Electrophoretic mobility and charge inversion of a colloidal particle

0 downloads 0 Views 1MB Size Report
(Received 3 April 2012; published 4 February 2013). Optical Tweezers are employed to study the electrophoretic and the electroosmotic motion of a single ...
PHYSICAL REVIEW E 87, 022302 (2013)

Electrophoretic mobility and charge inversion of a colloidal particle studied by single-colloid electrophoresis and molecular dynamics simulations Ilya Semenov,1 Shervin Raafatnia,2 Marcello Sega,2,* Vladimir Lobaskin,3 Christian Holm,2 and Friedrich Kremer1 1

Institute of Experimental Physics I, Leipzig University, Linn´estrasse 5, D-04103 Leipzig, Germany Institute for Computational Physics, Stuttgart University, Allmandring 3, D-70569 Stuttgart, Germany 3 School of Physics and Complex and Adaptive Systems Laboratory, University College Dublin, Belfield, Dublin 4, Ireland (Received 3 April 2012; published 4 February 2013) 2

Optical Tweezers are employed to study the electrophoretic and the electroosmotic motion of a single colloid immersed in electrolyte solutions of ion concentrations between 10−5 and 1 mol/l and of different valencies (KCl, CaCl2 , LaCl3 ). The measured particle mobility in monovalent salt is found to be in agreement with computations combining primitive model molecular dynamics simulations of the ionic double layer with the standard electrokinetic model. Mobility reversal of a single colloid—for the first time—is observed in the presence of trivalent ions (LaCl3 ) at ionic strengths larger than 10−2 mol/l. In this case, our numerical model is in a quantitative agreement with the experiment only when ion specific attractive forces are added to the primitive model, demonstrating that at low colloidal charge densities, ion correlation effects alone do not suffice to produce mobility reversal. DOI: 10.1103/PhysRevE.87.022302

PACS number(s): 82.70.Dd, 82.45.−h, 83.10.Rs, 87.10.Tf

I. INTRODUCTION

Charge stabilized suspensions play a prominent role in many industrial applications and in various biological processes. The description of electrostatic interactions in such dispersions is usually based on the Poisson-Boltzmann (PB) [1,2] or the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, which includes a linearized PB equation [3,4]. As the PB equation for spherical geometry has no closed form solution, in practice the ionic cloud formed at highly charged interfaces in liquids is often modeled as a layered structure consisting of two main ion populations: the condensed ions, directly bound to the surface, and the free ions. Similarly, for electrokinetic problems, one often sorts ions by their mobility with respect to the surface. The common naming convention distinguishes between (i) the Stern layer that consists of ions fixed at the surface and (ii) the diffuse layer that is characterized by the Debye length originating from the linearized mean-field PB description. To solve the electrokinetic equations, one needs to set a boundary, the shear plane, which separates the immobilized part of the fluid and the ionic cloud from the mobile diffuse part of the electric double layer (EDL). Such an approach is used, in particular, in the so-called standard electrokinetic model (SEM) [5,6], which employs the PB equation for the electrostatic part of the problem and the Stokes equation for the hydrodynamic part. The numerical solutions [6–8] of this model provide a firm basis for interpreting a variety of electrokinetic phenomena. In recent years a number of experimental and theoretical works have demonstrated that the mean-field description of the EDL can be insufficient for describing the electrokinetic phenomena in charged colloids. In particular, a reversal of the colloid electrophoretic mobility in external electric fields has been reported by several experimental groups [9–12]. The reversal is triggered by increasing the concentration of

* Present address: Department of Physics, Tor Vergata University of Rome, via della ricerca scientifica 1, I-00133 Rome, Italy.

1539-3755/2013/87(2)/022302(7)

