Article pubs.acs.org/Langmuir
Effect of Charge Inhomogeneity and Mobility on Colloid Aggregation Y. S. Jho,*,† S. A. Safran,§ M. In,§ and P. A. Pincus∥ †
Asia Pacific Center for Theoretical Physics, Pohang, Korea Department of Physics, POSTECH, Pohang 790-784, South Korea § Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot, Israel 76100 § Laboratoire Charles Coulomb - UMR5221 CNRS-Universite Montpellier 2, France ∥ Departments of Physics and Materials, University of California, Santa Barbara, California, United States ‡
ABSTRACT: The aggregation of inhomogeneously charged colloids with the same average charge is analyzed using Monte Carlo simulations. We find aggregation of colloids for sizes in the range 10−200 nm, which is similar to the range in which aggregation is observed in several experiments. The attraction arises from the strongly correlated electrostatic interactions associated with the increase in the counterion density in the region between the particles; this effect is enhanced by the discreteness and mobility of the surface charges. Larger colloids attract more strongly when their surface charges are discrete. We study the aggregation as functions of the surface charge density, counterion valence, and volume fraction.
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of the macroions.17 This effect is even larger for multivalent ions or higher surface charge density.5,14,18,19 Lobaskin and Linse2,20,21 simulated a model system that approximated the aggregation of charged colloidal particles for the case of trivalent counterions and where the surface charge density, with σ ∼ 1.2 e/nm2, for the case of monovalent Γ = 0.7, divalent Γ = 2.0, and trivalent Γ = 3.7. Aggregation was observed only for trivalent counterions and leads to phase separation. The authors mention that this aggregation may be analogous to the phase separation that arises from the attraction of highly charged colloids in the presence of trivalent counterions. To increase Γ, the diameter of colloidal particles simulated was taken to be ∼4 nm, which is much smaller than the situation typically studied experimentally where the colloid diameters are in the range of one to several micrometers.4,7,22 In the limit of high Γ, mean field theory no longer applies and strong coupling theory may be required to explain the attraction between macroions.8,13 In fact, many experiments are performed under conditions where the coupling parameter is not large, Γ ≲ 1, and for systems in which the colloid concentration is dilute. For example, Tata et al.7 observe attraction when Γ was equal to 1.29 and 0.43. There are theoretical difficulties in studying a system which has intermediate coupling parameters, since none of the perturbation expansions or mean field approximations are valid.15,23,24 Therefore, numerical approaches are very valuable when Γ is on the order of unity.
INTRODUCTION Attractive forces between similarly charged colloids in aqueous solution is a counterintuitive phenomenon that has been intensively investigated for over a decade.1−9 Mean field predictions based on the Poisson−Boltzmann theory or Derjaguin−Landau−Verwey−Overbeek (DLVO)10,11 theory cannot explain these phenomena, because these calculations (Poisson−Boltzmann theory or DLVO) assume that the thermal energies are larger than the electrostatic interaction energies. This results in uncorrelated counterions in solution that screen the macroion electric fields. The interaction between two screened, charged colloids is reduced but remains repulsive.12 Several new theoretical approaches have been proposed to treat the opposite limit in which the electrostatic interactions are larger than the thermal energies; these have been applied to simplified systems containing a few colloidal particles13−15 in the presence of multivalent counterions. In this regime, the dimensionless coupling constant Γ = Wc/kBT that balances the Coulomb and thermal energies is assumed to be large (Γ ≫ 1), where Wc (≡(q2e2/εaz)) is defined as the electrostatic energy of a 2D Wigner crystal, az is a Wigner-Seitz radius,16 e is the electronic charge, ε is a dielectric constant, and q is the counterion valence. Shklovskii14 predicted attractive interactions when Γ ≫ 1 in planar geometry; the attraction arises because the interactions between the correlated counterions and the surface charges dominate the repulsions between the correlated counterions, similar to the cohesion energy of an ionic crystal. Rouzina and Bloomfield5 showed that spontaneous attraction is possible when Γ > 2. The ion correlation drives for the strong condensation of counterions at the surface © 2012 American Chemical Society
Received: March 7, 2012 Revised: April 23, 2012 Published: May 9, 2012 8329
