Quasi-universal behaviour of shear relaxation times in simple fluids

A. G. Khrapak; S. A. Khrapak

REVIEW 3 major objections 4 minor 90 references

Quasi-universal behaviour of shear relaxation times in simple fluids

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper argues that the Maxwell shear relaxation time, reduced by interparticle spacing and thermal velocity, follows a single quasi-universal curve in simple fluids from dilute gas to freezing, with a common value near 0.18 at freezing.

desk verdict A useful, honest compilation showing quasi-universal reduced Maxwell relaxation times across four simple fluids, with the hard-sphere anchor as the one genuinely fragile input. read the letter →

arxiv 2412.07663 v1 pith:2IZMZ4PG submitted 2024-12-10 cond-mat.soft cond-mat.othercond-mat.stat-mechphysics.plasm-ph

S. A. Khrapak , A. G. Khrapak This is my paper · ORCID

classification cond-mat.softcond-mat.othercond-mat.stat-mechphysics.plasm-ph

keywords shearrelaxationtimeMaxwellquasi-universalitysimplefluidsfreezingdensityscalingviscosityinstantaneousmodulushard-spherefluid

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 Maxwell shear relaxation time $\tau_M = \eta/G_\infty$, the ratio of shear viscosity to instantaneous shear modulus, spans more than ten orders of magnitude in absolute terms across simple fluids. The paper asks whether this spread is due to the interaction potential or merely to different length and time scales. It computes $\tau_M$ for Lennard-Jones, Yukawa, soft-sphere, and hard-sphere fluids in common reduced units $\Delta = \rho^{-1/3}$ and $v_T = \sqrt{T/m}$, and finds that $\tau_M^* = \tau_M v_T/\Delta$ behaves the same way in all four: it decreases at low density, reaches a minimum at intermediate density, and then rises toward freezing. At freezing the values cluster around $\tau_M^* \simeq 0.18 \pm 0.04$ despite the enormous absolute spread. The upshot is that near their freezing point, simple monatomic melts share a universal viscoelastic time scale set mainly by density relative to freezing, not by interaction details.

What carries the argument

The load-bearing identity is $\tau_M = \eta/G_\infty$, evaluated in the system-independent reduced units $\Delta = \rho^{-1/3}$ and $v_T = \sqrt{T/m}$, so that $\tau_M^* = \tau_M v_T/\Delta$. The viscosity coefficients come from established fits and molecular-dynamics data for each fluid; the instantaneous shear modulus $G_\infty$ comes from the standard high-frequency elastic-modulus formula (a kinetic term plus an integral over the radial distribution function) for the Lennard-Jones, Yukawa, and soft-sphere fluids, and from a finite hard-sphere derivation that bypasses the divergence of that standard formula in the hard-sphere limit, using an approximate contact derivative of the radial distribution function. Normalizing density by the freezing density is what brings the four curves together.

What would settle it

Run a direct molecular-dynamics simulation of the hard-sphere fluid at packing fractions up to freezing and compute the instantaneous shear modulus from stress fluctuations, a route that stays finite, then form $\tau_M = \eta/G_\infty$ with the Green-Kubo viscosity. If the resulting $\tau_M^*$ at freezing lands outside $0.18 \pm 0.04$, the hard-sphere anchor, and with it the quasi-universal band, collapses.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the reduced Maxwell relaxation time is a quasi-universal function of density normalized by its freezing value. For the four model fluids, the same qualitative density dependence appears—decrease, minimum, increase—and the numerical values at both the minimum and the freezing point are comparable: $\tau_M^* \simeq 0.07$ to $0.15$ at the minima and $\tau_M^* \simeq 0.18 \pm 0.04$ at freezing. The authors read this as evidence that interaction softness or the presence of long-range attraction plays no systematic role over the investigated range. They then use the quasi-universal freezing value to derive further near-freezing regularities: a common reduced cutoff wave number $k_{\rm gap}^* \simeq 0.50 \pm 0.06$ for transverse shear waves, a large separation between the diffusion time and the Maxwell time ($\tau_D/\tau_M \simeq 27$ to $44$), and the time-scale ordering $1/\Omega_E < \tau_M \ll \tau_D$ consistent with a vibrational picture of transport in dense fluids.

Load-bearing premise

The hard-sphere shear modulus near freezing is the fragile input: because the standard formula diverges in the hard-sphere limit, the calculation relies on a specific finite-derivation route and on an approximate value for the derivative of the radial distribution function at contact; if that approximate value is off, the hard-sphere anchor of the universal band shifts.

Editorial extensions

If this is right

Near freezing, any simple monatomic fluid has $\tau_M^* \simeq 0.18 \pm 0.04$; given temperature and number density, this yields an absolute relaxation time estimate without knowing the interaction potential.
The reduced cutoff wave number for transverse collective modes is quasi-universal, $k_{\rm gap}^* \simeq 0.50 \pm 0.06$, so the Maxwell time directly fixes where shear waves begin to propagate.
The large separation $\tau_D/\tau_M \simeq 27$ to $44$ at freezing supports the vibrational model of dense-liquid transport and justifies treating atomic oscillations as temporarily solid-like.
The minimum of $\tau_M^*$ marks the gas-like to liquid-like dynamical crossover but sits deeper in the dense regime than other crossover indicators such as extrema of transport coefficients.
For the potentials studied, no systematic dependence on interaction softness or long-range attraction appears, so the same quasi-universal behaviour should hold for other simple monatomic fluids.

