Depletion Attraction of Sheet-like Ion Aggregates in Low-Dielectric Ionomer Melts Keran Lu, 1 Janna K. Maranas, 1 1, a)

Transcription

1 Sample Title Depletion Attraction of Sheet-like Ion Aggregates in Low-Dielectric Ionomer Melts Keran Lu, 1 Janna K. Maranas, 1 1, a) and Scott T. Milner The Pennsylvania State University 12 Fenske Laboratory, University Park, PA, 1682 USA (Dated: 3 December 216) Ionomers are polymers in which an ionic group is covalently bonded to the polymer backbone. Ion aggregates in ionomers have morphologies that allow for the packing of the attached polymer backbone. Using ion-only coarsegrained molecular dynamics, we observe that string-like ion aggregates become flat and sheet-like at lower dielectric constants. A consequence of the changing morphology is that the sheet-like aggregates self-assemble to form ordered, lamellar structures. We use a simple thermodynamic model to demonstrate that depletion attraction mediated by small aggregates can explain the observed order. Our results suggest depletion attraction can drive ions to form structures that have the size scale suggested by direct visualization, produce the commonly observed experimental correlation peak from X-ray and neutron scattering, and satisfy chain-packing constraints that have been demonstrated to be important in simulations. PACS numbers: Valid PACS appear here Keywords: Suggested keywords I. INTRODUCTION Ionomers are polymers that contain ionic functional groups. They can be used in a variety of applications such as coatings, packaging, fuel cell membranes, and solid state electrolytes. Because of the typically low dielectric constant of the solvating backbone, ionic groups in ionomers can aggregate strongly from electrostatic forces. These aggregates have a strong influence on material properties, and controlling aggregation is key to tailoring ionomers to their various applications. In this work, using coarse-grained molecular dynamics, we have identified depletion attraction of large, sheet-like aggregates driven by exclusion of small aggregate fragments as a possible additional driving force contributing to the nanoscale structure of ion aggregates. We observe that sheet-like ion aggregates further self-assemble into lamellar stacks with dimensions of approximately 2 nm. The depletion attraction of sheetlike structures has been widely reported in many systems such as those containing lipid bilayers, 1 nanoparticles, 2 and in biology 3. In these systems, the self-assembly is driven by the presence of the solvent or another solute. We demonstrate that depletion attraction can occur in ionomer melts containing only a polydisperse distribution of ion aggregates, in which the smaller aggregates produce depletion forces resulting in the aggregation of the larger aggregates. These structures provide a possible explanation for how ionic groups, and their attached backbone polymer, could pack to produce overall sizes greater than a few nanometers while still producing a strong correlation peak in the structure factor corresponding to nanometer length scales. Direct visualization of ion aggregates in ethylene- and styrene- based ionomers using electron microscopy show aggregates can have overall dimensions ranging from 1 nm to over 15 nm. 4 It is unlikely that ion aggregates larger than a few nanometers are spherical, because of the attached polymer backbone. In fact, vesicular 5 and sheet-like 6 morphologies have been a) Electronic mail: smilner@engr.psu.edu

