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Bendix
Petersen
^{ab},
Rafael
Roa
^{ac},
Joachim
Dzubiella
^{ad} and
Matej
Kanduč
*^{a}
^{a}Research Group for Simulations of Energy Materials, Helmholtz-Zentrum Berlin für Materialien und Energie, Hahn-Meitner-Platz 1, D-14109 Berlin, Germany. E-mail: matej.kanduc@helmholtz-berlin.de; joachim.dzubiella@helmholtz-berlin.de
^{b}Institut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, D-12489 Berlin, Germany
^{c}Departamento de Física Aplicada I, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos s/n, E-29071 Málaga, Spain
^{d}Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder Str. 3, D-79104 Freiburg, Germany

Received
27th February 2018
, Accepted 4th April 2018

First published on 10th April 2018

Metal nanoparticles are receiving increased scientific attention owing to their unique physical and chemical properties that make them suitable for a wide range of applications in diverse fields, such as electrochemistry, biochemistry, and nanomedicine. Their high metallic polarizability is a crucial determinant that defines their electrostatic character in various electrolyte solutions. Here, we introduce a continuum-based model of a metal nanoparticle with explicit polarizability in the presence of different kinds of electrolytes. We employ several, variously sophisticated, theoretical approaches, corroborated by Monte Carlo simulations in order to elucidate the basic electrostatics principles of the model. We investigate how different kinds of asymmetries between the ions result in non-trivial phenomena, such as charge separation and a build-up of a so-called zero surface-charge double layer.

In the case of a planar dielectric discontinuity, the electrostatic potential can simply be expressed as the electrostatic potential arising from a fictive “image charge” residing on the other side of the discontinuity. Therefore, in this context, the dielectric discontinuity effects are sometimes referred to as image charges. The dielectric effects in double-layer problems of planar geometry have been elaborated by Torrie, Valleau, and Patey,^{7} and by Bratko, Jonsson, and Wennerström^{8} using computer simulations, and by Kjellander and Marčelja^{9} and Outhwaite, Bhuiyan, and Levine^{10} utilizing various theoretical frameworks. The image-charge concepts have been adapted to spherical symmetry by Linse.^{1,11,12} He showed that approximating the exact mathematical expressions for the spherical geometry leads to a simplified picture in which the polarization is described by image charges as in planar cases. The image charges in the spherical geometry are of paramount importance, since a vast majority of the soft-matter electrostatics research in the recent decades has focused on colloidal and biological systems, where various macromolecular structures (e.g., colloids, proteins, polysaccharides, micelles) in water can be modeled as spherical entities with a lower dielectric interior ε′ (due to their predominantly hydrocarbon architectures) than the surrounding water environment (ε′ ≪ ε).^{1,13–24}

The other side of the spectrum, containing spherical bodies of a much higher dielectric interior than water (ε′ ≫ ε), such as for example small metal particles in aqueous environments, has been much less explored. However, the interest in this field has boosted with recent advances in metal nanoparticle chemistry and physics, which have emerged as a broad new discipline in a subdomain of colloids and surfaces.^{25,26} One of the most prominent discoveries was that gold nanoparticles (of a size 1–10 nm) are active catalysts for oxidation reactions.^{27} This has triggered a tremendous research activity in nanocatalysis, which presently remains one of the fastest growing areas of nanoscience.^{28–30} Furthermore, applications involving metal nanoparticles can for instance be found in electrochemistry for nanoelectrodes,^{31} photovoltaic cells^{32} electro-osmosis,^{33} or in biochemistry and nanomedicine for drug delivery, therapeutics, diagnostics, and bioimaging.^{34–38} At the same time, experimental findings pointed out cytotoxic features of some metal nanoparticles.^{39,40} Several studies suggested that metal nanoparticles interact with cell membranes in a complex way,^{41–44} governed by electrochemical potentials and ion distributions around the membrane and a nanoparticle. These achievements emphasize the importance of a deeper theoretical understanding of the interface between a nanoparticle and the solvent, which acts as a determining factor for many properties of the nanoparticle and its complexes in aqueous environments.^{45}

On a simplified level of theoretical description, a basic elucidation involves an implicit-solvent treatment of the electrostatic double-layer problem, adopted from the well-established framework of colloidal science. Now of course, the high dielectric interior inverses the role of the image charges as compared to the case of low-dielectric colloidal particles, which thus become attractive and can trigger completely new physics. The attraction between the metal nanoparticle surface and ions can lead to their accumulation and adsorption and thus to a build-up of an electric double layer surrounding the particle, which crucially impacts the colloidal stability^{46–50} and interactions with other molecules. This phenomenon is also of great importance in the catalysis by metal nanoparticles in liquid phase.^{30,51} The reaction rates of surface-catalyzed bimolecular reactions depend on the concentration at the nanoparticle surface of both reactants,^{52,53} which typically have asymmetric properties (charge, specific adsorption,^{30}etc.).

In this work, we employ theoretical approaches established in the colloidal electrostatics framework, and apply them to less investigated systems of neutral polarizable nanoparticles in different electrolytes. We corroborate the theoretical outcomes by Monte Carlo (MC) simulations, which enable us to assess their regimes of applicability. We show how different kinds of asymmetries between ions result in non-trivial phenomena, such as charge separation and a build-up of net electrostatic potential and effective surface charge.

