Article pubs.acs.org/Macromolecules
Molecular Connectivity and Correlation Effects on Polymer Coacervation Mithun Radhakrishna,† Kush Basu,‡ Yalin Liu,‡ Rasmia Shamsi,‡ Sarah L. Perry,‡ and Charles E. Sing*,§ †
Department of Chemical Engineering, Indian Institute of Technology Gandhinagar, Gandhinagar, Gujarat, India Department of Chemical Engineering, University of Massachusetts Amherst, Amherst, Massachusetts 01003, United States § Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana−Champaign, Urbana, Illinois 61801, United States ‡
ABSTRACT: Our ability to predict the thermodynamic phase behavior of a material system is a direct reflection on our understanding of the relevant interactions. Voorn−Overbeek (VO) theory, which combines Flory−Huggins polymer mixing with Debye−Huckel electrostatics, has been used to describe the associative liquid−liquid phase separation phenomenon known as complex coacervation since the 1950s. The long-standing utility of this theory stems from its simplicity coupled with its apparent agreement with physical systems. VO theory has also served as the starting point for a large class of field theories that predict similar phase behaviors. Recent work using new hybrid simulation methods demonstrates novel coacervate-driven self-assembly is strongly affected by molecular details. Liquid state (LS) theory suggests there are fundamental reasons for this observation and that agreement between VO and experiment is fortuitous. It is hypothesized that VO/experimental matching is due to a cancellation of errors arising from the neglect of monomer-level charge connectivity and excluded volume effects. In this article, we use Monte Carlo (MC) simulations to confirm the earlier predictions from LS theory. We directly observe effects related to connectivity-driven charge correlations. We also observe strong exclusion of salt from the polymer-rich coacervate phase, in direct opposition with VO theory and in near quantitative agreement with experimental results. Strikingly, a comparison of predicted phase diagrams using identical system parameters shows that VO overpredicts coacervate phase behavior and that previous agreement with experiments was likely due to the use of unphysical fitting parameters. This work provides new insights into the mechanisms driving complex coacervation and shows promise for predicting coacervate phase behavior based on resolving molecular level charge structure.
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ions.27,35−37 A number of theoretical investigations have sought to rationalize the experimental findings with some success.28 Classical Voorn−Overbeek (VO) theory, developed in 1957, treats coacervation by combining the mean field Debye− Hückel (DH) and Flory−Huggins (FH) theories.38−41 The resulting free energy of a mixture of polyelectrolytes and salt that can undergo coacervation FVO is predicted by VO theory28,38−40
INTRODUCTION Oppositely charged polymers and colloids can undergo associative liquid−liquid phase separation to form a polymerrich phase known as a complex coacervate.1−7 First observed experimentally in mixtures of gelatin and gum arabic by Bugenberg de Jong et al. in 1929,8 coacervate materials have continued as a subject of continuous inquiry. Interest in polymeric coacervates has heightened over the past decade, spurred by emerging applications in charge driven assembly.9−18 While initially useful as food modifiers due to their rheological properties19 and ability to be formed using natural polymers,6,20,21 polymeric complex coacervates now find use in a diverse array of applications such a tissue engineering,22 microencapsulation, sensors,17 drug delivery vehicles,14,15 and underwater adhesives.7,23 Despite extensive use in industrial and biological applications, a fundamental physical understanding of coacervation is still being developed. A broad parameter space has been explored with recent experiments investigating (for example) temperature,24−26 pH,25,27 ionic strength,28−31 polymer mixing ratio,25 chirality,32−34 and the identity and valency of the salt © XXXX American Chemical Society
ϕ FVOσ 3 = P ln(ϕP /2) + ϕS ln(ϕS /2) + (1 − ϕP − ϕS) VkBT NP × ln(1 − ϕP − ϕS) + χϕP (1 − ϕP ) −
2 π × [Γ(fP ϕP + ϕS)]3/2 3
(1)
Received: November 29, 2016 Revised: March 10, 2017
A