2 Sample Title 2 reported for aggregates in Zn-neutralized sulfonated polystyrene. X-ray and neutron scattering of ionomers reveal a characteristic correlation peak near q.2å 1, corresponding to a much shorter length scale of approximately 3Å. 7 9 This peak is attributed to interaggregate spacing. A variety of structural models 1 12 all fit the shape of the correlation peak, suggesting additional information is needed to corroborate the local structure of ion aggregates. Molecular dynamic simulations of ionomers support experimental observations by providing the details of ion aggregate morphology. Atomistic and bead-spring simulations of ethylene-based ionomers have been used to explore the role of dielectric constant 13, polymer architecture 13 15, and counterion type 16 on ion aggregate structure and ion transport 17. A similar analysis has been performed using atomistic models of poly(peo-cosulfoisophthalate) ionomers In general, ion aggregates become larger as the dielectric constant of the system is lowered. Larger ion aggregates in ionomer simulations generally are not spherical, because of chain packing restrictions. They generally exhibit cylindrical 13 or two-dimensional, sheetlike structures. 21,22 The challenge with using atomistic, or even bead-spring, simulations of ionomers is the slow relaxation time of ion configurations. In sodium-neutralized poly(peo-cosulfoisophthalate) ionomers, the local polymer segmental and ionic relaxation times (α and α 2 modes measured from dielectric spectroscopy) are separated by two orders of magnitude. For analogous PTMO-based ionomers, with a lower dielectric constant backbone, this separation in timescale reaches nine orders of magnitude. 23 Ion relax very slowly, particularly in low dielectric media, and molecular simulations have largely been limited to small box sizes because of the computational cost of extensively simulating the polymer backbone to obtain sufficient statistics on the aggregate structure. While local structure, measured through quantities such as the radial distribution function, are generally equilibrated in ionomer simulations, indications are that long-range structures are not equilbrated. Structure factor has traditionally been used as a measure of long-range order, but the plethora of structures that all can produce correlation peaks similar to those observed in X-ray scattering suggests simply comparing structure factors may not be sufficient. In particular, the larger aggregates observed from electron microscopy suggests higher order structure may be present that is not captured due to box size constraints in current molecular dynamics simulations. We have previously presented coarse-grained (CG) simulations of poly(peo-co-sulfoisophthalate) which have attempted to overcome these challenges through coarse-graining the atomistic simulation. Using using an ion-only CG simulation in which the effect of the backbone polymer is captured through atomistically-informed force fields derived through a g-ybg method 24,25 related to force-matching 26, we were able to reproduce ion structure 27 and dynamics 28 of the parent atomistic simulation 18 while increasing the simulation box size and dramatically improving statistics. For example, we sampled 1 5 times more independent ion aggregate configurations containing 1 or more ions compared to that from the atomistic simulation. 27 These improved statistics rely on avoiding explicit simulation of the backbone polymer, when only the structural or long-time dynamical properties of ions are of interest. At time scales two orders of magnitude greater than the local segmental relaxation of the attached backbone polymer, the net effect of the ensemble of local polymer segment configurations on ion configurations can be approximated by a short-ranged potential, statistically derived through coarse-graining. Long-range interactions between ions are captured by an electrostatic potential with a dielectric constant. By forgoing access to polymer structure, we are able to greatly improve ion aggregate equilibration and statistics. In this report, we present results from coarse-grained simulations using our previous CG force field in which the dielectric constant of the long-range electrostatic potential is artificially lowered. We are in effect simulating a hypothetical ionomer with a backbone polymer structurally identical to PEO but with a lower dielectric constant. We find that ion aggregates develop sheet-like morphologies, which makes sense when the polymer backbone packing constraints are imposed in a system with increased electrostatic interaction strength. Full equilibration of our ion aggregate sheets reveals that they further self-assemble because of depletion attraction. The self-assembled structures are 15 to 2 nm in size, and the self-