Reading between the lines

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

If the band survives closer scrutiny, the Maxwell relaxation time near freezing becomes a purely thermodynamic estimate, $\tau_M \simeq 0.18\Delta/v_T$, applicable to complex plasmas and colloidal suspensions where the interaction potential is poorly characterized.
The apparent universality suggests a corresponding-states principle for viscoelasticity: simple fluids at equal $\rho/\rho_{\rm fr}$ have equal reduced $\tau_M$; testing molecular liquids with anisotropic or bounded potentials would show whether the principle extends beyond monatomic pairwise-additive models.
The hard-sphere result is the least secure anchor; an independent stress-fluctuation simulation would settle whether the band is real physics or an artifact of the approximate contact derivative used for the hard-sphere modulus.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know this paper does something simple and useful: it puts the Maxwell relaxation time for LJ, Yukawa, soft-sphere, and hard-sphere fluids on a common reduced scale (Delta/vT) and shows they all follow the same trend, with freezing-point values clustered around tau*M ~ 0.18 +/- 0.04. The new content is the systematic cross-system comparison and the hard-sphere calculation, which nobody had done cleanly because the standard Zwanzig-Mountain modulus diverges in the HS limit. That part is legitimately new, and the qualitative picture--decrease at low density, minimum, then rise toward freezing--holds up across all four systems. The paper is also honest. It explicitly flags the HS modulus issue, uses established equations of state and simulation fits, and does not oversell the scatter. The implications section, especially the k-gap estimate and the time-scale hierarchy, follows naturally and gives the result practical value for people working on dusty plasmas or soft matter. The soft spots are real but not fatal. First, the HS anchor relies on Miller's derivation plus the Tao-Song-Mason approximation for g'(sigma), and there is no direct stress-autocorrelation simulation in the paper to check it. If that modulus is off, the 0.18 +/- 0.04 band loses one of its supports. Second, part of the apparent universality is inherited: the LJ and Yukawa curves use the authors' own prior freezing-density and excess-entropy scaling fits, so the comparison flatters the conclusion. Third, there are no error bars anywhere, and the freezing-point band is essentially a read-off of four numbers. These do not kill the claim--the qualitative behavior is consistent across independent sources--but they do mean the quantitative band should be treated as provisional, especially for HS. I would send this to peer review. It is a well-organized, clearly written paper that gives the community a convenient rule of thumb and identifies exactly where further verification is needed. For the right reader--someone working on viscosity, relaxation, or transport in simple fluids--it is a solid reference. A good referee should push for a direct HS stress-tau simulation, or at least a sensitivity analysis of the Miller/TSM input, before trusting the 0.18 +/- 0.04 value as more than an estimate.

Referee Report

3 major / 4 minor

Summary. This manuscript examines the Maxwell shear relaxation time τ_M = η/G∞ for four simple monatomic model fluids (Lennard-Jones, Yukawa, soft-sphere, hard-sphere) using the same reduced units based on the interparticle spacing Δ = ρ^{-1/3} and thermal velocity v_T = (T/m)^{1/2}. Using published equations of state, viscosity fits, and elastic-modulus formulas, the authors compute τ_M^* over the entire fluid range up to freezing. They report a common qualitative trend: τ_M^* first decreases with density, reaches a minimum, and then increases toward the freezing point. They further claim that the reduced relaxation times at both the minima and the freezing point are quasi-universal across the four systems, with freezing values τ_M^* ≈ 0.18 ± 0.04. The paper then derives implications for the transverse-mode k-gap, the Stokes-Einstein relation, and the hierarchy of relaxation times in dense fluids.

Significance. If the quasi-universal band is correct, it provides a robust, unit-independent estimate of the Maxwell relaxation time near freezing for simple monatomic fluids, and it connects the rheological Maxwell time to the k-gap and the vibrational model of transport. The paper's compilation is useful because it expresses previously scattered results in a common normalization and identifies the hard-sphere modulus as a delicate case. However, the central evidence is not fully independent: two of the four systems are built from existing quasi-universal fits, and the hard-sphere branch relies on an unvalidated approximate modulus. The paper would be strengthened by a direct test or uncertainty analysis of the hard-sphere input, and by correcting an evident inconsistency in the soft-sphere minimum. The claimed universality at freezing may well survive these tests, but as it stands the evidence is suggestive rather than conclusive.

major comments (3)

[II.D (hard-sphere fluid)] The hard-sphere shear modulus is the only input not obtained from a direct Zwanzig-Mountain integration or an established equation of state: it relies on Miller's derivation [66] with the contact derivative g'(σ) from the Tao-Song-Mason approximation [67], because Eq. (4) diverges in the HS limit. Since the HS freezing point (τ_M^* ≈ 0.17) is one of the four anchors of the claimed band τ_M^* ≈ 0.18 ± 0.04, the central claim is sensitive to this approximation. The paper provides no sensitivity analysis and no comparison against direct stress-autocorrelation MD data for G∞ or τ_M in hard spheres. I request such a test, or at least an estimate of the uncertainty of the TSM route, before the quasi-universal band can be considered secure.
[II.E (Summary)] The values of the minima listed for the soft-sphere fluid are inconsistent with the stated trend: the text gives τ_M^* ≈ 0.9 at the minimum, while Table I gives τ_M^* ≈ 0.16 at freezing for the same system. If 0.9 is the true minimum, then τ_M^* decreases from 0.9 to 0.16 on approaching freezing, contradicting the claim that it increases toward freezing and that minima are comparable across systems (LJ 0.07-0.08, Yukawa 0.12, HS 0.15). This appears to be a typo for 0.09, but as written it undermines the summary and must be corrected.
[II.A and II.B] The quasi-universal collapse for the LJ and Yukawa systems is in large part inherited from the empirical inputs: the LJ viscosity is generated by the modified excess entropy scaling of Ref. [22] (already a freezing-density-scaling collapse) and the Yukawa branch uses the practical viscosity formula of Ref. [44] and the quasi-universal G∞ of Ref. [48]. Thus, for these two systems, the near-constancy of τ_M^* across the dense-fluid regime is not a fully independent test of quasi-universality. The strongest independent evidence comes from the soft-sphere and hard-sphere branches, which use different sources. The paper would be more complete if it stated this explicitly and discussed how much of the spread in Figs. 3-4 reflects the uncertainty of the underlying fits.

