Guillermo Iván
Guerrero-García
*^{a},
Enrique
González-Tovar
^{b},
Martín
Chávez-Páez
^{b},
Jacek
Kłos
^{c} and
Stanisław
Lamperski
^{c}
^{a}CONACYT – Instituto de Física de la Universidad Autónoma de San Luis Potosí, Álvaro Obregón 64, 78000 San Luis Potosí, San Luis Potosí, Mexico. E-mail: givan@ifisica.uaslp.mx
^{b}Instituto de Física de la Universidad Autónoma de San Luis Potosí, Álvaro Obregón 64, 78000 San Luis Potosí, San Luis Potosí, Mexico
^{c}Faculty of Chemistry, Adam Mickiewicz University in Poznań, Umultowska 89b, 61-614 Poznań, Poland

Received
10th August 2017
, Accepted 21st November 2017

First published on 5th December 2017

The spatial extension of the ionic cloud neutralizing a charged colloid or an electrode is usually characterized by the Debye length associated with the supporting charged fluid in the bulk. This spatial length arises naturally in the linear Poisson–Boltzmann theory of point charges, which is the cornerstone of the widely used Derjaguin–Landau–Verwey–Overbeek formalism describing the colloidal stability of electrified macroparticles. By definition, the Debye length is independent of important physical features of charged solutions such as the colloidal charge, electrostatic ion correlations, ionic excluded volume effects, or specific short-range interactions, just to mention a few. In order to include consistently these features to describe more accurately the thickness of the electrical double layer of an inhomogeneous charged fluid in planar geometry, we propose here the use of the capacitive compactness concept as a generalization of the compactness of the spherical electrical double layer around a small macroion (González-Tovar et al., J. Chem. Phys. 2004, 120, 9782). To exemplify the usefulness of the capacitive compactness to characterize strongly coupled charged fluids in external electric fields, we use integral equations theory and Monte Carlo simulations to analyze the electrical properties of a model molten salt near a planar electrode. In particular, we study the electrode's charge neutralization, and the maximum inversion of the net charge per unit area of the electrode-molten salt system as a function of the ionic concentration, and the electrode's charge. The behaviour of the associated capacitive compactness is interpreted in terms of the charge neutralization capacity of the highly correlated charged fluid, which evidences a shrinking/expansion of the electrical double layer at a microscopic level. The capacitive compactness and its first two derivatives are expressed in terms of experimentally measurable macroscopic properties such as the differential and integral capacity, the electrode's surface charge density, and the mean electrostatic potential at the electrode's surface.

According to the linearized Poisson–Boltzmann theory of point ions, the spatial extension of the electrical double layer neutralizing an infinite planar electrode can be characterized by a single parameter, namely, the Debye length of the surrounding charged fluid in the bulk.^{16,17} In spite of the wide use of the Debye length as a practical measure to characterize the thickness of the electrical double layer (due probably to its simplicity), this iconic length does not take into account important characteristics of ionic fluids such as electrostatic ion correlations, ionic excluded volume, image charges, short-range van der Waals specific interactions, and colloidal charge.^{18–48} Thus, in order to incorporate several of these features, some authors have proposed mean field approaches including the influence of the colloidal charge and the excluded volume effects of ions.^{49} In this kind of study, the extension of the electrical double layer associated with a binary electrolyte in the presence of a planar electrode is characterized by an “effective thickness” x_{1/2}. The thickness x_{1/2} is defined therein as the distance from the electrode's surface at which the concentration of excess counterions drops to half of its value regarding its concentration at the electrode's surface. However, even if the excluded volume of ions and solvent molecules is taken into account via a three-dimensional cubic lattice using statistical mechanics methods, electrostatic ion correlations are missing in this approach. Another possibility to characterize the spatial extension of the electrical double layer could be via the effective charge of a colloid. If the colloidal charge and the diffuse charge of ions are integrated up to a distance at which this quantity is equal to the effective charge of the colloid, this distance could be used, in principle, as a measure of the thickness of the electrical double layer. However, it is worth mentioning two important limitations if this approach is used. In the literature, there is not a unique manner to define or calculate the effective charge of a colloid. For instance, the effective charge of a spherical macroion immersed in an electrolyte could be calculated as the net charge (of the macroion plus the ionic species) up to a radial distance at which either (i) the electrostatic energy of counterions is equal to the thermal energy of the electrolytic bath or (ii) the counterions' concentration is equal to the average ionic concentration in the bulk.^{50,51} In addition, there are other criteria to calculate the effective colloidal charge of macroions in salt-free and added salt systems, such as the Alexander prescription^{52} or the recently proposed Extrapolated Point Charge method,^{53} just to mention a few. Thus, such a definition of the spatial extension of the electrical double layer would be dependent on the criterion used to calculate the effective colloidal charge. Another more critical problem would arise if the profiles of counterions are nonmonotonic: in such a scenario, it would be possible to observe several distances at which the electrostatic energy of counterions is equal to the thermal energy or at which the counterions' concentration might be equal to the average concentration of counterions in bulk at high ionic volume fractions.

