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Maximilian
Mußotter
*^{ab},
Markus
Bier
*^{ab} and
S.
Dietrich
^{ab}
^{a}Max Planck Institute for Intelligent Systems, Heisenbergstr. 3, 70569 Stuttgart, Germany. E-mail: mussotter@is.mpg.de; bier@is.mpg.de
^{b}IV^{th} Institute for Theoretical Physics, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

Received
9th March 2018
, Accepted 29th March 2018

First published on 25th April 2018

The influence of a chemically or electrically heterogeneous distribution of interaction sites at a planar substrate on the number density of an adjacent fluid is studied by means of classical density functional theory (DFT). In the case of electrolyte solutions the effect of this heterogeneity is particularly long ranged, because the corresponding relevant length scale is set by the Debye length which is large compared to molecular sizes. The DFT used here takes the solvent particles explicitly into account and thus captures phenomena, inter alia, due to ion–solvent coupling. The present approach provides closed analytic expressions describing the influence of chemically and electrically nonuniform walls. The analysis of isolated δ-like interactions, isolated interaction patches, and hexagonal periodic distributions of interaction sites reveals a sensitive dependence of the fluid density profiles on the type of the interaction, as well as on the size and the lateral distribution of the interaction sites.

In recent years considerable theoretical interest has emerged in the effective interaction between two heterogeneously charged walls (which typically are the surfaces of colloidal particles) mediated by an electrolyte solution.^{16–26} In contrast to uniform substrates, this effective interaction can lead to lateral forces, in addition to the common ones in normal direction. However, all the studies cited above model the solvent of the electrolyte solution as a structureless dielectric continuum. This approach precludes coupling effects due to a competition between the solvation and the electrostatic interaction, which are known to occur in bulk electrolyte solutions.^{27–29} In particular, in the presence of ion–solvent coupling and far away from critical points, correlations of the solvent number densities in a dilute electrolyte solution decay asymptotically on the scale of the Debye length. Consequently, under such conditions, nonuniformities of the nonelectrostatic solvent–wall interaction can influence the structure of an electrolyte solution close to a wall and hence the strength and range of the effective interaction between two parallel plates immersed in an electrolyte medium on a length scale much larger than the molecular size. This mechanism differs from the one studied in ref. 16, 20, 22 and 26, in which the walls are locally charged but overall charge neutral.

In the present analysis a first step is taken towards a description of the structure of electrolyte solutions close to chemically and electrically nonuniform walls in terms of all fluid components. The natural framework for obtaining the fluid structure in terms of number density profiles of solvent and ion species is classical density functional theory.^{30–32} Here, the most simple case of an electrolyte solution, composed of a single solvent species and a single univalent salt component, is considered far away from bulk or wetting phase transitions. Moreover, the spatial distribution of nonuniformities of the chemical and electrostatic wall–fluid interactions can be arbitrary but their strengths are assumed to be sufficiently weak such that a linear response of the number density deviations from the bulk values is justified. This setup allows for closed analytic expressions which are used to obtain a first overview of the influence of ion–solvent coupling on the structure of electrolyte solutions in contact with chemically or electrically nonuniform walls. This insight will guide future investigations of more general setups within more sophisticated models.

After introducing the formalism in Section II, selected cases of heterogeneous walls are discussed in Section III. Due to the linear relationship between the wall nonuniformities and the corresponding number density deviations from the bulk values, the latter are given by linear combinations of elementary response features, which are discussed first. Next, two main cases are studied: wall heterogeneities, which are laterally isotropic around a certain center and wall heterogeneities, which possess the symmetry of a two-dimensional lattice; the study of randomly distributed nonuniformities^{16,18–21} is left to future research. For both cases various length scale regimes are discussed, which are provided by the bulk correlation length of the pure solvent, the Debye length, and a characteristic length scale associated with the wall nonuniformities. Conclusions and a summary are given in Section IV.

