# The Effect of adsorption phenomena on the transport in complex electrolytes

Ioulia Chikina, Michel Beaughon, Pierre Burckel, Emmanuelle Dubois, Ivan Lucas, Sawako Nakamae, Ozlem Sel, Hubert Perrot, Régine Perzynski, Thomas Salez, et al.

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Ioulia Chikina, Michel Beaughon, Pierre Burckel, Emmanuelle Dubois, Ivan Lucas, et al. The Effect of adsorption phenomena on the transport in complex electrolytes. Colloids and Interfaces, 2025, 9 (4), pp.44. 10.3390/colloids9040044. http://dx.doi.org/10.3390/colloids9040046

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Review

# The Effect of Adsorption Phenomena on the Transport in Complex Electrolytes

Ioulia Chikina <fnref n="1" /> ID, Michel Beaughon <fnref n="2" /> ID, Pierre Burckel <fnref n="3" /> ID, Emmanuelle Dubois <fnref n="4" /> ID, Ivan T. Lucas <fnref n="5" /> ID, Sawako Nakamae <fnref n="2" /> ID, Ozlem Sel <fnref n="5" /> ID, Hubert Perrot <fnref n="5" /> ID, Régine Perzynski <fnref n="4" /> ID, Thomas J. Salez <fnref n="2" /> ID, Blanca E. Torres-Bautista <fnref n="2" /> ID and Andrey Varlamov <fnref n="6" />,<fnref n="7" />,* ID

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check for
updates

Academic Editor: Rajinder Pal

Received: 20 April 2025

Revised: 21 June 2025

Accepted: 24 June 2025

Published: 7 July 2025

**Citation:** Chikina, I.; Beaughon, M.; Burckel, P.; Dubois, E.; Lucas, I.T.; Nakamae, S.; Sel, O.; Perrot, H.; Perzynski, R.; Salez, T.J.; et al. The Effect of Adsorption Phenomena on the Transport in Complex Electrolytes. *Colloids Interfaces* **2025**, *9*, 44. https://doi.org/10.3390/colloids9040044

**Copyright:** © 2025 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
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(https://creativecommons.org/
licenses/by/4.0/).

Abstract

Over the last decade, numerous impedance studies of the conductivity of suspensions containing colloidal (dielectric, semiconducting or metallic) particles have often led to the conclusion that the well-known Maxwell theory is insufficient to quantitatively explain the properties of these systems. We review some of the most characteristic results and show how the applicability of the Maxwell's theory can be restored taking into account the adsorption phenomena occurring during AC impedance measurements in nanoparticle suspensions. The latter can drastically change the capacitance of the metal-electrolyte cell boundaries from the standard value, making it strongly dependent on the nanoparticle concentration. This factor significantly affects conductivity measurements through RC circuit characteristics. We present an analysis of available impedance measurement data of the dependence of conductivity on the nanoparticle concentration in this new paradigm. In order to emphasize the novelty and the acute sensitivity of ac-diagnosis to the presence of adsorption phenomena at the metal-electrolyte interface, direct adsorption determinations at such interfaces by using two modern experimental techniques are also presented. The main result of this work is the restoration of Maxwell's theory, attributing the observed discrepancies to variations in cell conductance.

Keywords: adsorption; colloids; nano-suspensions

1. Introduction

In the early 2000s, there was an increased interest in the transport properties of electrolyte-based suspensions and polyelectrolytes. Thanks to advances in experimental AC-diagnosis and imaging techniques, it became possible to study the complex interface and surface phenomena in detail (such as nanoparticle adsorption, double layer formation, dynamic charge effects, etc.) revealing a need for deeper investigation. It turned out that the introduction of small additives of nanoparticles of one type ot another (e.g., metals, dielectrics, semiconductors, etc.) into a solvent (water, alcohol, stabilizing electrolytes)

<footnote n="1">¹ LIONS, NIMBE, CEA, CNRS, Université Paris-Saclay, CEA Saclay, 91191 Gif-sur-Yvette, France; julia.chikina@cea.fr</footnote>

<footnote n="2">² Service de Physique de l'Etat Condensé, SPEC, CEA, CNRS, Université Paris-Saclay, CEA Saclay, 91191 Gif-sur-Yvette, France</footnote>

<footnote n="3">³ Institut de Physique du Globe de Paris, CNRS, UMR 7154, 1 rue Jussieu, 75005 Paris, France</footnote>

<footnote n="4">⁴ Laboratory PHENIX, Sorbonne Université, CNRS, UMR 8234, 4 Place Jussieu, 75005 Paris, France</footnote>

<footnote n="5">5 Laboratory LISE, Sorbonne Université, CNRS, UMR 8235, 75005 Paris, France</footnote>

<footnote n="6">⁶ Institute for Superconductors, Innovative Materials, and Devices (CNR-SPIN), Via Fosso del Cavaliere 100, 00133 Rome, Italy</footnote>

<footnote n="7">7 Lombard Institute “Academy of Sciences and Letters”, Via Borgo Nuovo 26, 20121 Milan, Italy</footnote>

<footnote>* Correspondence: andrey.varlamov@spin.cnr.it</footnote>
changes significantly, and in some cases even dramatically, the transport properties of suspensions (see reviews [1–5]).

When describing the conductivity of a suspension of various stabilized solutions, and in particular, the transport phenomena in dilute electrolytes containing nanoparticles (of different origins), classic Maxwell’s theory [6] is successfully used. It provides the formula for the conductivity $\sigma(\phi)$ of an inhomogeneous medium, consisting of a matrix with a conductivity $\sigma_0$, containing randomly located spherical inclusions with a conductivity of $\sigma_\odot$:

$$ \frac{\sigma(\phi)}{\sigma_0} = 1 + \frac{3(\sigma_\odot/\sigma_0 - 1)\phi}{\sigma_\odot/\sigma_0 + 2 - (\sigma_\odot/\sigma_0 - 1)\phi} \quad (1) $$

The dimensionless volumetric concentration of the colloidal suspension $\phi$ is defined as

$$ \phi = \frac{4\pi}{3} n_{\odot} (\lambda_0 + R_0)^3 \ll 1, \quad (2) $$

where $n_\odot$ is the number concentration of dissolved particles, $R_0$ the radius of a bare particle (metallic, dielectric, etc.) and $\lambda_0$ the electric field screening length. The latter is different depending on the type of the suspension. For a colloidal suspension of dielectric/metallic nanoparticles, it is the characteristic length of the occurrence of electrostatic image forces in the vicinity of the dielectric-electrolyte interface [7]. In the case of some colloidal nanoparticles, it is the Debye length for the surrounding electrolyte [8]. Both $R_0$ and $\lambda_0$ are assumed to be much larger than the interatomic distances.

At low particle concentrations Equation (1) is able to explain the linearity on $\phi$ of the corresponding contribution to the conductivity for different types of suspensions. The slope in the concentration dependence is limited from above by the factor 3 for highly conducting inclusions ($\sigma_\odot/\sigma_0 \gg 1$):

$$ \frac{\sigma(\phi)}{\sigma_0} \bigg|_{max} = 1 + 3\phi \quad (3) $$

and by factor $-3/2$ from below for dielectric inclusions ($\sigma_\odot/\sigma_0 \to 0$):

$$ \frac{\sigma(\phi)}{\sigma_0} \bigg|_{min} = 1 - 3\phi/2 \quad (4) $$

By treating the dependencies of $\sigma(\phi)$ confined within the sector between rays (3) and (4) as the norm, any substantial deviation from this behavior—provided concentrations remain within the previously stated assumptions — can be regarded as anomalous.

**Dielectric inclusions.** The results of measurements of the ionic conductivity of diamond nanofluid versus volume fraction of the inclusions are presented in Figure 1 [9]. The solid line is the fit to the experimental results (black dots) according to the formula $\sigma(\phi)/\sigma_0 = 1 + 3734\phi$. The red dashed line represents the prediction of the linear Maxwell theory $\sigma(\phi)/\sigma_0 = 1 - 3\phi/2$ (see Equation (4)). The striking divergence between the observed values of conductivity and its predictions based on immediate application of the Maxwell formula is obvious. Even the sign of the slope is opposite. The situation is similar for aqueous suspensions of the oxides: CuO [10] and Al₂O₃ [11].
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Ionic conductivity enhancement of diamond nanofluid at 298.15 K [9]. Black asterisks represent the result of measurements, red dashed line represents the prediction of the linear Maxwell theory $\sigma(\phi)/\sigma_0 = 1 - 3\phi/2$ (see Equation (4)). The solid line is the fitting of experimental results according to the formula $\sigma(\phi)/\sigma_0 = 1 + 3734\phi$. For better visibility, the relative positions of the solid line and the dashed line in the vicinity of the origin, indicated by a circle, is shown in the inset. The latter allows to recognize better the striking divergence between the observed values of conductivity and its predictions based on immediate application of the Maxwell formula.</figcaption></figure>

*Charged colloidal inclusions*. The same striking contrast between the experimental results and predictions of the Maxwell theory is observed in the studies in dilute colloidal electrolytes. For instance, the authors of Ref. [5] study the electrical transport in charged colloidal suspensions of iron oxide nanoparticles (maghemite) dispersed in an aqueous medium (see Figure 2). These studies report that the conductivity of the dilute suspensions grows linearly with an increase of colloidal particle concentration, yet again the slope of its dependence exceeds the Maxwell predictions (see Equations (3) and (4) and Figure 2) by ten-fold.

Application of Maxwell’s formula to dilute colloidal electrolytes requires caution. Being immersed (or synthesized within) in an electrolyte solution, the colloidal particles of a bare radius $R_0$, acquire surface ions (e.g., hydroxyl groups, citrate, etc. Refs. [12,13] resulting in a very large structural charge $eZ$ ($|Z| \gg 10$). Its sign can be both positive and negative, depending on the surface group type. The latter in return, attracts counterions from the surrounding solvent creating an electrostatic screening coat of the size of the Debye length $\lambda_0$ with an effective charge $-eZ$ (see Figure 3). In these conditions, the approaching nanoparticles within a distance $r \le 2\lambda_0$ begin to repel each other without flocculation [14–16]. The region of an essential interaction between them corresponds to the condition $\frac{4\pi}{3}n_{\odot}^{(c)}(\lambda_0 + R_0)^3 = \phi^{(c)} \sim 1$, where the superscript c indicates the critical value. To maintain a homogeneous electrolyte, the screening shells around the colloidal particle cores must not overlap. This requires keeping the concentration sufficiently low to satisfy the applicability condition of the Deryagin–Landau–Verwey–Overbeek (DLVO) theory, as defined by Equation (2) [14–16]. When Equation (1) is applied to a colloidal suspension, the dimensionless parameter of the theory $\phi \equiv \phi_{\odot} \ll 1$ has to and, hence, only the linear regime in the concentration dependence of conductivity turns out to be accessible.
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Relative variation of conductivity of an aqueous solution of maghemite colloidal particles as a function of their volume fraction [5]. The dashed line is linear fit of the data. As can be seen the observed slope turn out to be an order of magnitude larger than the values following from Maxwell's Formula (3).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 3.** The schematic presentation of the multiply-charged colloidal particle surrounded by the cloud of counter-ions: (a) in the bulk of electrolyte; (b) in the vicinity of electrolyte-metal interface. In panel (a) is presented the standard scheme of the colloid structure assumed in the DLVO theory. Its core of the radius $R_0$ carries the charge $eZ$, while $\lambda_0$ appears in the DLVO theory as the characteristic screening length. The “Stern layer” is the auxiliary characteristics of the screening coat which determined in the early theories [17,18] the size of the region of elevate electrostatic energy $e\varphi \gg k_B T$. The screening coat of the volume $\frac{4\pi}{3}[(R_0 + \lambda_0)^3 - R_0^3]$ possesses the opposite charge $-eZ$. Hence, the colloidal particle in whole in the DLVO model is neutral. In panel (b) acquisition by colloidal particle of the effective charge $eZ^*$ (see Equation (34)) which, together with its mirror image, creates the electric dipole. Note that the effective charge considered here is that of static colloidal particles, to be distinguished from the dynamic effective charge of moving particles discussed in the Review [1].</figcaption></figure>

**Metallic inclusions.** Analogous discrepancies with the Maxwell formula were found studying the conductivity of the suspensions containing metallic nanoparticles and sin-
gle wall nananotubes [19]. It is seen, that for the suspension containing nanotubes the linear growth of the conductivity is extended to the wide range up to 2% of volumetric concentration, yet the slope is again at least one order of magnitude larger than the Maxwell prediction.

