Interplay between Depletion and Electrostatic Interactions in

Article pubs.acs.org/Macromolecules

Interplay between Depletion and Electrostatic Interactions in Polyelectrolyte−Nanoparticle Systems Victor Pryamitsyn Department of Chemical Engineering, University of Texas at Austin, Austin, Texas 78712, United States

Venkat Ganesan* Department of Chemical Engineering and Institute for Computational and Engineering Sciences, University of Texas at Austin, Austin, Texas 78712, United States S Supporting Information *

ABSTRACT: We use a numerical implementation of polymer selfconsistent field theory to study the effective interactions between two spherical particles in polyelectrolyte solutions. We consider a model in which the particles possess fixed charge density and the polymers contain a prespecified amount of dissociated charges. We quantify the polymer-mediated interactions between the particles as a function of the particle charge, polymer concentrations and particle sizes. We study the interplay between depletion interactions, which arise as a consequence of polymer exclusion from the particle interiors, and the electrostatic forces which result from the presence of charges on the polymers and particles. Our results indicate that for weakly charged and uncharged particles, the polymer-mediated interactions predominantly consist of a short-range attraction and a long-range repulsion. When the particle charge is increased, the interactions become purely repulsive. A longer range, albeit weaker, bridging attraction was also evident for some parametric regimes. We demonstrate that the short-range attraction and the longer-range repulsion can be modeled as a sum of a depletionlike attraction and an electrostatic Debye−Huckel like repulsion. However, the amplitude and range underlying the depletion and electrostatic interactions are shown to possess a complex relationship to the parameters of our system. We present scaling arguments and analytical theory to rationalize some of the dependencies underlying the parameters governing the interaction potentials.

I. INTRODUCTION Polyelectrolyte− (PE−) protein and PE−particle mixtures form an important class of systems in which their phase behavior and complexation characteristics play a crucial role in influencing their properties.1−9 Not surprisingly, experimental characterization of the phase behavior of polyelectrolyte−particle and polyelectrolyte−protein mixtures has attracted considerable attention.5,10−18 Despite the practical importance of PE− particle mixtures, the parameters governing the interactions and their phase behavior characteristics have had much less theoretical work compared to their neutral counterparts.19−22 Thermodynamic descriptions of colloidal particles in nonpolymeric electrolyte solutions are themselves known to involve a complex interplay of solvent effects, pH conditions, and salt concentrations.23−26 More recent developments have also identified new phenomena such as like-charge attractions, overcharging, charge-reversal of macroions, and the importance of multibody interactions, etc.23,27−31 Transitioning to a polymeric electrolyte brings in additional physics arising from the adsorption and depletion of polymers on the particles, the role of solvent quality, etc.19−21 Relatively little is known about the implications of the interplay between electrostatic © 2014 American Chemical Society

phenomena and such polymer physics aspects upon the phase behavior of PE−particle mixtures. Motivated by the above outstanding issues, theoretical and simulation studies have considered the physics underlying interactions and phase behavior of PE−particle systems.32−51 Despite the valuable insights furnished by such studies, utility of such approaches remain some what limited due to the significant computational expense involved in treating the particle, polymer, solvent molecules and counterions components on an equally rigorous footing in such methodologies. As a consequence, many of the studies have either relied on Debye− Huckel approximations for modeling electrostatic interactions (and thereby ignore the presence of counterions) and/or have studied only a relatively small parameter space within the possible spectrum of polymer charges, chain lengths, and concentrations. Field theory based approaches have emerged as a possible alternative for studying PE−particle mixtures.52−61 Such Received: May 15, 2014 Revised: July 21, 2014 Published: August 19, 2014 6095

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approaches rely on a continuum framework where the concentration of the different species are taken to be continuous functions of the spatial coordinates. By minimizing a free energy functional which embodies the composition and electrostatic field variations, the density, electrostatic potential profiles and the interaction characteristics between surfaces can deduced. In early studies, Muthukumar,62,63 Varoqui,64,65 and others61,66 used such approaches to investigate the adsorption of a polyelectrolyte chains to one or more surfaces. More recently, Andelman, Orland, and Netz,67−69 Wang,52,53,70,71 and Shi59 and their co-workers numerically solved the nonlinear version of the continuum field equations to study the adsorption characteristics of polyelectrolyte solutions onto planar and curved surfaces. As a first step toward clarifying the physics of interactions and phase behavior of polyelectrolyte−particle systems, in the present work we adapt a field-theoretic approach similar to the above studies to study the polymer-mediated pair interactions arising between spherical particles in polyelectrolyte solutions. We do note that there have been some earlier works which have used similar field-theory based approaches to study the effective interactions between surfaces in polyelectrolyte solutions.60,68,69 Below, we clarify the main features of our study and identify the aspects which distinguish it from earlier works: (i) Borukhov, Andelman, and Orland considered models and issues which are closely related to the system studied in the present article.68,69 Specifically, they considered both the adsorption characteristics and the resulting effective interactions between charged surfaces immersed in polyelectrolyte solutions. They used an approximation to the mean-field theory and quantified both numerically and by scaling laws the PE density profiles and the intersurface interaction energies. Further, by using the Derjaguin approximation,69 they translated their results to the case of spherical colloidal particles in PE solutions. Despite the important insights furnished by the above studies, there are still some unresolved issues. Specifically, the limit considered by Borukhov and co-workers corresponds to the situation in which the polymer concentrations are dilute and the surfaces are strongly charged.69 An outstanding question is the nature of interactions that arise in the context of weakly charged and uncharged particles immersed in more concentrated PE solutions. Indeed, in the limit of uncharged particles,72 we expect there to be an attractive effective interaction between the particles, commonly termed as the depletion interaction, arising from the exclusion of the polymers from the interior of the particles. The strength of such interactions are expected to increase with increase in polymer concentrations.72,73 On the other hand, electrostatic interactions are expected to depend on the charge of the particles and are likely to become weaker at higher polymer concentrations due to screening effects. As a consequence, one may expect interesting behaviors to result from the competition between depletion attractions and electrostatic repulsions when either the charge of the particles and/or polymer concentrations are modulated. A related issue is that Borukhov and Andelman studied the intersurface interactions in the limit where the surfaces were maintained at a constant electrostatic potential,69 and in such a limit found that intersurface interactions exhibited a strong short-range repulsion followed by an

attraction at larger distances. On the other hand, a different study by Forsman et al.42 used MC simulations to study the intersurface interactions in PE solutions, albeit at the conditions where a constant surface charge was maintained. Their study found a variety of possible interaction behaviors which included short-range attractions, a longer range repulsion and weaker oscillatory forces at even longer length scales. Such contrasting results reported in different studies motivate a comprehensive investigation of the effective interactions between charged surfaces in PE solutions and their dependence on different parameters. (ii) Many applications of PE−particle mixtures correspond to the nanoparticle limit, where the polymer size (i.e., the length scale controlling polymer physics) becomes comparable to or even larger than the size of the particles.73−77 In such limits, the relative sizes of the particle and the polymer, and the curvature of the particle are expected to become significant in influencing the interactions and phase behavior of the mixture.30,74,78 Moreover, in some situations, three and multibody interactions may overwhelm the influence of pair interaction potentials on the structure and phase behavior of the system.73,76,79,80 A second feature of our present work is the development of a numerical framework for the mean-field limit of PE solutions which can accommodate the presence of an arbitrary number of particles of specified curvatures and distribution of charges. In the present article we restrict our consideration to only pair interaction characteristics, but do discuss the role of particle curvature in influencing the pairinteraction characteristics. In a followup article, we use the same numerical framework to study the importance of three and multibody interaction characteristics in similar situations. Motivated by the above issues, in the present work, we study the pair interactions between uncharged and charged spherical particles in PE solutions. Specifically, we consider the infinite dilution limit and consider two spherical particles in a solution of PEs and counterions (see Figure 1). We simplify the parameters considerably by adopting a model in which the

Figure 1. Schematic of the model under consideration in this work. We consider two positively charged spherical particles of radius RC immersed in a solution of negatively charged polyelectrolytes, co- and counterions. The total charge of the particle is QC and is assumed to be distributed in a spherically symmetric and uniform manner. For the consideration of polymer segment, counterion, and charge density profiles, we consider the model of a single isolated particle in the solution. 6096