DOI: 10.1021/acs.macromol.6b02582 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules where ϕP is the total polyelectrolyte volume fraction, ϕS is the total salt fraction, NP is the polyelectrolyte degree of polymerization with a fraction of charged monomers f P, and Γ = e2/(4πϵσkBT) is a parameter related to the Bjerrum length that parametrizes the electrostatic interaction in an electrolyte solution with dielectric constant ϵ and charge diameter σ.28 The Flory χ-parameter is used to describe the noncharged, shortrange interactions.28,40 Agreement between VO theory and experimental measurements of complex coacervate phase behavior has been observed28 upon judicious choice of a number of fitting parameters. This has led to the widespread use of VO in both experimental and theoretical investigations. Despite this success, VO theory utilizes a number of wellknown approximations. In particular, the DH term is known to be inaccurate42 at the high salt concentrations typically considered in complex coacervate systems (0.5−4 M).28,36 Furthermore, this theory does not distinguish between the salt ions and the polyelectrolyte charges and thus does not account for chain connectivity except via the polymer mixing entropy.43,44 Theoretical work has sought to overcome the limitations of VO theory.44,45 Some sophisticated theoretical approaches systematically improve on VO using random phase approximation46,47 and field theoretic models16,48,49 to account for charge correlations. Field theoretic models capture charge fluctuations;49 however, small, molecular-length scale correlations remain difficult to resolve.16 An alternative picture of coacervation uses Manning−Oosawa counterion condensation as an approach to understanding coacervation.50−54 An entropic gain due to the release of condensed counterions can drive coacervation, which is consistent with thermodynamic data on coacervation.53,54 Unlike VO theory, counterion release requires that polymer charges are spatially nearby due to their connectivity along the chain backbone.52 The qualitative picture is well established,50 and recent efforts have made progress by combining counterion release and VO theory;55 however, counterion release has yet to be developed into a full theory of complex coacervation.44 Recently, the authors have also used new hybrid Monte Carlo/single chain in mean field methods to demonstrate that incorporation of molecular features into models of coacervatedriven block copolymer self-assembly can lead to qualitative matching with experiment. 56 This connection between molecular features and coacervation has been studied at a fundamental level using a polymer reference interaction site model (PRISM) calculation.43 PRISM is a liquid state theory that contrasts from prior approaches by directly calculating pair correlations and thus the macroscopic thermodynamics based on local charge structure,57,58 a general approach that has found success in a number of polyelectrolyte systems.59−62 PRISMbased coacervate theory shows that chain connectivity and excluded volume demonstrate opposing yet pronounced effects on coacervation.43 These theoretical results suggest that a cancellation of errors could lead to fortuitous agreement between VO (and related theories) and experiments.43 While PRISM theory can directly access correlation features, this implementation of the PRISM theory possesses weaknesses associated with the choice of closure relationship. The Debye− Hückel extended mean spherical approximation (DHEMSA) is the closure approximation used in this PRISM model due to its numerical stability and accuracy in electrolyte systems;63 however, its quantitative accuracy in PRISM remains untested. The results of this PRISM theory are in quantitative
disagreement with some literature values,28,43 likely due in part to the shortcomings of using the DHEMSA closure. Nevertheless, the qualitative trends remain intuitive and relevant and are important for understanding how molecular features dictate coacervation. It is therefore crucial that the disparities between PRISM and traditional VO theories be verified. This article assesses the hypotheses that arise from PRISM calculations: (1) correlations due to chain connectivity enhance coacervation, (2) polymer excluded volume suppresses coacervation, and (3) salt and polymer excluded volume drive salt ions to prefer the supernatant phase. We evaluate these hypotheses using Gibbs ensemble Monte Carlo simulations and experimental measurements of model coacervate systems. Our simulations and experiments conclusively show that both chain connectivity and excluded volume play a vital role in influencing the phase of polymer coacervates as demonstrated previously using PRISM theory calculations. These results present new exciting opportunities in Coulombically driven material design.