trivalent counterions. This behavior has been attributed to the phenomenon of the surface charge inversion predicted and measured for various types of multivalent ions and is due to counterion correlation effects that are not captured by mean-field theories like PB or DLVO; see Refs. [13–17] for reviews. Moreover, the same ion correlation mechanisms arising at high concentrations of multivalent ions are known to lead to like-charge attraction between colloids [18–20], macromolecules [21–23], and interfaces [24]. For electrokinetic phenomena, the SEM has the same problems, namely that it can quite successfully predict the electrophoretic mobility for monovalent salt solutions, but it is bound to fail for multivalent salt solutions since it neglects the necessary ionic correlations. In general, the interaction of a counterion with the colloidal surface includes electrostatic and nonelectrostatic (dispersion, entropic, bonding) contributions. The electrostatic part is universal and depends on the ion valency, surface charge density, and salt concentration, whereas the remaining part is usually ion- and surface-specific [25–27]. It has been suggested that either of these effects can lead to colloid charge and mobility reversal. Evidences of the ion specificity have been reported for various systems such as DNA [22,28], small peptides [29], anionic liposomes [30], silver iodide and mercury/aqueous interfaces [26,27], sodium montmorillonite particles [31], and electrostatically stabilized polystyrene beads [32]. Until now, however, the relative role of the various contributions to the charge and mobility reversal has not been carefully assessed. In the present article, a novel experimental method and computer simulations are used to examine the ion-surface interactions and the structure of the ionic double layer as a function of added KCl, CaCl2 , and LaCl3 salt. Mobility reversal of a negatively charged latex colloid is observed in the presence of trivalent La3+ ions. Moreover, these observations are corroborated by the results of molecular dynamics (MD) simulations of the restricted primitive model, which are used to calculate the EDL structure and the ζ potential, and then, augmented by the SEM, the colloid mobility. It is shown that the experimental mobilities in mono- and divalent salts can be well reproduced with the electrostatic mechanism alone, whereas

022302-1

©2013 American Physical Society

ILYA SEMENOV et al.

PHYSICAL REVIEW E 87, 022302 (2013)

for the trivalent La3+ ions an additional attractive term between ions and colloid surface is essential to reproduce the observed mobilities. The article is structured as follows: in Secs. II and III, the experimental and simulation methods are described, respectively. The results are discussed in Sec. IV, and finally, summary and conclusions are represented in Sec. V. II. EXPERIMENTAL METHOD

Optical tweezers (OT) are commonly used to manipulate microscopic particles and to measure the forces acting on them. Here, OT are employed to measure the electrophoretic mobility of a single colloid and study its dependence on ambient salt concentration and valency. Single colloid electrophoresis technique [33,34] based on OT [35,36] is used to measure independently the complex electrophoretic mobility of a single colloid and the complex electroosmotic response of the surrounding medium in a specially designed, symmetric microfluidic cell (Fig. 1). To extract the electrophoretic component of the motion, the very same colloid is placed with the OT in position A and for the latter in B. The small displacements of the colloid are measured by a CMOS high-speed camera [10 000 frames/s (fps)]. Image analysis is then used to determine the center positions of the colloid with a typical spatial resolution of ±2 nm [37,38]. A LED flash has been used to mark the change of the direction of the electric field. Furthermore, the amplitude and phase of the colloidal displacement for both effects are measured and deconvoluted according to Ae e−iφe = Atotal e−iφtotal + Aeo e−iφeo ,

(1)

where Atotal , φtotal , Aeo , φeo , and Ae , φe are the amplitudes and phases of colloidal oscillation at positions A and B and those corrected for the fluid motion (Fig. 2), respectively. AC E-Field

inlet

outlet A

B

Fluidic Cell IR-blocking filter Sync. Output for LED

FIG. 2. (Color online) Amplitude (bottom) and phase (top) vs. AC electric field strength at a frequency 500 Hz for a negatively charged PS colloid (diameter: 2.23 μm) in an aqueous solution of KCl molar concentration at pH = 5.8. The total electrokinetic (circles) and the electroosmotic (squares) responses (in amplitude and phase) are measured for the very same colloid. The dashed line represent a fit to the electrophoretic mobility (stars) using the overdamped harmonic oscillator model [33]; the error bars display the standard deviation over the data.

To a good approximation, the optical trap action on a dielectric particle can be described by a harmonic potential [39]. The trap force constant is proportional to the beam intensity, which is in the present study 0.039 pN/nm for a laser output power of 0.2 W. The electric field distribution at position A is calculated by finite element analysis [40]. For the flow velocity across the channel for both positions A and B a parabolic profile is found. Using impedance measurements, in which a reference electrode was inserted in position A, it was ensured that the voltage drop and the electric field distribution along the cell are symmetric and uniform, respectively, and that coincidence is achieved between the zero-crossing of the applied electric field and the LED flash. Amplitude and phase of the particle oscillations are measured always in the linear regime of the electrophoretic and electroosmotic motion (Fig. 2). The amplitude Ae and phase φe of the particle, corrected for the fluid motion Eq. (1), can be described by an overdamped harmonic oscillator model [33] Ae =

