Computing the Electrophoretic Mobility of Large Spherical Colloids by

Article pubs.acs.org/Langmuir

Computing the Electrophoretic Mobility of Large Spherical Colloids by Combining Explicit Ion Simulations with the Standard Electrokinetic Model Shervin Raafatnia,† Owen A. Hickey,† Marcello Sega,†,‡ and Christian Holm*,† †

Institut für Computerphysik, Universität Stuttgart, Allmandring 3, D-70569 Stuttgart, Germany Institut für Computergestützte Biologische Chemie, Währinger Strasse 17, 1090 Vienna, Austria

‡

S Supporting Information *

ABSTRACT: The electrophoretic mobility of large spherical colloids in different salt solutions of varying valency and concentration is studied via a combination approach of numerically solving the standard electrokinetic model with a ζ potential that has been obtained from explicit ion simulations of the restricted primitive model, thus going beyond the standard mean-field treatment. We compare our theoretical mobility curves to two distinct sets of experimental results and obtain good agreement for monovalent and divalent salt solutions. For the case of the trivalent La3+ salt, the experimentally obtained mobility reversal at high ionic strengths can be obtained only by adding an additional attractive interaction of 4kBT to the potential between the colloid and La3+, hinting at the presence of a nonelectrostatic binding term for this ion. It is also shown that, contrary to intuition, charge inversion does not necessarily result in mobility reversal.

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These are introduced in “The Standard Electrokinetic Model (SEM)” section. The value of μ thus provides information about the ζ potential, which in turn would yield the value of the surface charge density of the particle provided the underlying theory was fully understood. A common approach to solving this electrokinetic problem is to describe the electrostatic interactions via mean-field theories such as Poisson−Boltzmann (PB)1 or Derjaguin−Landau− Verwey−Overbeek (DLVO).2 The so-called standard electrokinetic model (SEM) uses the Poisson−Nernst−Planck (PNP) equation with an additional diffusion-convection term coupled to the Stokes equation. Solving this coupled set of electrokinetic equations allows one to calculate the electrophoretic mobility as a function of the ζ potential.3,4 This method is discussed in more detail in the Standard Electrokinetic Model section. In recent years, a number of experimental and theoretical studies have demonstrated the deficiencies of this approach in explaining some of the phenomena occurring in colloidal electrophoresis. An example of this is the reversal of the electrophoretic mobility at high concentrations of multivalent salt, which was reported by several experimental groups5−10 and also predicted through theory and simulations.10−17 This behavior is caused by the attraction of more counterions to the charged surface than necessary to neutralize it (overcharging), which can be due to the counterions’

INTRODUCTION Aqueous colloidal suspensions are ubiquitous in nature. Often, the colloids are highly charged because they possess dissociable surface groups that stabilize the solution against coagulation. Colloidal suspensions have long been the subject of scientific studies; nevertheless, their electrokinetic behavior in electrolytes is not yet fully understood.1 One method of characterizing the charge of a suspended colloid and thereby gauging the importance of charge effects is electrophoresis, in which a charged particle in solution moves in response to an applied electric field. The balance between the forces acting on the particle results in a drift speed |ν⃗| = μ|E⃗ |, where |E⃗ | is the strength of the applied electric field and μ is the electrophoretic mobility of the particle. To calculate μ analytically, one needs to solve a so-called electrokinetic problem, which requires a knowledge of the ionic distribution in the vicinity of the charged surface. Conventionally, this ionic cloud is separated into two distinct parts: (i) the Stern layer consisting of the ions strongly attached to the surface and (ii) the diffuse layer with a characteristic length scale called the Debye length λD. The shear plane (also referred to as the slip plane) separates these two layers, and in analytical calculations, this usually appears as a no-slip boundary condition for the fluid with respect to the object’s surface. The electrophoretic mobility is often converted to a ζ potential, defined as the electric potential drop between the bulk and the shear plane. This is the general definition of the ζ potential that is valid at all salt concentrations. Analytical formulas to convert the ζ potential to mobility and vice versa exist for the extreme limits of very low and very high salt concentrations (Hückel and Smoluchowski, respectively). © 2014 American Chemical Society

Received: October 16, 2013 Revised: January 18, 2014 Published: January 24, 2014 1758

dx.doi.org/10.1021/la4039528 | Langmuir 2014, 30, 1758−1767

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with Ψ being the electrostatic potential, ρ being the charge density, ε being the dielectric constant of the medium, E⃗ being the applied field, and rs being the position of the shear plane. For an incompressible Newtonian fluid with a small Reynolds number, the hydrodynamic part can be described by the Stokes equation combined with the incompressibility constraint

