Thermodynamic Descriptor Calculator

Computes mixing entropy, size mismatch, mixing enthalpy, valence electron concentration, the Yang–Zhang Ω parameter, and Miedema-model formation enthalpies for compounds, solid solutions, and amorphous alloys.

Pair enthalpies use the Miedema model with elemental parameters (φ*, nws1/3, V2/3) from de Boer et al. (1988). Compound, solid-solution, and amorphous enthalpies follow Zhang, R. F. et al., Comput. Phys. Commun. 209, 58–69 (2016).

Theoretical background

Empirical thermodynamic descriptors for multi-component alloys

This page derives every quantity the calculator produces. That covers the configurational entropy ΔSmix, atomic size mismatch δ, valence electron concentration VEC, the Miedema chemical mixing enthalpy ΔHmix, the Yang–Zhang stability parameter Ω, and Miedema-model formation enthalpies for compounds, disordered solid solutions, and amorphous alloys. Each section cites the primary literature.

01High-entropy alloys and the role of empirical descriptors

Cantor, Chang, Knight & Vincent (2004) and, independently, Yeh, Chen, Lin et al. (2004) showed that equimolar five-component alloys can form a single disordered fcc phase. That opened a new design problem. With the whole interior of an n-element phase diagram in play, exhaustive experimental screening is not practical. At five elements with 5 % composition steps, a brute-force survey needs about \(10^{4}\) compositions per crystal structure. Empirical thermodynamic descriptors are simple proxies for the entropic, enthalpic, and geometric driving forces, and they emerged as fast pre-screening tools.

They are not first-principles predictions. They are cheap surrogates that flag compositions unlikely to form a stable disordered solid solution before any DFT, CALPHAD, or experimental work happens. Across the published HEA database, surveyed by Yang & Zhang (2012), Guo & Liu (2011), and Zhang, Zuo, Tang et al. (2014), the descriptors used here correctly classify roughly four out of five cases. They are the ones most consistently reported in the HEA literature.

02Configurational mixing entropy

For a random ideal solid solution, the configurational entropy follows from the Boltzmann–Planck relation \(S = k_{\mathrm B}\,\ln\Omega_{\mathrm{cfg}}\), where \(\Omega_{\mathrm{cfg}}\) counts the microstates compatible with a given macrostate. With \(N\) atoms placed at random across an \(n\)-component substitutional lattice and \(c_i N\) atoms of element \(i\), \(\Omega_{\mathrm{cfg}}\) is the multinomial coefficient \(N! / \prod_i (c_i N)!\). Stirling's approximation \(\ln N! \approx N \ln N - N\) applied to every factorial, divided by Avogadro's number, gives the molar configurational entropy of mixing.

\[\Delta S_{\mathrm{mix}} \;=\; -R\sum_{i=1}^{n} c_i\,\ln c_i\]

The maximum sits at the equiatomic composition \(c_i = 1/n\), where \(\Delta S_{\mathrm{mix}} = R\ln n\). At \(n = 5\) this is \(R\ln 5 \approx 1.61\,R \approx 13.4\;\mathrm{J\,mol^{-1}\,K^{-1}}\). Yeh (2004) introduced these phenomenological cut-offs.

  • \(\Delta S_{\mathrm{mix}} > 1.5\,R\). High-entropy alloy.
  • \(R \le \Delta S_{\mathrm{mix}} \le 1.5\,R\). Medium-entropy alloy.
  • \(\Delta S_{\mathrm{mix}} < R\). Conventional dilute alloy.
Caveat

\(\Delta S_{\mathrm{mix}}\) here is purely configurational. Vibrational, electronic, and magnetic entropy are ignored. Their combined size can exceed the configurational term above the Debye temperature (Ma, Eisenbach & Yan, 2015). The 1.5 \(R\) cut-off is a convention, not a thermodynamic boundary.