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METHODS
Sample Preparation. Complex coacervate samples were prepared from lyophilized polyelectrolyte complexes, using the method reported by Schlenoff et al.30 This method was necessary to facilitate the formation of a significant volume of the coacervate phase relative to the supernatant phase (approximate ratio of 1 to 2) in order to provide an accurate measurement of salt partitioning between the two phases. Briefly, poly(4-styrenesulfonic acid, sodium salt) (PSS) was purchased as a 15 wt % solution in water (AkzoNobel, VERSA TL130, 70 000 g/ mol), and poly(diallyldimethylammonium chloride) (PDADMAC) was purchased as a 20 wt % solution in water (Hychem, Hyperfloc CP 626, 400 000 g/mol). Impurities in PSS were removed by 0.22 μm pore size filter (EMD Millipore, Billerica, MA), while PDADMAC was utilized as received. Stock solutions of 0.125 M PSS and PDADMAC (on a monomer basis) were prepared via dilution with Nanopure water (Barnstead Nanopure Infinity, Thermo Fisher Scientific, Waltham, MA) and adjusted to pH 7.2. 300 mL of 0.125 M PSS was added to a beaker containing 5 g of solid sodium chloride (NaCl, ACS grade, Fisher Scientific). 300 mL of 0.125 M PDADMAC was then added, and the mixture was allowed to stir for 30 min at room temperature. The mixture was then stirred slowly at 45 °C for an additional 10 min to facilitate agglomeration of the sample. The resulting polyelectrolyte complex solid was then cut into small pieces and placed in a beaker containing Nanopure water. The water was exchanged 12 times every 12 h. The resultant salt-free samples were then lyophilized (Labconco). This procedure means that a negligible amount of polyelectrolyte-associated counterions contribute to the overall salt concentration. Coacervate samples were then prepared by adding 3 g of lyohilized polyelectrolyte complex to a 50 mL Falcon tube and adding a preprepared solution of potassium bromide (1.2, 1.4, 1.5, 1.6, and 1.8 M KBr, ACS grade, Fisher) to a total sample volume of 30 mL. The final salt concentration was then determined based on the volume of salt added (1.09, 1.26, 1.34, 1.46, and 1.62 M). 1.8 M added salt was chosen as the upper limit for added salt based on previous experiments where turbidity and optical microscopy were used to determine the critical salt concentration for the system (approximately 1.9−2.0 M KBr, data not shown). Data at lower salt concentrations were not included because of a transition from a liquid coacervate to a solid precipitate phase below 0.9 M KBr. Samples (example in Figure 1a) were allowed to equilibrate overnight, followed by centrifugation for 20 min at 4000 rpm (Sorvall Legend X1R, Thermo Scientific). The supernatant was then extracted into a glass scintillation vial, and the conductivity was measured (Tetracon 325). Reported values are an average of three measurements; error bars represent the standard deviation. Conversion from conductivity to salt concentration was achieved using a calibration curve. The volume of the supernatant phase was determined by decanting into a graduated cylinder, with an B
DOI: 10.1021/acs.macromol.6b02582 Macromolecules XXXX, XXX, XXX−XXX
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ϵLJ((σ/r)12 − 2(σ/r)6). We use standard reduced units Utot * = Utot/ϵ, where ϵ = z2e2/4πϵ0ϵrσ, T* = kBT/ϵ, and Kθ/ϵ = 1. We use standard Gibbs ensemble Monte Carlo simulations to study the phase behavior of polymer coacervates.67 Two separate simulation boxes represent macroscopic phases that are far away from the interface between a polymer-rich (α-phase) and a polymer-deficient (β-phase) as shown in Figure 1b.67 We simplify our simulation by assuming that the polymer concentration is negligible in the βphase.28,43 Three trial moves are used: (1) particle displacement, (2) exchange of salt ions between the two boxes, and (3) volume exchange between the two boxes.67 The simulation considers protocols standard in the literature for trial moves.67 The exchange moves for both ion species are coupled such that electroneutrality is maintained in both boxes. Volume exchange moves rescale the cubic box dimension Li → Li + δLi of both boxes i = α, β such that overall volume is conserved, and all particle positions within each box are rescaled in the same fashion. The combined volume of both boxes is 5832σ3, and equilibrated box dimensions are typically Li ≈ 10 − 15σ. 2 × 107 equilibration cycles are performed, followed by thermodynamic averages calculated from the next 8 × 107 cycles. Figure 1c schematically shows all the species and their geometric parameters (σ and Δ). We consider the polymer-deficient β-phase to have negligible polymer concentration as validated by our experimental results; however, we can still characterize the structure of dilute chains in these simulations. We do so via NVT MC simulations that consist of a single polycation chain in a salt concentration corresponding to the β-phase. In a sufficiently large box, we can directly evaluate the ±/± and P+/± correlations.
Figure 1. (a) PSS and PDADMAC solution, phase separated into a coacervate phase (α, bottom) and supernatant phase (β, top). (b) Simulation snapshots of Gibbs ensemble Monte Carlo simulations. Equilibrium between the separate simulation boxes (α- and β-phases) is maintained via particle and volume exchange. (c) The α-phase consists of polycation (orange), polyanion (blue), cation (red), and anion (purple) species. This schematic demonstrates the bead diameter σ and polymer charge spacing Δ, which are molecular features of this restricted primitive model representation.
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RESULTS AND DISCUSSION Gibbs ensemble Monte Carlo simulations capture both the phase behavior and the local charge structure of complex coacervates. Figure 2 depicts standard phase diagrams of a
experimental error of ±0.25 mL. We also quantified the concentration of PSS present in the supernatant phase using UV−Vis spectroscopy at 262 nm. The measured polymer concentrations (∼5 mM on a monomer basis) should contribute only minimally to the total solution conductivity (∼1 mS/cm from polymer compared with ∼100−200 mS/cm from salt). Thus, we elected to neglect any polymer effects. Simulation. We model charged species using the restricted primitive model (RPM) with all charged species represented as charged spheres.64 Our coacervate system consists of polycations (P+), polyanions (P−), cations (+), and anions (−). The polycations and polyanions are modeled as chains of charged hard spheres of length NP+ and NP−, valency zP+ and zP−, and diameter σP+ and σP−, respectively. Salt cations and anions are similarly modeled as charged spheres of diameter σ+ and σ− and valency z+ and z−, respectively. The current study uses zP+ = zP− = z+ = z− = z = 1, NP+ = NP− = NP = 100, and σ+ = σ− = σP+ = σP− = σ = 4.25 Å. This parameter choice for σ represents a reasonable, but arbitrary value for charged species and their hydration shells. This does not represent any specific chemistry and carries the well-known assumptions inherent in any RPM representation of charged systems.65 The system is studied in an aqueous medium with dielectric constant ϵr = 78.5 and temperature (T) at 298 K. The total energy of the system is given by Utot = UE + UB + Uθ + Uint, with contributions due to electrostatics, bond potentials, bending potentials, and pairwise interaction potentials, respectively. Charges interact via the electrostatic contribution UE =
z 2e 2 , 4πϵ0ϵr r
Figure 2. Coacervate phase diagram as a function of polymer concentration cP and salt concentration cS. Points denote measurements of the binodal curve from Gibbs ensemble Monte Carlo simulations for a variety of different charge spacings Δ. The two-phase (2Φ) region in the bottom left represents a coexistance along the tie lines between a coacervate α and supernatant β phase. Smaller 2Φ regions are observed as Δ increases, demonstrating that linear charge density plays a large role in coacervation. Simulations become challenging at low values of cP, and dotted lines denote that the phase boundary may extend further. Experimental measurements (open purple circles) are a near-quantitative match for the Δ = 1.0 simulation binodals and tie lines. Experimental error is smaller than size of symbols.