(1) |

(2) |

−ε_{0}ε∇^{2}Ψ(r, [ϱ]) = eZ·ϱ(r) | (3) |

(4) |

The Euler–Lagrange equations, corresponding to the minimum of the density functional specified in eqn (1)–(4), can be written as

(5) |

(6) |

(7) |

The linear nature of the Euler–Lagrange eqn (5) and (6) tells that the quadratic (Gaussian) approximation of the underlying density functional in eqn (1) and (2) corresponds to a linear response approach. It is widely assumed and in some cases it can be even quantified (see, e.g., the quantitative agreement between the full and the linearized Poisson–Boltzmann theory in the case that the surface charges are smaller than the saturation value^{8,34}) that for sufficiently weak wall–fluid interactions linear response theory provides quantitatively reliable results.

(8) |

(9) |

(10) |

(11) |

(12) |

(13) |

(14) |

(15) |

(16) |

In the bulk local charge neutrality holds, i.e., Z·ϱ_{b} = 0. Hence the equilibrium bulk state is determined by the temperature T, the number density ϱ_{1,b} of the solvent, and the bulk ionic strength I = ϱ_{2,b} = ϱ_{3,b}.

In order to obtain expressions for the parameter b and for the coupling matrix in terms of experimentally accessible quantities, in a first step the pure, ion-free solvent is considered, the particles of which interact only nonelectrostatically. Here this nonelectrostatic interaction between solvent particles at a distance r is modeled by a square-well pair potential U(r) as displayed in Fig. 1(a). At small distances r < σ a hard core repulsion prevents the overlap of two particles. At intermediate distances r ∈ (σ, σ_{pot}) two particles attract each other with an interaction energy −ε_{pot}, and at distances r > σ_{pot} the nonelectrostatic interaction vanishes. According to the scheme due to Barker and Henderson,^{35} the interaction potential U can be decomposed as U = U_{hc} + U_{pot} into the hard core repulsion U_{hc} and the attractive well U_{pot} (see Fig. 1(b)). The microscopic density functional Ω^{mic}_{1}[ϱ_{1}] for the pure solvent (species 1) in the bulk can be approximated by the expression

βΩ^{mic}_{1}[ϱ_{1}] = βΩ^{hc}_{1}[ϱ_{1}] + βF^{ex,pot}[ϱ_{1}]. | (17) |

(18) |

(19) |

Fig. 1 The nonelectrostatic interaction between fluid particles is modeled by a square-well pair potential U(r) displayed in panel (a), where r denotes the distance between the centers of two spherical particles. For r < σ a hard core repulsion prevents the overlap of two particles. For r ∈ (σ, σ_{pot}) two particles attract each other with a constant interaction energy −ε_{pot} < 0. At distances r > σ_{pot} there is no nonelectrostatic interaction. Panel (b) sketches the decomposition U = U_{hc} + U_{pot} of the nonelectrostatic interaction potential U according to the scheme due to Barker and Henderson into the hard core repulsion U_{hc} and the attractive well U_{pot}, which is used in Section IIIA in order to obtain the parameters entering the Cahn–Hilliard square-gradient density functional in eqn (1). |

Following Cahn and Hilliard,^{33}eqn (19) can be approximated by a gradient expansion:

(20) |

(21) |

(22) |

(23) |

(24) |

(25) |

(26) |

(27) |

(28) |

The analogue of eqn (23) for the nonelectrostatic contribution of all three particle species is given by the first line of eqn (1) with the local contribution (compare eqn (24))

(29) |

(30) |

(31) |

Accordingly, from eqn (24) one obtains the following equation of state of the pure solvent:

(32) |

(33) |

(34) |

In the case of a pure solvent (δϱ_{2} = δϱ_{3} = Ψ = 0), in the bulk the density two-point correlation function G(r_{1}, r_{2}) = Ḡ(r_{1} − r_{2}) fulfills an equation similar to eqn (5):

b∇^{2}Ḡ(r) = M_{11}Ḡ(r). | (35) |

(36) |

(37) |

For common specular X-ray reflectivity measurements, i.e., for q_{‖} = 0, the normalized intensity reflected as function of the normal wave number q_{z} is given by^{39,40}