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In previous theoretical treatments, the surface charge distribution was characterized only by its average value. However, some experiments25−29 and some theories30−42 report that not only the average surface charge, but also its distribution on the colloidal surface can affect the interactions between macroions.25,30 However, some of these studies were restricted to the DH limit, which does not apply at low salt where the electrostatic interactions can be the most interesting.36−38,40 Furthermore, the mobility of surface charges allows dipolar fluctuations that have been shown to induce attraction.43 In this paper, we use numerical simulations to show that electrostatics alone can give rise to significant attraction that leads to aggregation of charged colloids even in the intermediate coupling regime (Γ ∼ 1) when the surface charges are discrete. When the surface charge density is around 0.01−0.1 e/nm2, the counterion valence is 2−3, and the size of the colloids is about 50 nm, we find aggregation of the colloidal particles when Γ ∼ 0.5 for divalent and Γ ∼ 0.7 for trivalent counterions. Increasing the mobility of the surface charges further enhances the attraction, because this allows the surface charges to adjust themselves in response to the charge (or potential) distribution in the system, thus allowing the system to lower its free energy by optimizing the attractions. Our simulations show that more counterions are condensed in the interstitial region between two neighboring colloids compared with the case of colloids with the uniformly charged surfaces. The angular distribution of the counterions is not isotropic, which induces the attraction. In particular, stronger attractions are found for larger particles, since there are more counterions between them to mediate these correlation attractions. We also find that larger colloidal volume fractions, higher surface charge densities, and higher counterion valence all enhance the aggregation. This paper is organized as follows: In section 2, we describe the simulation scheme and the model system. In section 3, we present the simulation results for both discrete and mobile surface charges. At the end of this section, we study the properties of larger particles up to 200 nm in size by performing simulation on two colloids with compensating counterions. In section 4, we discuss and summarize the physical meaning of the results.
Figure 1. Model colloidal particle is shown. The hardcore radius of the colloid has a diameter dc. Surface charges of radius rs are placed inside the colloid at a depth of Ds. Surface charges are densely packed on the surface and are governed by the Lennard-Jones interactions with a characteristic radius rs. We study two cases: (i) where the surface charges are fixed or (ii) where the charges can move on the surface on which they initially placed. rm is dc/2 − Ds. qiqj ⎧ r0i + r0j if rij ≥ ⎪ ∑ 2 ⎪ l , m , n ∈ A ε | rij⃗ + lLxx ̂ + mLyy ̂ + n Lzz|̂ ϕ(rij) = ⎨ ⎪ r0i + r0j ⎪∞ if rij < ⎩ 2
(
)
(1)
where l, m, n are integers; Lx, Ly, Lz are the lengths of the periodic cell; and x̂, ŷ, ẑ are the unit Cartesian vectors. qi is the charge of particle i of radius r0i. We consider the water as a dielectric continuum whose dielectric constant ε = 80. The molecular nature of water can be important, especially when the counterions are near the highly charged surface. However, in our system, because the volume fraction of counterion is much smaller than the volume fraction of water by a factor less than 10−4, we can consider water as a dielectric continuum. No ionspecific chemisorption processes are considered, and the counterions condense on the surface of colloidal particles only when Coulomb interaction dominates the thermal energy. The Ewald method is used to calculate the electrostatic potential energy.45 Note that hardcore repulsion between the colloidal particles is included. Motion. For uniformly charged colloids, only translational motion is considered in the simulation trial moves. For the case of nonuniformly charged colloids that have fixed charged groups, we include three additional rotational degrees of freedom. For the case of mobile, discrete surface charges, additional degrees of freedom associated with the surface motion are allowed. The large difference in size and charge of the colloids (50−200 nm) as compared to the counterions (3 Å) requires that we treat different time scales when modeling their respective motions. The fast counterions quickly adjust their position to the instantaneous configuration of the colloids so that the system is locally in equilibrium (adiabatic approximation). On the time scale on which the counterions diffuse, the colloidal particles hardly move, which allows the counterions to quickly diffuse and equilibrate. To treat the motion of the colloidal particles, we must include MC steps on a much longer time scale. We therefore include (in addition to the MC steps that characterize the fast counterion motion) trial
