Letter pubs.acs.org/macroletters
Comparing Solvophobic and Multivalent Induced Collapse in Polyelectrolyte Brushes Nicholas E. Jackson,†,‡ Blair K. Brettmann,† Venkatram Vishwanath,§ Matthew Tirrell,*,†,‡ and Juan J. de Pablo*,†,‡ †
The Institute for Molecular Engineering, The University of Chicago, Chicago, Illinois 60637, United States The Institute for Molecular Engineering and §Advanced Leadership Computing Facility, Argonne National Laboratory, Lemont, Illinois 06349 United States
‡
S Supporting Information *
ABSTRACT: Coarse-grained molecular dynamics enhanced by free-energy sampling methods is used to examine the roles of solvophobicity and multivalent salts on polyelectrolyte brush collapse. Specifically, we demonstrate that while ostensibly similar, solvophobic collapsed brushes and multivalent-ion collapsed brushes exhibit distinct mechanistic and structural features. Notably, multivalent-induced heterogeneous brush collapse is observed under good solvent polymer backbone conditions, demonstrating that the mechanism of multivalent collapse is not contingent upon a solvophobic backbone. Umbrella sampling of the potential of mean-force (PMF) between two individual brush strands confirms this analysis, revealing starkly different PMFs under solvophobic and multivalent conditions, suggesting the role of multivalent “bridging” as the discriminating feature in trivalent collapse. Structurally, multivalent ions show a propensity for nucleating order within collapsed brushes, whereas poor-solvent collapsed brushes are more disordered; this difference is traced to the existence of a metastable PMF minimum for poor solvent conditions, and a global PMF minimum for trivalent systems, under experimentally relevant conditions. from multivalent ions, most notably “bridging”.17 Past reports have examined the effects of multivalent ions on brush height, but the impact of multivalent ions on lateral or interior PE brush structure has not been examined in detail. Solvent quality plays a key role in the determination of polymer brush structure.18−20 Simulations by Dobrynin21 considered the range of accessible PE brush structure as a function of solvent quality in salt-free conditions, and elucidated the pinned micellar character of collapsed structures. Lee et al.22 examined the dynamics of collapse in PE solutions under various solvent-quality and multivalent-ion conditions. Notably, they reported that in polyelectrolyte solutions the delicate interplay of ionic effects and solvent quality can lead to the formation of Wigner crystals and Wigner glasses. To our knowledge, no work to date has examined the interplay of multivalent ion-induced collapse and poor solvent collapse in the context of PE brushes, despite the fact that both mechanisms could potentially lead to collapsed structures. Our simulations are based on a common primitive model of strong (quenched) polyelectrolyte brushes originally formulated for polyelectrolytes in solution.23 In that model, “M” PE chains comprising “N” spherical Lennard-Jones (LJ) charged
P
olymer brushes consist of a layer of polymer strands endtethered to a fixed surface.1 At high grafting densities, in a good solvent, these polymer strands extend perpendicular to the substrate, forming a “brush”. Polyelectrolyte (PE) brushes (charged polymer brushes in solution with their counterions) have risen to prominence due to their potential applications.2−4 Owing to the large number of tunable parameters in PE brushes (e.g., grafting density, salt concentration and valence, pH, solvent quality), understanding the interplay of polyelectrolyte brush structure and functionality under a range of solvent conditions is an important goal for nanoscale design. Specifically, poor solvent collapsed brushes have demonstrated potential for stimuli-responsive patchy particles,5 and dendritic collapsed structures from DNA brushes in multivalent ions have been used to form nanowire assemblies.6 These results forecast a need to better understand the mechanisms driving collapse and the methods used for tuning such nanostructures. The effects of salt valency are believed to be of considerable importance. Experiments7−11 and simulations12,13 on PE brushes have shown a sharp collapse in brush height with increased concentration of high valence salts, mimicking similar effects observed in solution.14−17 While this collapse has been attributed to a decrease in the osmotic pressure that occurs when multiple monovalent counterions in the brush are replaced by an individual multivalent ion, many studies have emphasized that there are other contributing factors arising © 2017 American Chemical Society
Received: November 5, 2016 Accepted: January 18, 2017 Published: February 3, 2017 155
DOI: 10.1021/acsmacrolett.6b00837 ACS Macro Lett. 2017, 6, 155−160
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Figure 1. Top-down view of polymer brush density profile in good solvent (ϵLJ = 0.4kBT) for (A) no salt, (B) cs = 10−3σ−3 monovalent salt, (C) cs = 10−3σ−3 trivalent salt, and (D) top-down trivalent ion density profile for cs = 10−3σ−3.