The presence of the dielectric inhomogeneity across the boundary of the sphere influences the electrostatic potential, which can be thus described by the Green's function connecting two points r and r′ outside the sphere as

u(r,r′) = u_{0}(r,r′) + u_{im}(r,r′) | (1) |

(2) |

(3) |

2.1.1 Onsager–Samaras self-energy.
The simplest theoretical treatment to calculate the self-energy of the above introduced model would be to ignore interactions between ions and considering only the image-charge attraction of an ion with the nanoparticle as dictated by eqn (3). The self-energy of the ion in this approach is given simply by the interaction potential of the ion with its own images, that is (1/2)e_{0}^{2}u_{im}(r,r). But due to the screening action of the “ionic atmosphere” caused by surrounding ions, the image force is considerable only within distances of the order of the Debye length from the surface, defined in terms of the screening coefficient κ (i.e., the inverse Debye length) as

where the sum runs over all ion species. In order to heal the impairments stemming from the surrounding ions, Onsager and Samaras^{55} proposed a “screening coefficient” to the image charge in the form of exp(−2κz), where z is the distance of the charge from the dielectric plane. The factor of 2 in the exponent arises because the total “action–reaction” screening distance from the ion to the surface and back to the ion is 2z. Besides, the distance 2z corresponds also to the distance between the ion and its virtual image charge in the planar geometry. Note that Onsager and Samaras originally proposed the correction for the planar geometry as a first approximation in order to simplify the laborious calculations by Wagner^{56} who used a spatially varying screening length. Their primary aim was to compute the excess surface tension of electrolyte solutions by integrating the Gibbs adsorption equation.^{55}

In the combination with the spherical non-screened image interaction u_{im} given by eqn (3), this yields a simple analytical expression (rescaled by the thermal energy β^{−1} = k_{B}T), which we will refer to as “Onsager–Samaras” (OS) image-charge interaction

This interaction can be also seen as the adsorption potential of a monovalent ion to the metal nanoparticle.

(4) |

In our first two approaches, we adopt the screening coefficient of Onsager and Samaras to derive an approximate image self-energy of a monovalent ion near a metal sphere. Yet, in the spherical geometry, we have at least two possibilities of adapting the screening distance 2z. In the first approach, we assume twice the distance between the ion and the sphere surface, 2(r − a), which gives the self-energy

w^{OS}_{0}(r) = (1/2)e_{0}^{2}u_{im}(r,r)e^{−2κ(r−a)} | (5) |

(6) |

In our second approach, we consider the screening distance as the separation between the ion and its images. In this case, each of the two induced image charges is screened by its own screening coefficient, which is exp(−κr) for the first and exp[−κ(r − a^{2}/r)] for the second image term in eqn (3). The resulting Onsager–Samaras* (OS*) expression of this approach (which we denote with an asterisk) is then

(7) |

Note that both expressions, OS and OS*, have not been self-consistently derived but obtained by an ad hoc “stitching” together the effects of dielectric discontinuity and ionic screening, and are therefore not exact. Consequently, it is also not a priori clear, which of the two approaches yields more accurate results.

2.1.2 Debye–Hückel self-energy.
In our third approach, we base the image self-energy on the exact Green's function u^{DH}(r,r′) of the Debye–Hückel (DH) equation in the presence of a metal sphere,^{50,57}viz.

The Green's function simultaneously accounts for dielectric and screening discontinuities at the surface of the metal sphere. The derivation details are provided in Appendix. The final result for the “DH image self-energy” of a monovalent ion reads

with

Here, the primes denote derivatives of the spherical modified Bessel functions of the first and second kind, which are defined as

where I_{l+1/2}(x) and K_{l+1/2}(x) are the conventional modified Bessel functions of the first and second kind, respectively.

where z is the distance of the ion from the wall. Exactly the same expression but with the opposite sign applies in the case of an ion near a planar wall with much lower dielectric interior than the electrolyte solution (ε′ ≪ ε).^{58}

(8) |

(9) |

(10) |

(11) |

In the limit of infinitely large radius a → ∞ (i.e., planar metal wall), eqn (6), (7), and (9) simplify to

(12) |

2.1.3 Boltzmann distribution.
In cases when cations and anions have symmetric properties, no electrostatic potential is generated, and their distribution around the nanoparticle is solely governed by the image self-energy. In a thermodynamic equilibrium, we therefore expect the ion densities to follow the Boltzmann distribution. Using the OS [eqn (6)] and OS* [eqn (7)] self-energies, this leads respectively to

and

which we will term as “Boltzmann–Onsager–Samaras” approximations (B–OS and B–OS*, respectively). Similarly, using the DH form (9), gives us

which we term the “Boltzmann–Debye–Hückel” (B–DH) approximation. Here, q_{i} is the valency and n^{(i)}_{0} the bulk concentration of species i.