which takes the
form of a Coulomb potential. We use a spherically averaged Ewald sum to account for the long-range nature of the Coulomb potential.66 Two connected polymer beads interact via a bond potential UB characterized by a charge spacing Δ: UB(r) = 0 if Δ < r < Δ + 0.1σ and UB(r) = ∞ otherwise. Three adjacently connected polymer beads interact via a bending potential Uθ = Kθ(θ − θ0)2 where Kθ is the bond stiffness constant and θ0 = 0 is the equilibrium value of the bond angle. Finally, we include pairwise interactions via a hard core potential (Uint(r) = 0 if r > σ and Uint(r) = ∞ otherwise). On a few specified occasions, we instead use a Lennard-Jones (LJ) potential Uint(r) = C
DOI: 10.1021/acs.macromol.6b02582 Macromolecules XXXX, XXX, XXX−XXX
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Figure 3. (a) Pair correlation functions (gij(r)) between species i and j for cP = 2.17 M and cS = 0.37 M (Δ = 1, α designated in Figure 2). The excess correlation of gP+/P− beyond gP+/− or g+/− partly drives coacervation due to increased electrostatic attractions. (b) Pair correlation functions (gij(r)) between species i and j for cS = 0.57 M and cP → 0 (Δ = 1, β designated in Figure 2). Excess correlation of gP+/− over g+/− indicates counterion condensation. (c, d) Pair correlations for a series of Δ, at same concentrations as for (a, b). In (c), gP+/P− decreases, leading to decreased electrostatic attractions in coacervates. In (d), gP+/− indicates decreased counterion condensation with increasing Δ.
understand this behavior, we consider coacervates at the α and β phase points circled in Figure 2. P+/P− correlation functions demonstrate strong association between oppositely charged polymers in the α-phase. Figure 3a plots gij(r) for species i and j (Δ = 1.0σ) at a point chosen to lie along the Δ = 1.0σ binodal (Figure 2). For a given charge along a polyelectrolyte, gP+/P−(r) describes the abundance of neighboring oppositely charged polymer species. Two distinguishable peaks for gP+/P−(r) show an abundance of oppositely charged polyelectrolyte beads at both r = 1.0σ and r ≈ 2.0σ. The first peak corresponds to the primary electrostatic interaction between oppositely charged beads on separate polymer chains. The connected nature of charges along the polyelectrolytes facilitates secondary interactions with neighboring beads. This connectivity contributes to an increased electrostatic attraction between polycation and polyanion and results in the second peak. In contrast to trends observed in the α-phase, an analysis of the P+/− and P−/+ correlation functions for the β phase demonstrates significant counterion condensation. This localization of oppositely charged counterions next to a polyelectrolyte with high linear charge density is evidenced by the abundance of anions around polycations in gP+/− that exceeds the population of anions around cations (g+/−). To contrast, in Figure 3a, gP+/− and g+/− do not show significant differences for anions next to polycations versus cations. This difference between the two phases supports a counterion release picture of coacervation, where the condensed counterions observed in the dilute β-phase are no longer localized around the polymer
coacervate in the plane of polymer concentration cP versus salt concentration, cS. A number of charge spacings Δ are considered. Lines are the binodal curves that demarcate the two-phase region (2Φ) in the lower left-hand corner of Figure 2 where there is coexistence between a polymer-dense coacervate phase (cαP > 0) and a salt-only supernatant phase (cβP = 0). Experimental measurements of the phase diagram provide a near-quantitative match to the simulation phase diagram for Δ = 1, including similar tie lines and binodal (open purple circles, Figure 2). Both experiments and simulations are consistent with experiments in the literature, primarily the decrease in cP = cαP with an increase in cS that continues until the system becomes miscible.1,28,39 We note that Gibbs ensemble simulations are computationally difficult near the critical salt concentration (i.e., at low cP) due to large box sizes; dashed lines in Figure 2 denote that the binodals continue beyond the observed data. The β-phase is also assumed to be infinitely dilute in terms of polymer concentration, and thus the binodal lies along this axis.43 Experiments support this assumption, with monomer concentration measured at ≈5 mM in the supernatant. Charge Spacing. The charge spacing along the polymer backbone Δ has a pronounced effect on the coacervate phase behavior. This is demonstrated in Figure 2, which varies the charge spacing Δ. This adjustment changes the linear charge density without also changing the degree of polymerization NP and thus the Flory−Huggins mixing entropy. We observe that an increase in Δ corresponds to a decrease in the tendency to form a coacervate, such that the two-phase region shrinks. To D