Figure 2. Top-down view of polymer brush density profile in very poor solvent (ϵLJ = 1.4kBT) for (A) no salt, (B) cs = 10−3σ−3 monovalent salt, (C) cs = 10−3σ−3 trivalent salt, and (D) top-down trivalent ion density profile for cs = 10−3σ−3.
beads (σ, ϵLJ) have one of their ends grafted to a surface of cubic lattice points separated by 10σ. Chains are initialized vertically from the brush in the “swollen” state. This work
considers M = 64 and N = 100. The spacing 10σ represents an experimentally relevant grafting density for poly(sodium styrenesulfonate) (PSS), and simulation parameters in this 156
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Figure 3. Polymer−polymer PCF in (A) good solvent (ϵLJ = 0.4kBT) and (B) very poor solvent (ϵLJ = 1.4kBT).
(Figure 2). Specifically, we examine the time-averaged brush density projected onto the xy-plane (a top-down view of the PE brush density). Figure 1A and B depict the polymer density in the cases of salt-free and monovalent “salted brush” (see SI) regimes, respectively. In both situations, the lateral brush structure is homogeneous, with high-density points only occurring directly where the polymer is grafted to the surface. In contrast, upon addition of trivalent salt, Figure 1C shows an inhomogeneous distribution of collapsed PE brush structures, where the “legs” of pinned/cylindrical micelles26 are clearly visible (at higher grafting densities “stripes and holes”27 are expected). By examining the spatial distribution of trivalent ions in Figure 1D, it is apparent that the trivalent ions are predominantly segregated within the collapsed polymer brush globules; they exhibit limited mobility, and they serve to neutralize the interiors of the collapsed micellar structure. The pinned micelle structure is consistent with a recent phenomenological theory for polyelectrolyte brush collapse in the presence of multivalent ions, and to our knowledge, this is the first verification of the validity of the predicted structure in the presence of multivalent ions.28 In ref 28, electrostatic attractions and poor solvent attractions contributing to collapse are represented by an effective poor solvent parameter that depends on ion valence and strength. An energy balance approach is used to capture brush behavior, with a sum of surface tension, leg stretching, and electrostatic contributions being minimized with respect to the number of polymer chains forming the pinned micelle. One key conclusion from this work is that a high number of uncompensated charges on the polyelectrolyte chain lead to instability of the pinned micelle structure and formation of an extended brush. It is clear by the colocalization of multivalent ions within the collapsed polymer shown in Figure 1 that the assumption made in ref 28 that a majority of polymer charges are compensated by condensed counterions in the pinned micelle is reasonable. While this simple model can accurately represent measured brush heights and, as shown with the comparison to the results in Figure 1 here, structure formation, the simple phenomenological theory is insufficient to distinguish bridging-driven collapse from
work are selected to mimic recently published work on PSS brushes9 that exhibit charge fractions of ≈1. This work assumes a charge fraction of 1 (all monomers charged). Each bead in a polymer chain is bonded to its nearest neighbor via a finite extensible nonlinear elastic (FENE) potential with a mean bond length of 1.1σ, along with a three-body cosine/delta bending potential between adjoining monomers enforcing a chain persistence length of ≈σ. These N × M polymer beads are in equilibrium with a solution of N × M counterion beads embedded in a dielectric continuum corresponding to a Bjerrum length (lB) of 3σ. Additional salt particles are added to the system corresponding to a specific monovalent or trivalent salt concentration, cs, of the solution. Electrostatic interactions are treated using a particle−particle−particle− mesh Ewald sum in the slab geometry. Solvent fluctuations are incorporated via a Langevin thermostat (τdamp = 100τ), with dynamics evolved using the velocity-verlet algorithm and a time step of 0.005τ. The monomer mass of our coarse-grained model roughly corresponds to the molar mass of a PSS monomer (≈230 g/mol), leading to a value of τ ≈ 2 ps. Structures are equilibrated for 2 × 106 time steps, after which 5 × 106 time steps are performed for sampling. Periodic boundary conditions are applied in the two spatial dimensions parallel to the substrate surface, and impenetrable walls are placed at z = 0 and z = 