In the present study, we would like to follow a different path to characterize the spatial extension of the electrical double layer. Specifically, we generalize here a previous proposal to quantify the spatial extension, or thickness, of the ionic cloud surrounding a small spherical macroion^{54} to study now the behaviour of the electrical double layer around an infinite planar electrode. Specifically, we propose here that the role of the Debye length, as a measure to characterize the thickness of the electrical double layer, be replaced by the corresponding capacitive compactness of the associated ionic fluid. The capacitive compactness has the advantage of being able to take into account consistently important characteristics of charged fluids such as electrostatic ion correlations, ionic excluded volume, image charges, and short-range van der Waals specific interactions, as well as the surface charge density of the solute. Even though our approach is general, we would like to study here the capacitive compactness of strongly coupled charged fluids. In this sense, molten salts are constituted by ionic particles with high electrostatic correlations in the liquid phase. Molten salts are relevant for several technological applications including electricity storage devices,^{55} coolants in nuclear reactors,^{56} or pyrochemical treatment of nuclear waste.^{57}

Thus, we have chosen to analyze the electric properties of a divalent model molten salt near a planar charged electrode via Monte Carlo simulations and integral equations theory, in an approach that goes well beyond the classical Poisson–Boltzmann picture. Our main aim is to demonstrate the advantages of using the capacitive compactness to characterize the spatial extension of the ionic cloud neutralizing a charged electrode, instead of the widely used Debye length of the supporting bulk ionic fluid. In particular, in this work we relate the behaviour of the integrated surface charge density (or net charge per unit area) of the electrode-molten salt system to the behaviour of the capacitive compactness, under several conditions of surface charge density and ionic concentration. We also establish here explicit connections between the capacitive compactness and its first two derivatives, and physically measurable quantities such as the differential and integral capacities, the colloidal charge density, and the mean electrostatic potential at the electrode's surface. On the other hand, counter-intuitive phenomena such as the inversion of the integrated charge in regions close to the colloidal surface can be observed under strong electrostatic coupling conditions. In this regard, the appearance and the behaviour of the maximum inversion of the integrated surface charge density are investigated here in different regimes.

The structure of the present study is as follows. First, the derivation of the capacity compactness, and the relationship of its first two derivatives with the integral and differential capacities in planar geometry are presented. With the purpose of illustrating the adequacy of the capacity compactness to characterize the thickness of a strongly charged fluid, a model system constituted by a divalent molten salt near an electrified planar electrode is introduced. A brief description of the Monte Carlo simulations and the integral equations theory used in this work is also provided. Then, the accuracy of our integral equations scheme is tested against simulation data collating (i) microscopic properties such as the ionic profiles and mean electrostatic potential close to the electrode's surface, and (ii) macroscopic properties of the electrical double layer such as the differential capacity and the capacitive compactness. Afterwards, the behaviour of the capacitive compactness is analyzed as a function of the microscopic ionic structure via the integrated surface charge density, and the net ionic volume charge density. The appearance and behaviour of the maximum inversion of the integrated surface charge density are analyzed as a function of the electrode's charge and the ionic concentration of the molten salt, to finish with some concluding remarks.

(1) |

(2) |

(3) |

Naturally, the compactness associated with an infinite charged electrode can be obtained from τ^{sphere}_{c} in the limit case of a spherical macroion with infinite radius. However, we would like to propose here an alternative and more simple derivation of the compactness in planar geometry. This will be done by using the concept of a parallel plate capacitor that is equivalent to a single charged electrode and its associated electrical double layer.

