The Relationship between Electrical Conductivity and Electromigration in Liquid Metals
Choong-un Kim
Round 1
Reviewer 1 Report
The manuscript must be carefully revised with respect to the real free randomly moving electron density in metals.
Comments for author File: 
Comments.pdf
The English language is sufficiently good.
Author Response
My answer to 1st review
Author Response File: 
Author Response.docx
Reviewer 2 Report
The paper is devoted for electrical conductivity analysis in liquid metals. The topic is generally interesting, however the paper contain unexplained places (below) and need major revisions.
Figs. 1, 2, Tables 1, 2 should be commented. In Table 1 is not clear meaning ‘’exp’’. The name of Table 1 is not clear.
For measurement described at lines 185-189 more details should be added.
Lines 278-281 ‘’Good agreement..’’ in this sentence I not find any hints to results presented in the work or in the external source.
Lines 269-270 ‘’However, the low accuracy of the experimental data’’, please indicate the accuracy of experiment.
Factor b, in the text is added value for copper, in Fig.6 is presented value for Te. Why only for these metals? How this parameter can be calculated? It is dependent on external parameters, for example temperature?
Why you present the data in Figs. 7-9?
Conclusions should be rewritten in more informative way.
Author Response
My corrections to 2nd reviewer
Author Response File: Author Response.docx
Reviewer 3 Report
This paper presents a theoretical analysis on the electron conduction in liquid metals (including alloys) and its connection to the effective valence of electromigration. A modified approach to the electron conduction in liquid metals (from Drude-Sommerfeld) leads to a new the correction factor b (to DS equation) in computing electrical conductivity. This correction factor b physically represents the varying degree of electron scattering at the Fermi surface of liquid metal. The resulting correction of effective valence of liquid metal (with the correction factor) fits very well to the experimental data than the factor proposed by Mott.
The idea of correction factor and treating the effective valence as the part of thermochemical potential (eq. 11 is an equilibrium condition in multicomponent system) is very refreshing and interesting. The results seem to match very with existing data and leading theory appears reasonable. One concern is the treatment on the electrostatic force (or true valence). The valance electrons even at fully ionized state may not be taken from the ionic valence in case of metallic system. The concern is the presence of the charge shielding by free electrons. Consideration on such effective is missing. When conductivity is under concern (wind force), the charge shielding is not playing role so overall theoretical frame presented in this paper is correct. However, when considering effective charge, then the electronic static force needs to be considered with inclusion of the shielding effect. Comment or analysis on this aspect seems necessary and may make the paper more valuable.
English is good except for occasional typos and grammatical errors. Careful proof reading is recommended.
Author Response
My corrections to 3 review
Author Response File: Author Response.docx
Reviewer 4 Report
8
Comments for author File: Comments.pdf
Author Response
Corrections - Review 4
All corrections are inserted in bold shrift
Reviewer 's mention:
- Introducton is too straight that could not able to catch the view of the author’s perceptonal
conceptualization about the proposed work.
Author's answer (p. 1-2):
Insert before the section "Calculation of electrical resistance of metal."
The basics of the theory of electromigration
These basics were formalized mainly in the 60s of the last century. Two forces act on a metal ion in the presence of an electric current - the field force and the electron wind force (Fix [3,18,19], Bresler and Pikus [44], Verhoeven [12], Mangelsdorf [45], Belashchenko [4,35] and many others). The field force is expressed in terms of the coulombic ("true") charge of the ion, and the wind force is described as the result of friction between the ion and the electron flow, or a consequence of electron scattering on the ion. There is no complete clarity regarding the true charge, but, according to V.B. Fix [31], this charge is equal to the number of electrons donated by the ion to the conduction band. This definition is accepted in many works (for example, Mangelsdorf [45], Belashchenko [4,6]). Most of the papers dealt with electromigration in crystalline metals with an activation diffusion mechanism, so that the properties of ions in the activated state were included in the discussion. For ions in the ground state, it was assumed that the total force acting on the ion is zero (Fix [3], Bresler-Pikus [44]).
Reviewer 's mention:
- I also intended to ask about formulas 1 to 4 in the introduction part. Which needs perfect references to support the statement that the authors have discussed. It seems that the author has considered the Columbic charge equation in their concept that trued applicable for the conductivity of metal ions in liquid and solid. I also got confused about how it could be directly implied to the theory. Please provide additional information that support your hypothesis and also attach the similar type of previous works that can provide basic information for your concept.
