Surfactant-Mediated Polyelectrolyte Self-Assembly in a Polyelectrolyte

Article pubs.acs.org/Macromolecules

Surfactant-Mediated Polyelectrolyte Self-Assembly in a Polyelectrolyte−Surfactant Complex Monojoy Goswami,*,†,‡ Jose M. Borreguero,§ Philip A. Pincus,∥ and Bobby G. Sumpter†,‡ †

Center for Nanophase Material Sciences, ‡Computer Science and Mathematics Division, and §Neutron Data Analysis & Visualization, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States ∥ Department of Materials Science, University of California, Santa Barbara, Santa Barbara, California 93106, United States

ABSTRACT: Self-assembly and dynamics of a polyelectrolyte (PE)−surfactant complex (PES) are investigated using molecular dynamics simulations. The complexation is systematically studied for five different PE backbone charge densities. At a fixed surfactant concentration the PES complexation exhibits pearl necklace to agglomerated double spherical structures with a PE chain decorating the surfactant micelles. The counterions do not condense on the complex but are released in the medium with a random distribution. The relaxation dynamics for three different length scales, the polymer chain, segmental, and monomer, show distinct features of the charge and neutral species; the counterions are fastest followed by the PE chain and surfactants. The surfactant heads and tails have the slowest relaxation due to their restricted movement inside the agglomerated structure. At the shortest length scale, all the charge and neutral species show similar relaxation dynamics confirming Rouse behavior at monomer length scales. Overall, the present study highlights the structure−property relationship for polymer−surfactant complexation. These results help improve the understanding of PES complexes and should aid in the design of better materials for future applications.

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INTRODUCTION Polyelectrolyte (PE)−surfactant complexes (PES-C) are the subject of major scientific research due to the general interest in understanding the fundamental physics of self-assembly at the nanoscale.1−4 The PES-C also have a broad range of applications in biological materials,5−9 drug delivery,10−12 surface modifications,13 colloid stabilization,1 and flocculation14 and numerous consumer healthcare products. The thermodynamic stability of PES-C is achieved by a balance of three different forces: (1) entropic contributions to the free energy from released counterions of the PE chain, (2) electrostatic interactions between the PE charges with the oppositely charged surfactant heads, and (3) hydrophobic interactions between the alkyl surfactant tails and the hydrophobic backbone of the PE chains. The formation of equilibrium structures in PES-C is controlled by a large number of parameters, e.g., PE chain length, dielectric constant of the medium, solvent quality, salt and surfactant concentration, charge distribution on the PE chain, etc. With so many parameters at hand, it is difficult to develop a comprehensive © XXXX American Chemical Society

theory of PES-C. Moreover, the long-range electrostatic interactions pose a rather difficult challenge for developing an exact analytical model. These challenges have throttled the progress toward thorough understanding of PES-C. In this work, we explain the effect of polymer backbone charge density on the structure and dynamics of a polymer−surfactant complex. Polyelectrolytes with oppositely charged surfactant molecules form highly stable, stoichiometric, water-insoluble complexes.15,16 Considerable effort has been dedicated toward understanding the structure and dynamics of polyelectrolytes with or without salt using theory/simulation and experiments. For example, Ou et al.17 studied uniformly charged flexible polymers with a variation of salt concentrations and electrostatic strengths. The same research group also investigated the role of PE chain segments, counterion valency, and temperature Received: September 30, 2015 Revised: November 16, 2015

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DOI: 10.1021/acs.macromol.5b02145 Macromolecules XXXX, XXX, XXX−XXX

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with variable backbone charge density in a surfactant dominated environment and a salt-free implicit solvent condition. The PE surfactant mixture forms a highly stable PES-C phase in dynamic equilibrium. Here, we focus on the effect of the variation of the PE backbone charges on the structure and dynamics of surfactant micelles and the scaling behavior of the PE-charge state on structural properties. The relaxation dynamics at different length scales is also investigated. The article is organized as follows: in the Method section, we describe the simulation model and its validity in the present context. In the Results and Discussion section, we present an explanation of the structure observed based on the contribution from different species in the PES-C. This is followed by details about the relaxation dynamics at different length scales and a representation of structure−property relationship in regard to variable PE charge states. In the Conclusion section we summarize the major findings and provide a brief discussion of future opportunities in PES-C research.

