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Atomistic Molecular Dynamics Simulations of Charged Latex Particle Surfaces in Aqueous Solution Zifeng Li,† Antony K. Van Dyk,‡ Susan J. Fitzwater, Kristen A. Fichthorn,† and Scott T. Milner*,† †

Department of Chemical Engineering, Pennsylvania State University, State College Pennsylvania 16802, United States Dow Coating Materials, The Dow Chemical Company, Collegeville, Pennsylvania 19426, United States

‡

S Supporting Information *

ABSTRACT: Charged particles in aqueous suspension form an electrical double layer at their surfaces, which plays a key role in suspension properties. For example, binder particles in latex paint remain suspended in the can because of repulsive forces between overlapping double layers. Existing models of the double layer assume sharp interfaces bearing fixed uniform charge, and so cannot describe aqueous binder particle surfaces, which are soft and diffuse, and bear mobile charge from ionic surfactants as well as grafted multivalent oligomers. To treat this industrially important system, we use atomistic molecular dynamics simulations to investigate a structurally realistic model of commercial binder particle surfaces, informed by extensive characterization of particle synthesis and surface properties. We determine the interfacial profiles of polymer, water, bound and free ions, from which the charge density and electrostatic potential can be calculated. We extend the traditional definitions of the inner and outer Helmholtz planes to our diffuse interfaces. Beyond the Stern layer, the simulated electrostatic potential is well described by the Poisson−Boltzmann equation. The potential at the outer Helmholtz plane compares well to the experimental zeta potential. We compare particle surfaces bearing two types of charge groups, ionic surfactant and multivalent oligomers, with and without added salt. Although the bare charge density of a surface bearing multivalent oligomers is much higher than that of a surfactant-bearing surface at realistic coverage, greater counterion condensation leads to similar zeta potentials for the two systems.

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INTRODUCTION Waterborne latex-based paints are of particular commercial importance, in part because of environmental restrictions on the use of volatile organic compounds (VOCs).1 In latex paints, synthetic polymer binder particles are dispersed along with inorganic pigment particles in an aqueous phase. The stability of the binder particles requires a balance between interparticle attractive van der Waals forces and repulsive electrostatic forces, which is typically achieved by manipulating the electrostatic forces. The electrostatic forces between particles arise from surface charge groups, which attract a cloud of counterions near to the surface, forming an electrical double layer.2 Because ions and counterions are concentrated in the Stern layer close to the bare surface, the electrostatic potential decays rapidly in the Stern layer, so that the surface appears more weakly charged. Beyond the Stern layer, the electrostatic potential in the diffuse layer decays to zero, and it decays faster with added salts, which help screen the potential. To understand how to tune the stability of paint suspensions at the molecular level, we address some fundamental questions regarding the double layer at the interface between charged polymer particles and water: (1) What is the charge distribution and electrostatic potential in the double layer? (2) In what region can the double layer be described by the Poisson− Boltzmann equation? (3) How do the double layers compare, © XXXX American Chemical Society

for surfaces with adsorbed surfactant and surfaces with multivalent grafted oligomers? (4) How does added salt influence the double layer? (5) How concentrated are the ions in the Stern layer? Experimental probes of charged particle surfaces in aqueous solution are rather limited in the information they can provide. Electrophoresis can measure the zeta potential, which can be interpreted as the value of the electrostatic potential at the apparent location of the no-slip boundary condition between particle and fluid. Thus, the zeta potential measurement provides only a single value of the electrostatic potential at a certain distance from the particle surface, not a continuous potential profile. Scattering techniques have been used to investigate the local environment of ions in solution, to infer whether the ions are fully or partially solvated.3,4 Only in special circumstances can scattering techniques be used to infer ion profiles near a charged surface, for example in recent work using resonant anomalous X-ray reflectivity to measure profiles of specific heavy ions near a flat interface.5 Analytical theories have been developed to describe the ion and potential profiles across the double layer but with their limitations. The Gouy−Chapman model with the Poisson− Received: October 24, 2015 Revised: December 11, 2015

A

DOI: 10.1021/acs.langmuir.5b03942 Langmuir XXXX, XXX, XXX−XXX

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are randomly end-grafted to the surface, but otherwise free to explore different conformations; the surfactants are physisorbed, and diffuse freely across the surface. The details of our system are specific to commercial latex binder particles, but the general features of a diffuse polymer−water interface, with mobile adsorbed anionic surfactants and randomly grafted multivalent hairs, are representative of a broad class of aqueous polymer colloids. In this work, we use atomistic MD simulations to study the double layer that results from (1) adsorbed ionic surfactant and (2) grafted hairs at a polymer−water interface, each with and without added salt, resulting altogether in four systems to explore. In particular, we will compare the interfacial width and structure, bound ion and counterion distributions, and extent of counterion condensation, for the four systems.

Boltzmann equation embedded was developed for pointlike ions so that it does not capture the competition in the concentrated Stern layer between electrostatic attraction and crowding of ions. Later approaches such as the Primitive model6 incorporate finite ion size by modeling ions as charged hard spheres, but this involves further approximations (mean spherical approximation) in addition to the electrostatic meanfield approximation at the heart of the Poisson−Boltzmann approach.7 Furthermore, both the Gouy−Chapman and the Primitive model assume a sharply defined and spatially uniform surface charge layer and a uniform dielectric constant throughout the double layer. Whereas, if the ion concentration in the Stern layer is high, or if the particle−water interface is diffuse, the dielectric constant varies significantly across the Stern layer.8−10 The triple layer model11 accounts for both the finite size of ions and a nonuniform dielectric medium; however, it assumes a sharp particle−water interface, at which bound surface charges are confined. Double layers for binder particles in latex paints are not well described by the existing models. The binder particle−water interfaces are not sharp but instead diffuse, with a characteristic interfacial width of several Angstroms. When multivalent grafted oligomers are present, this interfacial width increases further. As a result, the bound surface charges are not confined to a sharply defined plane, but intrinsically spread out over a region with a thickness comparable to the Stern layer itself. Also, because the composition varies over the region where bound ions are found, the dielectric constant in turn varies significantly. These consequences of a diffuse particle−water interface make realistic analytical modeling of the resulting double layer very challenging. By contrast, atomistic molecular dynamics (MD) simulation is a powerful tool to study such a system, since it mimics the real environment of each atom, given sufficiently realistic force fields. The finite size of ions enters MD simulations naturally through the short-range repulsive interactions between atoms. Diffuse particle−water interfaces arise naturally from the equilibrium fluctuations of a slab of polymer bounded by water. Likewise, fluctuations in the location of bound surface charges, whether contributed by adsorbed surfactant or multivalent oligomers, are a natural consequence of simulations. Although no atomistic simulations have been reported for charged polymer particle−water interfaces, atomistic simulations have been successfully used for a variety of systems,12−14 some of which have aspects in common with our system of interest, including chemically realistic surfaces, flexible surface charged chemical moieites, surface roughness, and random placement of surface charge groups. Bourg and Sposito12 studied a flat clay mineral surface, for which the surface charge is bound in the crystal lattice. Darve and Kim13 created a rectangular silica nanochannel with uniform surface charge, and a model roughness consisting of grooves and ridges. Heikkilä et al.14 investigated a spherical gold nanoparticle grafted with a sparse monolayer of charged flexible alkylthiol ligands. Our system is different than the systems studied with simulations in that the interface is diffuse, and the charged surface groups are mobile. Indeed, our entire substrate consists of polymer chains that are able to move in response to their surroundings, instead of being fixed in space as for a crystalline substrate. The diffuse interface results from equilibrium fluctuations of polymer into water and vice versa. Our system has two surface charge groups: anionic surfactants and grafted multivalent oligomers, which we refer to as “hairs”. The hairs

