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DOI: 10.1039/C7SC00848A
(Edge Article)
Chem. Sci., 2017, Advance Article

Pekka Peljo*^{a},
José A. Manzanares^{b} and
Hubert H. Girault^{a}
^{a}Laboratoire d’Electrochimie Physique et Analytique (LEPA), École Polytechnique Fédérale de Lausanne (EPFL), Rue de l’Industrie 17, CH-1951 Sion, Switzerland. E-mail: pekka.peljo@epfl.ch
^{b}Department of Thermodynamics, Faculty of Physics, University of Valencia, c/Dr. Moliner, 50, E-46100 Burjasot, Spain

Received
22nd February 2017
, Accepted 4th May 2017

First published on 9th May 2017

When a metallic nanoparticle (NP) comes in close contact with an electrode, its Fermi level equilibrates with that of the electrode if their separation is less than the cut-off distance for electron tunnelling. In the absence of chemical reactions in solution, the charge on the metallic nanoparticle is constant outside this range before or after the collision. However, the double layer capacitances of both the electrode and the NP are influenced by each other, varying as the function of distance. Because the charge on the nanoparticle is constant, the outer potential of the metallic NP and hence its Fermi level varies as the capacitance changes. This effect is more pronounced for small particles (<10 nm) in diluted supporting electrolyte solutions, especially if the metallic nanoparticle and the electrode have different potentials of zero charge. Nanoparticles were found to be more electrochemically active in the vicinity of the electrode. For example, the outer potential of a positively-polarized 2 nm radius NP was predicted to decrease by 35 mV or 100 mV (depending on the electrostatic model used to describe the electric double layer), when the NP moved from an electrode at 1 V (vs. its pzc) to the bulk. The force between the equilibrated NP and the electrode is always repulsive when they have the same pzc. Otherwise there can be an attraction even when the NP and the electrode carry charges of the same sign, due to the redistibution of surface charge density at both the NP and electrode surface.

An alternative approach is to consider the NPs as multivalent redox “molecules” with equally spaced, formal redox potentials. The condition of electrochemical equilibrium between an electrode and a solution of NPs and is equivalent to the Fermi level equilibration. The electrode potential then determines the relative populations of the different oxidation states, and hence the average oxidation state of the NPs in the solution. The average charge number of the NPs in the solution is a continuous variable analogous to the NP charge in our approach. The important point to be stressed is that the Fermi level of the electrons in the NP, analogous to the electrochemical potential of the electrons in a solution containing NPs, is a continuous variable that describes the tendency to exchange electrons in order to reach electrochemical equilibrium with the electrode. The Fermi level of the electrons in the NP is not necessarily equal to the energy of the NP in one of its oxidation states. In a solution of NPs, the transfer of just a single electron from the electrode to one NP in oxidized state z causes a dramatic change in the energy of this NP, but the electrochemical potential of the electrons in the solution does not undergo any dramatic change. The latter is a property of the solution that is equivalent to the electrochemical potential of the electrons in the single NP that we consider in this work.

Any change in the separation distance between the NP and the electrode affects also the capacitance of the NP, driving charge transfer between the two objects to adjust their Fermi levels. This equilibration can take place only when the NP is close enough for the electron tunnelling to occur. The charge of the particle will continue to change until the cut-off distance is reached. After this point, the charge is constant but the outer potential of the NP will change in response to the change in the capacitance. This is clear in vacuum or in air, but the purpose of this article is to show that this effect is also significant in electrolyte solutions, where the electrostatic interactions are screened by the electric double layer. With low supporting electrolyte concentrations typically used in NP impact experiments to avoid aggregation, the Debye length can be several nm, while the cut-off distance for electron tunnelling is shorter (<1.5 nm).^{11} Hence, the capacitance of a NP will be higher close to the electrode surface. When the NP moves away from the surface, the Fermi level of the electrons will actually vary even if the NP charge remains constant! Negatively charged NPs will have less negative outer potential (Fermi level decreases), while the Fermi level will increase for positively charged NPs!