03Atomic size mismatch

Hume-Rothery's rules for substitutional solubility (Hume-Rothery & Powell, 1934) say that mutual solid solubility is favoured when atomic radii differ by less than about 15 %. Zhang, Zhou, Lin et al. (2008) extended this to multicomponent systems with a root-mean-square fractional deviation about the mean radius.

\[\bar r \;=\; \sum_{i} c_i\,r_i,\qquad \delta \;=\; 100\,\sqrt{\,\sum_i c_i\!\left(1 - \frac{r_i}{\bar r}\right)^{\!2}\,}\;\;(\%)\]

\(\delta\) has no thermodynamic units. It is a geometric quantity. It tracks the elastic strain stored in a substitutional lattice when atoms of different sizes share the same site. Eshelby's (1957) inclusion model gives the underlying physics. A misfitting sphere of radius \(r_i\) in a matrix of radius \(\bar r\) carries a strain energy \(\propto (1 - r_i/\bar r)^2\). Across the HEA database, single-phase solid solutions are observed almost exclusively at \(\delta \lesssim 6.5\,\%\), as documented by Zhang et al. (2008) and Guo & Liu (2011).

04Valence electron concentration

The valence electron concentration (VEC) is the composition-weighted average of the valence electron count of each constituent.

\[\mathrm{VEC} \;=\; \sum_i c_i\,V_i\]

VEC extends Hume-Rothery's electron-to-atom ratio to many components. In the rigid-band picture of transition-metal alloys, the relative stability of close-packed (fcc) and open (bcc) structures depends on whether the Fermi level sits in a region of high or low d-band density of states. That density tracks the average number of valence electrons per atom. Guo & Liu (2011) tabulated phase occurrences across more than 100 HEAs and proposed three regimes.

  • VEC \(\geq 8.0\). Single-phase fcc dominates.
  • \(6.87 \le\) VEC \(< 8.0\). Mixed fcc and bcc.
  • VEC \(< 6.87\). Single-phase bcc dominates.

These thresholds are statistical, not exact. They ignore ordering, magnetism, and the distinct behaviour of the early refractory metals. Refractory HEAs in particular often disobey the rule (Senkov, Wilks, Miracle et al., 2010).

05The Miedema macroscopic-atom model

Miedema (1973), extended by Miedema, de Châtel & de Boer (1980) and tabulated in de Boer, Boom, Mattens, Miedema & Niessen (1988), treats each metal as a macroscopic atom. That atom is a Wigner–Seitz cell with two intensive electronic properties at its boundary.

  • \(\varphi^{*}\), an adjusted electronegativity (V). It is derived from the metal's work function and calibrated against measured dilute-alloy heats of solution.
  • \(n_{ws}\), the electron density at the Wigner–Seitz cell boundary in “density units” (one density unit equals \(6\times 10^{22}\) electrons cm\(^{-3}\)). It is inferred from bulk modulus and molar volume.

When atoms A and B come into contact across a unit area of common Wigner–Seitz boundary, two electronic mismatches need to be settled.

  1. An electronegativity difference \(\Delta\varphi^{*}\) drives charge transfer. This term is always attractive and lowers the enthalpy.
  2. An electron-density discontinuity \(\Delta n_{ws}^{1/3}\) cannot relax without an energy cost. This term is always repulsive and raises the enthalpy.

For transition and non-transition pairs an additional hybridisation correction \(R_{\mathrm{hyb}}\) applies. Together these give the Miedema interfacial enthalpy amplitude.

\[\Gamma_{AB} \;=\; \frac{-P\,(\Delta\varphi^{*})^{2} \;+\; Q\,(\Delta n_{ws}^{1/3})^{2} \;-\; R_{\mathrm{hyb}}} {\tfrac{1}{2}\!\left(n_{ws,A}^{-1/3} + n_{ws,B}^{-1/3}\right)}\]

The denominator is the harmonic-mean inverse boundary electron density, weighting both atoms equally. The empirical constants are fitted once to a reference dataset and held fixed across the periodic table.

ConstantValueApplies toSource
\(P\)14.2TM–TM pairsde Boer et al. (1988)
\(P\)12.35TM–NTM pairsde Boer et al. (1988)
\(P\)10.7NTM–NTM pairsde Boer et al. (1988)
\(Q/P\)9.4universalde Boer et al. (1988)
\(r/P\)1.9 (Al), 2.1 (Si)NTM partner onlyde Boer et al. (1988)

\(\Gamma_{AB}\) is the building block of every chemical enthalpy term in the model. Multiplying it by a volume factor and a composition-dependent mixing-degree function recovers the partial enthalpy at infinite dilution, the equiatomic mixing enthalpy, the compound formation enthalpy, and the chemical contribution to the solid-solution formation enthalpy.

