Conformational Properties of Comb-Like Polyelectrolytes: A Coarse

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Conformational Properties of Comb-Like Polyelectrolytes: A Coarse-Grained MD Study Mahdi Ghelichi, and Michael H. Eikerling J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b00568 • Publication Date (Web): 24 Feb 2016 Downloaded from http://pubs.acs.org on March 1, 2016

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The Journal of Physical Chemistry

Conformational Properties of Comb-like Polyelectrolytes: a Coarse-grained MD Study Mahdi Ghelichi†, Michael H. Eikerling†* †

Department of Chemistry, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A

1S6, Canada KEYWORDS: Ionomer molecules, conformational properties, coarse-grained molecular dynamics, backbone radius of gyration, persistence length

ACS Paragon Plus Environment

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Abstract

This article presents a coarse-grained molecular dynamics study of single comb-like polyelectrolyte or ionomer chains in aqueous solution. The model polymer is comprised of a hydrophobic backbone chain with grafted side chains that terminate in anionic headgroups. The combpolymer is modeled at a coarse-grained level with implicit treatment of the solvent. The computational study rationalizes conformational properties of the backbone chain and localization of counterions as functions of side chain length, grafting density of side chains, backbone stiffness, and counterion valence. The main interplay that determines the ionomer properties unfolds between electrostatic interactions among charged groups, hydrophobic backbone interactions, and steric effects induced by the pendant side chains. Depending on branching density, we have found two opposing effects of side chain length on the backbone gyration radius and local persistent length. Variation in comb-polyelectrolyte architecture is shown to have nontrivial effects on the localization of mobile counterions. Changes in Bjerrum length and counterion valence are also shown to alter the strength of Coulomb interactions and emphasize the role of excludedvolume effects on controlling the backbone conformational behavior. The simulation results are in qualitative agreement with existing experimental and theoretical studies. The comprehensive conformational picture provides a fundamental framework for future studies of combpolyelectrolyte systems.


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Introduction Polyelectrolytes are a class of polymers with a high fraction of periodically attached ionizable groups.1,2 Polyelectrolyte solutions exhibit intriguing physical properties and they find a wide range of applications as rheology modifiers and colloidal stabilizers, adsorbent agents, biocompatible coatings, and biosensors.1–5 The structure of polyelectrolyte chains in aqueous solution is dictated by a subtle interplay of electrostatic interactions between ionic species and hydrophobic interactions of the polymer backbone with the solvent.4,6–10 The ionizable groups can be located within the backbone chain or grafted to it. The first category, linear polyelectrolytes, forms the largest class of polyelectrolytes with paramount importance in biology and various commercial uses.11,12 Examples of linear polyelectrolytes include: poly(acrylic acid) and poly(methacrylic acid), poly(styrenesulfonate), DNA, RNA, and many other polyacids and polybases.1–3,11–14 Structural properties of linear polyelectrolytes have been explored extensively in theoretical studies4,6,8,9 and computer simulations.3,4,7,10,15–20 Graft polyelectrolytes, on the other hand, have received generally less attention in studies of basic ionomer properties. In graft polyelectrolytes, ionizable groups are either located along pendant side chains or at the termini of the side chains. Many synthetic and biological polymers belong to the latter group. Polypeptides containing amino acid monomers are prominent examples.21–24 The amino acid repeating unit comprises pendant side chains terminated with either positively (Lysine and Arginine) or negatively (Aspartic and Glutamic) charged units.21,22 Conformational properties of the polypeptide backbone control numerous biological functions of the protein. Polypeptides with longer side chains exhibit improved conformational backbone flexibility due to weakened electrostatic repulsion of ionic groups.21–24