(38) |

(39) |

(40) |

(41) |

h_{1}(r_{‖}) = h^{(0)}_{1}δ(r_{‖}), |

h_{2}(r_{‖}) = h_{3}(r_{‖}) = σ(r_{‖}) = 0. | (42) |

Fig. 2 Density distribution of the solvent (panel (a)) and of the ions (panel (b)) as function of the distance z from the wall and of the absolute value of the lateral Fourier wave vector q_{‖} in units of the inverse Debye length κ (see eqn (34)). The plane z = 0 is given by the positions of the fluid particle centers when the surface-to-surface distance between the hard wall and the hard particles vanishes. The data correspond to the boundary condition (see eqn (15) and (40)). The physical situation corresponding to this boundary condition is an attraction h_{1}(r_{‖}) = h^{(0)}_{1}δ(r_{‖}) (see eqn (42)) of the solvent particles by the wall located at the origin of the wall. Concerning the remaining relevant parameters see Section IIIA. |

Fig. 2 illustrates that for fixed |q_{‖}| the density deviations from the bulk value increase for smaller normal distances from the wall and, for fixed z, also upon decreasing the absolute value of the lateral wave vector |q_{‖}|. The behavior with respect to the normal distance from the wall can be anticipated, because the effect of the interaction between the wall and the fluid is expected to decrease with increasing distance from the wall. Moreover, also the behavior with respect to |q_{‖}| is as expected, because a strong attraction at the origin leads to a radially decreasing density deviation, which in Fourier space corresponds to a maximum at the origin. In order to allow for a quantitative analysis of the behavior of the density deviations, Fig. 3 shows various cuts through the data of Fig. 2 along several lines.

Fig. 3 Density profiles of the solvent (left column, panels (a), (c), and (e)) and of the ions (right column, panels (b), (d), and (f)) as functions of the normal distance z from the wall (top row, panels (a) and (b)), of the absolute value of the lateral wave vector |q_{‖}| (middle row, panels (c) and (d)), and of the wave number q_{z} in normal direction (bottom row, panels (e) and (f), with h^{(0)}_{1}κ^{3} being dimensionless) in corresponding units of the Debye length 1/κ and the inverse Debye length, respectively (see eqn (34)). Note that due to , in panels (e) and (f) the absolute values are shown. In each graph, there are three profiles shown corresponding to three values of the other relevant variable. Therefore the profiles correspond to cuts through Fig. 2(a) and (b) at various positions and in different directions. In this case the boundary condition is (see eqn (15) and (40)), corresponding to a δ-like nonelectrostatic attraction of the solvent particles at the origin of the wall (see Fig. 2 and eqn (42)). The graphs show, that the density deviations of the ions are proportional to the ones of the solvent, although different in sign. Since only the solvent particles are attracted by the wall, it is favorable for the system to increase their density close to the wall. However, due to the hard core nature of the particles and the equality of the interparticle attraction for all pairs of particles, the increase of solvent particles leads to an extrusion of ionic particles, leading to decreased ion densities at the wall. However, the density deviations of the ions are much weaker. For the remaining relevant parameters see Section IIIA. |

