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Ionic Correlations in Random Ionomers Boran Ma, Trung Dac Nguyen, Victor A Pryamitsyn, and Monica Olvera de la Cruz ACS Nano, Just Accepted Manuscript • DOI: 10.1021/acsnano.7b07432 • Publication Date (Web): 01 Mar 2018 Downloaded from http://pubs.acs.org on March 3, 2018
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Ionic Correlations in Random Ionomers Boran Ma1, Trung Dac Nguyen1, Victor A. Pryamitsyn1, Monica Olvera de la Cruz1 2 3 4 * 1. Department of Materials Science and Engineering, 2. Department of Chemistry, 3. Department of Chemical and Biological Engineering, and 4. Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208-3108, United States *E-mail:
[email protected] ABSTRACT Understanding the electrostatic interactions in ion-containing polymers is crucial to better design shape memory polymers and ion conducting membranes for multiple energy storage and conversion applications. In molten polymers, the dielectric permittivity is low, generating strong ionic correlations that lead to clustering of the charges. Here, we investigate the influence of electrostatic interactions on the nanostructure of randomly charged polymers (ionomers) using coarse-grained molecular dynamics simulations. Densely packed branched structures rich in charged species are found as the strength of the electrostatic interactions increases. Polydispersity in charge fraction and composition combined with ion correlations leads to percolated nanostructures with long range fluctuations. We identify the percolation point at which the ionic branched nanostructures percolate and offer a rigorous investigation of the statistics of the shape of the aggregates. The extra degree of freedom introduced by the charge polydispersity leads to bi-continuous structures with a broad range of compositions, similar to
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neutral A-B random copolymers, as well as to desirable percolated ionic structure in randomly charged – neutral diblock copolymers. These findings provide insight into the design of conducting and robust nanostructures in ion-containing polymers.
KEYWORDS ionomers, molecular dynamics, ionic clusters, morphology, electrostatic effects
Polymer electrolytes in the molten state are of great interest due to their potential for a broad range of applications including energy storage and conversion. In particular, ionomers, which are polymers with low content of charged groups, are desirable for their ion conducting properties,1 as well as for their excellent mechanical and thermal properties.2, 3 Ionomers, in addition, possess great potential for generating shape memory polymers (SMPs). SMPs are responsive materials capable of recovering from a programmed temporary deformation on application of an external stimulus.4,
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The formation of ionic aggregates driven by electrostatic interactions between
charged species leads to physical crosslinks. The thermal reversibility of such crosslinks is highly desirable in SMP applications. Moreover, ionomers can introduce additional beneficial properties to SMPs such as antibacterial capabilities, high temperature durability, and a broad glass transition temperature range.5-8 In addition, ionomers have been recognized as alternative materials for battery electrolytes. As single ion conductors, ionomers can avoid increase in internal resistance and reduction in the capacity of the battery caused by accumulation of anions at the electrode – electrolyte interface.9 Also, their excellent thermal and chemical stabilities and good mechanical properties can prevent leakages that lead to explosion hazards as sometimes seen in conventional electrolytes.1, 10-14 A better understanding of how ionic interactions impact
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the microstructure and phase behavior of ion-containing polymers is crucial to fulfill better design and enhanced properties of SMPs and battery electrolytes. Both theoretical and experimental studies have investigated the multiscale effects of ionic interactions in ion-containing polymers.13-18 It has been demonstrated that polymer charge is a powerful tuning parameter to achieve a variety of self-assembled nanostructures in ioncontaining polymer systems such as charged block copolymers and charged polymer blends.15, 19 Dielectric properties such as dielectric constant directly correlate to the strength of ionic interactions and therefore influence the morphological properties of ionic aggregates formed in ion-containing polymers.2, 20, 21 One striking feature of most amorphous ionomers is the so-called “ionomer peak”,2 which is a low angle scattering peak observed in the scattering function . This peak is caused2 by the formation of ionic aggregates, which are micro-segregated structures formed by neutral and charged monomers. Understanding the structure, shape and size distribution of such aggregates is important for understanding the conductive, mechanical and thermal properties of ionomers. Unfortunately, even modern experimental techniques do not provide all the information about the structure of the ionic clusters. Recently, an outstanding attempt made by a combination of Molecular dynamics (MD) simulations and experiments22-27 for the variety of precise and random ionomers, which have units of charged monomers linked to blocks of neutral monomers distributed periodically or randomly along the polymer backbone, elegantly shows that different block periodicities and chain architectures give rise to various morphologies and ionic clusters structures. These works provided insight into the relationship between the configuration of ionomers and the properties of the ionic clusters; for example, both discrete and percolated clusters were observed in those systems. However, many important questions about ionic clusters remain unanswered such as i) How do the fractions of ionic groups