The characteristic features of the experimental works [1,5], where the Maxwell theory was used for describing the obtained results, are the following:

* The validity of Equation (1) implicitly requires a uniformity of individual characteristics of inclusions (lack of dispersion in size, chemical composition, etc.). The methods of preparation of dilute charged-colloidal electrolytes containing maghemite nanoparticles in Refs. [1,5] allow to monitor the quality of *monodispersity* with good accuracy, and to control the fulfilment of the requirement $\phi_{\odot} \ll 1$. This is a qualitative difference of the studies [1,5] from the most of transport experiments with the suspensions prepared mechanically (see, for example, reviews [2,4]).

* In all experiments of Refs. [1,2,4,5] the linear change of conductivity versus relative volume occupied by the inclusions is observed in the region $\phi \ll 1$. This characteristic feature gives grounds to use the Maxwell theory in interpretation of the transport properties of various suspensions (see Figure 4 (taken from Ref. [3]) and the corresponding caption).

<figure><img src="image_7.png" /><figcaption>**Figure 4.** Comparative data on the conductivity of $\sigma(\phi)$ dilute liquid suspensions with different base liquids (water, alcohol) and nanoparticles: copper, its oxide, nanotube powder, aluminium oxide. In order of increasing linear slopes of $\sigma(\phi)$: red squares (CuO oxide in water [10]); blue triangles top-down (copper, Cu in water [10]); green triangles top-up (carbon nanotubes in ethylene glycol [19]); pink triangles top-side-down (Al₂O₃ oxide in water [11]). The figure is taken from Ref. [3] and we are grateful to Elsevier publishing company for permission to use it.</figcaption></figure>

Yet, it is necessary to attract attention to the fact, that in the most of such successful linear fits of the observed $\sigma(\phi)$ the slope exceeds dramatically that allowed by the Maxwell theory. Indeed, the maximal slope $\sigma_0^{-1} \frac{d\sigma(\phi)}{d\phi}$ of the linear approximation of Equation (1) is reached when $\sigma_{\odot} \gg \sigma_0$ and is equal to 3, while in the measurements over the years in mechanically prepared suspensions, this theoretical limit is exceeded by three orders of magnitude. For example, in the aqueous suspension of Al₂O₃ reported in Ref. [11], the slope value is as high as $3.68 \times 10^3$. There is an overwhelming inconsistency indicating the
inapplicability of Maxwell’s formalism to explain the citing experiments. There must be some other reasons which we will discuss below.

We begin by revisiting the fundamental principles of AC impedance diagnostics (Section 2), followed by an in-depth examination of how adsorption phenomena at the electrodes influence the electrolytic capacitance of the measuring cell (Section 3). Key topics include: the unique characteristics of ionic screening in dilute electrolytes, the influence of image forces on the formation of adsorption layers at the “metal–pure water” interface, and the interfacial structure between electrodes and either stabilized electrolytes or dielectric suspensions. Section 4 presents an analysis of impedance measurement results for various suspension types and addresses discrepancies with Maxwell’s theory. Section 5, which presents the current state of research on adsorption phenomena at metal–electrolyte interfaces, provides a clearer perspective on the novelty and significance of how adsorption processes manifest in the accurate application of AC diagnostics for analyzing the transport properties of dilute electrolytes. The review concludes with a discussion of experimental studies on aqueous dispersions of maghemite nanoparticles.

## 2. AC Impedance Diagnosis

To understand the large discrepancy between experimentally reported conductivity values from AC impedance measurements in nanoparticles suspensions and the predictions of Maxwell’s theory, we begin by examining the data processing methods used in these experiments.

The AC (alternative current) impedance diagnosis is regularly used in the study of transport phenomena in conductive media. A common reason for resorting to ac-complications rather than choosing relatively simpler possibilities in DC-mode (direct current) is the desire to exclude the influence on the I-V characteristics (current-volt) of the contact phenomena inherent in the dc measurements. In some cases (2d—electrons over helium), dc transport measurements are impossible by definition. In lightly doped semiconductors (analogous to dilute electrolytes), the situation is less critical, but problems with contact resistance persist (see [20–23]).

In the simplest case, impedance diagnosis deals with an RC circuit that obeys the equation [24]

$$R \frac{dQ(t)}{dt} + \frac{Q(t)}{C} = U(t), \qquad (5)$$

where $Q(t)$ is the value of the charge at the electrodes at the moment $t$ and $U(t) = U_0 \cos(\omega t)$ is the applied ac-voltage. Other parameters – the resistance of the electrolyte filling the volume between the electrodes of a flat capacitor $R = L/[\sigma(\phi_\odot)S]$, the distance between them $L$, the conductivity of the electrolyte $\sigma(\phi_\odot)$, the area of the electrodes $S$, the capacitance of a flat capacitor composed of plates $C$—are time independent. In electrical engineering, the resistor and the capacitor are usually considered as two separate elements of the circuit. Let us stress, that in the problem under consideration both these values characterize the same cell. The specifics of its capacitance $C$ is the essence of the present analysis and it will be discussed in details below.

Corresponding current $J(t)$ generated in the cell is frequency-dependent and equal to

$$J(t) = \frac{U_0 \cos(\omega t - \theta)}{\sqrt{R^2 + (\omega C)^{-2}}}, \qquad (6)$$

where $\omega$ is the frequency. AC lock-in amplifiers are capable of simultaneously measuring both real and imaginary components of the impedance providing the information on the phase shift $\theta$ between the current $J(t)$ and the voltage $U(t)$
$$ \tan \theta(\omega) = -1/(\omega RC). \qquad (7) $$

By measuring the dependence $\theta(\omega)$ one can extract the value of the product $RC$. Assuming that the capacitance $C$ is known, one can then determine the resistance $R$.

In “real” liquid electrolytes and in charged colloidal solutions, however, the measured complex impedance contains contributions such as the leakage resistance, slow diffusions (related to the micro-structure of the electrodes, and/or large ions and nanoparticles) in addition to the constant phase element, which refers to the ion motions within the Debye length (electronic double layer) at the electrode/liquid interface. Consequently, Equation (6) cannot fully describe such multi-electrochemical phenomena, and it is customary to use more phenomenological models such as the Havriliak-Negami model [25].

When the measurements are performed at so high frequencies that the phase shift becomes negligible, one can use an alternative way of the RC measurement. Let us follow the method of complex impedance and formally assume that the complex harmonic signal $\tilde{U}(t) = U_0 \exp(i\omega t)$ is applied to the cuvette filled by electrolyte in Equation (5). With such a “driving force” on the right-hand-side of the equation, one can easily see that after some relaxation period the stationary oscillations of the “complex charge” (which can be considered as the integral of complex current) $Q(t) = Q_0(\omega) \exp(i\omega t)$ will settle in the system. The amplitude $Q_0(\omega)$ itself is a complex function of frequency and the circuit properties, reflecting the existence of the charge oscillations retardation at the capacitor plates (cell electrodes) with respect to the voltage applied. The corresponding imaginary part

$$ \Im Q_0(\omega) = -C U_0 \frac{\omega \tau_0}{1 + \omega^2 \tau_0^2} \qquad (8) $$

with $\tau_0 = RC$ as the characterizing the inertial properties of the circuit time constant, demonstrates the maximum at the frequency $\omega_{max} = 1/RC$.

Here it is necessary to recall that we are interested in the dependence of the electrolyte resistance in function of the nanoparticles concentration in it. Hence, to be consistent in use of Equation (7), one also has to take into account the effect of the nanoparticle presence on the cell capacitance. When the cell filled with pure water functions as a capacitor, it exhibits characteristics well known in the radio engineering: the electric field penetrates between plates, only at the distances of the order of the Debye length $\lambda_0$, which is much less than the distance between the electrodes $L$ ($\lambda_0 \ll L$) and

$$ C_0 \approx \epsilon S / (8\pi\lambda_0), \qquad (9) $$

where $S$ is the capacitor plate area.

Hence, the effective capacitance turns out to be significantly greater (by $\sim L/\lambda_0$ times) than expected for a vacuum capacitor. When working with electrolytic suspensions, this fact requires special attention (see [26]). It is precisely the failure to take into account the dependence of the capacitance value on the properties of the inclusions contained in the suspension that leads to the apparent contradiction between conductivity measurements and Maxwell’s theory.

Applying Equation (5) to analysis of the electrolyte conductivity impedance measurements the authors usually make a series of assumptions.

* In the first term of Equation (5), Maxwell’s Formula (1) for the conductivity is used. The latter accounts for the current which occurs as the response to the electric field applied but ignores the diffusion contribution, related to the chemical potential gradient $\zeta(z)$ [24] (similar to the thermoelectric contribution, but proportional to $\nabla\zeta$ instead of $\nabla T$). It is not present in definition (1) but has a direct relevance to the capacitance contribution in the left part of Equation (5).
* The second term of Equation (5) also needs a comment. Different realizations of impedance circuits (Winston bridges, etc., see [21,27]) may contain a number of capacitances performing different functions in the measurement circuit. It is usually assumed that the main contribution to the capacitance, which appears in Equation (5), comes from the electrolyte-filled cell capacitance $C(\phi)$ (schematically represented in Figure 5). In the ac-measurements as a rule the value of such electrolytic capacitance is determined by the Debye length for the solvent (see, e.g., [21,28]).

<figure><img src="image_8.png" /><figcaption>**Figure 5.** Sketch of the circuit. A voltage $V$ is applied to suspension between parallel-plate electrodes separated by the distance $L$.</figcaption></figure>

However, we have already seen above how sharply this value can change due to the electrostatic image effects when the electric field is concentrated in the very narrow domains near the electrodes while in the bulk of the cell it remains zero. The value of the capacitance $C(\phi)$ is indeed determined by the relative thickness of these two domains, which we will estimate below.