3 Sample Title 3 assembly is driven by the presence of a smaller, polydisperse population of ion aggregates. Using simulations with well controlled aggregate sizes, we show that self-assembly appears as ion sheets become larger and/or the concentration of ion pairs increases, a characteristic of depletion-driven self-assembly. Our results suggest depletion attraction forces can drive ions to form structures that have the size scale suggested by direct visualization, produce the correlation peak in structure factor, and satisfy chain-packing constraints that have been demonstrated to be important in simulations. II. METHODS CG potentials for ion-only simulations of sodium-neutralized poly(peo-co-sulfoisophthalate) ionomers have been reported in a previous publication. 27 The CG simulation has two beads, one which represents sulfonated isophthalate, and one which represents sodium. The effect of the polymer backbone is captured implicitly in the effective potentials between ions. These force fields are derived from a force-matching procedure 24 and have been shown to successfully reproduce properties of ion aggregates from the parent atomistic simulations. In this report, we use CG potentials generated at a higher temperature of 423K using the same procedure. We found that the weak harmonic bonds between neighboring anions included in the previous study, representing the 13 PEO monomer units between anions, did not significantly impact ion aggregate structure when removed; for simplicity they were not included in this study. The removal of these weak harmonic bonded potentials prevents sheet-like ion aggregates from being interconnected, which can obfuscate the role of depletion attraction interactions. A physical interpretation of removing the harmonic bonds is that the PEO linker chains have been cut the middle, such that they are still present in the melt but no longer produce spring forces on their corresponding anions. The CG potentials can be separated into a short-ranged part, which contains local structural information specific to our polymer, and a long-ranged electrostatic potential with a dielectric constant. We modify the long-ranged portion of these potentials by changing the dielectric constant over a range from 1 to 1. The unmodified dielectric constant representing a PEO backbone is 6. We assume the short-ranged potential is relatively insensitive to dielectric constant. A similar approach has recently been used to generate partially transferable CG force fields for hydrated ions, 29 sulfonated polystyrene, 3 and PEO ionomers 31. In these CG systems, a constant short-ranged potential combined with a long-range electrostatic potential, with a state-dependent dielectric constant, was used to successfully reproduce structural distributions from atomistic parent simulations. By lowering the dielectric constant, we acknowledge that we are departing from simulating a specific ionomer chemistry. We are representing a hypothetical ionomer with a low dielectric constant backbone that packs like poly(peo-co-sulfoisophthalate). The short range potential we use here is a compromise between a completely generic repulsive interaction and a chemically specific one, which would be extremely computationally expensive to obtain. The coarse-grained potentials for poly(peo-co-sulfoisophthalate) simulations were generated from 1 ns of atomistic trajectory, which represents the lower limit on the statistics needed to generate reasonable potentials. That 1 ns trajectory took 6 months to generate using current computational resources. For lower dielectric constant systems, for which relaxation of local ion arrangements may be at least an order of magnitude slower, we would need at least 6 months to obtain a trajectory with comparable statistics, which would be too computationally expensive to obtain. The benefit of using our potential is that we are able to access simulation box lengths of 3Å with over 45 ions. These length scales are impossible to simulate at the atomistic level using standard high performance computing systems. The larger system size allows us to identify aggregation phenomena that may be hidden at smaller system sizes. Simulations were performed using LAMMPS 32 under the NVT ensemble using a Nosé Hoover thermostat. Long range Coulomb interactions were calculated using a particleparticle/particle-mesh algorithm. Simulation snapshots were generated using VMD. 33 Fig-

4 Sample Title 4 FIG. 1. Simulation snapshot for 3Å box length simulation at ε = 1 showing lamellar ordering of sheet-like ion aggregates. The blue box is marks the boundaries of the simulation. Partial periodic images are shown so that lamellar stacks are more easily visualized. r HÅL SHqL q HÅ -1 L FIG. 2. Combined ion structure factor for the ε = 1 simulation. ures were generated using Mathematica. III. RESULTS In this report, we will show that depletion attraction is the driving force for the selfassembly of sheet-like ion aggregates into ordered lamellae. Figure 1 is a 75Å thick crosssection from a 3Å box length simulation showing the spontaneous ordering of large, sheet-like aggregates at ε = 1 simulation. Three separate lamellar stacks are present in the box. Combined ion-only structure factors, in figure 2, show a strong correlation peak at q =.25Å 1, corresponding to a length scale of z = 26Å. Visual inspection shows that this distance corresponds to the preferred spacing of ion sheets. Higher order reflections at z/n indicate strong lamellar ordering of the ion sheets. The peaks at z = 3.3Å and z = 4.7Å correspond to nearest opposite-charge and like-charge coordination distances between ions in the sheets.