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and electrostatic interaction strength affect the percolation transition for ionic clusters? ii) How are the ionic clusters distributed in size below percolation point, at percolation point and above? iii) What is the shape and fractal dimension of the clusters? and iv) What is the effect of the chain composition polydispersity in random ionomers? Hall et al.24 analyzed scattering from precise and pseudorandom (random but with a fixed total number of charged groups on individual chains) ionomers and found that ionomer peak in pseudorandom ionomers shifts toward low and broadened, but at low remain low. However, in truly random ionomers, the population of chains with different fractions of charge is given by the binomial distribution, generating composition polydispersity, which we hypothesize generates nanostructures that are highly polydisperse in composition. In neutral random copolymer melts, composition polydispersity leads to interesting statistics and thermodynamics. Fredrickson and Milner28 have shown that the quenched disorder on the sequence of monomers in neutral A-B random copolymers with only a net short range van der Waals repulsion between the A and B monomers is equivalent to the annealed case in the thermodynamic limit. They analyzed the case of a neutral random copolymer melt with annealed disorder using field theory methods. A weakly first order transition to a highly dilute phase of chains with blocky A and B segments was found to coexist with a percolated structure of average size , which resembles the structure of a system undergoing spinodal decomposition.29 Further Monte Carlo simulation by Swift and Olvera de la Cruz30 revealed the local density distribution of monomers A (or B) within regions of size (where the peak of the scattering function occurs) in the case of a quenched random disorder. They found a decrease in the peak maximum intensity and a broadening of the peak width as the effective net interaction energy increases. For high enough values of , the local density in regions of size D is flat; that is,
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the probability of finding cubic cells of size D with a given fraction of A chains is broad and not peaked to some mean value as observed in strongly segregated block copolymer. Akin to neutral A-B random copolymers, which segregate into bi-continuous disordered structures when the degree of incompatibility between the A and B monomers increases,28 electrostatic interactions in random ionomers are expected to drive the system to segregate into bi-continuous nanostructures, whose size and composition can be tuned by simply changing the average fraction of charged group in the sample and the strength of the electrostatic interactions. Here we study random ionomers (with degree of polymerization ) with various fractions ( ) of charged monomers and under different electrostatic interaction strengths (represented by the ratio ⁄ of the Bjerrum length = ⁄4 and the Lennard-Jones (LJ) scale ), by MD simulations to determine the effect of ionic correlations on the nanostructures and its relation to their properties. The sequence of charged monomers along the ionomer backbones is random, which leads to a Gaussian distribution function describing charged monomer density, with a width of 1 − / /. The charge composition polydispersity is a key factor to determine the degree of physical crosslinking and therefore the properties that make ionomers of great scientific and industrial interest. Moreover, random ionomers are a general class of ionomers that can be synthesized experimentally. For example, the process of sulfonation of polystyrene leads to a multicomponent system of chains with different charge compositions. RESULTS AND DISCUSSION Phase-space Diagram Charged species in ion-containing polymers aggregate and form clusters due to electrostatic interactions, which have been demonstrated in both simulations and experiments.14, 22 Analysis of clustering was carried out by identifying charged species that fall within a certain cutoff
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distance (1.08σ) from each other as being in the same cluster. Snapshots of ionic clusters formed in ionomer melts taken by VMD software31 can reveal the morphologies of different clusters. As shown in Figure 1A, a phase-space diagram of ionic aggregates of different morphologies with a varying range of average charge fractions ( ) and under different electrostatic interaction strengths ⁄ is obtained with contributions from both visualization of ionic clusters and analysis of cluster size distribution (more details in later section). Only discrete ionic clusters exist in the region shaded by red, while in the region shaded by blue, a percolated ionic cluster is formed in the system; the boundary is drawn by connecting each state point where the probability of observing percolated clusters out of 10 – 20 snapshots (see Figure 1B) exceeds 50%. According to the phase-space diagram, a transition from discrete clusters to a percolated cluster can be achieved by either increasing the electrostatic interaction strength and/or increasing the average charge fraction. Figure 1C – 1E show snapshots of clusters with size larger than 20 beads formed in ionomer melts with different at different values. Discrete clusters are formed in systems with relatively low average charge fraction or under relatively weak electrostatic interaction, while a percolated cluster was observed in systems where the average charge fraction is higher and the electrostatic interaction is strong enough.