CMOS LED Indicator

Function Generator

FIG. 1. (Color online) Scheme of the experimental setup with the microfluidic cell and the optical tweezers (λ = 1064 nm). The electrophoretic response of a single spherical particle (diameter: 2.23 μm) and the electroosmotic response of the surrounding medium are measured separately by placing the identical colloid in positions A and B, respectively. For the colloid in position A, both the electrophoretic and the electroosmotic responses are superimposed, while in position B, no electric field is present and hence only the electroosmotic effect is observed. Inlet and outlet of the microfluidic channel (width: 0.3 mm, height: 1 mm) are open in order to avoid pressure-driven backflow of the solvent. The motion of the colloid under study is traced using an epifluorescence microscope accomplished with a high speed (10 000 frames/s) CMOS camera. Using a LED flash, the zero value of the external electric field is indicated in the sequence of the images.

μe E  , ωc 1 + (ω/ωc )2

φe = arctan

ω , ωc

(2)

where μe is the desired electrophoretic mobility and ωc is the angular frequency of the optical trap. Hence, the required mobility values are deduced from the slope of amplitude versus AC field strength dependence. In the current study, commercial polystyrene sulphonate latex (PS) particles (Microparticles GmbH, Berlin, Germany, diameter of 2.23 ± 0.05 μm) in a 4% stock solution were used. Excellent reproducibility of colloidal electrophoretic response is achieved for different colloids from the same batch [Fig. 3(a)] as well as for one single colloid in six subsequent runs measured [Fig. 3(b)]. The aqueous salt solutions have been checked by conductivity measurements in order to control the ionic concentration in the medium. High-quality deionized water (Millipore water with pH = 5.8) was used for dilution,

022302-2

ELECTROPHORETIC MOBILITY AND CHARGE INVERSION . . .

PHYSICAL REVIEW E 87, 022302 (2013)

The surface charge density of the colloid is σ = −0.31 μC/ cm2 as obtained by colloidal acid-base titration [41]. The reduced screening parameter, κa, where κ 2 = 8π lB NA I with 2 lB = 4πεreε0 kB T being the Bjerrum length, NA the Avogadro’s  2 number, and I = 12 N i=1 ci zi (ci is the concentration of the ith salt species and zi its valency) the ionic strength, is varied in the range from κa ≈ 11.5 to 1.5 × 103 . This corresponds to a variation of the Debye screening length κ −1 from 0.77 to 100 nm. III. SIMULATION METHOD (a)

(b)

FIG. 3. (Color online) Test of reproducibility. Amplitude of the electrophoretic response vs. salt ionic strength of KCl aqueous solutions for three identical negatively charged PS colloids (diameter: 2.23 μm) (a) taken from the same batch and (b) taken for the very same single colloid in six subsequent runs as indicated by different symbols.

showing a conductivity of about 5 μS/m. The conductivities of all three water solutions (KCl, CaCl2 , and LaCl3 ) were proven to be linearly dependent on the amount of salt added (Fig. 4). The pH of the prepared solutions was remeasured regularly during the experiment and found to be stable within the time scale of a full measurement cycle.

FIG. 4. (Color online) Conductivity vs. salt ionic strength of freshly prepared aqueous solutions of varying valency (KCl, CaCl2 , LaCl3 ).

