On the ac Measurements of the Electrical Conductivity of Dilute Colloidal Electrolytes

Round 1

Reviewer 1 Report

The article by Chinika et al. dealing with the ac measurement of electrical conductivity is interesting. However, I have some queries, which need to be addressed before its publication.

1.       Dilute colloidal electrolyte…Is it dilute suspension of colloidal particles.

2.       What is the type of colloids considered in the present study, is it rigid? Or any other type.

3.       I couldnot find any meaning to illustrate Fig. 2 as it is just reproduction of the results in  Ref. 13.

4.       Authors must rewrite the Introduction section to address the existing  knowledge gap and their contribution properly. The same must be reflected in conclusion section.

OK

Author Response

We are grateful to the Reviewer for careful reading of our manuscript and valuable comments and advises.

1. Dilute colloidal electrolyte…Is it dilute suspension of colloidal particles.

Yes, corresponding definition is added in line 10.

2. What is the type of colloids considered in the present study, is it rigid? Or any other type.

                        In the experiments discussed, the suspensions with all sorts of possible inclusions are used: metals, dielectrics, DLVO colloids. In our work, we deal in details with the suspensions containing the DLVO colloids: complex clusters of charged nucleous surrounded by the cloud of counterions. Suspensions containing the metallic and dielectric inclusions are discussed at the qualitative level which nonetheless allows elucidate the most important features of their properties  (see explanations to formulas (9), (10) ).

3. I couldnot find any meaning to illustrate Fig. 2 as it is just reproduction of the results in 13.

 Figure 2 (in the corrected text, it is denoted by Fig 1) is the basis for formulation of the problem which we study in our work. Indeed, the Maxwell's formula (1) is able to explain the linear dependence of $\sigma(\phi)$ and its slope when this slope does not exceed the critical value $[\sigma(\phi)-\sigma_0 ]/\sigma_0 \le 3\phi $. The data of Fig. 1 illustrate the fulfillment of this requirement providing the specific numbers.

The aim of this paper is to discuss possible causes contributing to the violation of the above requirement. One example of its serious break down is demonstrated in Figure 2 (old number Figure 1).

4. Authors must rewrite the Introduction section to address the existing knowledge gap and their contribution properly. The same must be reflected in conclusion section.

                       We have reorganized the Introduction and made the Fig.1 (ex Fig.2) as the central point for formulation of the goal of investigation. We extended the caption to it and hope that thanks to this rework the Introduction became clearer. We have extended also the Conclusions section positioning more clearly the place of our study.

Reviewer 2 Report

The authors have carried out a theoretical description of experimental results of other authors on the a.c. conductivity of nanoparticle suspensions. The study include dielectric and magnetic particles, and dielectric and non-dielectric solvents. Different concentrations of inert salts are included in some of the systems. The results are able to explain some of the discrepancies observed between the experimental results obtained for the slope of conductivity vs. volume fraction of particles, in the high dilution limit, and the predictions based on the use of Maxwell theory without taking into account the adsorption of counterions on the particle surfaces, and the possible overlaping of the double layers. The analysis is restricted to concentrations lower than that at which such overlaping starts to be produced. The topic of the manuscript is appropriate for the scope of the journal, but revision is necessary before it can be accepted for publication.

COMMENTS.

1.- The manuscript will be hard to follow for colloid researchers non expert on impedance spectroscopy, despite the topic is important for nanosuspensions, foams, etc. The authors overlook the fact that most researchers perform measurements of conductivity at constant frequency, usually 1 kHz. Therefore, they should give a short explanation about the main differences in an appendix.

2.- It has been shown in many articles dealing with dielectric spectroscopy that in the simplest cases in which impedance relaxations are coupled with conductivity and electrode polarization effects, a simple Maxwell mode and the Jong conductivity term (Eq.1) is not valid at low frequencies. Indeed, in a log-log plot of the complex part of the dielectric permitivity, two linear regions appear, one due to the conductivity and another to the electrode polarization. A relatively simple way to distinguish between them is the difference in the slope. Indeed, in these cases, it has been frequently shown that the use of electrical modulus is the best choice for a correct differentiation between conductivity and relaxation processes. Please, clarify this, and discuss whether the variables chosen for the analysis are the most appropriate.

3.- It would be important that the authors indicate which is the frequency range of the measurements analyzed. Do the data plot in figure 2 correspond to the extrapolation to zero frequency? In this case the decoupling between the conductivity and electrode polarization contributions would be crucial.

