Characterizing Atomic Interactions in Interstitial Non-Stoichiometric Compounds by Statistical Thermodynamics : Engineering Usage of Estimated Values of Statistical Thermodynamic Parameters

Statistical thermodynamics allows us to estimate atomistic interactions in interstitial non-stoichiometric compounds MXx through analysis of experimentally determined pressure-temperature-composition (PTC) relationships for MXx being in equilibrium with X2 in gaseous state ( ) , , or X H N P S = or for non-stoichiometric carbide MCx being in equilibrium with excess C. In case of analysis for MCx, chemical activity ( ) a C of C in place of partial pressure ( ) 2 p X of X2 gas must be known. On statistical modelling of crystal lattice structure for MXx, an a priori assumption of constant nearest-neighbour X X − interaction energy ( ) E X X − within a homogeneity composition range at arbitrary temperature T was accepted to determine number θ of available interstitial sites for occupation by X atoms per M atom. Values of interaction parameters estimated as such appear rational and realistic noting consistency of the values for M’s in the same group in the Periodic Table of the Elements and compatibility with enthalpy values evaluated by conventional thermodynamic approach. Engineering insights gained for MXx through analysis of atomistic interaction parameter values evaluated by the statistical thermodynamics are reviewed comprehensively in this paper. M might be substitutional alloy 1 y y A B − composed of constituents, A and B, or MZz containing another interstitial constituent Z besides X. Insights acquired from this line of statistical thermodynamic analysis appear to be of pragmatic use for advanced alloy design as shall be demonstrated hereafter. How to cite this paper: Shohoji, N. (2017) Characterizing Atomic Interactions in Interstitial Non-Stoichiometric Compounds by Statistical Thermodynamics: Engineering Usage of Estimated Values of Statistical Thermodynamic Parameters. Journal of Modern Physics, 8, 365-381. https://doi.org/10.4236/jmp.2017.83025 Received: November 22, 2016 Accepted: February 25, 2017 Published: February 28, 2017 Copyright © 2017 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access


, , or X H N P S
= or for non-stoichiometric carbide MC x being in equilibrium with excess C. In case of analysis for MC x , chemical activity ( ) of X 2 gas must be known.On statistical modelling of crystal lattice structure for MX x , an a priori assumption of constant nearest-neighbour X X − interaction energy ( ) E X X − within a homogeneity composition range at arbitrary temperature T was accepted to determine number θ of available interstitial sites for occupation by X atoms per M atom.Values of interaction parameters estimated as such appear rational and realistic noting consistency of the values for M's in the same group in the Periodic Table of the Elements and compatibility with enthalpy values evaluated by conventional thermodynamic approach.Engineering insights gained for MX x through analysis of atomistic interaction parameter values evaluated by the statistical thermodynamics are reviewed comprehensively in this paper.M might be substitutional alloy 1 y y A B − composed of constituents, A and B, or MZ z containing another interstitial constituent Z besides X. Insights acquired from this line of statistical thermodynamic analysis appear to be of pragmatic use for advanced alloy design as shall be demonstrated hereafter.

Introduction
Statistical thermodynamic analysis procedures were comprehensively summarized by Fowler and Guggenheim in a classical monograph published in 1949 [1].Statistical thermodynamics is considered as a bridge connecting between invisible atomistic scale microscopic world and experimentally observable macroscopic state for interstitial non-stoichiometric compound MX x possessing composition [ ] [ ] x X M = being in equilibrium with X 2 gas at partial pressure ( ) 2 p X at temperature T. By statistical thermodynamic analysis of equilibrium pressure-temperature-composition (PTC) relationships for MX x , nearest-neighbour atomic interaction energy ( ) between i and j atoms and atomic partition function i f of constituent i in MX x might be calculated ( ) Statistical thermodynamic parameters evaluated for extensive range of interstitial non-stoichiometric compounds including hydride, carbide, nitride, phosphide and sulfide were compiled in a monograph published by the author [2] that included calculation results reported by 2012 [3]- [46].All these analyses [2]- [48] were made accepting an a priori assumption of constant interaction energy ( )