correlations that are not captured in mean-field theories such as PB. See refs 18−20 for comprehensive reviews. There have been a wide variety of computational models developed to model the electrophoretic mobility of colloids;21,22 however, these methods have certain drawbacks that we circumvent here. Several studies have looked at the mobility of colloids by modeling the colloid as well as the ions via charged spheres coupled to a lattice-Boltzmann (LB) model for the fluid.23−25 This approach has the disadvantage that it is limited to colloids of a few tens of nanometers, which is much smaller than the colloids used in experiments that are typically on the order of micrometers. Other approaches26−29 are able to access much larger length scales by considering the ions on a mean-field level. This essentially produces the same results that one gets by solving the PB equation, which ignores the ion−ion correlations that are crucial for multivalent salt. In this work, we circumvent these deficiencies by employing molecular dynamics (MD) simulations of the restricted primitive model (described in detail in the Simulation Method section) to examine the ionic structure and to calculate the ζ potential. The electrophoretic mobility is computed by using the obtained ζ potential as an input for the numerical solver available for the SEM. In this manner, we take into account the ion correlations explicitly and produce a modified boundary condition for the mean-field treatment in the sense of Perel and Shklovskii.30 We compare our method to two independent sets of experimental data obtained in refs 6, 10, 31, and 32 where the mobility of latex colloids is measured as a function of monovalent, divalent, and trivalent salt concentrations. We find good agreement between our results and the experimental data. This proves the applicability of the SEM to systems with multivalent salt as long as it is fed the correct ζ potential. We also show that planar geometry in the simulations is an appropriate approximation for colloids whose size is much larger than the Debye length. Our simulations reveal that the restricted primitive model requires the addition of an adsorption term for the La3+ ions to be in agreement with the experimental data for latex colloids with low to moderate charge densities. Furthermore, we provide an example showing that the intuitive assumption that overcharging is equivalent to mobility reversal is not true. The paper is structured as follows: The Standard Electrokinetic Model (SEM) section consists of a short review of the SEM. The simulation method is described in the Simulation Method section, and the results are discussed in the Results section. We close with a short summary, and discuss possible further work in the Summary and Conclusions section.

= 0⃗

(5)

The balance of forces in the steady state results in a coupled nonlinear set of electrokinetic equations. This is analytically solvable only in a few selected limits. When there is no or very little salt in the system, namely, the so-called Hückel limit,33 the viscous drag acting on the spherical particle is given by Stokes’ law, and the drift velocity is vd⃗ = QE⃗ /(6πηR) with Q being the charge of the particle and R being its radius. This leads to an electrophoretic mobility of μ = Q/(6πηR). In the opposite limit of very high salt concentration, known as the Helmholtz− Smoluchowski limit, the Debye layer is very thin. When κR ≫ 1, with κ being the inverse Debye length, the geometry is essentially planar and the mobility is μ = εζ/η.1 Only numerical methods exist for the interesting intermediate range of salt concentration. The standard approach to calculating the electrophoretic mobility of spherical colloids is that of Wiersema3 and O’Brien and White.4 The model assumes a Boltzmann distribution of ions and takes into account the convective flow, ionic current, and relaxation effects by including the continuity and ionic force balance equations

∇⃗ ·(ni( r ⃗)vi⃗( r ⃗)) = 0

(6)

−λi(vi⃗( r ⃗) − u ⃗( r ⃗)) − zie∇⃗ψ ( r ⃗) − kBT ∇⃗ log ni( r ⃗) = 0 (7)

In these equations, ni(r)⃗ is the ion density of species i, vi⃗ (r)⃗ is their velocity field, and zi is their valency. The drag coefficient λi of ions of type i is obtained from the limiting conductance Λ0i of that ion species via λi = NAe2|Zi|Λ0i , where NA is Avogadro’s number. The authors focus on the case of low applied fields |E⃗ | ≪ κζ where the perturbation of the ionic cloud due to the external field is linear in |E⃗ |. This allows them to linearize the electrokinetic equations and decompose the problem into two independent problems, namely, a hydrodynamic part and an electrokinetic part.4 For the first part, the particle is held fixed in a flow field of −U⃗ with no applied electric field. The force required to hold the particle stationary in the linear regime is given by F⃗1 = ξU⃗ , where ξ is hydrodynamic friction coefficient independent of U⃗ . For the second part, an electric field is applied and the particle is held fixed in an electrolyte at rest far from the surface by the force F⃗2 = −ξμE⃗ . The superposition of the solutions to these two problems satisfies the boundary conditions in eqs 2 and 5. The electrophoretic mobility is then obtained from the zero net force requirement in the steady state, ξU⃗ − ξμE⃗ = 0⃗. With this method, O’Brien and White4 were able to calculate the mobility for a broad range of κR and surface charge densities without the limits and/or approx-

(1)

∇⃗Ψ(z → ∞) = −E ⃗ =ζ

(4)

u ⃗(z = rs)

and the boundary conditions

Ψ(z = rs)

∇⃗ ·u ⃗ = 0

u ⃗(z → ∞) = −μE ⃗

STANDARD ELECTROKINETIC MODEL (SEM) The simplest picture of electrokinetic effects is obtained by considering the following two main phenomena: (i) the electrostatic interactions that determine the distribution of the ions and (ii) the hydrodynamic interactions that cause the viscous dissipation of momentum. The electrostatic part is governed by Poisson’s equation ρ( r ⃗) ε