Fig. 3(a), (c) and (e) show the density profiles for the solvent and Fig. 3(b), (d) and (f) the ones of the positive ions, which in this case are the same as the profiles for the negative ions. This equivalence is due to the nature of the boundary conditions in this special case, which in real space lead primarily to an increased solvent density close to the origin at the wall. The ions, however, react only indirectly via the solvent, with which both ion types interact in the same way. Since the solvent particles get attracted by the wall, it is favorable to increase their density close to the wall. Due to the hard core nature of the particles, the space occupied by the solvent particles is blocked for the ions. Since the solvent is attracted by the wall and the interparticle attraction is the same for all pairs of particles, this leads to an extrusion of the ions in favor of an increased number of solvent particles. Fig. 3(a) and (b) show the density deviations as function of the normal distance z from the wall for three values of |q_{‖}|, i.e., Fig. 3(a) and (b) correspond to horizontal cuts through Fig. 2(a) and (b), respectively. For fixed |q_{‖}|, as in Fig. 2(a) and (b), Fig. 3(a) and (b) clearly show an exponential decay of the density deviation for increasing distances from the wall. In contrast, Fig. 3(c) and (d) show vertical cuts through Fig. 2(a) and (b), i.e., density profiles as functions of the absolute value of the lateral wave number |q_{‖}| for three normal distances z from the wall. The dependence of these profiles on the absolute value |q_{‖}| of the lateral wave vector q_{‖} implies a laterally isotropic decay of the density deviations in real space. The third pair of graphs, Fig. 3(e) and (f), shows the Fourier transforms of the density profiles of Fig. 3(a) and (b), being additionally Fourier-transformed with respect to the normal direction z, which leads to the Fourier transforms in terms of the lateral wave vector q_{‖} and the normal wave number q_{z}, respectively. All curves in Fig. 3(c)–(f) exhibit a Lorentzian shape as functions of |q_{‖}| and q_{z}, respectively. These Lorentzian curves in Fourier space correspond to exponential decays in real space in lateral or normal direction. The curves in Fig. 3(c) and (d) show widths of half height which decrease with increasing normal distance z, i.e., the lateral decay length in real space increases with increasing distance from the wall. This implies that the density distribution broadens upon moving away from the source of the perturbation. The curves in Fig. 3(e) and (f) exhibit widths of half height which increase with the lateral wave number |q_{‖}|, i.e., the normal decay length in real space decreases with increasing lateral wave number. Consequently, the range of influence of rapidly varying modes of wall heterogeneities onto the fluid is shorter than that of slowly varying modes. This relationship can also be inferred from Fig. 3(a) and (b). From the above discussions and from Fig. 3 one can conclude, that the response of all species to a simple attraction of nonelectrostatic type is the same up to a proportionality factor. This is confirmed by studying in addition the boundary conditions and ; these results are not shown here.

After having discussed the effects of the boundary condition ĥ ≠ 0 viaFig. 2 and 3, the following second type of boundary condition is analyzed:

ĥ(q_{‖}) = 0, |

(q_{‖}) = σ^{(0)}, i.e., σ(r_{‖}) = σ^{(0)}δ(r_{‖}), | (43) |

(44) |

(45) |

The three panels in Fig. 4 are obtained similarly as the ones in Fig. 3. Fig. 4(a) shows the density profiles as functions of the normal distance z from the wall for three values of the lateral wave number |q_{‖}|. Fig. 4(b) shows the same density profiles but as functions of |q_{‖}| for three distances z from the wall. Fig. 4(c) displays the double Fourier transform . Compared with the profiles in Fig. 3 for the previously discussed boundary conditions, all present profiles differ significantly from them. Fig. 4(a) reveals a much larger decay length in z-direction, i.e., normal to the wall. Also in Fourier space the decay in lateral direction occurs much more rapidly, i.e., on much longer length scales in real space than in the case of the nonelectrostatic wall–fluid interaction (cf.Fig. 3). This is also indicated by the much narrower peak in the double Fourier transform (see Fig. 4(c)). Furthermore, Fig. 4(a) shows a variation of the decay length in normal direction as function of |q_{‖}|. In Fig. 4(b) one observes that the lateral wave numbers |q_{‖}| at which the profiles decay to half of the maximum values decrease upon increasing the distances z from the wall, from which one infers that the lateral decay length in real space increases upon increasing z. The decay with respect to |q_{‖}| is much faster than in the previous case (compare Fig. 3(d)), indicating that in real space there is a slower decay in the lateral direction. Moreover, in Fig. 4(b) and (c) the functional form differs from the one shown in Fig. 2 and 3. These differences naturally occur due to the different form of the boundary condition. Since in the case of the boundary condition studied above (see Fig. 2 and 3) the relevant interaction is nonelectrostatic, the length scale dominating the decay is given by the corresponding short-ranged bulk correlation length ξ (see eqn (36)). In contrast, for the system shown in Fig. 4, due to the electrostatic nature of the corresponding interaction, the dominating length scale is the Debye length 1/κ (see eqn (34)), which is much larger than the correlation length ξ due to the nonelectrostatic interaction, giving rise to the much slower decay in Fig. 4(a) (on the scale of 1/κ) and the much faster decay in Fig. 4(b) and (c) (on the scale of κ).