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polymers are assumed to be present in a Θ solvent such that the excluded volume interactions between the polymer segments can be neglected.81 Also, we consider only the situation of strong PEs, in which the charge on the polyelectrolyte chains can be assumed fixed, independent of the solution conditions and determined only by the total number of acidic or basic groups on the polymer molecules.82 We also assume that the particles have homogeneously distributed charges and ignore the presence of patches and heterogeneities in charge distribution. Finally, we employ a mean-field approximation which neglects both fluctuation and charge correlation effects.27,31 While such effects may prove important for some situations, in this first study we wish to simplify the parameters and physics to clarify the main features of the interactions underlying our system. The rest of the article is arranged as follows: In Section II, we summarize the theoretical framework and the field theory model we use in this work. In Section III, we discuss the results for the model and the system considered in this article. In section III.A, we discuss the situation arising in uncharged particles in neutral polymer solutions. This situation, termed as the “depletion” case, is discussed in brief for the purpose of contrasting the influence of electrostatic interactions discussed in the subsequent sections. Following this, in parts B, C and D of section III, we discuss the qualitative features of the results for the case when uncharged and charged particles are present in the PE solution. We take recourse to the density profiles arising in the single body case to rationalize the qualitative trends we observe in our results. In section IV, we undertake a quantitative analysis of the potential by fitting the interactions to an analytical form inspired by our qualitative findings. We undertake a detailed analysis of the different parameters underlying the interaction potentials and clarify the physical bases for their dependencies.

relative to the two-body interactions, enabling one to retain just terms up to U2. The resulting pair interactions can then be combined with bare interactions between the particles and used either in MC, MD simulations and/or in theoretical approaches to discern the phase behavior and structural characteristics of the particle phase. On the basis of such considerations, in the present article, we present results for U2(Ri, Rj; Z) for the system of particles immersed in a polyelectrolyte solution. In a followup article, we will present results for U3 and higher order interactions to clarify the importance of such terms. B. Description of Model. As discussed above, discerning the PE-mediated pair interactions between the particles in the PE solution requires the calculation of the grand canonical free energies In Ξ2 and In Ξ1 for the mediating fluid in the presence of a fixed configuration of particles (cf. eq 1). As a consequence of translational invariance, Ξ2 is expected to depend only the distance r between the centers of the particles. To maintain brevity, henceforth we drop the subscript “2” from U2 and denote the pair interaction potentials as U(r). Toward the above objective, we considered a system of either one (for obtaining In Ξ1) or two (for obtaining In Ξ2) positively charged spherical particles of radius RC in a solvent containing a mixture of negatively charged polyelectrolytes, negatively charged co-ions and positively charged counterions (see Figure 1). The total charge of the particles is denoted as QC, and is assumed to be distributed in a spherically symmetric manner over the entire volume of the particle. Since the electrostatic field resulting outside a particle with spherically symmetric volumetric charge distribution with total charge QC is identical to that arising for a spherically symmetric surface charge with total charge QC, we adopted the former model as it provided better numerical performance. In our work, we used a mean-field framework for the PE solution to determine the free energies In Ξ1 and In Ξ2 as a function of the polymer solution chemical potential Z, or equivalently, the bulk polymer concentrations.54,81 We list some of the salient assumptions underlying our model: (i) We begin with a description where the degrees of freedom corresponding to the underlying solvent are assumed to be integrated out, and the particles and polymers are interacting by effective, solvent-averaged interaction potentials. In the present work, we adopt the model of a Θ solvent in which the solvent averaged interaction potentials are set identically to zero.81 (ii) We model the polyelectrolyte chains as continuous, flexible Gaussian chains. The polyelectrolytes are assumed to be strong acids with a prespecified overall charge which is independent of solvent or pH conditions. We further assume that the overall charge of the polyelectrolyte chain is smeared uniformly along the backbone of the chain and denote the charge fraction (i.e., fraction of charged groups relative to the total number of segments in the polymer) as α;54 (iii) The electrostatic interactions are modeled using a classical Coulomb potential, with a spatially constant dielectric value. (iv) We neglect all nonelectrostatic interactions (enthalpic and excluded volume) between the polymer monomers and ions. We also ignore all enthalpic interactions between the polyelectrolyte and particles and retain only electrostatic and excluded volume interactions between the particle and the polymer segments. We recall that in our earlier work,86 the presence of the particles and their interactions with the polymers were modeled explicitly by incorporating

II. THEORETICAL FRAMEWORK A. Statistical Mechanical Basis. A number of earlier researches have used the Mayer’s cluster expansion technique73,83−86 as a framework to identify the effective interactions between particles in complex fluids. In these approaches, it is customary to adopt a semigrand canonical framework in which the activity coefficient of the polymer is fixed (denoted as Z below), and the free energy of the system of one or more particles fixed at specific locations in the complex fluid is calculated. For the specific situation of particles in PE solutions, the polymermediated one and two body interaction potentials U1(Ri; Z) and U2(Ri, Rj; Z) can be written as U1(R i; Z) = ln Ξ1(R i; Z) − ln Ξ 0(Z)

(1)

U2(R i, R j; Z) = ln Ξ 2(R i, R j; Z) − 2 ln Ξ1(R i; Z) + ln Ξ 0(Z)

(2)

where ΞI(Ri, Rj, ...; Z) denotes the grand canonical partition function73,83−86 of the PE solution whose activity coefficient is fixed as Z and which contains I particles at positions Ri, Rj.... Similar formulas can be derived for any arbitrary I body interaction potential UI relating it explicitly to all the grand canonical partition functions ΞJ(Ri, Rj, ...; Z), J ≤ I. McMillan-Mayer theory based expansions for the free energy of particle−solvent system are known to be formally exact only in the limit of retaining all the multibody interaction terms UI(R1...RI;Z)(I = 1, 2, ..., ∞).83,84 Fortunately, however, in many situations, three and higher body effects prove negligible 6097

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The local densities of the co- and counterions ρk(r) are in turn obtained as

a steeply repulsive monomer-particle potential which ensured the impenetrability of polymer monomers into the particle core (denoted in ref 86. as wcp(r)). In the present context, we use a similar approach to model the excluded volume interactions between the particle and polymer monomers, as well as the interactions between the particle and the co-, counterion components. Explicitly, the steric interactions between the particle and polymer monomers, co-, and counterion components are modeled using the potential ⎛ σ ⎞ wcp(r ) = ⎜ ⎟ ⎝ r − Rc ⎠

ρk (r) = exp[−βμko − zk Φ(r) − βwcp(r)]

The polymer density field ρP(r) is in turn obtained from54,81 ρP (r) = Z

(3)

where r denotes the distance between the center of the particle and the monomer or counterion component, σ is a parameter chosen to minimize numerical issues arising from sharp interfaces. In a recent article, we outlined the field-theory framework and the resulting mean-field equations for the model of neutral particles immersed in a neutral polymer solution.86 Whence, we only present the salient modifications to our earlier model to capture the physics of the system considered here. With respect to the framework outlined in ref 86, the following two additional free energy contributions arise for the model considered in the present article: (i) The translational mixing entropies of the co- and counterions:87−89

∫ dr ∑ {ρk (r)[ln ρk (r) − 1 + βμko]} k

⎡

w*(r) = α Φ(r) + wcp(r)

∑ zkρk (r) − αρP (r) + q(r) k

(9)

(10)

(11)

represents the potential field acting on the polymer segments. C. Numerical Methods and Parameters. Note that in our notation, the Bjerrum length λB is defined as λB = βe2/ε. In the results we present in the following section, we worked in units in which the λB = 0.7 nm, the value corresponding to water. We adopted a numerical procedure to solve the above mean-field equations for different values of the chemical potentials μ0k and activity coefficient Z. Note that the condition of electroneutrality imposes that only two of the three chemical potentials corresponding to the polymer and co- and counterions can be chosen independently. In the units discussed above, we considered two different sized polymers with radius of gyration Rg = 12 and 24 nm (Rg corresponds to the radius of gyration of an ideal, neutral polymer). Since we considered a Θ solvent for the present work, Rg represents the correlation length for a solution of neutral polymers. We typically investigated polymer concentrations such that the monomer concentrations (and hence the charges) for the system of differing Rgs were maintained approximately the same. To investigate the effects of curvature, we considered two particle sizes of radii RC = 10 and 20 nm, respectively. The parameter σ in eq 3 was chosen as 2.7 nm. The Debye−Huckel screening length for the polyelectrolyte solution system λD is defined as