5 Sample Title 5 c = 2 2 < c < 3 c = 3 c = 4 FIG. 3. Schematic showing the approximate, representative structures of ion aggregates corresponding to different average ion coordination numbers. Filled red and blue circles represent anions and cations. Unfilled circles represent the rest of the aggregate, which continues beyond what is shown in the schematic. The gray lines represent polymer backbone chains. At c = 4, the polymer chains are not shown, but would extend orthogonally from the surface. A. Sheet-like Ion Aggregates Sheet-like ion aggregates form as the dielectric constant of the simulation is lowered. The reduced dielectric constant increases the electrostatic attraction between opposite charges, promoting aggregation. Figures 4(a) and 4(c) show the effect of lowering the dielectric constant on the size distribution of ion aggregates, and on the average coordination number of cations as a function of aggregate size. Figure 4(b) shows the aggregate size distribution out to larger sizes. As the dielectric constant decreases from ε = 1 to ε = 1, the average aggregate size increases, reflecting the increased binding energy between oppositely charged ions. At ε = 1 and ε = 2, there is a strong oscillation in the size distribution between even and odd-sized aggregates. Note that there are not two separate curves for the ε = 1 and 2 systems in figure 4(a) these are the oscillations between odd and even sized aggregates. Because odd-sized aggregates have net charges, they are energetically unfavorable and suppressed. As the dielectric constant is lowered, odd aggregates become progressively less favorable compared to even aggregates, resulting in the amplification of the oscillations seen in size distribution. From ε = 1 to ε = 2, the size distribution is exponential at large sizes, which we have attributed in a previous work to ion assembling into stringlike aggregates, analogous to worm-like micelles. 27 In these stringlike ion aggregates, the equilibrium size distribution is determined by the balance between the number of endcaps, which have an energetic cost, and the translational entropy of the system. The initial curvature in the distribution, which is shown on a log scale, is attributed to a size-dependent breaking energy, which becomes independent of size as ion aggregate sizes become larger than the electrostatic screening length. 27 The average coordination number of cations in aggregates for ε = 1 to ε = 2 system can provide insight into the morphology of ion aggregates in these systems. At large sizes, the cation coordination number levels off to values between two and three for these systems, confirming that the morphologies are string-like. Ions in defect-free chains have a coordination number of two. We have previously reported a detailed analysis of ion chain morphology for the PEO backbone system, at a lower temperature of 343K, in which we identified defects in the stringlike aggregate structure that increase the average ion coordination to values greater than Segments of ion chains can fold themselves such that ions located in the folded region become triple-coordinated, increasing the average coordination number of ions. As the dielectric constant decreases, these defects occur more often because of increased electrostatic attraction between ions, increasing the average coordination number. Figure 3 shows schematically how ion aggregate structure might change as a function of the average counterion coordination number. Large aggregates in the ε = 1 system are different from those in other systems. The size distribution of ion aggregates initially has worm-like micelle behavior, during which cations in the aggregates have an average coordination number of 3. In this regime, the ion aggregates are still string-like, but are crumpled and thicker compared to strings at higher dielectric constants. In contrast to systems at higher dielectric constants, the ion aggregate size distribution for at ε = 1 qualitatively changes, at sizes above 2. This

6 PHnL PHnL Sample Title n n (a) (b) Coordination Number n 1 (c) FIG. 4. (a) Ion aggregate number fraction as a function of size, (b) the distribution in (a) shown to larger sizes, and (c) average cation coordination number as a function of size for simulations with dielectric constants: (black) 1, (blue) 2, (red) 6 and (purple) 1. corresponds to a jump in the average cation coordination number for aggregates larger than 2, which increases from 3 to 3.5. This occurs from the presence of 4-coordinated ions, which are ions in the center of ion sheets. The formation of sheet-like ion aggregates at larger sizes corresponds to increased aggregate stability, as the size distribution levels off at sizes above 2. Our results suggest there must be additional physics at play in the ε = 1 system that causes the changes which occur at aggregate sizes above 2. The exact details of this transition are beyond the scope of this report, but we discuss the potential driving forces behind this transition in the conclusion. In the ε = 1 system, the sheet-like aggregates form regardless of the initial configuration. Simulations in which the initial ion configurations are random or in an ordered, 3D checkerboard, with ion arranged at their equilibrium distances, both converge to the same equilibrium ion aggregate morphologies. Sheets (and strings) form because the short-ranged portions of the CG anion-anion potential include repulsive interactions. The repulsions have physical origins in the fact that anions must find a way to pack in their associated polymer segments when they aggregate. The formation of sheets, but not crystals, is reasonable because chain packing constraints could prevent the latter. Although the chains are not explicitly present in CG simulations, these packing contraints are represented in the shortrange interactions obtained from force-matching. Our results suggest that, under strong electrostatic interactions, sheet-like aggregates may form in ionomer systems. We will demonstrate next that these sheet-like ion aggregates can further self assemble through depletion attraction forces, which may explain the observed transition in the aggregate size distribution and ion coordination numbers.

