AC-Diagnostics of Transport Phenomena in Dilute Suspensions

Abstract

Impedance diagnostics is commonly employed in the study of transport phenomena in conducting media of different sizes. A common reason for choosing the more complex method of exciting the conductive medium at finite frequencies ($ac$ mode) instead of the relatively simple method of excitation at zero frequency ($dc$ mode) is to eliminate the influence of contact phenomena on the current–volt charateristic (IVC) during $dc$ measurements. In this paper, we analyze relaxation phenomena in electrolytes with linear electrohydrodynamics in terms of dopant density $n_d$. It is shown that the requirement of linearity on $n_d$ of the electrohydrodynamics of dilute solutions cannot be satisfied by the Debye–Huckel–Onsager theory of electrolyte conductivity. A linear alternative based on the fundamental principles of the theory of transport in finely dispersed two-phase systems is proposed. This alternative is referred to in the literature as Maxwell’s formalism. It is noted that, in this case, there is a consistent possibility of treating the observed relaxation time, ${\tau }_c$, as impedance time ${\tau }_{rc}({\tau }_c\to {\tau }_{rc}=RC)$. Here, R is the resistance of the dilute electrolyte part of the cell, and C is the electrolytic capacitance of the same cell. This capacitance does not coincide with the traditional geometric one, $C_0<<C$, and has to be calculated self-consistently. Examples of the successful application of $RC$-consistent $ac$ diagnostics are discussed. This refers to the numerous instances in which the effective conductivity of various colloidal media deviates from the predictions of Maxwell’s well-known theory and to the correct interpretation of these anomalies in the RC representation.

1. Introduction

It is well known that direct current $dc$ transport measurements in various types of conducting media are not very effective, as the screening properties of conductors, particularly electrolytes, interfere [1]. A more consistent approach is the $ac$ diagnosis of electrolytes, which identifies the properties of different relaxation times, ${\tau }_c$, as discussed in work [1]. In fact, the relaxation problem formulation contains several requirements that are difficult to satisfy. The formalism [1] requires a linear Ohm’s law in terms of the density $n_d$ of the nanoparticles. It includes the effects of screening by the conducting medium of external perturbations, but it does not include the possibility of screening of centrally symmetric fields of each ion by the collective reaction of the surrounding charged nanoparticles. Moreover, this formalism ignores the intrinsic conductivity of dilute solutions, which significantly affects the final results.

A popular version of the theory that considers how ions interact with each other is explained by Debye, Huckel and Onsager [2,3,4] (see also the main tutorials on this subject [5,6,7,8,9,10,11]). In this version of the theory, the mobility of the ions ${\zeta }_{\pm }\left(n_d\right)$ involved in the formation of charged particle fluxes begins to depend on $n_d$ (see Formula (12) below for the details). Hence, Ohm’s law represented as ${\overrightarrow{j}}_{\pm }=\pm e{n}_d{\zeta }_{\pm }\left(n_d\right)\overrightarrow{E}$ ceases to be a linear function of $n_d$ and so is not appropriate for the theory [1]. An alternative to the theory [2,3,4] is proposed for the conductivity of electrolytes, with the force component defined according to Maxwell (see below for details). Within the scope of this review, we can discuss the physical significance of the time constant ${\tau }_c$, and the potential transition from the ${\tau }_c$ regime, characterised by a $\theta$-like shape of the perturbing potential $V\left(t\right)$, to the $RC$ time constant, which emerges in the impedance approximation for an electrolytic cell within a planar metal capacitor (see Figure 1). The current state of the art in electrolyte statistics and electrohydrodynamics of charged solutions allows for a comprehensive approach to solving problems arising in $ac$ diagnostics. This refers to the numerous instances in which the well-known Maxwell theory’s predictions regarding the effective AC conductivity of various colloidal media were violated (see reviews [12,13,14]) and to the correct interpretation of these anomalies within the RC impedance approximation.

2. Consistent ${\mathit{\tau }}_{\mathit{c}}$—Definition from Relaxation Theory

In relaxation theory, a self-consistent analysis involves solving a system of equations that includes the hydrodynamic continuity equation and Ohm’s law for the current ${\overrightarrow{j}}_{\pm }$ in the electrochemical representation.

$$e{\dot{n}}_{\pm }+div{\overrightarrow{j}}_{\pm }=0,\left|e\right|{\overrightarrow{j}}_{\pm }={\sigma }_{\pm }\nabla {\mu }_{\pm }$$

(1)

This analysis also considers the Poisson equation, which relates local values of the electric potential $\phi$ to the density of mobile charges $n_{\pm }$ in the same region. Here, ${\sigma }_{\pm }$ is the conductivity of the mobile charged components of the electrolyte, and ${\mu }_{\pm }(n_{\pm },\phi )$ is the explicit form of the electrochemical potential.

2.1. Relaxation Time ${\tau }_c$ in the Force Approximation

Let us start with a method known as the force approximation. The integral form of the continuity Equation (1) (integration is performed in the direction perpendicular to the planes of gate electrodes of the scheme in Figure 1 over the interval $0\le x\le L/2$) is given by

$$e{\dot{n}}_s\left(t\right)-[j\left(0\right)-j(+L/2)]=0,or,e{\dot{n}}_s\left(t\right)=\sigma j\left(0_{+}\right),$$

(2)

$$n_s\left(t\right)={\mathit{\int }}_0^{L/2}n(x,t)dx,{\dot{n}}_s\equiv d{n}_s/dt.$$

Equation (2) considers that the volume current $j(+L/2)$) ends to zero at the gate electrode boundary and converts it to surface charge density $n_s\left(t\right)$. The accumulation layer on the boundary corresponds to the layer of negatively charged particles on the right side of the cell (Figure 1). Similar transformations of continuity Equation (2) on the left side of the cell yield the following result:

$${\dot{n}}_s\left(t\right)=\sigma j\left(0_{-}\right),|j\left(0_{-}\right)|=|j\left(0_{+}\right)|$$