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The other notable class of polyelectrolytes with comb-like architecture corresponds to perfluorinated ionomers.25–28 Ionomers contain a smaller fraction of ionizable groups compared to polyelectrolytes. Ionomer applications span from paint suspensions27,29 to electrochemical energy applications25, 26,30,31,32–35. Ionomers are synthetized both in acid and basic forms.36–40 Acid-type ionomers are the key material in polymer electrolyte fuel cells (PEFCs).25,28,30,41 The basic forms are receiving escalating recent attention as anion exchange membranes (AEM) in alkaline membrane fuel cells.40,42 Excellent proton conductivity as well as mechanical and chemical stability render PFSA ionomers the material of choice in PEFC.43 PFSA-type ionomers consist of a PTFE backbone, to which the perfluorosulfonic acid side chains are grafted. Hydrocarbon-based ionomers, both in acidic and basic forms, have also been synthesized with controllable comb-like architectures and backbone rigidities, achieved mostly through incorporation of aromatic rings into the backbone chain.28,31,39,42,44,45–48 Variation in length and density of grafted side chains has been employed in both PFSA and hydrocarbon ionomers to tune membrane properties such as water sorption, mechanical and chemical durability, and ionic conductivity.49–51,28,39,44,47 Valency of counterion exerts a pronounced effect on mechanical and water sorption properties of the ionomer membrane.52,53 A recently developed poroelectroelastic model of water sorption in ionomer membranes relates these changes to the alteration of the elastic properties of the polymer walls formed by self-aggregated ionomer backbones.54 A coherent understanding of structure, properties and performance of ionomer membranes is complicated by the multiple scales involved. The three main scales are (i) molecular scale comprising ionomer chains, ionomer bundles, anionic charges and surrounding cloud of protons, (ii) pore level at which elastic properties of pore walls as well as proton distri-


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bution and mobility in the pore can be defined, and (iii) hydrated membrane scale at which sorption characteristics, proton conductivity and elasto-mechanical properties can be defined.55,56. Understanding basic structure and properties at the single chain level is of paramount importance. From a practical standpoint, it determines the ionomer solution behavior and elucidates how the structure and properties will evolve at all three levels. However, single chain studies are scarce; mostly because of the practically motivated focus on membrane-level investigations; but also due to spontaneous aggregation of hydrophobic chains that complicates single chain investigations. Recently, Minko and coworkers reported atomic force microscopy (AFM) imaging of singlechain adsorbed Nafion chains on mica and highly ordered pyrolitic graphite (HOPG) substrates.57 A compact globular structure was reported for adsorbed ionomer chains.57 Theoretically, conformational features of single chains of Nafion-type ionomers have been studied mainly through atomistic simulations. In the early 2000’s, Vishnyakov et al.58,59 studied the effect of aqueous and alcoholic solvents on the conformation of Nafion oligomers. Recently, density functional theory (DFT) combined with energy-loss spectroscopy (EELS) was employed to study the effect of different side chain chemistries on the conformational features of single PFSA backbone chains.60 Computational limitations associated with the incorporation of atomistic details along with illdefined simulations of inadequately short oligomeric chains have hindered drawing definite conclusions regarding the interplay of side chain length and density on the conformational features of comb-like ionomer molecules. Coarse-grained MD simulations, in which chemical details are foregone for the sake of increasing the time and length scale of simulations, have been able to capture structural changes in charged polymers. Changes in the radius of gyration of the backbone and in the persistence


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length of single linear polyelectrolyte chains have been explored in relation to variations in chain rigidity61,62, charge fraction10,65,47,49,50, and dielectric properties of the solvent.10,17,19,20,49 For branched polyelectrolytes with charged side chains, conformational properties of the hydrophobic backbone have been studied as a function of the length and density of the side chains.63,66 In neutral polymers with pendant side chains (bottle-brush or comb-polymers) it is known that steric hindrances due to side chain congestion increase the gyration radius and stiffness of the backbone.67–69 For comb-polyelectrolytes with charged moieties at side chain terminals more complex conformational responses are expected due to the presence of both electrostatic and excludedvolume interactions. To the best of our knowledge, for hydrophobic comb-like polyelectrolytes with charge-terminated side chains, there have not been any attempts in developing a coherent picture of their conformational properties. This article strives to rationalize the backbone conformation and the local rigidity of single chains of comb-polyelectrolytes as a function of changes in architectural and environmental parameters. The Section “Chain model” introduces the coarse-grained chain model, the type of interactions considered, and the simulation protocol. Section “Results and discussion” provides a systematic evaluation of backbone conformational changes as a function of variation in pendant side chain length and density, strength of electrostatic interactions, and counterion valence. Chain model In a coarse-grained model, the interacting species are groups of atoms or beads that interact via effective forces. We model a bead-spring polyelectrolyte chain consisting of a hydrophobic backbone and pendant side chains in implicit solvent with explicit counterions. Figure 1 is a schematic illustration of the branched polyelectrolyte chain considered in the model.