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SIMULATION Geometry. The charged macroions consist of large spherical colloidal particle of diameter dc, with a hard sphere boundary. The charge of uniformly charged colloidal particles is localized at the particle center. For nonuniformly charged colloids, the discretely charged groups are represented as small ellipsoidal charged particles of axes rs and Ds, whose center is located at a fixed distance rm from the macroion center, so that rm + Ds = dc/2 (Figure 1). rs is taken to be large enough so that the surface charges are fairly close-packed, i.e., rs ≈ [(rm − Ds)2/ Ns]1/2. To achieve this fairly closed-packed distribution on the surface, we performed simulations that allow the surface particles to arrange themselves via their mutual interactions including both electrostatic and Lennard-Jones interactions, which are repulsive at short distances.44 The mobile surface charges are represented a “soft spheres” that also interact via both electrostatic and Lennard-Jones potentials. The counterions are modeled as spherical particles of radius r, whose charge q is localized at their center. The pair potential energy between two charged particles i, j separated by a distance rij is written 8330
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Figure 2. The ratio n of free clusters to the total number of colloidal particles and the ratio α of colloidal particles in clusters to the total number of colloids are plotted as a function of the surface charge density. Three different surface charge distributions (uniformly charged; discrete, fixed surface charges; discrete, mobile surface charges) are compared. The line is for uniformly charged colloids; the circles are for colloids whose surface charges are discrete and fixed, and the squares are for colloids whose surface charges are discrete and mobile. dc is 50 nm; rs is 3 Å, and Ds is 3 Å. The volume fraction of colloids is 0.01. qs, the valence of the surface charge, is −1, and qc, the counterion valence, is 3.
specifically, we simulate a system in which this diameter is about 200 nm, using a description involving the potential of mean force.
moves of clusters that comprise both the colloids and the surrounding counterions. Two conditions must be enforced. First, the cluster trial moves must obey detailed balance before and after each trial move. The composition of the cluster must not change during the trial move; if this is violated, the trial is rejected. Second, there must be enough time for the counterions to relax or rearrange in response to new configurations of the colloids that are generated by the trial moves of the clusters. In order to accelerate the simulation, we consider two types of cluster moves: (i) clusters comprising a single colloid and its cloud of counterions, and (ii) cluster moves that involve multiple nearby colloids and their counterion clouds. Most of our simulations treat 50 colloidal particles. Overall, 2.4 × 105 steps are used for the equilibration, and an additional 8 × 104 steps (involving trial moves of the counterions) for the statistical averages appropriate to a temperature of 300 K. We repeat each simulation several times and present the average and standard deviation. The volume fractions of colloidal particles are in the range 0.05− 0.15, with the precise values indicated in each figure. The cluster moves occur about every 100−200 steps, and the composition of the cluster is renewed for every 1000−2000 steps. We first present our results for different distributions of surface charge: uniform; discrete, fixed; and discrete, mobile. The colloid diameter is chosen to be 50 nm which is close to the sizes range used in the experiments of refs 7,46,47. However, in order to accelerate the simulations, we actually use colloidal particles that are smaller than the experimental ones by about a factor of 2 or more. Much larger particles on the order of micrometers are often used in the experiments, but it is impractical to deal with these sizes in our simulations, because the number of counterions increases with the square of radius for fixed surface charge density. However, the size issue cannot be simply ignored because it turns out that the results depend not only the coupling constant, Γ, but also on the colloidal particle diameter, dc. Therefore, later on in this paper we discuss how the interactions between the colloids and their subsequent aggregation depend on the particle radius; more