* Finally, we note that the capacitance $C(\phi)$ is also affected by the nature of the ion movements within the cell volume under measure. Therefore, the properties of the accumulation layers leading to the main dependence of the cell capacitance on the nanoparticle concentration depend also on the current generated by the gradient of ionic chemical potentials mentioned above (see, for example, [21]). Nevertheless, in the wake of other authors, using Equation (5) in the *ac* regime, we shall accept the value of capacitance $C(\phi)$ in its static limit in the following discussion.

### 3. Effect of Adsorption Phenomena on Electrolytic Capacitance

In this section, we examine the specific features of electric field screening at the interface between a metallic electrode and an electrolyte containing different types of nanoparticles.

#### 3.1. *Electrolytic Capacitance of the "Metal-Pure Water" Boundary*

An electrolytic capacitor in radio-engineering as a rule has a flat geometry (a pair of metal strips wound into a coil with a narrow gap between them filled with a viscous electrolyte). Its properties are usually described in terms of the seminal works of Lewis & Randall [29], and Debye & Hückel [30], where the screening length $\lambda_D$, now called the Debye length, was introduced.

It should be noted, that the electrical properties of dilute suspensions containing dielectric inclusions differ strikingly from the predictions of Maxwell's theory for their pure
solvents (water, alcohols). In particular, the corresponding screening length $\lambda_{aq} \neq \lambda_D$ has been determined only recently in a consistent manner [31].

### 3.1.1. Peculiarities of Screening in the Bulk of Dilute Electrolyte

In order to explain the difference between $\lambda_{aq}$ and $\lambda_D$, we address the problem of Coulomb screening in a symmetrical (1:1) dilute electrolyte, which is the closest model for pure water. The electrochemical potentials for positive and negative ions $\mu_{\pm}$ can be presented in the form [29,30]

$$ \mu_{\pm}(z) = \eta_{\pm}e\varphi(z) + k_B T \ln A_{\pm}, \quad (10) $$

where $e = |e|$ is the absolute value of the electron charge, $\eta_{\pm}$ the algebraic values of the corresponding ion charges expressed in units of $e$, $T$ the temperature, $\varphi(z)$ the local electrostatic potential, and $A_{\pm}$ so-called activities of the ions (with volume concentrations of $N_{\pm}^{(d)}$) resulting from the decay process ($AB \leftrightarrow A_{+} + B_{-}$) of the dopant molecules (whose volume concentration is $N^{(d)}$).

The Debye- Hückel [30] theory allows us to find the explicit form of the activities $A_{\pm}$ in the case of complete decay of the dopant, provided that the average value of Coulomb interaction between ions satisfy $\bar{V}_C \ll k_B T$. It is also assumed that the inequality $eV \ll k_B T$, where $V$ is the voltage applied to the cell, is satisfied.

For a flat capacitor filled with (1:1) electrolyte, the Poisson equation with corresponding boundary conditions takes form

$$ \begin{gathered} \Delta\varphi(z) = -\frac{4\pi e}{\epsilon} \sum_{\pm} \eta_{\pm} N_{\pm}^{(d)}(z), \quad (11) \\ \varphi(z=0) = V, \quad \varphi(z \gg \lambda_D) \to 0. \end{gathered} $$

Here $\epsilon$ is the dielectric constant of the solvent (water). The concentrations $N_{\pm}^{(d)}(z)$ obey the Boltzmann distribution

$$ N_{\pm}^{(d)}(z) = N_{0,\pm}^{(d)} \exp \left[ -\frac{\eta_{\pm} e \varphi(z)}{k_B T} \right] \quad (12) $$

with $N_{0,\pm}^{(d)}$ being the concentration of the corresponding ions of the (1:1) bulk electrolyte far from the control electrodes, where the potential $\varphi(z) \to 0$. Expanding the Equations (11) and (12) with respect to $e\varphi/k_B T \ll 1$, one arrives to the linearized Poisson-Boltzmann equation

$$ \Delta\varphi = \kappa_D^2 \varphi, \quad (13) $$

$$ \kappa_D^2 = \frac{4\pi e^2}{\epsilon k_B T} \sum_a N_{0,\pm}^{(d)} \eta_{\pm}^2, \quad (14) $$

where $\kappa_D = \lambda_D^{-1}$ is the inverse Debye length. The neutrality condition reads as

$$ e \sum_{\pm} \eta_{\pm} N_{0,\pm} = 0. \quad (15) $$

According to Equations (13) and (14), not only the constants $N_{0,\pm}$ are involved in the formation of $\lambda_D$, but also the ion distributions (12) determining the properties of ionic fractions of the dopant within the framework of Boltzmann statistics.

The dopant ion concentrations $N_{\pm}$ can give an idea of the properties of the proton $H_+$ and the hydroxyl $OH_-$ fractions, which occur in the solution due to the reversible reaction
$H_2O \leftrightarrow (H_+ + OH_-)$. Yet, the product of their concentrations $N_+^{(aq)} \times N_-^{(aq)}$ fulfills the rule [20]:

$$N_+^{(aq)} \times N_-^{(aq)} \propto K^{(aq)}(T), \qquad (16)$$

where $K^{(aq)}(T)$ is so called ionic product of water. It is the equilibrium constant for the self-ionization reaction of water, i.e., the reaction between water molecules to form hydronium $H_3O^+$ and hydroxide $OH^-$ ions. The value of $K^{(aq)}(25 \, ^\circ C) = 1.0 \times 10^{-8} \, \text{mol}^2 \cdot \text{m}^{-6}$. As a consequence, the chemical component of the electrochemical potential of protons and hydroxyls cannot be presented in the form of simple logarithmic functions of $N_\pm^{(aq)}$, as it was done in Equation (10). Therefore, for pure water the described scheme (11)–(15) is relevant for determining the screening length $\lambda_{aq}$ only qualitatively.

The role of water’s own ions in the external field screening can be consistently treated in the framework of the theory of reversible chemical reactions, often called the theory of “ionic equilibria” [31,32]. It implies a deep analogy between the properties of dilute electrolytes and those of doped semiconductors [33,34]. Leaving apart the details of derivations (see Ref. [31]), here we only present the final results for the screening length $\lambda_{aq}$

$$\lambda_{aq}^{-2} = \frac{4\pi e^2}{\epsilon_{aq} T} (N_+^{(aq)} \times N_-^{(aq)})^{1/2}, \qquad (17)$$

$$N_+^{(aq)} \times N_-^{(aq)} = N_{0,+}^{(aq)} \cdot N_{0,-}^{(aq)} \cdot \exp(-\frac{E_g}{k_B T}). \qquad (18)$$

The last line is obtained within the validity of the Boltzmann approximation (Equation (12)). The temperature dependencies of $N_{0,+}^{(aq)}$ and $N_{0,-}^{(aq)}$ basing on the experimentally measured “ionic product of water” are presented in Figure 6.

<figure><img src="image_9.png" /><figcaption>Figure 6. Functions $N_{0,+}(T)$ (blue) and $N_{0,-}(T)$ (red). Here $n_{aq}$ is the concentration of water molecules. The data are taken from Ref. [31].</figcaption></figure>

To summarize, the screening length $\lambda_{aq}$, contrary to the Debye length $\lambda_D$, does not depend on the dopants density (while as it is seen from Equation (14) $\lambda_D^{-2} \propto N_0^{(d)}$).

Coming back to the concentration dependence of the cell capacitance $C(\phi)$, it is clear to see that in the absence of inclusions, the capacitance is determined by the value of screening length $\lambda_{aq}$:

$$C = \frac{\epsilon_{aq} S}{8\pi \lambda_{aq}}, \qquad (19)$$
where $\epsilon_{aq}$ is the dielectric constant of water. The factor “8” instead of “4” reflects presence of two plates in the cell. The same concerns properties of the individual ions near the electrolyte-vacuum boundary (see Refs. [35,36]).

### 3.1.2. Role of Image Forces in the Formation of Adsorption Layer at the “Metal-Pure Water” Interface

Above we have considered the features of screening in the (1:1) bulk dilute electrolyte and in particular, in the bulk of pure water. In the vicinity of the “metal-pure water” interface, the image forces (arising from the nanoparticles interaction with the metal electrode) act on protons and hydroxyls present in the water. These forces lead to the formation of an adsorption layer near the interface and significantly affects the value of the screening length, resulting in a strong renormalization: $\lambda_{aq}^* \ll \lambda_{aq}$.

The concentration of the ions ($q_{\pm} = \pm|e|$) in water at distance $z$ from the interface in the presence of an image electric field is governed by the Boltzmann distribution:

$$N_{\pm}^{(aq)}(z) = N_{0,\pm}^{(aq)} \exp\left(\frac{U(z)}{k_B T}\right), \qquad (20)$$

$$U(z) = \int_{\infty}^{z} F(\epsilon_{1}, \epsilon_{2}, z) dz. \qquad (21)$$

where $N_{0,\pm}^{(aq)}$ is the ion concentration in the bulk of water. The electrostatic force acting on a charged particle within the interface region between two semi-spaces with corresponding dielectric constants $\epsilon_1$ and $\epsilon_2$ is always attractive and has the form [24]

$$F(\epsilon_1, \epsilon_2, z) = \frac{\epsilon^2(\epsilon_1 - \epsilon_2)}{4\epsilon_1(\epsilon_1 + \epsilon_2)z^2}. \qquad (22)$$

In the case $\epsilon_1 = \epsilon_{aq}$, $\epsilon_2 \to \infty$, which corresponds to the situation at the “pure water-metal” interface, where the potential energy of an ion takes a simple Coulombian form:

$$U(\epsilon_{aq}, \infty, z) = U(z) = \frac{e^2}{4z\epsilon_{aq}}, \quad z > 0. \qquad (23)$$

When an external electric field is applied (for simplicity we assume it to be weak enough, so that the local values of the total electric potential at each point satisfies the condition $|e|\phi/k_B T \ll 1$), the Poisson equation can be linearized with respect to $\phi$, and reduces to

$$\frac{d^2\phi}{dz^2} \approx \exp\left(\frac{U(z)}{k_B T}\right) \frac{\phi}{\lambda_{aq}^2}, \quad \phi(z \to \infty) \to 0. \qquad (24)$$

Here $\phi(z)$ is the local value of the electric potential, and $\lambda_{aq}$ is determined by Equations (17) and (18).

Since the potential energy $U(z)$ enters in total $\phi(z)$, Equation (24) is hardly solvable exactly. This is the reason why we will use below the method of subsequent approximations that will allow us to estimate effect of image forces on the value of screening length without claiming quantitative results.