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SYSTEM COMPOSITION Modern latex paint has a complex formulation. It contains polymeric binder particles, which form a film on the substrate; inorganic pigment particles, which give the paint opacity and color; surfactants, which emulsified the monomer when the binder particles were synthesized; buffer, to maintain the pH; salts, to adjust the ionic strength; and rheology-modifying polymers, which confer shear thinning behavior to the paint. In this work, we focus on simulating the charged polymer binder particles in water. In our simulations we include the ions contributed by buffer and added salts, because they contribute directly to the electrical double layer of the binder particles; for simplicity, we do not simulate pigment particles or rheologymodifying additives. Latex binder particles are charged by two surface groups: anionic surfactants and grafted multivalent oligomer “hairs”. The anionic surfactant headgroup is solvated by water, which anchors the headgroup at the polymer−water interface. The multiple charges along these hairs are likewise solvated by water, which favors configurations in which the hair extends away from the polymer surface. Both surfactants and hairs contribute negative charges to the particle surfaces; the negatively charged surfaces in turn collect a counterion cloud, and the repulsive interactions between these clouds give rise to repulsive interactions between nearby particles. To simulate the surfaces of binder particles with their adsorbed and grafted surface charge groups, we must know the structure and surface concentration of these groups, i.e., the surface area per surfactant and per hair, and the chemical composition of the hairs. These parameters are determined from the recipe by which the particles were synthesized and formulated, combined with a variety of experimental characterizations, described in detail in this section. Particle Synthesis and Characterization. The acrylic latex particles used for this study are synthesized by conventional emulsion polymerization.15 This is a single stage copolymerization of butyl acrylate (BA, 46.2% by mass, 128.17 g/mol), methyl methacrylate (MMA, 52.9%, 100.121 g/mol), and methacrylic acid (MAA, 1.0%, 86.06 g/mol). Structures of MMA, BA, and MAA comonomers are shown in Figure 1. The reaction is thermally initiated by 0.40% (based on total monomer mass, BOM) ammonium persulfate (APS), which decomposes to form a pair of sulfate radical initiators. The growing particles are stabilized with 0.3% (BOM) sodium dodecyl sulfate surfactant (SDS, shown in Figure 2b). This SDS concentration is below the reported critical micelle concentration (CMC) for SDS before adding salts,16 so that B


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sodium carbonate buffer; the total ion concentration is 0.11 M. About half of this concentration arises from buffer ions (sodium cations and carbonate anions) and half from ammonium counterions of the neutralized hairs, with a smaller amount coming from the sodium counterions of the SDS. (In our simulations, for simplicity we replace ammonium cations from neutralized hairs with sodium cations, and carbonate anions from the buffer with chloride anions.) The CMC of SDS in 0.04 M sodium chloride is 2.53 mM; assuming this much SDS goes into solution, we arrive at an upper bound to the area per SDS of 3.6 nm2. Thus, the range of area per SDS is 2.6−3.6 nm2. Surface Area Per Hair. Multivalent oligomeric grafted “hairs” on the surface of latex binder particles are a consequence of the emulsion polymerization growth process.18 The water-soluble APS initiator breaks down to form sulfate free radicals, initiating the propagation of a chain. The chain starts to grow in the aqueous phase primarily by adding watersoluble monomers. The MAA monomer is highly soluble in water due to its polar group, the MMA monomer is less soluble, and the BA monomer is sparingly soluble. As a result, the starting sequence of a chain is an oligomer, rich in MAA monomer, although only 1% of the total monomers are MAA. As the chain grows, it becomes more hydrophobic by adding MMA monomers; as a critical number of less soluble MMA monomers are added, the chain drops out of solution, and adsorbs onto the surface a growing particle. The MAA-rich terminus of the chain tends to be anchored at the particle surface. The actively growing chain end continues to add MMA and BA monomers within the growing particle, which is swollen by the less-polar MMA and BA monomers. We assume there are no significant chain transfer reactions in this synthesisone chain grows from each initiator, and so we expect one chain in a particle for every hair on its surface. By this mechanism, each chain in a particle has a sulfate group (SO−4 ) at the starting end, followed by a short sequence rich in MAA. This oligomeric “hair” at the end of each long chain is solubilized by water, and therefore tends to be strongly anchored at the surface of the particle. Since we focus on interfacial properties, we only simulate these hairs at the surface, and neglect any remaining soluble oligomers in the aqueous phase. From the above description of emulsion polymerization synthesis of particles and chains, we expect one hair on the surface per chain in the particle, and one sulfate group per hair. Thus, we can determine the area per hairs on the particle surface in two ways: either “count the chains” or “count the sulfur” per particle to find the number of hairs per particle, which we divide into the particle surface area to find the area per hair. To count the chains, we compute the number of chains in a particle Nc as

Figure 1. Structural units of MMA, BA, and MAA (dissociated) comonomers.

Figure 2. System composition: (a) Bare polymer slab in water. (b) SDS. (c) Hair, containing 5 charged MAA and 10 MMA monomers in random order (oxygens in MAA are green).

there are no micelles present during synthesis, which if present would create new monomer droplets, resulting in a loss of particle size control. After synthesis is completed, the MAA acid groups are neutralized by adding ammonium hydroxide. The pH is maintained at pH = 9 by adding 0.35% (BOM) sodium carbonate buffer. The neutralized MAA groups can be regarded as soluble sodium carboxylate salt moieties, so that the final polymer bearing multiple MAA groups is a polyanion. This synthesis results in particles with a nominal diameter D = 306 ± 15 nm, a density ρ = 1.18 ± 0.06 g/cm3, and a zeta potential ζ = −45 mV. The mass fraction of particles in solution is 49.14%. The number-averaged molecular weight of a chain is Mn = 1 ± 0.3 × 105 g/mol, measured by gel permeation chromatography (GPC). The relatively large estimated error for Mn reflects uncertainties about the low end of the molecular weight distribution. Short chains in the latex suspension can be incorporated in particles, weakly adsorbed on particle surfaces, or solubilized in the aqueous diluent, which complicates the calibration of the GPC column. Surface Area Per SDS. The area per SDS on particle surfaces can be determined from the particle diameter and volume fraction, and the difference between the total amount of SDS present and the amount of SDS in solution. The total amount of SDS is known from the recipe, and the amount of SDS in solution is verified with a standard pinacyanol chloride test17 to be less than the CMC, from which we can bound the area per SDS on the surface above and below. Assuming first that all the SDS is adsorbed on particle surfaces with none remaining in solution, we compute the lower limit of the area per SDS on the surface as 2.6 nm2. Then, to find the upper bound of area per SDS on the surface, we assume that the SDS concentration in solution is just below the CMC. The CMC of SDS decreases with increasing salt concentration, because salt screens the electrostatic repulsion between charged surfactant head groups, thereby lowering the micelle formation energy. The CMC in sodium chloride solution has been measured as a function of salt concentration, and ranges from 8.11 to 0.58 mM for salt concentrations ranging from 0 to 0.4 M.16 In latex paint, the ionic concentration in solution arises from the sodium counterions of SDS and neutralized hairs, and the

Nc =

ρVNA = (1.07 ± 0.34) × 105 Mn

(1)

Here ρ and V are the density and volume of a particle, Mn is the number-averaged molecular weight of a chain and NA is the Avogadro number. The error bar for Nc reflects the errors in measured ρ, V, and M n , assumed independent. The corresponding surface area per hair is then 2.8 ± 0.9 nm2. To count the sulfurs, we first compute the number of sulfate groups from the APS initiator per particle: C


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Langmuir Ns,APS =

2ηfAPS ρVNA MAPS

= (2.24 ± 0.47) × 105

as the washing progresses. AFM is able to detect the mechanical response of SDS, which tends to accumulate in the boundaries between particles, and is mechanically more soft than the surrounding particles. From the mole fraction of sulfur reported by XPS, we compute the sulfur concentration in the sampling depth from the known composition and density of the copolymer. The comonomer consists of 58.7 mol % MMA, 40.0% BA, and 1.3% MAA. From the known copolymer density, sampling depth, and numbers of atoms in each monomer, the total atom concentration from copolymer in the sampling depth is 49.85 per nm3. The XPS mole fraction of sulfur is 0.14 ± 0.03%, so that the inferred concentration of sulfur within the sampling volume cS would be (7 ± 0.16) × 10−2 per nm3. [The XPS error bar represents contributions from the measurement itself (5%), film planarity (10%), and film deformations during casting (20%).] The above value of sulfur concentration cS assumes that the sulfur atoms are uniformly spread out over the sampling depth L = 10 nm; the corresponding number of sulfur atoms per area would be LcS, or 0.7 ± 0.016 per nm2. Actually, we expect the sulfurs to be concentrated at the top of the film. Because the XPS outgoing photoelectron beam is exponentially attenuated as it passes through the sample, atoms at the top of the film give a stronger signal than atoms buried in the film. A smaller number of atoms per area at the surface would give the same signal as some given number of atoms per area spread uniformly through the sampling depth. The naive estimate of 0.7 sulfur atoms per nm2 is thus too large, and must be corrected by taking account of the attenuation of the beam, described below. The probability of an outgoing photoelectron escaping the sample from a depth z is given by Beer’s Law, as P(z) = e−z/λ, in which λ is the mean escape depth, equal to the inelastic mean free path (IMFP) when the emission angle is zero (as in our measurements). The IMFP depends on the properties of the sample, and the photoelectron kinetic energy Ek. Ek can be calculated by Ek = hν − Eb, where Eb is a the binding energy of a core electron of the atom being counted. In the present experiment, we use Al Kα as the incident X-ray (hν = 1486.6 eV). Since Eb depends slightly on the compound the sulfur electron belongs to, we used Eb for the 2p electron in the SDS sulfur (Eb = 168.8 eV),19 which has a similar chemical structure to the terminal sulfur of the hairs; this gives Ek = 1317.8 eV. Our MMA/BA copolymer is chemically similar to PMMA, for which the IMFP as a function of Ek has been measured, and described by the modified Bethe equation.20 In this way, we computed the IMFP of sulfur 2p electron with Ek = 1317.8 eV as 3.37 ± 0.17 nm. Using this value for the IMFP and Beer’s Law, we can compute a corrected sulfur surface concentration, by multiplying the naive uniform sulfur concentration versus depth times probability at each depth that photoelectrons will emerge and be counted. The surface concentration of sulfur σS is therefore