A theoretical estimate of the electrophoretic mobility for colloids of micron size is difficult to obtain for several reasons. The validity of mean-field approaches like the PB equation is limited to the case of monovalent salts or low concentrations, since at room temperature ionic correlations close to the charged surfaces cannot be neglected for multivalent ions. These correlation effects can be easily included in computer simulations with explicit ions. The large size of the colloid compared to the size of an ion (dc ∼ 104 di ) makes the direct simulation unfeasible even at the coarse-grained level. Since in our system κa  1, the electrostatic interactions are well screened at distances much shorter than the colloid radius. Therefore, a locally flat surface can be safely assumed for the study of the EDL structure. This assumption has been tested numerically for the surface potential at the PB level by comparing the planar Gouy-Chapman (GC) solution (GC+LJ) to the spherical one (spherical PB+LJ), and seen to hold excellently [Fig. 5(a)]. MD simulations of the primitive model have been performed in the presence of a flat surface with the experimentally determined charge density σ ≈ −0.31 μC/cm2 using the ESPResSo package [42]. The solvent is modeled as a dielectric continuum with permittivity εr = 78.4 at a temperature of 298 K. The ions interact electrostatically as point charges with a value corresponding to the valency of the salt ions in units of the elementary charge e and otherwise with a standard Weeks-Chandler-Andersen (WCA) [43] potential ˚ (corresponding to the with interaction parameters σlj = 3.5 A hydrated diameter of ions) and εlj = 1 kB T . Ions are coupled to the colloid surface through electrostatic interactions (modeled as a homogeneous field) and through either a WCA potential ˚ and (in absence of ion-specific interactions) with σcoll = 3.5 A εcoll = 4 kB T or a full Lennard-Jones (LJ) potential (to model ˚ εcoll = 4 kB T and ion-specific attractions) with σcoll = 3.5 A, ˚ Besides the salt ions, an appropriate the LJ cutoff set to 7 A. number of counterions is added to balance the colloidal surface charge and keep the whole system charge neutral. To model the infinite surface, periodic boundary conditions are applied, and the electrostatic interaction is calculated using the ELC method [44]. The size of the central MD box is chosen to be at least five Debye lengths. The simulations are performed in the canonical ensemble using the Langevin thermostat. The large size of the colloid makes it impossible to include ion correlation effects by using explicit ions and hydrodynamic effects simultaneously, as has been done before; e.g., see Refs. [45,46]. Therefore, the electrophoretic mobility is computed as follows. First, the electrostatic potential along the simulation box is calculated from the charge distribution. Then, the ζ potential is determined, taking the electrostatic

022302-3

ILYA SEMENOV et al.

PHYSICAL REVIEW E 87, 022302 (2013)

potential drop between the bulk and the location of the shear plane, xsp = 1.025 σlj . It has been checked that small variations in the location of the shear plane do not alter the results significantly. The electrophoretic mobility is eventually calculated using the SEM, taking as input the ζ potential obtained from the simulation and the experimental κa value. IV. RESULTS

Experimental and simulation results for the electrophoretic mobility as well as the experimentally measured phase angle are shown in Fig. 5 as a function of the ionic strength. As can be seen in Fig. 5(b), at most conditions the colloidal motion is out of phase (phase offset is ∼ 225 degrees) with respect to the electric field due to its negative surface charge. The decrease of the electrophoretic mobility with increasing ionic strength predicted by the SEM for all three valencies is observed in the

(a)

experiments; see Fig. 5(a). For monovalent salt, a maximum in the mobility is observed at low ionic strengths of ∼10−4 mol/l, which is in agreement with the SEM and previous findings for similar monovalent ionic solutions [9,47–49]. A less pronounced monotonic decrease of the mobility with increasing ionic strength is observed for CaCl2 and LaCl3 . In the case of trivalent salt, the experimental results show a mobility reversal, as can be inferred from the 180 degrees phase offset change, i.e. the colloid oscillates in phase with the electric field at I > 10−2 mol/l [Fig. 5(b)]. Using the experimentally determined colloidal surface charge density in the simulations, a quantitative agreement with the experiment is achieved for the monovalent salt in the peak area and the full curves are at least qualitatively reproduced without further amendments to the restricted primitive model in the monovalent and divalent cases [see Fig. 5(a)]. However, no ζ potential (and hence mobility) reversal is observed in the trivalent case. This could have been expected, since in the strong coupling theory of Netz et al. [50] the Coulomb coupling parameter = 2π σ lB2 z3 , where z denotes the counterion valency, needs to be larger than 10 to produce charge inversion by pure electrostatic correlation effects, while for the investigated trivalent salt system we have only = 1.7. Obviously, the colloidal surface charge density is too low to bind a sufficient amount of trivalent ions electrostatically to produce the experimentally observed mobility reversal. This indicates the presence of other forces besides Coulombic attraction between the trivalent ions and the surface. In fact, specific interactions between La3+ ions and Latex colloids have been reported by other authors [9,51–53]. Therefore, to model this ion-specific attraction the WCA interaction between the colloidal surface and the counterions is replaced by a full LJ potential. The resulting reduced electric potential profiles is shown in Fig. 6 for different ionic strengths. An attractive well depth of εcoll = 4 kB T is found to be necessary in order to recover the mobility reversal in agreement with

(b)

FIG. 5. (Color online) The electrophoretic mobility (a) and phase angle (b) vs. salt ionic strength of aqueous solutions of varying valency (KCl, CaCl2 , and LaCl3 ). Squares represent the measured data, with error bars indicating the standard deviation. The measurements for each valency are carried out with the very same negatively charged PS colloid (diameter: 2.23 μm). The field strength is varied in the range 1–18 V/cm. The laser power is 0.2 W. The simulation results with and without LJ attraction are shown via stars and triangles, respectively, connected by dotted lines for a guide to the eyes. In the monovalent case, the solid line represents SEM calculations based on GC solutions, whereas the dashed and dotted lines indicate SEM calculations using GC and spherical PB solutions, including the LJ attraction, respectively.