(3)

where η is the viscosity of the solvent, u⃗ is its velocity, and p is the pressure. The boundary conditions of the hydrodynamic part are

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∇2 Ψ( r ⃗) = −

η∇2 u ⃗( r ⃗) − ∇⃗p( r ⃗) = ρ( r ⃗) ∇⃗Ψ( r ⃗)

(2) 1759


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imations used in the analytical solutions mentioned above. In this work, we use a computer program written by White, Mangelsdorf, and Chan that solves the electrokinetic equations based on the SEM and produces numerical results for the electrophoretic mobility versus the ζ potential for given values of the temperature, solvent viscosity, ion concentrations and valencies, and colloid radius. By solving the same equations as in O’Brien and White’s work,4 the program accounts for the ionic electric current, the convection of the ions, and the relaxation effect, with the latter being especially important at high ζ potentials. In some cases, we also employed an approximate formula for symmetrical electrolytes derived by O’Brien.35

direction. In the periodic directions, the minimum box size is one Debye length, and it is chosen such that there are enough particles in the box and that the ionic strength is not strongly changed by the wall counterions. In addition to the electrostatic interactions, in order to model the finite size of the ions, the latter interact via a Weeks−Chandler−Anderson (WCA) potential39 with each other as well as with the wall

SIMULATION METHOD We perform MD simulations, using the ESPResSo package,36 to calculate the ζ potential of a single colloid and use numerical solutions of the SEM to obtain the corresponding electrophoretic mobility. The colloidal particles considered in this work are quite large compared to the surrounding salt ions, having radii about 104 times larger. Direct modeling of the colloid as a sphere would thus require an unreasonably large number of counterions, even at a coarse-grained level. Because we are firmly in the κR ≫ 1 regime, the colloid can be safely modeled as a uniformly charged flat plate; see also the a posteriori discussion in the Results section. This model has the limitation that the mobility cannot be obtained directly by applying an electric field and measuring the particle velocity. However, we take advantage of the fact that in the limit of low applied field the distortion of the ionic cloud due to the field can be ignored. This means that the ζ potential calculated from the ion distribution in equilibrium can be assumed to be equivalent to that of a system subjected to a weak applied field, allowing us to perform equilibrium simulations (i.e., with no applied electric field). Using this potential as an input in the SEM numerical solver, we obtain the corresponding electrophoretic mobility. In this manner, we take into account ion correlations, crucial in the case of multivalent salt, which are ignored in the PB approach used in the SEM. We make use of the restricted primitive model, where ions are treated explicitly as charged spheres all having the same size and the solvent is considered to be a homogeneous dielectric medium having the electric permittivity of water at room temperature. This method is called primitive because it ignores the structural details of the fluid and restricted because all ion species have the same size. We simulate monovalent, divalent, and trivalent salts, with the co-ions having a valency of −1 and the counterions having valencies of z = 1, 2, and 3, respectively. The system is simulated at different salt ionic strengths, I = 1/2Σi N= 1cizi2, where ci is the molar concentration of the ith salt species and zi is its valency. To find the molar concentrations corresponding to the ionic strength, the relation cco‑ions = zccounterion is used. Charge neutrality of the whole system is maintained by inserting additional counterions of valency z to neutralize the charged plate, whose charge density is taken from the experiments. Periodic boundary conditions are applied parallel to the charged plate, and the system is confined in the perpendicular direction. The electrostatic interactions are calculated by taking into account the contribution from all periodic copies in the x − y plane using the electrostatic layer correction (ELC) method37 combined with the particle− particle−particle mesh (P3M) algorithm.37,38 The size of the MD box is at least five Debye lengths in the nonperiodic

where r is the distance between two ions or between an ion and the wall, σLJ is the sum of the radii of the two interacting particles or the distance of the closest approach between the center of the ions and the wall, εLJ is the interaction strength, and rcut = 21/6σLJ is the cutoff distance. The unit system is defined by σLJ = 3.5 Å and εLJ = kBT = εMD (εMD = kBT = 4.11 pN nm = 0.0256 eV at room temperature) and setting the mass of all particles equal to m = 1, corresponding to the mass of an ion (∼10−26 kg). The strength of the WCA interaction between the ions and the wall is εcoll = 4kBT. The simulations where we refer to the full Lennard-Jones (LJ) interaction also include an attractive part when we set rcut = 2.5σLJ and in this case εLJ = 4kBT for the trivalent La3+ counterions. The Bjerrum length, defined as lB = e2/(4πεkBT), with e being the elementary charge, takes on the value lB = 2σLJ = 7 Å, which corresponds to that of water at room temperature. Simulations are performed in the canonical NVT ensemble using a Langevin thermostat to keep the temperature constant. The time step is Δt = 0.01τ, where τ is the unit of time derived from the length, mass, and energy units as τ = σLJ(m/εLJ)1/2. Each system is first equilibrated for at least 1 000 000 MD steps, assuring that the ion distribution does not change with time, before running eight production simulations with different random number generator seeds for 10 000 000 MD steps each. We extract the average ion density and the total integrated charge density as a function of the distance from the wall from the MD simulation data. The reduced electric potential (Φ = eΨ/kBT, with Ψ being the electric potential) is then calculated via double integration of the latter