(2)

Here η is the initiator efficiency (η = 0.6 ± 0.12), i.e., the fraction of sulfurs in APS that actually initiate chain growth, and fAPS and MAPS are the mass fraction and molecular weight of the initiator. We find that Ns,APS is about twice Nc, indicating that some of the sulfate initiator does not end up in particles. It turns out that not all soluble chains initiated in the aqueous phase grow sufficiently to become insoluble, hence adsorbing onto a particle surface to form a chain in a particle; some remain as soluble oligomers in the aqueous phase of the final suspension. To account for the sulfurs on these soluble oligomers, we measured the serum polymer mass fraction faq (faq = 1.02 ± 0.05% of total monomer mass) and the number-averaged molecular weight Maq of these soluble oligomers (Maq = 777 ± 233 g/mol), using size exclusion chromatography coupled with mass spectrometry (SEC-MS). Each of these soluble oligomers carries one terminal sulfate group. In this way, we compute the number of oligomer sulfates in the aqueous phase per particle as Ns,aq =

faq ρVNV Maq

= (1.40 ± 0.44) × 105 (3)

which is about half of the total sulfur. We subtract these soluble sulfates from the total sulfur in the initiator, to determine the number of insoluble sulfate per particle, present on the hairs: Ns,p = Ns,APS − Ns,aq = (0.84 ± 0.64) × 105

(4)

The ratio of insoluble sulfate per chain Ns,p and number of chains per particle Nc is then 0.8, which is close to unity as we expect, since both are different ways to count the number of hairs per particle. Using Ns,p as an estimate for the number of hairs per chain, the surface area per hair is then 3.5 ± 2.7 nm2, which is consistent with the value estimated by the “count the chains” approach. Our two methods for counting the hairs have complementary potential shortcomings. In the “count the chains” approach, Mn of the chains in the particles is subject to uncertainties at the low end of the distribution. Whereas, in the “count the sulfur” approach, counting the insoluble sulfate may underestimate the area per hair, if significant numbers of sulfate groups are buried within the particles. Fortunately, these two approaches agree well, within experimental error: we obtain an area per hair of 2.8 ± 0.9 nm2 from counting chains, and 3.5 ± 2.3 nm2 from counting insoluble sulfates. A third way to obtain the area per hair is to measure the amount of sulfur on the surface of particles using X-ray photoelectron spectroscopy (XPS). Samples of latex were cast onto vinyl substrates and dried for 7 days at 30 °C (above the copolymer Tg), to produce continuous films of thickness about 40 μm. Because the particle suspensions contain sulfurterminated SDS and soluble oligomers, great care was taken to wash off these noncovalently bound species. After drying, the films were washed in deionized water for 40 days, with the water changed daily. The near complete removal of sulfur containing species and surfactant was confirmed by secondary ion mass spectrometry (SIMS) and AFM. We used SIMS to examine the concentration of sulfur in the films during the washing, which allows us to follow the decrease in surface sulfur

σS = cS

∫0

∞

P(z) dz = cSλ = 0.24 ± 0.06 nm−2

(5)

which corresponds to an area per hair of 4.2 ± 1 nm2. The XPS approach to “count the sulfurs” is consistent with results from our previous two approaches, (considering that despite our careful efforts to wash off all the SDS and soluble oligomers, D


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Figure 3. (a) Ethyl sulfate model molecule, with two directions of motion indicated for scanning potential energy with DFT (see main text for details). (b) Example of O3−S−O*angular potential obtained from DFT (points) and quadratic fitting (curve).

hairs, buried in chains in the interior of the particle, or in soluble oligomers. From the result that about 15% of the MAA are buried (i.e., somewhere along the chains, deep within the particle), we estimate that only about one MAA per chain are so buried. And since about 50% of the MAA is in the hairs, we can infer that there are about 4−5 MAA per hair, consistent with our earlier estimate based on the composition and molar mass of the largest soluble oligomers. From these considerations, we build the hairs in our simulations with five MAA monomers, 10 MMA monomers, and a sulfate group on the end, as shown in Figure 2c. In constructing the hairs, we assemble the MAA and MMA monomers in random order, corresponding to the simplest assumption of random copolymerization in aqueous solution. Thus, the hairs in our simulation are identical in overall composition, but random in sequence.

some sulfurs not from hairs may have remained behind to be counted by XPS). Composition of a Hair. Now we must specify the length and composition of the hairs, which we expect to be oligomeric, starting with a SO−4 group, and containing primarily MMA and MAA comonomers. We infer the hair structure by recalling how the hair grew as a multivalent oligomer in solution, and making use of characterization data on soluble oligomers, which may be regarded as oligomers that never added enough less-polar MMA monomers to adsorb onto a particle and grow a chain. (In reality, the hairs will have varying lengths and compositions, but for this work we are satisfied to select an average length and composition.) The solubility of an oligomer in solution is governed chiefly by the MMA/MAA ratio, and to some extent by the oligomer length. As a growing oligomer becomes rich in MMA overall, or develops a sufficiently long sequence in MMA, it will enter a growing particle. The typical MMA to MAA molar ratio in hairs can be inferred from SEC-MS measurements of extracted soluble oligomers. Because these soluble oligomers are essentially hairs that did not grow long enough to fall out of solution, the composition of hairs on particles surfaces should be similar to the longest of these, which have a MMA/MAA ratio of two. Likewise, the upper bound to the molecular mass of soluble oligomers is found by SEC-MS to be about 1500 g/mol, which we take as a lower bound and our best guide to total hair molar mass. Taken together, this corresponds to a hair composition of one sulfate group, about five MAA monomers, and about 10 MMA monomers. Another useful characterization method that gives information on the hair composition is conductometric titration, in which the amount of MAA in the soluble oligomers and on the surface of the particles is determined. Using this method, we find that about 35% of the MAA are in soluble oligomers, and about 50% of the MAA are in the hairs, from which we infer the remaining 15% are inaccessible, presumably buried in the particles.21 Given the recipe mass fractions and molar masses of MMA, BA, and MAA, we compute the average monomer mass as 111.2 g/mol; with the number-averaged molecular weight of a chain as Mn = 105g/mol, this means a typical chain has about 900 monomers. The corresponding mole fractions of MMA, BA, and MMA in the overall recipe are 0.4, 0.59, and 0.01, respectively. This means that for each chain, there are about nine MAA monomers somewhere in the producteither in