A note on terminology

\(\varphi^{*}\) and \(n_{ws}^{1/3}\) are not derived from alloy thermodynamics. \(\varphi^{*}\) is fitted to elemental work functions with a small offset that reproduces binary heats of solution. \(n_{ws}^{1/3}\) comes from the elemental bulk modulus and molar volume. The Miedema model therefore predicts alloy enthalpies from elemental data alone. It does not back-calculate from compound thermodynamics.

06Surface concentrations and the volume correction

The chemical interaction takes place across the Wigner–Seitz cell boundary. The surface area each species exposes (not its bulk mole fraction) controls the magnitude of \(\Gamma_{AB}\). The surface concentration of element \(i\) is defined as

\[c_{i}^{\,s} \;=\; \frac{c_i\,V_i^{2/3}}{\sum_{k} c_k\,V_k^{2/3}}\]

so that a larger atom presents a larger contact area per mole. \(V_i^{2/3}\) is Miedema's two-thirds-power molar volume in cm\(^{2}\), tabulated in de Boer et al. (1988).

\(V_i^{2/3}\) is not strictly constant. When an A atom dissolves in B, charge transfer driven by \(\Delta\varphi^{*}\) compresses or dilates its boundary. de Boer et al. (1988, Eq. 2.10) introduced a self-consistent volume correction.

\[V_A^{2/3*} \;=\; V_A^{2/3}\!\left(1 + a_A\,c_B^{\,s}\,(\varphi_A^{*} - \varphi_B^{*})\right)\]

The symmetric expression applies to B. The element-specific constant \(a_i\) is 0.04 for most transition metals and 0.07 for the noble and sp-bonded metals Cu, Ag, and Al. \(c_i^{\,s}\) depends on the corrected \(V_i^{2/3*}\) and vice versa, so the equations are solved iteratively. Convergence usually takes 3 to 5 cycles.

07Mixing enthalpy of disordered alloys

For a regular solid solution, the molar enthalpy of mixing is purely pairwise and quadratic in composition. \(\Delta H^{\mathrm{reg}}(c_A, c_B) = \Omega_{AB}\,c_A c_B\), where \(\Omega_{AB}\) is the binary interaction parameter. For an \(n\)-component system, Yang & Zhang (2012), following Takeuchi & Inoue (2005), wrote

\[\Delta H_{\mathrm{mix}} \;=\; \sum_{i

\(\Delta H_{ij} \equiv \Delta H_{AB}^{\mathrm{mix}}(c_A = c_B = \tfrac{1}{2})\) is the equiatomic binary mixing enthalpy. The factor of 4 absorbs the relation \(\Omega_{AB} = 4\,\Delta H_{AB}^{\mathrm{mix}}\), which follows from \(\Delta H^{\mathrm{reg}}(\tfrac{1}{2},\tfrac{1}{2}) = \tfrac{1}{4}\Omega_{AB}\). Each pair is counted once (\(i < j\)). Ternary and higher-order interactions are ignored.

Within Miedema, the equiatomic chemical mixing enthalpy comes from inserting \(c_A = c_B = \tfrac{1}{2}\) into the chemical SS expression of section 10.

\[\Delta H_{AB}^{\mathrm{mix}}(\tfrac{1}{2},\tfrac{1}{2}) \;=\; \frac{1}{2}\,\frac{V_A^{2/3*}\,V_B^{2/3*}}{V_A^{2/3*} + V_B^{2/3*}}\;\Gamma_{AB}\]

Tabulated \(\Delta H_{ij}\) values are commonly drawn from Takeuchi & Inoue (2005), who derived them from the same Miedema parameters. This calculator computes them on the fly, which keeps custom and off-table elements accessible. A manual override field lets you enter validated literature values.

Implementation note

The current calculator returns the interaction parameter \(2\,V_A^{2/3*}V_B^{2/3*}/(V_A^{2/3*} + V_B^{2/3*})\,\Gamma_{AB}\). That quantity is mathematically \(\Omega_{AB}\), four times the equiatomic mixing enthalpy. The multi-component sum then re-applies the factor of 4. Numerically this overstates \(\Delta H_{\mathrm{mix}}\) by a factor of four relative to the convention above. At equiatomic Cu–Ti the calculator gives about \(-33\,\mathrm{kJ\,mol^{-1}}\), the tabulated value in Takeuchi & Inoue (2005) is \(-9\,\mathrm{kJ\,mol^{-1}}\). Use the manual-override field if you need literature-consistent values.