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The model is devised to emulate PFSA-type ionomers. Each backbone bead resembles one (CF2-CF2) repeating unit with diameter of approximately 2.4 Å. The simulated backbone chain consists of 104 beads with regularly spaced pendant side chains, representing a real backbone contour length of ~ 25 nm. Anionic headgroups carry a -1e charge and they are represented by a single bead (red) with the same size as backbone and side chain beads. Ls denotes the number of beads in each side chain, and Nb is the number of backbone beads per ionomer repeat unit (containing 1 side chain), as illustrated in Figure 1. We explore values Ls = 1, 3, 5, and 7 and Nb = 1, 2, 3, 5. Since all side chains contain a charged unit, we define a charge fraction f = 1/Nb, for which we will consider values f = 1, 1/2, 1/3, 1/5, with the lower range resembling ionomers and the upper range corresponding to bottle-brush (f = 1) and polyelectrolyte systems (f = 1/2, 1/3). Electroneutrality is guaranteed by adding Nc = Ns/Zc counterions (yellow beads in Figure 1) where Zc is the counterion valence and Ns is the total number of side chains. No other electrolyte is added to the solution system. As a reference, we simulated all comb-like architectures in their uncharged form in electrolyte-free solution, for all considered combinations of f and LS. As a simplification, all beads are assumed to have the same diameter σ. Adjacent monomer beads in the backbone and side chains are connected by the finitely extensible nonlinear elastic (FENE) potential with spring constant k = 30 ε/σ2 and maximum extension R0 =1.5σ,70–72 U FENE

⎛ ⎛ ⎞2 ⎞ r 1 rij = − kR0 ln ⎜1− ⎜⎜ ij ⎟⎟ ⎟ ⎜ 2 R ⎟, ⎝ ⎝ 0⎠ ⎠

( )

(1)

The flexibility of the backbone is modeled with the bond angle potential

( )

(

(

Uθ rij = kθ 1− cos θ − θ 0

)) ,

(2)


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where kθ is the bending rigidity and θ0 the equilibrium angle that is set to 180º for linear segments along backbone. We considered values kθ = 0, 15, 30, 45 kBT for the backbone chain. Pendant side chains are freely rotating. Van der Waals interactions between monomer beads are modeled through a standard shifted and truncated Lennard-Jones (LJ), ⎧ ⎡⎛ ⎞12 ⎛ ⎞6 ⎛ ⎞12 ⎛ ⎞6 ⎤ σ σ σ ⎥ ⎪ ⎢ σ ⎪ 4ε LJ ⎢⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ ⎥, r r r r U LJ (rij ) = ⎨ ⎣⎝ ij ⎠ ⎝ ij ⎠ ⎝ c ⎠ ⎝ c ⎠ ⎦ ⎪ ⎪ , ⎩0

rij < rc rij > rc

(3)

where rij is the distance between beads i and j and σ is the bead diameter. The cut-off distance was set to rc = 2.5σ, for interactions between uncharged polymer beads (backbone and side chain beads), and rc = 21/6σ for all other pairwise interactions (neutral-anionic, neutral-cationic, anionic-anionic, anionic-cationic) resulting in the purely repulsive part of the Lennard-Jones (LJ) potential. The reference value of the Lennard-Jones interaction parameter is set to εLJ = 1 kBT. The units of energy, diameter and mass of all polymer beads are set to unity, i.e. kBT = 1, σ = 1, and m = 1. The θ solvent condition for the polymer backbone corresponds to εLJ = 0.33 kBT.19,73 The reference value of εLJ for interactions between uncharged beads was set to 1.0 kBT, representing an “intermediate” level of hydrophobicity. For interactions between charged beads, we used a Coulomb potential of the form,