Let us start by considering an infinite electrode immersed in a binary charged fluid with valences z_{+} and z_{−}, bulk concentrations ρ^{bulk}_{+} and ρ^{bulk}_{−}, temperature T, and dielectric constant ε. According to the bulk electroneutrality condition, the charged fluid satisfies the condition z_{+}ρ^{bulk}_{+} + z_{−}ρ^{bulk}_{−} = 0 in bulk. Formally, the inhomogeneous ionic profiles around the charged electrode can be written as

ρ_{i}(x) = ρ^{bulk}_{i}g_{i}(x) = ρ^{bulk}_{i}exp^{−Wi(x)/kBT}, | (4) |

In order to pose our alternative derivation for the capacitive compactness τ_{c} in planar geometry, let us consider for a moment a pair of infinite parallel electrodes separated by a distance τ_{c} and with surface charge densities σ_{0} and −σ_{0}, which are immersed in a continuous solvent with dielectric constant ε (in the absence of charged particles) as it is shown in Fig. 1(a). The difference in the mean electrostatic potential between both electrodes can be written as:

(5) |

(6) |

Notice that eqn (6) is a general result for a charged fluid in the presence of an infinite planar electrode. This can be seen directly starting from the definition of the mean electrostatic potential at the electrode's surface

(7) |

(8) |

(9) |

From the definition of the integral capacitance of a charged electrode

(10) |

(11) |

On the other side, let us ponder now the special case in which there is a common closest approach distance between all charged particles and the planar electrode's surface, as occurs in the restricted primitive model where all ionic species are equally sized. In this instance, the capacitive compactness and the integral capacity can be written as:

(12) |

(13) |

As an important illustration of the use of eqn (9), let us consider the case of the linearized Poisson–Boltzmann theory. In this mean field approach, the capacitive compactness reduces to the Debye length of the supporting charged fluid in bulk. This result can be easily obtained as follows. First, let us approximate the ionic potential of mean force by the corresponding electrostatic energy, W_{i}(x) ≈ z_{i}eψ(x), in eqn (4). If these ionic profiles are substituted in the Poisson equation, ψ′′(x) = −ρ_{el}(x)/(ε_{0}ε), one obtains the non-linear Poisson–Boltzmann equation:

(14) |

(15) |

(16) |

σ_{0} = ε_{0}εκ_{D}ψ_{0}. | (17) |

On the other hand, from the fundamental definition of the capacitive compactness (see eqn (6)), it is possible to write, in general, the differential capacity as

(18) |

C_{diff}^{−1} = C_{Δ}^{−1} + C_{int}^{−1}, | (19) |

(20) |

In general, τ_{c} depends on the colloidal surface charge density σ_{0}. However, if τ_{c} is independent of σ_{0} then and C_{diff} = C_{int}, which occurs precisely in the linearized Poisson–Boltzmann theory. Such an approximation is valid only in the limit of very dilute electrolytes with low electrostatic coupling and in the presence of very weakly charged colloids or electrodes.

On the other side, if the differential and integral capacities are known from experiments, simulations, or theory, the first derivative of the capacitive compactness can be calculated as

(21) |

Note that eqn (21) is trivially fulfilled in the linearized Poisson–Boltzmann theory given that C_{diff} = C_{int} in such a theoretical approach.

In addition, the second derivative of the capacitive compactness can be written as

(22) |

(23) |

The pair interaction potential between any pair of ionic particles, used in simulations and theory, is given by:

(24) |

The interaction potential between a hard ionic particle of type i and the infinite and impenetrable charged electrode is given by:

(25) |

(26) |

(27) |

(28) |

(29) |

(30) |

In terms of the total correlation functions, eqn (27) can be written as

(31) |

These equations are a complete set of integral equations that are solved numerically via an efficient finite element method.^{68}

Once the ionic singlet profiles g_{i}(x) have been determined, from theory or simulation, it is possible to calculate the mean electrostatic potential and the integrated surface charge density as a function of the distance to the electrode's surface, respectively, as:

(32) |

(33) |

The bare surface charge density of the electrode can also be calculated in terms of the total ionic adsorption of each ionic species, Γ_{i}, as

(34) |

(35) |

(36) |

(37) |

Based on the above, since the electric field is proportional to σ(x), the capacity compactness can also be recast as