minor comments (4)

[Fig. 2 caption] The phrase 'The dotes denote the original calculation' contains a typo: 'dotes' should be 'dots'.
[II.E (Summary)] The sentence 'This is not surprising, because the crossover is considered, and thus there is no "exact demarcation line"' is unclear; it should be reworded, for example as 'the crossover is not sharp, and thus there is no exact demarcation line'.
[II.C (Soft-sphere fluid)] The text states 'The freezing packing fraction is tabulated in the same work' without giving the numerical value; including the freezing packing fraction used would help the reader reproduce Fig. 5.
[Table I] The table would be easier to interpret if each row also reported the specific state point (e.g., T* for LJ, Γ/Γ_fr for Yukawa, ρ/ρ_fr for SS and HS) at which the tabulated values were evaluated.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the quasi-universal band is a numerical synthesis of independent EoS, MD, and virial inputs, not a definitional reduction.

full rationale

Walking the derivation chain, tau_M = eta/G_inf is applied to four systems using published inputs: the Thol equation of state for the Lennard-Jones modulus, MD-based viscosity fits (Bell et al., Pieprzyk et al., Daligault et al.), a virial equation of state plus Eq. (7) for soft spheres, and the Miller/Tao-Song-Mason route for hard spheres. None of these inputs defines tau_M to equal the claimed 0.18 +/- 0.04 freezing value; that band is a numerical outcome of the ratio eta*/G*. The LJ and Yukawa branches do import 'quasi-universal' scaling statements from earlier papers, some co-authored by the present authors (Refs. 23-27, 44, 48, and related work), but those papers supply independent MD/EoS data, so the import functions as evidence rather than a definitional reduction. The hard-sphere modulus is the least externally anchored input because the Zwanzig-Mountain integral diverges and the paper adopts the Miller route using Refs. 21, 55, 65 (self-authored) together with Refs. 66 and 67 (external). That is a robustness concern, not circularity: the numerical HS modulus is not constructed from the claimed universal tau_M, and no equation in the paper defines tau_M in terms of the final band. No fitted parameter is relabelled as a prediction, and no 'uniqueness theorem' from the authors' prior work is invoked to force the choice. Score 1 reflects minor self-citation in the input chain, not a load-bearing circular step.

Assumptions & free parameters 8 free parameters · 7 assumptions · 0 invented entities

The central claim rests on literature viscosity data and theoretical shear moduli, plus a chosen normalization. No new free parameters are introduced, but the curves inherit parameters and empirical fits from prior work, several by the same authors. The most fragile input is the hard-sphere shear modulus via Miller's derivation with an approximate contact derivative. The unit choice Δ/v_T is a convention, not a fitted parameter, but it is load-bearing for the claim of universality.

free parameters (8)

LJ viscosity fit parameters (modified excess entropy scaling) = not quoted in text
Used to compute η* for the LJ fluid along isotherms in Section II A (Fig. 1); the fit determines the LJ branch of the quasi-universal curve.
Yukawa viscosity practical formula parameters = not quoted in text
Used for η* in Section II B (Fig. 4), from Ref. [44] by one of the authors; fit to MD data.
Soft-sphere viscosity fit parameters = not quoted in text
Used for η* in Section II C (Fig. 5), from the Bell 2019 supplement based on Fomin et al.
Hard-sphere shear viscosity MD values = η*≈6.8 at freezing (Table I)
Taken from Pieprzyk et al. 2019 MD simulations in Section II D.
Thol et al. LJ equation of state parameters = not quoted in text
Used to obtain excess energy and pressure for computing G∞ of the LJ fluid in Section II A.
Eighth-order virial coefficients for soft spheres = not quoted in text
Used in Section II C to compute Z and thus G∞ for soft spheres via Eq. (7).
Tao-Song-Mason approximation for hard-sphere g'(σ) = not quoted in text
Used in Section II D to compute the HS shear modulus in Miller's approach; directly sets the HS anchor of the quasi-universal band.
One-component plasma shear modulus relation G*∞≈1+0.12Γ = 1+0.12Γ
Used for OCP symbols in Fig. 4 and for consistency with the Yukawa G∞; approximate fit from Ref. [45].

assumptions (7)