( )

U Coul rij = k BT

λB qi q j rij

,

(4)

where λB = e02/4πε εskBT is the Bjerrum length, e0 is the unit charge and ε0 and εs are the permit0

tivity of vacuum and the relative dielectric constant of the solvent. The baseline value of the Bjerrum length is λB = 3.0 σ, which corresponds to about 0.7 nm in aqueous electrolyte solution


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(σ ≈ 0.24 nm). In our parametric analysis, we explore effects of LS, f, λB, ZC, and kθ . Table 1 summarizes the baseline values and parameter ranges explored. The particle–particle particle–mesh (PPPM)74 was used for the calculation of electrostatic interactions. Simulations were carried out in the NVT ensemble with periodic boundary conditions in all three directions for a cubic simulation box with length L = 110 σ. The temperature was maintained by a Langevin thermostat,75 in which the motion of beads is governed by

mi

!" dV i t

()

dt

! !" ! ! d ri t !" = −∇U i − Γ + Fi t dt ,

()

()

(5)

where mi is the mass of bead i and Ui is the potential energy experienced by the bead as the result

! of bonded and nonbonded interactions. Fi t is a random force with zero average value. The

()

friction coefficient of Γ = τLJ-1 was calibrated to maintain a reduced temperature of T* = kBT/εLJ = 1 where τLJ is the standard LJ time, τLJ = σ (m/εLJ)1/2. The equations of motion for beads in the system were integrated using the velocity-Verlet algorithm with a time step of Δt = 0.01 τLJ. The following procedures were employed in simulations, which were performed with LAMMPS code.76 A single polyelectrolyte chain was placed in the middle of the simulation box and simulated for 7×106 time steps. The first 5.0×106 timesteps we used for equilibration, tracked through convergence in the total energy and the radius of gyration. For equilibrium state analysis, data collections were performed for the last 2×106 timesteps and data were saved every 2×104 timesteps, resulting in 100 regularly time-spaced snapshots. Before presenting simulation results, a remark on the implicit solvent treatment is in order. The effects of solvent molecules are implemented through the friction term of the Langevin thermostat. This specific treatment, which is a common practice in coarse-grained MD simulations on


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the polyelectrolyte systems, implies the absence of explicit solvent molecules. Absence of these molecules lowers the computational costs and thus longer simulation runs become practically possible, allowing for a proper equilibration of the system. However, as argued in Refs. [17,77,78], the absence of many-body effects imposed by solvent molecules can significantly influence the collapse and conformational dynamics of both neutral and polyelectrolyte solutions under poor solvent conditions. Since we aim at capturing the equilibrium conformational features of long comb-polyelectrolyte chains and not the relaxation dynamics of our model system, the use of implicit solvent molecules does not alter any of the reported conformational behaviors. We have also assumed a uniform polarizability of the implicit solvent by fixing the dielectric constant value. This assumption implies that the electrostatic interaction among any pair of charged groups depends only on their inter/intra separation distances. In reality, local charging effects result in polarization of solvent molecules and alteration of local solvent dipole moments. This implies subtle variations in dielectric constant and spatially dependent electrostatic interactions. Solvent polarization and local dielectric effects can be introduced to the studied system either through incorporation of explicit solvent molecules or by developing polarizable coarsegrained solvent models.65,79 Even though the consideration of polarization effects yields more realistic results for local conformational properties and needs to be the focus of future studies, the imposed computational complexity shortens the equilibration time period. For the sake of simulating long comb-like chains and capturing the global conformational trends, it seems that consideration of uniform dielectric media is a legitimate assumption. Results and Discussion Effects of side chain length and density. As discussed above, conformational behavior of a branched polyelectrolyte chain originates from both steric effects due to branching and the elec-