(38) |

The adsorption of counterions theoretically predicted by the HNC/MSA ionic singlets is compared with the corresponding Monte Carlo results in Fig. 2. Three cases are displayed in this figure: (i) divalent counterions for σ_{0} > 0, (ii) monovalent counterions for σ_{0} < 0, and (iii) the electrode is uncharged. The magnitude of the electrode's surface charge density is the same when the electrode is charged (|σ_{0}| > 0), and the ionic strength is also constant in all cases. A sensible agreement between theory and simulation is observed in a wide region near the electrode's surface, where the height and location of the extrema are similar in both approaches. The largest difference is observed very near to the closest approach distance between the ionic species and the electrode, or Helmholtz plane, where the ionic contact values predicted by the HNC/MSA integral equations are larger than the corresponding Monte Carlo values. This behaviour is a well known limitation of the HNC/HNC and HNC/MSA closures that usually appears in the presence of strong attractive interactions. On the other hand, an analogous comparison to that displayed in Fig. 2 is now shown in Fig. 3 for divalent and monovalent coions. Here, we observe again a sensible agreement between theory and simulations for all regions except close to the Helmholtz plane.

Fig. 2 Singlet distribution functions of divalent, and monovalent counterions near a charged electrode: g_{−}(x) corresponds to an electrode's surface charge density σ_{0} = 0.5 C m^{−2} (upward red triangles and solid line), and g_{+}(x) is associated to an electrode's surface charge density σ_{0} = −0.5 C m^{−2} (downward green triangles and dashed line). For comparison, the singlet distribution g_{0}(x) corresponding to an uncharged electrode (blue circles and dot-dashed line) is presented. Solid symbols and lines correspond to Monte Carlo simulations^{61} and integral equations theory results (this work), respectively. The contact values are g^{MC}_{−}(a/2) = 21.1, g^{MC}_{+}(a/2) = 12.1, and g^{MC}_{0}(a/2) = 3.9 for Monte Carlo simulations, and g^{IE}_{−}(a/2) = 31.727, g^{IE}_{+}(a/2) = 18.065, and g^{IE}_{0}(a/2) = 8.587 for integral equations. In all cases the ionic concentration of anions is 5.79 M. |

Fig. 3 Singlet distribution functions of divalent, and monovalent coions near a charged electrode: g_{−}(x) corresponds to an electrode's surface charge density σ_{0} = −0.5 C m^{−2} (upward red triangles and solid line), and g_{+}(x) is associated to an electrode's surface charge density σ_{0} = 0.5 C m^{−2} (downward green triangles and dashed line). For comparison, the singlet distribution g_{0}(x) corresponding to an uncharged electrode (blue circles and dot-dashed line) is presented. Solid symbols and lines correspond to Monte Carlo simulations^{61} and integral equations theory results (this work), respectively. The contact values are g^{MC}_{−}(a/2) = 0.01, g^{MC}_{+}(a/2) = 1.0, and g^{MC}_{0}(a/2) = 3.9 for Monte Carlo simulations, and g^{IE}_{−}(a/2) = 0.087, g^{IE}_{+}(a/2) = 2.246, and g^{IE}_{0}(a/2) = 8.587 for integral equations. In all cases the ionic concentration of anions is 5.79 M. |

