.macromol.7b02508 Macromolecules XXXX, XXX, XXX−XXX