domain assumption The reduced units Δ=ρ^(-1/3) and v_T=(T/m)^(1/2) are an appropriate system-independent normalization for comparing relaxation times across different fluids.
Used throughout Section II; the quasi-universal curves and the freezing-point value are defined in these units. A different normalization could change the apparent degree of universality.
domain assumption Freezing density scaling holds for the transport coefficients of the LJ fluid, i.e., η* is a quasi-universal function of R=ρ/ρ_fr along the isotherms studied.
Invoked in Section II A (Fig. 1) via the modified excess entropy scaling of Ref. [22]; this is what makes the LJ relaxation time quasi-universal.
standard math The Zwanzig-Mountain expression (Eq. 4) correctly describes the instantaneous shear modulus of LJ, Yukawa, and soft-sphere fluids.
Used in Section II A-C for the non-hard-sphere shear moduli; a standard statistical-mechanics result.
domain assumption For hard spheres, Miller's derivation rather than the Zwanzig-Mountain expression gives the correct finite elastic moduli.
Adopted in Section II D following Refs [21,55,65,66]; the paper explicitly notes the Zwanzig-Mountain result diverges in the HS limit.
domain assumption The Tao-Song-Mason approximation for the derivative of the hard-sphere RDF at contact is accurate enough for the HS shear modulus.
Used in Section II D to close Miller's formula; it sets the HS τM at freezing and is one of the fragile inputs.
domain assumption The Stokes-Einstein relation η*D*=α_SE with α_SE≈0.14 to 0.17 holds for the considered melts.
Used in Section III to derive D* and τD/τM from viscosity; affects the interpretation, not the central universal relaxation-time claim.
domain assumption The Lindemann criterion amplitude ⟨δr²⟩/Δ²≈0.01 applies to the considered melts at freezing.
Used in Section III to estimate ΩE τM; interpretation only.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Quasi-universal behaviour of shear relaxation times in simple fluids." pith.science (2026). https://pith.science/paper/2IZMZ4PG

@misc{pith2026241207663,
  author       = {Pith},
  title        = {Pith review of: Quasi-universal behaviour of shear relaxation times in simple fluids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2IZMZ4PG}},
  note         = {Machine review of arXiv:2412.07663}
}

read the original abstract

We calculate the shear relaxation times in four important simple monatomic model fluids: Lennard-Jones, Yukawa, soft-sphere and hard-sphere fluids. It is observed that in properly reduced units, the shear relaxation times exhibit quasi-universal behaviour when the density increases from the gas-like low values to the high-density regime near crystallization. They first decrease with density at low densities, reach minima at moderate densities, and then increase toward the freezing point. The reduced relaxation times at the minima and at the fluid-solid phase transition are all comparable for the various systems investigated, despite more than ten orders of magnitude difference in real systems. Important implications of these results are discussed.

Figures

Figures reproduced from arXiv: 2412.07663 by the authors.

**Figure 2.** FIG. 2. (Color online) Reduced instantaneous shear mod [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** FIG. 4. (Color online) Reduced shear relaxation time of the [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (1 more)

**Figure 6.** Figure 6: FIG. 6. (Color online) Reduced shear relaxation time of the [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

90 extracted references · 56 canonical work pages

[66]

Elastic moduli of a fluid of rigid spheres,

B. N. Miller, “Elastic moduli of a fluid of rigid spheres,” J. Chem. Phys. 50, 2733–2740 (1969)

work page 1969
[67]

Derivative of the hard-sphere radial distribution function at contact,

F.-M. Tao, Y. Song, and E. A. Mason, “Derivative of the hard-sphere radial distribution function at contact,” Phys. Rev. A 46, 8007–8008 (1992)

work page 1992
[22]

Modified entropy scaling of the transport properties of the Lennard-Jones fluid,

I. H. Bell, R. Messerly, M. Thol, L. Costigliola, and J. C. Dyre, “Modified entropy scaling of the transport properties of the Lennard-Jones fluid,” J. Phys. Chem. B 123, 6345–6363 (2019). 8

work page 2019
[44]

Practical formula for the shear viscosity of Yukawa fluids,

S. Khrapak, “Practical formula for the shear viscosity of Yukawa fluids,” AIP Adv. 8, 105226 (2018)

2018
[48]

Universal scaling of transverse sound speed and its iso- morphic property in Yukawa fluids,

N. Yu, D. Huang, S. Lu, S. Khrapak, and Y. Feng, “Universal scaling of transverse sound speed and its iso- morphic property in Yukawa fluids,” Phys. Rev. E 109, 035202 (2024)

work page 2024
[1]

Shear relaxation times of simple fluids,

R. D. Mountain and R. Zwanzig, “Shear relaxation times of simple fluids,” J. Chem. Phys. 44, 2777 (1966)

1966
[2]

On the relation between self-diffusion and viscosity of liquids,

R. Zwanzig, “On the relation between self-diffusion and viscosity of liquids,” J. Chem. Phys. 79, 4507–4508 (1983)

1983
[3]

Excess entropy and Stokes-Einstein relation in simple fluids,

S. A. Khrapak and A. G. Khrapak, “Excess entropy and Stokes-Einstein relation in simple fluids,” Phys. Rev. E 104, 044110 (2021)

2021

Show all 90 references

[4]

Cutoff wave number for shear waves and Maxwell relaxation time in Yukawa liquids,

J. Goree, Z. Donk´ o, and P. Hartmann, “Cutoff wave number for shear waves and Maxwell relaxation time in Yukawa liquids,” Phys. Rev. E 85, 066401 (2012)

2012
[5]

Collective modes and thermodynamics of the liquid state,

K. Trachenko and V. V. Brazhkin, “Collective modes and thermodynamics of the liquid state,” Rep. Progr. Phys. 79, 016502 (2015)

2015
[6]

Emergence and evolution of the k-gap in spectra of liquid and supercritical states,

C. Yang, M. T. Dove, V. V. Brazhkin, and K. Tra- chenko, “Emergence and evolution of the k-gap in spectra of liquid and supercritical states,” Phys. Rev. Lett. 118, 215502 (2017)

2017
[7]

Com- ment on “emergence and evolution of the k-gap in spectra of liquid and supercritical states

T. Bryk, I. Mryglod, G. Ruocco, and T. Scopigno, “Com- ment on “emergence and evolution of the k-gap in spectra of liquid and supercritical states”,” Phys. Rev. Lett. 120, 219601 (2018)

2018
[8]

