Examining the Contributions of Image-Charge ... - ACS Publications

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Examining the Contributions of Image-Charge Forces to Charge Reversal: Discrete Versus Continuum Modeling of Surface Charges Zhi-Yong Wang* and Zengwei Ma* †

School of Optoelectronic Information, Chongqing University of Technology, Chongqing 400054, China ABSTRACT: The effects of both repulsive and attractive image-charge forces on the structure of electric double layers are addressed by Monte Carlo determination, based on a primitive model of electrolytes in contact with two types of identically charged surfaces: one with a homogeneously smeared-out charge density and the other with discrete interfacial groups. It is shown that the behavior of ions is closely related to surface charge distributions. Moreover, charge reversal in the absence of image charges witnesses an initial enhancement and then follows a fast suppression with increasing valence of the interfacial groups. The situation is quite similar to what are observed in the presence of repulsive image charges, which can significantly facilitate counterion condensation by overcoming the electrostatic barrier presented by the low dielectric substrate. With transition to attractive image-charge interactions, however, charge reversal remains widely unaffected in different surface charge representations, which even becomes much weaker when compared to the corresponding cases of both no images and repulsive images, provided that the interfacial groups have adequate valences. The overall scenario is found to be independent of the surface charge density values under study. These findings clearly illustrate the enormous improvement in our quantitative understanding of the electric double layer structure and the associated charge reversal phenomenon at the interface of various substrates.

I. INTRODUCTION Considerable efforts have been devoted to studying the distribution of ions in aqueous solutions at charged surfaces on the theoretical, computational, and experimental frontiers as a consequence of great promise for a wide spectrum of applications in diverse areas covering electrochemistry, surface science, biology, materials science, and many others.1−8 For such applications, the electric double layers (EDLs) lie at the heart of our understanding of the interfacial structure. A traditional method to address EDLs has been the Poisson− Boltzmann theory, which is enjoying an exciting revolution so that one of its refined formalisms, by assuming a larger distance of closest approach to the particle surface for co-ions than for counterions,9 can predict charge reversal (CR) in the case of high electrolyte concentrations and high counterion valences, provided that the magnitude of the surface charges is sufficiently low. Despite this success, the refined formalism still breaks down when electrostatic correlations become significant. To overcome the deficiencies of the Poisson−Boltzmann type formalisms, numerous attempts based on the primitive model have been made to successfully capture CR considering correlation effects in ionic solutions (for comprehensive reviews, see refs 10−13), such as the HNC/MSA integral equation,14−16 density functional theory (DFT),17−19 field theory techniques,20−22 and the Wigner crystal approach.23−25 One common limitation of these theories is that they are built upon the assumption of idealized smooth surfaces with evenly © XXXX American Chemical Society

distributed net charges. However, a vast myriad of surfaces of particles are of a heterogeneous nature that is typically characterized by a certain topological roughness, because of the presence of different groups of active sites.26−28 While the idealistic theories mentioned above provide valuable insights into ion interfacial behavior and resulting CR, they nevertheless fall short of allowing a direct comparison with experiments in which the realistic surfaces are marked with heterogeneity.29 Therefore, lately, a growing attention from computer simulations has been paid to the analysis of surface characteristics by considering the discreteness and roughness of the surface charges.30−39 These studies have clearly confirmed the dominant impact of surface heterogeneity on the final structure of the EDLs and the associated CR phenomenon. Another problem of broader importance is the electrostatic properties at a dielectric interface frequently encountered in a wide range of industrial and biological systems. In contrast to a large amount of theoretical and simulation works only dedicated to the effect of dielectric discontinuity on counterion collapse,40−46 much less progress has been made in our grasp of CR through taking properly account of the combined action of image-charge forces and surface heterogeneity.34,35,47,48 It is self-evident that the dielectric behavior of such realistic systems is difficult to characterize theoretically, either analytically, numerically, or via simulations, and further complicated by Received: January 18, 2016

A

DOI: 10.1021/acs.jctc.6b00057 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

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Journal of Chemical Theory and Computation

