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arxiv: 2510.07353 · v2 · pith:YWEND432new · submitted 2025-10-08 · ❄️ cond-mat.mtrl-sci

General expression for the energy and the equation of state for polycrystalline solids

O. Bystrenko , B. Ilkiv , S. Petrovska , T. Bystrenko , O. Foia , O. Khyzhun (Frantsevich Institute for problems of materials science , Kyiv , Ukraine) This is my paper

Pith reviewed 2026-05-21 21:00 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords polycrystalline solidsequation of stateelasticity theorydensity functional theoryhigh pressureenergy expressionBirch-Murnaghan
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The pith

Extended classical elasticity theory supplies universal semi-empirical expressions for the energy and equation of state of polycrystalline solids.

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

The paper derives general analytical expressions for the internal energy and the pressure-volume relation in polycrystalline materials by extending classical elasticity theory. These expressions are semi-empirical, depending on a small set of material-specific parameters, and are intended to apply across different bonding types. The authors test the formulas against density-functional theory calculations for many inorganic compounds, including diamond, magnesium, sphalerite, boron, and rocksalt structures. The comparisons cover pressures up to 300 GPa and show agreement at the level achieved by the standard Birch-Murnaghan equation of state. If the expressions hold, they offer a simple closed-form route to high-pressure thermodynamics without repeated quantum-mechanical computations.

Core claim

On the basis of the extended classical elasticity theory, universal semi-empirical analytical expressions for the energy and the equation of state for polycrystalline solids are proposed. Validation by first-principles density functional theory simulations with the pseudo-potential approach and generalized gradient approximation demonstrates excellent agreement for a large number of inorganic crystalline compounds with metal, covalent and ionic bonding within the pressure range up to 300 GPa, with accuracy comparable to the Birch-Murnaghan approach.

What carries the argument

Universal semi-empirical analytical expressions for energy and equation of state derived from extended classical elasticity theory, providing a single functional form fitted by a few material parameters.

If this is right

  • The expressions furnish closed-form energy-volume and pressure-volume relations once a few parameters are determined for a given solid.
  • The same functional form applies to compounds with metal, covalent, and ionic bonding.
  • Accuracy remains comparable to the Birch-Murnaghan equation of state up to 300 GPa for the tested materials.
  • The relations can be used to model high-pressure behavior of polycrystalline solids without repeated full quantum simulations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the expressions prove robust, they could reduce computational cost for screening many candidate materials under compression in materials design.
  • The approach might be tested for thermodynamic derivatives such as thermal expansion or elastic constants derived from the same energy function.
  • Similar elasticity-based forms could be examined for disordered or amorphous solids to check whether the polycrystalline assumption is essential.
  • Direct comparison with experimental diamond-anvil cell data at overlapping pressures would provide an independent check beyond DFT.

Load-bearing premise

An extension of classical elasticity theory can supply a universal functional form that remains accurate for metal, covalent, and ionic solids once a small number of material-specific parameters are chosen.

What would settle it

A new polycrystalline compound or pressure above 300 GPa where the proposed analytical energy or pressure deviates from DFT results by more than the typical Birch-Murnaghan residual.

Figures

Figures reproduced from arXiv: 2510.07353 by B. Ilkiv, Kyiv, O. Bystrenko, O. Foia, O. Khyzhun (Frantsevich Institute for problems of materials science, S. Petrovska, T. Bystrenko, Ukraine).