FIG. 6. (Color online) Reduced electric potential ( keψ where ψ BT is the electric potential) profiles at different ionic strengths obtained from simulations of the trivalent case with full LJ interaction between the charged surface and the counterions. The position of the shear plane is taken to be at xsp = 1.025 σlj from the solid surface. It is seen that for ionic strengths I > 10−2 mol/l, the ζ potential changes sign and, consequently, the mobility is reversed. In the inset, the reduced potential profile is shown for the monovalent case at ionic strength 10−1 mol/l. No mobility reversal is observed in this case.

022302-4

ELECTROPHORETIC MOBILITY AND CHARGE INVERSION . . .

PHYSICAL REVIEW E 87, 022302 (2013)

FIG. 7. (Color online) Integrated charge density profiles for different surface charge distribution models at I = Iie = 0.026 M. The interaction between the ions and the interface is either the standard WCA potential or the full LJ, as indicated. For the values of the corresponding ζ potentials and electrophoretic mobilities, please see Table I.

FIG. 8. (Color online) Electrophoretic mobility vs. salt ionic strength of LaCl3 aqueous solutions for a negatively charged blank PS colloid (diameter: 2.23 μm) on addition of monovalent screening salt, as indicated. The symbols represent the mean value of four subsequent measurement runs; the error bars indicate the standard deviation. The lines are guides to the eye.

the experimental data [Fig. 5(a)]. The influence of correlations and ion-specific adsorption in the monovalent case is checked theoretically by computing the ζ potential from the solution of the nonlinear Gouy-Chapman (GC) equation, both in the presence and absence of the added LJ potential of the same strength; see Fig. 5(a). It is found that at the mean-field level neither of these factors influences the mobility curve much. The agreement between the mean-field solutions and the MD calculations is excellent at all conditions. Comparison between experiment and simulations for the divalent salt in the presence of the extra LJ attraction detects no specific adsorption in this case; see Fig. 5(a). It should be pointed out that the electrophoretic mobility in the presence of multivalent ions, and especially the mobility reversal, remain outside the SEM, as it uses the mean field approximation for the ion distributions. This problem is circumvented here by calculating the ζ potential using MD simulations that include all ion correlations. With this more accurate treatment of the Stern layer, the usage of the SEM becomes reasonable again since the diffuse layer should be more accurately described by the mean-field approach, which is similar in spirit to the procedure for the Poisson-Boltzmann description with a modified boundary condition outlined in Refs. [54,55].

example, if one considers instead of a homogeneously charged surface a finite-sized discrete surface charge distribution in the simulations, a stronger attraction of the counterions to the surface is expected due to ionic correlations [56–59], and this in turn can have a large impact on charge inversion. In order to ensure that such effects do not lead to significantly different results than those reported above, simulations with two types of discrete charges on the interface have been performed for the trivalent salt case at the experimental isoelectric point of the system, i.e., Iie ≈ 0.026 M. The first simulation was done with 100 monovalent charges disorderly distributed on a surface with area As = 200 × 200 σlj2 and having a minimum distance of 2 σlj . In the second set of simulations, 100 monovalent charges were regularly distributed over a square grid of lattice size 20 σlj , the surface area being again As = 200 × 200σlj2 . The charges were considered as spheres with an excluded ˚ interacting via a WCA potential volume set by σcoll = 3.5 A, of strength εcoll = 4 kB T with all ions. The integrated charge density profiles computed from the two discrete models are compared to that of the continuous surface charge models (with and without the additional LJ attraction) in Fig. 7, showing that the effect of the discretization is minimal. The values of the reduced ζ potential and the corresponding reduced electrophoretic mobility (μred = 32 εr εηe μe , with η being the 0 kB T dynamic viscosity of the medium) for the different surface charge distribution models reported in Table I also indicate that although a reduction of the mobility is observed as expected due to the discreteness, the effect is not strong enough to reproduce the experimentally observed isoelectric point. This is in agreement with previous studies [56,60–62], which also

A. Influence of discrete surface charge distribution

The actual distribution of the counterions around the macroion can also be influenced by the type of the surface charge distribution, the shape of the ions, or their size. For

TABLE I. Values of the reduced ζ potential and the corresponding reduced electrophoretic mobility obtained from simulations with different surface charge distribution models performed at I = Iie = 0.026 M of trivalent salt. The interaction between the surface and the ions is as indicated within the parentheses. For more details see the main text.