Onset of transverse (shear) waves in strongly-coupled Yukawa fluids,

S. A. Khrapak, A. G. Khrapak, N. P. Kryuchkov, and S. O. Yurchenko, “Onset of transverse (shear) waves in strongly-coupled Yukawa fluids,” J. Chem. Phys. 150, 104503 (2019)

2019
[9]

Deformations, relaxation, and broken symmetries in liquids, solids, and glasses: A unified topological field theory,

M. Baggioli, M. Landry, and A. Zaccone, “Deformations, relaxation, and broken symmetries in liquids, solids, and glasses: A unified topological field theory,” Phys. Rev. E 105, 024602 (2022)

2022
[10]

Shear flows and shear viscos- ity in a two-dimensional Yukawa system (dusty plasma),

V. Nosenko and J. Goree, “Shear flows and shear viscos- ity in a two-dimensional Yukawa system (dusty plasma),” Phys. Rev. Lett. 93, 155004 (2004)

2004
[11]

Highly resolved fluid flows: “Liquid plas- mas

G. E. Morfill, M. Rubin-Zuzic, H. Rothermel, A. V. Ivlev, B. A. Klumov, H. M. Thomas, U. Konopka, and V. Steinberg, “Highly resolved fluid flows: “Liquid plas- mas” at the kinetic level,” Phys. Rev. Lett. 92, 175004 (2004)

2004
[12]

Obser- vation of shear-wave Mach cones in a 2D dusty-plasma crystal,

V. Nosenko, J. Goree, Z. W. Ma, and A. Piel, “Obser- vation of shear-wave Mach cones in a 2D dusty-plasma crystal,” Phys. Rev. Lett. 88, 135001 (2002)

2002
[13]

Compressional and shear wakes in a two- dimensional dusty plasma crystal,

V. Nosenko, J. Goree, Z. W. Ma, D. H. E. Dubin, and A. Piel, “Compressional and shear wakes in a two- dimensional dusty plasma crystal,” Phys. Rev. E 68, 056409 (2003)

2003
[14]

High-frequency shear modulus and relaxation time of soft-sphere and Lennard-Jones fluids,

E. Keshavarzi, M. Vahedpour, S. Alavi, and B. Najafi, “High-frequency shear modulus and relaxation time of soft-sphere and Lennard-Jones fluids,” Int. J. Thermo- phys. 25, 1747–1762 (2004)

2004
[15]

In- vestigation of the density dependence of the shear relax- ation time of dense fluids,

M. Bamdad, S. Alavi, B. Najafi, and E. Keshavarzi, “In- vestigation of the density dependence of the shear relax- ation time of dense fluids,” Canad. J. Chem. 83, 236–243 (2005)

2005
[16]

Density dependence of the stress relaxation function of a simple fluid,

R. Hartkamp, P. J. Daivis, and B. D. Todd, “Density dependence of the stress relaxation function of a simple fluid,” Phys. Rev. E 87, 032155 (2013)

2013
[17]

Microscopic origin of shear re- laxation in a model viscoelastic liquid,

J. Ashwin and A. Sen, “Microscopic origin of shear re- laxation in a model viscoelastic liquid,” Phys. Rev. Lett. 114, 055002 (2015)

2015
[18]

High-frequency elastic moduli of simple fluids,

R. Zwanzig and R. D. Mountain, “High-frequency elastic moduli of simple fluids,” J. Chem. Phys. 43, 4464–4471 (1965)

1965
[19]

Elastic moduli of simple fluids with steeply repulsive potentials,

D. M. Heyes and P. J. Aston, “Elastic moduli of simple fluids with steeply repulsive potentials,” J. Chem. Phys. 100, 2149–2153 (1994)

1994
[20]

Thermodynamic, me- chanical and transport properties of fluids with steeply repulsive potentials,

D. M. Heyes and J. G. Powles, “Thermodynamic, me- chanical and transport properties of fluids with steeply repulsive potentials,” Mol. Phys. 95, 259–267 (1998)

1998
[21]

Elastic properties of dense hard-sphere flu- ids,

S. Khrapak, “Elastic properties of dense hard-sphere flu- ids,” Phys. Rev. E 100, 032138 (2019)

2019
[23]

Transport properties of Lennard-Jones fluids: Freezing density scaling along isotherms,

S. A. Khrapak and A. G. Khrapak, “Transport properties of Lennard-Jones fluids: Freezing density scaling along isotherms,” Phys. Rev. E 103, 042122 (2021)

2021
[24]

Freezing temper- ature and density scaling of transport coefficients,

S. A. Khrapak and A. G. Khrapak, “Freezing temper- ature and density scaling of transport coefficients,” J. Phys. Chem. Lett. 13, 2674–2678 (2022)

2022
[25]

Freezing density scaling of fluid transport properties: Application to liq- uefied noble gases,

S. A. Khrapak and A. G. Khrapak, “Freezing density scaling of fluid transport properties: Application to liq- uefied noble gases,” J. Chem. Phys. 157, 014501 (2022)

2022
[26]

Departures from perfect isomorph behavior in Lennard- Jones fluids and solids,

D. M. Heyes, D. Dini, S. Pieprzyk, and A. C. Bra´ nka, “Departures from perfect isomorph behavior in Lennard- Jones fluids and solids,” J. Chem. Phys. 158, 134502 (2023)

2023
[27]

Freezing density scaling of transport coefficients in the Weeks-Chandler-Andersen fluid,

S. Khrapak and A. Khrapak, “Freezing density scaling of transport coefficients in the Weeks-Chandler-Andersen fluid,” J. Chem. Phys. 160, 014501 (2024)

2024
[28]

Scaling of local density correlations in a fluid close to freezing,

F. Saija, S. Prestipino, and P. V. Giaquinta, “Scaling of local density correlations in a fluid close to freezing,” J. Chem. Phys. 115, 7586–7591 (2001)

