Article pubs.acs.org/Langmuir
Polyelectrolyte Mediated Interactions in Colloidal Dispersions: Hierarchical Screening, Simulations, and a New Classical Density Functional Theory Jan Forsman*,† and Sture Nordholm‡ †
Theoretical Chemistry, Chemical Centre, P.O. Box 124, S-221 00 Lund, Sweden Physical Chemistry, The University of Gothenburg, 412 96 Gothenburg, Sweden
‡
S Supporting Information *
ABSTRACT: The pair interaction between two charged colloidal particles, in the presence of a polyelectrolyte as well as simple salt, is analyzed theoretically. Of particular interest is the way in which such a combination of salts can be used to induce a strong, long-range attraction, with at most a minor free energy barrier. We show that the nature of the simple salt is highly relevant, i.e., 2:1, 1:1, and 1:2 salts generate quite different particle interaction free energies at the same overall ionic strength. We adopt several different theoretical levels of description. Defining simulations at the primitive model level with explicit simple salt as our reference, we invoke stepwise coarse-graining with careful evaluations of each approximation. Representing monovalent simple ions by the ionic screening they generate is one such simplification. In order to proceed further, with additional computational savings, we also develop a correlation-corrected classical density functional theory. We analyze the performance of this theory with explicit spherical particles as well as in a flat surface geometry, utilizing Derjaguin’s approximation. The calculations are particularly fast in the latter case, facilitating computational savings of many (typically 5−7) orders of magnitude, compared to corresponding simulations with explicit salt. Yet, the predictions are remarkably accurate, and considering the crudeness of the model itself, the density functional theory is a very attractive alternative to simulations.
1. INTRODUCTION The addition of a polyelectrolyte to a suspension containing charged colloidal particles will usually influence the stability of the latter. This is, for instance, utilized in paper production and wastewater treatments. The way in which polyelectrolytes mediate such interactions has been the focus of many experimental studies.1−13 Some of these were primarily concerned with flocculation rates, while in others attempts were made to monitor the full surface interaction curve (usually, at least one of the surfaces is then macroscopic). While results and conclusions are rather scattered, the consensus appears to be that, when polyions and colloidal particles are oppositely charged (note: this paper only considers cases where polyions and particles are oppositely charged, and this will henceforth be implicitly assumed), a maximum aggregation rate/attraction is found under conditions where the total polyion charge match that of the particles.12−14 In other circumstances, one will generally observe a reduced flocculation rate, commensurate with an observed long-ranged repulsion, as measured by SFA or AFM instruments.6−9 There have also been many reports of theoretical attempts to interpret and quantify these forces.1,15−29 Early studies highlighted the ability of the chain molecules to form attractive bridges between the colloidal particles at short-range.15−23 Ion correlation forces are © 2012 American Chemical Society
also important in this regime, and dendrimers often produce similar interactions, despite their inability to generate bridges.27,30,31 Among the correlation forces, we include those formed by the “patch-attraction” mechanism, that have been discussed in several works in this area.12,30,31 The latter refers to correlations between adsorbed charges, but rather than making such distinctions, we prefer here the less specific term “correlations”. Still, although we do not specifically analyze separate force contributions, we would anticipate that patchattraction is important in these systems. Recent simulation studies of equilibrium interactions between macroscopic and flat charged surfaces, immersed in a large bulk solution containing a polyelectrolyte, have demonstrated that there is a long-range repulsion prior to the attractive regime at short-range.26,32 This is in qualitative agreement with most AFM or SFA studies. Such equilibrium simulations are complicated by the requirement of a constant polyelectrolyte chemical potential, as the surface separation is changed. The origin of the long-range repulsion is a surface overcharging, which essentially always will occur, provided the Received: November 18, 2011 Revised: February 7, 2012 Published: February 9, 2012 4069
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to be negatively and positively charged, respectively. Here, the divalent ions have the same sign of the valency as the colloidal particles. Conversely, adding a 2:1 salt (rather than 1:2) can actually lead to an increased free energy barrier. We will also show how models with all-explicit ions can be simplified, at a small loss in accuracy but a significant computational gain, by integrating out the simple (unconnected) monovalent ions, using Debye−Hückel arguments. In addition to the simulation studies outlined above, this work also includes the development of a classical density functional theory, DFT, of the polyelectrolyte + colloidal particle solution, using the implicit monovalent ions model. Starting with a polymer DFT treatment of chain configurations, and screened Coulomb interactions between charged species, we construct a theory that, in an approximate manner, also takes ion correlations into account. The inclusion of such correlations, even in a rather crude manner, is crucial to the success of the theory. With DFT calculations, we take another important step in reducing the computational effort. If we furthermore make use of the Derjaguin approximation,33 and model our particles as flat surfaces, DFT predictions of full surface interaction curves can be established within a few seconds! While DFT is more approximate than simulations, the accuracy is quite satisfactory (shown below), especially considering the crudeness of the model itself. Also, harvesting the enormous computational savings it offers, compared to simulations, one can easily extend the scope to more realistic particle sizes and polymer lengths. Finally, DFT predictions are free from noise, and the option to switch off the correlationcorrection offers important insights to the relevant physics of the system under study.