(3)

Combining Equations (2) and (3) and considering that $j\left(0\right)\simeq {\sigma }_0(E_{\perp }-4\pi e{n}_s)$ and $E_{\perp }=V_g/\left(ϵL\right)$, we can derive the equation for ${\dot{n}}_s\left(t\right)$.

$$e{\dot{n}}_s\left(t\right)={\sigma }_0(E_{\perp }-4\pi e{n}_s),n_s(t=0)=0,{\displaystyle \frac{1}{{\sigma }_0}}={\displaystyle \frac{1}{{\sigma }_{+}}}+{\displaystyle \frac{1}{{\sigma }_{-}}}.$$

(4)

Here, ${\sigma }_0$ is the effective conductivity of the balk electrolyte. In the definition $E_{\perp }=V_g/ϵL$, the value $ϵ$ corresponds to the static part of the dielectric constant of water. At the initial moment of time, a potential difference $V_g$ is applied to the gate electrodes of the system Figure 1.

Equation (4) illustrates the general point that in the cell shown in Figure 1, it is impossible to organize stationary conditions for measuring the current–volt charateristic (IVC) of the electrolyte. Moreover, reasoning (2)–(4) shows that the relaxation time ${\tau }_c$ can be a rather simple function of ${\sigma }_0$. According to (3), ${\tau }_c\propto {\sigma }_0^{-1}$. This prediction is easy to verify, but so far, numerous studies on the subject (see reviews [12,13,14]) have not realized such verification. In this regard, let us examine the description of relaxation phenomena that accompany the development of instability in a charged liquid (water) surface under the influence of an external electric field [15]. The measurement results show the presence of saturation (Figure 1) and qualitatively confirm the predictions $Q(t\to \infty )\to Q_{\infty }$ of Formula (4). However, the characteristic relaxation time for water ${\tau }_c\gg {\sigma }_{aq}^{-1}$ is much longer than the predictions (3)–(4). Indeed, based on the known conductivity of water at room temperature ${\sigma }_{aq}$ [6,8], ${\sigma }_{aq}\simeq 50$ μS/cm, (${\sigma }_{aq}\simeq ({10}^2-{10}^3)$ s⁻¹ in CGSE), one would expect ${\tau }_c$ to be in the range of ${\tau }_c\sim ({10}^{-2}-{10}^{-3})$ s. This scale is not comparable to the observations shown in Figure 1. These data show ${\tau }_c>>10$ s. Thus, it becomes clear that Equation (1) should be studied using the electrochemical approximation. This was carried out (mainly numerically) by the authors of the seminal paper [1].

2.2. Relaxation Time ${\tau }_c$ in Electrochemical Approximation

The final results [1] follow from (1) for the charge $Q\left(t\right)$ with step-time excitation on the plates of the gate capacitor of the cell in Figure 1 with "blocking electrodes" (see the inset) in the bulk limit of interest to us (${\lambda }_D/L\ll 1$, ${\lambda }_D$ is the Debye length for a given electrolyte, and L is the dimension of the flat cell in the direction of the perturbing electric field), which indicate that the function $Q(t/{\tau }_c)$ starts from zero, is monotonic in time, contains one relaxation time ${\tau }_c$, and behaves exponentially in the region $t\ge {\tau }_c$,

$${\tau }_c\simeq {\lambda }_{D}L/D_{\pm },{\lambda }_D^2={\displaystyle \frac{ϵT}{2z^2{e}^2{n}_d}},{\zeta }_{\pm }=\left|e\right|D_{\pm }/T.$$

(5)

$$\delta Q(t>{\tau }_c)=[Q_{\infty }-Q(t\ge {\tau }_c)]\propto exp(-t/{\tau }_c),Q\left(t\right)=S{\int }_0^{L/2}{n}_{+}(x,t)dx.$$

(5a)

Here, T is the temperature in energy units, z is the number of elementary charges on a single ion, $D_{\pm }$ are the diffusion coefficients in the system of ± electrolyte ions, and ${\zeta }_{\pm }$ is their mobility related to $D_{\pm }$ by the Einstein relation (by definition, under these conditions, the mobility ${\zeta }_{\pm }$ should not depend on $n_{\pm }$).

The value $Q\left(t\right)$ characterizes the total charge of the accumulation layer. The integral in its definition is taken over the half of the interval $-L/2\le x\le +L/2$. The sign of the charge can be omitted, since $n_{\pm }(x,t)$ is an odd function of the coordinate x. $Q_{\infty }$ is the asymptotic value of function $Q\left(t\right)$ in the range $t\gg {\tau }_c$.

The origin of the time ${\tau }_c$ is explained by the authors [1] with the following estimates: (1) ${\tau }_c=\sqrt{{\tau }_L{\tau }_D}$ is ${\tau }_L=L^2/D$; (2) ${\tau }_L=L^2/D$ is the characteristic diffusion time at a distance of L between the gate electrodes; (3) ${\tau }_D={\lambda }_D^2/D$ is the characteristic time for forming the accumulation layer.

According to the definition of ${\tau }_c$ (5), information about the properties of $D_{\pm }$ (conductance ${\sigma }_{\pm }$) appears alongside with the details of the problem that determine the screening length ${\lambda }_D$. In other words, measurements of $Q(t/{\tau }_c)$ alone are insufficient to determine the properties of $D_{\pm }$ in the $ac$ mode. Independent information on ${\lambda }_D$, either experimental or calculated, is needed. Note that here, as in [16], ${\lambda }_D\ne {\lambda }_0$.