Electrical properties of a charged fluid such as the mean electrostatic potential or the integrated surface charge density (eqn (32) and (33)) are functionals (or spatial integrals) of the ionic density profiles. In Fig. 4, a comparison between the theoretical and simulation values of the mean electrostatic potential as a function of the distance to the electrode's surface is displayed for several surface charge densities. An excellent agreement is observed here at low surface charge densities, even though the theoretical description deteriorates at large electric fields. In spite of this limitation, we observe a semi-quantitative agreement between integral equations and Monte Carlo results, e.g., the theoretical positions of the maxima and minima are very close to those obtained via simulations, and we also observe both in theory and simulation that the magnitude of the mean electrostatic potential at the electrode's surface is larger in the presence of monovalent counterions (σ_{0} < 0) compared to divalent counterions (σ_{0} > 0) when the magnitude of the electrode's charge and the ionic concentration are large, at the same ionic strength. A more stringest test is the comparison of the differential capacity obtained via theory and simulation as a function of the electrodode's surface charge density and the ionic concentration of the molten salt. Such comparisons are displayed in Fig. 5, where a qualitative agreement between the HNC/MSA integral equations and Monte Carlo simulations is observed. The best agreement is seen at low ionic concentrations and close to the point of zero charge (that is, when the electrode is weakly charged). When the electrolyte concentration increases, integral equations theory is able to predict the bell-shape displayed by the Monte Carlo simulations. In this case, the agreement is only qualitative because the height of the maximum is overestimated and its location is shifted. At low ionic concentrations and low surface charge densities, integral equations theory reproduces qualitatively the U-shape differential capacity typically displayed by aqueous electrolytes very close to the point of zero charge. On the other hand, when the surface charge density increases, the simulation differential capacity increases until a maximum value or peak is reached. At higher surface charges, the differential capacity decreases and a double-humped camel shape can be observed according to molecular simulations and mechanical statistical theories.^{23,26,27} Notice that the criterion of inversion of the differential capacity concavity, as an indication of the local inversion of the electric field or the charge reversal, is proposed only for charge densities very close to the point of zero charge and not for the high colloidal charges at which the double-humped camel shape is observed. On the other hand, the HNC/MSA integral equations approach fails to predict the double-hump camel shape displayed by Monte Carlo simulations at low ionic concentrations, although this theory does show the single-hump maximum at high concentrations. This limitation has its origin very likely in the use of the Percus–Yevick direct correlation function for the hard sphere contribution. It is very well known that the pressure predicted by this approximation in a hard sphere system deviates from the Carnahan–Starling equation as a function of the volume fraction. We foresee that this limitation can be overcome by using a better hard sphere direct correlation function, or by using more sophisticated approaches such as the Modified Poisson–Boltzmann theory^{18–20,22,26} or improved versions of density functional theory.^{21,27}

Fig. 4 Mean electrostatic potential ψ(x) for several surface charge densities of the electrode. Solid symbols and lines correspond to Monte Carlo simulations^{61} and integral equations theory results (this work), respectively. In all cases the ionic concentration of anions is 5.79 M. |

Fig. 5 Differential capacitance, C_{diff}, of the electrical double layer of the molten salt as a function of the surface charge density, σ_{0}, for several concentrations of anions. Solid symbols and lines correspond to Monte Carlo simulations^{61} and integral equations theory results (this work), respectively. |

In order to interpret the behaviour of the capacitive compactness τ_{c} in terms of the microscopic ionic structure, the theoretical integrated surface charge density σ(x) is displayed in Fig. 7 as a function of the distance to the electrode's surface at a low salt concentration (0.4 M). In the presence of monovalent counterions (top panel), it is observed that the integrated surface charge density becomes more compact when the magnitude of the negative surface charge density of the electrode increases, that is, σ(x,σ_{0}′′)/σ_{0}′′ < σ(x,σ_{0}′)/σ_{0}′ if |σ_{0}′′| > |σ_{0}′| for all x. A similar behaviour is observed in the presence of divalent counterions (bottom panel). In addition, it is seen that the integrated surface charge density is more compact when counterions are divalent at the same ionic strength, that is, σ_{divalent}(x,σ_{0})/σ_{0} < σ_{monovalent}(x,σ_{0})/σ_{0} for all x and a given |σ_{0}|. On the other hand, an inversion of the integrated surface charge density is seen at large positive surface charge densities of the electrode in the presence of divalent counterions, that is, σ_{divalent}(x,σ_{0})/σ_{0} < 0 for sufficiently large charge densities on the electrode's surface. Let us define the maximum inversion of the integrated surface charge density as −σ*/σ_{0} ≡ −σ(x_{min},σ_{0})/σ_{0}, where σ(x_{min},σ_{0})/σ_{0} < 0 and σ(x_{min},σ_{0})/σ_{0} < σ(x,σ_{0})/σ_{0} for all x at a given value of σ_{0}. Accordingly, in Fig. 7(b) it is observed that the maximum inversion of the integrated surface charge density increases as a function of the electrode's charge. Moreover, notice that the inversion of the integrated surface charge density shown in Fig. 7(b) (for divalent counterions at the two highest electrode charges) is completely absent in Fig. 7(a) (for monovalent counterions at the same ionic strength when the electrode's charge has the same magnitude). Thus, the presence or absence of a local charge inversion near the electrode cannot be inferred a priori from the associated differential capacity or capacity compactness curves shown in Fig. 5 and 6, respectively, at low ionic concentrations.