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RESULTS Colloidal Aggregation. The first result of the simulation is that nonuniformly charged particles attracted each other. Figure 2 shows the state of aggregation of the system as a function of the average surface charge density. Two parameters characterize the aggregation: n and α. n is the ratio of the number of clusters (of aggregated colloidal particles) to the total number of colloidal particles. As described in the previous section, we define a cluster as containing colloidal particles and the surrounding counterions. So, 1/n is an average size of aggregates. α is the ratio of the number of colloidal particles that are associated with clusters to the total number of colloidal particles. In the thermodynamic limit of an infinite system, there would be a phase transition from a “gas” to a liquid or solid phase of aggregated colloids due to the correlationinduced attractions; in that case, the density would be the order parameter that distinguishes between the various phases. However, since our simulations treat a relatively small number (∼50) of colloidal particles; it is not appropriate to use the density as an order parameter. That is the reason we use n, which also contains information related to the density. If there is no aggregation, there are almost no clusters containing multiple particles and the number of clusters is about equal to the number of particles so that n ≈ 1. In the opposite limit, when all colloids aggregate into a single cluster, n approaches 0. While this dramatic change in n suggests the possibility of a phase transition, it does not necessarily predict one in the limit of an infinite system; that can be determined only by more detailed studies. As in the experiments, we consider colloidal particles to belong to the same cluster if two particles are separated by a distance smaller or equal to order dc.48 In the actual simulations, we vary this value somewhat empirically, but in most cases, we use a value of 0.5 dc. We treat the case in which the counterions are trivalent, and surface charge density varies from 0.01 to 0.1 e/nm2, in contrast to previous 8331
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Figure 3. Fraction of free clusters, n, is plotted as a function of the colloidal size, dc, for two cases in which the surface charges are both mobile and discrete. (a) The net charge per colloidal particle is fixed as 501e (This value is used to make sure of the charge neutrality in presence of trivalent counterions) with increasing colloidal size, and colloidal concentration is kept fixed at 0.2 μM as well. In (b) and (c), the surface charge density is kept fixed at 0.01 e/nm2. The other parameters are the same as in Figure 2. (b) The concentration of the colloidal particles (number per unit volume) is fixed. The volume fraction of colloidal particles increase as their size increases. (c) The volume fraction of colloidal particles is held fixed. Thus, the concentration of the colloid decreases as their size dc is increased.
simulations where σ ≈ 1.2 e/nm2, dc = 4 nm;2,49 our values of the coupling constant of Γ (≡Wc/kBT) that are associated with clusters vary from 0.36 to 1.14. Each graph in Figure 2 presents the results obtained for the three different surface charge distributions: uniformly charged, discrete and fixed, and discrete and mobile. The uniformly charged colloids do not aggregate.5 Since Γ < 2, they are not in the strong coupling regime and their behavior is well-described by mean field theory that predicts only repulsive interactions. Counterions are released to the solution, and the Debye− Huckel approximation is applicable,50,51 with a renormalized effective charge. If the electrostatic field is not strong (i.e., Γ ≪ 1), the behavior of the counterions is dominated by their thermal fluctuations and their distribution is determined by the locally averaged field. In such systems, each colloidal particle is locally screened, and all particles have the same net charge; therefore, their interactions are repulsive unless the volume fraction is very high when local correlations become important. In contrast to the existing theories but consistent with experiment, we find aggregation for inhomogeneously charged colloids even for relatively small values of the coupling constant. It is known that surface charge discreteness enhances the electrostatic correlations between macroions.31−34 Our simulation results suggest that aggregation may be enhanced by charge localization even for small values of the coupling constant. The circles in Figure 2 show the results when the surface charges are discrete and immobile. A relatively sharp change in the aggregation behavior is then found: the colloidal particles begin to aggregate when the surface charge is σ ≳ 0.03 e/nm2 and form large clusters above 0.05 e/nm2 for which case n approaches zero. For the case of discrete surface charges, more counterions are condensed near the surface31−34 than for a uniformly charged surface which may be described as a relative lowering of the chemical potential for counterions near the surface. In order to explore this further, consider the simple model system consisting of a single charged planar surface and its counterions. We assume that the condensed counterions are strongly correlated on the surface and are organized in an almost crystalline structure. The electrostatic contribution to the counterions chemical potential is expressed as