Figure 1
Figure 1. Figure 1: Behavior of the elastic energy per atom (left) and pressure (right) vs. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Same as in Fig.1, but for Magnesium, (α-Μg), hexagonal system, space symmetry P63/mmc [194], Nsites= 2; B= 385.61; α=0.428. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Same as in Fig.1, but for Gold, (Au), cubic system, space symmetry Fm-3m [225], Nsites=1; B=1575.07; α=0.642 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Same as in Fig.1, but for Rocksalt, (NaCl), cubic system, space symmetry Fm-3m [225], Nsites=2; B=263.4 ; α=0.518 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Same as in Fig.1, but for Calcium fluoride, (CaF2), cubic system, space symmetry Fm-3m [225], Nsites=2; B=833.49 ; α=0.507. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Same as in Fig.1, but for Diamond, (C), cubic system, space symmetry Fd-3m [227], Nsites=2 ; B=4398.9 ; α=0.415 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Same as in Fig.1, but for Zinc sulfide, (ZnS, Sphalerite), cubic system, space symmetry F-43m [216], Nsites=2 ; B=727.37; α=0.51. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 energy / atom, Ry ξ [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Same as in Fig.1, but for Zinc sulfide, (ZnS, Rocksalt), cubic system, space symmetry Fm-3m [225], Nsites=2; B=913.77; α=0.508. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Same as in Fig.1, but for Boron, (α-B12), trigonal system, space symmetry R-3m [166], Nsites=12; B=2216.4; α=0.392 [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Same as in Fig.1, but for Magnesium carboboride, (MgC2B12), orthorhombic system, space symmetry Imma [74], Nsites=30; B=2429.5; α=0.368 [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Same as in Fig.1, but for Topaz, (Al2SiO4F2) , orthorhombic system, space symmetry Pnma [62] Nsites=36; B=1403.3; α=0.471 8 [PITH_FULL_IMAGE:figures/full_fig_p008_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Behavior of enthalpies for rocksalt and sphalerite ZnS vs. pressure near the phase transition point calculated on the basis of Eqs. (10-11). From the intersection point, we determine the point of sphalerite to rocksalt phase transition in ZnS as PT=16.0 GPa. The respective experimental numbers are: 15.0-16.2 [13-16]; 15.0 [17]; 16.0 [18]. Thus, we have a perfect agreement with the experimental data availa… view at source ↗
read the original abstract

On the basis of the extended classical elasticity theory, we propose universal semi-empirical analytical expressions for the energy and the equation of state for poly-crystalline solids. The validation of the relations has been made by means of first principle density functional theory simulations with the use of pseudo-potential approach and generalized gradient approximation for the exchange-correlation energy. The calculations performed for a large number of inorganic crystalline compounds with metal, covalent and ionic bonding (including diamond, Mg, sphalerite, B, magnesium carboboride, topaz, rocksalt, etc.) within the pressure range up to 300 GPa demonstrated an excellent agreement with the predictions of the analytical theory comparable in accuracy with Birch-Murnaghan approach.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes universal semi-empirical analytical expressions for the energy and equation of state of polycrystalline solids derived from an extension of classical elasticity theory. These expressions are validated via DFT simulations (pseudo-potential, GGA) for numerous inorganic compounds spanning metal, covalent, and ionic bonding (e.g., diamond, Mg, sphalerite, B, rocksalt) up to 300 GPa, with reported agreement comparable to the Birch-Murnaghan EOS.

Significance. If the functional form is robust with a limited set of material-specific parameters and the derivation supports universality across bonding types, the work could supply a practical analytical alternative for high-pressure modeling in materials science and geophysics. The broad DFT validation across bonding types and pressure range is a clear strength, providing a reproducible computational benchmark; however, the semi-empirical character requires transparent parameter accounting to establish genuine generality beyond fitting.