Recently, there has been a discussion in the literature whether the dissolution of NPs takes place in one or multiple steps.^{7,12–18} For example, Unwin et al.^{16} have shown experimentally for the first time that large NPs partially dissolve in multiple collision events, and this observation was confirmed by Long et al.^{17} and White and Zhang et al.^{18} However, the exact mechanism of the NP collisions and subsequent leaching has not been clarified. For example, these large NPs seem to be consumed in a series of “bites”, with the NP dissolving closest to the electrode.^{16} However, it is not clear why the NP dissolution does not take place at the outer surface, as the dissolution would be most likely controlled by the mass transfer of dissolved ions away from the NP surface. If this is the case, dissolution events should be terminated by NP diffusion, and more likely by electrostatic interactions, as proposed recently by White and Zhang et al.^{18} However, the effect of contact electrification was neglected, and the electrostatic effects were not quantified. Further work is required to resolve the exact mechanism.

As stated above, the NP will equilibrate its Femi level with that of the electrode, but this does not mean that the outer potentials will be equal. For example, Ag and Au have different potentials of zero charge of −0.44 V and 0.18 V vs. SHE,^{19,20} so that at electrode potentials of 0.6 V vs. Ag/AgCl, typically used for Ag dissolution on a gold electrode, the outer potential of Au will be ca. 0.6 V while the outer potential of Ag will be ca. +1.2 V. This difference in outer potentials is due to the contact electrification (when two electrically-neutral metals are connected, electrons from the metal with the lower work function will flow into the metal with the higher work function, resulting in a Volta potential difference),^{21,22} modified by solvent–metal interactions and other surface modifications when the metal is brought into contact with the solvent. Trasatti and Lust have comprehensively reviewed this topic.^{19} Of course, the additional effects of ligands such as citrate and of the electric double layer affect the apparent potential of zero charge (pzc) of the NP.

ε_{0}ε_{r}∇^{2}ψ = −ρ
| (1) |

(2) |

(3) |

In a binary symmetric electrolyte (z_{+} = |z_{−}| = z) the Poisson–Boltzmann equation (PBE) is

∇^{2}φ = κ^{2}sinhφ
| (4) |

The Stern modification adds inner and outer Helmholtz planes next to the charged surfaces. The outer Helmholtz layer is the closest approach of solvated ions to the surface, while the inner Helmholtz layer consists of mostly organized solvent dipoles and may also contain some specifically adsorbed ions that have lost their solvation shell. In the absence of specific ion adsorption, the potential satisfies the Laplace equation ∇^{2}ψ = 0 within the Stern layer.^{23}

(5) |

(6) |

Including an uncharged Stern layer of thickness δ free of ions, the relation between the potential values at the boundaries of this layer is

(7) |

The NP capacitance is then obtained from 1/C_{NPS} = 1/C_{δ} + 1/C_{NP}(R_{NP} + δ) where C_{δ} = C_{∞}(1 + R_{NP}/δ) corresponds to a spherical capacitor of inner radius R_{NP} and outer radius R_{NP} + δ, and C_{NP}(R_{NP} + δ) is the capacitance of a NP of radius R_{NP} + δ (i.e., eqn (6) with R_{NP} + δ replacing R_{NP}).

The capacitance of weakly-charged NPs (φ_{NP} ≪ 1) simplifies to C^{φ≪1}_{NPS} = C_{∞}[1 + κR_{NP}/(1 + κδ)]. For a NP of radius R_{NP} = 2 nm in a 1:1 aqueous electrolyte of c^{b} = 10 mol m^{−3} (1/κ = 3.03 nm, ε_{r} = 78) with a surface potential RTφ(R_{NP})/zF = 50 mV, and a potential RTφ(R_{NP} + δ)/zF = 39 mV at the outer boundary of the Stern layer of thickness δ = 0.33 nm, these capacitances are C_{NP}(R_{NP} + δ) = 37.7 aF and C_{NPS} = 28.8 aF, and the areal value is C_{NPS}/4πR_{NP}^{2} = 0.57 F m^{−2}. For comparison, the areal capacitance of an electrode (with a Stern layer) at φ_{E}RT/zF = 50 mV is C_{E} = [1/C_{GC} + δ/ε_{0}ε_{r}]^{−1} = 0.28 F m^{−2}, where C_{GC} ≈ ε_{0}ε_{r}κcosh[φ_{E}/2(1 + κδ)] = 0.32 F m^{−2} is the contribution from the electrolyte solution.