Author's answer (p.1-2): Insert just after the 1st one
The discussion of the relationship between conductivity and electromigration was mainly reduced to the derivation of equations in which the effective charge of the ion during electromigration is expressed in terms of the electrical resistance increment due to an impurity in a crystalline or liquid solvent (see the review of J. Verhoeven [12]). For the electrical resistance in this case, the Drude-Sommerfeld equation generally is used (Bresler and Pikus [44], Mangelsdorf [45], Belashchenko [4,6] and others).
Reviewer 's mention:
- On page number 148 to 152 author stated that "We believe....." this statement needs further
justification for the readers better understanding.
Author's answer (p.5): After the words: " can't really exist."
It is convenient to consider this question on the example of an ideal one-component metal without defects. In this case, the condition of mechanical equilibrium means that the total force exerted by the electric current on each atom is zero. Therefore, the force of the field and the force of the electron wind in such a metal are exactly equal to each other.
Reviewer 's mention:
- Moreover, the authors have portraits the applicability of triangle rules in the theory to fulfill their
conceptual theory. Please elaborate their actual functionality in the proposed b factor analysis.
Author's answer (p.13): Insert before the line "Calculation of the factor g."
The inclusion of the factor b in Equation (9) leads to an increase in the mean free path of electrons by a factor of b. The feasibility of the consistency rule means that the factor b of a given metal does not depend on the method of its calculation, that is, this factor is a function of the state.
Round 2
Reviewer 1 Report
Comments for author File: Comments.docx
Author Response
Dynamisc-2384920
Answer to 1st reviewer
The 1st review provides basic information from the theory of conductivity of metals. The effective number of conduction electrons whose energy is at a distance of ~kT from the Fermi level is determined, and it is indicated that only a few percent of all valence electrons (neff) participate in the conduction. The conductivity is expressed in terms of neff and the electron mobility D/kT (D is the electron diffusion coefficient). As a result, the review results in the formula (C6) for the conductivity ϰ = (1/3)q2g(εF)vF2 τF. If we take into account that q = e, g(εF) = (3/2)n/εF, εF = (mvF2)/2 and vFτF = L, then we get exactly the Drude-Sommerfeld equation (3) from my articles. Thus, the reviewer refutes his own statement (at the beginning of the review) that formulas (1) - (4) for the conductivity of metals are wrong.
Further, the reviewer singles out neff atoms from their total number and proposes to consider them as partially screened ions in a periodic lattice. The remaining n - neff atoms behave as neutral. The valence electrons associated with them do not participate in the transfer process. An increase in temperature enhances lattice vibrations and stimulates an increase in the number neff. This interpretation means that it is assumed that there are two types of identical atoms - ionized and neutral. In addition, the review assumes the presence of a crystal lattice, while liquid metal solutions are discussed in my article.
As a result, the review discusses the picture of conductivity in crystalline metals in the concept of the reviewer, but it is not the topic of the article - the relationship between electrical conductivity and electromigration in liquid metals. Then I can't use the information of The Review 1. Therefore, I can only formulate the differences between the mechanisms of conduction in the metals of the reviewer and the author, but it is not up for discussion in the article.
1) The reviewer considers in detail the various stages of the proposed overall mechanism. But the article implements a phenomenological approach, which uses only the necessary parameters - the charge of the ion and the cross section of electron scattering on ions, averaged over an ensemble of particles. The separation of particles into mobile and immobile is considered unobservable in the experiment and therefore is not considered. The mention on phenomenological approach of the article inserted after the title of paragraph Modification of the Drude-Sommerfeld equation. Author's variant.
2) Electromigration has been investigated in a significant number of binary liquid metal systems.
3) For the first time, the "consistence rule" was established and experimentally confirmed, which connects electromigration in triangles of monogenic binary systems A-B, B-C and C-A.
4) In the author's scheme, as in most other cases, the question of the magnitude of the true charge of the ions remains unclear. Usually it is taken equal to the group number of the element in the Periodic system, but this is not necessary. Some possibilities are given by checking the satisfiability of the consistence rule with different choices of these charges.
Insert in the text after the section heading
Modification of the Drude-Sommerfeld equation. Author's variant
Let's apply the phenomenological approach and return to the equations (3)-(5), ….
Corrected text is written in rose color
Reviewer 2 Report
Authors make proper corrections according to reviewer remarks and I suggest to publish the paper as it is.
Author Response
Dynamisc-2384920
Answer to 2nd review
Corr. to Fig.1. According to Mott, the g factor is the ratio of the density of states N(ε) at the Fermi level (solid line) to the density of states in the free electron model (dashed line).