⎧ ⎛⎛ σLJ ⎞12 ⎛ σLJ ⎞6 1⎞ ⎪ ⎪ 4ϵLJ⎜⎜ ⎟ − ⎜ ⎟ + ⎟ , r < rcut ⎝ r ⎠ 4⎠ VWCA(r ) = ⎨ ⎝⎝ r ⎠ ⎪ ⎪0 , r ≥ rcut ⎩

■

(8)

ρ d2Φ 1 = − ⇒ Φ(z ) = − 2 ε ε dz

z

∫0 ∫0

z′

ρ (z ″ ) d z ″ d z′

(9)

Because the SEM requires the ζ potential as an input, we first need to define the shear plane. Because of the absence of explicit water and drag forces, it is not possible to determine directly the position of the shear plane as the location of the noslip boundary condition. However, the dependence of the noslip boundary condition as a function of several parameters such as surface corrugation, shear rate, and contact angle has been investigated in many studies (e.g. refs 40−43), also in the presence of charged surfaces.44 For hydrophilic surfaces, one can safely approximate this boundary with the position of the shear plane.44 This is applicable to the polystyrenesulfonate colloids that we study in this article because they are hydrophilic as a result of the presence of sulfonate groups. Note that while in the presence of explicit solvent the density profile of ions shows marked layering compared to that in the coarse-grained case, the electrostatic potentials computed in the two cases are very close to each other.44 On the basis of the 1760


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aforementioned studies, a distance slightly larger than one ion diameter from the surface, which encompasses the majority of ions in contact with the surface, should be a good approximation of the position of the slip plane. In our case, we have chosen a shear plane at a distance of zsp = 1.5σLJ from the wall, corresponding to 5.25 Å. Note that for our interaction definitions the distance of the closest approach of the center of an ion of diameter σLJ to the wall is σLJ. Thus, at zsp = 1.5σLJ we incorporate into the calculation the ions in the first layer. Slight changes in the location of the slip plane do not alter the results significantly, as can be seen in Figures S1 and S2 in the Supporting Information. This can also be seen in Figure 3: changing the position of the shear plane, represented by the vertical line, by a small amount (fractions of σLJ) does not change the ζ potential dramatically. It should be mentioned that in our previous work10 we used a shear plane at zsp = 1.025σLJ from the wall. The difference in the results is negligible. (See Figure S1 in the Supporting Information.) It is worth noting here that despite the fact that the SEM is based on mean-field theory (PB) we do take the ion correlations into account by obtaining the ζ potentials from simulations with explicit finite-sized ions. These correlations are crucial in systems with highly charged surfaces and/or multivalent salt.

Figure 1. Reduced PB ζ potential as a function of the ionic strength of the monovalent salt for both the planar geometry and the spherical geometry with different radii. The surface charge density is σ = −0.31 μC/cm2. The lines are a guide to the eye.

where the correlations are much weaker. On the basis of this and the excellent agreement of our results with the experimental data, we believe that the use of planar geometry is also valid for the non-mean-field cases that are not at equilibrium. However, strictly speaking we have no analytical proof for the validity of this approximation in the nonequilibrium case. In Figure 2, the experimental results of mobility as a function of ionic strength are compared to simulation results for the

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RESULTS In this section, the simulation results of the electrophoretic mobility for large colloids with different charge densities are compared to two different sets of experimental results6,10,31,32 that have been obtained with very different techniques but with exactly the same salts. Semenov et al. measured the electrophoretic mobility of a single colloid in solutions of varying salt concentration and valency (KCl, CaCl2, and LaCl3)10,32 using low applied electric fields to ensure the validity of the linear regime. The polystyrenesulfonate latex colloids used in their experiments have a diameter of 2.23 μm and a surface charge density of −0.31 μC/cm2 ≃ −0.02 e/nm2. These colloids are much larger than the experimental Debye lengths, and κR is sufficiently larger than 1. For the investigated systems, we have values of 11.6 < κR < 2.8 × 103; therefore, the planar approximation should be reasonable. To test the validity of this assumption, we calculate the ζ potentials in the monovalent salt case from the numerical results of the nonlinear PB in both spherical and planar geometries. Figure 1 shows the reduced ζ potentials as a function of ionic strength. The results for the spherical geometry using different radii are compared to the planar result. As can be seen, the difference between the two geometries decreases by increasing the ionic strength and/or the radius. For R > 400 nm, the two geometries are almost identical over the whole range of ionic strengths. It should be noted that this validates the use of the planar geometry for the mean-field case without an applied electric field. The convective flow and the ionic electric current might change the ion distribution in non-mean-field systems that are not at equilibrium. We are not able to investigate these changes directly in the simulations that are performed in equilibrium. However, as discussed in the Standard Electrokinetic Model (SEM) section, all of these effects are actually taken into account by the SEM numerical solver. However, by inserting the ζ potential obtained from the simulations, we include the important correlations that are strongest within the shear plane explicitly and the SEM equations are solved beyond this plane,