3. Relation of the Relaxation Time ${\mathit{\tau }}_{\mathit{c}}$ to the Time ${\mathit{\tau }}_{\mathit{R}\mathit{c}}$

A phenomenological theory for describing the relaxation properties of electrolytes has emerged since the publications of Gouy and Chapman (1909–1913), as noted in retrospect [1]. This formalism employs Equation (6) to calculate $Q\left(t\right)$.

$$R{\displaystyle \frac{dQ}{dt}}+{\displaystyle \frac{Q}{C_0}}=V_g\left(t\right),C_0\simeq S/{\lambda }_{solut}$$

(6)

Let us assume $V_g\left(t\right)=V_g\theta \left(t\right)$. In this case,

$$Q\left(t\right)=C_0{V}_g[1-exp(-t/{\tau }_{rc})],Q\left(0\right)=0,{\tau }_{rc}=R{C}_0,{\lambda }_{sout}\ll L.$$

(7)

Here, R is the resistance of the bulk of the cell with an electrolyte, $C_0$ is the static electrolytic capacitance at the metal–electrolyte interfaces, and S is the area of the contacting electrodes in plane-parallel geometry. ${\lambda }_{solut}$ is the effective screening length at the metal–electrolyte interface (in general, ${\lambda }_{solut}\ne {\lambda }_D$),

The properties of the function $Q\left(t\right)$ in Figure 1 are majorised by the dependence $Q\left(t\right)$ being the solution (7) of Equation (6) under the conditions $V_g\left(t\right)=V_g\theta \left(t\right)$. In this case, the time scales ${\tau }_c$ (5) and ${\tau }_{rc}=RC$ (7) are similar.

$${\tau }_c/{\tau }_{rc}=1.$$

(8)

This statement emphasised by the authors of Ref. [1] allows us to introduce the characteristic time ${\tau }_{rc}=RC$ and take advantage of new opportunities to gain a better understanding of what is happening.

It should be noted that the time constant ${\tau }_c$ arises from the analysis of the properties of Equation (1) in the context of Ohm’s law in the electrochemical representation. In this formalism, the force and diffusion components of the current are dependent on each other. The linearity of general Ohm’s law (1) with respect to the density $n_d$, as used in [1], is contradicted by the results of the theory [2,3,4], in which the mobility of electrolyte charges depends on the density $n_d$. In turn, the impedance time ${\tau }_{RC}=RC$ from (7) avoids difficulties when analysing the properties of Equation (1). This time contains the product of the force part R of the electrolyte conductivity and its diffusion component C, expressed as the electrolytic capacitance of the cell. There is a possibility to replace the conductivity’s force component from [2,3,4] in the definitions of the constants of Equation (6) with the force form of Ohm’s law presented by Equation (13) from Maxwell’s theory. This theory is generally used to describe the conducting properties of two-phase systems. By definition, the conductivity (13) of Maxwell’s theory is linear with respect to the nanoparticles density.

This combination is unique because the conductivity of the electrolyte, given by the relationship $\sigma \propto R^{-1}$, can be measured in $ac$ mode. The result (8) is only valid for solution (7) with the initial perturbation $V_g\left(t\right)=V_g\theta \left(t\right)$. Here, $\theta \left(t\right)$ is the step function. The difference between (1) and (6) lies in the model assumption that the capacity $C_0$ (6) exists over the entire time interval, including the initial moment of the perturbation $V_g\left(t\right)=V_g\theta \left(t\right)$. In this case, the behaviour of $Q\left(t\right)$ is determined by Equation (1). Solving this problem in the region $t\ge {\tau }_c$ answers the question of whether a single characteristic time, accessible to observations, exists. The answer for ${\tau }_c$ is positive, which means that the relaxation processes can be used to solve the problem of ohmic processes in electrolytes.

$AC$-methods using periodic pumping are popular. For a field of type $V_g\left(t\right)=V_{g}exp\left(i\omega t\right)$, we obtain

$$i\omega {\tau }_{rc}{Q}_0+Q_0=C_0{V}_g,or,Q_0\left(\omega \right)={\displaystyle \frac{C_0{V}_g}{1+i\omega {\tau }_{rc}}},Im{Q}_0\left(\omega \right)\propto {\displaystyle \frac{\omega {\tau }_{rc}}{1+{\omega }^2{\tau }_{rc}^2}}$$

(9)

The maximum of the imaginary part of $Im{Q}_0\left(\omega \right)$ occurs at the frequency point ${\omega }_{max}{\tau }_c=1$. As in (7), the definition of the parameter ${\tau }_{rc}$ in the periodic pumping problem (9) provides information about the combination ${\tau }_{rc}=RC$.

Another possible scenario is to use the phase relations between $V\left(t\right)$ and $J\left(t\right)$. If $V\left(t\right)=V_{g}cos\left(\omega t\right)$, then the corresponding current $J\left(t\right)$ is

$$J\left(t\right)={\displaystyle \frac{U_{0}cos(\omega t-\theta )}{\sqrt{R^2+{\omega }^{-2}{C}^{-2}}}}$$

(10)

The lock-in tester consistently measures the real and imaginary parts of the impedance. As a result, the following definition applies to the phase shift, $tan\theta$, between the current $J\left(t\right)$ and the control voltage $V\left(t\right)$.

$$-tan\theta \left(\omega \right)=1/\left(\omega RC\right).$$

(11)

The $RC$ combination can be extracted from the available data by measuring $\theta \left(\omega \right)$ and representing—$tan\theta \left(\omega \right)$ as a function of $1/\omega$ (11). As in Equations (7) and (9), the treatment of Equation (11) provides information about the $RC$ parameter.