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METHODS

Force Field Parameters. In this work, we use the all-atom Optimized Potentials for Liquid Simulations (OPLS-AA) force field22 for polymer chains (MMA and BA monomers), and the extended simple point charge (SPC/E) model for water.23 In our previous work,24 these potentials resulted in reasonable bulk and interfacial properties for a PMMA-co-BA polymer slab immersed in water, without surface charge groups. The SPC/E model is known to reproduce the dielectric constant of water reasonably well,25 which is important to properly represent electrostatic interactions between the various charged groups in our system. Atomistic force field parameters for SDS have been published, but unfortunately are either incompatible with OPLS-AA and SPC/E, or lead to unphysical behavior of surfactant micelles. The CHARMM potential includes all-atom parameters for SDS, but is designed to work with the TIP3P water model.26 OPLS-AA parameters for SDS have been published as well,27 in which the Berkowitz united atom parameters for the sulfate group28 were combined with OPLS-AA alkane parameters for the hydrocarbon tail. Unfortunately, Tang et al. found these extended OPLS-AA parameters for SDS yield unexpectedly strong attractions between the sulfate oxygens and sodium counterions, resulting in unphysical micelle structures.29 This finding motivated us to reparameterize the sulfate group, including its partial charge, angles, and dihedrals. We reparameterized the sulfate group on SDS using ethyl sulfate (Figure 3a) as a model molecule. We chose the ethyl sulfate since it includes the carbon and hydrogens adjacent to the sulfate, of which the parameters are expected to be different from those of the hydrocarbons. First, we optimized the geometry of ethyl sulfate using density functional theory (DFT). All DFT calculations were carried out using Gaussian 09,30 with the B3LYP functional and basis E


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Langmuir Table 1. Sulfate Head Group Potential Parametersa

set 6-311++g (3df,3pd), which includes diffuse functions to better represent the more loosely bound electrons of the anion.31 To assign partial charges for SDS, we compute partial charges for ethyl sulfate using the Merz−Singh−Kollman scheme,32 and transferred the values for the terminal OSO−3 to the SDS headgroup. To enforce the overall −1 charge for the SDS anion, we used the partial charge from DFT for atoms in OSO−3 group, and adjusted the charges of the C1 carbon next to OSO−3 so that the net charge of CH2OSO−3 is −1. We used existing OPLS-AA partial charges for atoms in the terminal CH3 group, CH2 groups along the alkane tail and hydrogen atoms attached to C1 carbon. We also used DFT to compute angular and dihedral potentials for two key angles (O−S−O, O−S−O*) and one key dihedral (O−S− O*-C1) of the sulfate group, indicated in Figure 3(a). To determine these potentials, we used DFT to calculate energy profiles by scanning the relevant angles or dihedral in a manner described in detail below, and fitted the resulting energy profiles to potential functions used in our simulation. For the angular potentials, we use a harmonic form: Va(θ) =

1 K θ(θ − θ0)2 2

n=0

q

O (ionic) S O* (ester) C (bonded to O) H (bonded to C) Na+ angle

3.15 0.8372 3.55 1.0465 3.00 0.7116 3.50 0.2761 2.50 0.1255 3.33 0.0116 Kθ (kJ mol−1 rad−2)

−0.657 (−0.654) 1.189 (1.284) −0.496 (−0.459) 0.158 0.060 1.000 θ0 (deg)

O−S−O O−S−O* S−O−Cb O−C−Hc

725.50 (853.54) 994.78 (853.54) 1040.14 292.88

115.0 (115.4) 103.1 (102.6) 112.6 109.5

C0 (kJ mol−1)

C1

C2

C3

C4

C5

O−S− O*−C

1.6915

−5.1916

0.0089

6.9270

−0.0035

−0.0098

b

Values in parentheses are united atom parameters for comparison.28 Berkowitz.28 cOPLS.22

charge group, to separately investigate the different interfacial behavior and electrical double layers arising from these two surface charge groups. For each system, we perform simulations with and without salt to explore the effects of additional screening, yielding a total of four systems to explore. We model the interface between polymer binder particles and water as a polymer slab immersed in water, as shown in Figure 2a. The polymer slab consists of 36 oligomeric random copolymer chains of 20 monomers of BA and MMA in a 1:1 molar ratio, consistent with the above synthetic recipe and described in our previous work.24 We employ periodic boundary conditions in our simulations, so that a polymer slab immersed in water is equivalent to a periodic alternation of polymer and water slabs, each extending indefinitely in the transverse directions. Thus, the top and bottom layers of the polymer slab may be regarded as interacting with each other across the intervening water layer. Our overall system dimensions are 5.62 × 5.62 × 11.69 nm in the x, y, and z directions, respectively, with a polymer slab thickness of 4.44 nm and a water layer thickness of 7.25 nm. For both the hair and SDS systems, we must select an areal density of charged groups, from the range of values determined in the previous section. In this work we focus on the lower limit of the estimated range of area per charged group, to give us somewhat more charged molecules in the system, and therefore somewhat better statistics as we perform time and transverse spatial averages to obtain the interfacial concentration and charge profiles. Hence for the SDS system, we take the area per SDS molecule as 2.6 nm2, and for hair system, the area per hair as 2.3 nm2. To generate the initial configurations for the SDS and hair systems, we used an intermediate configuration from our previous work equilibrating slab with walls, which features a slab with smooth surface and slightly larger surface area (6 × 6 × 4 nm). Thus, we have 14 SDS and 16 hairs per surface for the SDS and hair systems, respectively. We added the charged molecules (either SDS or hair) in all-trans conformations perpendicular to the surface. To avoid overlap between the added molecules, we first divided the polymer slab into a twodimensional grid in the x and y directions, with the grid size slightly larger than the transverse dimension of the all-trans configuration (about 0.77 nm). To prevent close overlaps between added SDS or hair molecules, we (somewhat arbitrarily) regard our square lattice of cells as a chessboard of black and white squares, and select cells at random from only the black squares, into which we add the molecules. SDS on the particle surfaces is physisorbed, whereas oligomeric hairs are grafted to the particle surfaces, as the hairs are the hydrophilic ends of long polymer chains that form the particles. We construct initial configurations for added SDS and hairs accordingly. For the system with added SDS, surfactant molecules are initially placed so that the end of the alkane tail is 4 Å above the surface of the polymer

5

∑ (−1)n Cn cosn ϕ

ϵ (kJ mol−1)

dihedral

a

(6)

in which θ0 is the equilibrium angle, and Kθ is the angular stiffness. For the dihedrals, we use the Ryckaert−Bellemans potential:

Vd(ϕ) =

σ (Å)

atom

(7)

in which the {Cn} are fitting parameters. Since the DFT potential includes Coulomb interactions, which contribute to the classical potential energy as the molecule moves, we fit the DFT results to the sum of angular, dihedral, and Coulomb potentials. To fit the two angular and one dihedral potential in a convenient way, we performed three different fit scans, in which we vary primarily one angle or the dihedral with the least involvement of the others. We first scanned the angle defined by O3 (ionic oxygen)−S (sulfur)−O* (ester oxygen) around its equilibrium value along motion 1 in Figure 3a, fixing the O3−S bond length. This motion also changes the angles O3−S−O1 and O3−S−O2, as well as the Coulomb potential, but does not affect the O−S−O*−C1 dihedrals. The potential during this scan from the DFT and its fitting using the above equations are shown as an example in Figure 3b. Next, we simultaneously scanned all three O (O1, O2, O3)−S−O* angles along motion 1, with all O−S bond lengths fixed. This motion also changes all three O−S−O angles and the Coulomb potential, but again does not affect the dihedrals. Finally, we scanned the dihedral O3−S−O*−C1 by moving atom O3 along motion 2, in which O3 remains in the O1−O2−O3 plane and the O3−S bond length is fixed. This motion also changes the O3−S−O1 and O3−S−O2 angles, as well as the Coulomb potentials. We fit these three DFT scans simultaneously to the simulation potential predictions, using the united atom parameters as the initial values, and iterating to minimize the sum of the square errors for all three predictions. Our fitted parameters for SDS are listed in Table 1, compared with the united atom parameters (in parentheses). The fitted partial charge and equilibrium angles turn out to be very similar to the united atom parameters. However, we find a larger spring constant for the O−S− O* angle, and a more rigid O−S−O*−C dihedral than united atom parameters. The more rigid the O−S−O* angle and O−S−O*−C dihedral are, the less the anionic headgroup geometry deforms in response to a nearby sodium cation, resulting ultimately in a weaker attraction between the anion and nearby cations. Together, these changes tend to overcome the unphysical behavior found by Tang using the previous extended OPLS−AA parameters.29 To simulate the hairs, we use the same parameters for the sulfate terminal group as for the SDS headgroup, and introduce one more new moiety, the MAA anion. For this new group, we use available OPLS-AA parameters for the carboxylate ion. Simulation Details. In this work, we simulate systems either with grafted multivalent oligomer hairs or with physisorbed SDS surface F