DOI: 10.1021/acs.macromol.6b02582 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules in the coacervate phase. These “released” counterions have greater translational entropy, which provides a driving force for coacervation. P+/P− correlations in the α-phase (Figure 3c) and P+/± correlations in the β-phase (Figure 3d) both show marked changes upon variations in the charge spacing Δ. In Figure 3c, the second gP+/P− peak associated with electrostatic attractions in the coacervate phase decreases in size and moves to higher r as Δ increases, leading to a lower electrostatic driving force for coacervation. In Figure 3d, the counterion condensation peak (gP+/−) decreases significantly with increasing Δ. This indicates that fewer counterions are condensed along a polycation chain of lower charge density, and thus less counterion release entropy is gained upon coacervation. The effect of Δ is thus a combination of two simultaneous effects: a decrease in the correlations between oppositely charged polymers (P+/P−) correlations in the coacervate and a decrease in polycation−anion (or polyanion−cation) correlations (P+/− and P−/+ ) in the supernatant. P+/P− correlations are driven by connectivity, such that the interaction between two polyelectrolyte charges forces interactions between their neighbors. As these neighbors become less proximate, it becomes increasingly less likely that they will interact, decreasing the significance of these cooperative interactions. Along similar lines, closely connected charges also drive significant P+/− and P−/+ correlations that lead to counterion condensation. The release of these counterions partially drives coacervation, so a decrease in connectivity leads to fewer available condensed counterions for subsequent release. This corresponds to the observed shrinking of the two-phase region observed in Figure 2. These charge spacing effects (increased chain/chain correlations, counterion release) qualitatively match results obtained from PRISM theory. In particular, correlation functions demonstrate similar features.43 This serves to verify the importance of correlation effects, with both theory and simulation exhibiting the same trend of decreasing coacervation with increasing Δ.43 Excluded Volume. The importance of excluded volume effects have been suggested by experimental results showing stable coacervation of hydrophobic polyelectrolytes at salt concentrations as high as 4 M and related PRISM calculations.25,36 In particular, excluded volume has been shown to suppress phase separation.43 The magnitude of this effect is challenging to determine in simulations due to the connection between the strength of the electrostatic interaction Γ = e2/(4πϵ0ϵrσkBT) and the monomer size σ. If the size of the charge is directly changed, both the excluded volume and the strength of the electrostatic potential change.43 Instead, we use an attractive short-range potential in an attempt to “cancel out” the effect of excluded volume. This cancellation is a frequent device for realizing idealized systems, such as the θ-point of a polymer or the Boyle temperature of a gas.41,42 We replace the hard core interaction between polyelectrolytes with a LJ potential with ϵLJ = 0.17kBT. This corresponds to the θtemperature of the polymer chains, which has been well established in the literature.68 Strictly speaking, this only cancels out the second virial coefficient (i.e., the two-body interactions),42 and therefore three-body excluded volume effects will still be present. Nevertheless, this approximates removing excluded volume interactions in coacervate systems. Figure 4 demonstrates that the phase behavior of θcoacervates is dramatically different than the hard core
Figure 4. Phase diagram for coacervates as a function of cP and cS. Binodals either have hard core interactions (black and dark blue symbols) or a Lennard-Jones potential at the θ-point for the polymer (grey and light blue symbols, ϵLJ = 0.17kBT). The binodal for Δ = 1.0σ and hard core interactions (black) is the same as in Figure 2. By effectively removing hard core interactions via application of the LJ potential, there is a significant increase in the two-phase (2Φ) region of the phase diagram. This enhanced coacervation also occurs with larger spacing between charges, Δ = 1.2σ upon inclusion of the LJ potential. The hard core Δ = 1.2σ has a significantly smaller region of coacervation, so the “effective” removal of excluded volume leads to a binodal that is similar to the hard core Δ = 1.0σ binodal. (Inset) VO parameters can be determined directly from simulation. The neglect of both excluded volume and connectivity in VO leads to unphysical results.