7 Sample Title 7 FIG. 5. Solid red and black represents the ionic group and the closest approach distance allowed by the backbone polymer. Grey regions indicate volumes in which other ionic groups are excluded. When two sheets aggregate, the total excluded volume of the system is reduced. 5 4 Kcalêmol r HÅL FIG. 6. [Black] CG short-ranged potential for cation-anion interactions. [Blue] Unbreakable modified potential used to create monodisperse ion sheets. B. Depletion Attraction Depletion attraction is a well known phenomenon in colloidal solutions. Each colloid has an exclusion volume where the center of mass of solvent molecules cannot enter. The aggregation of colloids reduces this volume, increasing the total entropy of the solvent at the cost of reduced entropy of the colloids. There is an equilibrium state of colloid aggregation which maximizes the total system entropy. In our system, the sheet-like ion aggregates self-assemble to increase the overall entropy of the system. Figure 5 is a schematic which demonstrates how the aggregation of ion sheets reduces the exclusion volume of the system. The red circles represent ionic groups, and the black circle represents the distance of closest approach between two ionic groups, because of the presence of the backbone polymer. The grey regions represent the volume in which the center of mass of the ionic groups are excluded from. As seen in the schematic, when two ion sheets aggregate, a significant amount of excluded volume becomes available (represented by the overlapping regions). To isolate depletion attraction as the driving force for sheet-like ion aggregate conglomeration, we performed simulations of specially prepared monodisperse ion sheets in various concentrations of ion pairs (see below for details on how these aggregates were prepared). As the concentration of ion pairs increases, the cost of an exposed ion sheet surface increases, increasing the driving force for the self-assembly of ion sheets. To make sense of these observations, we construct a simple thermodynamic model of depletion attraction in which we calculate the concentration of ion pairs at which it becomes energetically favorable for three ion sheets of a given size to self-assemble into a single aggregate. This model is described in detail below. We compare this qualitatively to the ion pair concentration at which ion sheets of various sizes clearly self-assemble into a single aggregate in simulation. We expect our calculation will underpredict the sheet size for a given concentration, but the dependence of ion sheet size on ion pair concentration should be similar. To prepare a system of well-defined aggregates, we modified our cation-anion short-ranged potential such that the energy barrier between the first and second minimums is dramatically

8 Sample Title Edge Length HÅL appleion Pairs, c FIG. 7. Shown is the transition from three unordered sheets to one ordered stack of three sheets. Red points are unordered, green points are ordered. The dotted line is the calculated edge length at which the transition becomes energetically favorable. increased. Using this potential, oppositely charged ions that are initially coordinated (e.g. within the first minimum of the potential) always remain coordinated. Oppositely charged ions that are not coordinated will never become coordinated. This force field makes the initial electrostatic coordination neighbors of ions essentially permanent over the simulation, and it allows us to set exactly the number of sheets, the size of each sheet, and the number of ion pairs. The blue dotted line in figure 7 shows the unbreakable potential compared to the original. However, because the shape of the potential wells are unchanged away from the barrier, ions still behave as they would using the original forcefield since they spend the majority of their time near potential minimums. The sheets are thus free to move and change shape, as as they would without the unbreakable bonds. Using this unbreakable potential, we are able to determine the threshold for ion sheet conglomeration as a function of the size of the sheets and the concentration of ion pairs. Shown in figure 7 is the transition from disordered ion sheets to an ordered stack for a system containing three sheets. Green points are systems in which sheets are clearly aggregated, and red points are systems in which sheets are not aggregated. Sheets are square in order eliminate charged edges that occur when a checkerboard ion configuration is cropped at oblique angles. Figure 8 shows example trajectories of aggregated and unaggregated sheets with an edge length of 63Å. The top pane shows a system containing 18 ion pairs in which the ion sheets are not aggregating. In the bottom pane, the system contains 194 ion pairs, and the ion sheets diffuse as one aggregate. As expected for depletion attraction, the minimum sheet size required for self-assembly decreases as ion pair concentration increases. Note that if the driving force were electrostatic, the aggregation of sheets should have the opposite trend with ion content. The increased number of ion pairs would screen electrostatic interactions, requiring larger sheets to maintain the same electrostatic force between sheets. In fact, the electrostatic interactions between sheets is essentially zero. With a cation-anion spacing of 3.3Å the checkerboard configuration of cations and anions in the ion sheets gives canceling interactions more than a few Angstroms away, so that the sheets appear essentially uncharged to each other at typical separations, of order the preferred separation distance of z = 26Å. Further evidence for depletion attraction is provided by comparing the observed transition in figure 7 to a simple thermodynamic model of depletion attraction. We can write an expression for the free energy change when 3 sheets are aggregated into 1, which is the transition depicted in figure 7. In the absence of interactions between the sheets, the free energy change contains translational and rotational entropy terms. The translational entropy change upon sheet aggregation is