(4)

(5)

In the above Φ(r) denotes the local electrostatic potential and ρe(r) represents the charge density at the location r (in units of unit electric charge). ϵ represents the dielectric constant of the medium (assumed a constant in this work) in units of 4πϵ0 where ϵ0 represents the permittivity of vacuum.81 The local charge density ρe(r) can in turn be written as ρe (r) =

ds q(r, s)q(r, N − s)

where

⎤

∫ dr ⎢⎣ρe (r)Φ(r) + 8πεe2 |∇Φ|2 ⎥⎦

N

∂q(r, s) = ∇2 q(r, s) − w*(r)q(r, s); q(r, s = 0) = 1 ∂s

where ρk(r) (k = +,−) denotes the local density of the co- and counterions, μok represents the chemical potential of the kth species, and β = (kBT)−1 where kB denotes the Boltzmann constant. (ii) The free energy arising from electrostatic interactions:54 β -elec =

∫0

where the field q(r,s) is commonly termed the single chain propagator and quantifies the probability of finding the “s”th segment of the chain at the location r (independent of its starting location). The latter can be obtained by solving a “diffusion-like” equation:81

12

β -mix =

(8)

λD = [4πλB(c+ + c − + mp 2cp)]−1/2

(12)

where c+, c− denote the concentration of positive and negative monovalent ions and mp denotes the overall charge of a polymer chain and cp denote the number concentration of the polymer chains. We choose polymer concentrations such that the λD ranged from 1.2 to 14 nm. The number concentration of polymer segments are in turn denoted as C, and in the results below we use a notation in which the segmental polymer concentrations are normalized by the overlap concentration C*. We solved the self-consistent mean-field equations within a three-dimensional Cartesian framework with periodic boundary conditions. We opted to solve the diffusion eq 10 numerically using a pseudospectral operator splitting method.81,86,90 We used a three-dimensional spatial box of size 200 × 200 × 200 (nm)3 with a discretization Δx = Δy = Δz = 1.5625 nm. Moreover, for the solution of eq 10 we used arc length discretizations (Δs) of 1/32, 1/64, 1/128, and 1/256, and used

(6)

In the above equation, zk denote the valency of the co- and counterions (assumed to be ±1 in the present article) and ρP(r) denotes the local segmental density of the negatively charged polymers. The last term above captures the charge contributions arising from the particles in which q(r) denotes the volumetric charge density of the particles. Minimization of the free energy terms with respect to the density fields ρk(r), ρP(r), and the electrostatic potential Φ(r), yields the equations governing the mean-field limit of our model. The fundamental equation underlying such a framework is the Poisson equation for the electrostatic potential:54 ε 2 ∇ Φ(r) = −ρe (r) (7) 4πe 2 6098

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Figure 2. Polymer densities and interaction potentials for the situation where both the particle and the polymers are uncharged. The legends indicate the bulk polymer concentration C/C* values: (a) Polymer density profiles near the particle for the case where Rg = 12 nm and the particle radius RC = 10 nm. r* represents the normalized distance from the particle surface defined such that r* = (r − RC)/Rg. Shown alongside (solid line) is a comparison to the analytical expression presented in ref 101, which accounts for particle curvature effects. (b) Pair interaction potentials for the case where Rg = 12 nm and the particle radius RC = 20 nm, and Δ denotes the normalized surface to surface distance between the particles such Δ = (r − 2RC)/Rg. Shown alongside (solid line) is a comparison to the analytical function U(r) ∝ (C/C*) exp(−1.687Δ) which was shown in our previous article86 to fit the depletion potentials for particles of sizes considered in the present work.

the results to extrapolate to the limit Δs → 0. The Poisson eq 7 was solved by an iterative procedure of the form: Φn + 1(r) = δ Φn(r) − (1 − δ)4πλB(∇2 )−1ρe (r)

potentials when the correlation length is set to Rg. These results are again consistent with the expectations for polymer depletion potentials in Θ solvents.72 B. Mixtures of Uncharged Particles and Charged Polymers. We begin by discussing the qualitative features of the interaction potentials for the case when the system is comprised of uncharged particles immersed in charged polymer solutions. Shown in Figure 3 are the results for the interaction potentials at different polymer concentrations for the case when RC = 10 nm and Rg = 24 nm. We observe that the interactions display a short-range attractive potential well followed by a barrier and a long-range repulsive decay of the interactions (for clarity, the repulsive portion of the potentials are enlarged and displayed in Figure 3b). It is also seen that both the strength of the short-range attractive well (Figure 3a) and the repulsive barrier (Figure 3b) increases with an increase in the bulk polymer concentrations. Further, with increasing polymer concentrations, the range of repulsive interaction is seen to decrease and the location of the peak in the potential is seen to move closer to the particle surface (Figure 3b). Interestingly, at higher polymer concentrations (see inset to Figure 3b) we observe there is also a small, but perceptible long-range attraction which follows the repulsive decay. Intuitively, in systems such as considered in this article, one may expect the interaction potential to possess an attraction arising from the polymer depletion effects. Moreover, in cases where the particles are charged, one would expect a repulsive force akin to the Debye−Huckel like interactions which manifest for colloidal macroions present in electrolyte solutions.23 While the results displayed in Figure 3 appear to conform to such intuitive expectations, several nontrivial features are also seen. Specifically, we observe that the range of the attractive portion of the potential is much smaller than the one seen for depletion interactions in neutral systems (compare Figure 3a with Figure 2b). A second surprising feature is the presence of a long-range repulsive component despite the fact that the particles are uncharged. Finally, the origin of the longer range attractive interaction manifest at higher polymer concentrations begs an explanation. In a number of other situations,92,94,95 the nature of polymermediated interparticle interactions have been observed to be strongly correlated to the density profiles of the polymer

(13)

where Φn(r) denotes the electrostatic potential at the nth step and δ < 1 is a numerical parameter tuned to provide convergence of the above equation. The inverse of the Laplacian operator ∇2 in the above equation was computed using fast Fourier transform routines (FFT).91

III. RESULTS AND DISCUSSION A. Mixtures of Uncharged Particles and Uncharged Polymers. We begin by discussing the results for the polymermediated interaction potentials for the case when neither the particles nor the polymers are charged. This scenario is commonly referred to as the “depletion” situation and the results for such a limit have been well-documented in the literature.72,73,86,92,93 The main intent of the section is to present the polymer density profiles and the pair interaction potentials so that they serve as a reference for comparing the results discussed in subsequent sections. In Figure 2a, we display the normalized polymer density profiles near the particle surface for different polymer concentrations. It can be seen that the polymer solution is depleted near the particle surface and that the range of depletion is independent of the concentration of the bulk polymers. Such a feature is consistent with our adoption of the Θ solution model in which the correlation length of the polymer solution is of the order of the polymer Rg. Indeed, it is seen that our numerical results agree quantitatively with the analytical prediction from ref 101 (which includes curvature corrections) when the correlation length is taken as Rg. The interparticle potentials corresponding to the uncharged situation are displayed in Figure 2b. It can be observed that the interactions are purely attractive in nature, and that the strength of the interaction varies linearly with the bulk concentration of the polymer solution. Moreover, the range of the potentials is seen to be independent of polymer concentration and scales with unperturbed radius of gyration Rg of the polymer chains. We also observe that an analytical form identified in our earlier work agrees quantitatively with our results for interaction 6099

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Figure 3. Polymer-mediated interaction potentials corresponding to the situation where the particles are uncharged. The results correspond to different normalized bulk polymer densities C/C* indicated in the legend. The polymer Rg = 12 nm and the particle radius RC = 10 nm, and Δ denotes the normalized surface to surface distance between the particles defined such that Δ = (r−2RC)/Rg. (a) Interaction potentials U as a function of Δ. (b) Portion of the interaction potentials is exaggerated to display the barrier and repulsive decay. The inset displays the long-range attractions for the four largest polymer concentrations. The points correspond to our numerical results and the lines are a guide to the eye.