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trostatic interactions of charged units. To be able to discriminate both of these effects, we simulated grafted polymers in both their charged and neutral states (UCoul = 0). We used the backbone gyration radius, Rg (σ), as the main conformational variable, which is defined as

1 N R = ∑ ri − rCOM N i=1 2 g

2

, rCOM

1 N = ∑ ri N i=1

,

(6)

where i is the backbone bead index, N is the number of backbone beads, ri is the position vector of the ith bead, and rCOM the center-of-mass (COM) of the chain. Angular brackets indicate averaging over chain configurations. Figure 2 shows the results for Rg obtained in the case of freely rotating chains. Zero bending rigidity implies that all conformational changes stem from excluded-volume and electrostatic effects. Figure 2a shows that increasing LS and f increases Rg. These effects become more prominent as f increases. No significant change in Rg with LS is observed for f = 1/5. The increase in backbone Rg upon increasing LS and f for the neutral polymer is due to enhanced excludedvolume interactions of side chain beads. This trend reflects the classical behavior that was previously captured through extensive Monte Carlo (MC)67–69,80 and MD63,81 simulations. Figure 2b shows the changes in backbone Rg for comb-polyelectrolyte chains in the presence of monovalent counterions at λB = 3 σ. For comb-polyelectrolytes, higher f increases the backbone Rg for all LS values explored. Higher density of side chains results in closer proximity of charged units and stronger electrostatic repulsion that causes greater stretching of the backbone. Charged polymers maintain larger Rg values compared to the uncharged equivalent polymer, clearly demonstrating that eliminating the charges leads to smaller and more compact conformations.


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Regarding the effect of side chain length, two different trends are observed depending on the fraction of side chains. At f = 1, increasing LS increases Rg, while f ≤ 1/2 results in the opposite trend. The change in LS induces a transition in the interplay of excluded-volume and Coulombic interactions. Increasing LS leads to a larger distance between charged headgroups and correspondingly weakened electrostatic repulsion, which favors a reduction in Rg. On the other hand, higher LS increases the steric congestion of side chains, which favors a stretching of the backbone chain, to increase Rg. At f = 1, the steric effect of side chain congestion outweighs the electrostatic effect and the increase in LS leads to enhanced Rg. The milder increase in Rg at f = 1 for the charged chain as compared to an uncharged chain reflects the opposite influence of increasing LS on the chain size. For f ≤ 1/2 excluded-volume effects are weakened due to the greater separation of pendant side chains and the electrostatic interactions among anionic headgroups exert the dominant effect on Rg. Weakened Coulombic repulsion at higher LS lowers the electrostatically induced stretching of the backbone and Rg decreases. The decrease in backbone size upon increasing the length of side chains was also experimentally reported for polypeptide chains bearing longer pendant side chains.21,22,24 Figure 2c shows the pure electrostatic contributions to the backbone Rg once the excluded-volume effects, Rg values in Figure 2a, are subtracted from Rg values presented in Figure 2b. There is a noticeable difference in the pure electrostatic contribution to Rg between short and long regimes of LS for all f values. This contribution is more pronounced for regimes of small LS values. The trend seen in Fig 2c shows that for short side chain polymers Coulombic effects prevail. This effect is owned to the closer proximity of anionic moieties at smaller LS values that strengthen the electrostatic