Reduced ζ potential Reduced mobility

Disordered discrete (WCA)

Regular discrete (WCA)

Continuous (WCA)

Continuous (full LJ)

−0.29 −0.42

−0.31 −0.46

−0.33 −0.49

0.07 0.04

022302-5

ILYA SEMENOV et al.

PHYSICAL REVIEW E 87, 022302 (2013)

FIG. 9. (Color online) Total integrated charge density distribution of simulations with trivalent salt at two different ionic strengths I = Iie = 0.026 M and I = 0.6 M in the absence of electrostatic interactions between the ions and the surface. A full LJ potential acts between the counterions and the surface while the interaction between the coions and the interface is provided via a WCA potential. It is seen that no charge inversion occurs within the shear plane, i.e., no mobility reversal.

show that interfacial charge discreteness is only relevant in the case of strong Coulomb interactions, i.e., when the surface is highly charged or the charged groups are comparable in valency with the counterions, none of them being applicable to the system under study. A comparison with the results of the continuous model with full LJ interaction between the surface and the counterions (see Fig. 7 and Table I) implies that the attractive part of the LJ potential (imitating the specific adsorption effects) is indeed necessary to obtain the isoelectric point correctly. B. Importance of electrostatic correlations

To demonstrate the importance of electrostatic correlations in the observed electrophoretic mobility reversal, the measurements in the presence of trivalent salt (LaCl3 ) are repeated with 10−2 M of added KCl. Since the monovalent salt screens the Coulomb interactions, a shift of the isoelectric point to higher ionic strengths is observed as expected; see Fig. 8. Contrary to some previous suggestions, there is also no enhancement (giant charge inversion) in the charge inversion seen, when salt is added, which is in line with our previous simulations [63]. This clearly demonstrates the important contribution of electrostatic interactions on the observed mobility reversal. Simulations of the extreme case where the ions interact only nonelectrostatically with the surface (counterions via the full LJ potential and coions via the WCA potential with parameters described in the previous sections) also confirm this point of view. Two sets of such supporting simulations were done

[1] G. L. Gouy, J. de Phys. 9, 457 (1910). [2] D. L. Chapman, Philos. Mag. 25, 475 (1913). [3] B. V. Derjaguin and L. D. Landau, Acta Physicochim. (USSR) 14, 633 (1941). [4] E. J. Verwey and J. T. G. Overbeek, Theory of the Stability of Lyophobic Colloids (Elsevier, Amsterdam, 1948).

for I = Iie = 0.026 M and I = 0.6 M of trivalent salt. The charge density of the surface and also the valency of the ions are taken into account in the analysis. The total integrated charge profiles are shown in Fig. 9. As is easily seen, no charge compensation or overcompensation occurs within the shear plane. In the case of I = Iie = 0.026 M, only about 0.85% of the surface charge is compensated within the shear plane, and for the case of I = 0.6 M about 10%. Therefore, we can safely conclude that no mobility reversal occurs in systems where the electrostatic interactions are completely screened. In the view of the results for the opposite case of electrostatic interactions without specific adsorption already represented in the previous subsection, it follows that both electrostatic and specific adsorption are needed to reproduce the experimentally observed mobility reversal. None of the interactions alone is sufficient. V. SUMMARY AND CONCLUSION

The electrophoretic mobility of a single latex colloid in various salt environments has been measured, using a single colloid electrophoresis technique. In addition, simulations of the restricted primitive model have been performed to determine the colloidal ζ potential, which was then used as an input parameter for the SEM to deduce the electrophoretic mobility. The measurements show a strong dependence on the ionic strength and valency, and we are able to achieve a quantitative agreement of the theoretical results with the experimental measurements for mono- and divalent salts. Mobility reversal is observed in the presence of trivalent ions. While, in principle, this can also be achieved for purely electrostatically ion-surface interactions, our molecular dynamics simulations provide compelling evidence that an additional ion-specific adsorption force acting on the La3+ ions is required to achieve quantitative agreement of simulations to experiments. We furthermore were able to show that neither discreteness effects of the surface charges nor a purely nonspecific attraction of the ions is able to reproduce the measurements quantitatively. Our results for the nonspecific adsorption for La3+ are in agreement with previous studies; however, the explanation of the origin of this attraction remains outside the scope of the present study and provides an interesting subject for further research. ACKNOWLEDGMENTS