on the various properties of polymer structure and counterion distribution.18 Complex formation between a single PE chain and surfactant molecules as a function of electrostatic strength, salt concentration, and surfactant tail length has been studied by von Ferber et al., which showed bottle-brush structures at low salt concentration exhibiting a nonmonotonic variation with increasing electrostatics.19,20 Uchman et al. studied the PES complexes using static and dynamic light scattering, smallangle neutron scattering, and small-angle X-ray scattering.21,22 They observed formation of crystalline PES cores in a mixture of poly(ethylene oxide)-block-poly(methacrylic acid) and Ndodecylpyridinium chloride (DPCl) surfactant in aqueous solution. Groot23 carried out an intensive mesoscopic simulation of polymer−surfactant complexation using DPD techniques. In that study the structures of the complexation as a function of surfactant concentration were studied, and it was concluded that the structure formation is dominated by two distinct modes, either continuous adsorption of surfactant on the PE chain or polymer−surfactant aggregation via discrete micelles, depending upon interactions. Wallin et al. 24 investigated PE−surfactant complexes using Monte Carlo simulation to understand the structure of the PES-C and changes in thermodynamic properties. Their results focused on the flexibility of the PE chain, linear charge density on the backbone,25 and surfactant tail length.26 There is also a vast literature on the effect of salt concentration, polymer chain length, and counterion size and valency on PE−surfactant complexation.17−19,27−29 Experimental investigation on the importance of the PE charge density in determining the cooperative activity of surfactant−PE complex formation has been carried out by Satake et al.30 as early as 1990. Recently there has been a renewed interest in the effect of PE backbone charge density on the structural and dynamical properties of PES-C both experimentally and theoretically. Li et al.31 established a semiempirical relationship for the strength of binding in PE−alkyl surfactant mixtures which follows a squared power law dependence. Association of oppositely charged surfactants with polyelectrolytes, their phase behavior, and dynamics have been investigated using small-angle X-ray diffraction, cryo-TEM, and PGSE-NMR32 as a function of varying charge density on PE chains, demonstrating increased steric repulsion with higher charge density. Variation of PE charge density was also explored by simulation.33 This work primarily looked into the effect of salt on the morphology of the complex, revealing extended structures for lower charge density and collapsed chain conformation for higher charge density depending on hydrophobicity or hydrophlicity of the PE chains.33 He et al.33 showed that the collapse of PE chains can also be achieved by the addition of optimum amount of surfactant which neutralizes the opposite charges on the PE backbone.34 The electrostatic self-assembly of block polyelectrolyte chains have also been explored using DPD simulations35 where it was found that the liberation of mobile counterions increases the entropy. This along with the solvent quality and incompatibility of different blocks plays an important role in self-assembly although the electrostatic interactions are crucial for self-assembly in block polyelectrolytes. Despite these efforts focused on understanding the thermodynamic and dynamic properties of PES-C, there is still a lack of systematic theoretical investigation on the effect of the PE backbone charge density. To address the complex structure−property relationship in PES-C as a function of PE charge density, we have simulated a single polyelectrolyte chain

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METHOD The LAMMPS molecular dynamics package36 was used to study the dynamics of a model system composed of a polyelectrolyte chain with different charge states on the backbone at a fixed surfactant concentration. In these simulations, a sodium dodecyl sulfate (SDS) type surfactant is modeled as a 12-mer polymer with 11 neutral tail groups and one anionic headgroup. A total number of 1000 surfactant molecules are used in a dilute solution of a single cationic PE chain under implicit solvent conditions. The PE chain length is 1000 monomers long, and a set of five different simulations are studied for 10%−50% of backbone charges with 10% increase in each step. To understand the effect of similar monomer sized systems, the monomer sizes (radius = σ) and mass (mi) are kept fixed. The simulations are performed within a periodic box of volume V = 100 × 100 × 100σ3 at number density ρ* ≈ 0.014σ−3. The critical micelle concentration (CMC) for Lennard-Jones surfactants with anionic/cationic heads was estimated to be within ρ*CMC = 10−5σ−3 to 10−6σ−3 range37,38 at T* = 1.0, where T* = kBT/ϵLJ and kB is the Boltzmann constant. Therefore, the number density ρ* ≈ 0.014σ−3 for our simulations is well above the CMC, allowing the system to form surfactant micelles. The highly dilute solution allows us to carefully investigate the dynamical transition associated with the PE−surfactant complexation. To preserve overall charge neutrality of the system, equal numbers of positive (+q for PE backbone charges) and negative (−q for surfactant heads) counterions are randomly distributed in the simulation box. The charges interact via explicit Coulomb interactions. The Kremer−Grest bead−spring model39 was incorporated to represent the polymer chain and surfactant molecules. The monomer beads are bonded by finitely extensible nonlinear elastic (FENE) springs Uij = − 0.5kR 0