Figure 2. Electrophoretic mobility as a function of ionic strength for monovalent (KCl), divalent (CaCl2), and trivalent (LaCl3) salt solutions. Experimental results, black squares, are from Semenov et al.10,31,32 for latex colloids with a diameter of 2.23 μm and a surface charge density −0.31 μC/cm2 ≃ −0.02 e/nm2. Simulation results are represented by blue circles (WCA interaction) and red filled circles (full LJ interaction). The dashed and dotted lines represent calculations based on the numerical solution to the nonlinear planar and spherical PB, respectively. In the trivalent case, the solid red line represents the PB solution, including the full LJ interaction between the counterions and the surface. Data are taken from ref 10.

three different valencies. The reduced mobility reported in this work is defined as μred = 3ηeμ/(2εkBT). It should be noted here that we use only experimentally measured parameters such as the surface charge density and salt concentration in our simulations and that there are no fitting parameters. An exception is the trivalent case where we added a specific 1761


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μred = −0.31 and 0.12, respectively. Therefore, we chose εcoll = 4kBT, and all of the reported results with full LJ use this value for the interaction strength. This is essentially a fitting parameter and is in line with previous studies of colloidal electrophoresis where a reasonable value for the energy of specific adsorption is found to be a few kBT.50 In Figure 3, the

adsorption potential of 4kBT, as discussed in detail further in this section. In the monovalent case, our simulation results are in excellent agreement with the numerical PB curves, where the ζ potentials are obtained from numerical solutions of the nonlinear PB equation and the mobilities are calculated using O’Brien’s formula.35 We tested the result of this formula for a few ionic strengths and found that the difference in the result obtained from the SEM numerical solver is negligible. The perfect agreement between the simulation results and those obtained via PB shows that for this set of parameters the ion correlations are not important and can be ignored and the mean-field treatment is an excellent match to the explicit ion simulations. The results show a mobility maximum of a similar height at roughly the same ionic strength as the experiment, and semiqualitative agreement with the experiment is achieved for all values of the ionic strength. The discrepancy between simulation and experiment increases at higher ionic strengths. As discussed later in this section, part of this might be due to systematic errors in the experimental determination of the mobilities. In the divalent case, we see semiquantitative agreement between simulation and experiment. Simulation data match the results obtained via the numerical solution to the nonlinear planar PB. This indicates that the ion correlations are weak even for divalent ions at this charge density. The numerical solutions show a maximum at an ionic strength of I ≈ 10−4 M. The experimentally observed mobility reversal in the trivalent case could not be reproduced, and the data are thus not even qualitatively correct. The electrostatic interactions alone seem not to be strong enough to attract sufficient counterions to the charged surface to produce the experimentally observed mobility reversal. The relative unimportance of the electrostatic correlations is corroborated by the good agreement between the simluation data and the results from the numerical nonlinear planar PB. As in the divalent case, the PB curve exhibits a maximum around ionic strength 10−4 M. It has been pointed out several times that using a discrete surface charge distribution in simulations can increase the ionic correlations and lead to a stronger attraction of the counterions to the surface.45−48 We found, however, that even the discreteness of the surface charge does not induce a mobility reversal in this system.10 The strong coupling theory of Netz et al.49 also supports the conclusion that the mobility reversal observed here cannot be caused by purely electrostatic interactions. This theory states that the Coulomb coupling parameter Ξ = 2πσlB2Z3 needs to be larger than about 10 to produce charge inversion purely by electrostatic effects. As will be discussed at the end of this section, overcharging is a necessary but not sufficient condition for mobility reversal. For the most strongly coupled system, the trivalent La3+ salt system, we have only Ξ = 1.7, which is well below the threshold for charge inversion; consequently, mobility reversal through electrostatic interactions alone does not occur. This also explains why the PB curves fit the simulation results so remarkably well. However, as a word of caution, the strong coupling theory is also not valid for this regime. We replaced the WCA potential between the counterions and the charged surface with a full LJ interaction, as described in the last section, implying a specific adsorption of the La3+ ions on the latex colloid. We used interaction strengths εcoll = 2kBT and 4kBT for a system at the experimental ionic strength for which the mobility is zero. The two strengths resulted in

Figure 3. Reduced electrostatic potential profiles for σ = −0.31 μC/ cm2 ≃ −0.02 e/nm2 at I = 0.6 M (trivalent ions) with and without the full LJ interaction between the counterions and the wall. The position of the shear plane, indicated by the vertical line, is taken to be at zsp = 1.5σLJ with respect to the solid surface.

