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Thermodynamic potentials of metallic alloys in the undercooled liquid and solid glassy states

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Pith's one-line read The paper shows that undercooled liquid and glassy metallic alloys give identical excess enthalpy, entropy, and Gibbs free energy in the supercooled liquid range, provided the melting entropy is computed as an integral rather than as…

desk verdict One clean test on Pd40Ni40P20 and two contingent ones; the enthalpy agreement is solid, the entropy agreement for Pt and Zr rests on an assumed correction the authors openly flag. read the letter →

arxiv 2507.02609 v1 pith:7GYF2PLA submitted 2025-07-03 cond-mat.dis-nn

A.S. Makarov , R.A. Konchakov , N.P. Kobelev , V.A. Khonik This is my paper · ORCID
classification cond-mat.dis-nn
keywords metallicglassesundercooledliquidexcessentropymeltingGibbsfreeenergydifferentialscanningcalorimetryorderparametercriticalcoolingrate
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tests whether the excess thermodynamic potentials of undercooled metallic melts and of solid metallic glasses are the same function of temperature in the supercooled liquid range between the glass transition and crystallization onset. Using three alloys, $\mathrm{Pd}_{40}\mathrm{Ni}_{40}\mathrm{P}_{20}$, $\mathrm{Pt}_{42.5}\mathrm{Cu}_{27}\mathrm{Ni}_{9.5}\mathrm{P}_{21}$, and $\mathrm{Zr}_{35}\mathrm{Hf}_{13}\mathrm{Al}_{11}\mathrm{Ag}_{8}\mathrm{Ni}_{8}\mathrm{Cu}_{25}$, the authors compare enthalpy, entropy, and Gibbs free energy computed from literature heat-capacity data for melts with values computed from their own differential scanning calorimetry of glasses. The enthalpy curves already agree; the entropy and Gibbs free energy curves agree only after the melting entropy is recalculated as an integral of $\delta Q/T$ over the melting range instead of the ratio $\Delta H_f/T_L$. This supports a thermodynamic continuity between the glassy state and the undercooled liquid across $T_g$. The paper also introduces an order parameter based on excess entropy that correlates with the critical cooling rate.

What carries the argument

The carrying object is the differential heat flow $\Delta W^{G-X}(T) = W_G(T) - W_X(T)$ between a relaxed glass and its crystalline counterpart, measured by differential scanning calorimetry at 3 K/min with a crystallized reference sample. Integrating this heat flow gives the glass excess enthalpy $\Delta H^{G-X} = (1/\dot{T})\int_T^{T_{cr}} \Delta W\,dT$, excess entropy $\Delta S^{G-X} = (1/\dot{T})\int_T^{T_{cr}} (\Delta W/T)\,dT$, and excess Gibbs free energy $\Delta \Phi^{G-X} = \int_T^{T_{cr}} \Delta S^{G-X}\,dT$ (Eqs. 6--8). The second load-bearing element is the integral melting entropy $\Delta S_{\rm melt} = (1/\dot{T})\int_{T_S}^{T_L} \Delta W(T)/T\,dT$, which replaces $\Delta H_f/T_L$ in the melt-side entropy formula, giving the corrected excess entropy used in Eq. (12). The order parameter $\xi = 1 - \Delta S_{\rm excess}/\Delta S_{\rm melt}$ then quantifies structural order, running from 0 for the fully liquid-like state to 1 for the crystal-like state.

What would settle it

Measure the melting DSC thermogram $\Delta W(T)$ for $\mathrm{Pt}_{42.5}\mathrm{Cu}_{27}\mathrm{Ni}_{9.5}\mathrm{P}_{21}$ and $\mathrm{Zr}_{35}\mathrm{Hf}_{13}\mathrm{Al}_{11}\mathrm{Ag}_{8}\mathrm{Ni}_{8}\mathrm{Cu}_{25}$, compute $\Delta S_{\rm melt}$ by Eq. (11), and check whether it equals $15.6$ and $9.8\ \mathrm{J\,mol^{-1}\,K^{-1}}$; if the measured values differ, the reported coincidence of excess entropy and Gibbs free energy curves in the supercooled liquid range for those alloys would not survive.