08The Yang–Zhang Ω parameter

Yang & Zhang (2012) proposed a single dimensionless parameter that contrasts the entropic stabilisation of the disordered state against the enthalpic push for ordering.

\[\Omega \;=\; \frac{\bar T_{m}\,\Delta S_{\mathrm{mix}}}{\lvert\Delta H_{\mathrm{mix}}\rvert}\]

The average melting temperature is \(\bar T_m = \sum_i c_i T_{m,i}\). \(\Omega\) is the ratio of the maximum entropic Gibbs-energy contribution (\(T\Delta S_{\mathrm{mix}}\) evaluated at \(\bar T_m\)) to the size of the chemical mixing enthalpy. When \(\Omega > 1\), entropy at the solidus dominates. When \(\Omega \ll 1\), enthalpy dominates. The empirical threshold for solid-solution formation is \(\Omega \gtrsim 1.1\), usually combined with \(\delta \lesssim 6.5\,\%\).

\(\Omega \to \infty\) when \(\Delta H_{\mathrm{mix}} \to 0\). The calculator reports this case explicitly to avoid numerical divergence.

09Compound formation enthalpy

For an ordered binary intermetallic, chemical short-range order enhances the mixing-degree factor relative to the random-solution case. Miedema introduced a phenomenological factor.

\[f_{AB}(c^{\,s}) \;=\; c_A^{\,s}\,c_B^{\,s}\,\bigl(1 + 8\,(c_A^{\,s}\,c_B^{\,s})^2\bigr)\]

The coefficient \(\mu = 8\) is fitted to fully ordered compounds (de Boer et al., 1988). The chemical part of the formation enthalpy is then

\[\Delta H^{\mathrm{Mied}} \;=\; f_{AB}(c^{\,s})\,\bigl(c_A V_A^{2/3} + c_B V_B^{2/3}\bigr)\,\Gamma_{AB}\]

Zhang, Zhang, He, Jing & Sheng (2016) refined the model with an atomic-size-difference (ASD) prefactor \(S_C\) that suppresses the formation enthalpy of compounds with poor geometric compatibility.

\[\Delta H_{\mathrm{form}} \;=\; S_C\;\Delta H^{\mathrm{Mied}} \;+\; \Delta H_{\mathrm{trans}}\]

Two functional forms appear in practice. The composition-dependent (“original”) form is \(S_C = 1 - |V_B^{2/3} - V_A^{2/3}| / (c_A^{\,s}V_A^{2/3} + c_B^{\,s}V_B^{2/3})^{2}\). The composition-independent (“recent”) form, used by default in this calculator, is \(S_C = V_A^{2/3}\,V_B^{2/3} / [\tfrac{1}{2}(V_A^{2/3} + V_B^{2/3})]^{2}\). \(\Delta H_{\mathrm{trans}}\) is the semiconductor-to-metal transformation enthalpy applied to elements that are non-metallic in their ground state. Si is the main case in this dataset, with \(\Delta H_{\mathrm{trans}} \approx 34\;\mathrm{kJ\,mol^{-1}}\) (de Boer et al., 1988). For all transition and noble metals \(\Delta H_{\mathrm{trans}} = 0\).

10Solid-solution formation enthalpy

For a disordered solid solution, where atoms occupy lattice sites at random, the formation enthalpy decomposes into three additive contributions.

\[\Delta H_{\mathrm{SS}} \;=\; \Delta H_{\mathrm{chem}} + \Delta H_{\mathrm{elast}} + \Delta H_{\mathrm{struct}}\]

Chemical contribution

With \(\mu = 0\) (no chemical short-range order), the mixing-degree factor reduces to \(f_{AB} = c_A^{\,s} c_B^{\,s}\). After symmetrisation,

\[\Delta H_{\mathrm{chem}} \;=\; c_A\,c_B\bigl(c_B^{\,s}\,V_A^{2/3} + c_A^{\,s}\,V_B^{2/3}\bigr)\,\Gamma_{AB}\]

Elastic mismatch contribution (Eshelby–Friedel)

A solute atom of molar volume \(W_A\) inserted into a matrix of molar volume \(W_B\) produces a strain field. Its energy comes from the Eshelby (1957) inclusion problem and was adapted to dilute alloys by Friedel (1954). For an elastically isotropic host,

\[E_{A\;\mathrm{in}\;B} \;=\; \frac{2\,K_A\,G_B\,(W_A - W_B)^{2}}{3\,K_A\,W_B + 4\,G_B\,W_A}\]

where \(K_A\) is the bulk modulus of the inclusion and \(G_B\) the shear modulus of the matrix. The symmetric expression \(E_{B\;\mathrm{in}\;A}\) describes the reverse case. The composition-weighted total elastic enthalpy (Niessen & Miedema, 1983) is

\[\Delta H_{\mathrm{elast}} \;=\; c_A\,c_B\bigl(c_B\,E_{A\;\mathrm{in}\;B} + c_A\,E_{B\;\mathrm{in}\;A}\bigr)\]