Figure 1. Laterally averaged density profile of ions, as a function of the distance from the surface at −σ0e = −0.04 C/m2, normalized with respect to their respective bulk concentrations for both (a) the uniform surface charge distribution and the discrete representation of charged groups with (b) ZG = −1, (c) ZG = −2, and (d) ZG = −3. The results for no images (ε< = ε> = 80), repulsive images (ε< = 2, ε> = 80), and attractive images (ε< = ∞, ε> = 80) are denoted by blue, red, and olive, respectively. Unless otherwise stated, the diameter of ions is taken as D = 4 Å throughout. The lines connecting the data points are merely present to guide the eye.

neglected in the above-cited theoretical studies. Ions are confined to watery phase of dielectric constant ε> of the half space z > 0 and modeled within the primitive model as equisized hard spheres of diameter DI with point charges at their centers. The total potential energy of the system involves the superposition of short-range contact repulsion and longrange Coulomb interactions between each pair of charge types. Complete details can be found in our most recent publication.51 The simulation box was a rectangular prism of dimensions W × W × 2L, with L being chosen so large that a substantial neutral bulk phase exists in the middle of the aqueous phase. The long-range nature of the Coulombic interaction was properly handled by an efficient modified Lekner sum with periodicity in two dimensions.51,52 Typically, we made use of the values of ε< = 2 and ∞, standing respectively for hydrocarbon matrices due to repulsive images and conducting materials to attractive images. The temperature was taken to be T = 298 K and the solvent dielectric constant was set to be ε> = 80. When ε< = ε> = 80, no dielectric difference is recovered between the substrate and solution phases. An asymmetric 3:1type electrolyte with multivalent counterions opposite to the polarity of surface charges was chosen to obtain an obvious CR at a constant concentration of 60 mM. More generally, we ignored the dielectric contrast between the hard-sphere cores and the external dielectric medium where they are embedded.53,54 Meanwhile, the effect of local changes of the dielectric constant in the EDL region could be negligible for typical aqueous solutions in the low salt regime.30,55,56 Unless otherwise stated, a common diameter of D = 4.0 Å was assigned to all of surface groups and each ion in solution on the basis of the fact that the ionic hydration numbers at interfaces are lower than those in the bulklike area.11,54,57 σ0e = −0.04 and −0.16 C/m2 were respectively used to represent the weak and strong coupling regimes in the presence of trivalent counterions.58

the involvement of surface characteristics. Therefore, one usually has a tendency to ignore it and handle the dielectric constant as spatially homogeneous throughout in studying aqueous electrolytes confined within various nanopores. Motivated by the general context described above, we have carried out a series of Monte Carlo simulations of an asymmetric 3:1 salt, with multivalent counterions holding opposite charge to the substrate, adsorbed on both the uniformly charged and energetically heterogeneous surfaces in the absence and presence of dielectric discontinuity. In the light of many excellent publications,30−33,36 heterogeneity is introduced in the form of a square lattice arrangement of equispaced discrete surface groups. This assumption can be further rationalized by the fact that, nowadays, advanced synthesis protocols offer the opportunity to produce nanoparticles with well-defined interfacial characteristics and tunable patchiness.49,50 Dielectric discontinuity can be treated through an image charge method. In more detail, a charge near an infinite planar dielectric boundary has an imaginary image charge located at the other side of the boundary at the same distance from it as the real charge. The sign and magnitude of the image are determined by the dielectric permittivities of the media involved.

II. MODELS AND COMPUTATIONAL DETAILS The present study uses the models developed in our earlier publication39 to further explore the ion behavior at the dielectric boundary. We consider a solid substrate in the half space z < 0, characterized by a dielectric constant ε = 80). Blue, magenta, and yellow spheres denote the co-ions, the counterions, and the groups, respectively. The cyan plane denotes the substrate surface.