The lack of justification at the microscopic level [1] for the phenomenology (6), (7) under $\theta \left(t\right)$ pumping conditions negates the attractiveness of techniques (9)–(11) (examples of their use are discussed below) compared to the more cumbersome method (7). So far, however, problem (1) with periodic pumping has not been solved.

4. Conductivity of Dilute Electrolytes According to Maxwell’s Theory

An important remark in the introduction and comments on Formula (1) concerns the real conductivity properties of the electrolyte. In the context of (1), electrohydrodynamics from [1] confirm the existence of Equation (6), provided that the conductivity ${\sigma }_{\pm }$ of electrolyte ions is a linear function of their density $n_{\pm }$ (otherwise, the second formula from (1) derived from Einstein’s rules does not work). The theory of Debye, Huckel, and Onsager [2,3,4] is more complex. In fact, the conductance ${\sigma }_{\pm }=e_{\pm }{n}_{\pm }{\zeta }_{\pm }$, where ${\zeta }_{\pm }$ is the mobility of the free ions, containing two corrections depending on the density $n_{\pm }$, both relaxation and electrophoretic. The second of them, ${\zeta }_{ef}$, has a simpler structure and is the same for all types of ions. It is equal to (see Formula (26.29) from [11])

$${\zeta }_{ef}=-1/\left(6\pi \eta {\lambda }_D\right)$$

(12)

where $\eta$ is the dynamic viscosity of the fluid, and ${\lambda }_D$ is the Debye length. This length decreases with increasing $n_d$. This determines the electrolyte’s overall drop in conductivity with increasing nanoparticle concentration. Returning to Equation (6), we see that the density of the nanoparticles should affect the ion mobility, resulting in a dependence on $R\left(Q\right)$. The nonlinearity of Equation (6) should significantly affect the structure of its solution. It may turn out to be a power law instead of an exponential function (see, for example, the comments in [17] regarding the properties of nonlinear relaxation equations). This change can be observed. For this reason, studying the asymptotics properties of $Q(t\ge {\tau }_{rc})$ and determining the properties of ${\tau }_{rc}$ enables us to evaluate the linearity of Equation (6).

The possibility of a linear dependence of electrolyte conductivity on volume fraction among the properties of Equation (1) suggests an alternative to the Debye–Huckel–Onsager charge transfer mechanism [2,3,4]. Maxwell’s mechanism [18] seems appropriate. The conductivity $\sigma \left(\varphi \right)$ of finely dispersed two-phase mixtures is determined by Maxwell’s equations [18]:

$${\displaystyle \frac{\sigma \left(\varphi \right)}{{\sigma }_0}}=1+{\displaystyle \frac{3({\sigma }_{\odot }/{\sigma }_0-1)\varphi }{{\sigma }_{\odot }/{\sigma }_0+2-({\sigma }_{\odot }/{\sigma }_0-1)\varphi }},\varphi =n_{\odot }{R}_0^3\le 1$$

(13)

where ${\sigma }_{\odot }$ is the effective conductivity of nanoparticles, ${\sigma }_0$ is the conductivity of base liquid (water, alcohols), and $\varphi$ is the volume fraction of nanoparticles. When deriving Equation (13), which is valid in the linear region $0<\varphi <1$, it is assumed that the nanoparticles do not move when the external field is introduced. Their presence distorts the electrolyte current lines (see Figure 2). This affects the effective conductivity $\sigma \left(\varphi \right)$ (13).

Figure 2 is related to nanoparticles of different nature.

In the case of spherical phase inclusions, the gray zone in Figure 2 occupies the entire volume of each nanoparticle. Small, point-like charged particles with a Bohr radius of $R_0\simeq R_{Bohr}$ are surrounded by a screening region with an increased density of counterions. According to Maxwell’s theory, this is equivalent to the including a phase with increased conductivity. In multi-charged systems (e.g., $DLVO$—colloids; the abbreviation is formed from the initials of the authors of the original papers [19,20]), the radius $R_0$ of the central charged nucleus is significantly larger than the Bohr radius $R_{Bohr}$ ($R_0\gg R_{Bohr}$). This nucleus is surrounded by a globular screening layer of thickness ${\lambda }_0$ with the following possible variants: $R_0\ge {\lambda }_0$, or $R_0\le {\lambda }_0$. The density of the counterions increases inside such a layer (exceeding the bulk density). Consequently, ${\sigma }_{\odot }/{\sigma }_0>1$.

In the case of suspensions, the nanocluster nucleus (either dielectric or metal, with a radius of $R_0$) has no ionic conductivity. However, there are image forces of electrostatic origin at the boundaries of these nanoparticles. Consequently, the dielectric nanoparticles are surrounded by a layer deficient in intrinsic solvent ions, resulting in a value of ${\sigma }_{\odot }/{\sigma }_0<1$. The metallic nanoparticles, which attract solvent ions, have an effective conductivity of ${\sigma }_{\odot }/{\sigma }_0>1$.

The additive $[\sigma \left(\varphi \right)-{\sigma }_0]$ to the base fluid conductivity is linear in $\varphi$ in the region of small $\varphi \ll 1$. This property distinguishes Maxwell’s transport theory from the Debye–Hückel approach. The behavior of $\sigma \left(\varphi \right)$ can be derived from the general Formula (13). Thus, if ${\sigma }_{\odot }/{\sigma }_0\to \infty$,

$${\displaystyle \frac{\sigma \left(\varphi \right)}{{\sigma }_0}}{|}_{max}\le (1+3\varphi ).$$

(14)

Otherwise, ${\sigma }_{\odot }/{\sigma }_0\to 0$

$${\displaystyle \frac{\sigma \left(\varphi \right)}{{\sigma }_0}}{|}_{min}\ge (1-3\varphi /2).$$

(15)

It is not immediately apparent how Maxwell’s formalism relates to charge transport in electrolytes. In the case of electrolytes, charges move under the influence of a homogeneous external field created within the conducting medium (1). The Maxwell transport Equation (13) describes how neutral inclusions with finite volume fraction affect the conductivity of an electrolyte (the mechanism of this influence is shown schematically in Figure 2). If the screening properties of the intrinsic electrolyte are taken into account, the problems (1) and (13) overlap. In this case, any charges generated in the solution by the dissociation of the dopant must be screened by the ions of its intrinsic electrolyte. As a result, each charge is transformed into a neutral formation, as shown schematically in Figure 2.