2001
[29]

Equation of state for the Lennard-Jones fluid,

M. Thol, G. Rutkai, A. K¨ oster, R. Lustig, R. Span, and J. Vrabec, “Equation of state for the Lennard-Jones fluid,” J. Phys. Chem. Ref. Data 45, 023101 (2016)

2016
[30]

Sound velocities of Lennard-Jones sys- tems near the liquid-solid phase transition,

S. A. Khrapak, “Sound velocities of Lennard-Jones sys- tems near the liquid-solid phase transition,” Molecules 25, 3498 (2020)

2020
[31]

Vibrational model for thermal conductivity of Lennard-Jones fluids: Appli- cability domain and accuracy level,

S. A. Khrapak and A. G. Khrapak, “Vibrational model for thermal conductivity of Lennard-Jones fluids: Appli- cability domain and accuracy level,” Phy. Rev. E 108, 064129 (2023)

2023
[32]

Hidden scale invariance in condensed mat- ter,

J. C. Dyre, “Hidden scale invariance in condensed mat- ter,” J. Phys. Chem. B 118, 10007–10024 (2014)

2014
[33]

Generalized hydrodynamics of the Lennard-Jones liquid in view of hidden scale invariance,

S. Knudsen, B. D. Todd, J. C. Dyre, and J. S. Hansen, “Generalized hydrodynamics of the Lennard-Jones liquid in view of hidden scale invariance,” Phys. Rev. E 104, 054126 (2021)

2021
[34]

Dusty plasmas,

V. E. Fortov, A. G. Khrapak, S. A. Khrapak, V. I. Molotkov, and O. F. Petrov, “Dusty plasmas,” Phys.- Usp. 47, 447 – 492 (2004)

2004
[35]

Complex (dusty) plasmas: Current status, open issues, perspectives,

V. E. Fortov, A. V. Ivlev, S. A. Khrapak, A. G. Khrapak, and G. E. Morfill, “Complex (dusty) plasmas: Current status, open issues, perspectives,” Phys. Rep. 421, 1–103 (2005)

2005
[36]

Ivlev, H

A. Ivlev, H. L¨ owen, G. Morfill, and C. P. Royall, Complex Plasmas and Colloidal Dispersions: Particle- Resolved Studies of Classical Liquids and Solids(World Scientific, 2012)

2012
[37]

Complex plasma—the plasma state of soft matter,

M. Chaudhuri, A. V. Ivlev, S. A. Khrapak, H. M. Thomas, and G. E. Morfill, “Complex plasma—the plasma state of soft matter,” Soft Matter 7, 1287–1298 (2011)

2011
[38]

Central collisions of charged dust particles in a plasma,

U. Konopka, L. Ratke, and H. M. Thomas, “Central collisions of charged dust particles in a plasma,” Physi. Rev. Lett. 79, 1269–1272 (1997)

1997
[39]

Measurement of the interaction potential of microspheres in the sheath of a rf discharge,

U. Konopka, G. E. Morfill, and L. Ratke, “Measurement of the interaction potential of microspheres in the sheath of a rf discharge,” Phys. Rev. Lett. 84, 891–894 (2000)

2000
[40]

Triple point of Yukawa systems,

S. Hamaguchi, R. T. Farouki, and D. H. E. Dubin, “Triple point of Yukawa systems,” Phys. Rev. E 56, 4671–4682 (1997)

1997
[41]

Universal scaling in complex (dusty) plasmas,

O. Vaulina, S. Khrapak, and G. Morfill, “Universal scaling in complex (dusty) plasmas,” Phys. Rev. E 66, 016404 (2002)

2002
[42]

Shear viscosity of strongly coupled Yukawa liquids,

Z. Donko and P. Hartmann, “Shear viscosity of strongly coupled Yukawa liquids,” Phys. Rev. E 78, 026408 (2008)

2008
[43]

De- termination of the shear viscosity of the one-component plasma,

J. Daligault, K. Rasmussen, and S. D. Baalrud, “De- termination of the shear viscosity of the one-component plasma,” Phys. Rev. E 90, 033105 (2014)

2014
[45]

Unified description of sound velocities in strongly coupled Yukawa systems of different spatial dimensionality,

S. A. Khrapak, “Unified description of sound velocities in strongly coupled Yukawa systems of different spatial dimensionality,” Phys. Plasmas 26, 103703 (2019)

2019
[46]

Instantaneous shear modulus of Yukawa fluids across coupling regimes,

S. A. Khrapak and B. A. Klumov, “Instantaneous shear modulus of Yukawa fluids across coupling regimes,” Physics of Plasmas 27, 024501 (2020)

2020
[47]

Elastic properties of Yukawa crys- tals,

A. A. Kozhberov, “Elastic properties of Yukawa crys- tals,” Phys. Plasmas 29, 043701 (2022)

2022
[49]

Monte Carlo study of a one-component plasma,

S. G. Brush, H. L. Sahlin, and E. Teller, “Monte Carlo study of a one-component plasma,” J. Chem. Phys. 45, 2102–2118 (1966)

1966
[50]

Statistical mechanics of sim- ple Coulomb systems,

M Baus and J. P. Hansen, “Statistical mechanics of sim- ple Coulomb systems,” Phys. Rep. 59, 1–94 (1980)

1980
[51]

Trapped nonneutral plasmas, liquids, and crystals (the thermal equilibrium states),

D. H. E. Dubin and T. M. O’Neil, “Trapped nonneutral plasmas, liquids, and crystals (the thermal equilibrium states),” Rev. Mod. Phys. 71, 87–172 (1999)