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⎡⎛ ⎞12 ⎛ ⎞6 ⎤ ⎡ ⎛ rij ⎞2 ⎤ σ σ ⎢ ⎥ ln 1 − ⎜ ⎟ + 4ϵLJ⎢⎢⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ ⎥⎥ + ϵLJ ⎢⎣ ⎥ R r r ⎝ 0⎠ ⎦ ⎝ ij ⎠ ⎦ ⎣⎝ ij ⎠ (1)

where R0 = 1.5σ is a finite extensibility, the spring constant k = 30ϵLJ/σ2, and σ is the monomer diameter. rij is the distance between two monomers, and ϵLJ is the energy parameter. The first term is the FENE potential with a maximum extent of the bond length R0. The second term is the Lennard-Jones (LJ) B


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Macromolecules potential with a repulsive cutoff rcut = 21/6 which creates an excluded volume interaction so that no two monomers can occupy the same position. Typically, ϵ = 0.35 refers to the θstate for neutral chains. We choose the polyelectrolyte energy parameter ϵLJ = 1.0. In this model, the surfactant neutral interactions are twice as strong as the neutral polyelectrolyte, and hence ϵLJ for surfactants is set to 2.0, representing strongly hydrophobic tails. The energetic interactions between the nonbonded monomers are modeled by a truncated and shifted LJ potential represented by the second term in eq 1. The interactions between the hydrophilic neutral monomers of the PE backbone is modeled by a repulsive LJ force. The hydrophilic neutrals of PE backbone and hydrophobic tails of surfactants repel each other. The anionic surfactant head and the cationic charges on the PE backbone are attractive due to the strong Coulomb interactions, given by UCoulomb (rij) = qiqj/ ij 4πε0εrij, where ε = 1.0 is the dielectric constant, ε0 is the vacuum permittivity, and qi, qj are the interacting electronic charges. The Coulombic interactions depend on the charge states of the PE backbone. We varied the charge states keeping the surfactant concentration fixed. The temperature for the system is set at T* = 1.0. In a dilute system with smaller aggregates, a short-range pairwise Coulomb interaction is sufficient to capture the effect of electrostatics. Therefore, instead of the computationally expensive Ewald summation technique, we incorporated a pairwise Coulomb interaction with a cutoff at 12.5σ, which is equivalent to a direct Coulomb energy calculation between two pairs within the cutoff distance of 1/8 of the box size. It should be noted that varying the PE charge states and number of negative counterions affects the dielectric constant of the solution,6 giving rise to a change in the Bjerrum length, lB = e2/4πε0εkBT, where e is the electronic charge. We will not elaborate on the effect of PE charge states on ε and lB as we are focusing on the structure−dynamics relationship as a function of PE charge states. For the MD simulation the reduced units are defined as U* = U/kBT and r* = r/σ.

Figure 1. Snapshots of the polyelectrolyte surfactant complexes for 100, 300, and 500 charge states of a single polyelectrolyte with a chain length of 1000 and 1000 surfactants with chain length of 12 in the central simulation cell of length L = 100σ. The blue spheres are the PE chain and the lime-color spheres are the charge sites. The yellow and red spheres are surfactant tail and head, respectively. The scattered tiny green and light blue dots inside the box are negative and positive counterions, respectively, plotted in smaller size for clarity. The PE chain appears to be broken, and broken surfactant micelles can be seen around the edges of the box. This is an artifact of presentation of simulation data due to the use of periodic boundary conditions (PBC). We apply PBC on all directions that allows the chain and surfactant coordinates to go out of the central simulation box. We wrap the coordinates back to the central simulation box, thereby giving a broken cluster and/or PE chain appearance. All the chain molecules are part of one single PE chain, and clusters are part of the single large micelle with the PE chain decorating the micelle at higher charge states.

form a micellar core while the PE chain decorates the surface, analogous to a highly stable core−shell structure where the PE chain forms the shell and the surfactant micelle forms the core. The self-assembly of the PE−surfactant complex into the core− shell structures occurs when the system attains a critical ratio, Z, defined as the ratio between the amount of the charges on the PE backbone to the total surfactant charge.41 In the present case, Z varies from 0.1 to 0.5. At Z > 0.3, the surfactant micelles aggregate to a large cluster, eventually forming a single large complex at a charge state of 500 (Z = 0.5). One noticeable feature in all the agglomerated states is the presence of more than one surfactant micelle separated by the PE chain. At NCharge = 500 (Figure 1e) two surfactant micelles are separated by a one layer shell of the PE backbone. This can be attributed to the high electrostatic surface energy of the anionic surfactant micelles leading to an attraction to the cationic PE chain which forms a layer between the boundary of two surfactant micelles. The PE chain decorating the surfactant micelles is similar to a Langmuir−Blodgett film42 where the cationic PE chains are attached to the anionic heads of the surfactants on the surface. In small-angle neutron scattering (SANS) experiments for poly(ethylene oxide) (PEO) and sodium dodecyl sulfate (SDS), Chari et al.43 also observed similar structures, but suggested that the polymer was more like a swollen cage instead of a pearl-necklace structure. We also observe swollen cage structures at higher PE backbone charges where the PE chain forms a cage-like structure surrounding the surfactant micelle.