\(\Delta H_{\mathrm{elast}}\) is identically zero for size-matched pairs and grows quadratically with \(\Delta W\). It is always non-negative. Elastic strain destabilises the solid solution.

Structural contribution

When the constituent elements adopt different ground-state crystal structures, forming a single-phase solution requires one or both elements to sit in a metastable lattice. The energy cost comes from the lattice-stability data of Niessen & Miedema (1983) and Loeff, Weeber & Miedema (1988). The structure-dependent energy \(E_{\alpha}(z)\) of a transition metal is tabulated as a function of valence-electron count \(z\), referenced to the bcc structure with \(E_{\mathrm{bcc}}(z) \equiv 0\).

The structural enthalpy of the alloy is the difference between the most stable structure at the composition-averaged \(\langle z\rangle = \sum_i c_i z_i\) and the composition-weighted reference energies of the pure elements.

\[\Delta H_{\mathrm{struct}} \;=\; \min_{\alpha}\,E_{\alpha}\!\bigl(\langle z\rangle\bigr) \;-\; \sum_i c_i\,E_{\alpha_i^{0}}(z_i)\]

\(\alpha_i^{0}\) is the equilibrium structure of pure element \(i\). The tabulated values are interpolated linearly between integer \(z\). For the late transition metals (\(z \geq 10\)) lattice-stability differences are taken to vanish, matching the original Niessen–Miedema treatment.

11Amorphous-alloy formation enthalpy

An amorphous alloy has no long-range order. Its chemical contribution matches that of a disordered solid solution (Bakker, 1998). The structural cost is replaced by a topological cost. That cost reflects the density-of-states penalty for non-crystalline packing (Loeff, Weeber & Miedema, 1988).

\[\Delta H_{\mathrm{AM}} \;=\; \Delta H_{\mathrm{chem}} \;+\; \Delta H_{\mathrm{topo}},\qquad \Delta H_{\mathrm{topo}} \;=\; \beta\,\sum_i c_i\,T_{m,i}\]

with \(\beta \approx 3.5\;\mathrm{J\,mol^{-1}\,K^{-1}}\) (Bakker, 1998). The topological term reflects an empirical observation. The enthalpy difference between an amorphous and a crystalline metal scales linearly with melting temperature, because both are controlled by the cohesive energy. \(\Delta H_{\mathrm{topo}}\) is always positive. Amorphisation is energetically unfavourable without a chemical driving force.

The crossover \(\Delta H_{\mathrm{AM}} < \Delta H_{\mathrm{SS}}\) is the basic glass-forming criterion in this picture. Senkov & Miracle (2001) added size-distribution descriptors, but the enthalpy crossover stays the dominant indicator.

12Phase-selection criteria

The descriptors above combine into a small number of phenomenological rules. The most widely cited are listed below.

  • Yang & Zhang (2012). A single-phase solid solution is likely when \(\Omega \geq 1.1\) and \(\delta \leq 6.5\,\%\).
  • Guo & Liu (2011). Within the SS field, the structure follows VEC. Fcc above 8.0, bcc below 6.87, mixed in between.
  • Zhang et al. (2008). A complementary cut on \(\Delta H_{\mathrm{mix}}\) between roughly \(-22\) and \(+5\,\mathrm{kJ\,mol^{-1}}\) separates SS from compounds and miscibility gaps.
  • Miedema-model crossover (Zhang et al., 2016). When \(\Delta H_{\mathrm{form}}^{\mathrm{compound}}\) is much more negative than \(\Delta H_{\mathrm{SS}} - T\Delta S_{\mathrm{mix}}\), an ordered intermetallic is expected. When \(\Delta H_{\mathrm{AM}} < \Delta H_{\mathrm{SS}}\), an amorphous phase is favoured.

13Limitations and scope

All descriptors implemented here share three structural limitations worth keeping in mind when reading the outputs.