9 Sample Title 9 (a)$ (b)$ FIG. 8. Shown are unwrapped 5ns trajectories of three monodisperse ion sheets with edge length of 63A in simulations with (a) 18 ion pairs and (b) 194 ion pairs. Each color represents a different point in time. The trajectory at the top shows uncorrelated sheets diffusing randomly and the trajectory on the bottom shows a self-assembled stack of three sheets diffusing together. Gtrans = c log kb T c c n n /n n c log + log n log V ν(n + 1)(n /n) V 2n ν n V V (1) where c is the number of small aggregates, n is the number of ion sheets, n is the average number of sheet in an aggregate, V is the volume of the system, and ν is the inaccessible volume per side of the sheet. For a stack of n sheets, there is a dead volume of ν(n + 1). We set ν = πd2 z/4, where d is the diameter of the sheet, and z is the preferred equilibrium spacing between sheets. The first two terms are the change in translational entropy for ion pairs based on the increase in accessible volume after sheet aggregation. The second two terms are the change in translational entropy of ion sheets after aggregation. In addition to changes in translational entropy, sheet rotation is also restricted when ion sheets self-assemble into a stack. If we consider a normal vector perpendicular to the surface of an ion sheet, the accessible solid angle of unrestricted rotation is 4π. When ion sheets are aggregated into stacks, the rotation angle off of this normal is restricted to be φ, assumed to be small. This results in an entropic penalty to the free energy because of restricted rotation of the ion sheets, 2 2 Grot πφ 4π φ = log + log = log kb T δ δ 4 (2) where δ is the discretization of the solid angle. The range of the rotation angle, φ, can be estimated as the angle which costs kb T in translational free energy. The cost originates from the extra dead volume that results from rotation of one sheet relative to its neighbor. The free energy change per sheet from sheet rotation is Gtrans,lost = c log kb T V V Vrot V V 1 (3) where V = ν(n /n)(n + 1) is the total zero rotation dead volume, and Vrot is the total additional dead volume because of sheet rotation. From this expression, we find, for large c,

10 Sample Title 1 d" z" φ" z" FIG. 9. A schematic representation of how V cyl represents the excess restricted volume produced from sheet rotation. V rot = (V V )(1 exp[ 1/c]) (V V ) c We define the extra volume that results from rotation of a sheet as the volume formed when the rotated sheet is projected onto its unrotated neighbors. This volume has the shape of a ellipsoidal cylinder with major radius: d/2, minor radius: (d/2) cos φ, and height: d sin φ. This is shown schematically in figure 9. The gray volumes are zero rotation dead volume of the sheet with its top and bottom neighbors. The additional volume from sheet rotation is highlighted in orange. We assume that φ is small enough such that this cylinder can adequately capture the shape of this additional volume. Note that φ is exaggerated in the schematic for purposes of clarity. The volume of the ellipsoidal cylinder is (4) V cyl = πd3 4 sin φ cos φ = πd3 8 sin(2φ). (5) For a stack of n sheets, there are n 1 of these cylinders which form because of sheet rotation. Each sheet in the center of the stack contributes one volume. Sheets at the ends each contribute half the volume. The average angle, φ, that costs k B T to the translational entropy can be found equating resulting in V rot = (n /n)(n 1)V cyl (6) φ = 1 [ ] 8(V 2 arcsin ν(n /n)(n + 1)) πcd 3. (7) (n /n)(n 1) The total free energy change from ion sheet conglomeration is G = G trans + G rot. (8) For the system depicted in figure 7, n = 3 and n = 3. This corresponds to three initial sheets conglomerating into one stack. The number of ion pairs, c, and sheet dimension, d are varied. The dotted line in figure 7 shows the sheet dimension at which the change in free energy, G, is zero. Given the simplicity of the thermodynamic model used, we are still able to capture, with a constant offset, the dependence of the sheet dimension needed for self-assembly on the number of ion pairs in the system.