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Figure 1. A): Phase-space diagram obtained by analyzing the distribution of ionic clusters. Only discrete ionic clusters are observed in the red region, a percolated cluster is formed in the blue region, B): Probability of observing percolated clusters out of 10 – 20 snapshots (5e5 – 1e6 MD timesteps) at each state point, the border line in A) is drawn by connecting the state points where ! 50%, C, D, E): Snapshots of ionic clusters (size larger than 20) formed in random ionomers with charge fractions = 0.2 under Bjerrum lengths = 10.0, 18.2, and 20.0, respectively. Beads of the same color are in the same ionic cluster. Discrete ionic aggregates are observed in ionomers at low , while at larger a percolated cluster (colored in light blue in D) and E) ) is formed.
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Visualization and analyses of ionic clusters formed by charged species in random ionomer melts with different under different values illustrate the electrostatic effects on morphologies of ionic clusters. Figure 1 suggests that the structure of ionic clusters can be tuned using an understanding of the electrostatic effects; that is, by either increasing the electrostatic interaction strength (decreasing dielectric constant) and/or increasing the average charge fraction, one can expect a transition from discrete clusters to the formation of a percolated cluster. Microstructure Characterization Pair correlation functions between negatively charged monomers and positively charged counter ions at different electrostatic interaction strengths are calculated using the built-in tool in VMD and are shown in Figure 2 (in which the average charge fraction is 0.1). The first peak that accounts for the nearest neighboring charged monomer - counter ion pair shifts towards left (from ~1.2σ to ~1.0σ) and the intensity increases with increasing . At relatively low , a shift in the second peak from ~2σ to ~3σ indicates the formation of linear structures of alternating positively charged counter ion and negatively charged monomer and the disappearance of Lennard-Jones model fluid feature. At higher Bjerrum length ( = 10σ), the second peak shifts to ~2.3σ (close to √5σ), indicating the formation of a closely packed lattice like structure of charged species, in which the second peak corresponds to the second nearest neighboring +/pair. At an even higher Bjerrum length ( = 30σ), the third peak occurs at ~3.8σ, corresponding to the third nearest neighboring distance between +/- pairs, and demonstrates a longer-range correlation arising from stronger electrostatic interaction. Increasing electrostatic interaction strength leads to a more closely packed and ordered structure of ionic clusters.
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3.0
lB/σ = 30 lB/σ = 10 lB/σ = 5 lB/σ = 0
2.5 2.0 g+-(r) 1.5 1.0 0.5 0.0 0.0
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Figure 2. Pair correlation function of charged monomers (-) and counter ions (+) in random ionomers with average charge fraction of 0.1 for a range of values. Pair correlation functions of charged species in random ionomers with other average charge fractions show similar trends and therefore are not included. X-ray and neutron scattering have been widely used to investigate the local structure in nanoaggregated polymers. A modified hard-sphere scattering (MHS) model is frequently used to interpret morphological information of ionic aggregates.32, 33 Detection of microphase separation and extraction of spacing between ionic aggregates and their dependence on ion content have been achieved by X-ray scattering.14, 20 However, the size or shape of the ionic clusters cannot be fully determined by scattering methods alone. To this end, MD simulations offer a direct and reliable approach for eliciting more detailed picture of aggregate morphology.34 From the Fourier transform (Eqn. 1) of pair correlation function +,, the structure factors of both charged species were calculated to further characterize the morphology and distribution of the ionic clusters,
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= 1 + 4. /5 ,+ , − 1
012 3 3
6,
(1)
where + , is the density-density correlation function of both charged species (including charged monomers and counter ions).