(8) |

φ(z) = 4artanh[tanh(φ_{NP}/4)exp(−κz)].
| (9) |

The electrostatic contribution to the force density on the plate

(10) |

If we place a second metal plate at a distance s from the plate at z = 0 their interaction can be attractive, repulsive or null depending on its potential φ_{E}. If φ_{E} is equal to φ(s) given by eqn (9), the charge density on the plate at z = s is

(11) |

If φ_{E} satisfies φ_{E} > φ(s) > 0, the charge density on the plate at z = s is more positive than σ_{E} in eqn (11) and the interaction between the plates is repulsive, even though σ_{NP} > 0 > σ_{E}. On the contrary, if φ_{E} satisfies φ(s) > φ_{E} > 0, its charge density is more negative than σ_{E} in eqn (11) and the interaction is attractive, even though ψ_{NP} > ψ_{E} > 0.

The capacitance matrix formalism of electrostatics^{21} can be used for conductors in electrolyte solutions provided that the potentials are small and the PBE can be linearized. When the plates interact at constant charge, their surface potentials decrease with increasing separation s.^{26} If ψ_{E∞} = σ_{E}/C_{GC∞} and ψ_{NP∞} = σ_{NP}/C_{GC∞} > ψ_{E∞} > 0 are the potentials at large s, with C_{GC∞} = ε_{0}ε_{r}κ, then the values at finite separation are

(12) |

(13) |

(14) |

(15) |

(16) |

(17) |

Additionally, the relative permittivity of the solution can be modified by the electric field, as described for example by the Booth model:^{34–37}

(18) |

(19) |

Scheme 1 Nanoparticle potential and charge after collision with the electrode. ψ_{E} = electrode potential, ψ_{NP} = nanoparticle potential, Q = charge and s = separation from the electrode surface. |

All potentials are given with respect to the potential of zero charge (pzc) of the electrode. Thus, for instance, the pzc of the NP is denoted by E_{pzc} and a value E_{pzc} = 0 indicates that the NP and the electrode have the same pzc. The electrode and the NP were considered to have the same Fermi level

ψ_{NP} + E_{pzc} = ψ_{E}
| (20) |

The capacitance of a NP in the vicinity of an electrode differs from that of an isolated NP discussed above because the potential distribution around the NP is affected by the electrode. The differential capacitance C_{NP} = ∂Q_{NP}/∂ψ_{NP} of the NP was calculated as the function of both the separation distance s and ψ_{E}. For each value of ψ_{E}, the NP was first placed at the cut-off distance of electron tunnelling (s_{t} = 1 nm) and was equilibrated with the electrode to obtain the NP charge; its potential ψ_{NP}(s_{t}) = ψ_{E} − E_{pzc} was given by eqn (20). For larger NP-electrode separation, s ≥ s_{t}, and the same ψ_{E}, the NP potential ψ_{NP} and the NP differential capacitance C_{NP} were unknown. The NP potential ψ_{NP} was varied from ψ_{NP}(s_{t}) to slightly lower values and the NP charge Q_{NP}(ψ_{NP},s,ψ_{E}) was calculated for every value of ψ_{NP}. From these values, C_{NP}(s,ψ_{E}) = (∂Q_{NP}/∂ψ_{NP})_{s,ψE} was evaluated and then, the actual value of ψ_{NP}(s) was estimated from the known NP charge, considered to be constant after the NP-electrode collision, that is, Q_{NP}(ψ_{NP}(s),s,ψ_{E}) = Q_{NP}(ψ_{NP}(s_{t}),s_{t},ψ_{E}).

The differential capacitance of the electrode was firstly evaluated for a planar surface in 1D geometry, considering the different models for the double layer, and a good agreement with the analytic and numerical results was obtained (see ESI†). Then, the capacitance of the electrode in 2D axis symmetrical geometry was evaluated, showing a good agreement with the 1D simulations.

The Fermi level of the electrons in the NP (and, hence its pzc) varies with the NP size, as shown in a recent review.^{40} Additionally, polycrystalline electrode materials have patches with different pzc values,^{21} which may also affect the exchange of electrons with the NP. For simplicity, these effects are not considered in this paper.