4.- Please, add error bars to the results and fitting parameters in the figures and tables.

5.- The RC circuit model used is too simple for complex systems over a broad frequency range (Eq.8), see, e.g., Encinar et al.

6.- In figure 4 the results have a high dispersion. To be sure that there is any trend as a function of Fi, the error bars should be given.

7.- 

There are typos that should be corrected.

Author Response

We are grateful to the Reviewer 2 for his careful reading of our manuscript and valuable comments and advises.

1.- The manuscript will be hard to follow for colloid researchers non expert on impedance spectroscopy, despite the topic is important for nanosuspensions, foams, etc. The authors overlook the fact that most researchers perform measurements of conductivity at constant frequency, usually 1 kHz. Therefore, they should give a short explanation about the main differences in an appendix.

The main part of our impedance formulas (definitions (5)-(8)) meets the basics of the AC measurements formalism widely presented in the university courses. We added corresponding reference (ref. [26]).  We do not see special sense to transfer this information to the Appendix. 

2.- It has been shown in many articles dealing with dielectric spectroscopy that in the simplest cases in which impedance relaxations are coupled with conductivity and electrode polarization effects, a simple Maxwell mode and the Jong conductivity term (Eq.1) is not valid at low frequencies. Indeed, in a log-log plot of the complex part of the dielectric permitivity, two linear regions appear, one due to the conductivity and another to the electrode polarization. A relatively simple way to distinguish between them is the difference in the slope. Indeed, in these cases, it has been frequently shown that the use of electrical modulus is the best choice for a correct differentiation between conductivity and relaxation processes. Please, clarify this, and discuss whether the variables chosen for the analysis are the most appropriate.

   We discuss the peculiarities of the Maxwell's model (1) because it is wrongly used in many articles dealing with dielectric spectroscopy to explain the properties of $\sigma(\phi_{\odot})$ in the DC-mode. However, its use in these conditions is practically impossible. After switching on of a constant potential difference $V$ between the control electrodes, the electric field in the cuvette volume is very quickly shielded due to the fields from the emerging accumulation layers along the metal-electrolyte boundaries, and no currents remain in the cuvette volume. So such measurements cannot be performed. 

       Yet, the formula (1) can be exploited in the AC - transport measurements by using corresponding Ohm's law in the impedance scheme (5). The capacitance $C$ in this case consists of different components, including the electrolytic capacitance of the metal-electrolyte boundary.  Comments to this effect are clarified in the text accompanying formula (5) .  

 3.- It would be important that the authors indicate which is the frequency range of the measurements analyzed. Do the data plot in figure 2 correspond to the extrapolation to zero frequency? In this case the decoupling between the conductivity and electrode polarization contributions would be crucial.

              That's a good question. But the authors of most of the papers labelled [11-16] do not ask such a question at all. It is believed that standard instruments, Oxford Instrument, are smart enough to give the final result for conductivity without burdening users with impedance details. We engage the impedance "kitchen" for a possible explanation of the observed paradoxes.

4.- Please, add error bars to the results and fitting parameters in the figures and tables.

During the August the CEA laboratories are closed and we cannot get the required information in the given for us by Editor 10 days term. We can do this in September.

5.- The RC circuit model used is too simple for complex systems over a broad frequency range (Eq.8), see, e.g., Encinar et al.

We are aware that the RC circuit model is too simple for complex systems over a wide frequency range (Eq. (8)). However, practically all experimental works on the subject do not talk about the details of measurements at all, processing their data in the framework of the Maxwell model (1), which as we already explained does not "work" in the DC-mode. Under these conditions, our task is not to discuss all possibilities of determining the conductivity of colloidal solution using impedance data, but to avoid the observed paradoxes by correctly interpreting the results of AC measurements. 

By the way, we are not aware of the work of Encinar et al. cited by Referee.    Having the reference to this publication, we could comment it.

6.- In figure 4 the results have a high dispersion. To be sure that there is any trend as a function of Fi, the error bars should be given.

Through the whole August the CEA laboratories are closed and we cannot get the required information in the given for us by Editor 10 days term. We can do this in September.

7.-  Comments on the Quality of English Language

All found typos are corrected.

Reviewer 3 Report

This work by Ioulia Chikina, Sawako Nakamae, and Andrey Varlamov proposed a theoretical argument using impedance diagnostics to resolve the discrepancy between the experimental data and the Maxwell’s theory regarding the relationship between colloidal fraction and effective conductivity of the colloidal suspension. However, I found that their arguments are not clearly conveyed, and I did not see how their conclusions could provide enough help to the further interpretation of experimental results. In addition, I also found that the use of English in this paper made it very hard to understand. Below are my comments towards the scientific aspect of this work. For comments on the language, please see another section.