Fig. 4 Density profiles of the ions as functions of the normal distance from the wall (a), of the absolute value of the lateral wave vector |q_{‖}| (b), and of the wave number in normal direction (c). Note that due to , in panel (c) the absolute value is shown. Each panel shows the profiles for three values of the other relevant variable. These profiles are cuts of the corresponding data (analogous to Fig. 2) along various directions. Here, the boundary condition is given by (see eqn (15) and (44)), which corresponds to a δ-like surface charge at the origin in real space (see eqn (43)). The profiles for the solvent are not shown, because the deviations linked to the two types of ions cancel out, , leaving the density of the solvent unchanged as if there were no ions. In comparison with Fig. 3, the profiles in (a) decay much slower on the scale of the Debye length 1/κ (see eqn (34)) instead of on the scale of the much shorter bulk correlation length ξ (see Fig. 3(b) and eqn (36)) due to the nonelectrostatic interaction. Accordingly, the profiles in (b) and (c) decay on the scale of κ more rapidly than their counterparts in Fig. 3(d) and (f). For the remaining relevant parameters see Section IIIA. |

Fig. 5 Physical configurations studied in Section IIID (a) and in Section IIIE (b). In Section IIID, a two-dimensional circular patch of radius R centered at the origin is studied (a). The dots in panel (b) correspond to the positions of the centers of the Gaussian interaction sites for the model used in Section IIIE, which form a two-dimensional hexagonal lattice with lattice constant Δ. The variance of the Gauss distributions is Δ^{2}_{peaks} (eqn (52) and (53)). Our results are based on the choice Δ = 5Δ_{peaks} (see, cf., Fig. 7). |

Due to the radial symmetry, the spatial structures in Fourier space depend only on the absolute value |q_{‖}| of the lateral wave vector q_{‖}. Fig. 6 discusses four distinct configurations.

Fig. 6 Density profiles of the solvent (panels (a) and (c)) and of the ions (panels (b) and (d)) as functions of the absolute value |q_{‖}| of the lateral wave vector for three normal distances z from the wall. The boundary condition corresponds to a circular interaction patch centered at the origin with radius R > 0. In the panels (a) and (b) the radius of the patch is R = 0.5/κ whereas in the panels (c) and (d) the radius is R = 2/κ, where 1/κ is the Debye length (see eqn (34)). All considered patch radii are much larger than the bulk correlation length, R ≫ ξ (see eqn (36)). In addition, there are different types of interaction. In panels (a) and (c) the interaction between the wall and the solvent particles is nonelectrostatic (h_{1}(r_{‖}) = ^{(0)}_{1}Θ(R − |r_{‖}|), h_{2}(r_{‖}) = h_{3}(r_{‖}) = σ(r_{‖}) = 0, see eqn (46) as well as Fig. 2 and 3), whereas in panels (b) and (d) the patch contains a constant surface charge and therefore interacts with the ions only (h(r_{‖}) = 0, σ(r_{‖}) = ^{(0)}Θ(R − |r_{‖}|), see eqn (49) and Fig. 4). Besides the profiles, all panels show also the lateral Fourier transform of the boundary condition (eqn (47) and (50)) displayed as a black solid line. In the case of the interaction of the wall with the solvent ((a) and (c)), the decay of the profiles as function of |q_{‖}| is proportional to the lateral Fourier transform of the boundary condition, which implies, that the density deviations in real space closely follow the shape of the patch. However, in the case of a charged patch at the surface the density distribution of the ions reflects the competition between the length scale R of the radius of the patch and the Debye length 1/κ. In the case of small patches (R < 1/κ, panel (b)), the Debye length dominates and therefore dictates the decay as function of |q_{‖}| without noticeable influence of the patch. In contrast, in the case of large patches (R > 1/κ, panel (d)), in which the radius of the patch is the dominating length scale, the shape of the profiles follows the Fourier transform of the charge distribution at the wall, i.e., the patch of radius R. For the remaining relevant parameters see Section IIIA. |