Let us integrate equation Equation (24) over the interval $(R_{\pm}, \infty)$, where the lower limit $R_{\pm}$ cut-off the singularity in $U(z)$. The physical meaning of these constants will be discussed below. Taking into account the boundary condition $\phi(z \to \infty) \to 0$, we arrive at

$$\lambda_{aq}^2 \left. \frac{d\phi(z)}{dz} \right|_{z=0} = - \int_{R_{\pm}}^{\infty} \exp\left(\frac{U(z)}{k_B T}\right) \phi(z) dz \qquad (25)$$
In the right-hand side of this equation let us substitute the potential $\varphi(z)$ in its unperturbed Debye form $\varphi_0(z) = \varphi(0) \exp(-z/\lambda_{aq})$. Introducing the screening length $\lambda_{aq}^*$, renormalized by image forces as $(\lambda_{aq}^*)^{-1} = \varphi'(0)/\varphi(0)$, one finds

$$ (\lambda_{aq}^*)^{-1} = -\frac{1}{\lambda_{aq}^2} \int_{R_{\pm}}^{\infty} \exp\left(\frac{U(z)}{k_B T} - \frac{z}{\lambda_{aq}}\right) dz, \quad (26) $$

relating $\lambda_{aq}^*$ to $\lambda_{aq}$. By substituting the explicit expression of $U(z)$ from Equation (23) and integrating, one finds

$$ \lambda_{aq}^* = \lambda_{aq} \left( \frac{\lambda_{aq}}{R_{\pm}} \right) \left( \frac{e^2}{4\epsilon R_{\pm} k_B T} \right) \exp \left( -\frac{e^2}{4\epsilon R_{\pm} k_B T} \right). \quad (27) $$

One can see that this value becomes exponentially small with respect to $\lambda_{aq}$ and $\lambda_{aq}^* \to 0$ in the limit $R_{\pm} \to 0$

The obtained result sheds light upon the meaning of the cut-off lengths $R_{\pm}$. An analogous problem already arose in the theory of 2D charged systems [37] where the properties of free electrons at the surface of liquid helium were studied [38]. In this case, the singular image force, together with a high energy barrier preventing electrons from entering the liquid helium, leads to the formation of surface electronic states.

In contrast to the problem with electrons on the helium surface, which has a quantum origin, the finiteness of the $R_{\pm}$ in electrolyte-metal interface problem should have a classical origin. Among the possible candidates for the cut-off parameter $R_{\pm}$ are: the non-ideality of the metal-pure water interface, the presence of uncontrolled impurities that prevent water from being considered perfectly pure, the proton $H_+$ and the hydroxyl $OH_-$ sizes, etc. In any case, this fundamental divergence is clearly evident in the impedance measurement interpretation of the conductivity anomalies in comparison to the predictions of Maxwell's theory (see Figures 1 and 2 and comments below).

The electrolytic capacitance (19) should also be renormalized respectively: the role of screening length $\lambda_{aq}$ in it passes to $\lambda_{aq}^*$, although we will see below that this is not the final adjustment.

## 3.2. Electrolytic Capacitance of the “Metal–Dilute Electrolyte” Boundary

Similar to the behavior of pure water molecules, the nanoparticles in a dilute electrolyte interact with the adjacent metallic interface. We illustrate the specifics of this interaction using a model example: the interaction between a colloid and a metallic plane (see Figure 3). This example will also help us understand the nature of similar interactions in suspensions containing metallic and dielectric nanoparticles (called here after «colloidal electrolytes»).

### 3.2.1. The Structure of the “Metal-Colloidal Electrolyte” Interface and Its Influence on Capacity

Let us start from the standard electrostatic problem of the interaction between a point charge $Q$, placed in an insulator semi-space (with dielectric constant $\epsilon$), with a metallic semi-space. The interaction force can be replaced by attraction of the charge to its mirror image [24]:

$$ F_{\epsilon}(z) = -\frac{Q^2}{4\epsilon z^2}, \quad (28) $$

where $z$ is the distance from the point charge $Q$ to the plane.

When the first semi-space is filled with an electrolyte, the electrostatic image force $F_{\epsilon}$ (Equation (28)) is screened at the distances of the order of the Debye length as one moves away from the plane. As it was demonstrated by Wagner, Onsager, and Samaras [35,36]:
$$F_{\text{WOS}}(z) = F_{\epsilon}(z) \exp\left(-\frac{2z}{\lambda_0}\right). \qquad (29)$$

The corresponding electrostatic energy of the charge Q in this configuration acquires the form

$$U_{\text{WOS}}(z) = -\int_{z}^{\infty} F_{\text{WOS}}(x)dx = -\frac{Q^2}{2\lambda_0\epsilon}\Gamma\left(-1, \frac{2z}{\lambda_0}\right), \qquad (30)$$

where $\Gamma(s, x)$ is the upper incomplete gamma function. In other words, a charged point-like particle located in the electrolyte at distances exceeding the Debye length $\lambda_0$ from the electrode interacts exponentially weakly with it.

Let us now face the behavior of a colloidal particle in the vicinity of the metal-electrolyte interface. Until the distance from the center of the colloidal particle to the interface $z$ noticeably exceeds the Debye length ($z \gg \lambda_0$), the latter keeps its integrity and electroneutrality. Approaching the interface ($z < \lambda_0$) this complex object loses a part of its screening counterions and acquires a finite number of net charges. The framework of the Poisson equation with infinite boundary conditions [8] is no longer applicable here. Thus the finite colloidal particle core-size $R_0$ serves as the natural cut-off of the image force potential singularity at small distances ($z < R_0$) [14].

As has been noted by the authors of [39] the value of the surface concentration $N_\odot$ of colloidal particles can be found. This is done by means of integrating the difference between the Poisson-Boltzmann distribution of the nanoparticles in the modified $U_{\text{WOS}}(z)$ potential (by Ohshima, see [40]) and their homogeneous density $n_\odot$:

$$N_{\odot} = n_{\odot} \left\{ \int_{R_0}^{R_0+\lambda_0} \left( \exp\left[ \frac{Z^2 e^2 \Gamma\left(-1, 2+\frac{2R_0}{\lambda_0}\right)}{\lambda_0 k_B T \epsilon} \exp\left(-\frac{z-R_0}{\lambda_0}\right) \right] - 1 \right) dz + \int_{R_0+\lambda_0}^{\infty} \left( \exp\left[ \frac{Z^2 e^2}{2\lambda_0 k_B T \epsilon} \Gamma\left(-1, \frac{2z}{\lambda_0}\right) \right] - 1 \right) dz \right\}. \qquad (31)$$

The conservation of the total number of colloidal particles in the dispersion implies that

$$N = 2N_{\odot} \cdot S + n_{\odot} \cdot S \cdot L = \text{const}, \qquad (32)$$

where S is the surface area of the interface (metal electrode), L the linear size of the sample cell, and N the total number of colloidal particles introduced in the solution. The latter retains its value in the process of mutual adjustments between two colloidal fractions approaching equilibrium.

The definition of $N_\odot$ (Equation (31)) together with the requirement (Equation (32)) contains all necessary information concerning the properties of the 2D colloidal fraction in terms of the 3D homogeneous density $n_\odot$ of colloids with an arbitrary ratio $\lambda_0/R_0$. Yet, the charge value $0 \le eZ^*(z) \le eZ$ acquired by a colloidal particle in result of the partial loss of counterions from the coat remains indefinite.

Far from the interface the colloidal gas is rarefied (the condition (2) is satisfied) and the bulk physical characteristics of the colloidal solutions (osmotic pressure [41], conductivity [8], Seebeck coefficient [39]) are found to be independent of the value of the core structural charge $eZ$. As the density of colloidal particles increases and the criterion (Equation (2)) approaches its upper limit, the effective charge of the particle core enters in gamble [42].

In the vicinity of the interface, however, the colloidal particles are localized at distances of the order of $R_0$ from it, and due to the partial loss of the counterions the role of effective charge $eZ^*(z)$ can no longer be ignored at any colloidal concentration $n_\odot$.
For further discussion it is crucial to find the binding energy of the colloidal particle $E_⊙$ in the vicinity of a metal surface. In Ref. [40] Ohshima performed a study of $E_⊙$ dependence on the distance (z) from the interface for an arbitrary ratio between $R_0$ and $\lambda_0$. One can see from Figure 4 of Ref. [40] that $E_⊙(z)$ remains negative for all z and reaches its minimum when the core of colloidal particle touches the plane: i.e., $z = R_0$ (see Figure 3b). As can be seen from the Ohshima’s numerical results, the analysis of the binding energy $E_⊙$ dependence on the particle core-size shows that this energy depends on $R_0$ much more significantly than it could be expected for the Coulomb interaction of a point charge Q with its electrostatic image ($E_Q = Q^2/2R_0$). This fact can be explained by taking into consideration that the effective charge $eZ^*$ acquired by the nanoparticles itself depends on $R_0$.

Operating in the geometrical terms one can identify $eZ^*$ with the charge of the spherical segment cut-off by the plane $z = R_0$ from the screening coat (see Figure 3b). The values of the segment chord a and the corresponding volume $V^*$ are determined as (see Figure 3b):

$$a = \sqrt{2\lambda_0 R_0 + \lambda_0^2}, \quad V^* = \pi\lambda_0^2(R_0 + \frac{2\lambda_0}{3}). \quad (33)$$

The corresponding value of the acquired effective charge is determined by the ratio between $V^*$ and the full volume of the counterions coat $\frac{4}{3}[(R_0 + \lambda_0)^3 - R_0^3]$ (see Figure 3a). Therefore in our simple geometric model the acquired effective charge $eZ^*$ indeed depends on the particle’s core radius:

$$eZ^* = eZ \frac{(3R_0 + 2\lambda_0)\lambda_0}{4[3R_0^2 + 3R_0\lambda_0 + \lambda_0^2]}, \quad (34)$$

and the binding energy takes form as:

$$E_⊙ = -\frac{e^2 Z^* 2}{2\epsilon R_0} = -\frac{e^2 Z^2}{32\epsilon R_0} \frac{(3R_0 + 2\lambda_0)^2 \lambda_0^2}{[3R_0^2 + 3R_0\lambda_0 + \lambda_0^2]^2}. \quad (35)$$

This expression demonstrates that the binding energy of large colloidal particles ($R_0 \gg \lambda_0$) in the vicinity of the metal surface decreases with increase of its size much more rapidly than for the point charge (compare $E_⊙ \sim R_0^{-3}$ and $E_Q \sim R_0^{-1}$).

One can see that the Debye length enters in the binding energy (Equation (35)) in the form of dimensionless parameter $\lambda_0/R_0$, exactly as it appears in Ohshima’s theory [40]. It is important to stress that $\lambda_0$ is *independent* on the structural charge value over a wide range of $eZ$.

To confirm the correctness of our model, we calculate $E_⊙(R_0)$ according to Equation (35) in the region $R_0 \le \lambda_0$ and find a good agreement with the results of Oshima’s numerical analysis for three values of $R_0 = 0.1\lambda_0; 0.5\lambda_0; 1.0\lambda_0$ (see Figure 7). Extrapolating this equivalence we apply below our model also to a range of colloidal particle core sizes exceeding the Debye length: $R_0 \ge \lambda_0$.