The behaviour of the theoretical integrated surface charge density, σ(x), as a function of the electrode's charge at a high concentration of the molten salt (5.0 M) is displayed in Fig. 8. In the presence of monovalent counterions, a non-monotonic behaviour is observed as a function of the distance to the electrode's surface for all negative surface charge densities as shown in Fig. 8(a). Moreover, the distances at which extrema occur increase when the magnitude of the electrode's charge augments. This behaviour is consistent with the expansion or swelling of the electrical double layer suggested by the capacitive compactness in Fig. 6. In addition, notice that the magnitude of the maximum inversion of the integrated surface charge density decreases when the magnitude of the electrode's charge augments. On the other hand, let us consider the case in which counterions are divalent as shown in Fig. 8(b). In this instance, the integrated surface charge density is non-monotonic and the magnitude of the maximum inversion of the integrated surface charge density decreases when the magnitude of the electrode's charge augments, as occurred in the presence of monovalent counterions at the same ionic strength. However, the position of the maximum inversion of the integrated surface charge density remains approximately constant as a function of the magnitude of the electrode's charge in the presence of divalent counterions. This behaviour is consistent with the arrest, or absence of a noticeable shrinking or swelling of the electrical double layer, displayed by the capacitive compactness in Fig. 6 as a function of the magnitude of the electrode's charge. In addition, the presence of the local charge reversal in Fig. 8(a) and (b) suggests that the concavity's reversion of the differential capacity at high ionic concentrations could be used, in principle, to detect the occurrence of a local inversion of the integrated surface charge density around the point of zero charge.

Another interpretation of the shrinking, swelling, or arrest of the electrical double layer predicted by the capacitive compactness can be elucidated from the behaviour of the net ionic charge density near the electrode. Due to the electroneutrality condition, it is easy to verify that the net ionic charge density is proportional to the difference of ionic singlets g_{+}(x) − g_{−}(x). This last quantity is plotted in Fig. 9 under the same conditions used in Fig. 8. Let us define x′ as the closest distance to the electrode's surface at which the difference of the ionic singlets is zero, i.e., g_{+}(x′) − g_{−}(x′) = 0; and let us define x′′ as the distance at which the ionic singlet difference has its minimum value, that is, g_{+}(x′′) − g_{−}(x′′) < g_{+}(x) − g_{−}(x) for all x. In Fig. 9(a), it is observed that both distances, x′ and x′′, increase as a function of the magnitude of the electrode's charge density. This microscopic behaviour is consistent with the swelling of the electrical double layer displayed by the capacitive compactness in Fig. 6, and with the expansion of the integrated surface charge density observed in Fig. 8(a). The arrest, or absence of a significant shrinking or swelling of the electrical double layer when the magnitude of the electrode's charge augments, can be also observed in the presence of divalent counterions as a function of the ionic singlet difference in Fig. 9(b). In this figure, it is evident that the location of extrema and the crossings with zero of the ionic singlet differences remain approximately constant when the electrode's charge increases. As a result, the main effect of increasing the surface charge density of the electrode in the presence of divalent counterions at high salt concentrations is to augment the net ionic volume charge density locally in specific fixed regions without causing a noticeable change in the magnitude of τ_{c}. These observations are again consistent with the behaviour displayed by the capacitive compactness in Fig. 6, and with the microscopic picture described by the integrated surface charge density in Fig. 8(b).

Fig. 9 Difference between the reduced density profiles of cations and anions obtained via integral equations theory. The concentration of anions in all cases is 5.0 M. |