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Here λ is the Gouy−Chapman length, which governs the decay of the potential away from the charged surfaces:

slab. For the system with hairs, we place the nonsulfate end of each hair near to a carbon atom on the slab surface, and chemically bond the hair to the surface, removing a pair of hydrogen atoms to create the bond. The initial dimensions of the slab itself are 6 × 6 × 4 nm; we choose the initial system dimension in the z direction to be 12 nm, so that the thickness of the region containing hairs or surfactant, water, and ions is about 8 nm. Finally, we added water molecules to fill the system volume, and add sodium counterions to neutralize the system. The Debye length just from the average counterion concentration in the water volume is about 1 nm, so that the water region width near each surface of 4 nm is about 4 Debye lengths wide. Thus, the water region is wide enough that the double layer of one side of the slab should be reasonably well screened by the counterions from the electrostatic effects of the periodic image of the other side of the slab, across the water region. To equilibrate the system, we utilized a standard workflow of energy minimization, followed by an NVT (constant number, volume, and temperature) and an NPT (constant number, pressure, and temperature) simulation. For energy minimization, we used steepest descents to approach the minimum and switched to the l-BFGS integrator33 to further minimize the energy, with a force tolerance of 10 kJ/(mol· nm2). To maintain the temperature, we used the velocity rescaling thermostat.34 To control the pressure, we used the Berendsen barostat35 to reach the target pressure and then switched to the Parrinello−Rahman barostat36 for further equilibration and data collection. We annealed the systems containing hairs at a relatively high temperature (500 K) to accelerate equilibration, and then quenched to 363 K (90 °C, which is 50 °C above Tg for the slab) for further equilibration and data collection. Using the equilibration workflow described above, we equilibrated for 10 ns at 500 K with a short time step of 0.5 fs, and then at 363 K for 20 ns with a longer time step of 2 fs (using bond constraint algorithms; see below). Finally, our production run is an NPT simulation for 500 ns with a time step of 2 fs, recording data every 10 ps. For systems with SDS, it turns out that equilibration is easier, because the physisorbed surfactant molecules are more mobile than the grafted hairs. We equilibrate systems with SDS as described above at 363 K for 20 ns with a 2 fs time step, and then collect data every 10 ps from an NPT simulation for 320 ns. Our total simulation times are relatively long (300−500 ns) for atomistic simulations, in order to average properly over the conformations and transverse motion of the charged surfactants and hairs, to obtain the equilibrium concentration and charge profiles and hence the double layer potential, discussed further below. We used the leapfrog algorithm for all simulations, and periodic boundary conditions as mentioned previously. The cutoff radius for nonbonded potentials is 11 Å. We used a shifted van der Waals potential, smoothly switched to zero beginning at 10 Å. We used particle-mesh Ewald to compute long-range electrostatic forces.37 To enable the use of a longer (2 fs) time step, we constrained water molecules with the SETTLE algorithm38 and all other bond lengths with the LINCS algorithm.39 All simulations were conducted using the GROMACS package.40 Modeling the Double Layer. Outside the Stern layer, where the counterion cloud becomes dilute and thus the finite size of ions is no longer important, we expect the charge concentration and potential profiles to be well-described by the Poisson−Boltzmann equation.2 For the case of counterions confined between two flat parallel charged surfaces and no added salt, the Poisson−Boltzmann equation has been solved exactly. The Gouy−Chapman solution takes the form

ψ (x) =

⎛x⎞ 2kT log cos⎜ ⎟ ⎝λ⎠ ze

λ2 =

2ϵ0ϵkT z 2e 2ρ0

(9)

Here ρ0 is the counterion concentration at the midplane, determined by normalization (integrating the concentration to recover twice the surface charge density). With added salt, the nonlinear Poisson−Boltzmann equation can no longer be solved analytically. However, sufficiently far from the Stern layer, the potential becomes low enough that the equation can be linearized, to yield the Debye−Huckel approximation. In fact, because counterion condensation tends to proceed until the binding free energy of ions in the Stern layer is of order kT, the potential just outside the Stern layer is typically low enough that the Debye−Huckel approximation holds. The corresponding solution for two flat surfaces a distance 2h apart is

ψ (x) = ψ0

cosh(κx) cosh(κh)

(10)

for which x = 0 is again the midplane between the surfaces. Here ψ0 is the potential at the surfaces, which may be regarded as an adjustable parameter, to incorporate the effect of counterion condensation. κ−1 is the Debye length, given by κ −2 =

ϵ0ϵkT z 2e 2ρ0

(11)

Here ρ0 is the total ion concentration in some region of space far from the plates (which we may think of as large but finite in the transverse dimensions), where the potential vanishes. (In writing the above, we have assumed all ions have the same charge magnitude ze.)

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RESULTS Density Profiles. To characterize the distributions of charged molecules relative to the polymer slab, and explore their influence on the polymer/water interface, we begin by calculating the concentration profiles of all major components, including polymer, water, charged molecules (SDS or hairs), sulfate groups, and sodium counterions. In particular, to compute the properties of the electrical double layer, we need the ion concentrations as a function of distance from the polymer/water interface. To obtain these concentration profiles, averaged over the x− y plane and 300−500 ns of simulation time, we divide the simulation box into 30 bins along the z direction (normal to the interfaces), and calculated the average number density of atoms within each bin. Representative snapshots of equilibrated system with SDS and with hairs are shown in Figure 4 and Figure 5 respectively, together with the concentration profiles for polymer, water, SDS, and sulfate and sodium ions. In Figures 4 and 5, concentration profiles are given in “atom concentration”, i.e., atoms per nm3 regardless of element. Thus, for the polymer concentration, any C, O, or H atom on the polymer contributes equally to the polymer atom concentration, and likewise for the water and SDS atom concentrations. Figures 4 and 5 both display a relatively diffuse polymer interface, extending over several Angstroms for both the SDS and hair systems. To quantitatively characterize the interfacial region, we fit the polymer density profile to a hyperbolic tangent function ρ(z) describing an interface located at z0 with a width parameter d:

(8)

in which ze is the charge of an ion with valence z. The zero of the potential ψ is taken for convenience at the midplane, here located at x = 0. G


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In the systems with SDS, the hydrophobic tails of SDS molecules penetrate completely into the polymer slab, with only the charged sulfate headgroup (SO−4 ) residing in the aqueous phase beyond the interfacial region, as is evident from the snapshot and concentration profiles shown in Figure 4. Because of the penetration of the SDS molecules into the polymer slab, the interfacial width increases about 25% compared with a bare polymer slab in water. The transverse dimensions of the slab increase slightly as well, by about two percent. Deep within the polymer slab, the bulk density remains the same as for a slab with no surfactant. By contrast, in systems with grafted hairs, the density profile of the hairs penetrates very little into the polymer slab itself. Most of the atoms in the hairs lie within the polymer interfacial region or immediately beyond, in an adjacent layer dense in counterions and containing significant amounts of water, as shown in the snapshot and concentration profiles of Figure 5. Evidently, the multivalent charged hairs are significantly solvated by water and attract a concentrated layer of counterions, both of which prevent the hairs from penetrating the less-polar polymer slab very much. Quantitatively, only three percent of atoms in the hairs reside in the slab region, with 40% in the interfacial region and the rest in the first 2 nm or so of the water region beyond. Because the hairs reside substantially in the interfacial region, the interfacial width increases by about 65% compared to a bare polymer slab. The transverse dimensions of the slab increase as well, by about 6%. The thickness of the hair-rich layer evident in Figure 5 is comparable to the RMS end-to-end distance of the hairs, which we find to be about 2.3 nm. For comparison, the fully extended length of a hair consisting of 15 monomers total (5 MAA and 10 MMA) and a sulfate group is about 3.8 nm. Thus, the hairs are not fully extended; however, they do tend to be oriented normal to the interface. Figure 1 of the Supporting Information displays the length and orientation distributions of hairs. Charge Distribution. In the customary discussion of electrical double layers, the bound surface charges are assumed to be uniformly distributed on a well-defined plane, called the inner Helmholtz plane. The half-space containing the counterions is divided into two layers. Immediately adjacent to the surface is a thin region of solvated counterions strongly interacting with or bound to the charged surface, called the Stern layer. The outer boundary of the Stern layer is the outer Helmholtz plane, with the diffuse layer of counterions beyond.2 In our systems, the bound surface charges from SDS headgroups or multivalent hairs are not confined to a welldefined plane, but are spread out over several Angstroms normal to the interface (see Figures 6 and 7). We must therefore redefine the locations of the inner and outer Helmholtz planes, maintaining the intent of the original definitions, which demarcate the Stern layer as a region of concentrated bound charges and condensed counterions. Here, we define the position of the inner Helmholtz plane as the inner boundary of the interfacial region between polymer slab and water, defined previously; it turns out that on this plane, the surface charge concentration starts to rise from zero. We take the outer Helmholtz plane to be the plane at which the net charge concentration starts to decrease as we progress from the bulk water region to the surface. The inner and outer Helmholtz planes are indicated with vertical lines in Figures 6 and 7.