The surface concentration $N_⊙$ of colloidal particles in this interface layer can be evaluated by equating the corresponding chemical potential with that of the colloids in the bulk

$$\mu_s = \mu_b. \quad (36)$$

Let us imagine a box with a face surface area S and a height $R_0 + \lambda_0$, built at the interface. We first fill the box with a set of colloidal particles without electrostatic interactions with the metallic semi-space (its density would be $n_⊙$). The corresponding chemical potential $\mu_b$ of the colloidal particles within the weak-electrolyte approximation is determined by their total number in the box [15]:
$$ \mu_b = k_B T \ln[2n_{\odot} \cdot S \cdot (R_0 + \lambda_0)] + \psi_b(P, T), \quad (37) $$

where $\psi_b(P, T)$ is some function of pressure and temperature (note that the factor 2 inside the logarithmic term takes into account the availability of two metallic surfaces, i.e., electrodes). Now let us switch on the electrostatic interactions. The surface concentration of the colloidal particles will become $N_{\odot}$, and consequently the chemical potential $\mu_s$ of those localized within the interface layer can be written as

$$ \mu_s = k_B T \ln(2N_{\odot} \cdot S) + \psi_s(P, T), \quad (38) $$

with $\psi_s(P, T)$ being another function of pressure and temperature different from the bulk. Equating Equations (37) and (38), and recognizing that the difference between the additive functions is determined by the colloidal particle binding energy in the interface layer, i.e., $\psi_s(P, T) - \psi_b(P, T) = E_{\odot}$, one finds:

$$ N_{\odot} = n_{\odot}(R_0 + \lambda_0) \exp\left(\frac{|E_{\odot}|}{k_B T}\right). \quad (39) $$

By the sign $|...|$ we stress that the binding energy $E_{\odot}$ is negative (see Equation (35)), i.e., the argument in the activation exponent is positive. One can see that Equation (39) qualitatively resembles the above relation Equation (31), but successfully avoids the cumbersome integration. The readers should keep in mind, however, that all these considerations are performed under the assumption $R_0 \ge \lambda_0$.

<figure><img src="image_10.png" /><figcaption>Figure 7. Binding energy $\tilde{E}_{\odot}$ (taken from Equation (35) and normalized to $E_{\odot}(R_0/\lambda_0 = 0.5)$) as the function of the ratio $R_0/\lambda_0$. The stars correspond to the numeric values of the binding energy taken from Oshsima's paper [40] at points $R_0/\lambda_0 = 0.1; 0.5; 1.0$.</figcaption></figure>

The normalization relation Equation (32) between $N_{\odot}$ and $n_{\odot}$ allows to track down the process of filling of the interface layer by partially stripped colloidal particles as their number in the cell increases (for instance, such process occurs at the beginning of the steady stage of the thermoelectric voltage measurements (Seebeck effect) in [39]). Their observed saturation effect $N_{\odot}(N) \to \text{const}$ is contained in Equations (36)–(39).

The adsorption of finite number of colloidal particles dramatically changes the value of cell capacitance. In fact, the presence of each adsorbed nanoparticle results in opening up a “hole” of the area of $\pi a^2$ at the metal boundary (see Figure 8, depicted as light-colored circles of the radius $a$ in the contact zone of each colloid with its mirror image).
<figure><img src="image_11.png" /><figcaption>**Figure 8.** The set of dipoles formed by the colloidal particles close to metallic surface. The nanoparticles partially lost their coats. The average distance between the dipoles is $L \sim N_{\odot}^{-1/2}$, where $N_{\odot}$ is the surface concentration of the colloidal particles in the vicinity of electrode. The light circles in the contact zone of each colloid with its mirror image have the radius $a = \sqrt{2\lambda_0 R_0 + \lambda_0^2}$.</figcaption></figure>

As a result, the cell capacitance $C$, filled with water containing colloids, considerably reduces with respect to that of a cell filled with pure water:

$$C = C_0 \frac{\lambda_0}{\lambda_{\odot}}, \quad \frac{1}{\lambda_{\odot}} = \frac{1}{\lambda_0} (1 - \pi a^2 N_{\odot}), \qquad (40)$$

where $N_{\odot}$ is the density of adsorbed colloidal particles. It was demonstrated (see Ref. [43]) that the latter is exponentially related to the bulk volume value $n_{\odot}$ through the nanoparticles’ binding energy $E_{\odot}$ (see Equations (33)–(35) and (39)). Clearly, therefore, the capacitance Equation (40) decreases sharply as $N_{\odot}$ increases.

Now let us turn to metallic nanoparticles. They do not have ionic mobility and in this sense are similar to dielectric nanoparticles. However, when such a nanoparticle is placed in a solvent, electrostatic images of the surrounding counterions appear on its surface, similar to the situation described in the previous subsection. As a result, metallic nanoparticles acquire a counterion coat of excess density. In other words, replacing $\lambda_{aq}$ with the effective thickness of such a counterion coat $\lambda_{met}$, one can find a complete analogy between their properties and the properties of colloids.

### 3.2.2. Electrolytic Capacitance of the “Metal–Dielectric Suspension” Boundary

The effect of dilute dielectric suspensions on the electrolytic capacitance of the control electrodes is much clearer, more essential and easier to describe. The anomaly here first implies a sign change in the derivative $d\sigma(\phi)/d\phi$, which is not the case in colloidal solutions. The solvents of dielectric nanoparticle suspensions are generally purer than that in the case of colloidal solutions. The latter circumstance contributes to a manifestation of anomalously strong screening properties of the “metal–solvent” interface. The data of Ref. [9] for alcohol-based diamond powder suspensions reliably prove the above statements (Figure 1).

The appearance of a layer of neutral nanoparticles at the “metal-electrolyte” interface radically changes its capacitance properties, especially if these are dielectric nanoparticles (for example, diamond powder). The presence of an adsorption layer of dielectric nanoparticles of effective thickness $d(\phi)$ increases the electric field penetration depth into the cell. As a consequence, the cell capacitance Equation (19) takes its final form:

$$C(\phi) = \frac{\epsilon_{aq}(\phi)S}{8\pi(d(\phi) + \lambda_{aq}^*)}, \qquad (41)$$

where $\lambda_{aq}^*$ is determined by Equation (27) (or its analogue $\lambda_{al}^*$ for alcohol), $d(\phi) \simeq 2R_0\Psi_{ai}(p(\phi))$ (the parameter $R_0$, introduced after Equation (2), here corresponds to the radius of the isolated
dielectric nanoparticle). Function $\Psi_{ai}(p)$, called the adsorption isotherm, characterizes the effective filling of energetically advantageous sites on the solid substrate (adsorbent). It depends on the partial pressure $p = p(\phi)$ arising due to the presence of nanoparticles. The scheme of multilayer adsorption is shown in Figure 9. The corresponding dependence $\Psi_{ai}(p)$ for this case was calculated in Ref. [44]:

$$ \Psi_{ai}(p) = \frac{\gamma \times \Psi_{ai}^{\infty} \times p/p_s}{(1 - p/p_s)[1 + (\gamma - 1)p/p_s]}, \quad (42) $$

and is called the Brunauer–Emmett–Teller isotherm. Here, $\Psi_{ai}^{\infty}$ is the maximum value of $\Psi_{ai}(p)$, $p_s$ the saturation pressure in the nanoparticle system filling the electrolyte volume above the plane, and $\gamma$ a numerical constant chosen from appropriate physical considerations (usually focusing on the explicit form of the adsorption isotherm). The influence of the constant $\gamma$ on the overall structure of such isotherms can be seen in the inset of Figure 9.

<figure><img src="image_12.png" /><figcaption>**Figure 9.** Details of the mechanism of reversible multilayer adsorption (Brunauer–Emmett–Teller adsorption) defined by Equation (42). The inset shows the effective thickness of the adsorbed layer at different variants of interaction between adsorbed nanoparticles and the substrate. Their structure to great extent depends on the constant $\gamma$, which is sensitive to the details of adsorbant–adsorbent and adsorbant–adsorbant interactions. If adsorbant–adsorbent interaction prevails in the system, a filled monolayer is first formed on the adsorbent surface, and then the following layers are filled. Such order of filling leads to the appearance of an S-shaped adsorption isotherm (Curve 1 in the inset). This isotherm shape is formed at $\gamma > 20$. When $\gamma \le 1$, the following layers start to form even before the complete filling of the first monolayer (Curve 3 in the inset). The interval of values $1 \le \gamma \le 20$ is obviously intermediate (Curve 2 in the inset).</figcaption></figure>

Lastly, Equation (41) presents the analogue of the expression for capacitance of a cell filled with a colloidal suspension renormalized by the accumulation layers formed in the vicinity of electrodes (see Formula (10) in Ref. [26]).

## 4. Analysis of Impedance Measurements Results and Elimination of Discrepancy with Maxwell’s Theory

Having dealt in Section 3 with the impact of the metal-electrolyte boundary on the electrolytic capacitance $C(\phi)$ we now return to the general RC-circuit description;
Equations (5)–(8). It is easy to see that the properties of such a boundary can indeed have a significant influence on the cell’s electronic relaxation time

$$ \tau_r(\phi) = R(\phi)C(\phi). \qquad (43) $$

Its role is especially clear in the ac-impedance experiments with dielectric suspensions.

It is natural to separate the discussion of the origin of discrepancies between the suspension’s measured conductivity as a function of colloidal inclusion concentration, and the predictions of Maxwell’s theory into two cases: the dielectric inclusions ($\sigma_\odot/\sigma_0 \ll 1$) and the well-conducting ones ($\sigma_\odot/\sigma_0 \gg 1$).

### 4.1. Suspensions Containing Dielectric Inclusions

The authors of Ref. [9] in their detailed work presents not only the experimental results on the conductivity and its dependence on the nanodiamond volume fraction in ethylene glycol, but also the dependence of the “loss factor” on the externally applied frequency $f$ for two different suspensions (see Figure 12 of Ref. [9]). They relate the position $f_{max}$ of the observed maximum (where the loss factor is the highest) to the relaxation time $\tau_r = 1/(2\pi f_{max})$. The latter decreases with the growth of the nanodiamond particle concentration. Table 1 shows the values of measured relaxation times $\tau_r$ as a function of the volume fraction of diamond nanoparticles for both suspensions.

In the framework of impedance analysis (see Equation (5)), the value of $\tau_r^{-1}$ determines the position of the maximum of Fourier image of the imaginary component, $\Im Q$ in the frequency domain. The relaxation time, $\tau_r$, in its turn, is determined by the product of the capacitance and the resistance of the suspension-filled cell [26] (see Equation (43)).

The analysis of the data shown in Table 1 performed in the assumption of $C(\phi) = \text{const}$ indeed leads to the anomalous behavior of conductivity reported in Figure 1 with an unreasonably large increase of the ratio $\sigma(\phi)/\sigma(0)$.

<table><caption>**Table 1.** Data on the values of $\tau_r(\phi)$ for a cell filled with two different nanodiamond suspensions (ND97-EG—Nano.Diamond 97-Ethylene Glycol; ND87-EG—Nano.Diamond 87-Ethylene Glycol). The two sets of diamond powders in the Table have approximately the same nano-particle size (~4 nm) but different Diamond content: 97% for ND 97 and 87% for ND 87. The data are taken from Ref. [9].</caption><thead><tr><th>φ</th><th>ND97-EG τ<sub>r</sub>(s)</th><th>ND87-EG τ<sub>r</sub>(s)</th></tr></thead><tbody><tr><td>0.0000</td><td>1.07 × 10<sup>-2</sup></td><td>1.07 × 10<sup>-2</sup></td></tr><tr><td>0.0032</td><td>2.64 × 10<sup>-4</sup></td><td>2.64 × 10<sup>-4</sup></td></tr><tr><td>0.0080</td><td>9.63 × 10<sup>-5</sup></td><td>1.89 × 10<sup>-4</sup></td></tr><tr><td>0.0163</td><td>6.68 × 10<sup>-5</sup></td><td>9.63 × 10<sup>-5</sup></td></tr><tr><td>0.0338</td><td>2.51 × 10<sup>-5</sup></td><td>9.63 × 10<sup>-5</sup></td></tr></tbody></table>

However, the relaxation time $\tau_r$ collected in Table 1 can be interpreted differently. The validity of Maxwell’s Formula (1) for the conductivity of a suspension of dielectric diamond nanoparticles will be restored if we attribute the strong dependence $\tau_r$ on $\phi$ not to the resistance $R(\phi)$ but instead, to the capacitance $C(\phi)$ in the corresponding product $R(\phi)C(\phi)$ (see Equation (41)).