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RESULTS AND DISCUSSION A series of MD simulations were performed for a system containing a single flexible PE chain with 100, 200, 300, 400, and 500 charges (NCharge) on the backbone. The number of negative counterions is varied accordingly to preserve overall charge neutrality. The simulations were carried out in the canonical ensemble (NVT) with time steps, Δt* = Δt/(miσ2/ εLJ)1/2 = 0.012 using a Langevin thermostat for temperature control. All simulations were run for 50 million LJ time steps to equilibrate; i.e., the structural properties do not change in successive runs. We emphasize that the system is in dynamic equilibrium; therefore, there are sufficient fluctuations to collect statistics for the calculations of dynamical properties. The analysis are based on the statistics of another 10 million times steps to achieve better statistics of the estimated thermodynamic and dynamic quantities. In Figure 1, we show snapshots of the PE−surfactant complexes at five different PE charge states, NCharge = 100−500. Charges on the PE chain causes agglomeration with the surfactant and stronger agglomeration occurs with an increase in the number of charges on the backbone. The lower charge states reveal pearl-necklace structures as can be seen in the snapshots. Similar structures have also been observed recently for poly(vinylpyrrolidone)−SDS and poly(diallyldimethylammonium chloride)−SDS systems.40 The surfactants also C


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observed in PE solutions even without surfactants. For example, in solution, PE charges and the oppositely charged counterions form a temporary f luctuating dipole, and the strong dipolar attraction leads to collapsed PE chains.29,46,47 Therefore, the two physical processes can be related to each other through the electrostatics whether that originates from the charges on the PE backbone as in the present case or the counterions in the case of normal PE solution. We will elucidate this further by incorporating a scaling analysis on structural parameters. For a freely jointed chain the end-to-end distance is Re2 = 6Rg2. In Figure 2, we plotted 6Rg2/N (blue diamond) and the calculated Re2/N (green circle) to understand the limits of PE chain conformations. As can be observed, from the 200 charge state onward, both the values fall on top of each other. The increase in charge density on the backbone with large number of surfactants greatly reduces the electrostatic effect by neutralizing the charges, leaving the PE chain to behave as a freely jointed neutral chain. The calculated Rg2/N and Re2 values are greater than the ideal chain values, Nb2/6 and Nb2 ≈ 0.21 and 1.25, respectively, for a chain of length, N = 1000 and b = 1.12. The anionic headgroup of the surfactant effectively neutralizes the charges on the PE chain at higher charge states, thereby reducing the rigidity of the PE chain even further.48 In Figure 3, we show the dependence of Rg on the PE backbone charge states, NCharge, of the polyelectrolyte. At the

To further analyze the structural properties, the radius of gyration (Rg) and the end-to-end distance (Re) dependence on the charge states of the PE chain are examined. Figure 2 shows

Figure 2. Dependence of radius of gyration (Rg) and end-to-end vector (Re) on the various charge states on the PE backbone for N = 1000. Green circles represent Re2/N, and red squares represent Rg2/N. The blue diamonds represent 6Rg2/N to show the expected behavior of an random-walk freely jointed chain of similar chain length. The dashed lines are for guide to the eye only.