Alluding to the insights gained in the previous section, Fig. 6(a) and (c) correspond to a homogeneous circular patch of radius R, which interacts with the solvent only, similar to Fig. 2 and 3. This amounts to the boundary condition (see eqn (15))

h_{1}(r_{‖}) = ℏ^{(0)}_{1}Θ(R − |r|), |

h_{2}(r_{‖}) = h_{3}(r_{‖}) = (r_{‖}) = 0 | (46) |

(47) |

(48) |

h(r_{‖}) = 0, σ(r_{‖}) = ^{(0)}Θ(R − |r_{‖}|), | (49) |

(50) |

(51) |

Distinct from the previous examples in Sections IIIC and D, the interaction strength around the individual interaction sites r_{peaks} is taken to form a Gaussian distribution, providing either a nonelectrostatic or an electrostatic interaction with equal amplitudes for all interaction sites:

(52) |

(53) |

(54) |

(55) |

Fig. 7 Density profiles of the solvent (panels (a) and (c)) and of the ions (panels (b) and (d)) for three lateral wave vectors q_{‖} = (q_{x},q_{y}) as functions of the normal distance z from the wall in units of the Debye length 1/κ. The boundary condition corresponds to a hexagonal lattice of interaction sites with a Gaussian charge distribution characterized by a standard deviation Δ_{peaks} = Δ/5. The lattice constant is denoted as Δ (see Fig. 5). In panels (a) and (b) the lattice constant and the variance are Δ = 0.5/κ and Δ_{peaks} = 0.1/κ, respectively, whereas in panels (c) and (d) the lattice constant and the variance are Δ = 2/κ and Δ_{peaks} = 0.4/κ, respectively, with the Debye length 1/κ (see eqn (34)). Panels (a) and (c) correspond to systems with a nonelectrostatic interaction between the wall and the solvent particles (see eqn (56)), whereas panels (b) and (d) correspond to systems with electrostatic interaction sites between wall and ions (see eqn (57)). The insets in (b) and (d) show a magnified version of the respective profiles in the main plot. In all cases, the profiles decay exponentially upon increasing the normal distance z from the wall. However, the decay length differs significantly between the two aforementioned types of interactions. In the case of the nonelectrostatic interaction, the decay length is set by the bulk correlation length ξ (see eqn (36)) of the fluid, whereas in the case of the electrostatic interaction it is set by the much larger Debye length 1/κ. This difference in the decay lengths, both in lateral and in normal direction, which leads to a much faster lateral decay in the case of the electrostatic interactions, is also responsible for the decreasing amplitude of the ion profiles for increased wave vectors (panel (b) and (d)). Another significant difference between the two interaction types is the variation of the decay length as function of the lateral wave vectors. In panels (a) and (c) all profiles decay exponentially on the same decay length ξ, whereas in panels (b) and (d) the decay length depends significantly on the wave vectors. This effect follows from the dependence of the eigenvalues on |q_{‖}| as discussed in eqn (A3), corresponding to a lateral decay proportional to . In principle this occurs for both types of interactions. However only in the case of the electrostatic interactions it is relevant, which again is due to the difference between the dominating length scales. For the remaining relevant parameters see Section IIIA. |

The four panels are arranged as in Fig. 6, with the boundary conditions corresponding to an interaction between the wall and solvent only in Fig. 7(a) and (c),

(56) |

(57) |

In summary, the present study provides a flexible framework to determine the influence of various surface inhomogeneities on the density profiles of a fluid in contact with that substrate. The resulting profiles are found to be sensitive to the type of interaction as well as to the size and the distribution of the interaction sites.

This framework is considered as a starting point for extensions into various directions, aiming for the analysis of more sophisticated and realistic models. First, the model used here to describe the fluid is a very simple one, chosen to lay a foundation for further research and to introduce the approach as such. Concerning future work, more realistic descriptions of the fluid and more elaborate density functional descriptions could be used. For instance, the present restriction to low ionic strengths and equal particle sizes can be removed along the lines of ref. 42. Second, for the systems studied here, the fluid is thermodynamically far from any bulk or wetting phase transitions. This is solely done for the sake of simplicity. In future studies of more realistic systems, taking into account the occurrence of phase transitions and their influence on the systems is expected to be rewarding. Third, this study is restricted to linear response theory. Whereas this allows for a broad overview of structure formation in terms of superpositions of only a few elementary patterns, the occurrence of nonlinear structure formation phenomena requires approaches beyond linear response theory. Finally, studying the influence of disordered surface structures within the present framework appears to be very promising.