  1. Pairwise additivity. The mixing-enthalpy and Miedema solid-solution formulae are sums over binary pairs. Ternary and higher-order interactions are ignored. That is exact for ideal regular solutions and a controlled approximation for sub-regular systems. It can fail when three- or four-body chemical interactions are large. Alloys that combine Al with several transition metals are a common example.
  2. Equilibrium-only. The descriptors describe the equilibrium driving force. They say nothing about kinetics, nucleation rates, diffusion-limited segregation, or processing-induced metastable phases. As-cast HEAs often disagree with equilibrium predictions for that reason.
  3. Empirical parameters. Miedema's \(\varphi^{*}\), \(n_{ws}^{1/3}\), and \(V^{2/3}\) values are calibrated against a fixed dataset of binary heats of formation. The model performs best for transition-metal-rich alloys. It degrades for systems with lanthanides, actinides, or strongly covalent partners (de Boer et al., 1988).

For quantitative phase prediction, supplement these descriptors with a CALPHAD free-energy minimisation (Lukas, Fries & Sundman, 2007) or first-principles methods such as the special quasirandom structure approach within DFT (Zunger, Wei, Ferreira & Bernard, 1990). The descriptors here remain useful as a fast, transparent pre-screen and as a sanity check on more expensive calculations.

Quick reference: equations at a glance

Size mismatch \(\delta\)

\[\bar{r} = \sum_i c_i\, r_i\]
\[\delta = 100\,\sqrt{\sum_i c_i \!\left(1 - \frac{r_i}{\bar{r}}\right)^{\!2}}\;\;(\%)\]

\(c_i\) = atomic fraction, \(r_i\) = atomic radius (pm)

Mixing entropy \(\Delta S_{\mathrm{mix}}\)

\[\Delta S_{\mathrm{mix}} = -R \sum_i c_i \ln c_i\]

\(R = 8.314\;\mathrm{J\,mol^{-1}\,K^{-1}}\)

Mixing enthalpy \(\Delta H_{\mathrm{mix}}\)

\[\Delta H_{\mathrm{mix}} = \sum_{i<j} 4\,\Delta H_{ij}\,c_i\,c_j\]

\(\Delta H_{ij}\) = Miedema pair enthalpy (kJ/mol)

Melting temperature and Omega

\[\bar{T}_m = \sum_i c_i\,T_{m,i}\]
\[\Omega = \frac{\bar{T}_m \;\Delta S_{\mathrm{mix}}}{\lvert\Delta H_{\mathrm{mix}}\rvert}\]

\(\Delta H_{\mathrm{mix}}\) in J/mol for this ratio (Yang and Zhang, 2012)

Valence electron concentration

\[\mathrm{VEC} = \sum_i c_i\,V_i\]

\(V_i\) = valence electrons of element \(i\)

Surface concentrations

\[c_i^{\,s} = \frac{c_i\,V_i^{2/3}}{\sum_k c_k\,V_k^{2/3}}\]

\(V_i^{2/3}\) = Miedema two-thirds molar volume (cm2)

Miedema interfacial enthalpy \(\Gamma_{AB}\)

\[\Gamma_{AB} = \frac{-P\,(\Delta\varphi^*)^2 \;+\; Q\,(\Delta n_{ws}^{1/3})^2 \;-\; R_{\mathrm{hyb}}}{\;\tfrac{1}{2}\!\left(\,n_{ws,A}^{-1/3} + n_{ws,B}^{-1/3}\right)}\]

\(\varphi^*\) = adjusted electronegativity (V), \(n_{ws}^{1/3}\) = electron density at Wigner-Seitz boundary (d.u.)
\(P\) = 14.2 (TM-TM), 12.35 (TM-NTM), 10.7 (NTM-NTM), \(Q/P = 9.4\)
\(R_{\mathrm{hyb}}\) = hybridization correction for TM-NTM pairs only

Volume correction (de Boer et al. 1988, Eq. 2.10)

\[V_A^{2/3*} = V_A^{2/3}\!\left(1 + a_A\, c_B^{\,s}\, (\varphi_A^* - \varphi_B^*)\right)\]
\[V_B^{2/3*} = V_B^{2/3}\!\left(1 + a_B\, c_A^{\,s}\, (\varphi_B^* - \varphi_A^*)\right)\]

\(a\) = volume correction constant (0.04 for most TM, 0.07 for Cu/Ag/Al), \(c^s\) = surface concentration of the other element
Solved iteratively: corrected \(V^{2/3*}\) values update \(c^s\) until convergence (de Boer et al., 1988)

Equiatomic pair enthalpy \(\Delta H_{AB}\)

\[\Delta H_{AB} = \frac{2\;V_A^{2/3*}\;V_B^{2/3*}}{V_A^{2/3*}+V_B^{2/3*}} \;\cdot\; \Gamma_{AB}\]

Uses volume-corrected \(V^{2/3*}\) from above. This feeds into \(\Delta H_{\mathrm{mix}}\) through the sub-regular solution model.