potential distribution is shown for I = 6 × 10−1 M, both with and without the full LJ interaction. As can be seen, ζ-potential reversal and consequently mobility reversal do not occur in the absence of specific adsorption. However, the simulation results including the attractive LJ interaction between the counterions and the wall exhibit very good quantitative agreement with the experiment. We also show the reduced mobility curve calculated via the nonlinear planar PB with the additional full LJ interaction in Figure 2. Again, the numerical curve matches our simulation results very well, indicating the weakness of the electrostatic correlations. In general, two models exist for explaining overcharging that is necessary for mobility reversal. One is based on ion-correlation theories and explains the phenomenon using purely physical arguments such as ion correlations and strong electrostatic coupling. The other model has a chemical nature, based on specific ion adsorption of the ions in the Stern layer to the surface. Our simulations show that correlations alone cannot describe the observed mobility reversal correctly and that extra forces are required that we tend to believe are a clue to the chemical part of the overcharging phenomenon. The existence of this specific interaction has also been reported by several other authors.6,51−53 Its origin is suspected to be the hydrolysis of lanthanum; it is known that lanthanum ions hydrolyze around pH ∼6,54,55 which leads to specific adsorption. Given the fact that despite the presence of ion correlations in our simulations no mobility reversal occurs at the experimentally observed ionic strength without modeling the specific adsorption, we strongly believe that the latter is the key to the reversal in this system. The coarse-grained nature of our model and the absence of explicit water makes it impossible for us to investigate directly the origin of this adsorption, and more detailed simulations ́ would be required for this purpose. Martin-Molina et al.8 reported the mobility reversal of sulfonated polystyrene latex in the presence of La3+ ions at pH 5.8. Using ion correlation theories to explain their results without taking into account specific adsorption, they are able to reproduce the mobility reversal at the same salt concentration as in the experiment. 1762


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Figure 4. Electrophoretic mobility as a function of ionic strength of monovalent (KCl), divalent (CaCl2), and trivalent (LaCl3) electrolytes. Simulation results, represented via blue (WCA) and red filled circles (full LJ), are compared to the experimental results of Elimelech et al. indicated by black squares. Experiments were done by Elimelech et al.6 with latex colloids of diameter 0.753 μm and a surface charge density −5.64 μC/cm2 ≃ −0.35e/nm2. The dashed and dotted lines represent calculations based on the numerical solution to the nonlinear planar and spherical PB, respectively. In the trivalent case, the solid red line represents the PB solution including the full LJ interaction between the counterions and the surface.

However, the overall agreement between theory and experiment is otherwise poor, which might be due to the use of the Helmholtz−Smoluchowski equation for converting their static results to mobilities. Moreover, the colloids used in their experiments are also much smaller than those investigated here (R = 98 nm), making the planar approximation more erroneous. The colloids also possess a much higher charge density (σ = −11.5 μC/cm2 ≃ −0.72 e/nm2), which increases the electrostatic correlations significantly. The Coulomb parameter in their case is Ξ = 62. Elimelech et al.6 have also performed experiments where they measured the electrophoretic mobilities of latex colloids in monovalent, divalent, and trivalent salts (KCl, CaCl2, and LaCl3, respectively) as a function of ionic strength using a Mark II microelectrophoresis apparatus. We compare our simulation results to their experimental data for a colloid of diameter of 0.753 μm and surface charge density −5.64 μC/cm2 ≃ −0.35e/ nm2 (Figure 4). Because of the large size of the colloid in comparison to the Debye length in the considered range of salt concentrations, 12 < κR < 3.91 × 102, the planar geometry inherent in our simulation technique is also appropriate for this system. For the monovalent case in Figure 4, we see fairly good qualitative agreement between our simulation results and the experiments of Elimelech et al. Importantly, our data reproduces the peak in the mobility at roughly the same value of the ionic strength as the experimental data. This peak is shifted toward higher ionic strengths, about 100 times higher compared to the data from Semenov et al. (Figure 2). Given that the colloids in this case are about 3 times smaller, this means that the peak occurs not only at higher ionic strengths but also at higher κR, about 3 times larger. This shift is due to the higher charge density of the colloid and is in line with the findings of Antonietti and Vorweg.56 Calculating the ζ potential from the solution of the nonlinear planar and spherical PB, we again found superb agreement between the two geometries. The numerical curves are in good agreement with the