n^{(i)}(r) = n^{(i)}_{0}exp[−βq_{i}^{2}w^{OS}_{0}(r)] (B–OS) | (13) |

(14) |

n^{(i)}(r) = n^{(i)}_{0}exp[−βq_{i}^{2}w^{DH}_{0}(r)] (B–DH) | (15) |

2.1.4 Modified Poisson–Boltzmann.
If cations and anions redistribute dissimilarly around the nanoparticle, the resulting charge separation can lead to a net electrostatic potential, and thus eqn (13)–(15) become inaccurate. As already mentioned, the standard PB equation, which relates electrostatic potential and charge distributions, lacks the image self-energy term. A simple heuristic “remedy” to account for the image effects is to insert by hand the self-energy correction into the Boltzmann factor, thus leading to a modified Poisson–Boltzmann equation^{59,60} in the form

Here, the summation runs over all ion species i. The self-energy term is in principle given either by eqn (6), (7), or (9). In our analyses, however, we will limit ourselves only to the DH-based form, eqn (9). Once the potential ϕ(r) is known, the ion densities can be evaluated as

which we term as the “Poisson–Boltzmann–Debye–Hückel” (PB–DH) approach in this paper. In the case of a symmetry between cations and anions, the electrostatic potential vanishes, ϕ = 0, and eqn (17) reduces to eqn (15).

(16) |

n^{(i)}(r) = n^{(i)}_{0}exp[−βq_{i}ϕ(r) − βq_{i}^{2}w^{DH}_{0}(r)] (PB–DH) | (17) |

Note again, that the obtained expressions, eqn (13)–(17), cannot be considered as mean-field results, because they do not follow from the PB equation. Even though many studies^{61–65} generalized the seminal work of Onsager and Samaras, it nevertheless remains widely misinterpreted what is the actual theoretical framework of their approach. In fact, these results extend beyond the mean-field level and can be deduced from the thermodynamic fluctuations of the instantaneous electric fields around the PB solution.^{66} Alternatively, the PB–DH equation can be derived from a self-consistent variational analysis^{67–70} by setting by hand the screening coefficient κ(r) to be location independent.

For very low ion concentrations (2.2 mM), the screening length is considerably large (κ^{−1} = 9λ_{B} for the monovalent and 4λ_{B} for the divalent case), such that the interaction near the surface is predominantly governed by the unscreened part of the image charge interaction. Moreover, in the limit of vanishing salt concentration, all three theories become equivalent and exact. In the cases shown in the figure, all the theories agree very well even for salt concentrations up to 220 mM.

The size of the nanoparticle is another important parameter that determines the strength of the image attraction. To demonstrate this effect, we plot in Fig. 3 the density profiles for a monovalent 1:1 electrolyte at 220 mM for different radii a of the nanoparticle. With an increasing size, the densities at the surface get higher. Larger metal nanoparticles have namely higher polarizability, thus attracting the ions more efficiently. From the plot it can also be observed that all three theories are becoming equivalent as the particle size increases. Fig. 3d shows a normalized ion density at the nanoparticle surface (i.e., at r = a + r_{0}) as a function of its radius a. In the limiting case of vanishing particle (a → 0), clearly, the polarizability vanishes and the density becomes bulk-like. The density increases with the radius and saturates at the limit of a planar wall (indicated by arrows), where the interaction is given by eqn (12). In the limit of vanishing ionic strength (blue solid line), both theories become exact, since in that case the ion–ion interactions become rare and negligible. For higher concentrations (220 mM), the particle of size of a ∼ 2λ_{B} already nearly reaches the planar-wall limit. Here, it can also be noted that the theories (except B–OS* in the case of small particles) tend to slightly underestimate the densities near the surface compared with MC simulations. This can be attributed to several effects. One of them might be the absence of screening in the ion-free layer of the width r_{0} around the surface, which has been for instance discussed by Levin and Mena.^{72}

Coming to the question of which of the three theoretical approaches is the most accurate: it is of course expected that B–DH should predict more accurate results than either B–OS or B–OS*, because it properly takes into account the spherical geometry of the problem on the DH level. As can be seen from Fig. 2 and 3, the results of both approximate theories are very close to the results of B–DH. Interestingly, B–OS seems to consistently yield a bit lower densities than B–DH, meaning that it underestimates the overall attraction of the ion to the metal sphere. On the contrary, B–OS* predicts consistently slightly larger results than B–DH. It seems that for small spheres, the B–OS* performs slightly better than B–OS. However, this cannot be claimed for larger spheres, as shown in Fig. 3c, where both B–OS and B–OS* are approximately equally off, yet in opposite directions. However, the advantage of the approximate OS and OS* expressions is their much simpler mathematical form than B–DH.

On the continuum-level description, the specific effects can be phenomenologically incorporated via various approximate approaches. In the simplest approximation, the specifically adsorbed ions in the Stern layer close to the surface can be, for instance, treated as a fixed pre-determined surface charge, which is a concept adopted in many theoretical approaches. The main shortcoming of this approximation is that it neglects the dependence of the adsorbed amount of ions on the bulk concentration. Furthermore, it also neglects the influence of surface polarizability and ion correlations. Another approach, which we will adopt here, is to assume an additional attractive potential U_{s}(r) between the ions and the nanoparticle. For simplicity, we use a square-well potential of depth ΔU = −2k_{B}T and the range of r_{s} = 0.3λ_{B} from the effective nanoparticle surface, as presented in Fig. 4. In order to introduce an asymmetry in our system, we apply this potential only to cations, while we assume no specific interactions for anions. We plug the potential U_{s}(r) into the Boltzmann factor of eqn (15) and (17). This break of the symmetry, assumed in the previous section, has far-reaching consequences as we will see in the following.