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repulsion among charged groups. The electrostatic contribution continuously diminishes as LS increases and the excluded-volume effect gains weight in determining Rg. Figure 3 shows the snapshots of the comb-polyelectrolyte chains at different LS and f values. The conformational behavior observed in Figure 2b is in line with the conformational trends in Figure 3. For f = 1/5 we can see the dominant effect of LS on the size of the chain: while chains with LS ≥ 3 exhibit a globular conformation, chains with LS = 1 assume an expanded coil-like conformation. For f = 1, increasing LS results in the transformation of an expanded chain at LS = 1 to a rod-like chain at LS = 7. Increase in the fraction of pendant side chains results in monotonic expansion of collapsed chains. All-atomistic15 and coarse-grained65 MD simulation studies of sodium polystyrenesulfonate (NaPSS) in water bearing different fractions of anionic-terminated branches demonstrated a similar conformational trend. Conformational structures obtained for f < 1/5 are indistinguishable from f = 1/5 and are not discussed in this work. The globular structure observed for f = 1/5 and LS ≥ 3 is in line with the globular structure of a single chain Nafion ionomer reported in Ref [57]. Distally located branches weaken the steric congestion of side chains and lower the Coulombic self-charge repulsions. Minimization of the surface energy of hydrophobic polymer triggers the collapse of the ionomer chain to form a globular structure. Figure 4 shows the effect of different bending rigidities, kθ , on Rg of both charged and uncharged comb-polymer for the explored range of f and LS. The value f = 1/2 closely resembles the charge fraction of the coarse-grained Nafion models employed by Malek et al82,83. Imposing bending rigidity to the backbone chain continuously increases Rg for both the charged and uncharged polymers.


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More rigid chains exhibit a smaller sensitivity of Rg to the structural changes for both charged and uncharged polymers. For very stiff chains, Rg is nearly independent of f and LS. Increased backbone rigidity introduces an energetic barrier for the conformational changes introduced either through topological effects (excluded-volume effect) or through electrostatic repulsions. In what follows, the effect of changes in length and density of side chains on the local orientational correlation or local persistence length, Lkp , of the backbone chain is explored. The local !" persistence length is the scalar projection of the end-to end vector, Rb , onto each bond in the

!"

backbone, rb , which is defined by63,67

Lkp

!" ! rb !" ! • Rb , = !" rb

(7)

where k is the backbone monomer or bead index and the angular brackets, , denote averaging over chain configurations. Figure 5 shows the results for Lkp or the persistent length (LP) as a function of monomer index, k, along the backbone chain for neutral and charged comb-polymers. For both charged and uncharged polymers, greater f results in higher Lkp . Backbones of a combpolyelectrolyte maintain a greater local rigidity than the uncharged polymer due to Coulombicinduced chain stiffening. Increasing LS results in a monotonic increase in Lkp at all f values for neutral polymers and at f = 1 for the comb-like polyelectrolytes. Increasing LS increases the backbone flexibility at f ≤ 1/2 for charged chains. No special features are observed in Lkp for f = 1/5 at LS > 1 in both uncharged and charged chains. In the regime of low and distant side chain


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contents, Lkp values are close to zero, which suggests that the entire backbone is in a compact globular conformation. The Lkp values of f = 1, 1/2, 1/3 for the charged polymer, shown in Figure 5 reveal a curving of Lkp upon increase in LS. This curvature arises from lower rigidity at both ends of the backbone chains. The monomers at the ends of a chain experience reduced steric repulsion from the adjacent grafts. Shorter branching, i.e., smaller LS, yields more homogenous backbone rigidity and diminished end effects as the long-range Coulomb forces gain in importance Overall, smaller LS and larger f result in greater rigidity of the backbone chain for f ≤ 1/2. These two main features, smaller LS and higher f, resemble the key characteristics of short side chain (SSC) ionomers compared to longer side chain ionomers such as Nafion. At the single-chain level, SSC ionomers are expected to maintain greater rigidity than their longer side chain counterparts. Localization of counterions We explore how the length and fraction of pendant side chains as well as the local rigidity of the backbone influences the localization of monovalent counterions. Figure 6a shows the radial distribution function (RDF) of anionic headgroups and counterions, gh-c(r) for LS = 3 at various f. For an isotropic system, an RDF gA-B(r) describes the probability of finding a particle A in a specific radial distance, r, of a reference particle B. It is defined as84

N

()

g A−B r =

N

A B V ∑∑ δ r − rij 4πr 2 N A N B i=1 j=1

(

)

,

(8)