simulation results except for the two points of lowest salt concentration. The deviation seems to increase with decreasing ionic strength, which we believe is due to the increased ionic correlations between ions very close to the colloid that are taken into account in our simulations but ignored in the PB theory. It is known that mean-field theory is exact only in the weak coupling regime, Ξ ≪ 1, but it breaks down in the strong coupling limit when Ξ ≫ 1, the Gouy−Chapman length is lGC = 1/(2πzσlB), and κlGC < 1. Because here lGC ≃ 6.45 Å and hence κlGC < 1 and Ξ = 1.13, some slight deviations from meanfield theory are expected.57−59 The divalent data in Figure 4 also shows good agreement between our simulations and the experimental results. In contrast to the previous experimental set (Figure 2), the divalent PB curve matches the simulations only qualitatively. Most notably at higher ionic strengths there is a systematic overestimation of the mobility. This is the result of stronger ion correlations in this system due to a greater surface charge density (more than 10 times higher than in the first experiment), which are not captured by PB. It should be noted that the good agreement between simulation and numerical results for low ionic strengths is partially due to the fact that the μ − ζ curves show a maximum around this ζ potential. More specifically, in the case of I = 0.003 M the difference between the ζ potential obtained from simulation and PB is about 25%, but the nonlinearity in mobility as a function of ζ potential results in smaller deviations in mobility, in this case about 10% (Figure S3 in the Supporting Information). This combination of colloidal diameter and surface charge density produces a maximum in the mobility for the divalent case. Our simulation results reproduce this maximum at approximately the same value of the ionic strength as seen in the experiment. It should be noted that in Figure 4 we appear to get quite accurate values for low ionic strengths whereas we systematically overestimate the experimental mobility values at high ionic strengths. This systematic overestimation of the mobilities at high ionic strength appears both in the monovalent and divalent cases and is in contrast to 1763


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of the surface charge distribution does not have a significant impact on the results and that additional LJ attraction is indeed required in order to reproduce the experimentally observed mobility reversal. The values of the ζ potentials and the corresponding electrophoretic mobilities are listed in Table 1.

the results in Figure 2, where we overestimate the mobility at low ionic strengths and underestimate it at high ionic strengths. In the absence of a better explanation, we speculate that at least some of the discrepancy could be due to systematic errors in the experiments. The trivalent results shown in Figure 4 show a strong resemblance to the previously presented results from Semenov et al. in Figure 2. No mobility reversal is observed in the trivalent case without the addition of the full LJ interaction with a 4kBT attractive potential between the counterions and the charged surface. The discrepancy between the PB curve and the simulation data is larger than in the divalent case because of stronger correlations that are absent in the mean-field theory. As noted above, the good agreement between simulation and numerical results at low ionic strengths partially stems from the nonlinearity of μ with respect to ζ. Remarkably, the same attractive interaction of 4kBT again yielded the best fit of the experimental data, suggesting that our added potential stems from an underlying specific physical interaction for La3+. When we include the full LJ interaction between the counterions and the surface in the nonlinear planar PB, a mobility reversal is obtained at I ≈ 0.3 M, which is about 10 times higher than the ionic strength for which mobility reversal is observed in the experiment. The PB results both with and without the LJ attraction show a maximum in the mobility at around I ≈ 10−3 M. To ensure that surface charge discreteness does not result in mobility reversal in this system without specific adsorption, we used two different types of discrete surface charge distribution in the presence of a 0.06 M trivalent salt: regular and disordered. For the first case, 484 monovalent charges were regularly distributed over a square grid with a lattice size of about 4.8σLJ. In the second case, the same number of surface charges were distributed in a disorderly manner over the surface with a minimum distance of 2σLJ. In both cases, the surface charges have a diameter of 1σLJ and interact only via the WCA potential and electrostatic interactions with the free ions. The area of the surface is chosen to be about (105 × 105)σLJ2, giving rise to the experimental surface charge density. The integrated surface charge density profile computed from the two discrete charge distributions is compared to the corresponding results of continuous surface charge, both with and without the additional LJ attraction, in Figure 5. It is clearly seen that the discreteness

Table 1. Values of the Reduced ζ Potential and the Corresponding Reduced Electrophoretic Mobility Obtained from Simulations with Different Surface Charge Distribution Models Performed for the System with a Surface Charge Density of −5.64 μC/cm2 ≃ −0.35e/nm2 for a I = 0.06 M Trivalent Salta

reduced ζ potential reduced mobility

disordered discrete (WCA)

regular discrete (WCA)

continuous (WCA)

continuous (full LJ)

−0.28

−0.35

−0.38

0.55

−0.41

−0.51

−0.57

0.82

a

The shear plane is positioned at zsp = 1.5σLJ. The type of interaction between the surface and the counterions is as indicated in the parentheses. For more details, see the main text.