bulk solution is large enough (as in SFA/AFM experiments). Hence, at long range, the surfaces will effectively carry charges with a reversed sign, but weaker in magnitude, and there will be a concomitant simple double-layer repulsion. In a colloid dispersion, such an overcharging naturally requires an excess of total polyion charge, as compared to the total particle charge. Collecting these theoretical and experimental observations, we conclude that, if a polyelectrolyte is gradually added to a dispersion containing oppositely charged particles, one would, at equilibrium, observe (1) an initial destabilization, up to some threshold concentration, where the total polyion charge matches the total charge of the colloidal particles; (2) restabilization beyond this threshold value, due to a doublelayer repulsion between overcharged layers; and (3) a reentrant destabilization regime, when the polyelectrolyte concentration is high enough to substantially screen this double-layer repulsion. Actually, this qualitative overall response is also expected upon the addition of dendrimers, multivalent salts, and so forth, as long as the highly charged component of the added salt is oppositely charged to the colloid particles. As already mentioned, ″charge matching″ generally ensures maximum aggregation rates. Still, in many practical situations, such a ″perfect match″ may be difficult to achieve. For instance, the problem of dispersing the polymer uniformly can be significant. In this work, we will consider the interaction between two charged colloidal particles in the presence of an excess amount of polyelectrolyte, i.e., the total polyion charge exceeds that of the particles (in fact, the system we try to mimic is two particles approaching at infinite dilution). Hence, in the presence of these two components alone, we anticipate a substantial free energy barrier, but a strong attraction at shortrange. However, since the long-range barrier is of double-layer origin (between overcharged surfaces), one would expect it to be reduced, or even removed, if simple salt is added. Salt addition is of course a common and general way to flocculate a dispersion of charged particles, but the approach suggested here differs in one important respect: the attractive forces are primarily provided by bridging and correlations, mediated by the polyions. These interactions are considerably stronger and more long-ranged than typical van der Waals forces, as we shall demonstrate below. Thus, the combined addition of polyelectrolyte and simple salt is anticipated to give rise to more efficient flocculation than either of these two components would generate alone (how much faster the aggregation process will proceed upon such an addition will also depend on the particle concentration, since diffusion often is an important limiting factor). We have mentioned that a high enough polyelectrolyte concentration would be another way to reduce the free energy barrier, but this approach has at least two disadvantages: polyelectrolytes are expensive, and the overall viscosity often increase substantially at high polymer concentrations. In addition, we expect that the range of polymermediated attractive contributions will decrease beyond the overlap concentration, where the correlation length drops (not investigated here). The combined effects from adding a polyelectrolyte, together with simple salt, will be illustrated via several explicit examples, using Metropolis Monte Carlo simulations of coarse-grained models, containing a minimum of parameters. We will demonstrate that a combination of polyelectrolyte + 1:2 salt (monovalent cations, divalent anions) provides a particularly efficient ″flocculation mix″, assuming the particles and polyions
2. MODELS AND THEORIES 2.1. A Hierarchy of Simplifications. This is a rather extensive study, where we evaluate various levels of coarsegraining and approximate theoretical approaches. Specifically, one can identify four different model/theory levels, in this work: (1) Simulations, in which all ions are explicitly modeled. For brevity, we shall denote this the “explicit model”. (2) Simulations, in which all simple monovalent ions enter implicitly, via the ionic screening they generate. This is our “implicit model”. (3) DFT calculations of the implicit model, using explicitly spherical colloid particles, and a cylindrical geometry. (4) DFT calculations, again within the implicit model, but where the particles are modeled as flat surfaces, while polyions are in equilibrium with an infinite bulk solution. The results are then mapped to the corresponding spherical particle system using Derjaguin’s approximation. Proceeding from top to bottom on the “ladder of levels” outlined above, the required computational effort drops quite dramatically, often by an order of magnitude or more, at each “step”. Thus, simulations of the explicit model are limited to the “concentrated system” and the “dilute system” at low concentrations of simple salt. Simulations at the all-explicit ion level are the approach that most closely resembles the “real” experimental system, in the sense that all charged species enter explicitly, and the model itself is in principle treated exactly (within statistical noise). Hence, this is our reference system, which we make various attempts to reproduce, using simplified approaches. We emphasize that a progressively increased simplification of the system also promotes a deeper understanding of the relevant physics. In the long run, this can be at least as important as the computational gain. 4070
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−1 Figure 1. Top: snapshot from a simulation of the concentrated system, with implicit unconnected monovalent ions, and κ−1 = κMC = 13 Å (see text). Particles and monomers are represented by large and small spheres, respectively. The simulation cylinder is not explicitly shown, but its confining effects are rather obvious. Bottom left: distribution of monomer density (n) along a ″tube″ of radius 8 Å that follows the simulation cylinder z-axis, penetrating the centers of both particles (i.e., from left to right, in the top graph). Bottom right: here the tube is thinner (2 Å), and ρ runs along the radial direction (perpendicular to z), from the center of a sphere toward the simulation cylinder surface. In the two bottom graphs, the estimated bulk monomer density, nb, is indicated by a horizontal dashed line.