The ion distributions, shown in Fig. 5, now exhibit a distinct accumulation of cations due to the specific adsorption potential. Notably, the simple Boltzmann-based approach B–DH [eqn (15)] already captures the densities sufficiently well at low concentrations, since the generated electrostatic potential has negligible influence on ions. But as we increase the concentration, the relative cation density n^{(+)}(r)/n^{(+)}_{0} near the surface starts to decrease and anion density slightly to increase. Namely, the potential generated by the adsorbed cations is hindering further accumulation of cations. This behavior is well captured by PB–DH [eqn (17)], whereas the simple B–DH starts breaking down. A crucial difference between PB–DH and B–DH shows up when zooming in to the far-field region [panel (d)] that extends beyond the specific adsorption potential. There, the ion distributions are considerably influenced by the generated potential. As can be noted, the anion density is higher than the cationic, since anions have to compensate the accumulated positive charge at the surface. An interesting comparison can be made when considering only a PB equation with the specific adsorption but without the image-charge self-energy,

(18) |

By integrating the density profiles, we obtain the cumulative charge Z(r) contained within a sphere of radius r around the nanoparticle,

(19) |

(20) |

Fig. 6 (a) Generated electrostatic potential at the nanoparticle stemming from the specific adsorption potential as predicted by the PB and PB–DH theories and MC simulations for 22 and 220 mM of 1:1 salt. (b) The corresponding surface potential, ϕ_{0} = ϕ(a), as a function salt concentration. (c) Linear fits of eqn (21) to the MC data points for 22 and 220 mM electrolyte concentrations. (d) The effective charge of the metal nanoparticle obtained from the fits of eqn (21) as a function of salt concentration. For the case of B–DH, the total cumulative charge is shown instead. |

As is well established in colloid science, we expect the generated potential ϕ(r) to follow a well-known DH law in the far-field,

(21) |

As before, we first look into the ion distributions, which are shown in Fig. 7 for asymmetric 2:1 and 3:1 electrolytes, and compare the theoretical approaches B–DH and PB–DH with MC simulations. As in the case of the ion-specific adsorption, the theories yield better results at low salt concentrations. At higher concentrations, they perform worse due to delicate ion–ion interactions, in particular for higher asymmetry (i.e., 3:1). This theoretical break-down is not unexpected, since multivalent ions are known for significant correlation effects, not accounted for on a mean-field level, a feature that is well established in the double layer literature.^{80–86} As such, Fig. 7 demonstrates a dramatic influence of the valency on the local densities of ions. The relative ionic density at the surface, n(a + r_{0})/n_{0}, scales namely as ∼exp(const. × q^{2}), which for low ionic strengths leads “only” to around 2-fold enrichment of monovalent ions in our system (Fig. 2), 16-fold (∼2^{4}) of divalent, and an enormous 512-fold (∼2^{9}) enrichment of trivalent ions compared to bulk. This implies high ability of metal particles to take-up multivalent ions from a solution. Cases of highly asymmetric electrolytes are very relevant also in catalytic science, where one of well-studied benchmark “model reactions” involves the reduction of trivalent hexacyanoferrate(III) ions by monovalent borohydride ions catalyzed by metal nanoparticles.^{30,87} The local density of the reactant at the surface is one of the governing factors that determines the reaction rate.^{52}

Fig. 7 Normalized ion density profiles for 2:1 and 3:1 electrolytes of concentrations 22 mM (left) and 220 mM (right) near a metal nanoparticle of radius a = λ_{B}. Cations are considered as the multivalent and anions as the monovalent components. Theoretical approaches B–DH [eqn (9)] and PB–DH [eqn (16)] are compared with MC simulation results. |

In Fig. 8a and b we plot the electrostatic potentials generated by asymmetric electrolytes. While PB–DH gives satisfactory agreement at 22 mM of 2:1 salt, it becomes poorer at 220 mM, where the deviation reaches a factor of 2. The situation significantly worsens for 3:1 case. The surface potential ϕ_{0} as a function of concentration is plotted in panel (c). The theoretical prediction, which is now only qualitative, predicts non-monotonic behavior. The surface potential first rapidly rises with concentration due to increased adsorption of ions. At larger concentrations, the rise of the adsorption slows down with increasing concentration due to electrostatic repulsion of already adsorbed ions. Additionally, increasing the salt concentration increases also the screening of the electrolyte, which eventually leads to a drop in the surface potential at high concentrations. Whereas PB–DH yields satisfactory agreement for the 2:1 case (deviating by a factor of 2 from MC at large concentrations), it fails considerably for the 3:1 case. As predicted by the MC simulations, a 3:1 electrolyte creates approximately 20 mV of surface potential in the range of 20–220 mM. This is comparable to the specific-adsorption model discussed in the previous section.