where μel is the electrostatic contribution to the chemical potential for a discretely charged surface. μ0 is the electrostatic contribution to chemical potential for a uniformly charged surface, and μ1 is the difference in the electrostatic contribution to chemical potential between discretely charged surface and uniformly charged surface. The major difference between the two surface charge distributions arises from the nearest surface charges in contact. When the counterions locally assembly into a hexagonal lattice structure on the uniformly charged planar surface, the approximate chemical potential is expressed as52
μ0 ≃ −1.65ΓkBT
(2)
μ0 contains the electrostatic contribution of counterions due to the strong correlation near the charged surface, and is proportional to Γ;14 μ1 represents the correction to the chemical potential considering the discreteness of surface charge. To approximate this correction, only the area of the smallest unit cell of the Wigner lattice of the surface charges is considered. The electrostatic interaction between the surface charges in this cell area and one counterion is kBT[(2q2c lB)/ az]{[1 + (rc + Ds)2/a2z ]1/2 − (rc + Ds)/az}. The total surface charge in this cell is the counterion valence, and the separation between the counterion and the surface charge is approximately rc + Ds. Since μ0 was related to the attractions of correlated counterions ions to a uniform surface charge density and the repulsion between correlated counterions, the difference these energies due to the discreteness of the surface charge can be estimated by replacing the uniform surface charge in one Wigner cell by a single, discrete charge allows us to estimate the correction term, μ1 ⎞ 2 2qc 2lB ⎛ ⎜ 1 + (rc + Ds) − rc + Ds ⎟ ≃− + 2 kBT rc + Ds az ⎜⎝ az ⎟⎠ az μ1
qc 2lB
(3)
Here, rc is the radius of counterion, and qc is a counterion valence. Thus, if μ1 is negative, more counterions are condensed on the nonuniformly charged surface than on the uniformly charged one. Since rc, Ds, az > 0, μ1 is always negative unless (rc + Ds)/az is unreasonably large, >5000. Therefore, we conclude that a discretely charged colloid attracts more counterions toward its surface than uniformly charged colloid. This was predicted more formally in refs 31−34,37,38.
μel = μ0 + μ1 8332
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In the same figure, we show that, in the case of discrete but mobile charges, the reduction in n, as the surface charge is increased, is even stronger; this occurs even if the motion is very limited. The surface charge motion gives the system additional freedom which allows the surface charges to reorganize in response to the counterion distribution; apparently, this reorganization increases the tendency of the colloids to cluster. Including the mobility of the surface charges results in aggregation of colloidal particles at even lower surface charge densities (by about 20−30%) compared with the discrete, fixed systems. In contrast to the case of fixed but discrete surface charges where aggregation occurred for σ > 0.03 e/nm2, mobile surface charges result in aggregation surface charge densities from 0.02 e/nm2. This surface charge density is comparable to recent experiments48 which shows aggregation of colloids when σ is about 0.019 e/nm2. As the surface charge is further increased, we find that n is almost 0 when σ > 0.04 e/ nm2 which suggests that, in the thermodynamic limit, the system might possibly show a phase transition to a condensed state. In the remainder of this section, we focus on the case of discrete and mobile surface charges which result in the strongest attractions. In Figure.3, n is plotted as a function of the colloid diameter when the counterion valence, qc = 3. If the total surface charge is fixed, DLVO theory for uniformly charged colloids predicts that increasing dc decreases the electrostatic potential by a factor 1/(1 + dc/λD) (λD is a Debye screening length). Interestingly, the opposite trends is observed in systems with discrete surface charges, i.e., stronger attractive interactions are seen. In Figure 3a, we vary the colloid diameter from 10 to 40 nm keeping the colloidal concentration fixed at 0.2 μM, and the colloidal charge fixed at 501e. As the colloid diameter is increased, the surface charge density decreases while the volume fraction increases. Our results show that the colloids begin to aggregate when the diameter is larger than 20 nm. In Figure 3b, we vary the diameter of the colloid from 10 to 50 nm while keeping the surface charge density constant at 0.1 e/nm2. The corresponding volume fractions