Point in fact, the assumption of a constant $R(\phi) \approx \text{const}$ is close to reality: according to Equation (1) $\sigma(\phi)/\sigma(0) \le 1 - 3\phi/2$. Then by combining $R(\phi)$ calculated from Equation (1) and $C(\phi)$ from Equation (5), one can reproduce the $\tau_r$ values as depicted in Table 1 as depicted in Figure 10.
<figure><img src="image_13.png" /><figcaption>**Figure 10.** Combination $R(0)C(0)/R(\phi)C(\phi)$ based on the data for ND97-EG (red circles) and for ND87-EG (blue squares) in the assumption that $R(\phi) \propto \sigma^{-1}(\phi)$ with $\sigma(\phi)$ determined by Equation (1). The non-exponential character of the observed dependence leads to the conclusion that adsorption at the metal-electrolyte interface in the experiments [9] is of the multilayer nature and it follows according to the scenario “2” (see the inset in Figure 9).</figcaption></figure>

The data presented in Figure 10 unambiguously confirm the validity of the hypothesis [26] about the importance of taking into account the properties of the multiplier $C(\phi)$ in the product $R(\phi)C(\phi)$ to explain the anomalous conductivity growth compared to the predictions of Maxwell's theory. The theory presented here allows to give a consistent interpretation which indicates marked decrease of the capacitance $C(\phi)$ with increasing bulk nanoparticle concentration $\phi$ as the adsorption of the latter at the metal-electrolyte interface also grows.

It is worth to note the high technological level of conductivity measurements of the suspensions containing insulating nano-particles. In the work of Zyla et al. [9], their size reaches $R_0 \approx 4$ nm. Today it is a record in research of the effect of diminishing of the nano-particles size on the increase of conducting properties of such solutions. Such tendency was still recognized by the authors of Ref. [10] where it was suggested to reduce the size of insulating nanoparticles in order to increase their effect on the suspension's conductivity. The physical understanding of such a tendency is still absent, raising further interest to the development of the Maxwell theory, which in its classic form does not answer these appeals.

## 4.2. Suspensions Containing Well-Conducting (Ionically Stabilized or Metallic) Inclusions

### 4.2.1. Colloidal Inclusions

Let us recall the significant contradictions arising in the interpretation of AC-conductivity measurements of dilute nanoparticle suspensions in the framework of Maxwell's theory.

- One of the most striking examples is that of nano-diamond suspensions in alcohol already presented in Figure 1. It seems rather intuitive that doping of alcohol with dielectric nanoparticles cannot increase the electrical conductivity of the suspension. However, the experimental data not only contradict this statement, but also show an increase by three orders of magnitude exceeding the maximum allowed by Maxwell's theory.

- The dependence of conductivity of dilute aqueous solution containing ionically stabilized nanoparticles (with high numbers of surrounding counterions) on their volume fraction shown in Figure 2 has a sign consistent with Maxwell's theory. However,
similar to the dependence shown in Figure 1, it exceeds the maximum allowed slope
by at least one order of magnitude.

The question arises: how, then, to explain the linear and huge dependences σ(φ)
clearly observed in the experiments?

This turns out to be possible within the framework of the theory of conductivity of dilute electrolytes constructed in the Mean Spherical Approximation (MSA) (see the review [1] and references therein). For example, when applied to a colloidal solution, the calculations within the MSA theory show that the colloidal particles lose their electroneutrality and acquire an effective charge $Z_{(eff-dyn)}$ already in a weak electric field. The effective charge, of course, depends on the electric field strength $\vec{E}$ and the density of the colloidal fraction $n_\odot$. The occurrence of $Z_{(eff-dyn)} \neq 0$ allows such nanoparticles to participate in the charge transfer process. Within the framework of the MSA model [1], it is possible to explain the observed dependence of the conductivity $\sigma(\phi)$ as it is presented in Figure 11.

The explicit dependence of $Z_{(eff-dyn)}$ on the electric field (shown in Figure 11) does not allow that the I-V characteristic of the colloidal solution obtained in the MSA model [1] to be linear. It is why the approach based on MSA theory cannot be used for interpreting the experimental data on ac conductivity measurements.

<figure><img src="image_14.png" /><figcaption>**Figure 11.** Conductivity of samples of maghemite nanoparticle colloidal solution prepared in HNO₃ at pH = 3.1 as a function of the volume fraction of nanoparticles (left scale). Dynamic effective charge Z<sub>(eff-dyn)</sub> of maghemite nanoparticles as determined from a fitting procedure by using the MSA theory to account for the experimental conductivity (right scale). Asymptote σ<sub>max</sub>(φ) (see Equation (3)) of Equation (1) would be presented here by the almost horizontal straight line at the height 0.04 S/m. The figure is taken from Ref. [1] and we are grateful to Taylor and Francis Ltd. (https://ww.tandfonline.com) for permission to use it.</figcaption></figure>

More correct (in our opinion) is the construction of the description of the transport
properties of polyelectrolytes on the basis of the DLVO theory [14,16], in which the colloid
has the view of a charged nucleus with radius R₀ and charge Z, surrounded by a completely
shielding shell of counterions (see Figure 3 of this review). In the framework of these
assumptions and approximation φ ≪ 1 under the applied electric field the solvent ions
flow around the fixed colloids (Figure 12).

Following this approach, the authors [8] have succeeded in explaining the linear de-
pendence of the conductivity of colloidal solution on the colloid concentration. The results
of these calculations agree with the prediction of Maxwell's theory [6] (see Equation (1) in the limit $\phi \ll 1$).

The theory proposed in [8] has to be used to analyze the impedance diagnostic data of suspensions containing dispersed nanoparticles. Unfortunately, such data are not available. Nevertheless, the information collected in Figures 2 and 11 helps to understand the possible descriptions of the conductivity properties of colloidal suspensions: either MSA theory or Maxwell's formalism in favor of Equations (1)–(4).

<figure><img src="image_15.png" /><figcaption>**Figure 12.** The sketch of ion current lines flow around a spherical colloidal particle. Their shape is determined by the continuity equation $\text{div}j = 0$ and relation between the local current and electric field $j(r) = \sigma_i E(r)$ accounting for the boundary conditions at the surface between the media with different conductivities $\sigma_{1,2}$.</figcaption></figure>

### 4.2.2. Metallic Inclusions

Let us turn to a suspension containing a suspension of metallic nanoparticles (Figure 4). When a voltage is applied to the electrodes of the cell, an electric field appears in its volume and a spherical layer of counterions of different signs is formed around the spherical nucleus of the metallic nanoparticle under the action of electrostatic image forces. Thus, the problem is reduced to the one discussed above of suspension of colloidal particles (see Figure 3).

## **5. Adsorption Phenomena in Complex Electrolytes—Experimental Investigation of Aqueous Dispersions of Maghemite Nanoparticles**

Now we pass the discussion to modern experimental techniques capable of detecting nanoparticle adsorption phenomena at the metal-electrolyte interface, allowing to connect with the high sensibility of the ac-diagnostics in the problems under discussion.

In Ref. [5], conductivity determinations, here reported in Figure 2, were conducted together with Seebeck determinations in aqueous dispersions of maghemite nanoparticles (also known as ferrofluids). Not only the slope of $\sigma(\phi)/\sigma_0$ as a function of $\phi$ is found ten-fold larger than the Maxwell prediction, but some Seebeck deviations were observed, letting suspect a nanoparticle adsorption on the electrodes. To demonstrate experimentally and to quantify the nanoparticle adsorption on platinum electrodes in such ferrofluids, two studies are here undertaken. The first one is a chemical technique—Inductively Coupled Plasma iron (ICP)—which determines the Iron quantity in the adsorbed layer. The second one is a physical technique—Quartz Micro Balance (QCM)—which determines in-situ the mass of adsorbed NPs on the electrode [45]. Samples similar to those of Ref. [5] are here used. Those aqueous dispersions at $pH \sim 7$ are based on citrate-coated maghemite nanoparticles of diameter 7–8 nm, with $TBuA^+$ counter-ions. The nanoparticles bear a
negative structural surface charge of 0.75 ± 0.22 elementary charge per nm² and present a zeta potential ~-40 mV. The colloidal stability of such dispersions has been extensively probed in [46,47] by Small Angle X-ray scattering, Dynamic Light Scattering and Rayleigh Forced Scattering (for more details see Section S1 of Supplementary Materials).

## 5.1. Inductively Coupled Plasma Determinations of NPs Adsorption on Pt Surface

The amount of adsorbed NPs is here deduced from the chemical determination of iron quantity on a platinum electrode by Inductively Coupled Plasma (ICP)-see Supplementary Materials. The series of measurements are realized in a teflon cell similar to the thermocell used for thermoelectric (Seebeck and a.c. conductivity) measurements (see [5]) with a polycrystalline Pt electrode at the bottom of the cell, of surface 0.302 cm². After a time of contact with the ferrofluid dispersion, the electrode is rinsed with the same electrolyte than the dispersion and the NPs remaining on the electrode are dissolved to determine the total amount of iron on the surface by ICP.

In practice, a monolayer of NPs on the electrode would correspond to 0.4 microgram of iron (40 ppb of iron if all the adsorbed iron is dissolved in 10 mL of acidic water). ICP can measure down to 100 ppt of iron, therefore the iron content here is far above the limit and the error on its determination is between 0.5 and 1.5%. Much larger errors very difficult to estimate can result from any pollution, which can originate from chemicals, equipment, etc. Consequently, an extreme care must be taken for avoiding the pollution from the chemicals and equipment used. The level of iron content of all chemicals (acid, distilled water, etc.) is established prior to the experiment with nanoparticles. Moreover, as few intermediate glassware or tubes as possible are used and their quality is also tested in advance (see Section S2 of Supplementary Materials for details of the experiment).

Three different durations of contact between the ferrofluid and the electrode were considered: 3 mn, 10 mn and 30 mn. Three steps are performed:

* Step 1—A blank is first realized with the pure electrolyte (aqueous solution of tetra-butylammonium citrate 34 mM) in contact with the electrode.

* Step 2—A measurement with the aqueous ferrofluid at 0.1 vol% of NPs (with a concentration of free TBuA₃Cit 34 mM) is performed; After the time of contact between the electrode and the ferrofluid, several rinsing with the electrolyte are performed to remove the free nanoparticles before the dissolution of the adsorbed nanoparticles with 30% HCl.