major comments (2)
  1. [Theory section] Theory section (opening paragraphs): The extension of classical elasticity is introduced to motivate the analytical energy/EOS expressions, but the specific functional form is posited rather than derived from microscopic considerations or shown to be necessarily independent of bonding character. This assumption is load-bearing for the universality claim across metal, covalent, and ionic solids and is not independently justified from first principles.
  2. [Validation section] Validation section and abstract: No error bars, average deviations, or quantitative metrics are supplied for the DFT comparisons, and there is no explicit statement of the number or identity of free parameters fitted per material. Without this, it is impossible to determine whether the reported accuracy comparable to Birch-Murnaghan reflects a controlled, low-parameter model or a flexible fit to the same data used for validation.
minor comments (2)
  1. [Abstract] Abstract: Replace the qualitative phrase 'excellent agreement' with at least one quantitative measure (e.g., mean relative error or maximum deviation) to allow immediate assessment of the validation strength.
  2. [Throughout] Throughout: Ensure all material-specific coefficients in the energy expression are explicitly tabulated for each compound examined, including their fitted values and any constraints applied during optimization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. We address the two major comments point by point below and indicate the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: [Theory section] Theory section (opening paragraphs): The extension of classical elasticity is introduced to motivate the analytical energy/EOS expressions, but the specific functional form is posited rather than derived from microscopic considerations or shown to be necessarily independent of bonding character. This assumption is load-bearing for the universality claim across metal, covalent, and ionic solids and is not independently justified from first principles.

    Authors: We agree that the functional form is motivated by an extension of classical elasticity rather than obtained from a microscopic electronic-structure derivation. The extension generalizes the quadratic elastic energy to a form that remains analytic and satisfies the correct limiting behavior at zero pressure and under strong compression; the resulting expression is therefore semi-empirical by construction. We do not claim that the specific shape is required by first-principles considerations independent of bonding type. Instead, we argue that the same macroscopic elastic assumptions apply across metallic, covalent, and ionic solids once the material-specific parameters (zero-pressure bulk modulus and its pressure derivative) are allowed to absorb the microscopic details. In the revised manuscript we will expand the opening paragraphs of the Theory section to state these limitations explicitly and to clarify that the claimed universality is empirical, supported by the breadth of the DFT test set rather than by a bonding-independent proof. revision: yes

  2. Referee: [Validation section] Validation section and abstract: No error bars, average deviations, or quantitative metrics are supplied for the DFT comparisons, and there is no explicit statement of the number or identity of free parameters fitted per material. Without this, it is impossible to determine whether the reported accuracy comparable to Birch-Murnaghan reflects a controlled, low-parameter model or a flexible fit to the same data used for validation.

    Authors: The current manuscript indeed omits quantitative error statistics and a clear statement of the fitting procedure. Each compound is described by exactly two free parameters (the zero-pressure bulk modulus B0 and its pressure derivative B0′), which are obtained by fitting the analytical energy expression to the DFT energy-volume points in the low-pressure regime (typically 0–10 GPa) and are then used without further adjustment to predict the entire pressure range up to 300 GPa. In the revised version we will (i) add a table listing B0 and B0′ for every compound together with the fitting range, (ii) report root-mean-square and mean-absolute deviations of both energy and pressure relative to the DFT data, and (iii) include representative error bars on the comparison plots. These additions will make transparent that the functional form itself is fixed and contains only the two elastic parameters per material. revision: yes

Circularity Check

0 steps flagged

Derivation chain from extended elasticity theory is independent of validation data

full rationale

The manuscript starts from an extension of classical elasticity theory to propose semi-empirical analytical expressions for energy and EOS in polycrystalline solids. These forms incorporate a small number of material-specific parameters and are then compared to independent DFT results across many compounds and bonding types up to 300 GPa. No quoted equations or steps in the provided text demonstrate that the proposed functional forms reduce by construction to a fit of the same DFT dataset used for validation. No self-citation chains, self-definitional loops, or ansatz smuggling via prior author work are evident in the abstract or description. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on an extension of classical elasticity whose precise additional terms are not visible in the abstract; at minimum one expects a small set of material-specific parameters that are fitted rather than derived.

free parameters (1)
  • material-specific coefficients in the extended elasticity energy expression
    Semi-empirical nature of the proposed formulas implies at least one adjustable parameter per material or per bonding class that is chosen to match data.
axioms (1)
  • domain assumption Classical elasticity theory can be extended by a small number of additional terms to describe polycrystalline solids universally across bonding types.
    Invoked in the basis statement of the theory section.

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