1. What is  in the denominator of equation 1?

2. I was wondering if the authors could explain more about the purpose of Figure 2. I did not see in detailed discussion of the results in Figure 2, especially the actual fitting values.

3. In section 2, the authors seem to attribute the discrepancy between experimental fitting values and theoretical predictions to the screen length lambda. I was wondering if the authors could talk more in the text about what the implications of this conclusion are and how their conclusions could help to interpret the experimental data. In particular, I was wondering if the authors could provide a concrete numerical example to demonstrate their approach and further show how their proposed quantities should be measured in experiments.

4. Similar for section 3, the conclusion seems to be hand wavy since the authors only acknowledge that there is no reason to not use Maxwell theory to interpret the experimental data. I was wondering if the authors could explain more about what the benefits of using Maxwell theory to interpret the experimental data while the discrepancy exists. And could the authors also discuss how to compensate for the discrepancy using theory so that the theoretical model could predict the actual experimental data?

1. Line 36 should be 'cited experiments.'

2. Close to line 101. 'When the role of capacitor plays the cell filled by pure water, it acquires well known in radio engineering literature specifics' this sentence is hard to understand.

3. Line 106, ' grows also their number in the vicinity of the electrode, grows the number of colloidal particles mirror images behind the metallic interface', not look correct to me.

4. There is an extra ')' in line 110.

5.  Close to line 114, 'In Ref. [25] was demonstrated' is not correct.

6. Line 115, 'The area S does not enter more in this expression' is not correct, maybe 'The area S does not enter this expression anymore'?

7. In line 125, 'As wee have' should be 'As we have'.

8. In line 134, 'The obtained in Refs. [13,26] results' is not correct.

Author Response

We are grateful to the Reviewer 3 for his careful reading of our manuscript and valuable comments and advises

1. What is in the denominator of equation 1?

              This is just coma at the end of formula. To avoid misunderstanding we insert the additional interval between the formula and coma in eq. (1).

2. I was wondering if the authors could explain more about the purpose of Figure 2. I did not see in detailed discussion of the results in Figure 2, especially the actual fitting values.

Figure 2 (in the corrected text, it is denoted by Fig 1) is the basis for formulation of the problem which we study in our work. Indeed, the Maxwell's formula (1) is able to explain the linear dependence of $\sigma(\phi)$ and its slope when this slope does not exceed the critical value $[\sigma(\phi)-\sigma_0 ]/\sigma_0 \le 3\phi $. The data of Fig. 1 illustrate the fulfillment of this requirement providing the specific numbers.

The aim of this paper is to discuss possible causes contributing to the violation of the above requirement. One example of its serious break down is demonstrated in Figure 2 (old number Figure 1).

We have reorganized the Introduction and made the Fig.1 (ex Fig.2) as the central point for formulation of the goal of investigation. We extended the caption to it and hope that thanks to this rework the Introduction became clearer. Bottom of Form

3. In section 2, the authors seem to attribute the discrepancy between experimental fitting values and theoretical predictions to the screen length lambda. I was wondering if the authors could talk more in the text about what the implications of this conclusion are and how their conclusions could help to interpret the experimental data. In particular, I was wondering if the authors could provide a concrete numerical example to demonstrate their approach and further show how their proposed quantities should be measured in experiments.

      The paper draws attention to the fact that Maxwell's theory in the form of formula (1) is not suitable for treating measurements of the conductivity of nanoparticles suspensions in the $DC$ -mode.   In this version of measurements, the perturbing external field is shielded by the internal fields that inevitably arise in the cuvette with electrolyte. It is more correct to use Ohm's law (1) in the $AC$-mode, when the electrolyte properties are described by Eq. (5) (impedance approximation). In this equation, the role of capacitance $C$ is played mainly by the electrolytic capacitance $C_0$ (see Eq. (9)) of the two metal-electrolyte boundaries, on which a perturbing voltage $U(t)$ is applied.   The measured quantity is, for example, the phase shift $tan(\theta(\omega)$ (see Eq. (7)).   Its definition includes the combination of $RC$.     As long as the capacitance $C_0$ is a known constant the measurements $tan(\theta(\omega)$ make it possible to determine the resistance $R$ of the electrolyte and its dependence on the density $N_{odot}$  of the nanoparticles. But in the case of suspensions of different origin, the capacitance $C$ itself turns out to be an essential function of $N_{odot}$ ( this is the essence of our communication) .   As a consequence, the measurements of the phased shift yields not $R(N_{odot})$ , but $R(N_{odot})C(N_{odot})$ .     And this is something to reckon with.      