* Step 3—A last rinsing with 30% HCl is performed to dissolve any iron remaining on the surface after step 2.

For each step, the amount of matter to titrate is diluted in 10 mL of acidic water for the titration by ICP.

The results of ⁵⁶Fe in ppb obtained at the end of these three steps are given in Table S1 of Supplementary Materials and their conversions in terms of mass of iron, which is very small, are given in Table 2.

<table><caption>**Table 2.** Adsorption test on the Pt electrode: total iron in the titrated samples × 10⁻⁸ g as deduced from ICP determinations at the different steps.</caption><thead><tr><td>Contact Duration (mn)</td><td>Blank (Step 1)</td><td>Ferrofluid (Step 2)</td><td>Concentrated HCl Rinsing (Step 3)</td></tr></thead><tbody><tr><td>3</td><td>2.34</td><td>41.5</td><td>5.59</td></tr><tr><td>10</td><td>0.98</td><td>20.2</td><td>4.18</td></tr><tr><td>30</td><td>1.81</td><td>20.1</td><td>4.81</td></tr></tbody></table>

We can note that the blank of step 1 is clean and that the order of magnitude for the ferrofluid (step 2) is significantly higher. Step 3 shows that the cleaning at the end of step 2 was not completely efficient. However, the next blank after step 3 is clean, so
cleaning of step 3 was efficient. The total quantity of iron to be determined is given by [(step 2 + step 3) – (step 1)]—see Table 3). Taking a compacity of the adsorbed layer of the order of 80%, this iron oxide mass can be converted in a surface coverage, here between 40 and 80%. This demonstrates that some NPs are strongly adsorbed on the Pt electrode, of the order of half to one monolayer in a few 10 min—in the experimental conditions of step 2 (namely with an aqueous ferrofluid at 0.1 vol% of NPs with a NP diameter ~8 nm and a concentration of free TBuA₃Cit of 34 mM).

<table><caption>**Table 3.** Adsorption test on the Pt electrode by ICP determinations at the different steps : Surface occupied by the adsorbed NPs deduced from the iron determinations of Table 2.</caption><thead><tr><th>Step 2 + Step 3<br>– Step 1</th><th>Mass of Iron Oxide<br>(g)</th><th>Number of NPs<br>on the Surface</th><th>Projected Surface<br>of the NPs</th><th>Fraction of Surface<br>Covered by the NPs</th></tr></thead><tbody><tr><td>44.74</td><td>6.39 × 10<sup>-7</sup></td><td>4.77 × 10<sup>+11</sup></td><td>0.24</td><td>0.80</td></tr><tr><td>23.4</td><td>3.34 × 10<sup>-7</sup></td><td>2.49 × 10<sup>+11</sup></td><td>0.125</td><td>0.42</td></tr><tr><td>23.1</td><td>3.30 × 10<sup>-7</sup></td><td>2.46 × 10<sup>+11</sup></td><td>0.124</td><td>0.41</td></tr></tbody></table>

Adsorption in 3 min is measured higher than at 10 and 30 min and no kinetic is observed. Adsorption in 3 min is measured higher than at 10 and 30 min and no kinetic is observed. The ICP measurements provide a good precision on these results, which can be estimated around 5%. The pollution of chemicals and vessels has been checked in advance and is very low. The remaining source of error is the preparation during which some non adsorbed particles could stay. Indeed 10 picoliters of the colloidal dispersion remaining in contact with the surface, despite washings, is only 0.05% of the 20 microliters introduced for these tests. However it would correspond to 10% coverage of the Pt surface. The amount of NPs on the surface can thus be overestimated. It nevertheless indicate that there is some adsorption on the platinum.

In situ QCM measurements are also conducted ahead to attest for this efficient NP adsorption on Platinum electrodes.

## 5.2. Quartz Microbalance (QCM) Adsorption Study of NPs on Pt Surface

### 5.2.1. QCM Principle

A quartz crystal microbalance (QCM) uses the piezoelectric properties of a quartz crystal to determine masses very accurately. Its principle is based on the coupling between the electrical and mechanical properties of quartz. Thus, if the electrical impedance of a homogeneous quartz crystal is measured as a function of the excitation frequency, a resonance is observed for frequencies multiple of the fundamental frequency. Here, commercial quartz crystals (AWSensors, Valencia, Spain) with a thickness of about 200 µm and a resonance frequency close to 9 MHz are used. The resonant frequency is measured using network analyzer (Agilent4294A, Agilent Technologies Inc., Santa Clara, CA, USA). The electrical excitation used is 0.1 V and frequencies from 8.5 to 9.5 MHz are scanned. Using the Agilent equipment, it is possible to follow a dispersion parameter, called Γ, corresponding to the half width at half height of the conductance peak related to the tested quartz resonator. The measurements are done in parallel with the measurements of the resonant frequency, *f*, during the nanoparticles adsorption.

If nanoparticles attach themselves to such a quartz crystal, the mass and thickness of the quartz crystal change, resulting in a shift of the measured resonance frequency. At the end of the 1950s, Sauerbrey [48] developed a model (see Supplementary Materials), still in use today, that allows to link the frequency variation of resonance $\Delta f$ to the variation of mass $\Delta m$ of quartz. This model treats the thickness $\Delta h$ of adsorbed nanoparticles as an extension of the thickness of the quartz crystal and is valid under several conditions. First, the nanoparticles must be attached to the crystal. In addition, the added mass
must be uniformly distributed over the crystal, i.e., the adsorbed nanoparticles must be sufficiently small in comparison to the size of the quartz crystal and the interactions between nanoparticles must be repulsive so that the measured mass change corresponds to a statistically homogeneous distribution of adsorbed nanoparticles on the surface. These two conditions are verified for the present electrostatically stabilized nanoscale particles. Moreover, the frequency variation of the quartz induced by the mass variation must be sufficiently small, typically less than 10%. This last condition is also verified in all experiments: the frequency variations being less than kHz, i.e., 0.01% of the initial frequency. Finally, in the following experiments, the $\Delta\Gamma$ changes are small compared with the changes in the resonant frequency $\Delta f$ confirming the hypothesis that the nanoparticles are firmly attached to the crystal (See Section S3.7 in Supplementary Materials). Then, the associated mass changes can be estimated here for our experiments.

Note that the roughness of the quartz is around 500 nm, which is much larger than the size of the NPs. These latter therefore do not modify the roughness of the surface. However the real surface of the quartz is larger than the macroscopic one, by up to 50%.

### 5.2.2. QCM Experiment

Quartz microbalance measurements are first validated by a first series of measurements with a gold electrode, reproducing the kinetic adsorption of NPs on gold surface observed in [49,50] by Lucas et al with an acidic dispersion of maghemite NPs; See Section S3 of Supplementary Materials for these results and for the details of the EQCM experimental apparatus. The platinum-coated quartz is prepared by the deposition of 20 nm of titanium followed by a 200 to 300 nm of platinum on the quartz (LISE laboratory, Sorbonne University, Paris, France). The titanium layer is a bonding layer that allows a good adhesion of the platinum layer to the quartz. The selected citrated ferrofluid is very close to that of Figure 2. It is composed of nanoparticles the average diameter of which is 7 nm. The mass increase for a monolayer of adsorbed NPs on the electrode is expected to be between 2 and 3 µg/cm², considering only the oxide for the low value and the whole layer with the shell of ligands and the surrounding liquid in the layer for the high value.

* Effect of concentration of the free ionic-species

At low concentration of free $TBuA_3Cit$, 5 mM, no particle adsorption on the platinum surface is observed in open circuit for $\phi = 2.8 \times 10^{-3}\%$—see Figures 13 and S3 of Supplementary Materials. These figures also show that at higher ionic concentration, 30 mM of $TBuA_3Cit$, nanoparticles adsorb spontaneously in open circuit. The mass adsorbed when the plateau is reached is about 2.8 µg/cm², which corresponds to 1 to 1.5 monolayers of nanoparticles, depending on the hypothesis. As $\Delta\Gamma \ll \Delta f$ (see Figure S3 of Supplementary Materials), these adsorbed nanoparticles are firmly attached to the quartz cristal.

The concentration of free ionic species of this ferrofluid is equal to that of the ferrofluid of Figure 2 and it can therefore be assumed, as a first hypothesis, that there is adsorption on the platinum electrodes when measuring the Seebeck coefficient, even when the cell is isothermal.

* Effect of potential

The effect of potential for citrated ferrofluid highlights several phenomena. First of all, the potential effect cannot be explained simply by the electrostatic attraction between the surface charge of the electrode and the nanoparticles. For example, citrated nanoparticles that are negatively charged do not adsorb at 5 mM of $TBuA_3Cit$ when the platinum potential is increased to 0.5 V and then 1 V relative to the reference electrode, i.e., when the surface charge of the platinum becomes more positive.
<figure><img src="image_16.png" /><figcaption>**Figure 13.** Open-circuit QCM measurements on platinum electrode as a function of ionic strength for citrated ferrofluid. The frequency change is shown on the left Y-axis in black and the corresponding mass change is shown on the right Y-axis in red (calculated with the macroscopic surface). Both measurements start with a nanoparticle-free solution. These are added after about ten minutes. (a) $\phi = 2.8 \times 10^{-3}\%$ and tetrabutylammonium citrate concentration of 5 mM. (b) $\phi = 10^{-2}\%$ and concentration of tetrabutylammonium citrate of 30 mM.</figcaption></figure>

Conversely, the adsorption of around half a monolayer (~1.5 µg/cm²) is observed when a negative potential of -0.1 V is applied (see Figure 14). This behavior tends to mean that non-electrostatic forces, e.g., from the electrical double layer on the platinum surface, are also modified by the application of a potential. Once the nanoparticles have been adsorbed at -0.1 V relative to the reference, if the potential is switched off and the electrode is opened again, there is a slow fall in mass corresponding to almost complete desorption of the nanoparticles. This result underlines, on the one hand, the quasi-reversibility of this adsorption, which had been assumed from the Seebeck coefficient (Se) measurements for this ferrofluid. On the other hand, the time constant associated with the desorption of nanoparticles, of the order of one hour, is compatible with the observed time constants for reaching the steady state of Se of about 2 h. Note that the potential of -400 mV relative to the open circuit potential applied in this experiment is 30 times greater than the potential difference of about -15 mV measured between the hot and cold electrode in a Seebeck coefficient measurement with $\delta T = 10$ K.

At a higher concentration of free TBuA₃Cit (30 mM), the application of the potential to the platinum electrode with already adsorbed nanoparticles reveals nontrivial effects of the potential. Thus, a slightly positive potential of 0.5 V with respect to the reference promotes additional adsorption, while potentials of 1 V and -0.1 V with respect to the reference cause desorption.

* Effect of NP's concentration

Analysis of the Seebeck coefficient measurements for such a ferrofluid showed a saturation volume fraction of about 0.1%, from which the difference between the initial and stationary Seebeck coefficients remains constant. Under this hypothesis, the effect is related to the saturation of the adsorption of nanoparticles on the platinum electrodes.