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Extended reading notes

Core claim

The central discovery is that the excess entropy of an undercooled melt, evaluated with the melting entropy taken as $\Delta S_{\rm melt} = (1/\dot{T})\int_{T_S}^{T_L} \Delta W(T)/T\,dT$, coincides with the excess entropy of the corresponding glass obtained from differential scanning calorimetry, and that the same corrected entropy makes the excess Gibbs free energies coincide. The conventional choice $\Delta S_f = \Delta H_f/T_L$ underestimates the melting entropy: for $\mathrm{Pd}_{40}\mathrm{Ni}_{40}\mathrm{P}_{20}$ it gives $10.6\ \mathrm{J\,mol^{-1}\,K^{-1}}$ while the integral gives $14.2\ \mathrm{J\,mol^{-1}\,K^{-1}}$, a 35 percent difference. With the corrected melting entropy, the melt and glass curves for $\Delta S$ and $\Delta \Phi$ coincide in the range $T_g < T < T_x$ for all three alloys. The corrected entropy also defines an order parameter $\xi = 1 - \Delta S_{\rm excess}/\Delta S_{\rm melt}$ whose value just below the crystallization onset, $\xi_{\rm scl}$, increases with the critical cooling rate $R_c$.

Load-bearing premise

The load-bearing premise is that the 35 percent difference between the integral melting entropy $\Delta S_{\rm melt}$ and the simple ratio $\Delta H_f/T_L$ measured for $\mathrm{Pd}_{40}\mathrm{Ni}_{40}\mathrm{P}_{20}$ applies unchanged to $\mathrm{Pt}_{42.5}\mathrm{Cu}_{27}\mathrm{Ni}_{9.5}\mathrm{P}_{21}$ and $\mathrm{Zr}_{35}\mathrm{Hf}_{13}\mathrm{Al}_{11}\mathrm{Ag}_{8}\mathrm{Ni}_{8}\mathrm{Cu}_{25}$, for which no melting thermograms are available.

Editorial extensions

If this is right

  • Entropy and Gibbs free energy of an undercooled melt can be recovered from DSC measurements on the glassy state alone in the supercooled liquid range, avoiding direct calorimetry on metastable melts.
  • The common approximation $\Delta S_f = \Delta H_f/T_L$ systematically underestimates the melting entropy, and the resulting error propagates into the melt excess entropy and Gibbs free energy; for $\mathrm{Pd}_{40}\mathrm{Ni}_{40}\mathrm{P}_{20}$ the underestimate is 35 percent.
  • The maximum of the excess Gibbs free energy curve does not correlate with the critical cooling rate, so this quantity should not be used as a glass-forming-ability indicator.
  • The order parameter $\xi_{\rm scl}$ at the end of the supercooled liquid range provides a thermodynamic proxy for glass-forming ability: smaller $\xi_{\rm scl}$ corresponds to smaller $R_c$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the equality of excess entropy between glass and undercooled melt holds generally, the structural (configurational) entropy is effectively fixed at $T_g$, so the glass can be treated as an isoconfigurational snapshot of the equilibrium liquid; this would link the present calorimetric result to entropy-based theories of glass formation.
  • The assumed transfer of the 35 percent melting-entropy correction from $\mathrm{Pd}_{40}\mathrm{Ni}_{40}\mathrm{P}_{20}$ to the other two alloys is directly testable: measuring their melting thermograms would either confirm the reported coincidence or show that it is an artifact.
  • The same comparison could be extended to non-metallic glass formers and to different heating rates, which would reveal whether the coincidence is universal or specific to these alloys and to the 3 K/min protocol.
  • A practical use of the $\xi_{\rm scl}$--$R_c$ correlation would be to estimate critical cooling rates from calorimetric data alone for newly synthesized glass-forming compositions, without rapid-quenching experiments.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This manuscript compares the excess thermodynamic potentials (enthalpy, entropy, Gibbs free energy) of undercooled melts and glasses for three metallic glass-forming alloys: Pd40Ni40P20, Pt42.5Cu27Ni9.5P21, and Zr35Hf13Al11Ag8Ni8Cu25. Melt quantities are computed from literature heat capacities and melting data using Eqs. (3)-(5); glass quantities are computed from differential scanning calorimetry using Eqs. (6)-(8). The authors report that the enthalpy agrees in the supercooled liquid range for all three alloys, while entropy and Gibbs free energy agree only after replacing the simple ratio ΔHf/TL by an integral melting entropy ΔSmelt defined in Eq. (11). They also introduce a dimensionless structural order parameter ξ = 1 − ΔSexcess/ΔSmelt and report a correlation between its value at the end of the supercooled liquid range and the critical cooling rate Rc.

