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This work presents a formalism to calculate the configurational entropy of mixing based on the identification of non-interacting atomic complexes in the mixture and the calculation of their respective probabilities, instead of computing the number of atomic configurations in a lattice. The methodology is applied in order to develop a general analytical expression for the configurational entropy of mixing of interstitial solutions. The expression is valid for any interstitial concentration, is suitable for the treatment of interstitial short-range order (SRO) and can be applied to tetrahedral or octahedral interstitial solutions in any crystal lattice. The effect of the SRO of H on the structural properties of the Nb-H and bcc Zr-H solid solutions is studied using an accurate description of the configurational entropy. The methodology can also be applied to systems with no translational symmetry, such as liquids and amorphous materials. An expression for the configurational entropy of a granular system composed by equal sized hard spheres is deduced.

For a long time, configurational entropy has been calculated using the same formalism,

The traditional methodology, however, has encountered severe restrictions in the development of expressions in multicomponent complex systems such as interstitial solid solutions or liquids and amorphous materials. Although a huge amount of works has been carried out in both fields of research, the deduction of accurate analytical expressions for the configurational entropy of mixing has been very elusive due to the complicated underlying physics. While the difficulties with interstitial solutions arise in the blocking effects, the difficulties with liquids, amorphous materials and glasses are derived from the almost insurmountable task of computing the number of configurations in a system without lattice periodicity. In addition, the number of configurations will depend on the characteristics of the interactions between the chemical elements in each particular system, thus adding more difficulties in multicomponent systems.

Although there are several expressions which can be used to compute the entropy of mixing in interstitial solid solutions, as ^{19} to 2 × 10^{7} in Ti-O according to Boureau and Tetot [

A similar situation to that in interstitial solutions, as regards the encoding of structural information into a compact expression, is found in liquids, amorphous materials and glasses. In these systems, the lack of periodicity of the atomic arrangements makes the deduction of a universal and unique structural description a very difficult task, leaving many open questions unanswered. Consequently, the deduction of an analytical expression for the configurational entropy of mixing of multicomponent systems, valid for any non-crystalline state of the matter, remains a largely unsolved problem.

There are numerous systems where the ideal entropy describes the experimental data sufficiently well. However, is the physics of these systems described correctly or is it just a model at hand fitting the configurational entropy data? The research on interstitial and amorphous systems shows that previous to the calculation of the configurational entropy, a detailed knowledge of the main physical features of the system is necessary. Nevertheless, this information will not be so useful unless an adequate theoretical tool is available. There are many examples of this issue in the scientific literature. The main purpose of this work is to present an alternative formalism for the calculation of the configurational entropy of mixing overcoming the previous limitation, and useful for application in complex systems even with no translational symmetry, such as liquids and amorphous materials. The methodology is based on the identification of the energy independent complexes in the mixture and the calculation of their corresponding probabilities.

This work is organized as follows: Section 2 presents a brief description of the methodology. The example of a linear polymer mixture is used to deduce the expressions due to Flory and to Huggins, and to illustrate the use of the current formalism once the main physical features of the system are identified. The method is applied to develop a general expression suitable for the study of SRO in interstitial solid solutions in Section 3. The final expression is implemented in the Fe-C system, and is also used to describe the SRO of H in bcc elements, such as Nb and Zr. Finally, in Section 4, the method is applied to an equal sized hard spheres system. Section 5 presents the Conclusions.

The entropy of a classical system with a discrete set of microstates is given by the Gibbs’ expression:
_{i}_{i}_{i}

where _{i}_{i}

The formidable problem of enumerating all the possible configurations available to the polymers on the lattice has been discussed by several authors. [_{o}

The approximations used by Flory to derive his well-known expression are the following:

The lattice is rigid: independence of lattice constant on composition,

Interchangeability of polymer segments,

All polymers are of the same size,

Average concentration of segments equal to overall average concentration.