2.2. Models and Simulation Method. In order to make the evaluations as straightforward as possible, we will use a simple theoretical model and keep several parameters fixed, such as monomer−monomer bond length, polymer molecular weight, and the number of simulated polymers. Specifically, our model solutions will contain as follows: (1) Two negatively charged colloidal particles, each having a net charge of −86e, with e denoting the elementary charge. These negative charges are uniformly distributed across the spherical surface, located Rc = 37 Å from the center. (2) Positively charged polyions in the form of flexible linear polymers, specifically 40-mers, where all monomers carry a unit charge. Hence, a colloidal particle carries more charge than two polyions. In the simulations, 28 such 40-mers are explicitly handled. Note that these are assumed to be abundant enough to form a “bulk solution” surrounding the particles (3) Simple spherical (unconnected) ions. These may or may not be explicitly modeled, as described below. Divalent ions, if present, are always modeled explicitly. We shall adopt the standard notation that for a 1:2 salt, the divalent ion is negative, i.e., +1e:−2e. Water is treated as a dielectric continuum, and a uniform dielectric constant of εr = 78.3 is assumed throughout the system. The bulk solution is included explicitly in the simulations. Specifically, particles, polyions, and simple ions are contained in a cylinder. The cylinder symmetry axis, z, runs through the centers of the colloidal particles. Furthermore, the particles are symmetrically distributed with respect to the center of the cylinder, along the z axis. In principle, the size of this cylinder also defines a particle concentration, primarily via the counterion screening. However, we actually wish to study the pair interaction of particles at infinite dilution. This is not really a significant problem, though, since the relative contribution to the total screening from the particle counterions is small in all systems studied (particularly so in the “dilute” system). We will consider two different sizes of this cylinder, corresponding to two different polyelectrolyte
concentrations (the number of polymers is fixed in the simulationssee below). The radius of the simulation cylinder is denoted ρcyl, while its length is Lcyl. Specifically, we define the two systems as follows: the “concentrated” system, with ρcyl = 160 Å, and Lcyl = 600 Å; and the “dilute” system, with ρcyl = 320 Å, and Lcyl = 1200 Å. The dilute system has the advantage of a low counterion concentration, i.e., it is a better model of infinite particle dilution. However, adding appreciable amounts of salt to the explicit system implies a very large increase of the number of ions present in the simulations. We have therefore only simulated the dilute system, within the explicit model, at low ionic strengths. Monomers, as well as simple ions (i.e., all mono- and divalent charges), are repelled at the particle surface by a soft potential, w(r): βw(r) = (σc/(r − Rc))6, with σc = 5 Å. The distance to the particle center is denoted r, while β = 1/(kBT), with kB and T denoting Boltzmann’s constant and the absolute temperature, respectively. The distance of divergence, Rc (where the particle charges are distributed), also defines the distance of closest approach for mono- and divalent charges. The use of a ″soft″, rather than ″hard″, model of a colloidal particle is in principle irrelevant, but the softness improves simulated force statistics considerably. The temperature is fixed at 298 K. As mentioned above, the polyelectrolyte is composed of positive 40-mers, together with their negative counterions. The bond distance, σ, separating neighboring monomers along a chain, is fixed at σ = 8 Å. The simulation cylinder (large or small) contains 28 such charged polymers and 1120 counterions. The latter are either treated explicitly or implicitly, as explained below. All charged species, except the particles, carry a hard-sphere potential, with diameter (distance of closest approach) of 4 Å. In the implicit model, this hard core is unimportant, and can be neglected. It is in fact neglected in the density functional treatment. We emphasize that “simple ions” denote unconnected ones (not monomers), excluding the particles. 4071
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more particles. Such cases are also interesting, but outside the scope of this study. 2.2.2. Implicit Simple Monovalent Ions. Here, we simplify the description further, by integrating out all simple and monovalent ions, while the remaining ions interact via a corresponding screened Coulomb interaction (see below). Note that, in cases where the solution contains divalent ions, these are treated explicitly and do not contribute to the inverse −1 screening length κMC that is adopted in the simulations. We emphasize that this screening length in these cases is different from the ″global″ one defined in eq 1. Specifically
Contrary to our previous simulation studies in this area,26,27,29 where we used flat charged surfaces to model the particles, spherical particles and the surrounding solution will be included explicitly here. There are a few advantages with this latter approach. First, chemical potentials are “automatically” kept constant, as we simulate in the canonical ensemble, with explicit bulk exchange. A grand canonical ensemble would be problematic with such relatively long polymers, although there are alternative techniques for the flat geometry.27,34 Second, since we calculate the net force acting between the particles, for a fixed particle surface separation S, the problem of establishing a complete force curve lends itself to ″trivial parallelization″. In other words, we can simply run a set of simulations, with different choices of S, allowing each processor to calculate the force for the specified separation. This ensures perfect linear scaling. Simulated interaction free energies were obtained by integrating a cubic-spline fit to the corresponding force curves. Figure 1 illustrates the model system. Computational limitations force us to use a model system containing rather small colloidal particles and short polymers. However, we will show that, while the Derjaguin approximation is far from perfect for such small particles, it is still accurate enough to facilitate reasonable estimates of the interactions in a ″scaled-up″ system. Furthermore, small colloidal particles are interesting in their own right, belonging to ″the protein limit″ side of colloid science. Another limitation of our model is of course that it only permits investigations of particle interactions at the pair level, i.e., we neglect the influence from other particles, which can be important in a concentrated suspension. This is, however, a drawback shared by several experimental approaches, such as the atomic force microscope, or second virial coefficient measurements (this quantity can in principle be obtained from our interaction curves). We will establish interactions at different salt concentrations, and hence different degrees of ionic screening. The latter can be crudely estimated via the overall Debye−Hückel screening length, κ−1, with κ2 =