A lB/σ = 30 lB/σ = 20 lB/σ = 15 lB/σ = 10 lB / σ = 2 lB / σ = 0
S(q)
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qσ
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lB/σ = 30 lB/σ = 20 lB/σ = 15 lB/σ = 10 l B/ σ = 2 l B/ σ = 0
S(q)
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3.0
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qσ Figure 3. A): Structure factors of charged species in ionomers with an average charge fraction of 0.1 (no stable percolated clusters formed at any Bjerrum length). The peaks marked by triangles for system at = 30 occur at 6.33 and 4.64 , those at = 20 occur at 7.98 and 5.24 . Inset in purple: snapshot of the clusters at = 30, inset in green: snapshot of the clusters at = 20; B): Structure factors of charged species in ionomers with an average charge fraction
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of 0.3 (stable percolated clusters formed at the highest two values). The peaks marked by triangles correspond to 7.67 and 6.89 for systems at = 30 and = 20, respectively. Inset in purple: snapshot of the clusters at = 30, inset in green: snapshot of the clusters at = 20. The scale bars in the inset figures are 10. Figure 3 shows the structure factors of both charged species in random ionomer melts with two different values of average charge fraction (0.1 and 0.3 in Figure 3A and 3B, respectively) at various Bjerrum lengths. The peak at ~7σ7 shifts toward smaller length scales with increasing Bjerrum length. This demonstrates a more closely packed ionic structure due to increasing electrostatic interaction strength, in agreement with the pair correlation functions shown in Figure 3. Peaks at smaller wave vectors reveal longer-range correlations between charged species. As demonstrated in Figure 3, the longer-range correlations become stronger with increasing Bjerrum length. As illustrated in Figure 3A and the inset, the shoulders at small wave vectors represent the characteristic distances between different clusters and the distance between different clusters reduces with increasing electrostatic interaction strength. Unlike random ionomers with an average charge fraction of 0.1, stable percolated clusters formed in random ionomers with average charge fraction of 0.3 at the highest two values (shown in Figure 3B and its insets). And within stable percolated clusters, the average distance between the branches increases with increasing electrostatic interaction strength. Note that the structure factor in Figure 3 shows a different trend at small wave vectors compared to that of pseudorandom ionomers studied by Hall et al.24 and reproduced by us as shown in the Supporting Information. An increase in with increasing and the trend of divergence at small values as shown in Figure 3 concur with structure factors of random
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copolymers,35, 36 indicating a similar effect that charge has on random ionomer nanostructures as miscibility of two different types of monomers does on that of random copolymers. To make sure that the trend of divergence at low is not caused by limited box size effect, we plotted the structure factors of larger systems at = 30 (shown in the Supporting Information) and the trend remains. Local density distribution (see Figure 4) of charged particles (including both charged monomers and counter ions) was generated by calculating the density of charged particles in cubic cells of size 3 that divided up the simulations box. Probability of finding a certain density is plotted against density of the cubic cells, it shows that the height of the peak decreases and the width broadens with increasing , which is similar to what was observed in random copolymers,30 indicating long range fluctuations in random ionomers.
0.25
lB/σ = 30 lB/σ = 20 lB/σ = 10 lB/σ = 0
0.20 0.15 P(ρ) 0.10 0.05
0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
ρ, σ-3 Figure 4. Local density distribution function of charged species in random ionomers with charge fraction = 0.3 for cell size of 3.
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While the size, size distribution, and shape of ionic aggregates in ion-containing polymers have been investigated using scanning transmission electron microscopy in experiments,37 MD simulations can provide more powerful insights in those characteristics of ionic aggregates by direct clustering analysis. Clustering Analysis 4
The cluster size distribution 8 is a normalized (9:; 8 = 1) probability of appearance of a cluster consisting of 8 + 1 ions in an ensemble of 8 for random ionomers with average charge fraction of = 0.05 and of = 0.1, respectively. Solid lines are the stretched
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exponential fit >8 given in Eqn. 3. Dashed line is the power fit >8 = 8 7N for = 30, μ ≈ 0.8; D): Cumulative distribution function >8 for random ionomers with average charge fraction of = 0.2. Solid lines are the stretched exponential fit >8 given in Eqn. 3. Dashed line is the power fit >8 = 8 7N for = 20σ, μ ≈ 1.1.
To reduce the noise, a cumulative distribution function >Q was introduced: ∑S TUV :
>Q = ∑S
(2)
TUW :
We found that the cumulative distribution function >Q has less noise than 8 (see Figure 5B), and for the case of low average charge fraction = 0.05, the best fit for this distribution can be approximated by a stretched exponential function38: >8 = X
7Y
TZW ^ ] [\
(3)
for all analyzed values of Bjerrum length . The parameters _ and