On the other side, one limitation of the differential capacity or the capacitive compactness is that they cannot provide information about the appearance of sign inversion of the integrated surface charge density or the electric field. Thus, in order to analyze this property as a function of the salt concentration and the electrode's charge density, the maximum inversion of the integrated surface charge density is plotted in Fig. 10 for several conditions. In this plot, it is observed that at the lowest salt concentration (0.4 M) there is no sign inversion of the integrated surface charge density in the presence of monovalent counterions (σ_{0} < 0). In contrast, a monotonic increase of the magnitude of the maximum inversion of the integrated surface charge density can be observed in the presence of divalent counterions (σ_{0} > 0) as a function of the electrode's charge at the same ionic concentration (0.4 M). When the salt concentration increases (to 1 M), we observe the appearance of an inversion of the integrated surface charge density very near the point of zero charge. The maximum inversion of the integrated surface charge density decreases when the magnitude of the surface charge density, or the electric field, increases in the presence of monovalent counterions (σ_{0} < 0). The opposite behaviour is observed in the presence of divalent counterions (σ_{0} > 0). When the ionic concentration further increases (to 2 M and higher concentrations), the maximum inversion of the integrated surface charge density displays a maximum value in the presence of divalent counterions (at some positive surface charge density on the electrode). Note that the magnitude of the maximum inversion of the integrated surface charge density displayed in Fig. 10 is larger in the presence of divalent counterions (σ_{0} > 0) compared to monovalent counterions (σ_{0} < 0) for the same magnitude of the electrode's charge at a given ionic strength. Moreover, notice that the local inversion of the integrated surface charge density around the point of zero charge at high ionic concentrations displayed in Fig. 10 coincides with the concavity reversion of the differential capacity around the point of zero charge displayed in Fig. 5. This suggests that a change of concavity in the differential capacity around the point of zero charge as a function of the ionic concentration, from a U-shape to a bell-shape, could be used to detect the appearance of a local inversion of the integrated surface charge density of a charged colloid or electrode.

Fig. 10 Maximum integrated surface charge density obtained via the integral equations theory as a function of the colloidal surface charge density, σ_{0}, and the concentration of anions. |

To finish this section, we would like to discuss some possible applications of the first two derivatives of the capacity compactness. In this sense, if the integral capacity is known experimentally as a function of the electrode's charge then, in principle, the first and second derivatives of the capacity compactness can be calculated numerically. On the other hand, if the differential and integral capacities are known experimentally, they have to fulfill eqn (21) and (23). Thus, these equations can be used to determine the quality and/or uncertainty of both capacities measured experimentally. Besides, if only the integral capacity is known, eqn (21) can be used to calculate the differential capacity of the system. By way of example, the typical behaviour of the first two derivatives of the theoretical capacity compactness at an ionic concentration of 0.4 M is displayed in Fig. 11 as a function of the charge density on the electrode's surface.

One limitation of the capacitive compactness is that this quantity is not able to provide precise information about the adsorption of ionic charge or the appearance of a local inversion of the integrated charge (or the electric field). However, notice that this limitation is also shared by the differential capacity that has a U-shape, that is, we have shown here that the numerical values of the differential capacity in these circumstances cannot be used to determine a priori the presence of a local inversion of the integrated charge near the electrode's surface. Contrastingly, we have observed that an inversion of the curvature or concavity of the differential capacity as a function of the electrode's charge, from a U-shape to a bell-shape, seems to be indeed related to the occurrence of an inversion of the integrated surface charge density or the electric field around the point of zero charge. This behaviour has been suggested in a previous study of an electrified nitrobenzene/water interface, in which a monovalent organic salt was present in the oil phase and a divalent inorganic salt was present in the aqueous phase at high electrolyte concentrations.^{62}

Experimental measurements of the capacity compactness and its derivatives are crucial to validate several of the predictions performed in this study. In order to calculate these quantities, it is necessary to determine the colloidal surface charge density, the differential capacity, and the mean electrostatic potential at the electrode's surface. The colloidal surface charge density can be determined in experiments via chemical titration. The differential capacity can be measured experimentally via impedance techniques. In contrast, the mean electrostatic potential at the electrode's surface poses important technical difficulties, which were thought impossible to overcome.^{73,74} Fortunately, it has been shown very recently that it is possible to measure experimentally the mean electrostatic potential at the surface of charged colloids directly by using X-ray photoelectron spectroscopy.^{75} The application of this kind of measurement is very promising to study fine details of the electrical double layer of multivalent aqueous electrolytes, molten salts, or ionic liquids near charged interfaces. Therefore, the usage of the above experimental techniques could be very appropriate to validate the physical reality and usefulness of the capacity compactness and its derivatives in colloidal systems. These novel concepts, introduced here, constitute a simple and robust set of tools that allow us to characterize more accurately strongly correlated charged fluids in external electric fields.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7cp05433e |

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