Fig. 1 shows that NP radius is another important factor. For R_{NP} = 10 nm, model II predicts that the potential changes only 3 mV at 0.2 V, in comparison with 35 mV for R_{NP} = 1 nm, and for R_{NP} = 20 nm the potential decreases by 1.1 mV. If the potential of the electrode is increased to 1 V, the potential decrease for the NP of R_{NP} = 20 nm when moving into the bulk is 14 mV, compared to 34 mV for R_{NP} = 2 nm. Further increase of the NP radius to 50 or to 100 nm results in a potential decrease of 7 mV and 3 mV, respectively (electrode at 1 V). Hence, slurry electrodes utilizing micrometer sized particles show hardly any effect from the change of the capacitance. This is because the double layer of the electrode perturbs only a small part of the double layer of the large NPs, while this perturbation is larger for small NPs (see Fig. S4 in ESI† to see the electric potential for 2 and 10 nm NPs). If the supporting electrolyte concentration is increased, the thickness of the diffuse double layer decreases, and the effects will be smaller, as shown in Fig. S4C.†

The NPs may contact the electrode and diffuse away. If the NP collision is followed by an electrochemical reaction in the solution, (for example in a system where the electrode is electrocatalytically inert for a given redox couple, while the NP is active), the Fermi level of the NP will equilibrate with the Fermi level of the redox couple in the solution. Interestingly, the NP is most electrochemically reactive (it has more oxidative potential if positively charged, and more reductive potential if negatively charged) close to the electrode surface. While the NP is still close to the surface, the redox reaction with the redox couple in the solution will take place, perturbing the concentration ratio of the redox couple at the NP surface. As the particle moves further from the electrode surface, its Fermi level will change, and the redox reaction can proceed to the opposite direction as a response for this change. Of course, it should be considered whether the process is controlled by kinetics or by mass transfer.

In Fig. 2, we have considered a value E_{pzc} = −0.5 V which is actually close to the situation of AgNPs with a glassy carbon or a Au electrode, as Ag has a pzc of ca. −0.7 V vs. GC and −0.6 V vs. Au.^{19,20,42} The largest shifts in potential when moving the NP from the vicinity of the electrode into the bulk are observed close to the pzc of the NP. Interestingly, the sign of the change in the potential of the NP changes at electrode potentials slightly above 0 V. Below these potentials, the NP potential increases when it moves to the bulk, with the maximum of ca. +90 mV close to the pzc of the NP. When ψ_{E} > 0, ψ_{NP} − ψ_{E} is negative and increases in magnitude when the NP separates from the electrode. The differential capacitance curve shows an asymmetric shape close to the electrode, with the maximum on the negative side of the pzc value, while the symmetry is recovered in the bulk.

Additionally, the capacitance of the electrode changes when the NP moves farther away from its surface. However, this effect is significant only with very small electrodes, and this double layer perturbation is compensated by the change in the electrode capacitance during the approach, although the magnitude of the change depends on the NP potential when it approaches the electrode. Generally, the baseline of the measured current response in impact experiments shows some variations. This paper suggests that some of these variations could be ascribed to the changes in the capacitance of the electrode due to the NPs perturbing the electric double layer of the electrode, but careful comparison of the experiments with and without NPs would be required. The behaviour of NPs approaching and colliding with the electrode is summarized in the Scheme 2, as well as in Schemes S1 and S2† for the cases where the metals have the same pzc.

Fig. 3B shows that pzc difference between the NP and the electrode material will have a significant influence on the electrostatic interaction between the NP and the electrode. Hence, the observed differences of the AgNP oxidation when changing the electrode material from Au to GC^{16} could be partly explained by this change in the electrostatic interactions. Fig. 3C shows also that positively charged AgNPs should not be able to approach close enough to the Au or GC electrode to be oxidized upon impact. In reality, the AgNPs are covered with a capping agent like citrate or tannic acid. These capping agents are adsorbed in the inner Helmholtz layer, screen the electrostatic effects of the positively charged core, and in some cases NPs covered with capping agents appear to have a negative charge (as measured with ζ-potential).^{7} Additionally, the double layer models used in this work do not consider specific adsorption of ions in the Stern layer.

All these effects add excess negative charge on the AgNP, resulting in attraction with the positively polarized electrode, and oxidation upon impact. However, if the contact with the electrode is enough to make the total charge of the NP positive enough so that is feels a repulsive force, then the NP will move away from the surface, resulting in a loss of reactivity as its capacitance decreases.

The force between an equilibrated NP and the electrode is always repulsive when they have the same pzc. Otherwise, there is a region of attraction when the NP and the electrode are oppositely charged. However, this region of attraction extends slightly also on the potentials where NP and the electrode have same sign of charge (i.e. attraction between two negatively charged or two positively charged objects), because the surface charge redistribution can result in formation of positively charged parts in an overall negatively charged object, and vice versa.