(A1) |

(A2) |

(A3) |

The expressions for s, t, u, and v can be obtained from the bulk quantities mentioned in Section IIIA and take on the forms (see eqn (25) and (30))

(A4) |

(A5) |

(A6) |

(A7) |

- V. S. Bagotsky, Fundamentals of Electrochemistry, Wiley, Hoboken, 2006 Search PubMed.
- W. Schmickler and E. Santos, Interfacial Electrochemistry, Springer, Berlin, 2010 Search PubMed.
- S. Dietrich, Wetting phenomena, in Phase Transitions and Critical Phenomena, ed. C. Domb and J. L. Lebowitz, Academic, London, 1988, vol. 12, p. 1 Search PubMed.
- M. Schick, Introduction to wetting phenomena, in Les Houches, Session XLVIII, 1988 – Liquides aux interfaces/liquids at interfaces, ed. J. Charvolin, J. F. Joanny and J. Zinn-Justin, North-Holland, Amsterdam, 1990, p. 415 Search PubMed.
- Protective Coatings, ed. M. Wen and K. Dušek, Springer, Cham, 2017 Search PubMed.
- N. Vogel, Surface Patterning with Colloidal Monolayers, Springer, Berlin, 2012 Search PubMed.
- A. Y. C. Nee, Handbook of Manufacturing Engineering and Technology, Springer, London, 2015 Search PubMed.
- W. Russel, D. Saville and W. Schowalter, Colloidal Dispersions, Cambridge University, Cambridge, 1989 Search PubMed.
- R. J. Hunter, Foundations of Colloid Science, Oxford University, Oxford, 2001 Search PubMed.
- Microfluidics, ed. B. Lin, Springer, Berlin, 2011 Search PubMed.
- F. J. Galindo-Rosales, Complex Fluids and Rheometry in Microfluidics, in Complex Fluid-Flows in Microfluidics, ed. F. J. Galindo-Rosales, Springer, Cham, 2018, p. 1 Search PubMed.
- D. Andelman, On the Adsorption of Polymer Solutions on Random Surfaces: The Annealed Case, Macromolecules, 1991, 24, 6040 CrossRef CAS.
- W. Chen, S. Tan, T.-K. Ng, W. T. Ford and P. Tong, Long-ranged attraction between charged polystyrene spheres at aqueous interfaces, Phys. Rev. Lett., 2005, 95, 218301 CrossRef PubMed.
- W. Chen, S. Tan, S. Huang, T.-K. Ng, W. T. Ford and P. Tong, Measured long-ranged attractive interaction between charged polystyrene latex spheres at a water-air interface, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2006, 74, 021406 CrossRef PubMed.
- W. Chen, S. Tan, Y. Zhou, T.-K. Ng, W. T. Ford and P. Tong, Attraction between weakly charged silica spheres at a water–air interface induced by surface-charge heterogeneity, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2009, 79, 041403 CrossRef PubMed.
- A. Naji, D. S. Dean, J. Sarabadani, R. R. Horgan and R. Podgornik, Fluctuation-induced interaction between randomly charged dielectrics, Phys. Rev. Lett., 2010, 104, 060601 CrossRef PubMed.
- D. Ben-Yaakov, D. Andelman and H. Diamant, Interaction between heterogeneously charged surfaces: surface patches and charge modulation, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2013, 87, 022402 CrossRef PubMed.
- A. Naji, M. Ghodrat, H. Komaie-Moghaddam and R. Podgornik, Asymmetric Coulomb fluids at randomly charged dielectric interfaces: anti-fragility, overcharging and charge inversion, J. Chem. Phys., 2014, 141, 174704 CrossRef PubMed.
- A. Bakhshandeh, A. P. dos Santos, A. Diehl and Y. Levin, Interaction between random heterogeneously charged surfaces in an electrolyte solution, J. Chem. Phys., 2015, 142, 194707 CrossRef PubMed.