the ratio of the square radii to the degree of polymerization of the PE chain. The Rg2/N (red squares) and Re2/N (green circles) show similar dependence with the number of charges on the chain except at low charge states, NCharge = 100 and 200. The decrease in NCharge leads to an overall weaker electrostatic interaction that results in chain swelling at low charges states. At higher charge states, the electrostatic interactions between the cationic charges and anionic surfactant is higher and hence the agglomeration is higher, leading to a collapsed chain. On the other hand, the strong electrostatic interactions at higher NCharge that helps create the Langmuir−Blodgett film reduces the surface energy42 of the agglomerated complex. The decrease in surface energy is responsible for the near-spherical shapes observed in Figure 1. Hajduova et al.44 used small-angle X-ray scattering and scanning transmission electron microscopy (STEM) experiments on a double-hydrophilic cationic polyelectrolyte and anionic surfactant, (poly[3,5-bis-dimethylaminomethyl-4-hydroxystyrene]-b-poly(ethylene oxide) (NPHOS-PEO) and sodium dodecyl sulfate (SDS)), to explain the electrostatically stabilized core−shell structures. They found that that these structures form as a result of anionic surfactant (SDS) binding to the cationic sites (PEO) of the polyelectrolytes.44 Similar structures are observed in our simulations, and these morphologies are ubiquitous in PE complexes including biomolecular complexes. For example, DNA complexes with ionic liquid surfactants ([C12mim]Br)34 shows an increase in charge ratio (surfactant to the PE charge) leads to an increase in the size of the PE−surfactant agglomerate and hence to a collapse of the PE chain observed from a rapid decrease of Rg. Additionally in polyelectrolyte solutions, charge doublets on the surface create an apparent temporary dipole effect on the polyelectrolyte chains due to density fluctuations of the counterions.45 In the present PE− surfactant complex, the fluctuations of the anionic surfactant causes an effective fluctuating dipole. The large number of dipole−dipole interactions gives rise to strong intrachain attraction leading to chain collapse, making the PE−surfactant complex densely packed as can be seen in Figure 2 (red squares around 1.0σ). Similar structural characteristics have been

Figure 3. A log−log plot of the Rg/σ vs NCharge used to determine the scaling exponent. The data are shown as black circles. The red line is a least-squares fit through the data with an exponent −0.194, and the −0.25 slope is shown as a blue line. The dotted red line at the low charge state regime is not a fit but is provided as a guide to the eye. “Only” two points for low NCharge values cannot be reasonably fitted for reliability/accuracy reasons.

higher charge states, NCharge > 200, the scaling law for PE chain as a function of charge states, Rg ∝ NCharge−α, shows a power law dependence with the exponent, α ≈ 0.194 ± 0.04. The decay of Rg can be approximated as Rg ∝ NCharge−1/4. For the lower charge states we could not fit the data to estimate the exponent due to the lack of data points (only two data points below NCharge = 200). The rigidity of the polyelectrolyte chain is an essential parameter in polyelectrolyte solutions and is strongly influenced by the PE concentration in the solution.48 Early SANS studies by Nierlich et al.49,50 on PE chain conformation as a function of polyion concentration, cp, have worked out the scaling laws at different concentrations. At high concentration the scaling law is represented by Rg ∝ cp−1/4. A similar scaling law was obtained by simulation51 for a polyelectrolyte solution with salt. The weak dependence on the concentration implies a more flexible polymer. In the present study, we also observe D


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Figure 4. Radial distribution function for (a) NCharge = 100 and (b) NCharge = 500. The black and red lines represent g(r) of only the negative and positive counterions. The green and blue lines represent PE charge and surfactant head g(r) with their corresponding counterions. The magenta line represent g(r) of PE charge with the surfactant heads.

Figure 5. Relaxation time as a function of charge states on the polyelectrolyte chains at long, medium and short length scales. (a) Q = 0.65σ−1, (b) Q = 1.3σ−1, and (c) Q = 6.48σ−1. The legends in (a) represent the color/symbol code of (b) and (c) the plots.

change in PE charges states in PES-C and the change in polyion concentration in a PE solution, essentially have an analogous physical mechanism that is affected by the total number of charges present in the solution, solely due to electrostatic interactions. Nierlich et al. have also shown Rg ∝ cp−1/2 at low concentration regime. As discussed earlier we did not reproduce the low concentration scaling behavior due to the lack of adequate data in that regime.

weak dependence on the PE charge states despite the present of the surfactants. The flexible limit of the PE chain was also observed in Re in Figure 2 where the PE chain behaves as a freely jointed chain at NCharge > 200. It is important to note that we have not altered the polyion concentration; alternatively, we varied the charge states on the PE backbone. Thus, the scaling behavior reinforces our claim in Figure 2 regarding the comparison between variation of charges on the PE backbone and polyion concentration in a PE solution. Therefore, the E


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Figure 6. (a, b) Mean-square displacement, ⟨|r⃗(t) − r⃗(0)|2⟩ (⟨Δr⃗2(t)⟩), after subtracting the center-of-mass motion for two different PE charge states, NCharge = 100 and 500. (c) Diffusion constant calculated using the Einstein relation D* = (1/6)limt→∞⟨Δr2⃗ (t)⟩. PE chain D* is not shown in the figure due to the lack of reliability of their MSD data in the diffusive regime. (d) Minimum of potential of mean force for all the species as a function of PE charge states. For all the plots the different species are shown in the legends. The lines are simply to guide the eye.