Significance. The Pd40Ni40P20 case provides a direct, calorimetry-based test: the integral melting entropy is measured from the same DSC protocol, and the resulting entropy and Gibbs free energy coincidence is a genuine finding. If the result extends to other alloys, the approach would offer a practical route to extracting melt thermodynamic potentials from glass calorimetry and a physically motivated parameter for glass-forming ability. However, for Pt42.5Cu27Ni9.5P21 and Zr35Hf13Al11Ag8Ni8Cu25, the central entropy coincidence is not an independent confirmation because the melting-entropy values are assumed rather than measured. The ξ-Rc correlation also builds on earlier work by the same group. The paper therefore merits publication only after the load-bearing assumption is either removed or explicitly tested.

major comments (3)
  1. [Sec. 3.3, Eq. (12), Figs. 3(d,f), 4(b,c)] The central claim that the melt and glass entropy and Gibbs free energy coincide for Pt42.5Cu27Ni9.5P21 and Zr35Hf13Al11Ag8Ni8Cu25 is not supported by independent data. In the paragraph beginning "Regretfully, DSC thermograms..." the authors state that melting-region thermograms for these two alloys are unavailable and assume the same 35% difference between ΔSf and ΔSmelt as in Pd40Ni40P20, yielding ΔSmelt = 15.6 and 9.8 J mol−1 K−1. Since Eq. (12) is linear in ΔSmelt, adjusting ΔSmelt by construction shifts the corrected melt entropy onto the glass entropy; the observed coincidence in Figs. 3(d,f) and 4(b,c) is therefore not an independent confirmation. Only the Pd40Ni40P20 case, where ΔSmelt is measured directly from the DSC thermogram via Eq. (11), provides a genuine test. The authors should either supply melting-region DSC data for the other two alloys or clearly rephrase the claim as conditional on the 35% scaling and quantify the sensitivity of the coincidence to that assumption.
  2. [Sec. 3.3, Eqs. (11)-(12)] The definition of ΔSmelt in Eq. (11) may not be the correct reference for Eq. (12). Equation (11) integrates the heat flow from the solidus TS to the liquidus TL, so it gives S_L(TL) − S_X(TS). Equation (12) then treats this value as the entropy at T = TL and subtracts ∫_T^{TL} ΔCp^{L−X}/T dT, which implicitly refers the crystal entropy to the same temperature T. The difference between the two reference states is ∫_{TS}^{TL} Cp^X/T dT. This term is not visible in the paper; please clarify why it can be neglected or include it in the corrected entropy. Without this clarification, the quantitative value of ΔScorr_{L−X} is not thermodynamically well defined.
  3. [Sec. 3.6, Eqs. (16)-(17), Fig. 7] The ξscl versus Rc correlation inherits the assumed ΔSmelt for Pt42.5Cu27Ni9.5P21 and Zr35Hf13Al11Ag8Ni8Cu25, so it is subject to the same circularity. Moreover, the correlation was already reported by the same group in Ref. [20]; the present contribution is essentially three additional points for which the normalization is not independently measured. The paper should state explicitly what new evidence is added beyond the earlier correlation, or restrict the claim to Pd40Ni40P20.
minor comments (5)
  1. [Footnote in Sec. 3.4] The statement that ΔΦ = ΔH − TΔS is valid only under isothermal conditions is incorrect; G ≡ H − TS holds for any thermodynamic state. If the authors intend to compare integration conventions, they should specify the integration constant in Eq. (15).
  2. [Sec. 3.2] The text says "The excess entropy ΔHG−X(T) for the glassy state ... was calculated using Eq.(6)"; Eq. (6) gives enthalpy, not entropy. The wording should be corrected to "excess enthalpy".
  3. [Sec. 3.6] The phrase "Pd40Cu30Ni10P20 exhibits the smallest ξscql" contains a typo: ξscql should read ξscl.
  4. [References] References [28] and [30] are identical (Inoue, Nishiyama, Kimura, Mater. Trans. JIM 38 (1997) 179); please consolidate them.
  5. [General notation] The symbol Tf is used for both the melting temperature and the liquidus temperature; please state explicitly that Tf = TL in Eqs. (3)-(5) and (12).

Circularity Check

1 steps flagged · score 6.0 of 10

For Pt42.5Cu27Ni9.5P21 and Zr35Hf13Al11Ag8Ni8Cu25, the claimed melt/glass entropy and Gibbs-energy coincidence is constructed from an assumed, unmeasured melting-entropy correction rather than independently predicted; only the Pd40Ni40P20 case is a genuine test.