Figure 4. Top: snapshot of SDS system with added salt; at left, polymer slab and all SDS molecules highlighted in red; at right, one representative SDS molecule highlighted. Bottom: concentration profiles of polymer, water, SDS, sulfate and sodium ions (both magnified 40x). Dashed lines bound the interfacial regions (see main text).

Figure 5. Top: snapshot of hair system with added salt; at left, polymer slab and all hairs highlighted; at right, one representative hair highlighted. Bottom: concentration profiles of polymer, water, SDS, sulfate, and sodium ions (both magnified 10x). Dashed lines bound the interfacial regions (see main text).

ρ (z ) =

⎛ z − z 0 ⎞⎞ 1 ⎛ ⎟⎟ ρ0 ⎜1 + tanh⎜ ⎝ d ⎠⎠ 2 ⎝

(12)

Here ρ0 is the bulk density of the polymer. Based on fits of the left and right interfaces in Figures 4 and 5, we define boundaries of the interfacial regions at z0 ± d, which are shown as dashed lines in the figures. H


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outer boundary. These distributions are expected for sulfate groups at the terminal end of multivalent hairs, which tend to orient normal to the interface. The bound negative surface charge of the hair system is evidently much larger than for the SDS system (compare Figures 7 and 6, bearing in mind the difference in vertical scale). Remarkably, however, the large peak in the hair system anion distribution within the Stern layer is largely canceled by the corresponding large peak in the cation distribution. This behavior, evident in Figure 7, is the result of counterion condensation, which ultimately results in rather similar zeta potential values for the two systems despite their very different bound surface charge density, as we discuss further below. The concentration profiles for the individual ionic species are all relatively smooth with single broad peaks near each interface. However, the resulting net charge concentration profile within the Stern layer is much smaller in amplitude, because of the near-cancellation between condensing cations and surface bound anions, and not singly peaked. This may be evidence of some structuring within the hair-rich region, or simply the result of difficulty in averaging the local net charge concentration in the Stern layer. A double layer is still present, as indicated by the net positive charge in the diffuse layer, where unbound sodium cations and chloride anions dominate the charge distribution. It is more challenging to obtain well-averaged charge concentration profiles for the system containing hairs than for the SDS system. The grafted hairs are hindered in their conformational relaxation compared to the fully mobile surfactants. Also, the hairs are permanently attached to atoms on the surface of the polymer slab. At the simulation temperature of 363 K, atoms in the slab (whether at the surface or in the bulk) move only a few Angstroms within 100 ns with respect to the center of mass of the slab. Thus, the grafting sites may be regarded as permanently quenched, and transverse averaging is necessary to obtain effective onedimensional concentration profiles. Because of these concerns regarding equilibration and averaging of the hair system, we have made several checks and estimates of the simulation time required to obtain adequate sampling of the concentration profiles. The correlation times of the end-to-end distance and end-to-end vector orientation are useful measures of the time required for grafted hairs to explore different conformations. Both these quantities relax in about 30 ns, as illustrated in Figure 2 of the Supporting Information. Thus, in our 500 ns simulations of hair systems, each hair can explore about 15 statistically independent configurations. With about 32 hairs total in the simulation volume, we are averaging over about 240 independent hair configurations in our measurement of the time- and transverse-averaged concentration profiles, which provides adequate sampling to obtain meaningful results. Another way we can estimate the number of independent system configurations our averages contain is to observe the time-dependent fluctuations of the transversely averaged concentration profiles. Because our system is spatially periodic, we can express the fluctuations of the concentration profile about its average in terms of Fourier modes in the direction normal to the slab-water interface. We computed the time autocorrelation function for various Fourier modes of the profile fluctuations, and found the relaxation times τ for the modes with the largest amplitudes to be on the order of τ = 1.5 ns (see Figure 3 in the Supporting Information). Within a t =

Figure 6. Charge concentration profiles for SDS system with added salt. Vertical lines mark the inner (IHP) and outer (OHP) Helmholtz planes.

Figure 7. Charge concentration profiles for hair system with added salt. Vertical lines mark the inner (IHP) and outer (OHP) Helmholtz planes.

For the SDS system, the charge concentration profiles display an evident “double-layer” structure (see Figure 6). The sulfate anions are spread out with a broad single-peak distribution centered in the Stern layer, rather than being confined to a narrow plane. The peak in the distribution of sodium cations is displaced away from the interface by about 2.4 Å from the peak of the sulfate distribution. The resulting net charge concentration goes from negative to positive in the Stern layer, indicating a negative surface charge layer with an adjacent positive bounded layer, consistent with the standard physical picture of a double layer. The peak in the sodium cation distribution is somewhat broader than the peak in the sulfate ion distribution (full width at half-maximum of about 6.6 Å), which is consistent with fact that the sulfate cations are bound to the interface by their surfactant alkane tails, while the sodium ions are free to move about. The concentration profile for chloride anions from the added salt is essentially excluded from the interfacial region, because of electrostatic repulsion from the bound sulfate anions. The chloride concentration begins to grow with distance from the interface near the plane on which the sodium concentration is at its maximum. Similar behavior has been found in simulations of the double layer near a clay surface in water.12 For the hair system, both the anions and the cations are more broadly distributed than in the SDS system, as shown in Figure 7. The Stern layer is evidently much wider, about 3.5 nm compared to 1.5 nm for the SDS system. The distribution of carboxylic anions (COO−2 ) from MAA monomers is spread out over most of the Stern layer, with the sulfate anion distribution more prominent near the Stern layer I


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the SDS systems, and Figure 9 for hair systems, with and without added salt. (The plots have been mirror-averaged with

500 ns simulation, we have roughly 150 statistically independent configurations of the concentration profile (estimated as t/(2 τ)), which again suggests our simulations have adequately sampled the ensemble of concentration profiles. Electrostatic Potentials. To obtain the electrostatic potential as a function of distance from the polymer−water interface, we integrate the Poisson equation to obtain the electric field from the charge concentration profile, and integrate the field to get the potential. In principle, we could carry out this calculation using the total charge concentration “bound” partial charges on the water molecules and polymer as well as explicit “free” charges on the anions and cationswith a dielectric constant of near unity, representing only the electronic polarizability of the constituent molecules. We prefer instead to relate the electrostatic potential to the explicit free charges only, accounting for the contributions of bound charges on water molecules and polymer in terms of the local dielectric constant. Likewise, we could in principle determine the local dielectric constant from simulation, by observing the dipole fluctuations in a set of narrow regions along the z axis normal to the interface. This approach is challenging, since it requires substantial sampling, particularly in the interfacial region. Instead, we estimate the local dielectric constant ϵ(z) in the interfacial region by using the Clausius-Mossotti relation, which states that for a medium consisting of different polarizable species, (ϵ − 1)/(ϵ + 2) should be a sum of contributions from different components, each proportional to their concentration. Using the Clausius−Mossotti relation, we write the local dielectric constant across the polymer−water interface as a function of the local concentration of polymer and water atoms and dielectric constants of the pure components, as ϵ − 1 ρ1(z) ϵ − 1 ρ2 (z) ϵ(z) − 1 = 1 + 2 ϵ(z) + 2 ϵ1 + 2 ρ1,0 ϵ2 + 2 ρ2,0

Figure 9. Electrostatic potentials for hair systems with added salt (filled squares) and without salt (open circles). Inset: potentials in diffuse layer (points) and analytical predictions (solid curves; see main text).

