International Journal of Materials and Chemistry

p-ISSN: 2166-5346    e-ISSN: 2166-5354

2019;  9(1): 13-22



Entropy Terms in Statistical Thermodynamic Analysis Formula for Non-stoichiometric Interstitial Compounds

Nobumitsu Shohoji

LEN - Laboratório de Energia, LNEG - Laboratório Nacional de Energia e Geologia I.P., Lisboa, Portugal

Correspondence to: Nobumitsu Shohoji, LEN - Laboratório de Energia, LNEG - Laboratório Nacional de Energia e Geologia I.P., Lisboa, Portugal.


Copyright © 2019 The Author(s). Published by Scientific & Academic Publishing.

This work is licensed under the Creative Commons Attribution International License (CC BY).


A series of statistical thermodynamic analyses were made since 1974 for different types of non-stoichiometric interstitial compounds MXx under simplifying a priori assumption of constant interaction energy E(X-X) between nearest neighbour interstitial atoms X within a homogeneity composition range of MXx at arbitrary temperature T [K]. Mode of distribution of X atoms in interstitial sites in MXx lattice is represented by number θ of available interstitial sites for occupation by X atoms per M atom and the value of θ is determined to fulfil the a priori assumption. Mode of atomic configuration would yield major contribution to entropy term ∆S that appears in conventional thermodynamic expression of Gibbs free energy of formation, ∆G, in form of TS. In the statistical thermodynamic formulation, contribution of tightly bound electron appearing in form of RT ln fX where fX refers to atomic partition function of X atom in the MXx lattice and R the universal gas constant. Judging from this mathematical form of the term, R ln fX is considered to represent entropic contribution from tightly bound electron to X atom in MXx lattice. In the published series of works on statistical thermodynamic analysis for non-stoichiometric interstitial compounds, calculated values for R ln fX were reported but they were not reviewed with serious attention because R ln fX was considered merely as a secondary factor compared to principal factor E(X-M) referring to interaction energy between X and M in MXx lattice that represents enthalpy ∆H in conventional thermodynamic term. In this review article, consideration is given exclusively to the factor R ln fX evaluated in statistical thermodynamic approach to non-stoichiometric interstitial compounds.

Keywords: Entropy terms, Statistical thermodynamics, Interstitial compound, Non-stoichiometry

Cite this paper: Nobumitsu Shohoji, Entropy Terms in Statistical Thermodynamic Analysis Formula for Non-stoichiometric Interstitial Compounds, International Journal of Materials and Chemistry, Vol. 9 No. 1, 2019, pp. 13-22. doi: 10.5923/j.ijmc.20190901.02.

1. Introduction

Statistical thermodynamics is a unique analysis tool to span a bridge between the experimentally measurable macroscopic thermodynamic state parameters such as temperature T, equilibrium partial pressure p(X2) of X2 gas and composition x in non-stoichiometric interstitial compound MXx consisting of metal M and interstitial element X and the non-measurable atomistic interaction energy parameters E(i-j) (i, j = M or X). Condensed phase MXx subjected to statistical thermodynamic analysis might be in either solid or molten state.
Basic principles of statistical thermodynamics as well as practical analysis procedures are summarized in a classical text book authored by Fowler and Guggenheim [1].
In a monograph published in 2013 entitled "Statistical Thermodynamic Approach to Interstitial Non-stoichiometric Compounds (Hydride, Carbide, Nitride, Phosphide and Sulphide)" [2], results of statistical thermodynamic analysis acquired on the basis of the simplifying a priori assumption of constant E(X-X) within a homogeneity composition range of MXx at arbitrary T were compiled [3-46] while a few more publications in this line of work arose thereafter [47-49]. In those publications, values for dissociation energy of X2 gas molecules, D(X2) [kJ·mol-1], characteristic temperature for rotation of X2 molecule, Θr [K], and characteristic temperature for vibration of X2 molecule, Θv [K], were taken from JANAF Thermochemical Tables [50] in the earlier works [3-43] or from its successor NIST-JANAF Thermochemical Tables [51] for the more recent calculations [44-49].
Metal constituent M in non-stoichiometric interstitial compound MXx might be a single metal but might be binary substitutional alloy to be represented by A1-yBy or even multi-component alloy with number of metallic constituents being greater than 2. For such alloys represented by A1-yBy, statistical thermodynamic analysis results demonstrated that, depending on difference in affinity of X atom to constituent A and to constituent B, atom cluster might develop in the alloy lattice of A1-yByXx [9, 11, 16-19, 23, 25, 29, 33, 35-38, 40, 44, 46-49]. Statistical thermodynamic analysis was also made for alloy systems represented by MZzXx in which Z is interstitial constituent whose affinity to M is stronger than that of X to M and composition z might be pre-determined arbitrarily before measuring equilibrium pressure - temperature - composition (PTC) relationships between MZzXx and X2 gas [13, 31, 32, 34].
Literature information for PTC data source used in the cited statistical thermodynamic analysis works is not given in the text below as the PTC data sources used for the calculations were cited in References section in each publication.

2. Statistical Thermodynamic Analysis Procedure

2.1. Fundamental Equations

Generalized fundamental formulae proposed for this line of analysis of interstitial non-stoichiometric condensed phase MXx are as follows [2, 3, 11, 12, 18, 20, 23, 25, 36, 37, 49].
Symbols used in the above formulae are listed in the APPENDIX according to the classification of quality of the parameters.
In right hand side of Eq. (2), term Z is included explicitly in the second term, RT ln ZfX(T). However, in the earlier publications on statistical thermodynamic analysis for non- stoichiometric interstitial alloy systems, explicit inclusion or not of this term was practised rather casually, except for analysis of very dilute interstitial alloy systems [11]. In reality, Z = 1 is valid for many MXx systems with x being close to 1. However, during the course of writing a monograph [2], it came to the author's mind that it is better to include the term Z in Eq. (2) explicitly. In fact, in the very first publication [3] in this line of work done for Cr sub-nitride, Cr2N, formulation was made with Z = 2 for θ = 0.50 (Z·θ = θ0 = 1 for octahedral sites (O-sites) occupation in hexagonal close-packed (hcp) lattice).

2.2. Partition Function

Procedure to derive fundamental equation (1) is reviewed briefly in the following.
Starting point of statistical thermodynamic analysis for non-stoichiometric interstitial alloy MXx is the compilation of partition function PF(MXx) for the condensed phase MXx and PF(X2) for ideal gas X2 at partial pressure p(X2) being in equilibrium with MXx at arbitrary T.
Formula for PF(X2) at partial pressure p(X2) and at temperature T is readily available in a classical text book authored by Fowler and Guggenheim [1]. On the other hand, PF(MXx) has to be composed taking into account lattice structure of M and mode of distribution of X atoms over interstitial sites. Generalized formula for PF(MXx) is represented by
where W(MXx) refers to number of configurations of constituent atoms in MXx lattice and E(MXx) lattice energy of MXx.
Expression for W(MXx) is
In most of MXx compounds with non-ionic (metallic) characteristics (X = H, C, N, P or S; but not O), proportion of vacancies in M lattice is empirically acknowledged to be negligibly small even at elevated temperatures close to melting point. On the other hand, X atoms occupy interstitial sites (either octahedral interstitial sites (O-sites) or tetrahedral interstitial sites (T-sites)) according to mode of atomic distribution defined by the parameter θ.
The term W(MXx), along with fM and fX, relates to entropy in conventional thermodynamic formulation as reviewed in some detail in [3].
By partial differentiation of PF(MXx) with respect to number ni of constituent i (i = M or X), chemical potential μi(MXx) in MXx is evaluated
As it is common to evaluate PTC relationship through isothermal experiment done for MXx being in equilibrium with X2 gas at partial pressure p(X2), statistical thermodynamic analysis is made for i = X rather than for i = M using equilibrium relationship
Expression for μX(X2) is readily drawn from the classical text book of Fowler and Guggenheim [1] and, as such, fundamental formula (1) is derived for statistical thermodynamic analysis of MXx in equilibrium with X2 gas.

2.3. 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 Eq. (1) by varying θ. As understood from Eq. (1), slope of isothermal A vs. x plot would become proportional to E(X-X). To fulfil the a priori assumption of constant E(X-X) within a homogeneity composition range of MXx at arbitrary T, θ yielding linear A vs. x relationship over the entire homogeneity composition range of MXx must be chosen for the subsequent calculations.
Then, from the intercept g(T) calculated using Eq. (1), K(T) vs. T relationship must be drawn using Eq. (2). Term Q on the right hand side in Eq. (2) refers to extent of stabilization of atom X in the MXx lattice due to formation of X-M bonds in the MXx lattice while the coefficient R ln [ZfX(T)] to T refers to electronic contribution to entropy term in thermodynamic sense. In fact, partition function fX(T) of X atom in the MXx lattice is a T-dependent factor as represented by Eq. (4) but, as the T range of statistical thermodynamic analysis for MXx is typically no wider than 500 K, it has been a common practice to approximate fX(T) as a T-independent constant factor [2-49].
For convenience of the readers, flow chart of the iterative determination procedure for a value of the parameter θ is presented as Fig. 1.
Figure 1. Flow chart of the statistical thermodynamic analysis procedure accepting the a priori assumption of constant E(X-X) within a homogeneity composition range of MXx at arbitrary T. (Reproduced from Fig. 1 in [49])
As represented by Eq. (5), term [Q + βxE(X-X)] refers to the net extent of stabilization of X atom in the MXx lattice taking into account additionally the contribution βxE(X-X) from the X-X interaction besides Q which represents contribution of the X-M interaction. In Eq. (5), E(MXx) refers to lattice energy of compound MXx calculated taking into account all nearest neighbour pair-wise atomic interactions E(i-j) for all combinations of i and j.
For pragmatic convenience of calculating K(T) using Eq. (2), [D(X2)/2 - RTC(T)] 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 [D(X2)/2 - RTC(T)] value at arbitrary T is calculated readily by interpolation.

3. Values of the Term R ln ZfX(T) in MXx Lattice Evaluated by Statistical Thermodynamics

Some representative calculation results for the term R ln ZfX(T) in MXx reported in the past publications are reviewed hereafter classifying M into different categories.

3.1. Estimated R ln ZfX(T) Values under Approximation of Constant fX with T

In the standard statistical thermodynamic analysis carried out by the author, T-dependence of atomic partition function of fX(T) was not taken into account and fX(T) was assumed to hold T-independent constant value. For some MHx systems in which values of vibration frequency ν are available, attempts were made to evaluate R ln fX(T) as a function of T. The tentative calculation resuls for R ln fX(T) values as a function of T are reviewed later in the text in 3.2.
3.1.1. Primary Solid Solutions MHx
Values of R ln ZfH and other parameters evaluated for primary solid solutions MHx are summarized in Table 1.
Table 1. Values of R ln ZfH and other parameters evaluated for primary solid solutions MHx
It is intriguing to note that R ln ZfH values as well as Q values in MHx for metals in the same group in the Periodic Table of the Elements are comparable to each other even when θ values chosen to fulfil the a priori assumption for constant E(H-H) within the given phase at arbitrary T were not the same; e.g. θ = 1.0 for ScHx and θ = 1.5 for BaHx (IIa-group metals) and θ = 0.55 for VHx and TaHx while θ = 0.75 for NbHx (Va-group metals).
Similarity in the values for R ln ZfH and Q, respectively, for MHx of metals belonging to the same group in the Periodic Table appeared realistic supporting indirectly the validity of the proposed statistical thermodynamic analysis procedure.
As presented in Table 2, expression for Q varies depending on whether the X atoms in MXx lattice are distributed over O-sites or T-sites as well as on whether the crystal lattice structure is close-packed (fcc or hcp) or open (bcc). It must be noted that, in relatively open bcc lattice, contribution from second-nearest neighbour would enter into the expression for interaction energy term Q.
Table 2. Expression for Q for representative crystal lattice structures. (Reproduced from Table 1 in [18])
3.1.2. Hyper-stoichiometric Di-hydrides MH2+δ of Lanthanides and Actinide
Analyses for hyper-stoichiometric di-hydrides MH2+δ of lanthanides (La, Ce, Pr, Nd, Sm) and actinides (Np, Pu, Am) with fcc lattice structure were made under assumption that H atoms up to H/M = 2.0 occupy the T-sites and H atoms exceeding H/M = 2.0 occupy randomly the O-sites [6]. Accepting this model, PF(MH2+δ) is represented by (in the original paper [6], the hyper-stoichiometric phase is represented by MH2+x instead of MH2+δ)
where fHO refers to atomic partition function of H atoms distributed over O-sites, fHT that of H atoms distributed over T-sites and E(MH2+δ) lattice energy of MH2+δ.
Using this expression for PF(MH2+δ), expression for chemical potential μH(MH2+δ) of H in MH2+δ crystal lattice is represented by
where E(H-M)O refers to interaction energy between the H atom in O-site and the M atom, E(H-H)O-T interaction energy between the H atom in O-site and the H atom in T-site and E(H-H)O-O interaction energy between the nearest neighbour H atoms in O-sites.
In Table 3, values of Q2 estimated from consideration for μM(MH2+δ) are listed besides Q1 evaluated from consideration for μH(MH2+δ). Unlike similarities in values of R ln ZfHO and Q1 among lanthanides (4f-group elements) and among actinides (5f-group elements), values of Q2 vary appreciably among lanthanides and among actinides. This aspect with Q2 was appreciated in terms of considerable extent of variation of molar volume of M [52] and MH2.0 [53] of 4f- and 5f-elements with respect to number m of the outer-shell f-electrons as represented in Fig. 2 and resultant variations of dissociation energy D(M) of these f-group metals with respect to number of the outer-shell f-electrons as listed in Table 3 [6].
Table 3. Values of R ln fHO and other parameters evaluated for hyper-stoichiometric MH2+δ. (Reproduced from Table 1 in [6])
Figure 2. Molar volume of some lanthanides (4f-group elements), actinides (5f-group elements) and their hydrides; M [52]: (○) 4f, (•) 5f. MH2.0 [53]: (□) 4f, (■) 5f. (Reproduced from Fig. 5 in [6])
3.1.3. Suppressed H Solubility for Va-group Alloys A1-yMyHx (A = V, Nb or Ta) by Alloying with Substitutional Metallic Constituent M (= Al, Co, Cr, Cu, Fe, Mo, Ni, Pd, Sn, Ru or W)
Statistical thermodynamic evaluations for parameters, θ, Q and R ln ZfH, were made for A1-yByHx alloys of Va-group metals (A = V, Nb or Ta) through alloying with substitutional metallic constituents M (= Al, Co, Cr, Cu, Fe, Mo, Ni, Pd, Sn, Ru or W). The evaluation results are listed in Table 4 [49]. Graphical presentation of values for R ln ZfH and Q for Nb1-yMyHx alloys with suppressed H absorption from that in Nb through alloying with M (= Pd, Sn, Al, Cu, Ni or Mo) are reproduced in Fig. 3 [9].
Figure 3. Values of R ln ZfH and Q plotted as a function of y in bcc Nb1-yMyHx (M = Pd, Sn, Al, Cu, Ni or Mo). (Reproduced from Fig. 4 in [9])
It is evident in Table 4 that value of θ for bcc A1-yMyHx in which solubility of H is suppressed from that in AHx became smaller than that for AHx. However, it is noticed in Table 4 that, for some A1-yMyHx in which H solubility was suppressed from that in AHx, Q(A1-yMyHx) became more negative than Q(AHx) meaning H in such A1-yMyHx lattice was more stable with regard to interaction energy between H atom and metal atoms in A1-yMy lattice than H in AHx in spite of the suppressed extent of H solubility; e.g. V0.95Fe0.05Hx, V0.948Co0.052Hx, Nb0.95W0.05Hx, Nb0.95Sn0.05Hx and Nb0.95Pd0.05Hx in Table 4 for which the Q values are presented in bold letters. It is intriguing to notice in Table 4 that R ln ZfH values for these alloys with suppressed H solubility are relatively small compared with those for the other alloys with suppressed H solubility listed in this table.
Table 4. Available statistical thermodynamic interaction parameter values for bcc A1-yMyHx that showed suppressed H solubility compared to that in bcc AHx where A refers to Va-group metals (V, Nb or Ta) (Reproduced from Table 2 in [49])
In Table 4, unique situation is encountered for Ta0.95W0.05Hx in which H solubility was suppressed compared for TaHx but θ(Ta0.95W0.05Hx) was 0.55 holding equal to θ(TaHx) and Q(Ta0.95W0.05Hx) was comparable to Q(TaHx) while R ln fH(Ta0.95W0.05Hx) < R ln fH(TaHx). 74W is neighbouring element to 73Ta in the Periodic Table of the Elements and the outer-shell electron structure of 74W, 5d46s2, is quite similar to that of 73Ta, 5d36s2, and this fact might have led to similar values for θ and Q between TaHx and Ta0.95W0.05Hx, while electronic surrounding around H atom was modulated somewhat from TaHx to Ta0.95W0.05Hx to lead to appreciable change in R ln fH from TaHx (62.4 [J·K-1·mol-1]) to Ta0.95W0.05Hx (32.1 [J·K-1·mol-1]).
Values of Q and R ln ZfH evaluated for Nb0.95M0.05Hx (M = Pd, Sn, Al, Cu or Ni) are plotted in Fig. 3 (in the original publication [9], R ln ZfH axis was indicated as R ln fH as seen above). It is seen in Fig. 3 that, at a fixed y = 0.05 for Nb1-yMyHx, R ln ZfH(Nb0.95M0.05Hx) as well as Q(Nb0.95M0.05Hx) and θ(Nb0.95M0.05Hx) (see Table 4) varied appreciably depending on the alloying constituent M reflecting difference in extent of modulation of electronic structure in the Nb0.95M0.05 metal sub-lattice depending on the electronic structure around the alloyed M to Nb.
It is seen in Table 4 that θ(Nb1-yMoyHx) decreased monotonically with increasing y from 0.75 at y = 0, through 0.45 at y = 0.10 and 0.30 at y = 0.20, to 0.20 at y = 0.3. Nevertheless, variation patterns for R ln ZfH(Nb1-yMoyHx) and for Q(Nb1-yMoyHx) are not at all monotonical with respect to variation of y (Fig. 3). The same is true for Nb1-yPdyHx; θ decreased monotonically with increasing y from 0.75 at y = 0 through 0.60 at y = 0.05 to 0.45 at y = 0.10 (Table 4) but patterns of variations of R ln ZfH(Nb1-yPdyHx) and Q (Nb1-yPdyHx) were not at all monotonical with respect to variation of y (Fig. 3).
Estimated values of θ, Q and R ln ZfC for fcc Fe1-yNiyHx and for fcc Co1-yNiyHx are plotted in Fig. 4 [35]. Composition axis of this plot corresponds to number of outer-shell 3d electrons; mean number m of 3d electrons being 6 for Fe, 7 for Co and 8 for Ni. Analysis for these alloys were done considering they were very dilute solutions of C in A1-yBy alloy lattice by taking an a priori assumption of
within the given phase at arbitrary T.
Figure 4. Estimated values of θ, R ln ZfC and Q for fcc Fe1-yNiyCx and for fcc Co1-yNiyCx plotted as a function of alloy composition (mean number m of 3d electrons was 6 for Fe, 7 for Co and 8 for Ni). (Reproduced from Fig. 2 in [35])
As seen in Fig. 4, variation patterns of θ for fcc Fe1-yNiyHx and for fcc Co1-yNiyHx with respect to m is not monotonical but it appears that, for both iron-group alloys, θ value took minimum at around m ≈ 7.25 and, correspondingly, peak values for R ln ZfC and Q emerged at around m ≈ 7.25.
3.1.4. Influences of Concentration Range x and Isotope Effect of Solute Element X in MXx (M = Ti or Zr; X = H or D) on R ln ZfX
Very dilute solution ranges of H and D (deuterium) in IVa-group metals, Ti and Zr, were analyzed under a priori assumption of
at arbitrary T where X refers to H or D [11].
Analysis results are summarized in Table 5 in which results of analysis for MXx in the moderate concentration range are also listed.
Table 5. Values of Q and R ln ZfX for the very dilute (x < 0.015) and the modest (0.1 < x < 0.8) solute concentration ranges in MXx (M = Ti or Zr; X = H or D). (Reproduced from Table 1 in [11])
On undertaking statistical thermodynamic analysis for MDx being in equilibrium with D2 gas, C(T) for D2 gas had to be calculated using Eq.(3). On this calculation, common approximations were employed for estimating values of Θr(D2) and Θv(D2), respectively, from available values of Θr(H2) and Θv(H2)
Values D(D2) of dissociation energy of D2 gas as a function of T were not available in JANAF Thermochemical Tables [50]. Hence, following approximation was used on the basis of the difference between D(D2) at 0 K and D(H2) at 0 K listed in [54] to estimate D(D2) at arbitrary T
Further, ρ was taken to be 2 for H and 3 for D. On the basis of these approximations, analysis results summarized in Table 5 were obtained.
In [11], brief review was made on isotope effect of H and D in Ti lattice for atomic partition function expressed by Eq.(4). The essence of the review is summarized in the following.
Accepting the expression for ln fX(T) represented by Eq.(4), difference between ln fD and ln fH is given by
Experimental data available on vibrational mode of H in M lattice were scarce [6, 8, 18, 55-57]; νH = 550 cm-1 for H at O-sites in fcc Pd and νH(1) = 970 cm-1 (singlet) and νH(2) = 1400 cm-1 (doublet) for H at T-sites in bcc V, Nb or Ta.
Thus, in the earlier works [6, 8, 18], very crude tentative approximations were made to assume that the vibration frequency νH in the metallic lattice was 750 cm-1 and that the multiplicity at this νH was triple. Results of approximate calculations accepting this very crude vibrational mode for H in MHx lattices are summarized in Table 6 as reviewed later in 3.2.
Accepting this crude model, Eq. (17) is reduced to
where νD might be estimated by approximating harmonic oscillator accepting νH = 750 cm-1
Accepting this value for νD, value of the ratio υ0(D)/υ0(H) was calculated to be 2.9 at T = 1200 K implying that the statistical weight υ0(D) of D atom in TiDx is different from υ0(H) of H atom in TiDx which seemed realistic (isotope effect). It is also noticed in Table 3 that Q(TiDx) and Q(TiHx) in both the very dilute and the moderate solution concentrations were different from each other by the same degree of about 10 [kJ∙mol-1] from each other in either the very dilute and the modest solution concentration range (isotope effect).
Looking at Table 5, it is noticed that difference between R ln ZfDd (= 78.8 [J∙K-1∙mol-1]) and R ln ZfHd (= 60.5 [J∙K-1∙mol-1]) in the very dilute solution range and the difference between R ln fD (= 50.4 [J∙K-1∙mol-1]) and R ln fH (= 33.2 [J∙K-1∙mol-1]) in the moderate solute concentration range were approximately the same being around 18 [J∙K-1∙mol-1]. It is also noticed that Z estimated for the very dilute TiHx was 27 which was comparable to Z = 30 estimated for the very dilute TiDx.
As such, parameter values estimated for TiHx and TiDx in both the very dilute and the moderate solution concentration ranges look generally consistent among themselves. In contrast, such consistency among the estimated parameters for ZrHx and ZrDx were not found in Table 5. Although the lacking inconsistency among the estimated statistical thermodynamic parameter values for ZrHx and that for ZrDx might be genuine origin and might be interpreted rationally but we cannot rule out the possibility that the results for ZrHx and for ZrDx were affected by inherent presence of Hf (hafnium) impurity in industrially pure Zr.

3.2. Attempts to Estimate R ln ZfX(T) Values Taking into Account T-dependence of fX(T)

As represented by Eq. (4), expression for ln fX(T) is a function of T
As pointed out above, there are scarce information on ν values for atoms X in non-stoichiometric interstitial solid solution MXx. Anyway, using a few available information for values of νH [55-57], very crude estimation was attempted to evaluate R ln ZfH(T) values as a function of T by taking νH = 750 cm-1 assuming the multiplicity of this vibrational mode of H being 3. The estimation results are summarized in Table 6.
Table 6. Estimated values for R ln ZfH(T) taking into account T-dependence of fH(T) for some MHx compared with estimated values for R ln ZfH according to standard statistical thermodynamic analysis procedure
In Table 6, it is noticed that R ln ZfH(T) range for any MHx system were lower than R ln ZfH value estimated from standard statistical thermodynamic analysis procedure under implicit assumption of T-independence of fH. There is, for the moment, no rational explanation for this.

4. Concluding Remarks

Characteristic features of entropy terms appearing in statistical thermodynamic analysis formula for non-stoichiometric interstitial compound MXx are reviewed.
On undertaking the statistical thermodynamic analysis for MXx, essential first step is to compose plausible partition function PF(MXx) to MXx by taking realistic distribution model for interstitial non-metallic constituent X in MXx lattice. For this criterion to be fulfilled, it was proposed in the first publication [3] of this series of work to choose the number θ of available interstitial sites per M atom that appears in fundamental formula Eq. (1) in a way to yield constant E(X-X) at arbitrary T over entire homogeneity composition range of MXx. The system analyzed in [3] was hypo-stoichiometric nitride of chromium, CrN0.50-δ, and θ to fulfil the a priori assumption of constant E(N-N) was determined to be 0.5 and the analysis results looked realistic holding rational compatibility with the available thermodynamic parameter values for Cr2N phase.
This choice of θ parameter value for analysis of CrN0.50-δ would mean that available sites for N atom occupation in hcp CrN0.50-δ lattice were O-sites and occupation of one O-site by N atom would block occupation by a neighbouring O-site by another N atom (i.e., Z = 2).
Geometrically available number θ0 of available interstitial sites per M atom in close-packed MXx lattice (fcc or hcp) would be 1 if the O-sites are occupied or 2 if T-sites are occupied
On the other hand, that for rather open bcc lattice might be represented by
Deviation of θ from θ0 might be quite significant for some MXx as experienced for analysis of primary solid solution MHx with bcc lattice structure for Va-group transition metals (M = V, Nb, Ta) [5]; θ = 0.55 was chosen for V and Ta (that is Z ≈ 5) while θ = 0.75 was chosen (that is, Z = 4) for M = Nb while θ0(bcc; O) = 3.
Without accepting the proposed a priori assumption of constant E(X-X) at arbitrary T within a given phase, conclusions drawn from statistical thermodynamic analysis would become with great arbitrariness and meaningless. For example, when θ = θ0(hcp; O) = 1 (that is, random distribution of N atoms over geometrically available O-sites in hcp lattice of CrN0.50-δ) is accepted for analysis of CrN0.50-δ, variation of E(N-N) with composition of x at arbitrary T is derived showing trend of increasing repulsive nearest neighbour N-N interaction with increasing x. Such situation is not realistic because if E(N-N) interaction changed with composition x, phase change should be induced at certain threshold x.
As such, choice of realistic θ representing configurational entropy term of MXx is the essential first step of initiating statistical thermodynamic analysis for MXx.
Then, as the consequence of the statistical thermodynamic analysis carried out on the basis of this simplifying a priori assumption, apparently realistic conclusion was drawn for value of the term R ln ZfH for MHx of Va-group metals in spite of different θ values assigned for M = V and Ta and for M = Nb (cf. Table 1) that fell at around 63.5 ± 2 [J∙K-1∙mol-1]. As also seen in Table 1, Q values evaluated for these MHx were also comparable to each other being around -227 ± 3 [kJ∙mol-1].
Discussion on contribution of electronic term R ln ZfX of tightly bound electrons in MXx lattice remains still rather primitive and imperfect. It is hoped that this aspect is exploited further being assisted by compiled data on vibrational frequency ν of interstitial constituent X in MXx lattice.
As reviewed in this paper, entropy terms in multi-component alloys including non-stoichiometric interstitial alloys are consisted of contribution from tightly bound electrons besides contribution from configuration of constituent atoms over crystal lattice points.
Unlike enthalpy term, entropy term is not straightforwardly measurable by experiment for alloys. For example, in spite of strategic importance of entropy term in characterizing fashionable high-entropy alloys, report appears to be scarce on values of entropy terms in published works on high-entropy alloys.
The concept of high-entropy alloy was born in 2004 by Cantor et al. [58] being defined originally as “equi-atomic multi-component alloy”. After being christened with distinctive name “high-entropy alloys” [59], research on this category of alloys became highly fashionable. Nevertheless, in available works published on high-entropy alloys [60], explicit quantitative evaluation for entropy term did not seem to be made very seriously. This is somewhat peculiar and surprising.
High-entropy alloys are classified in substitutional alloys and thence the analysis procedure used in this work for interstitial alloy systems cannot be applied straightforwardly. However, if effort is invested to evaluate somehow contribution from electronic terms as well as from atomic configuration term, rate of progress in high-entropy alloy might be raised.


Late Dr. Masahiro Katsura (2017.01.31) is sincerely acknowledged for his patient guidance for the author to get acquainted with principles and analysis procedures in statistical thermodynamics. Although most of the works in this line were done by the author himself, there were occasional assistances provided for calculations by collaborators including Mrs. Teresa Marcelo, Mrs. Maria Cândida Monteiro Dias and Ms Sofia Figueiredo Marques. The author would like to thank an institutional colleague, Dr. Cristina Sena Ferreira, for her painstaking careful reading of the manuscript to point out several typographic errors contained in the original text. Long-term research collaborator, Dr. José Brito Correia, is sincerely acknowledged for his provision of information on chronological development and current status of research in high-entropy alloys.

Appendix / Symbols

universal constants>
R: universal gas constant (= 8.31451 [J·mol-1·K-1]),
h: Planck constant (= 6.6260755 × 10-34 [J·s]),
k: Boltzmann constant (= 1.380658 × 10-23 [J·K-1]),
materials constants>
mX: mass of X atom,
ρ: nuclear spin weight,
Θr: characteristic temperature for rotation of X2,
Θv: characteristic temperature for vibration of X2,
υ0*: electronic state of normal state of X2 molecule,
D(X2): dissociation energy of X2 molecule per mole,
β: factor determined from crystal structure consideration,
θ0: geometrically available number of interstitial site per M in MXx,
υ0: statistical weight of tightly bound electrons around X in MXx,
ν: vibrational frequency of X atom in MXx lattice,
g(ν): distribution function,
experimentally measurable macroscopic parameters>
p(X2): equilibrium pressure of ideal gas X2,
T: absolute temperature [K],
x: composition (X/M atom ratio = nX/nM) in MXx,
nX: number of X atoms in MXx lattice,
nM: number of M atoms in MXx lattice,
atomistic parameters to be evaluated>
Q: degree of stabilisation of X atom in MXx lattice with reference to isolated X and M atoms in vacuum,
E(i-j): interaction energy between i and j atoms in MXx lattice,
E(MXx): lattice energy,
fX(T): partition function of X atom in MXx lattice,
fM(T): partition function of M atom in MXx lattice,
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 MXx lattice,
a resultant model parameter referring to extent of blocking of interstitial sites>
Z: extent of blocking of interstitial sites by X in MXx lattice (= θ0/θ); that is, when one interstitial site in MXx is occupied by an X atom, (Z - 1) neighbouring interstitial sites are blocked from occupation by other X atoms.


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