In order to verify this effect, the mass adsorbed on the quartz is measured as a function of the volume fraction of ferrofluid in the solution containing 30 mM of TBuA₃Cit. The mass increase rate of adsorbed nanoparticles relative to $\Phi$ is then calculated. These results show that as the concentration of nanoparticles increases, fewer and fewer additional nanoparticles are adsorbed on the surface. This indicates the existence of a saturation phenomenon, the nanoparticles already present on the platinum prevent new nanoparticles from adsorbing, and the plateau at $\Phi = 0.1\%$ for which the rate of increase of Seebeck coefficient tends towards 0 is also observed by microbalance. Moreover, at this concentration the influence of the potential on the adsorbed mass also becomes very small. These results confirm the hypothesis proposed from the measurements in [5], that the observed
steady state of Seebeck coefficient is not the Soret equilibrium but an apparent steady state related to adsorption phenomena on the platinum electrodes. It also confirms that the huge deviation to Maxwell law of the determined conductivity in Ref. [5] is due to the adsorption of the nanoparticles on the electrodes which strongly modifies the measurements.

<figure><img src="image_17.png" /><figcaption>**Figure 14.** EQCM measurement with platinum electrode for citrated ferrofluid with 5 mM $TBuA_3Cit$. The applied potential is shown in the upper graph while the lower graph shows the frequency variation and the corresponding mass variation. The open circuit voltage is 0.3 V with respect to the reference electrode (ECSaq). During the entire duration of the measurements presented here the solution contains $\phi = 2.8 \times 10^{-3}\%$ nanoparticles.</figcaption></figure>

### 5.3. Summary

Thus two independent experiments performed with comparable well-stabilized aqueous ferrofluids (based on citrate coated maghemite NPs with similar NP diameter (~7–8 nm), volume fraction ($\phi \sim 10^{-3}$), negative structural surface charge (0.75 elementary charge/nm²), zeta potential (~40 mV) and same counterions $TBuA^+$ with free $TBuA_3Cit$ concentration (~30 mM)) demonstrate that some NPs remain adsorbed on the Pt electrode in these conditions:

- Chemical ICP determinations of iron quantity found on the surface after a strong rinsing lead to half to one monolayer of NPs;

- Physical QCM measurements in situ, of the mass attached to the electrode lead to 1 to 1.5 monolayer.

These two measurements are fully compatible taking into account the difficulty of both experiments and the difference in the nature of the measurements. The existence of this adsorbed NPs layer on the platinum electrodes can be directly correlated with the observations of Figure 2, showing experimental conductivity measurements of a similar ferrofluid dispersion, which sensibly deviate from Maxwell prediction.

Lastly, we attempt to make a quantitative comparison between the *measured* amounts of nanoparticle adsorption on metallic surface to the theoretically expected values derived in Section 3; i.e., the Equations (34), (35) and (39). The values of the average particle diameter (between 7 and 8 nm) and the effective charge ($-33 \pm 7$, effective dynamic charge at the infinite dilution limit) are taken from [46] where the authors studied the electrophoretic mobility of NPs in a ferrofluid nearly identical to that reported in Figure 2 [5]. The binding energy is calculated at $T = 300$ K (with $\epsilon_{aq} = 77$ for water) for the bulk NP volume fraction at 0.002, corresponding to the highest concentration used in the conductivity measurements performed as shown in Figure 2. Here, the effective static charge is considered to be equal
to the effective *dynamic* charge at the infinite dilution limit, and the screening length is taken as ~1.0 nm (see [51], Table 1 in ref. [51]). Within the estimated values of the fractional surface area coverage by the NPs calculated for different combinations of the NP diameter and the effective charge number values within the experimentally determined uncertainty limits are summarized in Table 4.

<table><caption>**Table 4.** Estimated adsorption of maghemite NPs on the Pt electrode using the Equations (34), (35) and (39). The values are calculated for different combinations of particle diameter and effective surface charge values within the experimentally defined uncertainty limits. Note that here the effective charge number of the NPs are approximated to the dynamic effective charge number at the infinite dilution limit. Bold numbers in the Table corresponds to values comparable to the observed experimental absorption—See text for more details.</caption><thead><tr><th>NP Diameter (nm)</th><th>Z* = -26 (Lower Limit)</th><th>Z* = -33 (Median Value)</th><th>Z* = -40 (Upper Limit)</th></tr></thead><tbody><tr><td>7 (lower limit)</td><td>0.072</td><td><strong>0.68</strong></td><td>11.0</td></tr><tr><td>7.5 (median value)</td><td>0.040</td><td><strong>0.25</strong></td><td>2.50</td></tr><tr><td>8 (upper limit)</td><td>0.025</td><td>0.12</td><td><strong>0.80</strong></td></tr></tbody></table>

With the median values of experimentally determined effective charge and the average particle size, the predicted surface area coverage by the NPs coincides well with those observed via ICP and QCM measurements reported above, providing evidence of NP adsorption phenomena of dispersed nanoparticles on metallic electrodes. The lack of knowledge on the Brunauer-Emmet-Teller isotherm (and its associated physical parameters and constants) does not allow a direct examination of the capacitance change in the presence of adsorbed NPs here (see Equations (41) and (42)). Rather, the experimental observation of ionic conductivity deviation from Maxwell's theory can be used to estimate how the electric field penetration depth into the thermocell, which should follow the same $\phi$ dependence as $\Delta\sigma(\phi)$; i.e., ~56$\phi$.

## 6. Conclusions

As was mentioned in the Introduction, the main goal of this review is to demonstrate the applicability of the approach proposed in Ref. [26] to suspensions containing dispersed nanoparticles of different natures: dielectric, semiconducting and metallic.

We demonstrated that the treatment of the ac measurements of the suspension conductivity has to be performed with caution because of the enormous dependence of the sample-cell capacitance on the concentration of nanoinclusions. The variation of the latter strongly affects the thickness of the adsorption layers occurring in the vicinity of the electrodes, which results in corresponding changes in total sample-cell capacitance. It is the product $RC$ which is in reality measured by impedance technique. Consequently, to correctly extract from it the $R(\phi)$ dependence, which then needs to be compared with the predictions of Maxwell's theory, a non-trivial dependence $C(\phi)$, which takes into account the adsorption of nanoparticles, must be taken into account.

Finally, we presented the results obtained with two modern techniques able to investigate adsorption phenomena at the metal-electrolyte interface and allowing to reach the record sensitivity of the ac-diagnostics in the problems under discussion. These are namely—chemical Inductively Coupled Plasma determinations to reach the adsorbed quantity of iron on an electrode, after rinsing, and—physical Quartz Micro-Balance measurements in situ of the mass adsorbed on the electrode. They both evidence a nanoparticle adsorption in the experimental range of ac conductivity measurements.
The main results of our discussion are summarized in Table 5. One can easily see that the behavior of $\epsilon_{\text{eff}}(\phi)$ in the limiting cases of metallic ($\epsilon_{\text{met}} \to \infty$) and dielectric ($\epsilon_{\text{diel}} \ll \epsilon_{\text{aq}}$) nanoparticles coincides with that one of conductivity (see Equations (3) and (4)).

<table><caption>**Table 5.** Influence of nanoparticles on physical properties of suspensions. Here $\sigma_{\text{met}}$ and $\epsilon_{\text{met}}$ are own conductivity and dielectric constant of metallic nanoparticles, $\sigma_{\text{diel}}$ and $\epsilon_{\text{diel}}$ are own conductivity and dielectric constant of dielectric nanoparticles.</caption><thead><tr><th>Type of Nanoparticles</th><th>Conductivity of Suspension</th><th>Capacitance of the Interface with Metal</th><th>Effective Dielectric Constant</th></tr></thead><tbody><tr><td>dispersed nanoparticles, size 10<sup>-7</sup> &divide; 10<sup>-5</sup> cm</td><td>Equation (1)</td><td>Equation (40)</td><td>&epsilon;<sub>eff</sub>(&phi;) = &epsilon;<sub>aq</sub> + 3(&epsilon;<sub>&odot;</sub> - &epsilon;<sub>aq</sub>)/(&epsilon;<sub>&odot;</sub> + 2&epsilon;<sub>aq</sub>) &phi;</td></tr><tr><td>metallic beads, size 10<sup>-7</sup> &divide; 10<sup>-5</sup> cm</td><td>Equation (1) with replacement &sigma;<sub>&odot;</sub> &rarr; &sigma;<sub>met</sub></td><td>Equation (40) with replacement N<sub>&odot;</sub> &rarr; N<sub>met</sub></td><td>&epsilon;<sub>eff</sub>(&phi;) = &epsilon;<sub>aq</sub> + 3(&epsilon;<sub>met</sub> - &epsilon;<sub>aq</sub>)/(&epsilon;<sub>met</sub> + 2&epsilon;<sub>aq</sub>) &phi;</td></tr><tr><td>dielectric (latex) beads, size 10<sup>-7</sup> &divide; 10<sup>-5</sup> cm</td><td>Equation (1) with replacement &sigma;<sub>&odot;</sub> &rarr; &sigma;<sub>diel</sub></td><td>Equation (41)</td><td>&epsilon;<sub>eff</sub>(&phi;) = &epsilon;<sub>aq</sub> - 3(&epsilon;<sub>aq</sub> - &epsilon;<sub>diel</sub>)/2(&epsilon;<sub>aq</sub> + &epsilon;<sub>diel</sub>) &phi;</td></tr></tbody></table>

**Supplementary Materials:** The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/colloids9040044/s1, Table S1: Adsorption test on the Pt electrode; Figure S1: EQCM device with a diluted ferrofluid solution; Figure S2: Open circuit QCM measurements on gold electrode, Figure S3: QCM measurements on platinum electrode—associated to Figure 13; Figure S4: QCM measurements on platinum electrode—associated to Figure 14.

**Author Contributions:** I.C.: Methodology, Formal analysis, Writing review and editing, Visualization, Analysis of a.c. conductivity experimental data. M.B.: PhD involved in ICP measurements, Analysis of experimental data. P.B.: ICP measurements Investigation, Formal analysis, Writing, review and editing. E.D.: ICP and QCM measurements, Methodology, Analysis of experimental data, Writing review and editing, Visualization. I.T.L.: QCM measurements, methodology, Analysis of experimental data. S.N.: Methodology, Formal analysis, Writing review and editing, Visualization. O.S.: QCM measurements, methodology, Analysis of experimental data. H.P.: QCM measurements, methodology, Analysis of experimental data. R.P.: Formal analysis, Writing review and editing, Visualization. T.J.S.: former PhD involved in QCM measurements, Analysis of experimental data. B.E.T.-B.: post-doc involved in ICP measurements, Analysis of experimental data. A.V.: Methodology, Formal analysis, Writing review and editing, Visualization. All authors have read and agreed to the published version of the manuscript.

**Funding:** We acknowledge the European Union's Horizon 2020 research and innovation programme under grant agreement No. 731976 (MAGENTA) as well as the French National Research Agency (ANR) under grant agreement ANR-22-CE50-0030 (WH-RECOLTE).

**Data Availability Statement:** The data supporting the findings of this study are available from the corresponding author upon reasonable request.

**Acknowledgments:** The authors are grateful to Valeriy Shikin for valuable discussions and permanent interest to this work. A.V. is grateful to Xiamen University for hospitality.

**Conflicts of Interest:** The authors declare no conflict of interest.

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