In this study, we have used a rather complicated model for the electric double layer, and we expect that the trends of the results will be general. Further improvements could be obtained utilizing more sophisticated methods to describe the double layer structure, and by modelling the electron tunnelling more carefully. However, these results highlight that the effect of the electric double layer of the electrode upon the Fermi level of the NP can be significant, especially with small NPs of different material than the electrode.

- S. Zaleski, A. J. Wilson, M. Mattei, X. Chen, G. Goubert, M. F. Cardinal, K. A. Willets and R. P. Van Duyne, Acc. Chem. Res., 2016, 49, 2023–2030 CrossRef CAS PubMed.
- T. J. Anderson and B. Zhang, Acc. Chem. Res., 2016, 49, 2625–2631 CrossRef CAS PubMed.
- Y. Fang, H. Wang, H. Yu, X. Liu, W. Wang, H.-Y. Chen and N. J. Tao, Acc. Chem. Res., 2016, 49, 2614–2624 CrossRef CAS PubMed.
- J. Kim, J. E. Dick and A. J. Bard, Acc. Chem. Res., 2016, 49, 2587–2595 CrossRef CAS PubMed.
- V. Brasiliense, P. Berto, C. Combellas, G. Tessier and F. Kanoufi, Acc. Chem. Res., 2016, 49, 2049–2057 CrossRef CAS PubMed.
- M. V. Mirkin, T. Sun, Y. Yu and M. Zhou, Acc. Chem. Res., 2016, 49, 2328–2335 CrossRef CAS PubMed.
- S. V. Sokolov, S. Eloul, E. Kätelhön, C. Batchelor-McAuley and R. G. Compton, Phys. Chem. Chem. Phys., 2017, 19, 28–43 RSC.
- Y. Wang, X. Shan and N. Tao, Faraday Discuss., 2016, 193, 9–39 RSC.
- Z. A. Kostiuchenko, B. Zhang and S. G. Lemay, J. Phys. Chem. C, 2016, 120, 22777–22783 CAS.
- M. D. Scanlon, P. Peljo, M. A. Méndez, E. Smirnov, H. H. Girault, M. A. Mendez, E. Smirnov and H. H. Girault, Chem. Sci., 2015, 6, 2705–2720 RSC.
- C. M. Hill, J. Kim and A. J. Bard, J. Am. Chem. Soc., 2015, 137, 11321–11326 CrossRef CAS PubMed.
- V. Brasiliense, A. N. Patel, A. Martinez-Marrades, J. Shi, Y. Chen, C. Combellas, G. Tessier and F. Kanoufi, J. Am. Chem. Soc., 2016, 138, 3478–3483 CrossRef CAS PubMed.
- V. Brasiliense, P. Berto, C. Combellas, R. Kuszelewicz, G. Tessier and F. Kanoufi, Faraday Discuss., 2016, 193, 339–352 RSC.
- A. N. Patel, A. Martinez-Marrades, V. Brasiliense, D. Koshelev, M. Besbes, R. Kuszelewicz, C. Combellas, G. Tessier and F. Kanoufi, Nano Lett., 2015, 15, 6454–6463 CrossRef CAS PubMed.
- T. R. Bartlett, S. V. Sokolov and R. G. Compton, ChemistryOpen, 2015, 4, 600–605 CrossRef CAS PubMed.
- J. Ustarroz, M. Kang, E. Bullions and P. R. Unwin, Chem. Sci., 2017, 105, 1025–1102 Search PubMed.
- W. Ma, H. Ma, J.-F. Chen, Y.-Y. Peng, Z.-Y. Yang, H.-F. Wang, Y.-L. Ying, H. Tian and Y.-T. Long, Chem. Sci., 2017, 129, 9610–9612 Search PubMed.
- S. M. Oja, D. A. Robinson, N. J. Vitti, M. A. Edwards, Y. Liu, H. S. White and B. Zhang, J. Am. Chem. Soc., 2017, 139, 708–718 CrossRef CAS PubMed.
- S. Trasatti and E. Lust, in Modern Aspects of Electrochemistry No. 33, ed. R. E. White, J. O. Bockris and B. E. Conway, Kluwer Academic Publishers, New York, 1999, pp. 1–215 Search PubMed.
- S. Trasatti, J. Electroanal. Chem., 1971, 33, 351–378 CrossRef CAS.
- P. Peljo, J. A. Manzanares and H. H. Girault, Langmuir, 2016, 32, 5765–5775 CrossRef CAS PubMed.