There are a number of ways in which the processes of neutralization of ions resulting from the dissociation of nanoparticles can be manifested. Recall the results of Wagner, Onsager, Samaras [21,22] (WOS) regarding the behavior of charges near the interface of two media with different dielectric constants, the structure of DLVO colloids in dilute electrolytes, and the metal–electrolyte interface [23,24]. Recall also the direct Maxwell theory calculations of conductivity for dilute colloidal solutions performed in these works.

In cases [21,22], we are referring to the statement

$$F_{wos}\left(z\right)\simeq F_{ϵ}\left(z\right)exp(-2z/{\lambda }_{solut}),F_{ϵ}\left(z\right)={\displaystyle \frac{q^2({ϵ}_1-{ϵ}_2)}{4{ϵ}_1({ϵ}_1+{ϵ}_2)z^2}},$$

(16)

which the authors of these papers discovered. Here, $F_{ϵ}\left(z\right)$ is the classical image force for a test point charge q [25], and ${\lambda }_{solut}$ is the effective screening length. The values of the dielectric constants ${ϵ}_i$ are the dielectric constants values of the contacting media. The charge q, which manifests classically in zone $z<{\lambda }_{solut}$, is self-screened and does not interact with the boundary in zone z > ${\lambda }_{solut}$. The force $F_{wos}\left(z\right)$ appears when studying the properties of surface tension at the vacuum–electrolyte interface, where the value of surface tension is significantly renormalized [25,26].

In order to apply the solution of DLVO electrostatic problem [23,24], we reformulate the result (16). If these integrally neutral formations approach the interface from the electrolyte bulk, they must acquire a finite charge by partially losing ions to reach the state $Q_{*}=Z^{*}e$, where $Z^{*}\ll Z$ and $R_0\gg {\lambda }_0$. A typical DLVO colloid of radius $R_0$ has a charged nucleus with a charge $Q=Ze,Z\gg 1$, which is completely screened by counterions of the electrolyte. The $DLVO$ colloids at the metal–electrolyte interface are shown in Figure 3.

As of Maxwell’s formalism, it is only natural to present some successful examples of its use. Figure 4 contains data from [12] and shows the existence of the transient region ${\varphi }_c$, which is defined as $n_{\odot }{R}_0^3\sim 1$. In the region $n_{\odot }{R}_0^3\le 1$ to the left of this area, adjacent screening spheres do not touch each other. In this region, according to (14), ${\sigma }_{\odot }/{\sigma }_0$ should be linear in $\varphi$. To the right of this transition region ${\varphi }_c$, the overlap of the screened regions of DLVO colloids increases, and the laws of the proposed MSA (mean spherical approximation) formalism [27,28,29,30] are self-consistent. This theory is consistent with the results of the Debye–Hückel–Onsager theory [2,3,4]. The observed data (black squares) in Figure 4 is explained quantitatively by the appearance of a dynamic charge, $e{Z}_{din}$ (white circles, right-hand ordinate), on each of the moving particles. The values of $e{Z}_{din}$ increase as the volume fraction $\varphi$ decreases. To the right of this transition region, the overlap of the DLVO colloids screening regions increases, according to the authors [27,28,29,30]. As mentioned above, the effective conductivity of such a medium cannot be a linear function of the volume fraction of nanoparticles.

In the region where $\varphi \ll 1$, which is represented by the $\sigma \left(\varphi \right)/{\sigma }_0$ ratio data in Figure 5 of a colloidal medium from [31] (similar to that in [13]), $\sigma \left(\varphi \right)/{\sigma }_0$ is linear in $\varphi$. This behaviour is commonly observed in the linear transport of colloidal media, which contrasts sharply with the well-known percolation theory of conductivity in dilute solutions (i.e., weakly doped media) under conditions of $\sigma \left(\varphi \right)\to 0$ [32].

5. Impedance Diagnostics of Dilute Electrolytes

The success of the Maxwell conductivity theory of colloidal solutions (Figure 4 and Figure 5) has coincided with a growing interest in Maxwell conductivity anomalies in dilute colloidal solutions (see reviews [12,13,14]).

Experiments on the transport of nanoparticles in dilute nanosuspensions are of great interest for a variety of applications. One area of research is the study of Maxwell conductivity anomalies in these types of medium. Studies conducted by various authors using different nanosuspensions have shown that traditional methods for measuring the electrical conductivity of diluted media can yield unexpected results (see reviews [13,14]). According to Maxwell’s theory (estimations (13)–(15)), the conductivity values $\sigma \left(\varphi \right)$ increase linearly with an increase in the parameter $\varphi )$. However, this increase (or decrease) is anomalously large compared to the theoretical estimations (14) and (15).

For example, Figure 4 shows the conductivity data of the $DLVO$ water-based colloidal solution from [12]. The experiment, conducted in the region left of ${\varphi }_c,(\varphi <{\varphi }_c)$, demonstrates that $\sigma \left(\varphi \right)/{\sigma }_0\simeq (1+50\varphi )$ with a slope $d\sigma \left(\varphi \right)/d\varphi$ is ten times greater than what Maxwell’s theory can account for. For comparison, the corresponding MSA method data are also shown in the same figure. These theories, developing the ideas of Debye, Huckel and Onsager [2,3,4], provide a much better explanation of the data in Figure 4 than the Maxwell approximation. However, as will be seen below, the Maxwell anomaly for the data to the left of the region where ${\varphi }_c\simeq 1$) is explained by the $RC$ interpretation.