To clear up these details we have added at the end of Section 2 the following text:   

Note that the details of the description of experiments in the cited papers [11–16]  devoted to measurements of the conductivity of suspensions of different types of nanoparticles are almost identical. Thus, the type of the used device is indicated, sometimes the operating frequency at which this device works optimally, and the results of σ(φ⊙) measurements are given, including the value of σ0. All authors assume that the algorithms required to extract the dependence of σ(φ⊙) from the impedance formalism are already  included in the instrument software. However, as we have shown above, this is not the case when measuring the transport characteristics of nanoparticles suspensions.

It is necessary to evaluate beforehand which situation we are talking about based on the behavior of the function (σ(φ⊙) − σ0)/σ0. If this value remains less than 3φ⊙ for all investigated concentrations, as it is the case for the data presented in Fig. 1, then the  instrument together with its standard software can be trusted. If, however, when working with oxides (read, dielectrics) we get instead of (σ(φ⊙) − σ0)/σ0 ≈ −3φ⊙/2 an inversion of this combination by sign and (σ(φ⊙) − σ0)/σ0 ≫ φ⊙ (as it takes place in Fig. 2), the interpretation of the data using only Maxwell’s formula (1) becomes meaningless.

In this case for the treatment of the data presented in Fig. 2 using Eq. (13) the  product σ(φ⊙)λ(φ⊙) becomes essential. The first multiplier takes into account the decreaseof the suspension conductivity with increase of the volume fraction (φ⊙ of dielectric nanoparticles, the second considers the influence of the latter’s adsorption on the metal-electrolyte interface (see Eqs. (10)-(11)) on the capacitance value (see Eq. (9)).  

The proposed treatment of the anomaly in the behaviour of (σ(φ⊙)/σ0 for aluminium nitride granules in alcohol suspension (see Fig. 2) is only qualitative and probably not the only one. But no alternative to our proposal is seen in the hundreds of papers on the  subject, collected in reviews [15,16 ].

4. Similar for section 3, the conclusion seems to be hand wavy since the authors only acknowledge that there is no reason to not use Maxwell theory to interpret the experimental data. I was wondering if the authors could explain more about what the benefits of using Maxwell theory to interpret the experimental data while the discrepancy exists. And could the authors also discuss how to compensate for the discrepancy using theory so that the theoretical model could predict the actual experimental data?

                  We already answered on this question above.

Comments on the Quality of English Language

1. Line 36 should be 'cited experiments.'

DONE

2. Close to line 101. 'When the role of capacitor plays the cell filled by pure water, it acquires well known in radio engineering literature specifics' this sentence is hard to understand.

Thank you. We have excluded useless statement and now the sentence reads as “When the role of capacitor plays the cell filled by pure water 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 electrodes $L$ ($\lambda_0 \ll L$)”.

3. Line 106, ' grows also their number in the vicinity of the electrode, grows the number of colloidal particles mirror images behind the metallic interface', not look correct to me.

Thank you. We have rewritten the corresponding paragraph as follows:

“Indeed, as the concentration of colloidal particles in the bulk of the electrolyte increases, their number in the vicinity of the electrode also grows, and, accordingly, the number of associated mirror images increases. Thus, an increasingly dense dipole gas of colloidal particles with corresponding images is formed in the vicinity of the electrode”.

4. There is an extra ')' in line 110.

DONE

5. Close to line 114, 'In Ref. [25] was demonstrated' is not correct.

DONE

6. Line 115, 'The area S does not enter more in this expression' is not correct, maybe 'The area S does not enter this expression anymore'?

DONE

The area $S$ is no longer included in this expression

DONE

7. In line 125, 'As wee have' should be 'As we have'.

DONE

8. In line 134, 'The obtained in Refs. [13,26] results' is not correct.

DONE: The results for compressibility obtained in Ref [13,27]

Round 2

Reviewer 1 Report

Acceptable in present form

Reviewer 2 Report

The authors have taken into account my comments to the previous version, thus I consider that it can be accepted for publication.

There still are some typos.

Reviewer 3 Report

The authors have addressed my questions. Now I am recommending the publication of their manuscript.