Figure 3. Snapshots for the distribution of ions near a surface of −σ0e = −0.04 C/m2 at varying group valence in the absence of dielectric images (ε< = ε> = 80). The coloring scheme is the same as that described for Figure 2.

which was evaluated in slabs 0.4 Å thickness along the z direction. Typical results are shown in Figure 1 of the ion distribution next to an impenetrable wall of −σ0e = −0.04 C/m2 with different representations of surface charges in the absence and presence of dielectric images. A closer look in the light of the olive symbols with line reveals that the dependence of the ionic densities on the distance from the surface in the zdimension shows no changes for different surface charge distributions under the action of attractive image-charge forces. However, it should be pointed out that the x and y coordinates of the ions proximal to the substrate will be strongly dependent on the way that the charges are distributed over the surface, as can be observed in the snapshots shown in Figure 2, where the introduction of the interfacial groups of varying valence leads to significant variations in the local EDL configurations. Without any loss of generality, Figure 1 shows that, because of the attractive image-charge contributions, there is a region of much higher concentration of co-ions adjacent to the surface, in comparison to the ideal case of no images. As far as the counterions are concerned, however, their behavior does not necessarily follow the same pattern. At ZG = −2, the counterions in the absence of attractive images instead exhibit a more significant propensity for the surface, as evidenced by a higher contact peak. This is because the larger dielectric constant of the substrate weakens the direct association between the counterions and the interfacial groups, because of the electrostatic repulsion of their dielectric images with the

Monte Carlo simulations were done in the NVT ensemble, using periodic boundary conditions and minimum image convention.59 Each simulation with global charge neutrality being satisfied started from an arbitrary initial configuration of the N ions ranging from 800 to 1800, determined by the box’s size. Trial configurations were generated from the old ones by moving the center of mass of a randomly selected ion, which were accepted or rejected according to their probability given by the Metropolis criterion. Adjustment of the maximum displacement was tackled based on the standard procedure described elsewhere.59 The acceptance ratio was kept between 0.4 and 0.5 in this procedure. Each system was thermalized, lasting at least 105 Monte Carlo steps, with a step covering N single-ion translation attempts. After equilibration, our simulations were extended over 106 Monte Carlo steps, saving to disk one every 20 steps for subsequent statistical analyses. To ensure that our results were not affected by finite size effects in the lateral direction, for each W × W, we considered different areas and observed no detectable differences.

III. RESULTS AND DISCUSSION Critical in our study is the local number density profile, which is defined as ρα (z) =

∑ ⟨δ(z − ziα)⟩ i

(α = −1, +3) (1) C


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Figure 4. Snapshots for the distribution of ions near a surface of −σ0e = −0.04 C/m2 at varying group valence in the presence of repulsive images (ε< = 2, ε> = 80). The coloring scheme is the same as that described for Figure 2.

Figure 5. Mean electrostatic potential as a function of the distance from the surface at −σ0e = −0.04 C/m2 in the presence of (a) no images, (b) repulsive images, and (c) attractive images where additional simulations for D = 6.4 Å are performed to support the generality of the predictive properties. The results for different surface charge distributions are denoted by black (uniform), red (ZG = −1), olive (ZG = −2), and blue (ZG = −3), respectively. The vertical dashed lines represent the location of the shear plane separating immobilized charges (surface charges and adsorbed ions) from mobile charges, which is identified by means of the fact that the variations of the potential are taken to be linear from the surface to the characteristic plane.65,66 The lines connecting the data points are merely present to guide the eye.

surface ion distribution is critically sensitive to the valence of the interfacial groups, which can further be clarified from the representative snapshots of Figure 3, where electrostatic binding of counterions at the interface for the unpolarized substrate becomes more pronounced with the increase of valence of the interfacial groups so that, for ZG = −3, all sites pair with counterions. This feature is contrary to what has been seen in atomistic simulation, where Freund observed that, in a salt-free aqueous solution, the uniformly charged wall causes more collapse of counterions than the discretely charged wall.30 The difference is mainly a consequence of the ion-specific adsorption. More important in Figure 1 is the fact that there is an anomalous enrichment of counterions in the near-surface region in the presence of repulsive images, which even exceeds the counterion collapse of the situation of no dielectric discontinuity. The counterintuitive behavior is determined by the fact that the interfacial groups serve effectively as bridges linking the counterions and their own images of the same polarity.51 It is easy to visualize, by comparing the typical snapshots shown in Figures 3 and 4, especially at ZG = −1, more association of counterions with groups develops when repulsive image charges are present. For higher valences of the interfacial groups, the effect is less apparent, since (almost) all of the groups are prone to pairing with counterions, even when there is no dielectric discontinuity. Note that, although we claim that the surface groups effectively bridge the counterions and their own images, the repulsion still exists between the real ions and their mirror charges. This can be clearly appreciated by the snapshots, where, in the absence of dielectric images, the