  1. fitted input called prediction [Section 3.3, Eq. (12) and the paragraph on missing DSC thermograms for Pt and Zr alloys]
    "Regretfully, DSC thermograms for Pt42.5Cu27Ni9.5P21 and Zr35Hf13Al11Ag8Ni8Cu25 alloys covering the melting region are not available. ... Assuming the same difference for Pt42.5Cu27Ni9.5P21 and Zr35Hf13Al11Ag8Ni8Cu25, one can accept the melting entropy for these alloys to be Smelt = 15.6 and 9.8 J mol−1 K−1 ... These ΔSmelt-values together with ΔCL−XP(T)-data from Refs [25, 26] were further used to recalculate the melt excess entropy using Eq.(12). Figure 3(d,f) shows temperature ΔSL−X(T)-dependences thus determined ..."

    Eq. (12) defines the corrected melt entropy as ΔS_corr_L−X(T) = ΔSmelt − ∫_T^Tf ΔC_p/T dT. For Pd40Ni40P20, ΔSmelt is directly measured by integrating the DSC melting peak, so the resulting agreement with the glass entropy is a genuine, nontrivial test. For the other two alloys, ΔSmelt is not measured; instead it is assumed to be 35% larger than ΔHf/TL because that is the ratio found for Pd. This single assumed number fixes the vertical offset of the entire corrected melt-entropy curve. Since the glass-entropy curve is measured independently, choosing ΔSmelt determines whether and where the two curves coincide. Thus the reported coincidence in Figs. 3(d,f) and 4(b,c) for Pt and Zr is not an independent prediction but a consequence of an unverified input.

full rationale

The paper contains one genuine, non-circular comparison: for Pd40Ni40P20, the integral melting entropy ΔSmelt = 14.2 J mol−1 K−1 is obtained directly from DSC heat-flow data via Eq. (11), and inserting it into Eq. (12) makes the corrected melt entropy coincide with the independently measured glass entropy. This is a real test and does not reduce to an input. However, for Pt42.5Cu27Ni9.5P21 and Zr35Hf13Al11Ag8Ni8Cu25 the same entropy correction is not measured but assumed by scaling the Pd result by 35%. Because ΔSmelt enters Eq. (12) as the vertical offset of the melt entropy curve, the claimed melt/glass entropy and Gibbs-free-energy coincidence for these two alloys is constructed from the assumption rather than established by data. The paper is transparent about this, but the central generalization nonetheless rests on an unverified fitted-style input. The secondary order-parameter correlation (Eqs. (16)-(17), Figs. 6-7) inherits the same assumed ΔSmelt values for the same two alloys, and the correlation was already reported by the same group in Ref. [20], so it does not provide independent confirmation for those compositions. The enthalpy comparison is not affected by the entropy correction and remains independent. Overall, the derivation is partially circular: the headline coincidence is forced for two of the three alloys by the assumed melting-entropy offset, while the third alloy provides genuine support.

Assumptions & free parameters 2 free parameters · 4 assumptions · 1 invented entities

The paper's central comparison rests on one measured entropy correction for Pd and one assumed scaling for the other two alloys, plus literature Cp polynomial coefficients and standard thermodynamic integration. The order parameter xi is derived from the same adjusted entropy, so the ledger is dominated by the assumed melting-entropy scaling.