To apply _{0}_{0} + rn

where _{i}

The computed probability of finding a polymer is an approximation in Flory’s model, but their number is correctly counted. It is important to note that

A better approximation for the probability of finding a polymer was computed by Huggins, which also takes the surface fraction σ into account. In this way the size and number of bonds of the polymer are included in the description, but not its shape, e.g., numbers of kinks. For Huggins, the probability of finding a polymer is computed by:

where _{i}

The probabilistic approach of

Due to interstitial repulsion, the set of vacancies is divided into two different species:

Therefore, the original problem could be re-interpreted as a random mixture of independent complexes of different sizes: free vacancies of one lattice site size and _{i}_{i}

where

These complexes could be free vacancies, individual interstitials, pairs and double pairs of interstitial atoms, among others. The probability of finding each one is computed by:

where the indices

where _{i}=n_{i}/N

The partial entropy of mixing for a random mixture of vacancies and one kind of interstitial atom

The unknown variables in

a linear function:

a quadratic function:

a sigmoid function:

In these functions, _{I}_{I}_{c}

One interesting example to illustrate the methodology outlined in Section 3.1 is the hcp Ti-O solid solution. In the hcp structure, there is only one octahedral interstitial site per metal atom, thus β = 1. The compound Ti_{2}O is the solubility limit of the solid solution. Thus, the occupancy of one site excludes two near neighboring sites with a soft blocking character. Consequently, the number of blocking sites per interstitial atom is (

The Fe-C in the austenitic phase is another useful system to apply _{4}C. This author found that, although the expression has no rational grounds, the model gave a satisfactory, result close to the values used to fit the experimental data [

It is necessary to identify the structure of the compound corresponding to the solubility limit of the solid solution in order to apply _{4}C compound with the Pm3m space group is assumed. For this system, β = 1 as there is only one octahedral interstitial site per iron atom. The C atoms occupy the octahedral sites and block all their twelve near neighbors belonging to the interstitial lattice. The structural analysis reveals that the blocking effect has a soft character. Indeed, each blocked site is shared by four C atoms located in the second coordination sphere of the interstitial lattice. Consequently, the number of blocked sites per interstitial atoms is (

where x is the C molar fraction. The final expression for the C activity computed from

One important issue to be noted here is that the Fe_{4}C compound was selected as the solubility limit of the austenitic phase. This compound could not be a correct model for the real solubility limit and, consequently, the blocking model and the expression for the configurational entropy should be revisited once more experimental information is obtained. Previous modeling effort [

The H-Nb system is an amazing system due to the still unsolved controversies relating to the experimental results, such as: (1) inconsistencies between the solubility limit and the size of the basic complex,

Although various models have been proposed in the past for Nb-H, the most precise analytical expression for the partial configurational entropy using the RBM is due to Boureau, as was shown by Oates

It is clear from a comparison of the models shown in _{0}

The experimental solubility limit of the solid solution is H/Nb = 1.21 ± 0.04 [_{0}

The configurational entropy derived from

In this expression _{1}_{2}_{I}_{1}_{2}

The dependence of the number of pairs

These results highlight that it is possible to solve the controversies between the size of the basic complex, _{0}

The good description of the partial configuration entropy achieved by including H pairs in the solid solution gives enough confidence in the model to compute the critical composition of the miscibility gap located at _{c}_{c}_{c}_{0}_{I} = θ_{1} + 4θ_{4}

_{c}

It is also proposed that the double pair configuration could be interpreted as the seed for the ordered phases observed experimentally for H compositions greater than

Zr-H is another very interesting system with some unexplained results and unsolved controversies regarding the behavior of the H atoms in the bcc phase. It is currently assumed that the H atoms are located randomly in tetrahedral positions [

There are several models [_{0}

The same methodology applied in Nb-H will be used in this system to look for a phenomenology different to the accepted RBM behavior. The first step is to apply

It is clear from _{I}

The examples in Section 3 show the difficulties arising in counting the number of configurations if the assumption of no superposition between chemical species is abandoned. One solution has been proposed in this work using a methodology different from the traditional one.