The authors in ref 11 used a hypernetted chain/meanspherical approximation (HNC/MSA), which includes electrostatic and excluded volume correlation effects, to fit the experimental data of Elimelech et al.6 for the trivalent salt. Although they were able to observe a mobility reversal, they obtained only qualitative agreement with the experimental results. The mobility reversal predicted by their theory occurs at much higher salt concentrations than in the experiment. In fact, their results are comparable to our MD simulations in the absence of the full LJ interaction. Extrapolating our WCA results to higher ionic strengths would also produce a mobility reversal at an ionic strength of >1 M. This strongly supports our argument that pure electrostatic correlations are not strong enough to cause a mobility reversal at the experimentally observed ionic strength in the system under study, and an additional specific adsorption term is required. It is interesting that the coupling parameter in this case is Ξ = 30, and as discussed before, electrostatic interactions can lead to overcharging for Ξ > 10. This might sound contradictory, but as shown in Figure 6, the system is overcharged at high ionic strengths yet the ζ potential does not change sign and thus no mobility reversal occurs. This shows that although the reversal of the electrophoretic mobility is related to the overcharging of the charged particle, overcharging does not always lead to mobility reversal. All-atom MD simulations have shown similar behavior for DNA.60 The electrophoretic mobility of DNA is reversed at high concentrations of trivalent (spermidine3+) and quadrivalent (spermine4+) salt whereas charge inversion can occur at lower concentrations. For divalent salt (Mg2+), only charge inversion of the DNA is observed. The mobility reversal of the DNA is demonstrated to be a complex interplay of electrostatics and hydrodynamics; charge inversion is characterized by the change in sign of the cumulative radial charge density, that is, the sum over all ions within a certain distance. It is clear that the whole charge of each ion contributes to the sum. Mobility reversal, however, is due to the electroosmotic flow produced by the counterions, which applies a shear force to the DNA. If this force is strong enough, then the DNA will change direction. Because only a fraction of the counterions’

Figure 5. Total integrated surface charge density as a function of the distance from the wall of charge density −5.64 μC/cm2 ≃ −0.35e/nm2 in the presence of an I = 0.06 M trivalent salt. Different surface charge distribution models are considered as described in the text. The results of the discrete charge models are very similar to the continuous charge with WCA interaction between the wall and the counterions. Only the full LJ interaction results in overcharging. The corresponding values of the reduced ζ potential and reduced mobility are listed in Table 1. 1764


Langmuir

Article

In this article, we presented in detail our method for studying the electrophoretic mobility of colloids whose radii are much larger than the Debye length. The method we presented in this article represents a unique approach to predicting the electrophoretic mobility of colloids in electrolytic solutions of different valencies for colloids with diameters from 100 nm up to the order of micrometers. Although this is the regime in which most experiments are conducted, most of the existing methods used to model colloidal electrophoresis are incapable of capturing it. The good agreement between our simulations and two independent sets of experimental results shows that our method represents an excellent technique for the quantitative prediction of colloidal electrophoretic mobilities.

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ASSOCIATED CONTENT

* Supporting Information S

The effect of changing the position of the shear plane from zsp = 1.5σLJ to 1.025σLJ on the reduced electrophoretic mobility obtained from simulations for the two experimental sets of parameters studied in this work is shown in Figures S1 and S2. An example of the reduced mobility as a function of the ζ potential obtained from the numerical SEM solver is provided in Figure S3. This material is available free of charge via the Internet at http://pubs.acs.org.

Figure 6. Total integrated charge density (a) and reduced potential (b) as a function of the distance from the wall for σ = −5.64 μC/cm2 ≃ −0.35e/nm2 and the trivalent salt of stated ionic strength. The inset in plot a shows an enlargement.

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momentum is transferred to the DNA, it can happen that despite overcharging the mobility does not reverse.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

SUMMARY AND CONCLUSIONS In this work, we have computed the electrophoretic mobility of large colloids (κR ≫ 1) in salt solutions of varying valency and concentration. We first calculated the ζ potential of the colloids by means of static (no applied electric field) MD simulations of the restricted primitive model using planar geometry. This ζ potential was then used as an input into the SEM continuum description to extract the corresponding mobility. The results were compared to two independent sets of experimental results6,10,31,32 for latex colloids in monovalent (KCl), divalent (CaCl2), and trivalent (LaCl3) counterions, and good agreement was found between our simulations and experimental data. The results demonstrate the validity and usefulness of the planar approximation for studying the electrophoretic behavior of colloids whose radius is much larger than the Debye length, where modeling the whole colloid still remains far beyond what is possible using modern supercomputers. Furthermore, the good agreement between simulation results and experimental data shows that the SEM can be used to obtain the electrophoretic mobility in the presence of multivalent salt, given that the important ion correlations are taken into account when calculating the ζ potential. We further showed that in the trivalent cases that we considered, electrostatic attraction alone is not enough to reproduce the experimentally observed mobility reversal of latex colloids and an additional attractive LJ potential acting between the counterions and the colloidal surface is needed. This further supports the existence of ion-specific adsorption between La3+ ions and latex colloids, which has also been suggested by several other authors.6,51,52 Furthermore, we provide examples where the charged surface is overcharged by the counterions in its vicinity but no mobility reversal occurs, showing that overcharging does not always lead to mobility reversal.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the German Science Foundation within the SFB 716 and by the Volkswagen Foundation. The computing time was provided by the HLRS, and the code to solve the SEM was provided by D. Chan. We thank F. Kremer, V. Lobaskin, I. Semenov, and J. de Graaf for helpful discussions. M.S. acknowledges FP7 IEF p.n. 331932 SIDIS for support.

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REFERENCES

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Computing the Electrophoretic Mobility of Large Spherical Colloids by

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