Compound formation enthalpy

\[\Delta H_{\mathrm{form}} = S_C \;\Delta H^{\mathrm{Mied}} + \Delta H_{\mathrm{trans}}\]
\[\Delta H^{\mathrm{Mied}} = f(c^s)\!\left(c_A V_A^{2/3} + c_B V_B^{2/3}\right)\Gamma_{AB}\]

\(S_C\) = atomic size difference factor, \(\Delta H_{\mathrm{trans}}\) = semiconductor to metal transformation energy
\(f(c^s) = c_A^s c_B^s(1 + 8(c_A^s c_B^s)^2)\) (Zhang et al. 2016, Eq. 3-6)

Solid solution enthalpy

\[\Delta H_{\mathrm{SS}} = \Delta H_{\mathrm{chem}} + \Delta H_{\mathrm{elast}} + \Delta H_{\mathrm{struct}}\]
  • \(\Delta H_{\mathrm{chem}} = c_Ac_B(c_B^sV_A^{2/3} + c_A^sV_B^{2/3})\,\Gamma_{AB}\)
  • \(\Delta H_{\mathrm{elast}} = c_Ac_B(c_B E_{A \text{ in } B} + c_A E_{B \text{ in } A})\) (Eshelby-Friedel, using \(K\) and \(G\))
  • \(\Delta H_{\mathrm{struct}}\) from lattice stability vs. average valence electrons (Niessen and Miedema, 1983)

Elastic mismatch energy

\[E_{A \text{ in } B} = \frac{2\,K_A\,G_B\,(W_A - W_B)^2}{3\,K_A\,W_B + 4\,G_B\,W_A}\]

\(K\) = bulk modulus (GPa), \(G\) = shear modulus (GPa), \(W\) = molar volume (cm3/mol)

Amorphous alloy enthalpy

\[\Delta H_{\mathrm{AM}} = \Delta H_{\mathrm{chem}} + \Delta H_{\mathrm{topo}}\]
\[\Delta H_{\mathrm{topo}} = \beta\!\left(c_A T_{m,A} + c_B T_{m,B}\right)\]

\(\beta \approx 3.5\;\mathrm{J\,mol^{-1}\,K^{-1}}\)

Composition

ElementPercent (%)

Pair enthalpies (ΔHᵢⱼ)

Select at least two elements to show pairs.

Unique pairs (i<j) for the currently selected elements.

Calculated from the Miedema model. Override manually if needed.

Results

Normalized composition

xᵢ
Cu
0.2000
Co
0.2000
Mn
0.3500
Ni
0.2000
Fe
0.05000
mole fraction

Mixing entropy

ΔS mix

12.33

J/mol·K

Atomic size difference

δ

3.674

%

Mean melting temperature

Tm

1593

K

Mean melting temperature

Tm

1320

°C

Mixing enthalpy

ΔH mix

2.777

kJ/mol

Valence electron concentration

VEC

8.850

—

Omega parameter

Ω

7.071

—

Phase-prediction rules

What each canonical empirical rule predicts for this composition, applied to the descriptor values above.

Miedema-model formation enthalpies

Auxiliary outputs. Compound, solid-solution, and amorphous formation enthalpies and their decompositions.

Compound enthalpy (Miedema+ASD)

ΔH compound

1.185

kJ/mol

Solid solution enthalpy

ΔH SS

-6.954

kJ/mol

SS chemical contribution

ΔH chem

-0.6507

kJ/mol

SS elastic mismatch

ΔH elast

0.2222

kJ/mol

SS structural contribution

ΔH struct

-6.525

kJ/mol

Amorphous enthalpy

ΔH AM

21.65

kJ/mol

AM topological contribution

ΔH topo

22.30

kJ/mol

Saved results

Elements Composition (at. %) δ (%) Tm (°C) ΔS mix ΔH mix VEC Ω ΔH comp ΔH SS ΔH AM Action
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