∑ i
βcie 2z i2 εr ε 0
κMC2 =
where the primed sum only runs over simple monovalent ions. The explicit inclusion of multivalent species is motivated by the important ion correlations these display, which are lost if they are implicitly treated. We emphasize that, as we do handle the multivalent species in an explicit manner, the screening from these ions is automatically captured, i.e., including them to the −1 definition of κMC would amount to overcounting their effects SC on screening. As indicated, the electrostatic interaction, uαγ , between two of the remaining (explicit) charges, α and γ, is now screened βuαγSC =
lBz αz γ e−κMCr (4)
r
where the superscript indicates that these are screened Coulomb interactions. We will, by making several comparisons, demonstrate that this way to integrate out the monovalent ions leads to predictions that agree well with corresponding ones, obtained with an all-explicit ion treatment. The abundance of monovalent ions in many of the systems makes this simplification a convenient alternative, with substantial computational savings. Using screened interactions, we must take into account the region that is excluded by the particles. Assuming ionic screening also inside the particles, and a uniform surface charge density at r = Rc from the particle center, we can describe the screened electrostatic interaction with another mono- or divalent ion, α, at a distance r (between origins) as36
(1)
βuc αSC = lBzcz α
sinh(κMCR c) e−κMCr κMCR c r
(r > R c) (5)
The corresponding expression for the two interacting particles is ⎡ sinh(κMCR c) ⎤2 e−κMCr βuccSC = lBzc 2⎢ ⎥ r ⎣ κMCR c ⎦
lBz αz γ r
(3)
i
where the sum runs across simple ionic species i, with concentration ci and valency zi. The dielectric permittivity of vacuum is denoted by ε0, while e is the elementary charge. Recall that polyions and particles are not regarded as ″simple″ ions, i.e., they are excluded from our definition of the overall screening length.35 2.2.1. Explicit Simple Monovalent Ions. Here, charged species α and γ interact electrostatically via a full Coulomb C interaction, uαγ (r) C βuαγ (r ) =
2 ′ βe ci(mono) εr ε 0
∑
(2)
(r > 2R c) (6)
where r is the separation between the centers of the charged species, while lB = βe2/(4πε0εr) is the Bjerrum length (7.16 Å in this study). In principle, we assume that the charge of a particle is uniformly distributed at a distance Rc from the center, but elementary electrostatics allows us to conveniently collect this charge at the origin, with zc = −86. Recall that neutralization of a particle requires complete adsorption of more than two chains (40-mers). A different regime would be systems where the charge of a single chain is, in absolute terms, larger than two or
We will only consider particle−particle separations well exceeding 2Rc. 2.3. Density Functional Theory. Simulations are in principle able to establish the properties of our model system exactly. Still, even within the implicit simple ion approach, simulations are computationally expensive, which motivates some efforts to establish an alternative theoretical approach. Another, sometimes forgotten, motivation is the deeper understanding of the physics that often results from the 4072
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hole”, we will instead assume that the radial distribution function displays an exponential approach to unity
development and use of approximate theories. Here we outline a correlation-corrected density functional theory (DFT) for the implicit simple ion model, utilizing a combination of Woodward’s general treatment for polymers37 and a simple approach to ion correlations. The latter borrows ideas from Nordholm’s “hole corrected Debye-Hückel theory”,38,39 and Forsman’s “correlation-corrected Poisson-Boltzmann theory”.40,41 Initially, the theory will be tested on systems identical to those simulated. In a second step, we will utilize the Derjaguin approximation and perform calculations on a corresponding system with flat surfaces, where the confined region is in equilibrium with a bulk solution. The reason is that, contrary to simulations, DFT calculations run considerably faster for systems with flat surfaces, since this geometry allows us to integrate out two dimensions (rather than just one). Woodward’s polymer density functional theory37,42 has become quite standard in recent years, so we will only provide a condensed description here. In order to keep things as simple as possible, we will not add any hard-sphere exclusion free energy. We have checked that this only gives small and irrelevant contributions in our system. Instead, the DFT for ideal chains is simply augmented by the presence of the charged particles (or flat surfaces) and the screened Coulomb interactions acting between monomers. The latter part will include a correction for self-interactions. This correction is our way to estimate ion correlations in this system. While approximate, we shall demonstrate that the correction has a crucial impact on predicted particle−particle interactions. 2.3.1. Polymer Configurations. In this section, we briefly recapitulate the exact DFT for ideal chains,37 devoting the subsequent section to the approximate part, where monomer− monomer interactions are included. Denoting by ri the coordinate of monomer i, an r-mer polymer configuration can be written as R =(r1, ..., rr). The ideal chain free energy, - id[N(R)] can be exactly expressed as a functional of the density distribution N(R) β-id[N (R)] =
∫ N(R)(ln[N(R)] − 1) dR + β ∫ N (R)[VB(R) + Vex(R)] dR
n(r′) → n(r′) 1 − e−λ r − r ′|
(
1 2
∫ n(r′)φ(|r − r′|) dr′
(9)
where λ is a measure of the inverse ″hole″ size, i.e.,
∫ n(r′)e−λ|r − r ′| dr′ = 1
(10)
We then obtain a corrected interaction energy per particle, ep(c), with 1 e(pc)(r) = n(r′)(1 − e−λ|r − r ′|)φ(|r − r′|) dr′ (11) 2 (c) which implies that the total interaction energy, U , is given by 1 U (c) = n(r) n(r′)(1 − e−λ|r − r ′|)φ(|r − r′|) dr′ dr 2