Fig. 8 Electrostatic potentials for 22 and 220 mM of (a) 2:1 and (b) 3:1 electrolytes. (c) Surface electrostatic potential as a function of concentrations of 2:1 and 3:1 electrolytes as obtained from the PB–DH theory (lines) and MC simulations (symbols). (d) Effective charge Z_{eff} of the nanoparticle evaluated from fitting eqn (21) to the potential curves [same legend as in (c)]. In addition, the total accumulative charge Z_{tot} from the B–DH theory is plotted by dotted curves. |

We now fit the DH theory [eqn (21)] to the long-distance potential, which gives us the effective charge Z_{eff}, shown in (d). Contrary to the specific-adsorption model in the previous section, the effective charge in this case is notably a non-linear function of concentration. It first shows a rapid increase with the concentration that turns into a more gradual trend at higher concentrations. Consistently with the results for ϕ_{0} in (c), the PB–DH theory underestimates the effective charge. Similarly as in the previous section, the total cumulative charge Z_{tot} from the B–DH approach is very similar to Z_{eff} from MC and PB–DH, with an exception for high concentrations of the 3:1 electrolyte.

The last plot is revealing an immense influence of the valency asymmetry on the effective charge. According to the MC result, a neutral nanoparticle of a radius λ_{B} gains an effective charge of around 1 e_{0} at 220 mM of 2:1 electrolyte, and an impressive 6 e_{0} in a 3:1 electrolyte of the same concentration. Here we note that the expected effective charge scales with an increasing nanoparticle size faster than its surface, since, as we have seen in Fig. 3, larger particles adsorb ions more effectively due to their higher polarizability. In the limit of large nanoparticle sizes, we then expect Z_{eff} ∼ a^{2}. That means that in the case of a polydisperse solution with various particle sizes, larger ones gain significantly larger charges than smaller ones.

The presented model points to a practical relevance in the physical chemistry, namely the build-up of an electric double layer even in the absence of surface charge, solely because of the difference in cation and anion concentrations in the surface vicinity. The so-called “zero surface-charge double layer”, a concept introduced by theoretical models a few decades ago,^{88,89} helped to interpret several experimental facts, such as electrokinetic effects of uncharged colloids.^{33,90,91} A charged nanoparticle surface enhances its chemical reactivity and consequently has a strong impact on its growth.^{92} In reality, metal nanoparticles can also possess an intrinsic charge. Partially because nanoparticles can be contaminated with various compounds from electrolytes and oxidized material.^{92,93} On the other hand, some syntheses techniques of gold nanoparticles (e.g., pulsed laser ablation) lead to partial oxidation (3.3–6.6%^{94}) of surface atoms, forming a pH-dependent equilibrium of Au–OH/AuO^{−} terminal groups, which thus contribute to the overall negative charge of gold nanoparticles.

Focusing first on the case of symmetric electrolyte, we found very good agreement between the theoretical approaches and MC simulations. Here, the polarizability effects lead to sizable ion accumulation near the nanoparticle surface, which further depends on the ionic strength as well as on the nanoparticle size. In addition, we investigated how an asymmetry in the adsorption affinities for cations and anions influences their distributions. We separately considered two different kinds of asymmetries, in one case stemming from an additional specific adsorption potential to one ionic species, and in the other case stemming from an asymmetric electrolyte (i.e., 2:1 and 3:1). The asymmetries, which give rise to asymmetric distributions of ionic profiles, engender a net electrostatic potential and an effective charge of the nanoparticle. Here, even the most simple approaches that neglect the generated potentials can nevertheless very satisfactorily predict local ion densities (i.e., in the surface vicinity). Of course, at larger distances, where ions tend to neutralize the accumulated charge, it is necessary to invoke a Poisson–Boltzmann description with implemented image-charge corrections. For very high charge asymmetries, such as in a 3:1 electrolyte, the theories face difficulties when compared with the “exact” solutions of MC simulations. The difficulties may be associated with correlation effects between multivalent ions, which are not captured within our theoretical framework. Still, the theories are able to capture the qualitative behavior considerably well and thus help to elucidate basic principles of electrostatics of metal nanoparticles in electrolyte solutions.

Finally, we need to be aware of various conceptual challenges that occur in such systems containing metal-like particles in aqueous solutions. Due to high ionic adsorption affinities, the surface details become very important. This is in stark contrast to low-dielectric macromolecules, where ions are typically repelled from the surfaces, and therefore their molecular structure becomes less relevant. One of such details is for instance the exact geometry of the nanoparticles, which typically possess a well-defined atomic arrangement (e.g., resembling the face-centered cubic structure^{95}) and seems to be critical for the nanoparticle's activity.^{96} A deeper understanding of fine details of metal nanoparticles calls for approaches beyond the idealized continuum model. In this context, in particular atomistic models that take the granularity of the nanoparticle surface and solvent into account are nowadays becoming the focus of sophisticated simulation approaches.^{45,97,98}