The example of this section highlights the difficulties encountered in computing the configurational entropy of mixing when there is no periodicity of the lattice. The model of Section 2 can be applied as there is no reference to any lattice in the deduction of

The quantitative morphological description of local structure is a requirement when studying amorphous systems, such as granular matter or glasses. While granular structures are disordered, metallic glasses displays various degrees of structural ordering beyond the short range. This means that a unique ideal structure where all the grains positions are uniquely assigned does not exist. There are a very large number of structures that have equivalent global properties (packing fraction, mechanical properties,

There is a huge amount of research being carried out trying to answer that question. Of these, the results of Aste

The Aste _{min}_{min}

The formalism of Section 2 is applied in this work to this idealized system. The identification of the basic complex or motif is fundamental to the application of the current methodology to compute the configurational entropy. For this system such structures are the Voronoi cells with _{T}

The volume distribution of the Voronoi cells for a given _{min}_{i}_{i}

The _{min})/k_{min}_{i}_{i}

This expression should be put in a compact form in order to get a useful expression to compare with previous expressions and to deduce the equation of state of this idealized system. In any case, the hard spheres system is a nice example showing the benefit of using

This work presents a formalism to calculate the configurational entropy of mixing alternative to the usual method of counting the number of atomic configurations. The traditional methodology found important restrictions to encode the physical information into compact expressions in complex systems such as interstitial solid solutions or liquids and amorphous materials. The methodology presented in this work is based on the identification, through a careful analysis of the main physical features of the system, of the energy independent complexes in the mixture and the calculation of their corresponding probabilities. The examples presented in this work, related to interstitial solutions and equal sized hard spheres fluid, show that accurate and general expressions for the configurational entropy of mixing can be developed, even in systems with no translational symmetry.

The effects of H SRO on the structural properties of the bcc-H solid solutions, such as Nb-H and Zr-H, are analyzed using an accurate description of the configurational entropy developed in this work. This expression, suitable for the treatment of interstitial clustering and SRO, can be applied to tetrahedral or octahedral interstitial solutions in any crystal lattice. The structural model for the Nb-H solid solution presented in this work is based on the existence of H pairs in the α–phase and double pairs in α′-phase. The model explains unsolved controversies in this system related to the length of the H-H interaction, the critical composition of the miscibility gap and the relation between the disordered and the ordered phases. The same expression is applied to the bcc Zr-H system. The unusual shape of the partial configurational entropy measured in this system can be accurately described if pairs and a small amount of double pairs are included in the model. The expression deduced in this work, by including the effect of SRO, opens up new possibilities in the calculations of phase diagrams and the thermodynamic modeling based on an accurate physical description of the alloy.

Regarding blocking effects, it is important to note here the distinction between chemical and elastic blocking and their influence on the configurational entropy. The difference between these kinds of interaction cannot be distinguished with previous models,

In some systems, such as Nb-H, it is not easy to distinguish between both kinds of blocking effects. Indeed, the chemical and elastic interactions between basic complexes result in a new kind of complex, e.g., double and triple pairs. However, a clear distinction between the blocking effects can be found in hcp TM-H solid solutions (TM = Sc, Y, Lu). In these systems, the behavior of H atoms at intermediate temperatures is not well described by a random distribution of H atoms over tetrahedral sites. Instead, a random distribution of pair H-TM-H along the c-axis of the cell, where a TM is located in the middle of the pair, is more appropriate for intermediate temperatures. These pairs are formed due to the chemical interactions between H atoms. In addition to chemical blocking effects, there is another one related to the relaxation of the host metal around the H-TM-H pairs providing support for the existence of a coherent stress field whose consequence is a chain formation along the c-axis of the hcp cell, observed only in Sc, Y and Lu but not in Ti and Zr [

If there is no formation of new complexes,

A similar situation to that of interstitial solutions in terms of deducing a general analytical expression for the configurational entropy of mixing is found in liquid, amorphous materials and glasses, due to the lack of lattice periodicity. In this work, an analytical expression based on the model of Aste

The author declares no conflict of interest.

_{2+x}phase

The partial configurational entropy of the Zr-H system. Comparison between experimental data and theoretical models available in the literature. The theoretical results are adjusted to the lowest experimental values. The data for the McLellan model is taken from [

The partial configurational entropy of Nb-H system. Comparisson between experimental and theoretical models available in the literature. The theoretical results are adjusted to the lowest experimental values. The O’Keeffe model [

Blocking spheres for: (_{0}

The partial configurational entropy of Nb-H. Comparison between experimental data and theoretical models presented in this work: a random mixture of vacancies, isolated H atoms and pairs with hard blocking of size _{0}

Location of the critical composition of the miscibility gap with the amount of pair clustering. An extreme is found at _{c}_{c}

The partial configurational entropy of Zr-H system. Comparison between the experimental data and theoretical models. Solid line: the model from _{0}_{0}

(_{0}_{0}