Corr. to Fig.2. The Bi impurity in dilute Cd-based solutions has a large negative effective charge.
After words "They are shown in Table 1":
The true charges of the ions were taken equal to the group number of the element in the Periodic system
Corr. to Table 1. Abbreviation exp changed to experim
The title of Table 1 changed to Checking the correspondence between the ratios σ2/σ1 from the equations of electrical conductivity (5) and electromigration (14)
The title of 2nd column in Table3 1 changed to σ2/σ1 using (14)
Just after Table insert Significant discrepancies are seen between the experiment and the ratios of the resistances in pair of metals calculated from the electromigration data.
The expression just before Table 2 "However, the low accuracy of the experimental data should also be noted" changed to: However it should be noted that the accuracy of electromigration paraneters (10-20%) is lower than at the conductivity data.
The expression "Good agreement between calculation and experiment was obtained only for Pb, Bi, Sb; for Ag, Cu, Sn, the discrepancy reaches 50–100%, and for Hg and Al, it is even greater. In [7], especially noticeable discrepancies - by a factor of 2 - 3 - were obtained for Cd, Hg, Ga, In." changed to:
Good agreement between calculation and experiment was obtained only for binary systems Cd-Pb, Cd-Bi, Cd-Hg, In-Tl, Sn-Sb (discrepancy less than 10%). For systems Cd-Sn, Ga-Hg, Hg-Tl, Hg-Sn, discrepancies reach 2- 3 times [7].
Before Fig.6: the maximum volumetric heat of evaporation between the metals considered. The smooth dependence of b factor on the value ΔH/V allows to predict the factors b for some metals not studied up to now (Li, Rb, Au, Be, Mg, Ca, Ba, Si) [6].
After Figure 8. "… at X1 ≈ 1 it is equal to σ2/σ1 = 10.74. It means that this ratio changes non-linearly in ~140 times. Such cases are common when the resistivity isoterms go through high maximum. The new investigations of systems like Na-K and similar ones Na-Cs and K-Cs (Figures 7-9) allow to verify our hypothesis that the cross-sections σ1 and σ2 are partial values with respect to mean cross-section .
In conclusions. After words "between conductivity and electromigration":
Let us repeat again, that some principal points were invented and applied in the discussion: 1) ion cross-sections for conductivity and electromigration coincide, 2) cross-sections of the components are partial values with respect to the mean cross-section, 3) the basic equation of electromigration is established for monogenic solutions, 4) factor b is included in the Drude-Sommerfeld equation for conductivity. As a result, the consistency rule is discovered that connects the electromigration parameters in triangles of every three binary metallic systems A-B, B-C and C-A. The fulfillment of this rule allows to confirm the important properties of metallic systems.
After this insert the regular text begins: "The calculated values of factor b ……."
All corrections in article are given in red color for clarity.
Reviewer 4 Report
Authors have done a good revision. Thus this manuscript needs publication warrant.
Round 3
Reviewer 1 Report
The author for calculation used the Drude model. This model is not applicable to metals (see Attachment). The experimental part can be published as independent work.
Comments for author File: Comments.pdf
Author Response
Dynamisc-2384920
Answer to 1st reviewer, his 3rd comment
The reviewer still believes that the Drude-Sommerfeld equations (1)-(4) are unsuitable for calculating the conductivity of metals, and proposes another equation - (C6) in the first review and CV in the third review. He did not pay attention to my objection in response to the first review that this equation can easily be reduced to the Drude-Sommerfeld equation for conductivity. However he agrees that for a hundred years the Drude-Sommerfeld equation has been applied continuously. For example, it is stated in the textbook of N.W. Ashcroft and N.D. Mermin "Solid State Physics" (Chapters 1 and 2).
Perhaps he believes that the DS equation is not applicable to real metals due to deviations from the free electron approximation? However, my calculations regarding the relationship between conductivity and electromigration refer specifically to the free electron model (that is, to elemental metals), which is clearly seen from the text of the article. To avoid confusion, I amended the text of the article after equation (3):
These equations are valid in the free electron approximation. This approximation will be taken into account in what follows.
This eliminates the rest of the reviewer's comments on the density of states. As a result, I can only repeat the reviewer's phrase:
The referee does not convince me that Drude-Sommerfeld free electron model can't be applicable for describing electron transport in metals. Really it has been used by many authors during the last hundred years.
Round 4
Reviewer 1 Report
There I can once more repeat that the Drude-Sommerfeld model is not applicable to metals.
Comments for author File: Comments.pdf
