_{1}

Statistical thermodynamics allows us to estimate atomistic interactions in interstitial non-stoichiometric compounds
*MX*
_{x} through analysis of experimentally determined pressure-temperature-composition (
*PTC*) relationships for
*MX*
_{x} being in equilibrium with X
_{2} in gaseous state
*(X=H,N,P or S)* or for non-stoichiometric carbide MCx being in equilibrium with excess C. In case of analysis for
*MC*
_{x}, chemical activity
*a(C)* of
*C* in place of partial pressure
*p(X*
_{2}
*)* 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 parame-ters 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
*A*
_{1-y}
*B*
_{y} 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.

Statistical thermodynamic analysis procedures were comprehensively summarized by Fowler and Guggenheim in a classical monograph published in 1949 [_{x} possessing composition _{2} gas at partial pressure _{x}, nearest-neigh- bour atomic interaction energy _{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 [_{x} lattice at arbitrary T. Parameter values estimated for M’s in the same group in the Periodic _{x} at any T although there is no rigorous first-principle-based justification for this a priori assumption. Further, statistical thermodynamic parameter values for _{2}N [

Besides analysis for pure M, analysis was made also for substitutional alloy with _{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 [

In the early stage of this line of work to characterize nature of atomistic interaction in interstitial non-stoichiometric compound MX_{x} [

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 [

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 [

Among literatures cited in References section of this paper, [

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:

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}),

m_{X}: mass of X atom,

ρ: nuclear spin weight,

_{2},

_{2},

_{2} molecule,

_{2} molecule per mole,

β: factor determined from crystal structure consideration,

θ_{0}: geometrically available number of interstitial site per M in MX_{x},

_{x},

ν: vibrational frequency of X atom in MX_{x} lattice,

_{2},

T: absolute temperature (K),

x: composition (_{x},

n_{X}: number of X atoms in MX_{x},

n_{M}: number of M atoms in MX_{x},

Q: degree of stabilisation of X atom in MX_{x} lattice with reference to isolated X and M atoms in vacuum,

_{x} lattice,

_{x},

_{x},

K & g: parameters determined by Equations (1) & (2), from the experimental PTC data for an assigned value of θ,

θ: number of the interstitial sites per M atom available for occupation by X atoms in MX_{x},

Z: extent of blocking of interstitial sites by X in_{x} is occupied by an X atom,

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,

On the other hand, geometrical factor β to

Value of θ to fulfill the a priori assumption of constant _{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 _{2}N [_{0} for O-site occupation of N in the hcp lattice),

_{x}, it is more natural to accept phase change to occur rather than to hold the same crystal lattice structure [

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 become proportional to_{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} lattice while the coefficient

_{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

For convenience of the readers, flow chart of the calculation procedure is presented below as

As represented by Equation (5), term _{x} lattice, _{x} calculated taking into account all nearest neighbour pairwise atomic interactions

For pragmatic convenience of calculating K(T) using Equation (2),

As might be understood from expressions for fundamental equations reviewed in 2.1., reference state of energy in the statistical thermodynamic analysis is each constituent atom in infinite separation in vacuum whereas the reference state of constituent in conventional thermodynamic analysis is the pure substance in standard state. That is, by conventional thermodynamic analysis, enthalpy of formation of MX_{x} from M and X_{2} represents the difference in energy between the reaction product MX_{x} and the reactants, M and

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

The more stable the X in Fe lattice the more negative would become _{x} lattice. That is, according to

implying that C is the most stable and H is the least stable in Fe lattice.

Further, it is notice in

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 _{x} is not specified uniquely. This is due to inherent difficulty of determining exactly the value of θ for statistical thermodynamic