2Nσ. A separate model system is defined to analyze the potential of mean force (PMF) of strand−strand interactions. In our model, two short oligo-electrolytes (five repeat units) are examined in solution in the presence of salt. We compute the PMF between two strands as a function of the distance between their centers of mass (COM) using umbrella sampling simulations along with a weighted histogram analysis (WHAM).24 All simulations are performed in LAMMPS;25 additional simulation details and example input files are provided in the Supporting Information. The internal structures of collapsed PE brushes are examined through the use of a pair correlation function (PCF). Analysis begins with the examination of the time-averaged PE brush density in a good solvent (Figure 1) and a poor solvent 157
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Figure 4. Free-energy profiles from Umbrella Sampling for two brush strands in (A) good solvent (ϵLJ = 0.4kBT) w/trivalent salt ions and (B) very poor solvent (ϵLJ = 1.4kBT) w/monovalent salt ions.
neighbor and nearest-neighbor correlations. Upon addition of low concentrations of trivalent salt, the brush begins to aggregate, resulting in an overall increase in the magnitude of the PCF over all distances. When a threshold trivalent ion concentration is reached, the intrachain PE neighbor correlations disappear as the brush collapses, forming a disordered aggregate structure. These results are in contrast to the poor solvent case of Figure 3B. In Figure 3B one can appreciate that the salt-free collapsed brush also exists in a disordered aggregated state akin to that in the good solvent trivalent salt case. While one might expect that the further addition of salt should lead to a larger disordered structure via the inclusion of more chains, one observes the opposite effect, with the formation of noticeable correlations out to five neighbors in the PCF, as is most clearly observed at c3+ s = 2 × 10−3σ−3. Concurrent with this appearance of longer-range structural order in the PCF is a corroboration of the decrease in feature size observed in Figure 2, evidenced by the rapid decay of the PCF at long distances. The formation of a denser, more ordered structure is in agreement with the computed trends in brush heights for these systems (see SI), despite brush height not being an ideal order parameter in the case of laterally heterogeneous brush systems. While the previous results have focused on surface-scale PE brushes, here we focus on the similarity of solvophobic and multivalent ion-induced interactions between two individual brush strands. Figure 4A shows the PMF between two strands under good solvent conditions and lB = 3σ; it is apparent that a global minimum in the PMF forms at an optimum trivalent ion concentration which corresponds to the neutralization of all PE charges. As the trivalent ion concentration increases beyond this point, electrostatic screening increases and the minimum transitions from global to metastable, leading to dissolution of the aggregate, which has been observed on several occasions.14,15 In Figure 4B, the PMF for poor solvent conditions shows that the local aggregated structure is always a metastable minimum when trivalent salt is absent (though this will not be true at higher lB). However, when trivalent salt is added, the
collapse driven by poor solvent interactions following strong counterion condensation and neutralization. Here, we utilize simulations to differentiate these mechanisms further. We now examine the PE brush density under very poor solvent conditions (ϵLJ = 1.4kBT) in Figure 2. The results in Figure 2 show that solvophobic-induced collapse also leads to heterogeneously distributed collapsed PE brush structures, even under salt-free conditions; similar effects are observed in monovalent salt concentrations that would constitute the “salted brush” regime for good-solvent conditions. At this point, it is useful to consider whether solvophobically collapsed PE brush structures are distinguishable from good solvent multivalent ion-induced collapsed structures. This point is further illuminated by examining the collapsed poor-solvent PE brush structure when trivalent ions are added. Notably, Figure 2 demonstrates that the characteristic feature sizes of the heterogeneously distributed brush structure clearly decrease upon addition of trivalent ions (see SI for quantitative aggregation number statistics). This effect is repeatable: feature sizes decrease when trivalent salt is added as a function of worsening solvent quality in all systems considered