This work was supported by the German Science Foundation, under Grants No. SFG FOR877 and No. SFB 716, and the Volkswagen Foundation. Helpful discussions with A. Delgado, O. W. Hickey, J. Lyklema, T. Palberg, and F. Dommert are gratefully acknowledged. Computing time was provided by the HLRS, and the code to solve the SEM was provided by D. Chan.

[5] P. H. Wiersema, A. L. Loeb, and J. T. Overbeek, J. Colloid Interface Sci. 22, 78 (1966). [6] R. W. O’Brien and L. R. White, J. Chem. Soc. Faraday Trans. 74, 1607 (1978). [7] C. S. Mangelsdorf and L. R. White, J. Chem. Soc., Faraday Trans. 86, 2859 (1990).

022302-6

ELECTROPHORETIC MOBILITY AND CHARGE INVERSION . . . [8] R. J. Hill and D. A. Saville, Colloids Surf. A 267, 31 (2005). [9] M. Elimelech and C. R. O’Melia, Colloids Surf. 44, 165 (1990). [10] M. Quesada-Perez, A. Martin-Molina, F. Galisteo-Gonzalez, and R. Hidalgo-Alvarez, Mol. Phys. 100, 3029 (2002). [11] A. Martin-Molina, J. A. Maroto-Centeno, R. Hidalgo-Alvarez, and M. Quesada-Perez, Colloids Surf. A 319, 103 (2008). [12] C. Labbez, A. Nonat, I. Pochard, and B. Jonsson, J. Colloid Interface Sci. 309, 303 (2007). [13] C. Holm, P. K´ekicheff, and R. Podgornik, eds., Electrostatic Effects in Soft Matter and Biophysics, Vol. 46 of NATO Science Series II—Mathematics, Physics, and Chemistry (Kluwer Academic Publishers, Dordrecht, NL, 2001). [14] Y. Levin, Rep. Prog. Phys. 65, 1577 (2002). [15] A. Y. Grosberg, T. T. Nguyen, and B. I. Shklovskii, Rev. Mod. Phys. 74, 329 (2002). [16] A. Naji, S. Jungblut, A. G. Moreira, and R. R. Netz, Physica A 352, 131 (2005). [17] A. V. Dobrynin, Curr. Opin. Colloid Interface Sci. 13, 376 (2008). [18] A. Mart´ın-Molina, M. Quesada-P´erez, F. Galisteo-Gonz´alez, and ´ R. Hidalgo-Alvarez, J. Phys.: Condens. Matter 15, S3475 (2003). [19] W. Lin, P. Galletto, and M. Borkovec, Langmuir 20, 7465 (2004). [20] M. Quesada-Perez, E. Gonzalez-Tovar, A. Martin-Molina, M. Lozada-Cassou, and R. Hidalgo-Alvarez, Colloids Surf. A 267, 24 (2005). [21] V. A. Bloomfield, Biopolymers 44, 269 (1997). [22] K. Besteman, M. A. G. Zevenbergen, and S. G. Lemay, Phys. Rev. E 72, 061501 (2005). [23] I. Mela, E. Aumaitre, A.-M. Williamson, and G. E. Yakubov, Colloids Surf. B 78, 53 (2010). [24] F. Plouraboue and H.-C. Chang, Phys. Rev. E 79, 041404 (2009). [25] B. W. Ninham, Adv. Colloid Interface Sci. 83, 1 (1999). [26] J. Lyklema and T. Golub, Croatica Chemica Acta 80, 303 (2007). [27] E. Wernersson, R. Kjellander, and J. Lyklema, J. Phys. Chem. C 114, 1849 (2010). [28] F. H. J. van der Heyden, D. Stein, K. Besteman, S. G. Lemay, and C. Dekker, Phys. Rev. Lett. 96, 224502 (2006). [29] A. Kub´ıcˇ kov´a, T. Kˇr´ızˇ ek, P. Coufal, M. Vazdar, E. Wernersson, J. Heyda, and P. Jungwirth, Phys. Rev. Lett. 108, 186101 (2012). [30] A. Martin-Molina, C. Rodriguez-Beas, and J. Faraudo, Phys. Rev. Lett. 104, 168103 (2010). ´ V. Delgado, and J. Lyklema, Langmuir 28, [31] M. L. Jim´enez, A. 6786 (2012). [32] C. Schneider, M. Hanisch, B. Wedel, A. Jusufi, and M. Ballauff, J. Colloid Interface Sci. 358, 62 (2011). [33] I. Semenov, O. Otto, G. Stober, P. Papadopoulos, U. F. Keyser, and F. Kremer, J. Colloid Interface Sci. 337, 260 (2009). [34] I. Semenov, P. Papadopoulos, G. Stober, and F. Kremer, J. Phys.: Condens. Matter 22, 494109 (2010).

PHYSICAL REVIEW E 87, 022302 (2013) [35] A. Ashkin, Proc. Natl. Acad. Sci. USA 94, 4853 (1997). [36] T. A. Wood, G. S. Roberts, S. Eaimkhong, and P. Bartlett, Faraday Discuss. 137, 319 (2008). [37] O. Uebersch¨ar, C. Wagner, T. Stangner, C. Gutsche, and F. Kremer, Polymer 52, 1829 (2011). [38] C. Wagner, T. Stangner, C. Gutsche, O. Uebersch¨ar, and F. Kremer, J. Optics 13, 115302 (2011). [39] G. S. Roberts, T. A. Wood, W. J. Frith, and P. Bartlett, J. Chem. Phys. 126, 194503 (2007). [40] G. Stober, L. J. Steinbock, and U. F. Keyser, J. Appl. Phys. 105, 084702 (2009). [41] L. H. Mikkelsen, Water Res. 37, 2458 (2003). [42] H. J. Limbach, A. Arnold, B. A. Mann, and C. Holm, Comp. Phys. Comm. 174, 704 (2006). [43] J. D. Weeks, D. Chandler, and H. C. Andersen, J. Chem. Phys. 54, 5237 (1971). [44] A. Arnold, J. de Joannis, and C. Holm, J. Chem. Phys. 117, 2496 (2002). [45] V. Lobaskin and B. D¨unweg, New J. Phys. 6, 54 (2004). [46] V. Lobaskin, B. D¨unweg, M. Medebach, T. Palberg, and C. Holm, Phys. Rev. Lett. 98, 176105 (2007). [47] A. Delgado, F. Gonz¨alez-Caballero, M. A. Cabrerizo, and I. Alados, Acta Polymerica 38, 66 (1987). [48] T. Palberg, M. Medebach, N. Garbow, M. Evers, A. B. Fontecha, H. Reiber, and E. Bartsch, J. Phys.: Condens. Matter 16, 4039 (2004). [49] H. Reiber, T. Koller, T. Palberg, F. Carrique, E. R. Reina, and R. Piazza, J. Coll. Interface Sci. 309, 315 (2007). [50] R. R. Netz and H. Orland, Europhys. Lett. 45, 726 (1999). [51] M. L¨obbus, H. P. van Leeuwen, and J. Lyklema, Colloids Surf. A 161, 103 (2000). [52] D. Bastos and F. J. Nieves, Colloid Polym. Sci. 271, 860 (1993). [53] D. Bastos and F. J. Nieves, Colloid Polym. Sci. 274, 1081 (1996). [54] B. I. Shklovskii, Phys. Rev. E 60, 5802 (1999). [55] V. I. Perel and B. I. Shklovskii, Physica A 274, 446 (1999). [56] R. Messina, C. Holm, and K. Kremer, Eur. Phys. J. E 4, 363 (2001). [57] A. G. Moreira and R. R. Netz, Europhys. Lett. 57, 911 (2002). [58] D. B. Lukatsky, S. A. Safran, A. W. C. Lau, and P. Pincus, Europhys. Lett. 58, 785 (2002). [59] R. Messina, Physica A 308, 59 (2002). [60] R. Messina, C. Holm, and K. Kremer, Phys. Rev. Lett. 85, 872 (2000). [61] C. Calero and J. Faraudo, Phys. Rev. E 80, 042601 (2009). [62] S. Madurga, A. Martin-Molina, E. Vilaseca, F. Mas, and M. Quesada-Perez, J. Chem. Phys. 126, 234703 (2007). [63] O. Lenz and C. Holm, Eur. Phys. J. E 26, 191 (2008).

022302-7