∫
∫
∫
(12)
Thus far, we have assumed a position-dependent inverse hole size. While this is a more comprehensive, and possibly more accurate, route, we shall nevertheless simplify the problem by using a constant value throughout the entire system. Specifically, λ is estimated in the bulk region, where the density is uniform and equal to nb nb
∫ e−λ|r − r ′| dr′ = 1
(13)
Since the bulk fluid is uniform, this equality is independent of r, and λ = (8πn b)1/3
(14)
Now we can apply this general approach to our specific model. Specifically, we let n(r′) denote the monomer density, while βϕ(r) = lB e−κMCr/r. This completes the total free energy functional, - , i.e., - = -id + U (c)
(15)
The external field from the particles, Vex(r), is identical to that used in the simulations, except when Derjaguin’s approximation is utilized (see below). The grand potential, Ω, is obtained by subtracting a term containing the polymer chemical potential, μp
(7)
where Vex(R) is an external potential, in this case due to the particles (or surfaces). Note that N(R) is defined such that N(R) dR is the number of polymers having configurations between R and R + dR. Neighboring monomers are connected by a bond potential, VB(R), with a fixed bond length σ: e−βVB(R) ∝ Πi =r‑11 δ(|ri+1 − ri| − σ). 2.3.2. Self-Interaction Correction, Using an Exponential “Hole”. We start by a rather general discussion on selfinteraction corrections, specializing to screened Coulomb interactions at the end of this section. Consider the meanfield expression for the interaction energy per particle, ep(mf)(r), given some pairwise additive interaction potential ϕ(r) e p(mf)(r) =
)
Ω = - − μp
∫ n(r) dr
(16)
The grand potential was minimized in a standard manner, utilizing Picard iterations. In the cylindrical geometry, the longrange interactions are conveniently handled by subtracting the grand potential for a pure bulk solution, Ω0. In this way, we arrive at what we shall denote as the “net free energy”, W = Ω − Ω0. More details concerning DFT in a cylindrical geometry can be found in ref 43. One advantage of the DFT approach, compared to simulations, is that we are able to model a truly infinite system, avoiding spurious effects from the cylinder boundary. In other words, densities approach their bulk value far away from the particles. Using simulations, there are inevitable surface effects due to the finite size of the simulation cylinder (cf. Figure 1). An important and sensitive test of DFT codes and theories is that the calculated forces should match discrete derivatives of corresponding interaction free energy curves, i.e., that the contact value theorem is fulfilled. We have repeatedly made such checks, confirming this crucial con-
(8)
where n(r) denotes the density at r. This expression contains a spurious part, which is due to the particle itself. One way to correct for this would be to place a spherical exclusion volume around the particle, with a radius σ. Here, σ would be chosen such that exactly one particle is excluded: ∫ θ(σ; |r − r′|)n(r) dr′ = 1. This approach would, however, impose a rather dramatic onset of the self-correction. Anticipating a softer “Coulomb 4073
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Table 1. System Parameters for the Concentrated System, Using Explicit and Implicit Models, at Three Different Ionic Strengthsa model
κ−1
explicit implicit explicit implicit explicit implicit explicit implicit explicit implicit
13.0 13.0 13.0 13.0 10.3 10.3 10.3 10.3 20.4 20.4
−1 κMC
Å Å Å Å Å Å Å Å Å Å
13.0 16.7 10.3 14.5 20.4
Å Å Å Å Å
z
Npoly(+)
Npoly(−)
Nsalt(+)
Nsalt(z−)
Ncounter(+)
−1 −1 −2 −2 −1 −1 −2 −2 −1
1120 1120 1120 1120 1120 1120 1120 1120 1120 1120
1120 0 1120 0 1120 0 1120 0 1120 0
942 0 628 0 1884 0 1256 0 0 0
942 0 314 314 1884 0 628 628 0 0
172 0 172 0 172 0 172 0 172 0
a
Npoly(+) is the total number of (positively charged) monomers in the system, while Npoly(−) is the corresponding number of simple monovalent counterions. Nsalt(z) is the number of negatively charged simple salt ions, with valency z, while Nsalt(+) is the number of positive counterions of this 1:−z salt. Finally, Ncounter(+) is the number of positive and monovalent counterions to the particles.
Figure 2. Particle−particle pair interactions, in the concentrated system, at various concentrations of simple 1:1 salt. (a) Forces, F, explicit model. (b) Free energies, explicit model. The crosses show a typical van der Waals attraction (Hamaker constant: 1e−20 J). (c) Free energies, implicit model.
principle more clearly illustrated, in the dilute system, but the number of salt pairs required to generate an appreciable increase of the simple salt concentration is very high. 3.1. Concentrated System: Simulations of Implicit and Explicit Models. This section is primarily devoted to comparisons between simulations of the concentrated system, using the explicit and implicit model, respectively. We shall also investigate effects from adding multivalent salts, again including evaluations of the cheaper implicit model. In order to avoid misunderstanding, Table 1 provides system parameters, for all the simulated particle−particle interactions, that are included in this section (i.e., for the concentrated system). 3.1.1. Adding 1:1 Salt. In Figure 1, we display interaction free energies between our particles, as predicted by the explicit and implicit models, respectively. In the former case, forces are also displayed. Recall that the free energies are obtained via integration of such force curves. Let us first focus on the explicit model results. We see that, in the absence of additional simple salt, there is a considerable free energy barrier, prior to adhesive
sistency. One example is provided in the Supporting Information. Note that our DFT approach is based on a mean-field ansatz, with a correlation correction that enters at the monomer level. A potential problem is correlations between entire polyelectrolyte chains, which are not expected to be accurately captured by this method. However, this is actually not a severe problem, since the monomer density is relatively uniform, and high, within the adsorbed layers, i.e., the polymers overlap and ″lose their identity″, so to speak. Correlations in the bulk regime are perhaps less well captured, but these are not very important, since the polymer concentration is low, i.e., even as ideal chains they would, on average, stay far apart.