here (see SI). These structural transitions in the presence of trivalent ions are not reproduced by making the solvent quality arbitrarily poor in monovalent systems (see SI). The observation of decreasing feature size with worsening solvent quality and increasing trivalent ion concentration provides another piece of evidence that solvophobic collapse and multivalent-induced collapse are not identical. We note that this feature could be influenced by our Lorentz−Berthelot mixing rules for LJ parameters. However, poor solvents for PE brushes are also poor solvents for added salts, and consequently, the notion of having enhanced salt-brush LJ interactions in poor solvent conditions is a physically reasonable expectation. We now examine the interior of the collapsed brush structure using a pair correlation function (PCF) between brush particles. Figure 3A shows the PCF for brush particles in good solvent under a variety of salt conditions. For salt-free or monovalent salt conditions, the PE brush exhibits intrachain 158
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Figure 5. ϵLJ = 0.0kBT brush simulations in trivalent salt (A) top-down density view, (B) side-density view, (C) PMF for strand−strand interaction in monovalent lB = 9σ, and trivalent lB = 3σ, and monovalent poor solvent conditions.
good solvent conditions, which resulted in collapsed structures, it appears that the renormalization of the lB is not wholly capable of describing all structural repercussions of trivalent ions, though the topic certainly warrants future study. It is worthwhile to consider whether the trivalent ion induced ordering could be replicated by making the solvent quality arbitrarily poor. To further explore this point, we have run monovalent simulations of the full brush structure using a solvophobic LJ parameter of 5.0kBT (see SI), and Umbrella Sampling simulations in monovalent salt with a parameter of 3.0kBT (Figure 5D). The full brush simulations show noticeable differences in collapsed feature sizes when compared to trivalent collapsed structures, and the PMF of approach in these two cases are starkly different, with the trivalent case exhibiting a smaller free-energy barrier and a different position of the global minimum in the PMF. Such differences likely arise from fundamental differences in the length scales of LJ and electrostatic interactions. While the maximum magnitude of LJ and electrostatic interactions can be similar, leading to collapsed structures, the potential energy surface at all points cannot be matched, and thus the mechanisms and precise details of collapse should be different. Based on solution studies,22 we hypothesize that the ordered structures of trivalent collapsed systems observed in Figure 3B rely upon the formation of a cocrystal of trivalent ion and PE monomers, and indeed such ordered structure is observed in a combined brush-trivalent ion PCF as well (see SI), suggesting the fundamental role of the trivalent ion within the globule. This feature was also observed in PE globules in solution.22 The minimum in the PMF at smaller values of separation distance (Figure 5C) further supports the formation of these potential cocrystals under trivalent conditions. It is also possible that the intervening trivalent ions help relax three-body intrachain bending interactions that might otherwise be unfavorable in a homogeneous collapsed state. As a final note, we would like to comment on the role of kinetic and thermodynamic effects in these systems. Previous
metastable minimum transitions to a global minimum, and becomes significantly deeper than in the good solvent case. The magnitude of this minimum is likely responsible for the correlations observed in the PCF of Figure 3B. These results suggest that the underlying mechanisms for solvophobic and multivalent ion-induced interactions are different. An important mechanistic consideration regards whether the trivalent mechanism of collapse is unique when compared to a renormalized lB, monovalent system. In monovalent systems, collapse occurs at high lB when PE charges are neutralized by counterions, and the solvophobic character of the backbone emerges, leading to collapse. We test these considerations by setting ϵLJ = 0.0kBT and run a trivalent simulation (Figure 5) at lB = 3σ. According to the previous work of Carrillo et al.,21 simulations at ϵLJ = 0.0kBT and lB = 9σ result in a uniform film and possess no heterogeneous structures. The density plots in Figure 5A,B demonstrate that trivalent heterogeneous collapse still occurs even when using a noninteracting polymer