Q(r) displays a weak, but perceptible charge inversion at large distances. As may be expected, increasing the polymer concentrations results in both a larger peak and charge inversion in Q(r). Together, the results displayed in Figure 4 serve to rationalize the qualitative trends of the interaction potentials seen in Figure 3. Explicitly, the short-range attractive portion of the potential can now be understood to be a consequence of the polymer depletion from the particle surfaces. Since the amount of polymer depletion increases (in absolute terms) with increasing polymer concentrations, the depth of the attractive interaction is also seen to increase with increasing polymer concentrations. The smaller range of such potentials (relative to the neutral systems) and its variation with increasing polymer concentrations can now be rationalized as a consequence of the attraction of the polymer monomers to the counterion cloud around the particle. The long-range repulsive interactions seen in Figure 3 can be understood to be a consequence of the interparticle electrostatic interactions arising from the charge clouds Q(r). We defer a discussion of the origins of long-range attraction to a later section (section III.E). C. Charged Particles and Charged Polymers: Particle Charge and Polymer Concentration Effects. In this section, we discuss the qualitative features of the interaction potentials for the case when the particles are charged. In Figure 5a, we display the results for the case where the polymer concentration is fixed (C/C* = 0.0926) and the particle charge is varied. For weakly charged particles, we observe that the interaction potentials display features which are similar to those noted for uncharged particles. Explicitly, the interactions are seen to be comprised of a short-range attraction and a long-range repulsion. With an increase in the charge on the particles, the strength of the repulsive portion is seen to increase at the expense of the attractive portion of the potential, and for strongly charged particles, the potentials are seen to become purely repulsive. Interestingly, as displayed in the inset to Figure 5a, the decay characteristics of the repulsive portion of the potential is seen to be independent of the particle charge. In Figure 5b, we depict the interaction potentials for the situation in which the particle charge is fixed (QC = 10) and the bulk polymer concentration is changed. Overall, we observe characteristics which are similar to those discussed in the context of uncharged particles (Figure 3). Explicitly, we observe that

segments around an isolated particle. Motivated by such considerations, we consider the density profiles of the polymer and counterion components around an isolated spherical particle immersed in the polyelectrolyte solution (Figure 4, parts a and b). In accord with our intuitive expectations, the polymer densities are seen to display a depletion near the particle arising as a consequence of the exclusion of polymer molecules from the interiors of the particles. Surprisingly, we observe that the range of polymer depletion decreases with an increase in polymer concentration and that the magnitudes of the length scales are much smaller compared to the neutral systems (contrast with Figure 2a). Further, despite the uncharged nature of the particle, the polymer density is seen to exhibit an “adsorption” like characteristic which manifests as an enhanced density (relative to the bulk density) near the particle surface. The results of Figure 4a can be rationalized with the help of the corresponding counterion density profiles displayed in Figure 4b. In this physical picture, the inclusion of charges on the polymer and ions leads to an electrostatic attraction between the ions and the polymer segments. Such interactions draw the ions away from the surfaces, thereby leading to the ion depletion seen in Figure 4b (note that point sized uncharged ions which do not otherwise interact with the surface of the particle are not expected to display any density modulations near the particle). Such electrostatic attractions are also expected to draw the polymer monomers toward the counterion cloud near the surface, and thereby leads to both a smaller range of depletion and the enhanced polymer densities (relative to the bulk value) seen in Figure 4a. Increasing polymer concentrations enhances the electrostatic attractions between the counterions and polymers and as a consequence the polymer monomers are seen to move closer to the surface while the ions move further away from the surface. In Figure 4c, we present results for the corresponding integrated charge density Q(r) around a single isolated spherical particle in the PE solution. Q(r) corresponds to the total charge present within a sphere of radius r with its center at the origin of particle. It is seen that near the surface of the particle, Q(r) displays a positive value due to the presence of the counterions. As we move radially outward from the surface of the particle, the adsorption of the polymer results in a decay of Q(r) to the charge neutral bulk conditions. Moreover, as a consequence of the polymer “adsorption” and excess densities, it is seen that 6100

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decay, characteristics which are similar to those seen in the context of uncharged particles. For dilute polymer concentrations, we observe that an increase in particle charge leads to a significant increase in the polymer “adsorption” on the particle. In contrast, at higher polymer concentrations, we observe that the polymer charge profiles are only mildly influenced by the increase in particle charge. To understand the above results, we recall our arguments in section III.B, which implicated the electrostatic interactions between polymers and the counterions as the origin of the adsorption peak in the polymer density profiles. When the particles are charged, the polymers experience an additional attraction to the particle surface arising from the electrostatic interaction between the polymer monomers and the charged surface. The peak in the resulting polymer densities is qualitatively related to the amount of polymer needed for the neutralization of the charges arising from the counterion cloud and the charge carried by the particle. For dilute polymer concentrations, the corresponding counterion densities are small and hence the particle charge is expected to be the dominant contribution to the total positive charges. In such a case, the polymer densities exhibit a significantly enhanced value relative to the bulk polymer concentration due to the amount of polymer charges required to neutralize the charge carried by the particle. At higher polymer concentrations, the counterion density cloud is expected to be the dominant contribution to the total positive charge near the particle, and hence the peak in polymer densities are comparable to the bulk polymer concentrations and are not significantly influenced by the particle charges. The above polymer density profile characteristics are also seen to be reflected in the integrated charge density profiles Q(r) displayed in Figure 6b. At low polymer concentrations, we observe that an increase in the particle charge leads to significant increase in the overall charge density and the effective positive charge around the particle. In contrast, at higher polymer concentrations, an increase in the particle charge leads to only a smaller increase in the peak of the charge distribution around the particle. Similar to our results for uncharged particles, at larger polymer concentrations we observe a weak charge inversion at far distances from the particle. Together, the density and charge profiles presented in Figure 6 allow us to rationalize the results presented in Figure 5. Explicitly as suggested in section III.B, the short-range attractive component in the pair-interaction potentials can be understood to arise from polymer-depletion induced whereas the longrange repulsion can be attributed to the electrostatic interactions between the charge clouds near the particle surface (see section III.E for a discussion of the long-range attraction). The former is expected to increase with an increase in the bulk polymer concentrations, and the latter is expected to be depend both on the inherent particle charge as well as the charge arising from the counterion cloud surrounding the particle. At low polymer concentrations and/or for strongly charged particles, the electrostatic repulsions arising from the charge clouds are expected to be dominant, and hence the pair interactions are purely repulsive. At higher polymer concentrations, the depletion effects are expected to become more important and hence the pair interactions display a short-range depletion attraction and a long-range electrostatic repulsion. D. Particle Curvature Effects. Parts a and b of Figure 7 display the influence of curvature upon the interaction potentials for uncharged and charged particles, respectively.

Figure 4. (a) Polymer (ρ(r)), (b) counterion (ρi(r)), and (c) integrated charge densities (Q(r) in electron units) for the situation where the particles are uncharged, polymer Rg = 12 nm and the particle radius RC = 20 nm. r* represents the normalized distance from the particle surface defined such that r* = (r − RC)/Rg. The results correspond to different bulk polymer densities C indicated in the legends by the corresponding values of C/C*.

except for the lowest polymer concentrations, the potentials possess an attractive well and a repulsive tail, in which both the repulsive barrier and attractive strength of the interaction potential increases with increasing polymer concentration. Moreover, the range of the repulsive decay (inset to Figure 5b) is seen to decrease with increasing polymer concentrations. For the lowest polymer concentrations, the potentials are seen to be purely repulsive over the entire range of interparticle separations. In Figure S1 of the Supporting Information, we present results that indicate that the long-range portion of the pair potential displays a weak attraction at higher polymer concentrations. To understand the physics underlying the above results, we again consider the polymer density variations and the charge distributions around an isolated spherical particle in the polyelectrolyte solution. The polymer density profiles (Figure 6a) are seen to display an adsorption peak followed by a long-range 6101

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Figure 5. Pair interaction potentials for the situation where both the particle and the polymers are charged. The points correspond to our numerical results and the lines are a guide to the eye. The polymer Rg = 24 nm and the particle radius RC = 20 nm, and Δ denotes the normalized surface to surface distance between the particles such Δ = (r−2RC)/Rg. (a) Results for different particle charges QC with the polymer concentration fixed at C/C* = 0.0926. The inset displays the repulsive portion of the pair potentials for QC = 0,5 and 10 normalized to their peak values Um and the intersurface coordinates shifted to the location of the peak value Δm. (b) Results for different polymer concentrations C/C* with the particle charge fixed at QC = 10. The inset displays the repulsive portion of the pair potentials normalized to their peak values Um and the intersurface coordinates shifted to the location of the peak value Δm.