The charge/counterion distribution along the PE chain and surfactant head were analyzed by investigating the pair correlation function for the different groups of charges shown in Figure 4a for NCharge = 100 and Figure 4b for NCharge = 500. The negative and positive counterions are mostly dispersed in the system as can be seen from the black and red curves of the g(r) plot. No peak is observed for the counterions regardless of the charge states which indicates there is no substantial agglomeration. The counterions and the NChargeg(r) (green lines) show a weak peak at 4.1σ, far away to form charge doublets (1.8σ) and hence cannot condense on the PE backbone. In Figure 4b, the peak shifts closer at 1.8σ for higher charge states, NCharge = 500, and the peak height also increases, representing an increase in the counterion density in close proximity to the PE chain. The second small peak in Figure 4b (green lines) is the position of the next charge sites of the PE chain, indicating there is a thin interfacial layer of counterions that is clustered in the micellar aggregation seen in Figure 1e. The counterions are mostly released with only a small fraction of them near the PE chain. The surfactant head and the counterions (blue lines) show a higher peak height at shorter distances, indicating a larger accumulation of positive counterions around the surfactant head. It appears that the

positive counterions reside next to the surfactant head within the same interfacial charge layer. The surfactant head and PE charge (magenta lines) show strong agglomeration that increases as much as 5 times with increase in charge states from NCharge = 100 to 500. Recently, Juarez et al.52 performed dissipative particle dynamics (DPD) simulations and obtained similar results for polyelectrolyte complexes. For equally spaced charges, the charge separations on the PE backbone for NCharge = 100 and 500 are 10 and 2 monomers, respectively. This amounts to 10σ and 2σ distances. However, the peak position, r = 1.5σ does not change, reflecting that the PE charges reside close to the surfactant head and decorate the surface of the micelles. For headgroup dominated PE−surfactant complexes, the micellar phase is more likely than that of bottle-brush conformations as have been observed previously by mesoscopic simulation of polymer−surfactant aggregation.23 To this end, we consider the dynamical behavior of different species of the PES-C. In a PES complex, relaxation can occur on a variety of length scales, from the relaxation of monomers and molecular segments to relaxation of entire chains. We therefore investigate the structural relaxation time, τ*(τ/τ0), derived from the self-intermediate structure factor, S(q,t). The S(q⃗,t) was calculated with the Sassena package.53 The F


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Macromolecules orientation averaged S(q⃗,t) was then obtained by averaging it over 1000 randomly chosen q⃗ vectors of modulus, |q⃗|. In these highly dilute systems, the relaxation time is defined by the time required for S(q,t) to reach 1/e times the maximum, S(q,t*=0). Figure 5 shows τ/τ0 for three different length scales, q = 0.65σ−1 (Figure 5a), q = 1.3σ−1 (Figure 5b), and q = 6.48σ−1 (Figure 5c) corresponding to long, medium, and short length scales. Associated with these q values, the length scales (l) are 9.67σ, 4.83σ, and 0.97σ, respectively, representing the entire chain to molecular segments to monomeric length scales. Except at short length scales, q = 6.48σ−1, an increase in relaxation time with increase in charge states was observed for all the species except the counterions. For the largest length scales (Figure 5c), only the tail monomer shows a slight enhancement of τ*. Polyelectrolyte theory predicts a decrease in PE relaxation time with an increase in polyelectrolyte concentration,6,54 cpolymer, and follows τ* ≈ cpolymer−1/2 scaling. The theoretical predictions have also been verified by previous MD simulations on semidilute and dilute PE solutions.47,55 From these works, it is clear that increasing charge density in the PE solution have a strong effect on the relaxation time. In our simulations, we vary the charge density by replacing c with NCharge. The results show increase in τ as NCharge increases as opposed to the decreasing relaxation time with NCharge. Therefore, these results do not correlate exactly with semidilute or dilute polyelectrolyte solutions because of the presence of the surfactants. In our case, the increase in PE charge states leads to stronger electrostatic attractions between the charges and surfactant and hence slow the relaxation process. This is akin to an increase of monomeric friction coefficients with increasing polymer concentration.55 In the present case the increase in electrostatic interactions with higher NCharge can be related to an increase in monomeric friction. The absolute values of relaxation time decreases an order of magnitude as the length scale decreases, as can be seen from Figure 5a−c except for the counterions (Figure 5d). The reason for little or no change in counterion relaxation can be related to the release of counterions from the chain backbone. As discussed earlier, most of these counterions are released from the agglomerated states and hence are free. The counterions move freely in a random manner which gives rise to a faster relaxation. At the longest length scale (Figure 5a), the surfactant tail/head groups show the slowest relaxation followed by the PE chain. Both the neutrals and charges of the PE chain relax at the same rate due to the high conformational entropy associated with the long PE chain of length, N = 1000. At the shortest length scale, l = 0.97σ (q = 6.48σ−1), the relaxation time does not depend on the charge states, and all of the species relax at the same rate. This length scale (l = 0.97σ) corresponds to the monomeric length scale (σ). The monomer beads impart friction for all the species with the same amount of mechanical drag with respect to the all the other species and with the implicit solvent, leading to the weaker motion for all the charged and neutral monomers. This can be explained from the Rouse model where the friction factor of the monomer beads are connected to the Gaussian entropy of the spring48,55,56 comparable to the short-length scale relaxation properties of polyelectrolyte gels at swelling equilibrium.57,58 Therefore, even in the PES-C, the relaxation dynamics of all the charge and neutral species at monomer length scales maintain the Rouse dynamics of a polyelectrolyte at short lengths.