free parameters (2)
  • 35 percent melting-entropy scaling for Pt and Zr = 15.6 and 9.8 J mol^-1 K^-1, i.e. 35 percent above literature Delta Sf values
    No melting DSC data for these two alloys; the same relative correction as for Pd40Ni40P20 is assumed to make entropy curves coincide.
  • Specific-heat polynomial coefficients a, b, c, d for melt and crystal = e.g., for Pt42.5Cu27Ni9.5P21: a = 11.5852e-3 J mol^-1 K^-2, b = 5.1317e6 J mol^-1 K, c = -7.2702e-3 J mol^-1 K^-2, d =…
    Taken from literature Refs. [23-26]; the computed melt excess potentials depend on these fitted coefficients.
assumptions (4)
  • domain assumption Specific heat of melt and crystal can be represented by polynomials (1) and (2) over the whole undercooled range.
    Used to build Delta Cp and integrate Eqs. (3)-(5); polynomial fits from literature are extrapolated between Tg and Tf.
  • domain assumption Differential heat flow Delta W_GX = W_G - W_X measured against a crystallized reference equals the heat-flow difference between glass and crystal of equal mass.
    Eqs. (6)-(8) assume the run-3 DSC signal with a crystallized reference cell is a clean difference; drift and mass mismatch are neglected.
  • domain assumption The thermodynamic relation Delta Phi = Delta H - T Delta S is applicable at varying temperature, and Eq. (13) is close to the exact integral dDelta Phi = -Delta S dT.
    Authors acknowledge in the footnote to Section 3.4 that Eq. (14) is valid only for isothermal conditions but state numerical calculations are close; no quantitative error bound is given.
  • ad hoc to paper The 35 percent discrepancy between Delta Sf and Delta Smelt measured for Pd40Ni40P20 applies to Pt42.5Cu27Ni9.5P21 and Zr35Hf13Al11Ag8Ni8Cu25.
    Needed to compute corrected entropies for two alloys without melting DSC data; not independently verified.
invented entities (1)
  • Dimensionless structural order parameter xi = 1 - Delta S_excess / Delta Smelt independent evidence
    purpose: Quantify evolution of structural order from liquid-like (0) to crystal-like (1) and correlate with glass-forming ability.
    It is computable from independent calorimetry and compared with Rc; however, it is a reparameterization of excess entropy and for two alloys depends on the assumed melting entropy.

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Pith. "Pith review of Thermodynamic potentials of metallic alloys in the undercooled liquid and solid glassy states." pith.science (2026). https://pith.science/paper/7GYF2PLA

@misc{pith2026250702609,
  author       = {Pith},
  title        = {Pith review of: Thermodynamic potentials of metallic alloys in the undercooled liquid and solid glassy states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7GYF2PLA}},
  note         = {Machine review of arXiv:2507.02609}
}
abstract

We first present a comparative analysis of temperature evolution of the excess thermodynamic potentials (state functions), the enthalpy $\Delta H$, entropy $\Delta S$ and Gibbs free energy $\Delta \Phi$, determined for \textit{i}) undercooled melts using literature data and \textit{ii}) solid glassy state calculated on the basis of calorimetry measurements using an approach proposed recently. Three metallic alloys were taken as an example for data analysis. It is found that temperature dependences $\Delta H(T)$, $\Delta S(T)$ and $\Delta G(T)$ calculated with both approaches coincide in the supercooled liquid range (i.e. at temperatures $T_g<T<T_x$, where $T_g$ and $T_x$ are the glass transition and crystallization onset temperatures, respectively). However, the necessary conditions for this coincidence is the introduction of important changes to the above approach \textit{i}), which are related to the calculation of the melting entropy. We also introduce and calculate a dimensionless order parameter $\xi$, which changes in the range $0<\xi<1$ and characterizes the evolution of the structural order from liquid-like to crystal-like one. It is shown that the order parameter $\xi_{scl}$ calculated for the end of the supercooled liquid range (i.e. for a temperature just below $T_x$) correlates with the melt critical cooling rate $R_c$: the smaller the order parameter $\xi_{scl}$ (i.e. the closer the structure to that of the equilibrium liquid), the smaller $R_c$ is.

Figures

Figures reproduced from arXiv: 2507.02609 by the authors.

Figure 1
Figure 1. Temperature dependences of the excess enthalpy ∆ [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. (a) Differential heat flow ∆WG−X(T) of glassy Pd40Ni40P20 up to the temperature of the complete crystallization Tcr. (b) DSC thermogram ∆W(T) of the same alloy after crystallization up to the liquidus temperature TL. (c) The same data as in panel (a) but divided by the absolute temperature, i.e. ∆WG−X(T)/T. (d) The same data as in panel (b) but divided by the absolute temperature, i.e. ∆W(T)/T. The characteristic te… view at source ↗
Figure 3
Figure 3. Temperature dependences of the excess enthalpies [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Temperature dependences of the excess Gibbs free e [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: (a) – Temperature dependences of the excess Gibbs fr [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Temperature dependences of the order parameter fo [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Critical cooling rate Rc for indicated melts as a function of the order parameter ξscl calculated with Eq.(17) for the end of the supercooled liquid region (see Fig.6). glass ξG−X as ξ corr L−X(T ) = 1 − ∆S corr L−X(T ) ∆Smelt , (16) ξG−X(T ) = 1 − ∆SG−X(T ) ∆Smelt , (…

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Pith tools

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