Figure 6. (a) Polymer densities (ρ(r)) and (b) integrated charge densities (Q(r) in electron units) for the situation where the particles are charged (particle charge values are denoted as QC), polymer Rg = 12 nm and the particle radius RC = 20 nm. r* represents the normalized distance from the particle surface defined such that r* = (r − RC)/Rg. The results correspond to different bulk polymer densities C/C*.

Figure 7. Curvature dependence of interactions potentials for uncharged and charged particles. (a) Comparing U(r) for uncharged particles of radius RC = 10 and Rc = 20. Polymer Rg = 12 nm and C/C* = 0.1. (b) Comparing U(r) for charged particles (QC = 10) of radius RC = 10 and Rc = 20 at two different polymer concentrations indicated. Polymer Rg = 12 nm.

the interaction potential is seen to be shifted further from the surface. Figure 7b displays the corresponding curvature effects for charged particles (QC = 10) at a fixed polymer concentration.

In Figure 7a, it can be observed that for uncharged particles, increasing the curvature of the particle decreases the strength of both the attractive and repulsive portion of the interactions. Additionally, for smaller particles the location of the peak in 6102

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Figure 8. (a) Polymer densities (ρ(r)) and (b) integrated charge densities (Q(r) in electron units) for the situation where the particles are uncharged, polymer Rg = 24 nm. r* represents the normalized distance from the particle surface defined such that r* = (r − RC)/Rg. The results correspond to different bulk polymer densities C/C*.

When the polymer concentrations are dilute, the potentials are purely repulsive, and we observe that the smaller particles exhibit significantly stronger repulsive interactions. In contrast, at higher polymer concentrations we observe that the different particle sizes display comparable strengths of attraction (contrast with the results displayed in Figure 7a for uncharged particles) but the smaller particles exhibit a smaller repulsive barrier. In the next section, we provide a quantitative analysis of the interaction potentials which rationalizes the results of Figure 7. However, for the purposes of completeness, we provide a qualitative picture to help understand the results presented in this section. In Figure 8a, we display the corresponding curvature effects on the polymer density profiles near uncharged particles. We observe that while the range of depletion remains approximately the same for the different sized particles, the magnitude of the adsorption peak is slightly mitigated for the smaller particles. These results, in conjunction with the smaller volume and surface area of smaller particles, suggests that the entropic costs of polymer depletion and the resulting attractions are expected to be weaker for smaller particles. Moreover, as a consequence of polymer density characteristics, the ion depletion (not displayed) and the corresponding integrated charge densities (Figure 8b) near the surface of the particle are also seen to be of lower magnitude for smaller particles. Such effects can be understood to underlie the weaker repulsive interactions seen in Figure 7a. To explain the results displayed in Figure 7b, we display the Q(r) profiles in Figure 9 for particles of different curvatures at two different polymer concentrations. We observe that at low polymer concentrations, the peak in Q(r) is approximately the same magnitude for different particle sizesa fact which is rationalized by noting that for dilute polymer concentrations, the inherent particle charge is expected to be the dominant contribution to the charge distribution around the particle. Since electrostatic interactions between charged particles depend on the center-to-center distances (rather than the surface-tosurface distance), we expect the repulsions to be stronger for smaller particles at the same corresponding intersurface distances. Such a reasoning serves to explain the stronger repulsions seen in Figure 7b for smaller particles at dilute polymer concentrations. For larger polymer concentrations, we observe in Figure 9 that the magnitude of the peak in Q(r) is much larger for the larger particles. Indeed, for such situations, the charge

Figure 9. Integrated charge densities (Q(r) in electron units) for the situation where the particles are charged (QC = 10), polymer Rg = 24 nm. r* represents the normalized distance from the particle surface defined such that r* = (r − RC)/Rg. The results correspond to different bulk polymer densities ρb. The bulk polymer densities ρb are in-turn normalized by the polymer overlap concentration and are indicated in the figure through the corresponding C/C* values.

distribution around the particle is influenced more significantly by the ion and polymer clouds around the particles. Since the volume occupied by such a cloud is larger for bigger particles, the corresponding Q(r) is also seen to be of significantly higher magnitude for larger particles. For larger polymer concentrations, we hypothesize that the repulsive component of the interaction between the particles is influenced more significantly by the magnitudes of charge densities Q(r). Hence, the electrostatic repulsions are expected to be stronger and correspondingly the depletion attractions to have a weaker influence for larger particles. Such expectations are seen to be consistent with the behaviors seen in Figure 7b. E. Polymer Bridging and Long-Range Attractions in U(r). In the previous sections, we discussed the qualitative features of the short-range attraction and long-range repulsion portions of the interaction potentials. However, many of our interaction potentials also displayed a weak, but perceptible long-range attraction following the electrostatic repulsion. Such attractive interactions were seen to become more prominent at higher polymer concentrations and/or particle charges. A number of earlier studies on polyelectrolyte-mediated interactions between particle surfaces have noted the existence 6103

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IV. FUNCTIONAL FORM OF INTERACTION POTENTIALS The preceding sections presented the results for the polymermediated pair interaction potentials between charged and uncharged particles and identified the salient qualitative features and their dependence on polymer concentrations, particle charges and their curvatures. In the present section, we formalize the results of the preceding section by probing whether the interaction potentials can be quantitatively fitted through a combination of a depletion-like attractive contribution and an electrostatic repulsion. Specifically, we undertook a fit of the potentials by the following functional form: U (r ) = AU Figure 10. Interaction potentials U(r) for different bulk polymer concentrations C/C* and particle charges QC for Rg = 12 (closed symbols) and Rg = 24 (open symbols). Particle radius Rc = 20 nm. Only the long-range attractive portion of the interactions are displayed and the lines are meant to be a guide to the eye.

exp( −κU r ) − BU exp[− (r − 2R C)/δU ] r

(14)

where AU, BU, δU, κU > 0, and r denotes the center-to-center distance between the particles (the subscript U is to denote that the parameters correspond to the potential fits). In the above, the first term represents a “Yukawa”-like screened electrostatic repulsion in which κU denoting the fitted “screening length” and A U represents the “effective” charge of the particles.23 The second term represents the depletion-like attraction, with BU and δU representing respectively the strength and the range of the interaction (the above specific form for the depletion portion was inspired by the results of our earlier work 86 ). Note that the above potential form ignores the presence of a weak long-range attraction. As discussed in section III.E, due to the extremely weak nature of the long-range attraction and the difficulties involved in nonlinear fitting with multiple parameters, we made no attempt to correct for the long-range attractions. In Supporting Information Figure S2a−d, we display the parameters AU, κU, BU, and δU for a few of the systems as a function of the concentrations C/C*. In Supporting Information Figure S3a−d, we display the fits corresponding to eq 14 alongside our numerical values for the potential. In such a context, it can be seen that the above functional form provides an excellent representation of our numerical results. In the discussion which follows in the sections below, we clarify on a systematic basis the parametric dependencies of the parameters underlying our interaction potentials. A. Connection Between Interaction Potentials and Density Profiles. Considering the strong qualitative similarities noted between the polymer density profiles and the

of a long-range attraction and have attributed it to the presence of polymer bridging between the surfaces.57,66,69,96 In Figure 10, we provide tentative evidence that polymer bridging effects are also likely the cause of long-range attractions in our system. Explicitly, we display the influence of polymer Rg on the interaction potentials for different polymer concentrations and particle charges. Polymer bridging effects are expected to be strongly correlated to the physical Rg of the chains. It is evident from the results in Figure 10 that the long-range attraction is always more significant for the larger Rg chains, thereby lending support to the hypothesis that such interactions arise from polymer bridging effects. For most parameters corresponding to the results presented in the article, the bridging-induced long-range attraction was seen to be much smaller in magnitude compared to the shortrange attraction and the electrostatic repulsions. Such a trend could likely be a result of the parametric range of polymer concentrations and particle charges investigated in this work. Moreover, polymer bridging effects are also expected become mitigated due to the finite curvature of the particle. Due to the weak magnitude of such interactions, we refrain from further discussion of the quantitative features of the long-range bridginginduced attractions.