Compound | Range of T | θ | Q^{b} |
---|---|---|---|

[K] | [kJ・mol^{−1}] | ||

α-FeH_{x} (bcc) | <1175 | >0.01 | −171 |

γ-FeH_{x} (fcc) | 1185 - 1590 | (0.50)^{a} | −138 |

δ-FeH_{x} (bcc) | 1665 - 1715 | (0.50)^{a} | −100 |

α-FeC_{x} (bcc) | <1000 | >0.10 | −648 |

α-FeC_{x} (bcc) | 1025 - 1150 | >0.10 | −613 |

γ-FeC_{x} (fcc) | 1175 - 1575 | 0.20 | −679 |

FeC_{x} (molten) | 1550 - 1775 | 1.00 | −699 |

α-FeN_{x} (bcc) | <1115 | >0.05 | −420 |

γ-FeN_{x} (fcc) | 1250 - 1625 | (0.50)^{a} | −455 |

δ-FeN_{x} (bcc) | 1675 - 1715 | (0.50)^{a} | −352 |

FeN_{x} (molten) | 1850 - 2000 | >0.10 | −427 |

FeP_{x} (molten) | 1475 - 1775 | 0.50 | −347 |

FeS_{x} (molten) | 1775 - 1875 | 1.00 | −307 |

a. Value of θ used for convenience on calculating Q value for the very dilute interstitial solution. b. Q corresponds to partial molar enthalpy of solution h(X) of X into Fe. Some h(X) values reported by McLellan and co-workers (da Silva, J. R. G and McLellan, R. B. (1976) The Solubility of Hydrogen in Super-pure-ron Single Crystals. J. Less Coomon Met., 50, 1 - 5.; McLellan, R. B. and Farraro, R. J. Thermodynamics of the Iron-Nitrogen System. (1980) Acta. Metall., 28, 417-422.) were in good accord with the corresponding values of Q. h(H)^{α} = −177 kJ・mol^{−1}: Q(H)^{α} = −171 kJ・mol^{−1}, h(C)^{α} = −603 kJ・mol^{−1}: Q(C)^{α}(T > T_{C}) = −613 kJ・mol^{−1}, Q(C)^{α}(T < T_{C}) = −648 kJ・mol^{−1}, h(C)^{γ} = −650 kJ・mol^{−1}: Q(C)^{γ} = −699 kJ・mol^{−1}, h(N)^{α} = −424 kJ・mol^{−1}: Q(N)^{α} = −420 kJ・mol^{−1}, h(N)^{γ} = −460 kJ・mol^{−1}: Q(N)^{γ} = −455 kJ・mol^{−1}, where T_{C} refers to Curie temperature 1043 K for Fe.

analysis in very dilute interstitial solution under certain circumstances as discussed in some detail in [

First cases of statistical thermodynamic analysis for very dilute interstitial solutions was made in [

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. _{X} varied as a function of θ in the range of θ where Q value became constant with θ.

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 [

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 _{1−y}CryNx as reproduced in

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 _{N} with respect to y looked quite “regular” (_{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 0.2, θ varied following

For this range of θ (<0.2), the interpretation was quite simple. That is, N atom in an O site was assumed to become surrounded by one Cr atom and 5 Fe atoms (1 Cr/5 Fe cluster or Cr-N dipole) as depicted in

in the range of y smaller than 0.20 was consant with y being represented approxi- mately by

where ^{−1}]) in molten CrN_{x} and ^{−1}]) in molten FeN_{x} [

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

implying formation of 4 Cr/2 Fe cluster as depicted in

Detected deviation of θ vs. y relationship from the one represented by

in

To explain why θ vs. y relationship in range of y between 0.4 and 0.9 in

It is intriguing to note that no evidence of existence of 2 M/4 Fe cluster as depicted in _{1−y}B_{y}X_{x} type interstitial non-stopichio- metric compounds analyzed so far.

Yukawa and collaborators at Nagoya University [

investigated H permeation behaviors as well as H absorption behaviors for Va-group metal-based alloy membranes. The author [_{H}, as summarized in

According to Yukawa and co-workers, Va-group metal-based alloys identified as favuorable H permeation membrane includes V_{0.95}Fe_{0.05} [_{0.95}Ru_{0.05} and Nb_{0.95}W_{0.05} [_{0.95}W_{0.05} [_{1−y}M_{y} type alloys containing Va-group metal (represented by A) in _{0.95}W_{0.05}. Thus, it was proposed [

for screening of H permeation alloy membrane from among the candidate alloys

bcc A_{1−y}M_{y}H_{x} | θ | Q^{a} | R ln Zf_{H} |
---|---|---|---|

[kJ・mol^{−1}] | [J・K^{−1}・mol^{−1}] | ||

VH_{x} | 0.55 | −223.6 | 64.7 |

V_{0.96}Cr_{0.04}Hx | 0.525 | −220.0 | 26.9 |

V_{0.916}Cr_{0.084}Hx | 0.475 | −207.2 | 54.5 |

V_{0.949}Mo_{0.051}Hx | 0.525 | −207.3 | 37.5 |

V_{0.95}Fe_{0.05}Hx | 0.45 | −229.4 | 23.7 |

V_{0.948}Co_{0.052}Hx | 0.45 | −230.1 | 21.5 |

NbH_{x} | 0.75 | −229.5 | 61.7 |

Nb_{0.95}Ru_{0.05}Hx | 0.55 | −221.8 | 41.2 |

Nb_{0.95}W_{0.05}Hx | 0.725 | −243.1 | 14.1 |

Nb_{0.90}Mo_{010}Hx | 0.45 | −225.5 | 65.4 |

Nb_{0.80}Mo_{0.20}Hx | 0.30 | −228.2 | 61.3 |

Nb_{0.70}Mo_{0.30}Hx | 0.20 | −220.8 | 67.7 |

Nb_{0.95}Al_{0.05}Hx | 0.60 | −225.8 | 36.9 |

Nb_{0.95}Cu_{0.05}Hx | 0.60 | −223.5 | 41.2 |

Nb_{0.95}Sn_{0.05}Hx | 0.60 | −231.8 | 30.6 |

Nb_{0.95}Ni_{0.05}Hx | 0.60 | −219.2 | 45.0 |

Nb_{0.95}Pd_{0.05}Hx | 0.60 | −231.8 | 26.1 |

Nb_{0.90}Pd_{0.10}Hx | 0.45 | −213.9 | 45.5 |

TaH_{x} | 0.55 | −229.4 | 62.4 |

Ta_{0.95}W_{0.05}Hx | 0.55 | −228.6 | 32.1 |

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.

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 _{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.

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 [_{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

understanding for inherent nature of interstitial non-stoichiometric compounds like γ-FeC_{x}.

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 [

The reviewed standardized statistical thermodynamic analysis procedure accepting an a priori assumption of constant _{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.

The author would like to thank sincerely Prof. Dr. Masahiro KATSURA who introduced the statistical thermodynamic analysis procedure to the author during the years of apprenticeship at Department of Nuclear Engineering, Faculty of Engineering, Osaka University in 1970s through reading together the classical text book on statistical thermodynamics authored by Fowler and Guggenheim.