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where NA and NB are the numbers of A and B particles in the simulated system, respectively, and the triangular brackets around the δ-function indicate averaging over the equilibrium system trajectory. Greater f increases the surface charge of the chain and creates a stronger surface potential, thereby causing greater Coulombic attraction of the counterion, a trend that was found in other simulation studies as well [15,65,79]. Figure 6b demonstrates the effect of LS on the localization of counterions for f = 1/2. Counterions have a stronger affinity to anionic headgroups for ionomers with shorter side chains. The enhanced surface charge density at lower LS is responsible for the creation of stronger surface potential and greater localization of counterions. The changes in the localization of counterions in the vicinity of anionic headgroups as a function of f and LS can also be seen in the snapshots depicted in Figure 3. Figure 6c shows how the intrinsic rigidity of the backbone chain influences the localization of counterions to anionic groups for f = 1/2 and LS = 3. Affinity of counterions to the headgroups decreases as the bending rigidity increases. Greater kθ results in more stretched backbone chains with farther separated side chains. This greater distance among charged moieties results in lower effective surface charge of the chains and thus, weaker electrostatic attraction to the counterions. Counterion localization exhibits a smaller sensitivity to kθ than to f and LS. These observations highlight the nontrivial effects of architectural variation in comb-polyelectrolyte chains on the distribution and possibly the dynamics of mobile counterions. Effect of electrostatic interaction strength In this section, we explore how the change in strength of electrostatic interactions introduced through the variation in λB influences Rg. Figure 7a shows Rg plotted over λB for f = 1/2 and for different LS. Several trends are noticeable in this plot. First, a nonmonotonic dependence of Rg on


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the λB is evident for all LS. Secondly, the sensitivity of the nonmonotonic dependence decreases as LS increases. A nonmonotonic dependence of Rg on λB was reported in Refs.[19,20,61,62] for linear polyelectrolyte chains. It was attributed to the initial expansion of the chain due to enhanced electrostatic repulsion upon increasing λB. Further increase in λB results in strong localization of counterions in the vicinity of the chain that screens anionic charges and lowers their effective electrostatic repulsion. In the regime of large and small λB values, i.e., λB = 0.5 σ and λB ≥ 9 σ, side chains with larger LS assume a greater radius of gyration. This behavior can be rationalized through the dominance of excluded-volume effects in the case of weak or screened electrostatic interactions. In the midrange of λB = 1 to 6 σ, enhanced Coulombic repulsions along with shorter distance between the charged units of chains with small LS results in higher Rg. Thus, we can see that increasing the side chain length does not always result in enhanced chain size and this behavior is sensitive to the strength of electrostatic interactions. Increasing LS results in weaker sensitivity of Rg to λB, which can be attributed to the reduced impact of Coulomb interactions. Figure 7b shows how comb-polyelectrolytes (f = 1/2, LS = 3) with different backbone rigidity behave upon variation of λB. We can see the nonmonotonic dependence for chains with different kθ values. However, the sensitivity to the change in Bjerrum length, and the extent of chain ex-

pansion and shrinkage, diminishes for more rigid backbones. Ou et al.62 also observed similar behavior and reported that increasing the backbone rigidity diminishes the dependence of Rg on λB. Effect of counterion valency The results for the change in Rg upon change in ZC for chains of different f and LS are shown in Figure 8. Figure 8a shows the change in Rg of comb-polyelectrolytes with different fraction of