11 Sample Title 11 IV. CONCLUSIONS Because of the effect ion aggregation has on the properties of ionomers, it is important to understand the forces that drive ion aggregation. We modify our CG potentials derived at higher dielectric constants, to generate well equilibrated ion structures of a hypothetical ionomer in which the PEO backbone dielectric is artificially lowered. For a sufficiently low dielectric constant, ions form into sheet-like structures. In the presence of smaller aggregate fragments, these sheet-like ion aggregates self-assemble into ordered states, because of depletion attraction. These attractions can produce order at length scales beyond the range of electrostatic interactions, and can be significant in highly-aggregated ionomer systems. One interesting phenomenon we did not analyze in detail is the apparent change in morphology of the ion aggregates, which may be driven in part by depletion attraction. There are many factors contributing to the free energy change of transitioning from collapsed chain structures (see c = 3 schematic in figure 3) to sheet-like structures. We will only discuss how depletion attraction forces may affect this transition. The restricted volume of ion aggregates, that excludes small aggregate fragments, grows linearly with mass for ion chains and as the square root of mass for ion sheets. There exists a threshold size upon which the free energy gains from transitioning from an ion chain to an ion sheet is greater than whatever free energy cost was preventing such a transition at smaller sizes. These costs include configurational entropy loss and increased packing density of the associated polymer chain. In addition, the depletion attraction of sheets in the aggregated, lamellar structure may reduce restricted volume more than aggregating random walks of ion chains could, due to more efficient packing. It is not clear without a detailed analysis whether the formation of sheet-like aggregates is independent from the formation of the lamellar stacks. In other words, it is not clear whether ion sheets can exist outside of the lamellar stack. An early theoretical model did predict disklike and lamellar aggregates as the lowest energy structures that minimize unfavorable ion/polymer interfacial area given polymer packing restrictions. 34 However, the net effect of the formation of the lamellar stacks is a reduction in the total free energy, by increasing the translational entropy of the remaining, smaller aggregates. Our results offer a possible explanation for the discrepancy between the large, apparently spherical aggregates observed in electron microscopy, and the highly unfavorable polymer backbone packing large spherical aggregates would entail. It may be that the spherical objects are in fact stacks of ion sheets self-assembled through depletion attraction. On physical grounds, we may expect that the boundaries of finite, self-assembled aggregates have a free energy cost, which acts an an interfacial tension, to drive equilibrium aggregate morphologies towards spherical shapes. The morphologies observed in our 3 nm box dimension simulation (which contains over 45 ions) are not spherical but even this simulation is not large enough to fully capture the equilibrium shape of these structures, as the self-assembled stacks are larger than half the box dimension. Of course, the question arises of why we do not observe signs of lamellae in the electron micrograph visualizations of these spherical aggregates. Two reasons could be: 1) the ion aggregates are not flat sheets and have some curvature, 2) there is insufficient contrast and or resolution to see lamellae. An interlamellar spacing of a few nanometers is on order of the resolution of electron microscopy techniques. 35 The evidence for this conjecture is far from sufficient, but the demonstration of depletion attraction in simulation does raise the question of whether it could play a significant role in higher-order ion aggregation in ionomers with low dielectric constant backbones. V. ACKNOWLEDGEMENTS K.L. would like to acknowledge funding for this project from the Pennsylvania State University Diefenderfer Graduate Fellowship and DOE Basic Energy Sciences, Grant #21954

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13

14 SHqL r HÅL q HÅ -1 L

15 c = 2 2 < c < 3 c = 3 c = 4

16 1.1 PHnL n

17 1.1 PHnL n

18 4 Coordination Number n

19

20 5 4 Kcalêmol r HÅL

21 12 1 Edge Length HÅL appleion Pairs, c

22 (a)$ (b)$

23 d" z" φ" z"