backbone (ϵLJ = 0.0kBT), in contrast to the renormalized lB results of Carrillo.21 This result suggests that the mechanism of trivalent collapse is not a simple hydrophobic effect resulting from charge neutralization, and is likely due to some combination of “bridging” and stronger multipolar interactions between condensed counterions. We further utilize umbrella sampling simulations to examine the issue of lB renormalization in accounting for trivalent effects, running at lB = 9σ monovalent, and lB = 3σ trivalent conditions in Figure 5C. While qualitatively similar at long-range, at shortrange the well depths deviate by greater than 2 kBT, and the position of the minimum is significantly shorter for the trivalent system. The deviation at short distances is likely due to the significantly different multipole moments of condensed trivalent and condensed monovalent systems at short distances. Specifically, the ability of a single trivalent ion to “bridge” two monovalent ions is an effect not anticipated in the monovalent system, and could account for the differences observed at shortrange. Given our previous trivalent brush simulations run at 159
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experimental results7 show that the force distance profile of two PSS brushes in low ionic strength trivalent solutions is exposure time dependent, indicating that collapsed brush structures in the presence of trivalent counterions are likely kinetic structures. This is supported by our simulations in which chain exchanges between micellar aggregates are rarely observed. To investigate this point further, we have compared brush simulations of trivalent systems in poor solvent directly, as well as by first equilibrating a good solvent trivalent system, and then changing the solvent quality to poor (see SI). Encouragingly, the feature sizes in both cases are very similar, however the issue of kinetically trapped collapsed structures is an important issue to be carefully considered in future work. While ostensibly similar, solvophobic and multivalent ioninduced collapse possess many distinguishing features. Our comparisons of trivalent systems with renormalized l B monovalent systems demonstrates that the mechanism of collapse in trivalent systems is different from the traditional poor solvent collapse transition, and this difference is likely attributable to multivalent “bridging”. At experimentally relevant values of simulation parameters, we observe the effect that trivalent ions induce a denser, more ordered structure, potentially templated by multivalent ions; no such features are observed in the case of poor solvent collapsed structures. By examining free-energy simulations of the fundamental interaction between two brush strands, we observe that the PMFs of poor solvent and trivalent collapse are qualitatively distinct, with poor solvent leading to a metastable free-energy minimum, and multivalent-ion interactions leading to a more stable global minimum. These differences in the mechanisms of collapse have the potential to be utilized for the tailoring of specific nanostructures in future work.
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ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy Office of Science, Program in Basic Energy Sciences, Materials Sciences and Engineering Division. Dr. Jackson would like to thank the Argonne National Laboratory Maria Goeppert Mayer Named Fellowship for support. We thank Dr. Jing Yu for useful discussion.
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REFERENCES
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ASSOCIATED CONTENT
* Supporting Information S
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.6b00837. Definition of radial distribution function, brush height, and dimensionless parameters, system brush heights, simulation descriptions, and example input files, PE brush density plots for all concentrations and solvent qualities studied, energy conservation and convergence plots for PE brush simulations, brush/trivalent pair correlation function, proof of charge neutralization in the osmotic brush regime, aggregation number analysis for trivalent systems, and Debye lengths of simulated systems (PDF).
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AUTHOR INFORMATION
Corresponding Authors
*E-mail:
[email protected]. *E-mail:
[email protected];
[email protected]. ORCID
Blair K. Brettmann: 0000-0003-1335-2120 Juan J. de Pablo: 0000-0002-3526-516X Notes
The authors declare no competing financial interest. 160
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