The anomaly is more pronounced for the diamond powder in alcohol (the average size $R_0$ of the diamond nanoparticles is about $R_0\simeq 4$ nm). According to Figure 6 of the author’s data [33], doping the alcohol with dielectric nanoparticles increases the conductivity of the suspension by approximately three orders of magnitude rather than decreasing it, as predicted by Equations (11) and (13) (see Figure 6, where $\sigma \left(\varphi \right)/{\sigma }_0\simeq 1+3734\varphi$!)

Clearly, it is not only the conductivity properties (14) in the limit of ${\sigma }_{\odot }/{\sigma }_0<<1$ that are affected. Additional factors must be contributing to the formation of the anomaly. Constructive suggestions are contained in [34,35]. They noted that in linear $ac$ diagnostics (4)–(9), it is not the conductivity of the medium that is measured but rather the time ${\tau }_{rc}=RC$. Not only the resistance $R\left(\varphi \right)$ but also the electrolytic capacitance $C\left(\varphi \right)$ of the metal–alcohol interfaces that are inevitably present in the measurement can both be sensitive to $\varphi$. The very definition of time, ${\tau }_{rc}$, in the form ${\tau }_{rc}\simeq RC$ was already taken into account by the pioneers of methods for accurate conductivity measurements of ${\tau }_{rc}\simeq RC$ (see details in the classic book on this topic, [5]). The nature of the dependence $C\left(\varphi \right)$ and its influence on the properties of ${\tau }_{rc}$ have not yet been discussed. The dependence $C\left(\varphi \right)$ is most evident in experiments with dielectric nanoparticles. Base liquids (e.g., water or alcohol) favor the formation of a large electrolytic capacitance, $C_0\propto S/{\lambda }_{aq}$, at the metal–electrolyte interface, where ${\lambda }_{aq}$ is the characteristic screening length for a given solvent (water, for example). For more details on the formation of ${\lambda }_{aq}$ in this case, see [34,35]. The appearance of nanoparticles with a density of $\varphi$ in the bulk of the suspension is accompanied by their adsorption at the metal–electrolyte interface.

For neutral nanoparticles, the development of such a reversible process (single-layer adsorption according to Langmuir theory or its multilayer generalization BET theory [36]) leads to the appearance of a dielectric layer at the metal boundary with a thickness $d\left(\varphi \right)$, which increases monotonically with the increase $\varphi$ in the suspension.

Identifying the origin of the dielectric layer $d\left(\varphi \right)$ between the metal and the electrolyte enables us to trace its impact on the effective electrolytic capacitance $C\left(\varphi \right)$ of the metal–electrolyte interface. By definition,

$$C\left(\varphi \right)V_g^2/2={\int }_0^{\infty }{\displaystyle \frac{E^2(x,y,z)}{8\pi }}dxdydz,$$

(17)

where $E(z,y,z)$ is the distribution of fields in the volume of the parallel-plate capacitor, and $V_g$ is the external voltage on the control electrode. Assuming

$$C\left(\varphi \right)=S/\Lambda \left(\varphi \right),E(x,y,z)\to E\left(z\right),0\le {\phi }_1\left(z\right)\le dandd\le {\phi }_2\left(z\right)\le \infty ,$$

we have from (17)

$$V_g^2/(2\Lambda )={\displaystyle \frac{1}{8\pi }}\left[{\int }_0^d{(d{\phi }_1/dz)}^{2}dz+{\int }_d^{\infty }{(d{\phi }_2/dz)}^{2}dz\right].$$

(18)

Here, $\Lambda$ is the effective penetration length of the field into the electrolyte volume, $0\le z\le d$ is the interval of the electrolyte volume occupied by the ${ϵ}_1$ dielectric layer, $d\le z\le \infty$ is the volume occupied by the electrolyte with a dielectric constant ${ϵ}_2$, and the distributions ${\phi }_1\left(z\right)$ and ${\phi }_2\left(z\right)$ arise from the solution of the system of equations in the sandwich approximation.

$$\Delta {\phi }_1=0,{\phi }_1\left(0\right)=V_g,{ϵ}_{1}d{\phi }_1/dz{{|}_{z=(d-0)}={ϵ}_{2}d{\phi }_2/dz|}_{z=(d+0)}$$

(19)

$$\Delta {\phi }_2=={\phi }_2/{\lambda }_{aq}^2,{\phi }_1{{|}_{z=(d-0)}={\phi }_2|}_{z=(d+0)},{\phi }_2(\infty )\to 0$$

(20)

In the limit $d\to 0$, the quantity $C\left(\varphi \right)$ is reduced to

$$C_1\simeq {ϵ}_{2}S/{\lambda }_{aq}$$

(21)

and has an electrolytic origin. Otherwise, if $d\gg {\lambda }_0$, the quantity $C\left(\varphi \right)\to C_2$

$$C\left(\varphi \right)\simeq {ϵ}_{1}S/d$$

(22)

is determined by planar electrostatics.