3. RESULTS We will first discuss results from simulations of the fully explicit model. As these simulations are quite expensive, we have focused on the concentrated system. The dilute system is only simulated at low concentrations of simple salt, using this level of description. Salt effects are actually more pronounced, and in 4074
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Another effect, also promoting a stronger adsorption at a spherical particle, is a nearly “omnidirectional” screening, as compared to the “one-sided” screening established at a flat surface. 3.1.2. Adding 1:2 Salt. Here, we investigate how the system responds if we add a 1:2 salt, and compare these results with a corresponding 1:1 salt addition. By ″corresponding″, we mean that the respective salt contents generate the same overall screening length, κ, as defined by eq 1. One could argue that divalent ions often carry a hydration shell, and our model may indeed be extended to include such considerations (for instance, by increasing the size of these ions). However, we should keep in mind that this is only one out of several possible extensions, many of which probably are required for any specific experimental comparison. Here, we keep things as simple and generic as possible. In Figure 4, we see that the divalent salt reduces the free energy barrier in an efficient mannerand more so than simple Debye−Hückel screening would predict. According to standard Debye−Hückel theory of bulk electrolytes, where all simple ions are treated implicitly and thus contribute to κ, the results with 1:1 and 1:2 salt would be identical, since we have chosen concentrations such that κ (but not κMC) is the same in both cases. Clearly, our suggested approach, wherein all multivalent species enter explicitly, is much more accurate. At a screening length of about 10 Å, the barrier has essentially vanished. Note that a 2:1 salt would generate a completely different response. This will be further analyzed below, using the implicit salt model. So why does a 1:2 salt so efficiently reduce the free energy barrier? Given the origin of the barrier, one might anticipate that the divalent ions reduce, or even remove, the overcharging, since they would adsorb rather strongly on any overcharged (i.e., positive) surface. This hypothesis is tested in Figure 5, where we compare the relative overcharging in the presence of a 1:1 salt, and a corresponding concentration of a 1:2 salt. We see that the overcharging is smaller in the latter case, but rather slightly so. Based on this, it seems likely that other mechanisms are at play as well. Presumably, ion correlation forces between the overcharged surfaces, generated by the oppositely charged divalent ions, also provide a significant reduction of the free energy barrier. However, we should also consider the screening of the polyions, which in this system may be at least as important as particle screening. Divalent ions, oppositely charged from the polyions, will effectively screen the repulsion between the latter, which is another way of stating that the repulsion between overcharged layers is substantially reduced. 3.1.3. Adding 2:1 Salt. The overall conclusion from the comparisons presented above is that we can safely rely on predictions from our implicit monovalent ion model, at least in terms of the overall response to changes of relevant system parameters. Armed with some confidence, we now proceed to investigate responses from the addition of a 2:1 salt, rather than 1:1 or 1:2. Implicit model predictions are given in Figure 6. The difference is quite remarkable! Recall that the overall ionic strength, κ−1 is conserved between these three systems. Hence a simple DLVO approach would predict identical interactions, but according to these simulations, the barrier is more than three times higher in the presence of a 2:1 salt, as compared with a 1:2 salt at the same ionic strength. The positive divalent ions will promote, rather than reduce overcharging at the particle surfaces. Furthermore, the overcharged (positive) particles are probably not efficiently screened by positive divalent ions remaining in the “bulk” solution. Still, at high enough salt
bridging and correlations, which dominate at short range. For such small particles, the barrier is not exactly hugeless than 4 kBTbut adopting Derjaguin’s approximation we anticipate a linear increase with particle radius. The barrier drops substantially as monovalent salt is added, but a rather large amount would be required to completely remove the barrier. Still at about 30 mM concentration of a 1:1 salt, the barrier is reduced to less than 20% of it is ″original″ height, and the interaction curve is overall attractive. It is important to emphasize that this scenario is drastically different from merely adding salt to a colloid suspension, without the presence of any polyelectrolyte. Then, attraction is provided by van der Waals forces, but it is clear from Figure 2 b that these interactions are very weak and short-range compared to the ones mediated by the polyelectrolyte. As we use a continuum model, results at short-range should be considered with some caution, since specific interactions, and the molecular nature of the solvent, could be relevant in this regime. Keeping this caveat in mind, there is nevertheless little doubt that ion correlations and bridging will bring about a strong and relatively (compared to dispersion) long-range attraction. Hence, by adding polyelectrolyte and salt of an appropriate kind and amount, we anticipate a much more rapid flocculation than we could achieve by just adding simple salt. All these features are captured, qualitatively and semiquantitatively, by the implicit model. The latter seems to overestimate the impact of salt somewhat, and the barrier maxima are located at slightly too short separations, using the explicit model as reference. Still, these are minor shortcomings of little practical interest. Overall, the results from the two models agree surprisingly well. As we have mentioned, the repulsion at long-range is associated with an overcharging at each particle. This is illustrated in Figure 3, where we also display how the relative
Figure 3. Overcharging at a single particle, with the explicit model, for various particle sizes. The surface charge density is the same in all systems, as is the concentration of simple 1:1 salt (and polyelectrolyte).
overcharging depends on particle size. These results were obtained by simulations of a single particle in a spherical cell, also containing polyelectrolyte and simple salt. For our reference particle, with Rc = 37 Å, and in the presence of 16 mM explicit 1:1 salt, the overcharging is about 25% of the nominal particle surface charge. This value decreases with particle size, and is about half as large in the limit of a corresponding (same surface charge density) flat macroscopic surface. This is obviously a curvature effect. At a small particle, there is a reduced repulsion between the parts of the adsorbed polyions that are not in direct contact with the particle surface. 4075
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Figure 4. Interaction free energies between the particles for the concentrated system, in the presence of 1:1 and 1:2 salt, respectively. Comparisons are made at identical overall screening lengths, κ−1, as defined in eq 1. Note that the particles are negatively charged. (a) Explicit model. (b) Implicit model.
3.2.1. Simulations Results and DFT Predictions, Explicit Particles. Here, the spherical particles, and the cylindrical geometry, is retained in the DFT formulation. The DFT calculations are still much faster than the simulations, but a full interaction curve can take a few days with a single CPU core (compared to about a month, using single core simulations). In practice, of course, we have devoted one core to each separation. Before we evaluate predictions on interactions, we shall briefly consider structure. A calculated monomer density profile, using DFT, is compared with corresponding simulation data in Figure 7. This illustrates an advantage of the DFT
Figure 5. Overcharging at a single spherical particle, in the presence of 1:1 and 1:2 salt, respectively. The overall Debye screening length, κ−1, is identical for the two cases.