E X X −
between nearest-neighbour interstitial atoms X within a homogeneity composition range of MX x lattice at arbitrary T. Parameter values estimated for M's in the same group in the Periodic Table of the Elements for given X were comparable to each other.This evidence appeared to support validity of the a priori assumption of the constant ( ) E X X − within a homogeneity composition range of MX x at any T although there is no rigorous first-principle-based justification for this a priori assumption.Further, statistical thermodynamic parameter values for ( ) such yielded enthalpy values comparable to those determined by the conventional thermodynamics for Cr 2 N [3] as well as for several hydrides [6] [18].
Hence, the atomic interaction parameter values evaluated as such by statistical thermodynamics must be considered realistic as well as rational.
Besides analysis for pure M, analysis was made also for substitutional alloy with as well as for ternary alloy MZ z X x containing another interstitial constituent Z besides X in which affinity of Z to M was stronger than that of X to M [13] [31] [32] [34].
In the early stage of this line of work to characterize nature of atomistic interaction in interstitial non-stoichiometric compound MX x [2]- [47], attention was not paid explicitly on engineering significance of the parameter values evaluated by the statistical thermodynamic analysis.However, after the analysis was made N. Shohoji to evaluate interaction parameters for H absorption behaviours for Va-group metal-based alloy membranes [47], it occurred to the author that it might be of pragmatic convenience if the correlation was established between the estimated values of the interaction parameters by statistical thermodynamic analysis and the reported H permeation performance for the Va-group metal-based alloy membrane materials.This led the author to summarize somewhat speculative paper [48] soon after [47].The background idea for this attempt of correlating the statistical thermodynamic parameter values evaluated for Va-group metal-based alloy membrane to the H permeation performance of the alloy membrane was to screen promising ones from candidate Va-group metal-based alloys so that the number of H permeation experiments could be minimized.H permeation experiment is time-consuming and the results are dependent on setting of ( ) p H on the inlet side and that on the outlet side.
This special issue of Journal of Modern Physics bears title "Engineering Thermal Physics" with "statistical thermodynamics" being included as one of the possible fields of concern.Thus, the author decided to summarize this manuscript to review comprehensively the engineering significances of the interaction parameters estimated by statistical thermodynamics reported in the published works during the last four decades [2]- [48].
As the main purpose of this manuscript is to demonstrate potential usefulness of evaluated atomic interaction parameters for MX x by statistical thermodynamics for advanced alloy design, PTC data sources used in the analyses are not cited as the References.Statistical thermodynamic analysis procedures for interstitial non-stoichiometric compound MX x shall be reviewed in the next Chapter although they might be referred to elsewhere [2] [3] [6] [44] [47] [48] as the statistical thermodynamic analysis procedure, unlike mathematically well-defined conventional thermodynamic analysis procedure, is not yet widely accepted as a standard analysis tool among materials researchers.
Among literatures cited in References section of this paper, [22] [23] [25] and [28] were works in which results of statistical thermodynamic analysis were made use of for interpretation of the discussed matters rather than works of statistical thermodynamic analysis itself and [41] was a preparative work to convert the equilibrium data format reported for Fe-Cr-P system by the original authors of the experimental work to a format adequate for straightforward statistical thermodynamic analysis that was published eventually as [42].

Fundamental Equations
Generalized fundamental formulae proposed for this line of analysis of interstitial non-stoichiometric condensed phase MX x are as follows.
Symbols used in the above formulae are classified as follows: <universal constants> R: universal gas constant (=8.31451J•mol −1 p X : equilibrium pressure of ideal gas X 2 , T: absolute temperature (K), x: composition ( X M atom ratio) in MX x , n X : number of X atoms in MX x , n M : number of M atoms in MX x , <parameters to be evaluated> Q: degree of stabilisation of X atom in MX x lattice with reference to isolated X and M atoms in vacuum, ( ) E i j − : interaction energy between i and j atoms in MX x lattice, ; that is, when one interstitial site in MX x is occupied by an X atom, ( ) terstitial sites are blocked from occupation by other X atoms.For example, in case that X atoms in MX x occupy octahedral interstitial sites (O-sites) expression for Q in close packed lattices like fcc (face centred cubic) and hcp (hexagonal close packed) is simply, but that for bcc (body centred cubic) lattice is expressed as taking into account second nearest neighbour interactions, ( ) to openness of the atom packing in the bcc crystal lattice [2] [8] [18].
Value of θ to fulfill the a priori assumption of constant ( ) homogeneity range of MX x at arbitrary T is usually close to the solubility limit of X in the MX x .For example, in the statistical thermodynamic analysis of hypo-stoichiometric Cr 2 N phase, θ was chosen to be 0.50 to fulfill the condition of constant ( ) When θ was chosen to be 1 (=θ 0 for O-site occupation of N in the hcp lattice), ( ) x showing trend of increasing positive (repulsive) ( ) with increasing x.If such variation of ( ) , it is more natural to accept phase change to occur rather than to hold the same crystal lattice structure [2].