In the transition region $d\ge {\lambda }_{aq}$, the general structure of $C\left(\varphi \right)=S/\Lambda$ is formed from the asymptotic expressions (21) and (22) according to the rules for calculating the total capacitance C of two capacitors $C_1$ and $C_2$ connected in series: $C^{-1}=C_1^{-1}+C_2^{-1}$. The general calculations (17)–(22) confirm this rule and provide the definition of $\Lambda$,

$$\Lambda \simeq {\lambda }_{aq}+d$$

as used in [34,35], but only asymptotically (in the sense of (21) and (22)). Nevertheless, this is sufficient to draw qualitative conclusions about the properties of $C\left(\varphi \right)$ (17) and (18). Therefore, we can use the estimation (23) for $C\left(\varphi \right)$ following from Equations (17) and (18)

$$C\left(\varphi \right)\simeq {\displaystyle \frac{S}{{\lambda }_{aq}+d\left(\varphi \right)}}.$$

(23)

According to (23), the capacitance $C\left(\varphi \right)$ decreases monotonically with increasing film thickness $d\left(\varphi \right)$ in the region $d\left(\varphi \right)\ge {\lambda }_{aq}$. The presence of a drop is suitable for observation if the initial capacitance $C_0\propto S/{\lambda }_{aq}$ is large enough and the quality of the metal–electrolyte interface is good. Additionally, the sizes $R_0$ of the nanoparticles must be relatively small, $R_0\le {\lambda }_{aq}.$ The current state of the art in nanosuspension preparation [13,14] makes creating the necessary conditions quite possible.

Among the numerous publications and reviews on the subject [13,14], work [33] provides data that confirms the hypothesis [34,35] that the electrolytic capacity of the metal–water interface depends on the variable volume density $\varphi$. The primary source of information (data Figure 7) for the final results presented in Figure 6 is preserved [33]. The authors of the papers discussed in reviews [13,14] seem to consider the information in Figure 7 [33] as auxiliary or intermediate material. These are the relaxation resonances in linear $ac$ diagnostics, based on Equation (6), in the presence of periodic pumping. Such resonances were studied in detail by the authors of Ref. [33] and are shown in Figure 7. Among the available scenarios for determining the relaxation time ${\tau }_{rc}$, i.e., (7), (9) and (11), method (9) seems to be the clearest. However, its justification from first principles (1) is still missing. The ${\tau }_{rc}$ values resulting from the data in Figure 7 for $ND97$ and $ND87$ [33] samples are summarised in

Table 1. Data for $t_{rc}\left(\varphi \right)$ values (in s) in a cell filled with two different suspensions. Corresponding properties of diamond powders: (ND97-EG-Nano.Diamond 97-EthilenGlycol; ND87-EG-Nano.Diamond 87—EthilenGlycol) data [33].

$\mathit{\varphi }$	ND97-EG ${\mathit{\tau }}_{\mathbf{rc}}\left(\mathit{s}\right)$	ND87-EG ${\mathit{\tau }}_{\mathbf{rc}}\left(\mathit{s}\right)$
$0.0000$	$1.07·{10}^{-2}$	$1.07·{10}^{-2}$
$0.0032$	$2.64·{10}^{-4}$	$2.64·{10}^{-4}$
$0.0080$	$9.63·{10}^{-5}$	$1.89·{10}^{-4}$
$0.0163$	$6.68·{10}^{-5}$	$9.63·{10}^{-5}$
$0.0338$	$2.51·{10}^{-5}$	$9.63·{10}^{-5}$

. To demonstrate how sensitive the technique is to the composition of the nanoparticles used (diamond powder), Figure 7 shows two sets of relaxation resonance data for suspensions with slightly different nanoparticles, as presented in Table 1.

For both nanosuspensions, the data in Table 1 shows a sharp decrease in the ${\tau }_{rc}$ time as the parameter $\varphi$ increases. Following convention (reviews [12,13,14]), the impedance capacity does not change significantly when $\varphi$ is varied, i.e., $C\left(\varphi \right)\simeq const$. In this case, as in most of the works collected in the reviews [12,14], we have the result in the form of Figure 6, which reveals a giant conductivity anomaly compared to the results of Maxwell’s theory. The possible alternative is obvious. We assume that there are no anomalies in the conductivity of the electrolyte. In this case, the main reason for explaining the behaviour of ${\tau }_{rc}\left(\varphi \right)$ from the Table 1 is that $C\left(\varphi \right)\ne const$. The graph of this dependence considering the asymptotic $\sigma \left(\varphi \right)$ in the form (15) is shown in Figure 8.

It becomes evident that the significant dependence of the parameter ${\tau }_{rc}\left(\varphi \right)$ on the diamond powder density, as shown in Table 1, arises mainly not from anomalies in the Maxwell conductivity behavior (as interpreted in most publications from [12,13,14]) but from the strong dependence of $C\left(\varphi \right)$ shown in the graphs of Figure 8.

The reason for the strong dependence of capacitance on the concentration of nanoparticles in suspension (including diamond nanoparticles [33]) remains to be explained. The answer to this question appears to be one of the main achievements of [34,35] and can in general be explained by Formula (23). The proposed addition concerns observable phenomena that help explain what happens to the conductivity of a dielectric suspension when the size $R_0$ of individual nanoparticles changes. This refers to the data presented in Figure 9 [37], which preceded the publication of [33]. This encourages further research in the region of smaller nanoparticle sizes, $R_0$. In experiments by Zila et al. [33], this threshold shifts to $R_0\simeq 4$ nm (Figure 6 and Figure 7). The single-layer Langmuir adsorption theory and its multilayer generalization ($BET$ theory [36]) show that the thickness of an equilibrium-formed adsorbent layer depends on the size $R_0$, ($<d\left(R_0\right)>\sim R_0$). The results in Figure 9 show the opposite: the smaller the $R_0$, the thicker the $<d\left(R_0\right)>$. This question deserves attention and can be resolved by applying existing concepts from the theory of surface phenomena at the metal–electrolyte interface (see the discussion in [23,24] regarding the data in Figure 3).

If a nanoparticle in solution interacts with the interface according to a $V\left(z\right)$ law, then the Boltzmann distribution of such particles $n\left(z\right)$ in the vicinity of the interface has the following form:

$$n\left(z\right)=n_{0}exp[\pm V(z)/T],V\left(z\right)\ge 0$$

(24)

The density $n_0$ corresponds to its value in the volume of the solution, where $V\left(z\right)\to 0$. In this expression, the plus sign indicates attraction of particles to the boundary, and the minus sign indicates repulsion.

The model we would like to use originates from the work of Wagner, Onsager and Samaras (WOS) [21,22]. In this model, the surface density $N_s$ of nanoparticles is introduced by the following formula:

$$N_s={\int }_0^{\infty }dz[n\left(z\right)-n_0]$$

(25)

This integral must be calculated accurately, since the resulting characteristic potentials $V\left(z\right)$ are singular at the interface: $V\left(z\right)\propto z^{-n};n=1,2,3$. Approximation (25) has been successfully used by the authors [21,22] to describe the effect of electrolyte properties on the surface tension of the electrolyte–vacuum interface. In this case, the V(z) interaction between the charges and the boundary is repulsive. Expression (25) converges at the boundaries of the integration interval. We use this method to describe the metal–electrolyte interface, as discussed in References [23,24]. In this scenario, the charge interaction V(z) is attractive, and expression (17) diverges at the lower limit. There is only one way to overcome the non-integrable singularity in the definition of $N_s$ (25). It is necessary to assume that nanoparticles have finite minimum dimensions $R_0$. In this case, we have from (25)

$$N_s\simeq {\int }_{R_0}^{\infty }dz[n\left(z\right)-n_0],V\left(z\right)=\gamma /z^n$$

(26)

$$<d\left(r_0\right)>\simeq n_0{R}_0[exp[+V(z)/T]-1]/n_0=R_{0}exp[+\gamma /\left(T{R}_0^n\right)]\gg R_0$$

(27)

Equation (27) (with the Boltzmann basis) resolves the question of the role of the parameter $R_0$ in the formation of a layer with a thickness of $<d\left(R_0\right)>$, confirming the consistency of the observations presented in Figure 9 from [37].

6. Summary

A unified system of equations has been proposed that enables the consistent analysis of relaxation phenomena in dilute colloidal solutions and nanosuspensions. It has been shown that, given the presence of Ohm’s law definitions in the system, a modification of Maxwell’s theory in finely dispersed two-phase media can be applied to the problem. It is shown that within the framework of the linear (Maxwell) relaxation theory, the characteristic time ${\tau }_{rc}$ can be extracted from the experimental data in the combination ${\tau }_{rc}\left(\varphi \right)\simeq R\left(\varphi \right)C\left(\varphi \right)$. Here, $R\left(\varphi \right)$ is the ohmic resistance of the dilute electrolyte volume, and $C\left(\varphi \right)$ is the electrolytic capacitance of the metal–electrolyte interface, depending on the pumping mode (either the $\theta \left(t\right)$-shaped or periodic excitation) and the dimensionless density $\varphi$ of the nanoparticle density in the dilute electrolyte. By monotonically increasing the density $‘\varphi ’$ of nanoparticles, we stimulate not only a change in $R\left(\varphi \right)$ but also a change in the value of capacitance $C\left(\varphi \right)$. Based on the theory outlined above along with modern knowledge of screening processes at the metal–electrolyte interface, it is possible to understand the reasons of the observed conductivity anomalies in colloidal media compared to Maxwell’s theory, especially in diluted nanosuspensions with dielectric nanoparticles. As it turns out, increasing the concentration of nanoparticles in solution monotonically not only changes $R\left(\varphi \right)$ (the expected effect) but also changes the capacitance $C\left(\varphi \right)$ (more significantly than $R\left(\varphi \right)$). As a consequence, the procedure for extracting information about $R\left(\varphi \right)$ from the data for ${\tau }_{rc}$ requires independent data about the properties of $C\left(\varphi \right)$. This problem is relatively easy to solve for suspensions containing dielectric nanoparticles.

This scenario is considered using the example of the suspension “alcohol + nanodiamond powder”. A consistent system of equations is proposed, which makes it possible to interpret relaxation phenomena in dilute colloidal solutions and nano-suspensions. It is shown that in the role of Ohm’s law, which is necessarily present in the system of definitions, its modification from Maxwell’s theory for transport in finely dispersed two-phase media can be used.

A retrospective [1] of activity on diffusion kinetics in electrolytes contains references dating back to the end of the 18th century. Special interest in “Maxwell anomalies” in the conductivity of liquid suspensions emerged around 2010–2022, mainly due to potential applications (as even weak doping with impurities could probably significantly increase the conductivity of the suspension). Publications on this topic continue to appear (see [38,39,40,41,42,43]).

From a general perspective, a promising area for future research is studying (in situ) the phenomenon of adsorption of various impurities at the metal–electrolyte interface using transport methods, as discussed in this review.

Of particular interest is a detailed study of the cooperative effect associated with the formation of colloidal droplets at the metal–electrolyte interface filled with DLVO colloids, as shown in Figure 3. This is a first-order phase transition associated with the tendency of a system of interacting particles (three- or two-dimensional) to pass from the gaseous to the liquid state. The effect can be consistently interpreted within the framework of van der Waals theory [44]. This was experimentally observed in a series of studies with two-dimensional colloidal systems at the metal–electrolyte interface [45,46,47,48]. Moreover, the data [47] indicate that the resulting droplets have a crystalline rather than a liquid structure (see Figure 10 of [47]). This result allows us to better understand the problem of the existence of Coulomb crystallization in colloidal $DLVO$ solutions of finite density. The existence of such a crystal is discussed by Alexander et al. [49] when referencing data from [50,51,52,53,54,55,56,57,58]. In fact, they are discussing a droplet phase transition with droplets in the form of individual crystallites (as can be seen in Figure 10 of [47]).