respect to the midplane between the two interfaces.) The positions of the inner and outer Helmholtz planes, determined by the criteria given in the previous section, are indicated with vertical lines (labeled IHP and OHP). The zero of potential is chosen to be the potential at the midplane between the two polymer−water interfaces for systems with no salt. For both SDS and hair systems, the potential displays clearly distinct behavior in the Stern layer (between the inner and outer Helmholtz planes) and in the diffuse layer (beyond the outer Helmholtz plane). Inside the Stern layer, the potential varies rapidly, starting at a fraction of a volt at the inner Helmholtz plane and reaching a much smaller value on the order of tens of millivolts at the outer Helmholtz plane. For the SDS system especially, the potential in the Stern layer varies nearly linearly with distance, similar to a parallel-plate capacitor. The widths of the Stern layers for SDS and hair systems are markedly differentabout 15 Å for the SDS system compared to 35 Å for the hair systemwhich results from the broader distribution of carboxylate bound charges on hairs compared to the relatively narrow distribution of surfactant headgroups. The outer Helmholtz plane for the hair system resides roughly at the end of the hair-rich region, which is displaced from the polymer−water interface by 25 Å or so, the typical end-to-end distance for the hairs. The surface potential for the hair systems (about −0.7 V) is nearly 3 times larger than for the SDS system (about −0.25 V), consistent with a much larger bound surface charge density (2.53 per nm2 for the hair system, compared to 0.40 per nm2 for the SDS system). Both systems display substantial counterion condensation, which greatly reduces the potential at the outer Helmholtz plane, to about −50 mV for the SDS system and −20 mV for the hair system. Indeed, although the bare surface charge for the hair system is larger than for the SDS system, counterion condensation is highly effective at compensating the multivalent oligomer charges, so that the potential at the outer Helmholtz plane is weaker for the hair system than for the SDS system. In the diffuse layer, the potentials for all four systems are well fit by analytical solutions of the Poisson−Boltzmann equation: the Gouy−Chapman solution to the Poisson−Boltzmann equation for systems without salt, and the Debye−Huckel

(13)

Here ρ1(z) and ρ2(z) are the concentration of polymer and water atoms respectively, and ρ1,0 and ρ2,0 the corresponding pure-phase limits. Likewise, ϵ1 and ϵ2 are the dielectric constant of pure polymer and pure water at 363 K, here obtained from the dipole fluctuations in simulated bulk systems (ϵ1 = 1.61 and ϵ2 = 53.35). For comparison, the experimental dielectric constant of water at 363 K (90 °C) is 58.3.48 Our results for the electrostatic potential as a function of z position (normal to the interfaces) are shown in Figure 8 for

Figure 8. Electrostatic potentials for SDS systems with added salt (filled squares) and without salt (open circles). Inset: potentials in diffuse layer (points) and analytical predictions (dashed curves; see main text). J


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Langmuir solution for systems with added salt. In the fits, both the surface potentials and the characteristic lengths (Gouy−Chapman and Debye lengths) have been adjusted to fit the simulation results. The insets to Figure 8 for SDS systems and Figure 9 for hair systems show the analytical fits (curves) which coincide with the simulation results (points). From the fits of analytical solutions to the observed simulation potentials, we obtain the potentials at the outer Helmholtz plane as well as the characteristic lengths, presented in Table 2. We can compare these fitted lengths to what we

naive ion concentration is much more than the observed concentration at the midplane; this is a consequence of counterion condensation, which is a substantial effect in all four systems, discussed further below. We can extract an effective surface charge density from the fitted potentials in the diffuse layer, by applying Gauss’ law to the electric field at the outer Helmholtz plane. Table 2 reports the fitted surface charge densities for SDS and hair systems with and without added salt. By comparing the effective surface charge density to the “bare” surface charge density (arising only from the bound anions on the SDS and hairs), we can quantify the amount of counterion condensation. From Table 2, the effective surface charge density for the SDS system is about 30% of the bare surface charge, contributed by the bound surfactant anions. For the hair system, counterion condensation is even more effective at neutralizing the bare surface charge; the effective surface charge density as seen from the diffuse layer is only 3.4% of the bare surface charge density contributed by the grafted multivalent hairs. Even though the bare surface charge of the hair system is about 6 times larger than that of the SDS system, the effective surface charge is actually less, about half as large as for the SDS system. The polyvalent hairs are evidently very effective in holding a compensating cloud of sodium ions. Adding salt to either the SDS or hair system should reduce the electrostatic screening length, so that the potential in the diffuse layer decays to zero more quickly. Adding salt might also be expected to increase counterion condensation to some degree, because more ions are available to condense onto the bare charge. In our simulations, we added 40 mM NaCl to the water region, or equivalently an increase of 80 mM in the average total ion concentration. From Table 2, we see that 80 mM is a modest amount of salt compared to the naive ion concentrations (particularly for the hair system), but a significant amount compared to the observed midplane ion concentrations. Indeed, the SDS and hair systems with and without salt differ in their observed midplane ion concentrations by about 100 mM, which is close to an additional 80 mM ion concentration as estimated above. The effect of added salt in our simulations is mainly to change the nature of the screening in the diffuse layer, with only small or no effects on the amount of counterion condensation. Without added salt, the screening is described by the Gouy− Chapman result, which for an isolated surface gives a power-law decay of the potential, rather than exponential screening as in the presence of salt. As described above, a fair comparison of the Gouy−Chapman and Debye lengths can be made by comparing λ/√2 and κ−1. From Table 2, we see that adding salt increases the midplane total ion concentration significantly (by a factor of 4.0 for the SDS and 2.7 for the hair system). This results in a significant decrease in the corresponding screening lengths (by a factor of 2.0 for the SDS and 1.7 for the hair system). The overall effect on the shape of the potential curves is visible in the insets to Figures 8 and 9, but hardly dramatic. The effect of added salt on counterion condensation in our simulations is minimal; in retrospect, this may be expected because the amount of ions added is small compared to the total available ions, i.e., compared to the naive ion concentrations (see Table 2). Because we have added relatively few ions compared to the total available for condensation, the equilibrium between free and condensed ions should be perturbed only slightly.

Table 2. Potentials at the Outer Helmholtz Plane and Characteristic Lengths (λ/√2 for No Salt, κ−1 with Salt), for SDS and Hair Systems with and without Added Salt SDS systems no salt

salt

outer Helmholtz plane potential −50.9 −41.5 (mV) characteristic length (nm) 1.90 0.95 ion concentrations (mM) fitted 42.7 170.2 observed 42.2 149.9 naive 167 251 surface charge density (e/nm2) fitted −0.139 −0.139 bare −0.40 −0.40

hair systems no salt

salt

−16.6

−20.1

1.47

0.89

71.0 70.8 1111

193.8 178.3 1207

−0.076 −2.53

−0.093 −2.53

would predict, based on eq 9 for the Gouy−Chapman length in terms of the midplane counterion concentration ρ0, and eq 11 for the Debye length in terms of the total ion concentration ρ0. Based on the fitted lengths, we extract fitted midplane ion concentrations for the systems with and without salt. (Note that the Debye length is related to the reference ion concentration far from any charges. However, the Debye− Huckel theory is linearized, so the variations in ion concentration should be small for the theory to be valid. Thus, for the Debye length we compare the fitted total ion concentration to the observed midplane total ion concentration.) In the table, the Gouy−Chapman lengths λ in the SDS and hair systems without salt are presented as λ/√2, for a more direct comparison with the Debye lengths κ−1 in the systems with added salt. The reason for this can be seen by comparing eqs 9 and 11. Because of the different conventions in defining the Gouy−Chapman and Debye lengths, both λ/√2 and κ−1 are equal to ϵ0 ϵ k T/(z2 e2 ρ0), where ρ0 is the total midplane ion concentration (which for systems without salt includes only counterions, and for systems with salt includes both counterions and ions from added salt). Table 2 presents the fitted Gouy−Chapman and Debye lengths for SDS and hair systems without and with salt, as well as the fitted ion concentrations as described above. The table also presents the corresponding observed midplane ion concentrations for the four systems. We see that the fitted and observed midplane ion concentrations are very close for the SDS and hair systems, both with and without salt. Table 2 also supplies values for the “naive” average ion concentration in the four systems, under the idealized assumptions that (1) there is no counterion condensation, and (2) all the mobile ions are uniformly distributed throughout the water region (bounded by the inflection points in the concentration of water atoms). For all four systems, the K


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variety of techniques,43,44 and is reported as a hydration radius, which includes an average shell of strongly interacting water molecules. To assess the ion concentration in the Stern layer for our simulations, we report the volume fraction of ions, with each ion assigned its hydration radius taken from the literature (see Table 3).