u^{DH}(r,r′) = u^{DH}_{0}(r,r′) + u^{DH}_{im}(r,r′) | (22) |

(23) |

A general Ansatz for the second term in eqn (22) is^{50,57}

(24) |

(25) |

(26) |

(27) |

(28) |

- P. Linse, J. Chem. Phys., 2008, 128, 214505 CrossRef PubMed.
- C. Holm, P. Kékicheff and R. Podgornik, Electrostatic Effects in Soft Matter and Biophysics: Proceedings of the NATO Advanced Research Workshop on Electrostatic Effects in Soft Matter and Biophysics Les Houches, France 1–13 October 2000, Springer Science & Business Media, 2012, vol. 46 Search PubMed.
- D. S. Dean, J. Dobnikar, A. Naji and R. Podgornik, Electrostatics of Soft and Disordered Matter, CRC Press, 2014 Search PubMed.
- C. Böttcher, O. Van Belle, P. Bordewijk and A. Rip, Theory of Electric Polarization, 1973 Search PubMed.
- S. Gavryushov and P. Linse, J. Phys. Chem. B, 2006, 110, 10878–10887 CrossRef CAS PubMed.
- B. Hess, C. Holm and N. van der Vegt, Phys. Rev. Lett., 2006, 96, 147801 CrossRef PubMed.
- G. M. Torrie, J. P. Valleau and G. N. Patey, J. Chem. Phys., 1982, 76, 4615–4622 CrossRef CAS.
- D. Bratko, B. Jönsson and H. Wennerström, Chem. Phys. Lett., 1986, 128, 449–454 CrossRef CAS.
- R. Kjellander and S. Marčelja, Chem. Phys. Lett., 1984, 112, 49–53 CrossRef CAS.
- C. W. Outhwaite, L. B. Bhuiyan and S. Levine, J. Chem. Soc., Faraday Trans., 1980, 76, 1388–1408 RSC.
- P. Linse, J. Phys. Chem., 1986, 90, 6821–6828 CrossRef CAS.
- P. Linse and L. Lue, J. Chem. Phys., 2014, 140, 044903 CrossRef PubMed.
- R. Messina, J. Chem. Phys., 2002, 117, 11062–11074 CrossRef CAS.
- J. Zwanikken and R. van Roij, Phys. Rev. Lett., 2007, 99, 178301 CrossRef PubMed.
- J. Reščič and P. Linse, J. Chem. Phys., 2008, 129, 114505 CrossRef PubMed.
- G. I. Guerrero-García, E. González-Tovar, M. Chávez-Páez and M. Lozada-Cassou, J. Chem. Phys., 2010, 132, 054903 CrossRef PubMed.
- A. P. dos Santos, A. Bakhshandeh and Y. Levin, J. Chem. Phys., 2011, 135, 044124 CrossRef PubMed.
- A. Bakhshandeh, A. P. dos Santos and Y. Levin, Phys. Rev. Lett., 2011, 107, 107801 CrossRef PubMed.
- V. Pryamitsyn and V. Ganesan, J. Chem. Phys., 2015, 143, 164904 CrossRef PubMed.
- Z. Gan, X. Xing and Z. Xu, J. Chem. Phys., 2012, 137, 034708 CrossRef PubMed.
- A. P. dos Santos, M. Girotto and Y. Levin, J. Phys. Chem. B, 2016, 120, 10387–10393 CrossRef CAS PubMed.
- R. Wang and Z.-G. Wang, J. Chem. Phys., 2016, 144, 134902 CrossRef PubMed.
- G. I. Guerrero García and M. Olvera de la Cruz, J. Phys. Chem. B, 2014, 118, 8854–8862 CrossRef PubMed.
- V. B. Tergolina and A. P. dos Santos, J. Chem. Phys., 2017, 147, 114103 CrossRef PubMed.
- M.-C. Daniel and D. Astruc, Chem. Rev., 2004, 104, 293–346 CrossRef CAS PubMed.
- R. Sardar, A. M. Funston, P. Mulvaney and R. W. Murray, Langmuir, 2009, 25, 13840–13851 CrossRef CAS PubMed.
- M. Haruta, Catal. Today, 1997, 36, 153–166 CrossRef CAS.
- C. Burda, X. Chen, R. Narayanan and M. A. El-Sayed, Chem. Rev., 2005, 105, 1025–1102 CrossRef CAS PubMed.
- R. Ferrando, J. Jellinek and R. L. Johnston, Chem. Rev., 2008, 108, 845–910 CrossRef CAS PubMed.
- P. Herves, M. Perez-Lorenzo, L. M. Liz-Marzan, J. Dzubiella, Y. Lu and M. Ballauff, Chem. Soc. Rev., 2012, 41, 5577–5587 RSC.
- C. M. Welch and R. G. Compton, Anal. Bioanal. Chem., 2006, 384, 601–619 CrossRef CAS PubMed.
- W. J. Beek, M. M. Wienk and R. A. Janssen, Adv. Mater., 2004, 16, 1009–1013 CrossRef CAS.
- M. Z. Bazant and T. M. Squires, Phys. Rev. Lett., 2004, 92, 066101 CrossRef PubMed.