Figure 7. Comparing a simulated monomer density profile with corresponding DFT predictions. The implicit model is used, and the system is dilute (particle surface separation: 46 Å). The displayed profile is analogous to the one shown for the concentrated system, on the bottom right graph of Figure 1.
Figure 6. Interaction free energies between the particles for the concentrated system, in the presence of 2:1, 1:1, and 1:2 salt, respectively. The implicit model is used.
concentrations, the barrier is expected to be reduced, on account of an increased ionic screening, even when the added salt contained positive divalent ions (not investigated, though). 3.2. Dilute System, Implicit Model: Simulations and DFT Calculations. Recall that “dilute” here refers to the polyelectrolyte concentration. If we take one of our concentrated systems and dilute it, simply by adding water, the free energy barrier increases rather rapidly, as we shall see. One might therefore anticipate a more dramatic response to an increased ionic strength, in the dilute system. Having established the influence of added multivalent ions, we will restrict ourselves here to how the free energy barrier is affected by the addition of monovalent salt, and to evaluations of our DFT approaches. Given the large amount of ions that would be required to model salt addition, we have only included explicit model simulation results for cases where the concentration of simple salt is relatively low.
approach, in addition to the computational savings, namely, that boundary effects essentially are absent. We see how the calculated density reaches, and retains, its bulk value, whereas the finite cylinder generates inevitable surface effects in the simulations. If we switch focus to the region close to the particles, we note that the DFT predictions are reasonably accurate, albeit with some underestimation of the adsorbed layer, in terms of height as well as width. The strong response to salt addition in the (polymer-) dilute system is illustrated in Figure 8, where we also compare predictions from the correlation-corrected DFT with simulation results. Note that the systems with κ−1 = 36.8 Å can be obtained via an 8-fold dilution (adding water) of the concentrated system, with κ−1 = 13 Å (i.e., 13(8)1/2 = 36.8). At high ionic strengths, such simulations would become very 4076
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Figure 8. Simulations and DFT predictions of interaction free energies for the dilute system. (a) Implicit model simulations. (b) Explicit model simulations. (c) DFT predictions, utilizing the implicit model.
demanding. Analyzing these DFT predictions, we see that we arrive at slightly different conclusions, depending on which system we use as our reference. Given that the DFT is based on the implicit model, one might be inclined to choose the corresponding simulations, displayed in Figure 8a, as reference. In that case, we see that the DFT clearly underpredicts the barriers and overpredicts the separation of the barrier maxima. A better agreement is observed at high ionic strength, which could reflect an underestimation of ion correlations. However, this might also be a simple consequence of the overall stronger salt response predicted by the simulations, finally ″catching up″ with the overall lower barriers predicted by DFT. If, on the other hand, we choose the, presumably more accurate, explicit model simulations as our reference, the DFT predictions are remarkably accurate, in fact, more so than the implicit model simulations. This circumstance seems somewhat fortuitous, but it does highlight that one must keep in mind the crudeness of the model itself, when analyzing approximate theories, such as DFT. Our models, including the explicit one, are simple and coarse-grained, aiming to be generic. In any ″real″ scenario, comparing or predicting the outcome of specific experiments, there are a number of additional complications: the surface charge density is generally not known with high accuracy, there are often hydrophobic interactions to take into account, and so forth. Thus, the DFT calculations are, at least for practical purposes, sufficiently accurate. Note, however, that in absence of the correlation correction, the DFT becomes highly inaccurate. This is exemplified in Figure 9. It is obvious that the performance of the DFT is severely deteriorated in the absence of the correlation-correction. Strictly speaking, the correlation-free version does predict a barrier (though not visible on the scale of the graph), but this is underestimated by about 2 orders of magnitude! Similar findings apply at other ionic strengths (not shown), except of course if enough salt has
Figure 9. Predictions of interaction free energies in the dilute system, using DFT with and without the correlation-correction.
been added to remove the barrier, even when correlations are accounted for. 3.2.2. DFT Predictions in a Flat Surface Geometry: The Derjaguin Approximation. Thus far, we have investigated two alternative ways to full-scale explicit salt simulations. First, we treated the simple monovalent ions in an implicit manner. As a further reduction of the computational effort, we have used DFT calculations. This already amounts to a huge computational savings. Still, DFT calculations are relatively slow in a cylindrical geometry, since there are two dimensions to integrate across. A complete force curve typically requires a few days on a single core. By utilizing Derjaguin’s approximation33 (DA), we can treat the particles as flat surfaces, which allows us to integrate away two dimensions, rather than just one. This makes an enormous difference, allowing a full surface force curve to be calculated within a few seconds! According to the DA, the interaction free energy, W(S) between two particles of radius Rc, at a surface separation S, can 4077
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Figure 10. Evaluating predictions based on DFT calculations in a flat surface geometry, utilizing the Derjaguin approximation, eq 17. (a) Predictions of how the free energy barrier responds to the addition of simple salt. (b) Comparing simulations and DFT predictions of the relatively long-range attraction that is obtained at high ionic strengths. The dashed line shows a typical van der Waals attraction, with a Hamaker constant of 1e − 20 J.