Figure 11. Comparison of (a) κU (solid symbols), κρ (open symbols) and (b) δU (solid symbols), δρ (open symbols) for two different parameter combinations indicated (more comparisons are presented in Supporting Information Figure S5, parts a and b). 6104

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interaction potentials, and the fact that in many other situations the length scales embodied in the density profiles are usually also representative of those characterizing pair-interactions, we also sought to fit the density profiles by a functional form similar to eq 14: ρ(r ) = Aρ

exp( −κρr ) r

− Bρ exp[−(r − R C)/δρ]

representing the single chain structure factor of an unperturbed Gaussian chain. To reduce the above expression to the model considered in the present article, we consider the case with negligible salt concentration (c− ≃ 0) and introduce a notation in which the Debye length for just the counterion component is denoted as λ+ = (4πλBc+)−1/2. In such a case, the expression for Sq(k) becomes

(15)

2 2

In Supporting Information Figure S4a−d, we display a few representative fits corresponding to the above function alongside our numerical results for the densities where it is seen that the above functional form provides an excellent representation of our numerical results. Of specific interest is the correlation, if any, between the length scales κU, δU characterizing the interaction potentials and the corresponding length scales κρ, δρ of the density profiles. In parts a and b of Figure 11 (and Supporting Information Figure S5, parts a and b), we compare the respective length scales as a function of concentration for a few different parameters. Strikingly, we observe that within the statistical errors, the length scales embodying the density and potential variations are quantitatively identical to each other over the range of concentrations. This suggests that an understanding of the origins and physical parameters underlying κU and δU can be derived by considering the origins and mechanisms underlying the corresponding density profile parameters κρ and δρ. We exploit this premise in the following sections, and unravel the physical underpinnings of the length scales κU and δU by considering the behaviors of the corresponding length scales κρ and δρ. B. Decay Range of the Repulsive Portion of the Potential and Density Profiles. Our hypothesis is that the long-range decay of the repulsive part of the potential above is related to a Debye−Huckel like electrostatic interaction between two macroions within an electrolyte medium of PEs and counterions.23 In such a situation, the decay behavior of the potential is expected to be related to the correlations embodied within the static structure factor of the PE solution. When viewed within this perspective, the quantitative equivalence noted between κU and κρ is straightforwardly rationalized. Below, we first outline an analysis of the correlations in the PE solution to extract the underlying length scales. Subsequently, we compare our fitted parameters κU with the resulting length scales to verify our hypothesis. To extract the parametric dependencies underlying the static structure factor of the PE solution, we employ the framework of random phase approximation (RPA).97−100 Within the framework of RPA, the charge−charge correlation function is given as Sq(k) = ⟨ρq (k)ρq ( −k)⟩

Sq(k) =

1 c+ + c − + qp 2cpf (bp 2k 2qp /6)

(19)

Sq(k) =

k2 k 2λdp2 + 1

(20)

in which λ+2 1 + qp

λdp 2 =

The above expression eq 20 can be recognized to be the classical Debye expression for the charge correlation function of a simple fluid electrolyte solution. The decay length of the correlation function for such a regime can be identified to be λdp and represents the limit in which the polymer molecule is assumed to be just a “point” multivalent ion. (ii) The other limit of the expression eq 19 occurs for semidilute and concentrated solutions (C/C* ≫ 1): Sq(k) =

k 2(b2k 2 + 12) 4πλ+(b2λ+2k 4 + b2k 2 + 12)

(21)

Identification of the decay behavior of the above correlation length entails the computation of the roots of the denominator of eq 21. Explicitly, the roots kr are given as kr = ±

⎛ λ ⎞2 ± 1 − 48⎜ + ⎟ − 1 ⎝b⎠

1 λ+ 2

Due to the presence of both real and imaginary parts characterizing kr, the above analysis suggests that the polymer correlations exhibit a combination of an exponential decay and an oscillatory component. A length scale rRe which is inverse of the imaginary part of kr is expected to be responsible for the long-range decay of correlations, and a length scale rIm which is inverse of the real part of kr is expected to characterize the scale of the oscillation of correlation functions. In the limit when λ+ ≳ b, we can identify these length scales as

(16)

(17)

In the above, qp denotes the number of charges on a single PE chain and bp is defined such that qpbp2 = Rg2 and represents the spatial distance between the charges along the PE chain. The function f(x) in the above equation denotes the Debye function: 2(x + e−x − 1) f (x ) = x2

2 2

b4k6qpλ+2 + b4k 4qp + 12b2k 2qp − 72 + 72e−b k qp /6

A general analysis of the above expression (and the relevant length scales) proves too complicated due to the nonlinear nature of the wave vector dependence. However, two relevant limits can be extracted from eq 19: (i) For dilute PE solutions such that C/C* ≪ 1, we have

where ρq(k) denotes the Fourier transform of the local charge density and k denotes the wave vector. The RPA expression for such a correlation function is given as97−100 Sq−1(k) = 4πλBk−2 +

k 2λ+2(b4k 4qp + 12b2k 2qp − 72 + 72e−b k qp /6)

2 4 3 λ+

rRe = ±

12λ+ b

4

6105

3

2λ+

rIm = ± (18)

+

3

4λ+ b

−

1 3

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As an additional test of our hypothesis, we rescaled the spatial scale of the potential by the length scales embodied in the correlation length above. In parts a−d of Figure 14, we display the repulsive portion of the potentials normalized by the length scales λdp and rIm. It is seen that at low polymer concentrations the potentials are collapsed by the length scale λdp, but such a length scale fails to collapse the potentials at higher concentrations. For higher concentrations the other length scale rIm appears to collapse the potentials. We do note that a strict adherence to such a scaling is not expected to work due to the presence of two length scales in the correlation functions and the interaction potentials. In summary, the above results demonstrate that the length scales and the behaviors characterizing the long-range behavior of the potentials and the density profiles can be understood by considering the charge correlation functions. We presented explicit expression for the parametric dependencies of the length scales underlying the charge correlation lengths and demonstrated their relevance in characterizing the long-range decay behavior of the interaction potentials. We note that an analysis along the above lines, albeit using the ground state dominance approximation (GSA), was effected by Johanny and Chatelier60 to study the density profiles near a flat surface. However, since we use the RPA framework, we are able to recover both the dilute and semidilute limits of the correlation function from an unified expression. Our results presented in this section, serve to explain the insensitivity of the long-range repulsion to the charge density of particles. In contrast, due to the presence of the center-to-center distance r in the potential, the interactions are expected to depend on the size of the particles (for the same intersurface distance). Such dependencies were hypothesized and used to explain the curvature dependencies of the interaction potentials in section III.D. C. The AU Parameter. The second parameter we consider for discussion is AU. The preceding section presented evidence that the long-range decay term in our potential behaves similar to what may be expected based on electrostatic interactions between macroions in an electrolyte solution. In such a context, the AU parameter is typically viewed as an “effective charge” of the macroion which encompasses the effects of the bare charge of the particle and the screening arising from the counterions in the solution. For instance, within the classical DLVO theory, the effective charge Qeff underlying such interactions is usually

Figure 12. Comparison of the different length scales arising from the RPA treatment of charge density fluctuations displayed as a function of the polymer concentrations C/C* (refer to the text for the meanings of λdp, rIm, and rRe). The results correspond to Rg = 24.

In Figure 12, we display λD, and the decay and oscillation length scales rIm, rRe as a function of the concentration C/C* for Rg = 24. We propose that the above behavior of the structure factor serves to explain the long length scale behavior of the density profiles and the interaction potentials. In parts a and b of Figure 13, we display the parameters κU as a function of C/C* and compare it with the length scales λdp, rIm, and rRe discussed above. Within the limits of statistical errors present in such nonlinear fits, it is seen that κU approximately matches the length scale rIm. However, since rRe and rIm are seen to be almost proportional to each other within the studied range of concentrations, our decay length scales can be equally well taken to be as rRe. More interestingly, from the results for Rg = 12, we observe a signature of a transition of κU to the length scale λdp at dilute polymer concentrations. Since the concentrations probed for Rg = 24 are larger, such a crossover is not seen in explicitly in such a case. Moreover, while κU is seen to be by and large independent of the charge on the particles and dependent only on the polymer concentrations, Figure 13a indicates that there is tangible curvature dependence in κU. Such a dependence cannot however be captured within the simple RPA analysis presented above.