- N. Holmberg, K. Laasonen and P. Peljo, Phys. Chem. Chem. Phys., 2016, 18, 2924–2931 RSC.
- H. H. Girault, Analytical and Physical Electrochemistry, EPFL Press, Lausanne, 2004 Search PubMed.
- H. Ohshima, T. W. Healy and L. R. White, J. Colloid Interface Sci., 1982, 90, 17–26 CrossRef CAS.
- J. A. Stratton, Electromagnetic Theory, McGraw-Hill Book Company, Inc., New York, 1941 Search PubMed.
- S. Bhattacharjee and M. Elimelech, J. Colloid Interface Sci., 1997, 193, 273–285 CrossRef CAS PubMed.
- B. V. Derjaguin, Discuss. Faraday Soc., 1954, 18, 85 RSC.
- R. Hogg, T. W. Healy and D. W. Fuerstenau, Trans. Faraday Soc., 1966, 62, 1638–1651 RSC.
- J. Cervera, P. Ramírez, J. A. Manzanares and S. Mafé, Microfluid. Nanofluid., 2009, 9, 41–53 CrossRef.
- J. Cervera, V. García-Morales and J. Pellicer, J. Phys. Chem. B, 2003, 107, 8300–8309 CrossRef CAS.
- J. Cervera, J. Memb. Sci., 2001, 191, 179–187 CrossRef CAS.
- J. Cervera, J. A. Manzanares and S. Mafé, Phys. Chem. Chem. Phys., 2001, 3, 2493–2496 RSC.
- M. Z. Bazant, M. S. Kilic, B. D. Storey and A. Ajdari, Adv. Colloid Interface Sci., 2009, 152, 48–88 CrossRef CAS PubMed.
- F. Booth, J. Chem. Phys., 1951, 19, 391–394 CrossRef CAS.
- F. Booth, J. Chem. Phys., 1955, 23, 453–457 CrossRef CAS.
- K. Lebedev, S. Mafé, A. Alcaraz and P. Ramírez, Chem. Phys. Lett., 2000, 326, 87–92 CrossRef CAS.
- M. Aguilella-Arzo, A. Andrio, V. M. Aguilella and A. Alcaraz, Phys. Chem. Chem. Phys., 2009, 11, 358–365 RSC.
- E. R. Nightingale, J. Phys. Chem., 1959, 63, 1381–1387 CrossRef CAS.
- J. G. Simmons, J. Appl. Phys., 1963, 34, 1793–1803 CrossRef.
- M. D. Scanlon, P. Peljo, M. A. Méndez, E. Smirnov and H. H. Girault, Chem. Sci., 2015, 6, 2705–2720 RSC.
- D. C. Grahame, J. Am. Chem. Soc., 1954, 76, 4819–4823 CrossRef CAS.
- H. R. Zebardast, S. Rogak and E. Asselin, J. Electroanal. Chem., 2014, 724, 36–42 CrossRef CAS.
- W. B. Russel, D. A. Saville and W. R. Schowalter, Colloidal Dispersions, Cambridge University Press, Cambridge, 1989 Search PubMed.
- J. H. Masliyah and S. Bhattacharjee, Electrokinetic and Colloid Transport Phenomena, John Wiley & Sons, Inc., Hoboken, NJ, USA, 2006 Search PubMed.
- J. Lekner, Proc. R. Soc. A, 2012, 468, 2829–2848 CrossRef.
- J. Lekner, Am. J. Phys., 2016, 84, 474–477 CrossRef.

## Footnote |

† Electronic supplementary information (ESI) available: Finite element model description, justification of model assumptions, schematic descriptions of the Fermi level changes upon NP collision with an electrode, the simulations of double layer capacitance of an electrode calculated with different double layer models, 3D plots corresponding to the contour plots of Fig. 1, the effect on the electrolyte concentration and NP radius on the potential distribution, surface charge density plots on the electrode and NP and their comparison with those calculated analytically in the absence of electrolyte, details of the calculations of the interaction between dissimilar parallel plates, and a further discussion of the approximate analytical expressions for the force between NP and electrode in the presence of electrolyte. See DOI: 10.1039/c7sc00848a |

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