1999
[52]

Internal energy of the classical two- and three-dimensional one-component- plasma,

S. A. Khrapak and A. G. Khrapak, “Internal energy of the classical two- and three-dimensional one-component- plasma,” Contrib. Plasma Phys. 56, 270–280 (2016)

2016
[53]

Probing the link between residual entropy and viscosity of molecular fluids and model potentials,

I. H. Bell, “Probing the link between residual entropy and viscosity of molecular fluids and model potentials,” PNAS 116, 4070–4079 (2019)

2019
[54]

Transport coefficients of soft sphere fluid at high densi- ties,

Yu. D. Fomin, V. V. Brazhkin, and V. N. Ryzhov, “Transport coefficients of soft sphere fluid at high densi- ties,” JETP Lett. 95, 320 (2012)

2012
[55]

Collective modes in simple melts: Transition from soft spheres to the hard sphere limit,

S. Khrapak, B. Klumov, and L. Couedel, “Collective modes in simple melts: Transition from soft spheres to the hard sphere limit,” Sci. Rep. 7, 7985 (2017)

2017
[56]

Thermo- dynamic properties and entropy scaling law for diffusivity in soft spheres,

S. Pieprzyk, D. M. Heyes, and A. C. Bra´ nka, “Thermo- dynamic properties and entropy scaling law for diffusivity in soft spheres,” Phys. Rev. E 90, 012106 (2014)

2014
[57]

Mulero, ed., Theory and Simulation of Hard-Sphere Fluids and Related Systems(Springer Berlin Heidelberg, 2008)

A. Mulero, ed., Theory and Simulation of Hard-Sphere Fluids and Related Systems(Springer Berlin Heidelberg, 2008)

2008
[58]

Mean-field theory of hard sphere glasses and jamming,

G. Parisi and F. Zamponi, “Mean-field theory of hard sphere glasses and jamming,” Rev. Mod. Phys. 82, 789– 845 (2010)

2010
[59]

Theoretical perspective on the glass transition and amorphous materials,

L. Berthier and G. Biroli, “Theoretical perspective on the glass transition and amorphous materials,” Rev. Mod. Phys. 83, 587–645 (2011)

2011
[60]

Struc- tural properties of dense hard sphere packings,

B. A. Klumov, S. A. Khrapak, and G. E. Morfill, “Struc- tural properties of dense hard sphere packings,” Phys. Rev. B 83, 184105 (2011)

2011
[61]

Simple liquids’ quasiuniversality and the hard-sphere paradigm,

J C Dyre, “Simple liquids’ quasiuniversality and the hard-sphere paradigm,” J. Phys.: Condens. Matter 28, 323001 (2016)

2016
[62]

Thermodynamic and dynamical properties of the hard sphere system revisited by molecu- lar dynamics simulation,

S. Pieprzyk, M. N. Bannerman, A. C. Bra´ nka, M. Chu- dak, and D. M. Heyes, “Thermodynamic and dynamical properties of the hard sphere system revisited by molecu- lar dynamics simulation,” Phys. Chem. Chem. Phys. 21, 6886–6899 (2019)

2019
[63]

High frequency linear response of classical fluids,

H. L. Frisch, “High frequency linear response of classical fluids,” Phys. 2, 209–215 (1966). 9

1966
[64]

Dynamic viscoelas- tic modulus of associative polymer networks: Off-lattice simulations, theory and comparison to experiments,

R. D. Groot and W. G. M. Agterof, “Dynamic viscoelas- tic modulus of associative polymer networks: Off-lattice simulations, theory and comparison to experiments,” Macromolecules 28, 6284–6295 (1995)

1995
[65]

From soft- to hard-sphere fluids: Crossover evidenced by high-frequency elastic moduli,

S. Khrapak, N. P. Kryuchkov, L. A. Mistryukova, and S. O. Yurchenko, “From soft- to hard-sphere fluids: Crossover evidenced by high-frequency elastic moduli,” Phys. Rev. E 103, 052117 (2021)

2021
[68]

The Widom line as the crossover between liquid-like and gas-like be- haviour in supercritical fluids,

G. G. Simeoni, T. Bryk, F. A. Gorelli, M. Krisch, G. Ruocco, M. Santoro, and T. Scopigno, “The Widom line as the crossover between liquid-like and gas-like be- haviour in supercritical fluids,” Nature Phys. 6, 503–507 (2010)

2010
[69]

Two liquid states of mat- ter: A dynamic line on a phase diagram,

V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, and K. Trachenko, “Two liquid states of mat- ter: A dynamic line on a phase diagram,” Phys. Rev. E 85, 031203 (2012)

2012
[70]

Where is the su- percritical fluid on the phase diagram?

V. V. Brazhkin, A.G. Lyapin, V. N. Ryzhov, K. Tra- chenko, Y. D. Fomin, and E. N. Tsiok, “Where is the su- percritical fluid on the phase diagram?” Phys.-Usp. 182, 1137–1156 (2012)

2012
[71]

Liquid- gas

V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, E. N. Tsiok, and K. Trachenko, “Liquid- gas” transition in the supercritical region: fundamental changes in the particle dynamics,” Phys. Rev. Lett. 111, 145901 (2013)

2013
[72]

Dynamics and thermodynamics beyond the critical point,

F. A. Gorelli, T. Bryk, M. Krisch, G. Ruocco, M. San- toro, and T. Scopigno, “Dynamics and thermodynamics beyond the critical point,” Sci. Rep. 3, 1203 (2013)

2013
[73]