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side chains at fixed LS = 3. Increasing the valency of counterions, from ZC = 1 to ZC = 2 and further to ZC = 3 leads to a decrease in Rg. Increasing the counterion valence yields stronger correlation of counterions with charged headgroups. Enhanced counterion localization triggers an effective charge screening that results in decreased electrostatic repulsion. Weakened Coulomb interactions along with attractive short-range hydrophobic effects causes the comb-polyelectrolyte chain to collapse to a neutral-like structure. The reduction in Rg is more pronounced for chains with higher f, owed to the prevailing electrostatic contribution to Rg at high f (Figure 2c). Figure 8b shows the effect of an increase in ZC for f = 1/2 and different LS. The extent of the decrease in Rg is more pronounced with ZC upon decreasing LS. At ZC = 1, shorter branching yields greater Rg due to enhanced Coulombic repulsion. This trend is opposite of that observed at ZC = 2, 3 where excluded-volume effects primarily control the backbone Rg and higher LS yields larger backbone Rg. Conclusions The presented coarse-grained molecular dynamics study scrutinizes the effects of different parameters on the conformational properties of hydrophobic comb-like polyelectrolytes. Observed trends can be rationalized through the fine interplay of hydrophobic interactions of individual backbone units, excluded-volume effects of side chains, and Coulombic interactions among charged moieties. It is observed that increasing the fraction of pendant side chains increases the radius of gyration and rigidity of comb-polyelectrolyte chains, a trend similar to the neutral counterpart. For high density of side chains, increasing the length of side chains leads to backbone stretching. For lower density of side chains, increasing side chain length yields a smaller backbone radius of gyration, resulting in the formation of collapsed globular structures. Counterions possess stronger affinity to anionic moieties for shorter side chains and higher grafting den-


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sities. Chain radius of gyration is shown to demonstrate a nonmonotonic dependence on the strength of electrostatic interaction. In the regime of weak Coulomb interaction steric effects of side chains control the radius of gyration and backbone conformation. Multivalent counterions are shown to decrease the radius of gyration through the screening of anionic charges. Qualitative agreement in the conformational trends with both experimental and similar theoretical studies highlights the ability of the generic coarse-grained model in capturing the subtle conformational behavior of comb-polyelectrolytes and ionomer chains. Conformational understandings at the level of single chain are particularly essential for characterization of ionomer solutions and the underlying process of self-assembly. Finally, we hope that this study will trigger further fundamental and single chain studies of fluorinated and hydrocarbon ionomers. Author information Corresponding Author *E-mail: [email protected]. Notes The authors declare to competing financial interest. Acknowledgment The work reported in this manuscript was performed with the financial support from an NSERC Discovery Grant, entitled Materials for Electrochemical Energy Conversion: From Fundamental Physics to Advanced Design. Simulations were performed using the Grex computational facility at the University of Manitoba, provided through WestGrid and Compute Canada.


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Insert Table of Contents Graphic and Synopsis Here.


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Figure 1. Schematic illustration of comb-polyelectrolyte architecture. Backbone and side chain hydrophobic beads are shown in blue and green, respectively. Red beads represent anionic headgroups and yellow beads represent the counterions.


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Table 1. List of system paramters and their baseline values along with the explored range.

Parameter

Baseline value

Parameter range explored in this work

LS

3

1-7

f

1/2

1 - 1/5

λB

3.0 σ

0.5 - 12 σ

kθ

0 kBT

0 - 45 kBT

ZC

1 e-

1 - 3 e-


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Figure 2. Backbone Rg for (a) neutral comb-like polymers, (b) comb-polyelectrolyte chains, and (c) pure electrostatic contribution to the backbone Rg.


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Figure 3. Equilibrated comb-polyelectrolyte chains at different values of f and LS. Color-coding is the same as Figure 1.


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Figure 4. Effect of

kθ on backbone Rg. Comb-polyelectrolyte: (a) LS = 3 (b) f = 1/2. Neutral counterpart: (c) LS = 3 (d) f = 1/2. All

the plots share the legend of (a).


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Figure 5. Local persistent length for the comb-polyelectrolyte and the neutral branched polymer at different side chain fractions and lengths.


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Figure 6. Radial distribution functions between anionic headgroups and counterions for (a) f = 1-1/5 and LS = 3 (b) f = 1/2 and LS = 1-7 (c) f = 1/2, LS = 3, and

kθ = 0 - 45 kBT.


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Figure 7. Rg values as a function of Bjerrum length (a) f = 1/2 and LS = 1-7 (b) f = 1/2, LS = 3, and

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kθ = 0 - 45 kBT.


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Figure 8. Values of square root of backbone radius of gyration at different ZC (a) f = 1-1/5 and LS = 3 (b) f = 1/2 and LS = 1-7.


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Conformational Properties of Comb-Like Polyelectrolytes: A Coarse

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