interfacial groups. The repulsion is drastically suppressed when the groups’ valence decreases or the charges are smeared uniformly over the surface, so that more counterions concentrate around the surface than the situation where no image charges are present. It should not be surprising that, when compared to the case of ZG = −2, the stronger repulsion at ZG = −3 does not cause a larger difference in the contact density of the counterions for both cases with and without attractive image charges. The reason rests with the fact that, for the identical surface charge density, the higher the valence of the interfacial groups, the larger the lattice constant becomes, meaning that the counterions can trap in a broader region near the surface but away from the groups of ZG = −3, because of the direct attraction of their images. All of these observations can be readily recognized by comparing the characteristic snapshots, shown in Figures 2 and 3, of the positions of the ions very close to the surface. As seen from Figure 1a, the ions experience an obvious depletion from the uniformly charged surface when the dielectric constant of the bulk medium is greater than that of the substrate medium, consistent with the earlier theoretical predictions and computer simulations.8,40,43−45 This is since the ions see image-charge reflections of their same polarity in the region of low dielectric constant. Once surface charges occupy fixed lattice points in a two-dimensional array, on the other hand, the density profiles of counterions and co-ions become qualitatively different, in comparison to the case of the equivalent uniform charge density, regardless of whether repulsive image charges are considered. Moreover, the nearD


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Figure 6. Same as Figure 1 but for a highly charged surface of −σ0e = −0.16 C/m2.

repulsive image forces, which favor counterion condensation. The features show that the effects are intertwined of dielectric discontinuity and surface charge representation, complementary to a similar investigation performed for a highly charged surface of a low dielectric constant, where Taheri-Araghi and Ha using a two-state model and a matching method at the mean-field level drew the same conclusion through comparing the monovalent interfacial groups and the equivalent uniformly charged surface near a mixture of NaCl and CaCl2 electrolytes.47 Moreover, our simulations for both ZG = −1 and −2 are very consistent with those expected from experiments where different authors observed a reversal in the electrophoretic mobility of latex colloids in the presence of trivalent La3+ ions.62−64 As seen, the potential at the shear plane serving as an essential indicator to characterize the electrophoretic mobility exhibits a positive value at the negatively charged surface when there are repulsive image charges. Unfortunately, the image charge effect is ignored without exception in elucidating the experimentally observed mobility reversal. On the other hand, Figure 5 indicates that, for ZG = −3, the potential at the shear plane becomes smaller when dielectric discontinuity is present, implying that image repulsion works although the group bridging attraction takes effect. Overall, our observations show that theories developed on the basis of the assumption of homogeneous surfaces will miss important behavior if the dielectric constant of the surface medium is lower than that of the aqueous environment. With transition to the highest dielectric permittivity of the substrate, we see that all of the curves in Figure 5c are indistinguishable for distinct surface charge representations in both cases of the weakly and strongly hydrated sizes of ions, which is completely different from what are observed in the presence of no images and repulsive image charges. This simple but important characteristic considerably facilitates future theoretical development by assuming the homogeneous nature of the surfaces and ignoring the action of attractive image charges, effectively circumventing the technical difficulty imposed by both surface charge heterogeneities and dielectric discontinuity, to elucidate the mechanisms underlying the experimentally observed CR phenomenon of the planar

orientation of condensed counterions is random, whereas upon polarizing the substrate, the repulsion allows the counterions to orient in a more orderly array toward the normal direction of the surface. According to the local densities of eq 1, we can calculate the mean electrostatic potential, using the following formula: Ψ(z) =

