Richard Kerner’s Path Integral Approach Aims to Understand the Self-Organized Matter Agglomeration and Its Translation into the Energy Landscape Kinetics Paradigm

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript reviewed Richard Kerner’s path integral method and a few applications for self-
organized matter agglomeration. In this article the author reformatted the theory and showed it’s
equivalent to the energy landscape kinetics paradigm when the energy barrier is much smaller than
thermal energy, validating the seemingly non-physical Boltzmann factor used in the transition matrix.
This manuscript provided a simple yet clear clarification of Richard Kerner’s path integral method. I
would recommend accepting the paper after the following questions are addressed.
1. At the beginning of section 2, the author pointed to 3 common objections to this method, can the
author provide a few citations to relevant publications? Section 3 is mostly relevant because of
those common objections; thus a few citations can further support its relevance.
2. The author summarized a few systems where this method was successfully applied in the
introduction and showed that Boltzmann factor here is justified in the limit of minimal
transitional barrier. Could the author comment on the validity of the assumptions on other
systems, and provide guidelines when this method can be used, or suggest a few other systems
where this method might be relevant.
3. In line 83-85, it states that “The only condition is to assume contact with a thermal bath with a
well-defined temperature”. But in conclusion, it claims that a small transitional barrier comparing
to temperature is required. They appear to conflict with each other.
4. In line 141 to 144, appeared in all four equations, but physically, they seem should be
𝜂, 𝛼, 𝜂, 𝛼, respectively, since they correspond to As-Se and As-As bounds.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The author introduced the RK model for the network glassy structure evolutions. The RK model is based on the valence structure and each atom have a certain coordination number. The network structures of glasses can be controlled by changing the composition of atoms with different coordination numbers. So this theory have a certain value historically. However, the reality is not so simple, as described by the following standard textbooks. (1) F. A. Cotton and G. Wilkinson, "Advanced Inorganic Chemistry" 5th edition (John Wiley & Sons, 1988) (2) Shriver and Atkins "Inorganic Chemistry" 3rd edition (Oxford 1999).

In p. 171 of ref. (1), structures of several borate ions are shown. The B(O-)4 and B(O-)3 structures coexist. Actually the pressure crystal phase of B2O3 consists of B(O-)4 units. (3) Acta Cryst. B24, 869 (1968).

In p. 403-404 of ref. (1), several structures of molecular compounds (P)n(S)m and (As)n(S)m are shown. S can form double bonds with P, so that the composition of the molecular chalcogenides varies from (P)4(S)3 to (P)4(S)10. The  coordination numbers vary 3 and 4 for P and 1 and 2 for S.

In p. 70 of ref. (2), the term "hyper-valence" is introduced. P and S have larger coordination numbers than 4, which is given by the valence bond theory.

The recent PAW DFT molecular dynamics calculations of liquid B2O3 shows the existence of the two-fold coordination structure OB(O-) at normal pressure and the B(O-)4 structure at high pressure, which contribute the diffusion of atoms.

(4) Rhys. Rev. B78, 224206 (2008); ibid. B81, 014208 (2010)

Therefore fixing the coordination numbers are sometimes dangerous.

Your Figure 1 assume that the liquid glassy melt consists of atoms. It is also very unrealistic. I think that the electronic MD calculations like ref. ($) may be necessary to understand the bond breaking events in the liquid state.

As you explained in Sec. 2  and 3, the results of MD calculations of glasses can be simplified by using the Markov state model. The problem is that the RK model only includes the solidification reaction, cluster(j) + monomer -> cluster(j+1). We also include the decomposition reaction, cluster(j+1) -> cluster(j) + monomer and also  the rearrangements of the cluster surface, cluster(j) <-> cluster(j).

I also found an error in your Eq. (9). It should be [(1-p(t))exp(-E1/T(t))-p(t)]. So, Eq. (10) also should be corrected.

By using the Markov state model, not only the activation energy E(i,j), but also the activation entropy S(i,j) can be obtained according to W(i,j) = exp[-E(i,j)/T+S(i,j)], where E(i,j) and S(i,j) are separated from the temperature dependence. The equilibrium energy difference E1 is obtained by considering the reverse reaction and by E(i,j)=E1+V; E(j,i)=V. Experimentalists have been using this method to analyze the chelate reactions.

(5) Coordination Chemistry Reviews 133, 39-65 (1994).

You may be able to discuss the residual entropy in this way.

The RK theory may have contributed much enabling many students to write their thesis, but it may be your tusk to step over the model and find new jobs for future students. These are my opinions. I hope that you can get some ideas to include above references to improve your paper.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf