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 MX_x 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 MX_x 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 MX_x 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, Θ_r [K], and characteristic temperature for vibration of X₂ molecule, Θ_v [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 MX_x might be a single metal but might be binary substitutional alloy to be represented by A_1-yB_y or even multi-component alloy with number of metallic constituents being greater than 2. For such alloys represented by A_1-yB_y, 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 A_1-yB_yX_x [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 MZ_zX_x 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 MZ_zX_x 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 MX_x are as follows [2, 3, 11, 12, 18, 20, 23, 25, 36, 37, 49].

(1)

(2)

(3)

(4)

(5)

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 Zf_X(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 MX_x 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(MX_x) for the condensed phase MX_x and PF(X₂) for ideal gas X₂ at partial pressure p(X₂) being in equilibrium with MX_x 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(MX_x) 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(MX_x) is represented by

(6)

where W(MX_x) refers to number of configurations of constituent atoms in MX_x lattice and E(MX_x) lattice energy of MX_x.

Expression for W(MX_x) is

(7)

In most of MX_x 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(MX_x), along with f_M and f_X, relates to entropy in conventional thermodynamic formulation as reviewed in some detail in [3].

By partial differentiation of PF(MX_x) with respect to number n_i of constituent i (i = M or X), chemical potential μ_i(MX_x) in MX_x is evaluated

(8)

As it is common to evaluate PTC relationship through isothermal experiment done for MX_x 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

(9)

Expression for μ_X(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 MX_x 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 MX_x 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 MX_x lattice due to formation of X-M bonds in the MX_x lattice while the coefficient R ln [Zf_X(T)] to T refers to electronic contribution to entropy term in thermodynamic sense. In fact, partition function f_X(T) of X atom in the MX_x lattice is a T-dependent factor as represented by Eq. (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 f_X(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 MX_x 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(MX_x) refers to lattice energy of compound MX_x 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 Zf_X(T) in MX_x Lattice Evaluated by Statistical Thermodynamics

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

3.1. Estimated R ln Zf_X(T) Values under Approximation of Constant f_X with T

In the standard statistical thermodynamic analysis carried out by the author, T-dependence of atomic partition function of f_X(T) was not taken into account and f_X(T) was assumed to hold T-independent constant value. For some MH_x systems in which values of vibration frequency ν are available, attempts were made to evaluate R ln f_X(T) as a function of T. The tentative calculation resuls for R ln f_X(T) values as a function of T are reviewed later in the text in 3.2.

3.1.1. Primary Solid Solutions MH_x

Values of R ln Zf_H and other parameters evaluated for primary solid solutions MH_x 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 Zf_H values as well as Q values in MH_x 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 ScH_x and θ = 1.5 for BaH_x (IIa-group metals) and θ = 0.55 for VHx and TaHx while θ = 0.75 for NbH_x (Va-group metals).

Similarity in the values for R ln Zf_H and Q, respectively, for MH_x 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 MX_x 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 MH_2+δ of Lanthanides and Actinide

Analyses for hyper-stoichiometric di-hydrides MH_2+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(MH_2+δ) is represented by (in the original paper [6], the hyper-stoichiometric phase is represented by MH_2+x instead of MH_2+δ)

(10)

where f_H^O refers to atomic partition function of H atoms distributed over O-sites, f_H^T that of H atoms distributed over T-sites and E(MH_2+δ) lattice energy of MH_2+δ.

Using this expression for PF(MH_2+δ), expression for chemical potential μ_H(MH_2+δ) of H in MH_2+δ crystal lattice is represented by

(11)

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 μ_M(MH_2+δ) are listed besides Q₁ evaluated from consideration for μ_H(MH_2+δ). Unlike similarities in values of R ln Zf_H^O 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 MH_2.0 [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 A_1-yM_yH_x (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 Zf_H, were made for A_1-yB_yH_x 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 Zf_H and Q for Nb_1-yM_yH_x 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 A_1-yM_yH_x in which solubility of H is suppressed from that in AH_x became smaller than that for AH_x. However, it is noticed in Table 4 that, for some A_1-yM_yH_x in which H solubility was suppressed from that in AH_x, Q(A_1-yM_yH_x) became more negative than Q(AH_x) meaning H in such A_1-yM_yH_x lattice was more stable with regard to interaction energy between H atom and metal atoms in A_1-yM_y lattice than H in AH_x in spite of the suppressed extent of H solubility; e.g. V_0.95Fe_0.05H_x, V_0.948Co_0.052H_x, Nb_0.95W_0.05H_x, Nb_0.95Sn_0.05H_x and Nb_0.95Pd_0.05H_x in Table 4 for which the Q values are presented in bold letters. It is intriguing to notice in Table 4 that R ln Zf_H 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 Ta_0.95W_0.05H_x in which H solubility was suppressed compared for TaH_x but θ(Ta_0.95W_0.05H_x) was 0.55 holding equal to θ(TaHx) and Q(Ta_0.95W_0.05H_x) was comparable to Q(TaH_x) while R ln f_H(Ta_0.95W_0.05H_x) < R ln f_H(TaH_x). 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 TaH_x and Ta_0.95W_0.05H_x, while electronic surrounding around H atom was modulated somewhat from TaH_x to Ta_0.95W_0.05H_x to lead to appreciable change in R ln f_H from TaH_x (62.4 [J·K^-1·mol^-1]) to Ta_0.95W_0.05H_x (32.1 [J·K^-1·mol^-1]).

Values of Q and R ln Zf_H evaluated for Nb_0.95M_0.05H_x (M = Pd, Sn, Al, Cu or Ni) are plotted in Fig. 3 (in the original publication [9], R ln Zf_H axis was indicated as R ln f_H as seen above). It is seen in Fig. 3 that, at a fixed y = 0.05 for Nb_1-yM_yH_x, R ln Zf_H(Nb_0.95M_0.05H_x) as well as Q(Nb_0.95M_0.05H_x) and θ(Nb_0.95M_0.05H_x) (see Table 4) varied appreciably depending on the alloying constituent M reflecting difference in extent of modulation of electronic structure in the Nb_0.95M_0.05 metal sub-lattice depending on the electronic structure around the alloyed M to Nb.

It is seen in Table 4 that θ(Nb_1-yMo_yH_x) 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 Zf_H(Nb_1-yMo_yH_x) and for Q(Nb_1-yMo_yH_x) are not at all monotonical with respect to variation of y (Fig. 3). The same is true for Nb_1-yPd_yH_x; 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 Zf_H(Nb_1-yPd_yH_x) and Q (Nb_1-yPd_yH_x) were not at all monotonical with respect to variation of y (Fig. 3).

Estimated values of θ, Q and R ln Zf_C for fcc Fe_1-yNi_yH_x and for fcc Co_1-yNi_yH_x 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 A_1-yB_y alloy lattice by taking an a priori assumption of

(12)

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 Fe_1-yNi_yH_x and for fcc Co_1-yNi_yH_x 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 Zf_C and Q emerged at around m ≈ 7.25.

3.1.4. Influences of Concentration Range x and Isotope Effect of Solute Element X in MX_x (M = Ti or Zr; X = H or D) on R ln Zf_X

Very dilute solution ranges of H and D (deuterium) in IVa-group metals, Ti and Zr, were analyzed under a priori assumption of

(13)

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 MD_x 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 Θ_r(D₂) and Θ_v(D₂), respectively, from available values of Θ_r(H₂) and Θ_v(H₂)

(14)

(15)

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

(16)

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 f_X(T) represented by Eq.(4), difference between ln f_D and ln f_H is given by

(17)

Experimental data available on vibrational mode of H in M lattice were scarce [6, 8, 18, 55-57]; ν_H = 550 cm^-1 for H at O-sites in fcc Pd and ν_H(1) = 970 cm^-1 (singlet) and ν_H(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 ν_H in the metallic lattice was 750 cm^-1 and that the multiplicity at this ν_H was triple. Results of approximate calculations accepting this very crude vibrational mode for H in MH_x lattices are summarized in Table 6 as reviewed later in 3.2.

Accepting this crude model, Eq. (17) is reduced to

(18)

where ν_D might be estimated by approximating harmonic oscillator accepting ν_H = 750 cm^-1

(19)

Accepting this value for ν_D, 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 TiD_x is different from υ₀(H) of H atom in TiD_x which seemed realistic (isotope effect). It is also noticed in Table 3 that Q(TiD_x) and Q(TiH_x) 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 Zf_D^d (= 78.8 [J∙K^-1∙mol^-1]) and R ln Zf_H^d (= 60.5 [J∙K^-1∙mol^-1]) in the very dilute solution range and the difference between R ln f_D (= 50.4 [J∙K^-1∙mol^-1]) and R ln f_H (= 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 TiH_x was 27 which was comparable to Z = 30 estimated for the very dilute TiD_x.

As such, parameter values estimated for TiH_x and TiD_x 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 ZrH_x and ZrD_x were not found in Table 5. Although the lacking inconsistency among the estimated statistical thermodynamic parameter values for ZrH_x and that for ZrD_x might be genuine origin and might be interpreted rationally but we cannot rule out the possibility that the results for ZrH_x and for ZrD_x were affected by inherent presence of Hf (hafnium) impurity in industrially pure Zr.

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

As represented by Eq. (4), expression for ln f_X(T) is a function of T

(4)

As pointed out above, there are scarce information on ν values for atoms X in non-stoichiometric interstitial solid solution MX_x. Anyway, using a few available information for values of ν_H [55-57], very crude estimation was attempted to evaluate R ln Zf_H(T) values as a function of T by taking ν_H = 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 Zf_H(T) range for any MH_x system were lower than R ln Zf_H value estimated from standard statistical thermodynamic analysis procedure under implicit assumption of T-independence of f_H. 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 MX_x are reviewed.

On undertaking the statistical thermodynamic analysis for MX_x, essential first step is to compose plausible partition function PF(MX_x) to MX_x by taking realistic distribution model for interstitial non-metallic constituent X in MX_x 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 MX_x. The system analyzed in [3] was hypo-stoichiometric nitride of chromium, CrN_0.50-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 CrN_0.50-δ would mean that available sites for N atom occupation in hcp CrN_0.50-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 MX_x lattice (fcc or hcp) would be 1 if the O-sites are occupied or 2 if T-sites are occupied

(20)

(21)

On the other hand, that for rather open bcc lattice might be represented by

(22)

(23)

Deviation of θ from θ₀ might be quite significant for some MX_x as experienced for analysis of primary solid solution MH_x 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 CrN_0.50-δ) is accepted for analysis of CrN_0.50-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 MX_x is the essential first step of initiating statistical thermodynamic analysis for MX_x.

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 Zf_H for MH_x 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 MH_x were also comparable to each other being around -227 ± 3 [kJ∙mol^-1].

Discussion on contribution of electronic term R ln Zf_X of tightly bound electrons in MX_x 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 MX_x 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.

ACKNOWLEDGEMENTS

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>

m_X: mass of X atom,

ρ: nuclear spin weight,

Θ_r: characteristic temperature for rotation of X₂,

Θ_v: 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 MX_x,

υ₀: statistical weight of tightly bound electrons around X in MX_x,

ν: vibrational frequency of X atom in MX_x 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 = n_X/n_M) in MX_x,

n_X: number of X atoms in MX_x lattice,

n_M: number of M atoms in MX_x lattice,

atomistic 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,

E(MX_x): lattice energy,

f_X(T): partition function of X atom in MX_x lattice,

f_M(T): partition function of M atom in MX_x 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 MX_x lattice,

a resultant model parameter referring to extent of blocking of interstitial sites>

Z: extent of blocking of interstitial sites by X in MX_x lattice (= θ₀/θ); that is, when one interstitial site in MX_x is occupied by an X atom, (Z - 1) neighbouring interstitial sites are blocked from occupation by other X atoms.