Analysis Procedure
At the onset of the analysis, isothermal A vs. x plots must be prepared from available isothermal PC relationship at arbitrary T using Equation (1) by varying θ.As understood from Equation (1), slope of isothermal A vs. x plot would be- within homogeneity composition range of MX x at arbitrary T, θ yielding linear A vs. x relationship over entire homogeneity composition range of MX x must be chosen for the subsequent calculations.
Then, from the intercept g(T) calculated using Equation (1), K(T) vs. T relationship must be drawn using Equation (2).Term Q on the right hand side in Equation (2) refers to extent of stabilization of atom X in the MX x lattice due to formation of X M − bonds in the MX x lattice while the coefficient ( )  to T refers to electronic contribution to entropy term in ther- modynamic sense.In fact, partition function ( ) X f T of X atom in the MX x lattice is a T-dependent function as represented by Equation (4) but, as the T range of statistical thermodynamic analysis for MX x is typically no wider than 500 K, it has been a common practice to approximate ( ) X f T as a T-independent constant term [2]- [48].For convenience of the readers, flow chart of the calculation procedure is presented below as Figure 1.
As represented by Equation ( 5), term ( )  refers to the net extent of stabilization of X atom in the MX x lattice, ( ) ( ) for all combinations of i and j.For pragmatic convenience of calculating K(T) using Equation ( 2), ( ) ( )  values for X = H and N are presented in tabulated form in [2] and [37] at 100 K interval from 0 K up to 3000 K so that  value at arbitrary T is calculated readily by interpolation although values of ( )

Some Insights Drawn from Statistical Thermodynamic
Analysis Results for Interstitial Non-Stoichiometric Compounds

Stability of X in FeXx Lattice
As might be understood from expressions for fundamental equations reviewed

N. Shohoji
given M lattice difficult through conventional thermodynamic analysis.In contrast, statistical thermodynamic analysis results allow us to compare straightforwardly the relative stability of different X's in a given M as seen in Table 1 for M = Fe and X = H, C, N, P and S [20].
The more stable the X in Fe lattice the more negative would become ( ) in FeX x lattice.That is, according to Table 1, the stability of X in Fe lattice would decrease in the order of implying that C is the most stable and H is the least stable in Fe lattice.Further, it is notice in Table 1 that, for given X, stability in Fe lattice would vary depending on the lattice structure of Fe implying that the most stable state of C in Fe is realized in molten state, that of N in γ phase and that of H in α phase.
In Table 1, θ value of some MX x is not specified uniquely.This is due to inherent difficulty of determining exactly the value of θ for statistical thermodynamic Table 1.Estimated values of Q in miscellaneous FeX x (reproduced from Table 1 in [20]).analysis in very dilute interstitial solution under certain circumstances as discussed in some detail in [15].
First cases of statistical thermodynamic analysis for very dilute interstitial solutions was made in [11] in which a priori condition of constant ( ) noting the reality that, in the very dilute interstitial compound, there must be no neighbouring interstitial atom around any interstitial atom.
However, when solutions of H, C and N in α-Fe was investigated in terms of statistical thermodynamics, unambiguous specification of θ to fulfill condition (12) was difficult but, instead, when θ value was taken to be greater than certain threshold value, estimated value of Q converged to a constant level whereas, in the range of θ smaller than the threshold level, estimated value of Q showed steady variation with varying θ (cf. Figure 2 in [15]).On account of this situation, unique specification of θ was given up for some very dilute interstitial compounds and, as a compromising solution, θ value which must have been greater than the threshold level was used for the analysis because, by so doing, realistic value for Q was evaluated as discussed in [15] although value of the product Zf X varied as a function of θ in the range of θ where Q value became constant with θ.