Finally, we explored the effect of PE charge states on the translational motion of the different monomer species of the PE-complex. We plotted mean-square displacement (MSD) after subtracting the center-of-mass drift motion for the two extreme charge states 100 and 500 in Figure 6a,b. Figure 6c shows the diffusion coefficients, D*, calculated from the MSD using Einstein’s relation (shown in the figure caption). It can be seen from Figure 6a,b the counterions (blue and magenta lines) are the fastest and highly diffusive. The head−tail (black and red lines) and the PE chain−PE charges (green and orange lines) follow each other at long times. Both the positive and negative counterions (blue and magenta lines) also follow each other’s motion. The counterions motions are 2−3 orders of magnitude faster than the PE chain and surfactant for NCharge = 100 and 500, respectively. As discussed in Figure 4, the counterions are not condensed on the PE chain. The released counterions, as expected from the theory of flexible polyelectrolytes,18 move “freely” as Brownian particles despite being in the polyelectrolyte surfactant complex. The presence of the anionic surfactant heads acts as a local neutralizer for the PE charges. The cationic PE charge and anionic head electrostatic interactions are strong enough to compensate electrostatics attraction of the counterions to either the chain backbone or the surfactant heads, thereby releasing the counterions in the solvent instead of a Manning-type condensation.59 The effective dielectric constant is high, ϵ* = 1.0, which makes the Coulomb interaction strengths between the charge species weak, therefore the dominating interactions for the PE chain and the surfactants would be due to the hydrophobic interactions. That explains the 2−3 orders of magnitude decrease of the MSD for the PE chain and surfactants. The MSD of the PE charges (and also the PE chain) decreases an order of magnitude from NCharge = 100 to NCharge = 500 as they form larger agglomeration and hence a slower motion. The diffusion constant D* is shown in Figure 6c for all the species except the PE backbone charge and PE chain because the MSD data shown as green and orange lines in Figure 6a,b for these two species at long times does not reach the diffusive regime. The head and tail D* (black circles and red crosses) decreases as a function of NCharge. This can be explained in terms of the effect of net charge Q on the mobility of the charge species. For diffusion in a dilute solution, the electrophoretic mobility μE is inversely proportional to the net charge on the system and is represented by the Einstein−Smoluchowski relation D = μEkBT/Q. The electric field in this case represents the “local” electric field created by the constantly fluctuating charge cloud.60 Following this relation, we expect, as Q increases, the diffusion constant (D*) decreases which is observed in Figure 6c for the surfactant head and tail. The PE chain diffusion is not shown on this plot due to the unavailability of PE chain MSD data. In Figure 6a,b we observe a change in the overall motion of the PE chain with increasing charge density The MSD values at long time are reduced by an order of magnitude at NCharge = 500. The minimum (potential well) of the potential of mean force (PMF) is plotted in Figure 6d. We show the thermodynamic quantity, PMF, purposefully instead of the activation energy derived from the diffusion constant (D*) as the PMF can capture all the different species of the system from the structural information, whereas the activation energy cannot be estimated for some of the species due to lack of adequate MSD data. It is important to note that the PMF underestimates the G