- T.-H. Chung, S.-H. Wu, M. Yao, C.-W. Lu, Y.-S. Lin, Y. Hung, C.-Y. Mou, Y.-C. Chen and D.-M. Huang, Biomaterials, 2007, 28, 2959–2966 CrossRef CAS PubMed.
- P. AshaRani, G. Low Kah Mun, M. P. Hande and S. Valiyaveettil, ACS Nano, 2008, 3, 279–290 CrossRef PubMed.
- C.-A. J. Lin, T.-Y. Yang, C.-H. Lee, S. H. Huang, R. A. Sperling, M. Zanella, J. K. Li, J.-L. Shen, H.-H. Wang and H.-I. Yeh, et al. , ACS Nano, 2009, 3, 395–401 CrossRef CAS PubMed.
- C. J. Ackerson, P. D. Jadzinsky, J. Z. Sexton, D. A. Bushnell and R. D. Kornberg, Bioconjugate Chem., 2010, 21, 214–218 CrossRef CAS PubMed.
- M.-C. Bowman, T. E. Ballard, C. J. Ackerson, D. L. Feldheim, D. M. Margolis and C. Melander, J. Am. Chem. Soc., 2008, 130, 6896–6897 CrossRef CAS PubMed.
- A. Verma, O. Uzun, Y. Hu, Y. Hu, H.-S. Han, N. Watson, S. Chen, D. J. Irvine and F. Stellacci, Nat. Mater., 2008, 7, 588 CrossRef CAS PubMed.
- P. R. Leroueil, S. A. Berry, K. Duthie, G. Han, V. M. Rotello, D. Q. McNerny, J. R. Baker, B. G. Orr and M. M. Banaszak Holl, Nano Lett., 2008, 8, 420–424 CrossRef CAS PubMed.
- Y. Roiter, M. Ornatska, A. R. Rammohan, J. Balakrishnan, D. R. Heine and S. Minko, Nano Lett., 2008, 8, 941–944 CrossRef CAS PubMed.
- R. Chen, T. A. Ratnikova, M. B. Stone, S. Lin, M. Lard, G. Huang, J. S. Hudson and P. C. Ke, Small, 2010, 6, 612–617 CrossRef CAS PubMed.
- A. Verma and F. Stellacci, Small, 2010, 6, 12–21 CrossRef CAS PubMed.
- J. Lin, H. Zhang, Z. Chen and Y. Zheng, ACS Nano, 2010, 4, 5421–5429 CrossRef CAS PubMed.
- E. Heikkilä, A. A. Gurtovenko, H. Martinez-Seara, H. Häkkinen, I. Vattulainen and J. Akola, J. Phys. Chem. C, 2012, 116, 9805–9815 Search PubMed.
- F. Mafuné, J.-y. Kohno, Y. Takeda, T. Kondow and H. Sawabe, J. Phys. Chem. B, 2000, 104, 8333–8337 CrossRef.
- A. Roucoux, J. Schulz and H. Patin, Chem. Rev., 2002, 102, 3757–3778 CrossRef CAS PubMed.
- T. Kim, K. Lee, M.-S. Gong and S.-W. Joo, Langmuir, 2005, 21, 9524–9528 CrossRef CAS PubMed.
- G. S. Perera, G. Yang, C. B. Nettles, F. Perez, T. K. Hollis and D. Zhang, J. Phys. Chem. C, 2016, 120, 23604–23612 CAS.
- L. Javidpour, A. Lošdorfer Božič, R. Podgornik and A. Naji, 2018, preprint.
- D. Astruc, F. Lu and J. R. Aranzaes, Angew. Chem., Int. Ed., 2005, 44, 7852–7872 CrossRef CAS PubMed.
- R. Roa, W. K. Kim, M. Kanduč, J. Dzubiella and S. Angioletti-Uberti, ACS Catal., 2017, 7, 5604–5611 CrossRef CAS PubMed.
- S. Roy, A. Rao, G. Devatha and P. P. Pillai, ACS Catal., 2017, 7, 7141–7145 CrossRef CAS.
- H. L. Friedman, Mol. Phys., 1975, 29, 1533–1543 CrossRef CAS.
- L. Onsager and N. Samaras, J. Chem. Phys., 1934, 2, 528–536 CrossRef CAS.
- C. Wagner, Z. Phys., 1924, 25, 474 CAS.
- R. A. Curtis and L. Lue, J. Chem. Phys., 2005, 123, 174702 CrossRef CAS PubMed.
- M. Kanduč, A. Naji, J. Forsman and R. Podgornik, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2011, 84, 011502 CrossRef PubMed.
- B. Corry, S. Kuyucak and S.-H. Chung, Biophys. J., 2003, 84, 3594–3606 CrossRef CAS PubMed.
- A. Onuki, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2006, 73, 021506 CrossRef PubMed.
- H. Diamant and D. Andelman, J. Chem. Phys., 1996, 100, 13732–13742 CrossRef CAS.
- D. Dean and R. Horgan, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2004, 69, 061603 CrossRef CAS PubMed.
- Y. Levin, A. P. dos Santos and A. Diehl, Phys. Rev. Lett., 2009, 103, 257802 CrossRef PubMed.
- N. Schwierz, D. Horinek and R. R. Netz, Langmuir, 2010, 26, 7370–7379 CrossRef CAS PubMed.