Figure 13. Comparison of κU (solid symbols) with the length scales λdp, rim, and rRe for the different parameter combinations indicated: (a) Rg = 24; (b) Rg = 12. 6106

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Figure 14. Test of the collapse of the repulsive portion of the interparticle potentials. r* denotes the position of the peak in the potential: (a) QC = 20, Rg = 24; (b) parameters correspond to part a, except that rIm is used to normalize the length scales; (c) QC = 10, Rg = 12; (d) parameters corresponding to part c, except that rIm is used to normalize the length scales.

related to the bare charge QC, the screening length κ and the particle radius RC through23 ⎛ e κR C ⎞2 Q eff = Q C 2⎜ ⎟ ⎝ 1 + κR C ⎠

surprisingly, except for the case of QC = 0, we observe that Q* is approximately proportional to the bare charge of the particles themselves. Even for the case of QC = 0, we observe that while Q* is nonzero, the value is much smaller in magnitude, consistent with the uncharged nature of the particle. Together, our results indicate that the Q* which appears in eq 23 is consistent with the classical DLVO like framework and is indeed proportional to the “bare charge” of the particles. This result is somewhat surprising considering the complicated nature of electrostatic interactions in our system. Unfortunately, we do not have a good explanation for justifying this occurrence. Now we discuss the behavior of R*, and for the moment consider only QC ≠ 0 cases. Overall, we observe R* is dependent strongly on RC and varies only to a minor extent with Q*. More interestingly, we observe that R* is much larger than bare radius of the particles. Among the different length scales and behaviors discussed in parts B and C of section III, we hypothesize that R* is most likely to be correlated to the peak in the integrated charge distributions Q(r). To probe this further, in Figure 15b, we display the location of the peaks in the charge density profiles (denoted as Rp in the figure). We observe that within statistical errors, the Rp is approximately independent of the polymer concentration and the overall charge on the particles. More interestingly, we observe that Rp exhibits a magnitude which is comparable to the R* deduced in our fits (however, it can be noted they are not identical). For QC = 0, we observe in Figure 15b that the Rp displays a strong variation with concentration and exhibits far higher magnitudes than the results corresponding to the charged particles of the same particle size. These results are consistent

(22)

A number of subsequent researches have examined the validity of the above expression and have proposed corrections and improvements which more accurately account for the nature of counterion screening in different situations. On the basis of our discussion in parts B and C of section III, we expect the scenario accompanying our model to be substantially more complex. Indeed, electrostatic interactions in our system involve a combination of the charge cloud consisting of the macroion and the associated counterions. Indeed, as we saw in section III.B, even uncharged macroions exhibited a longrange repulsion as a consequence of the counterion and polymer density profiles. Notwithstanding these caveats, we probed whether we can indeed fit our AU parameter by a function form: ⎛ e κUR* ⎞2 AU = (Q )2 ⎜ ⎟ * ⎝ 1 + κU R ⎠ (23) * where Q* and R* were treated as adjustable parameters. In Figure 15a, we display the fits of AU based on eq 23, and the results for our fitting parameters are displayed in Table 1. We observe an excellent conformance of the AU parameter to a DLVO like functional dependence on the screening length κU. Based on the fits we observe that Q* is independent of the RC of the particles and is only dependent on the QC. More 6107

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Figure 15. (a) Plot of the AU values as a function of κU for the parameters indicated in the figure. Also displayed are the best fit lines corresponding to the function eq 23. For clarity, we do not display separate fits for the parameters RC = 20, Rg = 24, QC = 10 and RC = 20, RG = 24, QC = 20. (b) Location of the peak in the integrated charge density profiles Q(r) as a function of C/C* for the parameters displayed in part a. The legends are the same as in part a.

Table 1. Q* and R* Parameters Corresponding to a fit of AU to the Functional Form Equation 23 Rc

Rg

QC

Q*

R*

10 10 20 20 20 20

24 12 12 24 12 24

0 10 10 10 20 20

0.75 5.04 5.12 5.72 8.96 8.88

47.2 29.4 45.1 44.6 41.5 41.8

ϕ* σ ≃1+ ϕb Dmαϕb

(24)

From the above expression, we observe that for the case when the particle charge σ is identically zero, ϕ* ≃ ϕb, a result which was seen in section III.B. In the case when the charge of the particle is nonzero, the last term above indicates that ϕ* ≫ ϕb for small ϕb, whereas, ϕ* ≃ ϕb for large ϕb. These observations are consistent with the results seen in section III.C. In a series of articles, Andelman, Orland, and co-workers considered systems consisting of highly charged surfaces in contact with dilute concentrations of polyelectrolytes.67−69 They considered the entropic costs arising from the polymer density depletion and balanced it against the favorable electrostatic energy resulting from the interaction between polyelectrolytes and the surfaces. Using scaling and numerical results, they clarified the parametric dependencies underlying the length scale Dm. The system considered in the present study differs slightly from the one considered in the above-discussed articles in the fact that we have considered more weakly as well as noncharged surfaces. In such situations, the presence of counterions and their interactions with the polymer cannot be ignored. Despite these differences, we hypothesize that the physics of the polymer depletion in our case can still be understood to arise from an interplay of the entropic costs arising from the gradients in polymer density and the favorable electrostatic interactions. However, we propose that the electrostatics involve the interactions between the polymer and the counterion cloud in addition to those between the polymer and the charged surface. Below, we adapt the arguments of Andelman while accounting for such additional features and test the resulting scaling predictions against our numerical results. In the following, we ignore the effects of curvature and the variations evident in the counterion density profiles. For the case of charged particles, we assume that the Dm primarily arises due to the interplay between entropic costs of polymer depletion balanced against the electrostatic energies arising from interactions between the polymer and the charged surfaces. If we estimate the former as ϕ*/Dm and the latter as (see ref 68): ϕ*σDm2, we get the result that Dm scales as σ−1/3 and should be independent of the polymer bulk concentration, ϕb. For the case of uncharged particles, we utilize the same scaling expression

with the extraneously large values of R* noted for QC = 0. However, we are unable to come up with a transparent correlation between Rp and R* for the uncharged particles. In summary, the above results indicate that the long-range repulsion arising in the polymer-mediated interactions, can be likened to an“electrostatic” repulsion involving an “effective” charge which exhibits a DLVO-like functional dependence on κU. The “bare” charge was approximately proportional to the charge on the particles and the “radius” R* exhibited a strong correlation to the peak in the integrated charge density profiles. The case of uncharged particles also conformed to such a fit, but however the underlying parameters did not admit such a transparent interpretation. D. δU Parameter. In this section, we discuss the physical underpinnings of the δU parameter. Unlike the previous section, however, we confine ourselves to scaling arguments (which ignore the curvature effects) and demonstrate broad consistency between our scaling proposals and the results from our potential fits. At the outset, we discuss a few aspects regarding the depletion characteristics of the polymer density profiles. The results presented in parts B and C of section III indicated that the polyelectrolyte densities profiles exhibited a depletion zone near the particles followed by an adsorption like peak which eventually decays to the bulk concentration. For the purposes of scaling arguments below, we visualize the polymer density profiles as possessing a peak value which is denoted as ϕ* and the location of the peak as measured from the surface of the particle denoted as Dm. Further, we denote the bulk polymer concentration as ϕb. Since we ignore curvature effects, in this section we work with charge density of the particle surface which is denoted as σ. If the density variations of the counterions are neglected, the peak value ϕ* can be related to the other parameters by simple charge conservation arguments: 6108

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Figure 16. (a) Dm values as a function of the bulk concentrations (displayed here as C/C*). The legend indicates the parameters (QC,Rg,RC) corresponding to the result. The best fit power law for the case of (QC = 0, Rg = 24, RC = 10) which indicates an exponent of −0.17 is depicted. (b) Parameter δρ from our fits displayed as a function of Dmϕ*/ϕb. The legend indicates the parameters (QC, Rg, RC) corresponding to the result. The best fit power law fit for all the combined results is depicted in the figure and indicates an exponent of 0.95.