- M. Ghodrat, A. Naji, H. Komale-Moghaddam and R. Podgornik, Ion-mediated interactions between net-neutral slabs: weak and strong disorder effects, J. Chem. Phys., 2015, 143, 234701 CrossRef PubMed.
- M. Ghodrat, A. Naji, H. Komaie-Moghaddama and R. Podgornik, Strong coupling electrostatics for randomly charged surfaces: antifragility and effective interactions, Soft Matter, 2015, 11, 3441 RSC.
- R. M. Adar and D. Andelman, Electrostatic attraction between overall neutral surfaces, Phys. Rev. E, 2016, 94, 022803 CrossRef PubMed.
- R. M. Adar and D. Andelman, Osmotic pressure between arbitrarily charged surfaces: a revisited approach, 2017, arXiv:1709.02114.
- R. M. Adar, D. Andelman and H. Diamant, Electrostatics of patchy surfaces, Adv. Colloid Interface Sci., 2017, 247, 198 CrossRef CAS PubMed.
- S. Ghosal and J. D. Sherwood, Screened Coulomb interactions with non-uniform surface charge, Proc. – R. Soc. Edinburgh, Sect. A: Math., 2017, 473, 20160906 CrossRef.
- S. Zhou, Effective electrostatic interactions between two overall neutral surfaces with quenched charge heterogeneity over atomic length scale, J. Stat. Phys., 2017, 169, 1019 CrossRef.
- A. Onuki and H. Kitamura, Solvation effects in near-critical binary mixtures, J. Chem. Phys., 2004, 121, 3143 CrossRef CAS PubMed.
- M. Bier, A. Gambassi and S. Dietrich, Local theory for ions in binary liquid mixtures, J. Chem. Phys., 2012, 137, 034504 CrossRef PubMed.
- M. Bier and L. Harnau, The structure of fluids with impurities, Z. Phys. Chem., 2012, 226, 807 CrossRef CAS.
- R. Evans, The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids, Adv. Phys., 1979, 28, 143 CrossRef CAS.
- R. Evans, Microscopic theories of simple fluids and their interfaces, in Les Houches, Session XLVIII, 1988—Liquides aux interfaces/Liquids at interfaces, ed. J. Charvolin, J. F. Joanny and J. Zinn-Justin, North-Holland, Amsterdam, 1990, p. 1 Search PubMed.
- R. Evans, Density functionals in the theory of nonuniform fluids, in Fundamentals of inhomogeneous fluids, ed. D. Henderson, Marcel Dekker, New York, 1992, p. 85 Search PubMed.
- J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 1958, 28, 258 CrossRef CAS.
- L. Bocquet, E. Trizac and M. Aubouy, Effective charge saturation in colloidal suspension, J. Chem. Phys., 2002, 117, 8138 CrossRef CAS.
- H. P. Hansen and I. R. McDonald, Theory of simple liquids, Academic, San Diego, 2nd edn, 1986 Search PubMed.
- S. J. Suresh and V. M. Naik, Hydrogen bond thermodynamic properties of water from dielectric constant data, J. Chem. Phys., 2000, 113, 9727 CrossRef CAS.
- D. R. Lide, Handbook of Chemistry and Physics, CRC, Boca Raton, 79th edn, 1998 Search PubMed.
- J. Y. Walz, Measuring particle interactions with total internal reflection microscopy, Curr. Opin. Colloid Interface Sci., 1997, 2, 600 CrossRef CAS.
- S. Dietrich and A. Haase, Scattering of X-rays and neutrons at interfaces, Phys. Rep., 1995, 260, 1 CrossRef CAS.
- J. Als-Nielsen and D. McMorrow, Elements of modern X-ray physics, Wiley, New York, 2001 Search PubMed.
- J. D. Jackson, Classical Electrodynamics, Wiley, Hoboken, 1999 Search PubMed.
- A. C. Maggs and R. Podgornik, General theory of asymmetric steric interactions in electrostatic double layers, Soft Matter, 2016, 12, 1219 RSC.

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