1 ε0ε>

∫z

∞

dz′(z − z′)

∑ α =−1, +3

Zαeρα (z′) (2)

More generally, CR takes place when Ψ(z) changes the sign from negative to positive, since the surface charge densities under study are always less than zero throughout,51 which is consistent with the claim by Diehl and Levin that the maximum of the electrostatic potential can be used to appreciate CR.60 In Figure 5, we present the distance dependence of the potential near the charged surface along the z-direction for the same systems described in Figure 1. In the absence of no image interactions, the effect of the discreteness of surface charges can be clearly seen. The magnitude of the surface potential for ZG = −1 is significantly smaller than that for the uniform charge density, implying that discrete interfacial charges enhance counterion condensation. In both cases, the mean electrostatic potential witnesses an inversion of sign and a nonmonotonic evolution, which shows that the adsorbed counterions have reversed the native surface charges. The effect is more impressive in the case of ZG = −2, where the surface potential can become positive, even for the negative surface charge values. Our findings are in qualitative agreement with what were predicted by different theoretical approaches and computer simulations.31−34,48,61 However, when ZG = −3, the maximum of the potential curve is found to approach zero, implying that CR vanishes but full counterionic pairing occurs with the interfacial groups. As repulsive image charges are considered, we observe a more negative surface potential on the assumption of a uniform charge distribution, revealing that dielectric discontinuity diminishes counterion condensation. In contrast, the discrete surface charges for ZG = −1 reverse the sign of the surface potential and yield a more positive electrostatic surface potential in the case of ZG = −2, because of the action of E


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Figure 7. Snapshots for the distribution of ions near a surface of −σ0e = −0.16 C/m2 at varying group valences in the presence of attractive images (ε< = ∞, ε> = 80). The coloring scheme is the same as that described for Figure 2.

Figure 8. Snapshots for the distribution of ions near a surface of −σ0e = −0.16 C/m2 at varying group valence in the presence of repulsive images (ε< = 2, ε> = 80). The coloring scheme is the same as that described in Figure 2.

substrates of large permittivity, despite the fact that the introduction of interfacial group of various valences remarkably influences the distribution of ions at the charged surface, as shown in Figure 2 by typical equilibrium configurations. In what follows, we will characterize the associative behavior of ions for a highly charged surface of −σ0e = −0.16 C/m2 but keep other parameters fixed. The laterally averaged density profiles of counterions and co-ions are shown in Figure 6, with respect to the center of the watery phase in the direction normal to the plane of the interface. Within the model of the assumption of uniformly smeared-out charges, the more negatively charged surface attracts and repels ions under the action of attractive and repulsive image-charge forces, respectively, although the accumulation and depletion of ions in the EDL region are decreased in strength, when compared to the weakly charged surface. These behaviors are in accord with our traditional knowledge that the image-charge effect gets gradually dampened with an increase in the surface charge value. When the surface charges are localized at discrete sites, by and large, there is no difference in the distributions of counterions and co-ions, because of the action of attractive image-charge forces, irrespective of the valence of interfacial groups. Even so, the presence of inhomogeneities in the dielectric constant is still expected to have an important impact on transverse spatial distribution of ions as shown by the representative snapshots in Figure 7 where the counterions are gradually pushed away from the surface sites with an increase of the groups’ valence due to their enhanced image repulsion with

the interfacial groups, which is consistent with what are observed in the case of −σ0e = −0.04 C/m2. A careful inspection of Figure 6c and Figure 1c indicates that, for three cases under study at ZG = −2, different scenarios occur in the magnitude of the contact value of co-ions as the surface charge density increases. In detail, when the surface is weakly charged, the sets of co-ion density profiles follow the same pattern as the situation where the surface charges are uniform, conforming to the default rule of action of image charges. However, the trend is gradually reversed with a rise in the magnitude of surface charge density. We explain this anomalous behavior as follows. In the absence of dielectric discontinuity, the association of counterions with the interfacial groups gets stronger in the higher surface charges, since the number increases of groups per unit area, which, in return, yields more surface enrichment of co-ions in the EDL region. It should be mentioned that our observation for the counterion contact pairing with the interfacial groups can be further corroborated by coarse-grained molecular dynamics simulations of colloidal systems with multivalent counterions,38 as well as by large-scale atomistic simulations of DMPA2− monolayers57 and anomalous X-ray reflectivity measurements of biomimetic membranes with DMPA− headgroups67 in contact with aqueous solutions containing BaCl2. When the substrate has a lower permittivity than the aqueous environment, this co-ion enrichment becomes more pronounced, in virtue of more counterion association, as seen from Figure 8b ,where all sites pair with counterions in the presence of repulsive image F