model. Two different values of Δ = 1.0σ and 1.2σ are considered, for both hard core and LJ potentials. The effective removal of two-body excluded volume interactions strongly enhances coacervation, with the width of the two-phase region increasing as much as 1 M for θ-coacervates, as compared to the phase boundary observed with a hard core excluded volume. This qualitatively agrees with the results obtained from the PRISM theory, which hypothesized that excluded volume strongly suppressed coacervation.43 We also note that nearby, connected charges and finite charge size are both neglected in Voorn−Overbeek, yet their effects can cancel out. In Figure 4, this is seen by nearly identical binodal curves for both chargedense, hard core polyelectrolytes (Δ = 1.0σ, hard core potential) and low charge-density, θ-polyelectrolytes (Δ = 1.2σ, LJ potential). The use of the LJ potential to cancel out the second virial coefficient enables direct comparison between VO theory and simulation. Three VO parameters are typically fit to experimental results: the fraction of charged monomers along the chain f P, the size of the monomer unit σ, and the value of the Flory χ-parameter.28 These are typically chosen to match data due to practical challenges in the experimental determination of these parameters.28 If weak polyelectrolytes are used, then f P becomes a variable quantity whose value is determined by a complex combination of pH and the local charge environment. 25,69 σ requires knowledge of the “molecular size” of the monomer, which includes the ill-defined hydration shell around any charged species.70 The Flory χparameter represents local short-range interactions, which requires the assumption of a model.41,61 All of these parameters are unambiguously defined in our simulation, f P = 1 and σ = E
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Macromolecules 4.25 Å. The LJ potential at the θ-condition corresponds to χ = 0.5 (i.e., the second virial coefficient is zero). The inset of Figure 4 plots the VO result predicted from these parameters and demonstrates that accurate choices of typical fitting parameters vastly overestimates the size of the coacervation region. Salt Partitioning. In addition to the boundaries of the twophase region, salt partitioning between the coacervate and supernatant phases is a measurable value71 that can be used to distinguish between VO and competing models of polymer complex coacervation. VO theory predicts that the salt concentration in the supernatant phase will be lower than in the coacervate phase (see Figure 4 inset); however, PRISM43 and some RPA models72 predict that the opposite should be observed. We also observe this trend, which is apparent from the tie lines in Figures 2 and 4. In all situations, a lower value of cS is observed in the polymer-rich α-phase compared to the polymer-deficient β-phase. Figure 5 plots the ratio of salt
our simulation and PRISM theory but in disagreement with Voorn−Overbeek theory. This is in agreement with literature data for the same system available from Wang and Schlenoff,53 who do not calculate this value but provide enough information to do so. This is also plotted in Figure 5 (purple open squares). Furthermore, the trends we observe are quantitatively consistent with simulations, falling in between Δ = 1.0σ and Δ = 1.1σ. The critical salt concentration c*S is the salt concentration at which a dilute solution of coacervate-forming polyelectrolytes undergoes the coacervation transition. Despite not being able to observe c*S directly in simulation, visual observation of Figure 2 would place c*S ≈ 1.7 M for Δ = 1.1 and c*S ≈ 2.5 M for Δ = 1.0σ. This is similarly consistent with the experimental observation, with cS* ≈ 1.9 M for PSS/PDADMAC (1.9 M). We note that the matching between experiment and simulation does not mean that this is a predictive model. In particular, there remain well-known ambiguities in how real chemical systems with molecular water are mapped to RPM representations that treat charged species as beads in a continuum solvent.65 Nevertheless, the quantitative consistency between both experiment and simulation for both λ and c*S values demonstrates that physically reasonable choices of molecular parameters (charge separation, size) in this model can capture experimental observables.