The zeta potential, obtained from electrophoresis measurements41 or electroacoustic measurements,42 is the most common way to assess the effective surface charge of colloidal particles. Theoretically, the zeta potential corresponds to the electrostatic potential at the “slip plane”, at which the hydrodynamic no-slip boundary condition appears to be enforced. To determine the location of the no-slip boundary definitively from simulations, we would need to simulate shear flow at a polymer−water interface. However, we expect the no-slip boundary to be close to the outer Helmholtz plane, because inside the Stern layer, ions are concentrated and strongly interacting with the surface, which should tend to impede flow. Hence the potential at the outer Helmholtz plane should be a reasonable proxy for the zeta potential. Indeed, previous analyses of the Primitive Model and Triple Layer Model assume that the potential at the outer Helmholtz plane is equivalent to the zeta potential.6,11 The measured zeta potential for acrylate polymer particles made by the recipe we used to construct our simulated polymer−water interfaces is about −45 mV. This experimental zeta potential is comparable to the potential at the outer Helmholtz plane for the SDS system (−41.5 mV), and about twice that for the hair system (−20 mV). Of course, the experimental particles have both SDS and hairs on their surfaces, whereas we considered these two sources of surface charge separately, in order to understand their individual effects on the interface structure and double layer. Based on these results, we may expect the simulated potential for a system with both SDS and hairs would be comparable to the experimental zeta potential. Stern Layer. In the Stern layer, the concentration of ions is high and varies rapidly in space, and the specific details of ionic size, shape, and short-range interactions matter most. Because of this, describing the Stern layer quantitatively with analytical theory is very challenging. Likewise, experimental information about the local variations of concentration and structure within the Stern layer is very limited. In contrast, atomistic simulations of the charged polymer−water interface are particularly well suited to investigating the properties of the Stern layer, because simulation potentials can account reasonably well for ion size, shape, and short-range interactions, and the resulting local structure is directly accessible. The most basic properties of the Stern layer are its overall width (described above for SDS and hair systems), the fraction of ions of various species that reside there, and their average concentrations. For both the SDS and hair systems, the Stern layer contains essentially all the bound surface anions, contributed by the surfactant headgroups, hair sulfate end groups, and MAA carboxylate groups. Additionally, the Stern layer contains sodium ions that substantially neutralize the bare surface charges, as described previously. For the SDS system, there are 70% as many sodium ions as surfactant headgroups, while for the hair system there are 96.6% as many sodium ions as charges on the hairs; to first approximation, the hairs are completely neutralized. The Stern layer for systems with added salt contains almost none of the chloride anions, which mostly reside in the diffuse layer. A simple way to visualize the Stern layer is as a concentrated salt solution. The concentration is high because counterions are strongly attracted by the bound surface charge, limited only by their finite size in how closely they can approach the bound charges. The finite size of aqueous ions has been measured by a

Table 3. Volume Fraction Percent of Ions within the Stern Layer SDS system

hair system

ion

radius (Å)

no salt

salt

no salt

salt

SO−4 Na+ CO−2 Cl− total

5.54 2.85 3.5045 3.1646

19.1 1.75 − − 20.8

19.1 1.75 − 0.05 20.9

8.62 6.83 10.86 − 26.3

8.58 6.95 10.83 0.15 26.5

From Table 3, we see that for both SDS and hair systems, the total volume fraction of ions is large (21 and 26%, respectively). For comparison, a 1.9 M sodium chloride solution has a total volume fraction of ions of about 25%. For the SDS system, the much larger sulfate anions (hydration radius 5.54 Å) contribute most of the volume fraction, compared to the smaller sodium ions (radius 2.85 Å). Coincidentally, the total anion volume fraction in the SDS and hair Stern layers are similar (about 20%), though the layers are of different thickness, and the hair system has a much larger bound charge density. The higher concentration of sodium in the Stern layer of the hair system reflects the nearly complete neutralization by counterion condensation of this more heavily charged system.

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CONCLUSIONS In this work, we have used atomistic MD simulations to characterize the charged surface of latex polymeric binder particles used in waterborne coatings. These binder particle suspensions are typical of a broad class of soft colloidal aqueous suspensions, stabilized by some combination of surfactants and grafted charge-bearing groups. We simulated the charge distributions and electrostatic potentials near the polymer− water interface, which was charged either by adsorbed anionic surfactant SDS, or grafted multivalent oligomer “hairs” containing charged monomers, both of which are present in many commercial latex particles. To construct a realistic model of the double layers, we estimated the surface concentration of surfactants and hairs, as well as a plausible set of molecular structures for the hairs, from the emulsion polymerization recipe for typical commercial binder particles and a variety of experimental characterizations. In particular, several complementary approaches to estimate the surface area per hair gave encouragingly consistent results. The two charge-bearing groups give qualitatively different structures at the polymer−water interface. In the SDS system, the hydrophobic surfactant tails penetrated completely into the polymer slab, leaving only the anionic sulfate group exposed to water. This structure results in well-defined double layer. For the hair system, the multivalent hairs do not penetrate into the slab at all. The carboxylate anions locate randomly along the length of the hairs are hydrophilic, which keeps the hairs in the water and out of the polymer. As a result, the hairs form a layer atop the polymer slab, of width about 2 nm. L


Langmuir

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The effect of added salts is relatively small given the amount of salts present in typical coatings recipes. Salts yield a slightly shorter characteristic length in the diffuse layer, but the concentration of added salt (40 mM) is similar to or smaller than the ion concentration without added salt at the midplane (43 mM for SDS and 73 mM for hair). For both the SDS and hair systems, the Stern layer is broader than just a single solvated ionabout 15 Å wide for the SDS system and 35 Å wide for the hairs. The width of the Stern layer for the hair system is determined by the end-to-end distance of the hairs themselves, which are oriented away from the surface but not fully stretched into the water. In both cases, the Stern layer contains all the surface-bound anions, as well as a substantial number of sodium cations, but almost no chloride anions from added salt, which are electrostatically expelled into the diffuse layer. Although the hair system has 6 times more surface-bound anions than the SDS system, counterion condensation is so effective in neutralizing the multivalent hairs that the potential at the outer Helmholtz plane is about half of that for the SDS system. For both systems, the outer Helmholtz plane potential, a widely used proxy for the experimental zeta potential, is in the range 20−50 mV or 1−2 kT per ion, as expected from counterion condensation. Yet in practice, hairs are useful in preventing particle aggregation, even though the effective surface charge is not increased. We hypothesize that the important effect of the hairs is to produce thicker Stern layers (about 2 nm thick). When two particles approach closely, their Stern layers overlap and interact repulsively, because each layer is concentrated with hairs and ions. If the Stern layers are thick and concentrated enough, this repulsion can prevent particles from aggregating by van der Waals attractions. The extensive counterion condensation especially in the hair system suggests our simulations may be rather sensitive to the short-range repulsive potentials between ions. Our results may overestimate counterion condensation, as recent simulation evidence in our group suggests that sodium and chloride ions modeled with the standard OPLS-AA potentials tend to associate too strongly at high concentrations relative to experimental osmotic coefficients. A modified potential with a slightly larger repulsion may weaken the counterion condensation, resulting in somewhat larger zeta potentials. To directly test potential parameters used for ions, we are simulating osmotic compressibilities of salt solutions, and tuning the potentials to reproduce the experimental osmotic coefficients. In future work, we can use our microscopically realistic model of the charged polymer−water interface to determine the free energy of binding to the particle surfaces of various additives, including surfactants and rheology modifiers. The surface binding free energy for surfactants can be compared to the binding free energy of surfactants in a micelle, to investigate the equilibration between surfactants adsorbed on particle surfaces, surfactants in micelles, and free surfactants in solution. The relative strength of surface binding for surfactants and rheology modifiers governs the competition between these additives for accessible surface area. How strongly the ends of the rheology modifiers bind to the surface affects the rheological behavior of paint formulations.47 Results of the present work can also supply key interaction parameters for coarse-grained simulations of colloidal interactions in paint formulations.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.5b03942. Size and orientation of hairs, and their relaxation time; relaxation of the net charge density profile for the hair system (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by the Dow Chemical Company. We thank Kevin Henderson, Beth Cooper, and Pu Luo for synthesis of latex particles; Michaeleen Pacholski, Johnpeter Ngunjiri, Tianlan Zhang, Wei Gao, and Kebede Beshah provided data, discussions and assistance with interpretation of the XPS, AFM, SEC-MS, GPC, and PFGNMR experiments, respectively. We thank Valeriy Ginzburg for helpful discussions.

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REFERENCES

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