Atom Clustering in Fe1−yMyXx around X Atom
During the course of statistical thermodynamic analysis of PTC relationships reported for N solution in molten Fe 1−y M y in which affinity of M to N is stronger than that of Fe to N, it was concluded that certain types of atom clustering might develop around interstitial N atom [19] [24] [26].This aspect shall be reviewed in the following.
As always in this line of statistical thermodynamic analysis, θ parameter values on analysis of molten Fe 1−y Cr y N x for varying y were determined accepting an a priori assumption of constant ( )  3.57 in [2] or Figure 4 in [24]).
N. Shohoji analysis done with the θ values determined as such, values of R ln Zf N (Fe 1−y Cr y N x ) (a) and Q(Fe 1−y Cr y N x ) (b) were obtained as a function of y as reproduced in Figure 3.In spite of somewhat peculiar variation pattern of θ with y (Figure 2), variation patterns of Q and R ln Zf N with respect to y looked quite "regular" (Figure 3).In this analysis, molten Fe 1−y Cr y N x at temperatures close to liquidus temperature above Fe 1−y Cr y N x solid phase possessing fcc structure was assumed to hold fcc structure.In the range of low y not exceeding 0.  (reproduced from Figure 3.59 in [2] or Figure 3 in [26]).
in the range of y smaller than 0.20 was consant with y being represented approximately by ( ) ( ) ( ) where ( ) On the other hand, it was felt difficult to appreciate rationally the variation pattern of θ with y in the range of y higher than 0.4 at first glance.However, as seen in Figure 3, Q values determined in the range of 0.4 ≤ y < 1 was consant with y being represented approximately by implying formation of 4 Cr/2 Fe cluster as depicted in Figure 4(c).
Detected deviation of θ vs. y relationship from the one represented by 4 y θ = (16) in Figure 2 was interpreted to be the consequence of Guinier-Preston zone type planar extensiton of 4 Cr/2 Fe clusters as detected in Figure 4(c).
To explain why θ vs. y relationship in range of y between 0.4 and 0.9 in Figure 2 deviated from the relationsip defined by Equation ( 16), model Guinier-Preston zone type planar extensions of 4 M/2 Fe clusters for a fixed numnber 12 of M atoms leading to different values of θ are depicted in Figure 5.As seen in Figure 5(b) and Figure 5(c), increased degree of planar extensiton would yield higher value of θ than the one anticipated from Equation ( 16) defined for the isolated 4 M/2 Fe clusters depcited in Figure 5(a).
It is intriguing to note that no evidence of existence of 2 M/4 Fe cluster as depicted in Figure 4(b) was detected for A 1−y B y X x type interstitial non-stopichiometric compounds analyzed so far..60 in [2] or Figure 4 in [26]).

N. Shohoji investigated H permeation behaviors as well as H absorption behaviors for
Va-group metal-based alloy membranes.The author [44] [47] analyzed the reported PCT relationships by Yukawa and collaborators [49] [50] [51] and obtained values for parameters, Q and R ln Zf H , as summarized in Table 2.
According to Yukawa and co-workers, Va-group metal-based alloys identified as favuorable H permeation membrane includes V 0.95 Fe 0.05 [49], Nb 0.95 Ru 0.05 and Nb 0.95 W 0.05 [50] as well as Ta 0.95 W 0.05 [51].Looking at values of θ and Q for these A 1−y M y type alloys containing Va-group metal (represented by A) in Table 2, it is noticed that θ was smaller and Q was more negative in this group of alloys than those in pure Va-group metal A except Ta 0.95 W 0.05 .Thus, it was proposed [48] to use the simultaneous fulfillment of conditions ( ) ( ) ( ) ( ) for screening of H permeation alloy membrane from among the candidate alloys based on Va-group metal.
On H permeation process, H 2 gas pressure p(H 2 ) in on the inlet side of the membrane is set higher than p(H 2 ) out on the outlet side.On the inlet side of the membrane, adsorbed H 2 gas over the membrane surface must be subjected to dissociation into adsorbed monatomic H atoms before being absorbed into A 1−y M y alloy lattice Then, by concentration gradient along the membrane thickness, absorbed H in the A 1−y M y lattice is subjected to diffusion towards the outlet side of the membrane.On the reaction (20) to proceed at the inlet side of the membrane, condition (18) is certainly favourable to suck faster the H atoms into the A 1−y M x lattice from the inlet side surface.
Then, on release of the transported H atoms through the outlet side surface of the A 1−y M y membrane, successive inverse reactions, (19) and (20) in this order, must proceed to recombine the absorbed monatomic H atoms in the A 1−y M y X x alloy lattice to be released in form of diatomic H 2 gas molecules.For this process of H 2 release to take place faster on the outlet side of the membrane surface, condition ( 17) is considered to be of convenience.
As such, simultaneous fulfillment of conditions, ( 17) and ( 18), was appreciated as rational for the alloy design guideline for Va-group metal-based H permeation membrane although this criterion did not seem to apply to Ta 0.95 W 0.05 alloy.
Among Va-group metal-based alloys listed in Table 2, V 0.948 Co 0.052 , Nb 0.95 Sn 0.05 and Nb 0.95 Pd 0.05 fulfill the conditions, ( 17) and (18), simultaneously although the H permeation performance of these alloys remains unknown.