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CONCLUSIONS In this article we have demonstrated that charge correlations at short, molecular length scales dictate the macroscopic phase behavior of polymer coacervates. We use simulations to explore the extent to which the charge spacing and excluded volume of the polyelectrolytes alter the phase behavior of complex coacervates and demonstrate that both effects have a strong and opposing effect on coacervation. Classical VO theory specifically neglects molecular features such as charge connectivity and non-mean-field excluded volume.38,39 VO theory can also be directly compared to simulation results to reveal a profound quantitative difference between predicted and observed behaviors. These features are needed to fully reflect the physical state of complex coacervation, and we demonstrate that charge expulsion from the coacervate is an experimental observable that reflects the weaknesses of VO theory. The ability for our simulations to accurately reflect molecular features such as excluded volume and charge spacing and capture observable thermodynamic properties like salt partitioning sets the stage for using this approach to extend our understanding of coacervation. In particular, this method has the potential to capture molecular-level details such as complex polymer architectures and charge patterning effectsaspects of complex coacervation that cannot be addressed by field theory methods that do not resolve molecular details. In combination with traditional routes to controlling complex coacervate phase behavior (i.e., salt concentration, pH, stoichiometry),1,25,26,29,73 we envision that control over local architecture and the sequence of charged monomers represents a new paradigm in charge driven self-assembly.
Figure 5. Value of the salt partitioning ratio λ = cαS /cβS as a function of the salt concentration in the supernatant phase cβS . Solid points represent values taken from Gibbs ensemble Monte Carlo simulations, with colors corresponding to Figures 2 and 3 for different values of Δ and ϵ. Lines are a guide to the eye. All points are well below λ = 1, indicating that salt prefers to be in the β-phase. Indeed, stronger coacervation (see Figures 2 and 4) tends to lead to more significant exclusion of salt from the coacervate phase. We also include experimental data points (purple, open circular symbols) for λ determined by ion conductivity measurements and points calculated from data in Wang and Schlenoff.53 These values are well within the range observed in our simulation, and correspond to situations in our model where 1.0σ < Δ < 1.2σ.
concentration in the α and β phase, cαS /cβS = λ, as a function of cβS . For all values of Δ and for θ-coacervates, we demonstrate that the coacervate phase is salt-deficient (λ < 1). This deficiency is largest at low cS due to the high concentration of polymer cαP. We are able to determine the value of λ experimentally for a series of coacervate samples prepared from PSS and PDADMAC in varying concentrations of KBr. The salt concentration cβS was determined via conductivity measurements on a known volume of supernatant, enabling subsequent back-calculation of cαS . The corresponding data for λ are plotted in Figure 5 (open circles). Strong preferential partitioning of salt into the supernatant phase was observed, in agreement with
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Sarah L. Perry: 0000-0003-2301-6710 F
DOI: 10.1021/acs.macromol.6b02582 Macromolecules XXXX, XXX, XXX−XXX
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Charles E. Sing: 0000-0001-7231-2685 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank Tyler K. Lytle for computational help, Xiangxi “Zoey” Meng for assistance in sample preparation, and Michael Leaf for help with conductivity measurements. Acknowledgment is also made to the Donors of the American Chemical Society Petroleum Research Fund for support of this research.
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