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Figure 9. Snapshots for the distribution of ions near a surface of −σ0e = −0.16 C/m2 at varying group valence in the absence of dielectric images (ε< = ε> = 80). The coloring scheme is the same as that described in Figure 2.

Figure 10. Same as Figure 5 but for a highly charged surface of −σ0e = −0.16 C/m2.

profiles show significant variations where CR becomes more apparent than the uniform surface charge approximation and has a tendency to be gradually strengthened with an augmentation of the interfacial group’s valence at low values. In particular, it has been firmly established that the low dielectric substrate further boosts the CR magnitude, as shown by higher peak values in the Ψ(z) curves. Also, the counterintuitive observation can be clearly seen from Figure 8 in which, for both ZG = −1 and ZG = −2, more contact pairs emerge between the counterions and the surface sites, as a result of the group bridging attraction than the ideal case of no dielectric images of Figure 9. As the group’s valence has a higher value of ZG = −3, however, there is only a very small amount of CR near the interface of no dielectric discontinuity with Ψ(z) somewhat overshooting the isoelectric line. In the presence of repulsive images, we see no indication of CR. Note that the group bridging effect still exists for ZG = −3. The cancellation of CR lies in image repulsion making the paired counterions with the interfacial groups as far as possible away from the phase boundary, as evidenced by comparing Figure 9c with Figure 8c. In this regard, our finding appears to contradict earlier simulations where Gan and Xu observed that the mirror charge effect significantly enhances CR of a spherical nanoparticle immersed in divalent salt solutions.68 The contradiction demonstrates that surface curvature of dielectric objects could have a drastic effect on the potential of an ion69 and affect the resulting CR. If the surface charge density is reduced but the concentration of electrolyte in the bulk is maintained, the potential curves in Figures 5a and 5b display the similar characteristic features in the different surface charge representations. On the other hand, when attractive image charges are properly incorporated, our simulations again predict a

charges, when compared to the case of no dielectric discontinuity shown in Figure 9b. With transition to attractive image-charge forces, on the other hand, the contact density of co-ions for −σ0e = −0.16 C/m2 is closer to the bulk concentration, which becomes significantly suppressed, relative to −σ0e = −0.04 C/m2. The larger the magnitude of the surface charge density, the smaller the lattice constant becomes. This fact makes available space narrower for co-ions pairing directly with their own images at the interface of −σ0e = −0.16 C/m2, thus reducing their adsorption, as can be identified by comparing Figure 2b with Figure 7b. What’s worse, the repulsion between the co-ions and the finite-sized groups is more unfavorable to the association of co-ions with their own images. Overall, the anomalous trend in Figure 6c becomes less conspicuous at ZG = −1 and returns to the traditional scenario at ZG = −3. Figure 10 shows variations of the mean electrostatic potential for those systems considered in Figure 6 from an integration of the ionic density profiles. For the uniformly charged case, it can be seen that the surface potential decreases when a low dielectricity of the surface medium is taken into account, since the counterions are driven to depart from the immediate surface region, because of the repulsive interactions with their mirror charges. Compared to Figure 5, an increase of the magnitude of the surface charge density at a fixed salinity leads to a stronger CR effect and a reduction of the surface potential difference in both cases with and without repulsive images, implying that the image-charge forces at surfaces with the homogeneous charge distribution become insignificant for strongly correlated ionic systems, in agreement with previous theoretical analyses.40,42,43,45 Once the discreteness of surface charges is taken into consideration, which is of wide practical interest, the potential G