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Macromolecules activation barrier for transport across flexible polyelectrolytes;61 however, an underestimate of the activation barrier will not alter the important physics in the present coarse-grained simulations. The minima of PMF shown in this plot are also representative of the minima of free energies with contributions from all the interactions, e.g., van der Waals, bond (FENE), and electrostatic. Since the Coulomb energy is the strongest for the charged species, most of the PMF contribution for the head and PE charges comes from electrostatics. The difference in free energies between the counterions (blue and magenta squares) and their respective charges (black and green squares) is noticeably higher compared to the positive and negative counterions and head-charge free energies. This results in the strong agglomeration between the surfactant and PE charges, thereby releasing the loosely bound counterions, consistent with the generally accepted theory of a favorable energy landscape for counterion release to instigate polyelectrolyte− surfactant aggregation.62 Surfactant heads, tails, PE charges, and neutrals show large negative U(r)min with PE neutrals (orange squares) showing as low as −8kBT for NCharge = 100. The large negative free energies stipulate a relatively stable agglomerated phase for these species. As NCharge increases, the PMF of all of these four species reaches around −5kBT. Since these four species show small variation in free energy minimum at the highest PE charge states, it establishes the stability of the agglomerated PE complex.

time on charge states, except at the lowest length scale. The surfactant head and tail, being in the agglomerated micellar form, always show the longest relaxation time. The PE chain, being on the surface of the micelle, shows intermediate relaxation time. The relaxation time drops by an order of magnitude with each change of length scales, i.e., from PE chain to segmental to monomer length scales. At the longest length scale, relaxation for all the species is decoupled except for the “free” counterions. At the shortest length scales (q = 6.48σ−1) all the species show nearly identical relaxation time. This is due to a frictional drag at the monomer length scale that is nearly equal for all the monomers (as the monomer sizes are the same). This is comparable to Rouse dynamics where the friction factor of the monomer beads are connected to the Gaussian entropy of the spring.48,55,56 This is also comparable to the short-length scale relaxation properties of polyelectrolyte gels at swelling equilibrium.57,58 This implies that the relaxation dynamics at the monomer length scale in PES-C could be similar to the relaxation dynamics of polyelectrolyte solution. With an increase in charge states the PE chain and surfactant both show decreased relaxation dynamics due to strong aggregation of the PE chain and surfactant. Overall, these results agree well with earlier studies that demonstrated the presence of two important driving forces in PE complexation: (i) electrostatic interactions and (ii) the favorable change in counterion entropy which is related to lowmolecular-weight counterion release. The present study considers only a single PE chain. However, it would be interesting to see the effect of large number of PE chains and polyelectrolyte−neutral block copolymer on the PES-C. We are currently looking into that direction. Also, it would be interesting to see the effect of surfactants on saloplastics64 composed of cationic and anionic PE chains that can be important in designing biocompatible materials. Finally, experiments in understanding the role of surfactants in PESC dynamics using neutron scattering would be immensely beneficial for the polyelectrolyte community due to its importance in drug delivery mechanism.

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CONCLUSIONS Molecular dynamics simulations were used to show that the PE backbone charge density affects the structure and dynamics of a PE surfactant complex. The structures show pearl-necklace to nearly spherical shapes with an increase in charge density on the polymer chain. For the highest charge state studied, the complex shows two connected nearly spherical structures with the PE chain decorating the surface of the surfactant micelles. The Rg of the PE chain follows the scaling law Rg ∝ cp−1/4 at higher concentration, resembling the scaling behavior of a classical polyelectrolyte solution,49 thereby indicating the surfactant acts like a solvent for the complex. The counterions (both positive and negative) do not condense on the PE chain or the surfactant; alternatively, they are released from the chain. The anionic surfactant neutralizes the cationic PE chain, thereby imparting little or no electrostatic attractions with the counterions. This results in the release of the counterions from the PE chain. The radial distribution function (Figure 4) indicates the presence of a thin interfacial layer of counterions, but it does not alter the structures of the PES complex. The increase in charge states lead to larger complex structures. Similar behavior has been observed in a variety of biopolymers. It is important to note that vesicle bilayers have been observed in small-angle neutron scattering (SANS) experiments on amphiphilic biopolymers;63 however, we have not observed these in our simulations. We believe this is due to the use of a “single” PE chain in our model simulation, whereas the SANS experiment used a large number of PE chains. The relaxation dynamics for three different length scales, monomer, segmental, and a stretched chain, were also investigated. The counterion’s relaxation is the fastest and does not depend on PE charge states or length scales. They always show relaxation at the monomer level which evidently comes from the “free” released counterion of size equivalent to a single monomer bead. On the other hand, the PE charges and surfactant heads show a strong dependence of the relaxation

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (M.G.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences, Materials Science and Engineering Division (MSED). This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract DEAC05-00OR22725. Research by M.G. and J.M.B. is supported by the Center for Accelerated Materials Modeling (CAMM) funded by the U.S. DOE, BES, MSED. This manuscript has been authored by UT-Battelle, LLC, under Contract DE-AC0500OR22725 with the U.S. Department of Energy.

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REFERENCES

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Surfactant-Mediated Polyelectrolyte Self-Assembly in a Polyelectrolyte

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