be expressed in terms of the interaction free energy per unit area, gs(S), in the flat surface geometry W (S) = −πR c
∫S
∞
dxgs(x)
(17)
This relation also implies linear scaling of the interaction free energy with the particle radius. The DA is, however, not expected to be very accurate if the particles to be modeled are small, i.e., if the particle charge is similar to the charge of a polyion. This is indeed the condition prevailing for the particles that we have studied in this work. In Figure 10a, we nevertheless compare results from applying the DA on DFT predictions for a flat surface geometry, to corresponding simulation results. The agreement with implicit model simulations is actually better than what we found with DFT for explicit particles in the cylindrical geometry. This appears to be yet another fortuitous circumstance, related to the DA being somewhat inaccurate for such small particles (see below). In Figure 10b, we illustrate how the combination of polyelectrolyte and simple salt produces a long-range attraction, in a similar way as we found for the concentrated system (Figure 2). Note that the very weak barrier found with simulations has a height close to the statistical detection limit, i.e., it is barely significant. By comparing the DFT predictions displayed in Figures 8c and 10a, we see that the DA is not very accurate in these cases. On the other hand, one should keep in mind that our particles are quite small, chosen so in order to facilitate simulation approaches. In most experimental systems, one would use larger particles, which should improve the accuracy of the DA. This is illustrated in Figure 11, where we compare DFT predictions for small spheres, corresponding (same surface charge density) larger spheres, and the limiting case of flat surfaces, connecting the results via the DA. For simplicity, we have used our “standard” particle size, with surface charges located 37 Å from the center, as our reference. This means that our flat surface calculations are mapped to particles of the “reference size”, by utilizing eq 17, with Rc = 37 Å. Similarly, the interaction free energies obtained with twice as large colloids are mapped to the smaller ones simply via a division by 2. We note that the DA limit is not reached, even with twice as large particles. On the other hand, the approach to this limit seems rather rapid, so for particles with a radius larger than, say, 10 nm, one can use flat surface calculations with some confidence (at least for this polymer length and ionic strength).
Figure 11. DFT evaluation of the Derjaguin approximation, at κ−1 = 36.8 Å. In order to facilitate comparisons with other figures, the results are mapped on our “reference particle” size, i.e., Rc = 37 Å. For instance, from calculations with Rc = 74 Å, we obtain a net interaction free energy, W(74), say. This is then mapped to the interaction free energy displayed in the graph, W, simply via W = 0.5 × W(74).
4. SUMMARY We have analyzed our model at four different levels. Switching from one level to the next implies an increase of “crudeness” in the model description, but also a dramatic drop of the required computational effort. These levels are as follows: (1) simulations at the primitive model level, with explicit representation of all ions; (2) simulations, in which simple monovalent ions only enter implicitly via ionic screening; (3) correlation-corrected DFT calculations, with implicit simple monovalent ions, but explicit particles (i.e., in the cylindrical geometry); (4) correlation-corrected DFT calculations in a flat surface geometry, utilizing the Derjaguin approximation for predictions of corresponding particle−particle interactions. For reasonably large particles, the last option seems to be almost as accurate as the first, at least considering the overall crudeness of our model. Computationally, however, it can in many cases be more than 6 orders of magnitude faster! Also, simplified models and theories allows us to more readily isolate important mechanisms in the systems. Switching focus from models and theories to the results in this work, one observes several findings that should be of interest to colloid scientists, in particular, if a rapid flocculation is desirable. Specifically, we have established how a combination of polyelectrolyte and salt can generate an attraction with a strength and range that would otherwise be difficult to obtain, at least for the studied case of dilute colloidal particles. We have 4078
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(26) Turesson, M.; Åkesson, T.; Forsman, J. Langmuir 2006, 22, 5734. (27) Turesson, M.; Åkesson, T.; Forsman, J. Langmuir 2007, 23, 9555. (28) Turesson, M.; Woodward, C. E.; Åkesson, T.; Forsman, J. J. Phys. Chem. B 2008, 112, 5116. (29) Turesson, M.; Åkesson, T.; Forsman, J. J. Colloid Interface Sci. 2009, 329, 67. (30) Lin, W.; Galletto, P.; Borkovec, M. Langmuir 2004, 20, 7465. (31) Trulsson, M.; Forsman, J.; Åkesson, T.; Jönsson, B. Langmuir 2009, 21, 6106. (32) Turesson, M.; Forsman, J.; Åkesson, T. Phys. Rev. E 2007, 76, 021801. (33) Derjaguin, B. V. Kolloid Zeits. 1934, 69, 155. (34) Turesson, M.; Woodward, C. E.; Åkesson, T.; Forsman, J. J. Phys. Chem. B 2008, 112, 9802. (35) Beresford-Smith, B.; Chan, D. Y. C.; Mitchell, D. J. J. Colloid Interface Sci. 1985, 105, 216. (36) Beresford-Smith, B., Thesis, Australia National University, Canberra, Australia, 1985. (37) Woodward, C. E. J. Chem. Phys. 1991, 94, 3183. (38) Nordholm, S. Chem. Phys. Lett. 1991, 105, 302. (39) Nordholm, S.; Penfold, R.; Jönsson, B.; Woodward, C. E. J. Chem. Phys. 1991, 95, 2048. (40) Forsman, J. J. Phys. Chem. B 2004, 108, 9236. (41) Forsman, J. J. Chem. Phys. 2009, 130, 064901. (42) Woodward, C. E. J. Chem. Phys. 1992, 97, 695. (43) Forsman, J.; Woodward, C. E. Langmuir 2010, 26, 4555.
also, in this study, shown that the nature of the added salt has a dramatic influence on interactions and colloid stability. An alternative route to rapid flocculation might be to reverse the charge of the polyions, relying on depletion. One could also add divalent positive ions to this latter system, which might lead to long-range bridging and correlation interactions between overcharged particles. We postpone such investigations to future work.
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ASSOCIATED CONTENT
S Supporting Information *
Additional figure. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank Christophe Labbez and Martin Turesson for valuable comments. J.F. acknowledges the Swedish Research Council for financial support.
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REFERENCES
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