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Journal of Chemical Theory and Computation qualitatively different behavior. It is clear that the curves in Figure 10c coincide with each other, regardless of how the charges are distributed over the surface. Still, the observation is independent of the size of hydrated ions. Another interesting effect is that, in striking contrast to the point of view of entropy,12,70 the larger diameter of the ionic species does not yield a more appreciable CR, as shown in the potential curves with a smaller peak. A similar anomaly has also been reported in a most recent publication for ZG = −1 in the presence of repulsive image charges.51 In addition, there is a much stronger CR, as shown in Figure 10b, by a higher peak value in the vicinity of low dielectric substrate of ZG = −1, because of the group bridging attraction, in comparison to the case where the dielectric constant of the substrate is larger than that of the aqueous environment. The situation becomes more noticeable at ZG = −2 for both cases, with and without repulsive images being taken into consideration. In contrast, when one assumes that the surface charges have a homogeneous nature, the degree in CR has a tendency to decrease in the presence of no images and repulsive images relative to the case of attractive images, returning to our conventional knowledge that repulsive imagecharge forces between ions and a low dielectric boundary give rise to preferential desorption of salt from the interface and that attractive image-charge forces between ions and a high dielectric boundary result in favorable adsorption of salt to the interface. Also, we note that the characteristics for ZG = −3 show an identical trend with the homogeneous approximation of surface charges. Finally, it should be stressed that the same regularities above also hold true for the systems of low surface charge density. Overall, the results presented here provide novel insights into the electrostatic-driven adsorption of ions at aqueous interfaces of various substrates and pave the way for future studies toward exploring these more-complex systems, which have prevalence in more realistic systems.

adsorbate. This behavior will make it easy to develop adequate theories in predicting charge reversal of materials of high dielectric constants by ignoring interfacial polarization effects and electrostatic surface heterogeneity. In particular, charge reversal with attractive image charges in action unconventionally gets diminished in strength than the cases of no images and even repulsive image charges, as long as the interfacial groups remain adequate valences. At any rate, this work provides a step forward toward studying the more realistic and experimentally relevant systems and may help us to control colloidal selfassembly by decorating the surface of colloidal particles with functional groups of optimal valence.49,50 Of course, more work is clearly needed in order to put this study in the context of a deeper understanding of the double layer structure and the charge reversal phenomenon. Further investigations including discrete solvent effects are presently underway in our laboratory.

IV. CONCLUDING REMARKS The current study complements our earlier works on the contributions of image charges to the distribution of ions in the immediate vicinity of charged interface between two different phases. Just as before, we choose the canonical Monte Carlo approach to investigate charge reversal. In addition to the case of repulsive image-charge forces, the results presented here involve the consideration of attractive images in the restrictive primitive model approximation (ions of equal size) for different electrostatic coupling regimes in diverse representations of surface charges. Generalizing the results of performed studies, we can observe that the discreteness of surface charges, which is more realistic than the uniform surface charge approximation, plays an important role in determining the formation of the ionic atmosphere of macroions. On the one hand, the repulsive image charges, in conjunction with the types of surface charge representations, may cause a tremendous deviation in both the magnitude of charge reversal and the experimentally associated reversal of electrophoretic mobility at moderate valences of the interfacial groups. Therefore, it is necessary for theoretical development to include interfacial polarization effects and electrostatic surface heterogeneity to predict adsorption characteristics of ions. On the other hand, the proper consideration of attractive image charges brings about the collapsed potential curves, indicating that charge reversal remains the same, despite the fact that different surface charge distributions might dramatically impact the final structure of the

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (Z.-Y. Wang). *E-mail: [email protected] (Z. Ma). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research is funded by the National Natural Science Foundation of China (Nos. 11104364 and 21304110) and by Chongqing Research Program of Basic Research and Frontier Technology (No. cstc2015jcyjBX0056) and partially by the Scientific and Technological Research Program of Chongqing Municipal Education Commission (No. KJ1400921).

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