- T. Markovich, D. Andelman and R. Podgornik, J. Chem. Phys., 2015, 142, 044702 CrossRef PubMed.
- R. Podgornik and B. Žekš, J. Chem. Soc., Faraday Trans., 1988, 84, 611–631 RSC.
- Z.-G. Wang, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2010, 81, 021501 CrossRef PubMed.
- R. Wang and Z.-G. Wang, J. Chem. Phys., 2015, 142, 104705 CrossRef PubMed.
- R. Wang and Z.-G. Wang, J. Chem. Phys., 2016, 144, 134902 CrossRef PubMed.
- R. R. Netz and H. Orland, EPL, 1999, 45, 726 CrossRef CAS.
- N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, J. Chem. Phys., 1953, 21, 1087–1092 CrossRef CAS.
- Y. Levin and J. E. Flores-Mena, EPL, 2001, 56, 187–192 CrossRef CAS.
- K. D. Collins, Biophys. Chem., 2012, 167, 43–59 CrossRef PubMed.
- M. Manciu and E. Ruckenstein, Adv. Colloid Interface Sci., 2003, 105, 63–101 CrossRef CAS PubMed.
- D. F. Parsons, M. Boström, P. L. Nostro and B. W. Ninham, Phys. Chem. Chem. Phys., 2011, 13, 12352–12367 RSC.
- A. P. dos Santos and Y. Levin, Faraday Discuss., 2013, 160, 75–87 RSC.
- N. Schwierz, D. Horinek, U. Sivan and R. R. Netz, Curr. Opin. Colloid Interface Sci., 2016, 23, 10–18 CrossRef CAS.
- C. Pfeiffer, C. Rehbock, D. Hühn, C. Carrillo-Carrion, D. J. de Aberasturi, V. Merk, S. Barcikowski and W. J. Parak, J. R. Soc., Interface, 2014, 11, 20130931 CrossRef PubMed.
- V. Merk, C. Rehbock, F. Becker, U. Hagemann, H. Nienhaus and S. Barcikowski, Langmuir, 2014, 30, 4213–4222 CrossRef CAS PubMed.
- G. M. Torrie and J. P. Valleau, J. Phys. Chem., 1982, 86, 3251–3257 CrossRef CAS.
- A. Y. Grosberg, T. Nguyen and B. Shklovskii, Rev. Mod. Phys., 2002, 74, 329 CrossRef CAS.
- Y. Levin, Rep. Prog. Phys., 2002, 65, 1577 CrossRef CAS.
- H. Boroudjerdi, Y.-W. Kim, A. Naji, R. R. Netz, X. Schlagberger and A. Serr, Phys. Rep., 2005, 416, 129–199 CrossRef CAS.
- A. Naji, M. Kanduč, J. Forsman and R. Podgornik, J. Chem. Phys., 2013, 139, 150901 CrossRef PubMed.
- L. Bhuiyan and C. Outhwaite, J. Phys.: Condens. Matter, 2017, 20, 33801 CrossRef.
- M. Kanduč, M. Moazzami-Gudarzi, V. Valmacco, R. Podgornik and G. Trefalt, Phys. Chem. Chem. Phys., 2017, 19, 10069–10080 RSC.
- S. Carregal-Romero, J. Pérez-Juste, P. Hervés, L. M. Liz-Marzán and P. Mulvaney, Langmuir, 2009, 26, 1271–1277 CrossRef PubMed.
- S. S. Dukhin and A. E. Yaroshchuk, Kolloidn. Zh., 1982, 44, 884–895 CAS.
- B. V. Derjaguin, S. S. Dukhin and A. E. Yaroshchuk, J. Colloid Interface Sci., 1987, 115, 234–239 CrossRef.
- A. Dukhin, S. Dukhin and P. Goetz, Langmuir, 2005, 21, 9990–9997 CrossRef CAS PubMed.
- M. Z. Bazant and T. M. Squires, Curr. Opin. Colloid Interface Sci., 2010, 15, 203–213 CrossRef CAS.
- J.-P. Sylvestre, S. Poulin, A. V. Kabashin, E. Sacher, M. Meunier and J. H. Luong, J. Phys. Chem. B, 2004, 108, 16864–16869 CrossRef CAS.
- M. Mucalo and C. Bullen, J. Mater. Sci., 2001, 20, 1853–1856 CAS.
- H. Muto, K. Yamada, K. Miyajima and F. Mafuné, J. Phys. Chem. C, 2007, 111, 17221–17226 CAS.
- V. Petkov, Y. Peng, G. Williams, B. Huang, D. Tomalia and Y. Ren, Phys. Rev. B: Condens. Matter Mater. Phys., 2005, 72, 195402 CrossRef.
- M. A. Mahmoud, R. Narayanan and M. A. El-Sayed, Acc. Chem. Res., 2013, 46, 1795–1805 CrossRef CAS PubMed.
- X. Chen, A. Munjiza, K. Zhang and D. Wen, J. Phys. Chem. C, 2014, 118, 1285–1293 CAS.
- X. Li and H. Ågren, J. Phys. Chem. C, 2015, 119, 19430–19437 CAS.
- A. H. Boschitsch, M. O. Fenley and W. K. Olson, J. Comput. Phys., 1999, 151, 212–241 CrossRef CAS.

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