E. BU Parameter. In this final section, we clarify the mechanistic underpinnings of BU parameter. Our discussion is in the spirit of the previous section, and relies primarily on scaling arguments. Instead of relating BU explicitly to the other parameters in our model, we focus on the contact value of the interaction potential. From a mathematical perspective, in the notation of eq 14, the contact value of the potential U(r = 2RC) is given as

for the entropic cost of depletion, however, for electrostatics interactions, we modify the energy as (ϕ*)2 D3m. By balancing these factors, we obtain, Dm ∼ (ϕ*)−1/4. Furthermore, for uncharged systems, as discussed in the context of eq 24, ϕ* ≃ ϕb and hence we obtain Dmϕb−1/4. In Figure 16a, we display the peak location of the densities, Dm, from our numerical results as a function of the bulk polymer concentrations. Broadly, it is seen that the results agree with our predictions above. Specifically, we observe that systems of lower charge density display a larger Dm (a result which is consistent with the prediction Dm ∼ σ−1/3). Moreover, it is seen that Dm for charged systems is almost independent of the bulk polymer concentrations. In contrast, for uncharged particles, the Dm is seen to display an explicit dependence on bulk polymer concentration. The scaling exponent is seen to be −0.17, which is in reasonable agreement with the predicted −1/4 (small discrepancies are to be expected since we ignored curvature effects). In many works, it has been common to define a “depletion thickness” based on the total amount of polymer depleted.73,74,101 Based on such considerations, we speculatively ascribe δρ (and thereby δU) to a depletion thickness which corresponds to the layer width which accounts for the total amount of the polymer depleted. We estimate the latter such that δρϕb ≃ Dmϕ*, or in other words, δρ ∼ Dmϕ*/ϕB. To verify this scaling, in Figure 16b, we display the δρ as extracted from our density fits as a function of Dmϕ*/ϕb (we obtain Dm and ϕ* from our density profiles, but they are virtually identical to those extracted from the fits). The shown best fit power law is seen to display an exponent of which is consistent with the predicted linear dependence. In the Supporting Information, Figure S6, we demonstrate that the corresponding parameters δU also obey a scaling with the same exponent when considered as a function of Dmϕ*/ϕb. In summary, the depletion portion of the potential was shown to involve a thickness which accounted for both the peak in polymer density and its location. The location of peak was argued to be determined by an interplay between electrostatic attractions and the entropic costs of polymer density variations whereas the peak value was suggested to be determined by charge conservation arguments. The resulting scaling prediction for δρ and δU was shown to be consistent with our fitted parameters.

U (r = 2R C) = AU

e−2κUR C − BU 2R C

(25)

Hence, knowledge of AU and κU and the contact value of the interaction, U(r = 2RC), allows one to deduce BU. At a physical level, we focus on the contact value of the potential since it has a transparent connection to the free energy of the polymer layer near a surface, or equivalently, the surface tension of the polyelectrolyte solution. To deduce the latter, we can directly utilize the free energy factors discussed in the previous section. We recall that at a scaling level, the free energy contribution of the polymer layer near the surface was suggested to involve an interplay between the entropic terms arising from the polymer depletion and the electrostatic attractions between the PE and the oppositely charged counterions and the surface. At equilibrium, the resulting free energy contribution scales as −ϕ*/Dm, and at a scaling level, represents the surface tension of the polyelectrolyte in contact with the charged surface. To deduce the interaction potential at contact, we note that in addition to the above surface tension term, there is expected to be an additional contribution arising from the direct electrostatic interaction between the particles. However, the latter is expected to be independent of the polymer concentration and only dependent on the surface charge density of the particles. Together these considerations predict that U(r = 2RC) to scale as −ϕ*/Dm+Fe, where Fe is suggested to depend only on the charge and size of the particles. Moreover, beyond the implicit dependencies embodied in ϕ* and Dm, the first term is expected to be independent of the charge on the particle and the Rg of the chains. In Figure 17, we test the above hypothesis by displaying the contact values based on eq 25 as a function of the ϕ*/Dm. We observe excellent agreement with the above predictions. Explicitly, from Figure 17a, we observe that for a specified 6109

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Figure 17. (a) Contact values of the interaction potential (U(r = 2RC)) as a function of ϕ*/Dm for the particle size Rc = 20. The legend indicates the parameters ((QC,Rg,RC) corresponding to the result. The best linear fits are depicted. For the case of QC = 20, the linear fit corresponds to the equation y = −115.7x + 1.9. For the case of QC = 10, the linear fit corresponds to the equation y = −121.4x + 0.6. (b) Contact values of the interaction potential (U(r = 2RC)) as a function of ϕ*/Dm for the particle size Rc = 10. The legend indicates the parameters (QC, Rg, RC) corresponding to the result. The best linear fits are depicted. For the case of QC = 10, the linear fit corresponds to the equation y = −41.6x+1.7. For the case of QC = 0, the linear fit corresponds to the equation y = −57.3x.

electrostatic repulsions. We demonstrated both qualitatively and quantitatively that the interactions between the particles in PE solutions can be expressed as a sum of a depletion-like attraction and an electrostatic Debye−Huckel like repulsion. However, the strength and scales of the depletion and electrostatic interactions were shown to possess a complex relationship to the parameters underlying our model. We presented scaling arguments and analytical theory to rationalize the parameters underlying the interaction potentials. Our results serve to resolve some of the contradictions regarding the nature of interactions noted in the earlier works.42,68,69 Explicitly, our results indicate that depending on the particle charges and polymer concentrations, the interactions can be either purely repulsive or possess a short-range attraction. Moreover, we also demonstrated the presence of a long-range attraction at larger polymer concentrations. The more interesting and nontrivial aspects of our results relate to the fact that the interaction potentials can be fitted quantitatively to a combination of depletion-like attraction and an electrostatic repulsive interactions and the intriguing DLVO-like connections noted between the effective charge underlying the electrostatic interactions and the screening length of the solution. In future articles, we propose to build on the present study in a number of different directions. Specifically, we wish to use the numerical framework developed herein to study the influence of multibody interactions, polyelectrolyte rigidity,102 and the effect of particle anisotropies.85 Also, systems possessing a short-range attraction and long-range repulsions have been suggested to form equilibrium cluster phases.103 We propose to investigate the phase behavior of PE-particle systems to probe the occurrence of such characteristics.

radius and charge of the particle, the contact values obey the same linear fits as a function of ϕ*/Dm. Moreover, we observe that systems with differing particle charges differ only in the intercepts and not in the slopes (see caption of Figure 17 for the functions underlying the fits). These results are consistent with the above expectation that beyond the dependencies embodied in ϕ* and Dm, the slopes are expected to be independent of the charge of the particles and Rg. Moreover, for a given size, Fe is seen to depend only on the charge density of the particles. As expected, particles with larger charges exhibit larger intercepts consistent with the increased repulsion at contact. The results for smaller particles (Figure 17b) do exhibit some small deviations from the above predictions. However, we do observe that the intercept for the uncharged particle is identically zero (as expected from the above arguments), and that the slopes for the charged and uncharged particles are relatively close in magnitude. This can be construed as a reasonable agreement for a theory which made no allowance for curvature effects. In summary, the results presented in this section serve to demonstrate that strength of the depletion portion of polymermediated interaction potential scales as the free energy arising from the interplay between electrostatic attractions and the entropic costs of polymer density variations.

V. SUMMARY In summary, in this article we presented results for the interactions between uncharged and charged particles in PE solutions. We considered a model in which the particles possessed a fixed charge density and examined the resulting interactions in the presence of both dilute and semidilute PE solutions. Within such a framework, we studied the interplay between depletion interactions which arise as a consequence of polymer exclusion from the particle interiors and the electrostatic forces which arise as a result of the presence of charges on the polymers and particles. Our results indicated that for weakly charged and uncharged particles, the interaction potentials predominantly consisted of a short-range attraction and a long-range repulsion. When the particle charge was increased, the interaction potentials became purely repulsive. A longer range attraction attributed to polymer bridging effects was also evident for some parameters. The latter was much weaker relative to the short-range attraction and the

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ASSOCIATED CONTENT

S Supporting Information *

Results supplementing our discussion of the article. This material is available free of charge via the Internet at http:// pubs.acs.org/.

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AUTHOR INFORMATION

Corresponding Author

*(V.G.) E-mail: [email protected]. 6110

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Notes

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported in part by grants from Robert A. Welch Foundation (Grant F1599) and National Science Foundation (DMR-1005739 and DMR-1306844) and the US Army Research Office under grant W911NF-13-1-0396. The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing computing resources that have contributed to the research results reported within this paper.

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