Constant-a(C) Curves in γ-FeCx Phase
On account of pragmatic industrial importance of steel materials, intensive efforts have been invested on characterizing basic phase relationship for Fe-C binary system in equilibrium state.Taking advantage of abundance of equilibrium data for binary Fe-C system with high qualitative precision, statistical thermodynamic analysis for Fe-C system [20] was done choosing experimental data reported by Ban-ya et al. [54] in which chemical activity a(C) of C in equilibrium with γ-FeC x was varied widely through control of p(CO)/p(CO 2 ) ratio instead of using C in solid state.
In common experimental equilibrium study of metal carbide, excess graphite (reference state of C) is arranged to co-exist in the synthesized carbide MC x .Under such condition, a(C) is fixed to be 1 and, as such, influence of a(C) on x in MC x cannot be evaluated.
From the statistical thermodynamic analysis, values of θ and Q listed for γ-FeC x in Table 1 were calculated and constant-a(C) curves as reproduced in Figure 6 were drawn [20].This presentation of Figure 6 might be of no practical industrial importance but must be of fundamental significance towards profound

Conclusions
A few example cases of estimating properties of interstitial non-stoichiometric compounds with potential industrial applications on the basis of atomic interaction parameters evaluated by statistical thermodynamic analysis were demonstrated in this review article.Looking at the variation pattern of θ parameter value referring to number of available interstitial sites per metal atom M (M might be pure M, A 1−y B y type substitutional alloy or AZ z type compound containing another interstitial constituent Z besides interstitial constituent X) with respect to change of y or z, significant insight in atom clustering tendency in the condensed phase might be gained.There are several other materials properties predictable by referring to statistical thermodynamic analysis results including interstitial site occupation information for intermetallic alloys.Interested readers are advised to refer to original papers by the author [8] [12] [18] to look into further details.
The reviewed standardized statistical thermodynamic analysis procedure accepting an a priori assumption of constant ( ) E X X − in MX x within homogeneity composition range at arbitrary T was proved applicable to interstitial compound holding metallic characteristics but this analysis procedure is not applicable to non-stoichiometric compounds with ionic bonding characteristics like non-stoichiometric oxide.
Compared with standardized conventional thermodynamic analysis procedure to determine enthalpy, entropy and a few types of free energies through well-established mathematical procedure, statistical thermodynamic analysis is quite tedious demanding reliable PCT data set at least at three different T levels over certain range of p(X 2 ) and additional necessity for composing realistic statistical model.This is certainly a drawback of statistical thermodynamic analysis compared with conventional thermodynamics but this feature of statistical thermodynamic approach might be considered as a merit in some sense as the evaluated interaction parameters possess unambiguous physical significance provided that the statistical model used for the analysis is a valid one.
partition function of X atom in MX x , partition function of M atom in MX x , K & g: parameters determined by Equations (1) & (2), from the experimental PTC data for an assigned value of θ, <a factor to be assigned a priori> θ: number of the interstitial sites per M atom available for occupation by X atoms in MX x , <a resultant model parameter referring to extent of blocking of interstitial sites> Z: extent of blocking of interstitial sites by X in ( ) taking into account the contribution of the X X − interaction besides Q which represents contribution of the X M − interaction alone where ( ) x E MX refers to lattice energy of compound MX x calculated taking into account all nearest neighbour pairwise atomic interactions

Figure 2 .
Figure 2. Relationship between θ and y in molten Fe 1−y Cr y N x to fulfill the a priori condition of constant ( ) E N N − over homogeneity composition range at arbitrary T (reproduced from Figure 3.57 in[2] or Figure4in[24]).

Figure 4 .
Figure 4. Possible atom clusters formed in fcc Fe 1−y Cr y N x lattice in which affinity of M to N is stronger than that of Fe to N. (a) 1 M/5 Fe cluster (composed of one M atom and five Fe atoms around N); θ = y, (b) 2 M/4 Fe cluster; θ = y/2 and (c) 4 M/2 Fe cluster, θ = y/4

Table 2 .
[48]lable statistical thermodynamic interaction parameter for bcc A 1−y M y H x that showed suppressed H solubility compared to that in bcc AH x where A refers to Va-group metals (V, Nb or Ta) (reproduced from Table1in[48]).
a. Q values of θ for A 1−y M y H x that were evaluated to be more negative than that for AH x are displayed with bold letter.