are 0.00216 for 30 nm and 0.01 for 50 nm. Our simulation results show that the colloids tend to aggregate under these conditions. Colloids of diameter of ∼10 nm already show some degree of aggregation, and a relatively sharp increase in the tendency to aggregate is seen for sizes in the range of 20 nm. In Figure 3c, the volume fraction of colloidal particles in the system is fixed at 0.01. The other conditions remain the same as Figure 3b. If the radius of a colloidal particle is increased, the colloid concentration decreases as d−3 c . Even under these conditions, the colloids tend to aggregate and the ratio of the number of aggregates to the number of colloidal particles, n, decreases, going to very small values for sizes of about 50 nm. Figure 4 shows n as a function of the volume fraction, Φ, for the case of divalent counterions. The charge density is 0.1 e/ nm2. For the divalent case, with all other conditions the same, more counterions are required to enforce charge neutrality (for a fixed value of the colloidal surface charge), compared with the case of trivalent counterions. In order to limit the computational cost, the colloid size is chosen to be 30 nm, which results in a value of the coupling parameter, Γ ≈ 0.5. A relatively sharp decrease is seen in the behavior of n around Φ = 0.01. We did not investigate the case of monovalent counterions, because this requires even more counterions, which is computationally prohibitive. We extrapolate and estimate that attractions may be
Figure 4. Fraction of free clusters, n, related to the tendency of the colloids to aggregate is plotted versus the volume fraction of colloidal particles in the system in presence of a divalent counterions. dc is 30 nm. The surface charge density is 0.1 e/nm2. The other parameters remain the same as in Figure 2.
expected in the monovalent case for very large colloidal particles. For uniformly charged colloidal particles, the interaction can only be attractive when the surface charge density is very high, ≳1 e/nm2,53 but for a discretely charged colloid, attractions may exist even when the surface charge density is significantly smaller. In this case, the intercolloid force is not precisely known, but we expect that it mainly depends on the amount of counterion bridging. Assuming that the critical bridging charge for monovalent counterions is the same as that of the divalent case, we can determine the critical colloid size for attraction. As a rough estimate,14 for a planar geometry, the number of free counterions in the bulk depends on ∼e−|μel|. In a spherical geometry, the force depends on the area of the sphere as well, i.e., the total number of condensed counterions scales as e−|μel|d2c . Therefore, eq 3 yields the result that, in the monovalent case, attractions may be seen for colloidal particles that are about 4−5 larger than those that yield attractions for the case of divalent counterions. Mean Force. The results in Figure 3 indicate that larger colloids show a stronger tendency to aggregate. It is difficult to perform multiple particle simulations with very large colloids because of the computational limit on the number of counterions required by charge neutrality. Instead, in this section we study systems with even larger colloidal particles by analyzing the effective mean force between two colloidal particles. In addition, we obtain more detailed information about the origin of the attractive interaction by studying the anisotropic counterion distribution for the two particle case. Figure 5 presents the spatial distribution of the counterions between both colloids when the surface charge is (a) uniform and (b) discrete, mobile; here, θ and ϕ are the polar and the azimuthal angle (radians), respectively . The diameter of the particles is 200 nm. The surface charge density is 0.04 e/nm2 and the counterions are divalent corresponding to Γ = 0.32. The distance between the colloidal particle centers is kept fixed at 220 nm. The volume fraction is 10−4. The geometry is defined in the inset of the figure. The calculation of the counterion distribution counts the total number of counterions within a distance r < Rd/2, from the particle center (see inset of 8333
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Figure 5. Contour plot of the density distribution of the counterions for the case of two colloidal particles separated by a distance Rd is presented for the two cases in which the surface charge is (a) uniform and (b) discrete, mobile. The x-axis is the azimuthal angle φ, and y-axis is the polar angle θ. Only counterions whose distance from the center of the colloids is less than Rd are included. The densities are measured from the center of two colloidal particles and averaged. The diameter of the colloid is 200 nm. The valences of the colloidal particles and counterions, respectively, are qc = 2 and qs = 1.
distance d is composed of three parts, the electrostatic force between macroions, the electrostatic force between macroion and counterions, and the hardcore force. The hardcore force arises from the excluded volume of particles.54 Figure 6 shows that the effective interaction between colloids is attractive for discrete charges that lie a small distance below the surface (the plot given by squares in Figure 6) when the distance between colloids is larger than approximately 25 nm. It is worthwhile to contrast the attractions seen for discrete charges with the case of the uniformly charged system under identical condition. Because the surface charge density is moderate and we are not in the strong coupling regime, the force for the uniformly charged case is repulsive. We also compare our results for the case when Ds is very high (=30 nm). The result is closer to the uniform surface charges than the Ds = 3 Å case. An interesting observation is that the attractive force requires that the separation of two colloidal particles be greater than approximately 25 nm (but diminishes with increasing separation). This demonstrates that the use of the renormalized surface charge alone is inadequate to describe the force in the regime of distances studied here.
Figure 5). The distribution of counterions around each particle should be spherically symmetric if the two macroions are separated by a large distance. The figure shows that the distribution has maximum at ϕ = 0 and θ = π/2; i.e., there are relatively more counterions in the bridging region between the two particles. Furthermore, discrete, mobile surfaces attract even more counterions to the bridging region compared with uniformly charged surfaces. It is symmetry breaking in the counterion distribution that is the origin of the stronger attraction in the case of discrete and mobile surface charges. The force between the two colloidal particles as a function of their spacing is shown in Figure 6. After equilibration, we measured the mean force on each of the colloidal particles. The effective force between colloidal particles separated by a
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DISCUSSION Experiments7 show that charged colloids tend to aggregate not only in the strong coupling regime (Γ ≫ 1), but even in the intermediate coupling (Γ ≲ 1) regime, when the surface charge is discrete. This occurs under dilute conditions with relatively low counterion valence, with a surface charge density less than 0.1 e/nm2 and a volume fraction smaller than 0.05. However, previous theories and numerical simulations that assume that the surface charge is uniformly distributed find only repulsion between the colloidal particles and cannot explain the observed attraction in this intermediate regime. We find that an inhomogeneous distribution of surface charges can describe the observed attractions, even in the relatively weak coupling, dilute limit, at least for the case of zero salt. Discretely charged colloids attract more counterions to the surface than uniformly charged particles. Interestingly, when several colloidal particles are in the same region, this attracts even more counterions to the bridging region and increases the
Figure 6. Effective force between two colloidal particles is plotted versus their separation. Positive and negative values represent repulsion and attraction, respectively. The circles show the results for uniformly charged colloids, and the squares for a discretely charged surface for which the surface charges are placed at a depth of Ds = 3 Å. The triangles are for the case of discrete surface charges with Ds = 30 nm. 8334
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intercolloidal aggregation. We also find that the surface charge mobility plays an important role in the attraction. Discrete, mobile surface charges enhance the number of counterions in the bridging region and show strong attraction. Although the surface charges are only able to move a small extent, they can reorganize in response to the bulk counterion distribution and further lower the free energy of the system, thus increasing the attractions.
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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 S.A.S. and P.A.P. thank the US-Israel Binational Science Foundation and the Schmidt Minerva Center for their support. S.A.S. acknowledges the ISF and the historical generosity and support of the Perlman Family Foundation. P.A.P. acknowledges a World Class University Visiting Professor of Physics appointment supported by the National Research Foundation of Korea, funded by the Ministry of Education, Science and Technology grant no. R33-2008-000-10163-0. P.A.P. also acknowledges the support of the National Science Foundation NSF DMR1101900. Y.S.J. acknowledges the Max Planck Society (MPG), the Korea Ministry of Education, Science and Technology (MEST), Gyeongsangbuk-Do and Pohang City for the support of the Independent Junior Research Group at the Asia Pacific Center for Theoretical Physics (APCTP), and was supported by the National Research Foundation of Korea grant funded by the Korea government (MEST) (NRFC1ABA001-2011-0029960, and 2012R1A1A2009275).
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