Transition from gas-like to liquid-like behavior in supercritical N 2,

J. E. Proctor, C. G. Pruteanu, I. Morrison, I. F. Crowe, and J. S. Loveday, “Transition from gas-like to liquid-like behavior in supercritical N 2,” J. Phys. Chem. Lett. 10, 6584–6589 (2019)

2019
[74]

An entropy scaling demarcation of gas- and liquid-like fluid behaviors,

I. H. Bell, G. Galliero, S. Delage-Santacreu, and L. Costigliola, “An entropy scaling demarcation of gas- and liquid-like fluid behaviors,” J. Chem. Phys. 152, 191102 (2020)

2020
[75]

Minima of shear viscosity and thermal conductivity coefficients of classical fluids,

S. A. Khrapak and A. G. Khrapak, “Minima of shear viscosity and thermal conductivity coefficients of classical fluids,” Phys. Fluids 34, 027102 (2022)

2022
[76]

Re- vealing the supercritical dynamics of dusty plasmas and their liquidlike to gaslike dynamical crossover,

D. Huang, M. Baggioli, Sh. Lu, Z. Ma, and Y. Feng, “Re- vealing the supercritical dynamics of dusty plasmas and their liquidlike to gaslike dynamical crossover,” Physical Review Research 5, 013149 (2023)

2023
[77]

Self-diffusion in simple liquids as a ran- dom walk process,

S. A. Khrapak, “Self-diffusion in simple liquids as a ran- dom walk process,” Molecules 26, 7499 (2021)

2021
[78]

Elementary vibrational model for trans- port properties of dense fluids,

S.A. Khrapak, “Elementary vibrational model for trans- port properties of dense fluids,” Phys. Rep. 1050, 1 (2024)

2024
[79]

Revealing the mechanism of the viscous-to-elastic crossover in liquids,

D. Bolmatov, M. Zhernenkov, D. Zav’yalov, S. Stoupin, Y. Q. Cai, and A. Cunsolo, “Revealing the mechanism of the viscous-to-elastic crossover in liquids,” J. Phys. Chem. Lett. 6, 3048–3053 (2015)

2015
[80]

Critical wave vectors for transverse modes in strongly coupled dusty plasmas,

M. S. Murillo, “Critical wave vectors for transverse modes in strongly coupled dusty plasmas,” Phys. Rev. Lett. 85, 2514–2517 (2000)

2000
[81]

Wave dispersion relations in Yukawa fluids,

H. Ohta and S. Hamaguchi, “Wave dispersion relations in Yukawa fluids,” Phys. Rev. Lett. 84, 6026–6029 (2000)

2000
[82]

Non-hydrodynamic transverse collective excita- tions in hard-sphere fluids,

T. Bryk, A. Huerta, V. Hordiichuk, and A. D. Trokhym- chuk, “Non-hydrodynamic transverse collective excita- tions in hard-sphere fluids,” J. Chem. Phys. 147, 064509 (2017)

2017
[83]

Simple dispersion relations for Coulomb and Yukawa fluids,

S. Khrapak and A. Khrapak, “Simple dispersion relations for Coulomb and Yukawa fluids,” IEEE Trans. Plasma Sci. 46, 737–742 (2018)

2018
[84]

Collective modes in a strongly coupled dusty plasma,

P. K. Kaw, “Collective modes in a strongly coupled dusty plasma,” Phys. Plasmas 8, 1870 (2001)

2001
[85]

Excitation spectra in fluids: How to analyze them properly,

N. P. Kryuchkov, L. A. Mistryukova, V. V. Brazhkin, and S. O. Yurchenko, “Excitation spectra in fluids: How to analyze them properly,” Sci. Rep. 9, 10483 (2019)

2019
[86]

Revisiting the Stokes-Einstein relation without a hydrodynamic diameter,

L. Costigliola, D. M. Heyes, T. B. Schrøder, and J. C. Dyre, “Revisiting the Stokes-Einstein relation without a hydrodynamic diameter,” J. Chem. Phys. 150, 021101 (2019)

2019
[87]

Physical properties of soft repulsive particle fluids,

D. M. Heyes and A. C. Bra´ nka, “Physical properties of soft repulsive particle fluids,” Phys. Chem. Chem. Phys. 9, 5570 (2007)

2007
[88]

Communication: Correlation of the instantaneous and the intermediate-time elasticity with the structural relaxation in glassforming systems,

F. Puosi and D. Leporini, “Communication: Correlation of the instantaneous and the intermediate-time elasticity with the structural relaxation in glassforming systems,” J. Chem. Phys. 136, 041104 (2012)

2012
[89]

Is the structural relaxation of glasses controlled by equilibrium shear viscosity?

R. F. Lancelotti, D. R. Cassar, M. Nalin, O. Peitl, and E. D. Zanotto, “Is the structural relaxation of glasses controlled by equilibrium shear viscosity?” J. Amer. Ce- ramic Soc. 104, 2066 (2021)

2021
[90]

The calculation of molecular vibration frequencies,

F. Lindemann, “The calculation of molecular vibration frequencies,” Z. Phys. 11, 609 (1910)

1910

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [66]

Elastic moduli of a fluid of rigid spheres,

B. N. Miller, “Elastic moduli of a fluid of rigid spheres,” J. Chem. Phys. 50, 2733–2740 (1969)

work page 1969

[2] [67]

Derivative of the hard-sphere radial distribution function at contact,

F.-M. Tao, Y. Song, and E. A. Mason, “Derivative of the hard-sphere radial distribution function at contact,” Phys. Rev. A 46, 8007–8008 (1992)

work page 1992

[3] [22]

Modified entropy scaling of the transport properties of the Lennard-Jones fluid,

I. H. Bell, R. Messerly, M. Thol, L. Costigliola, and J. C. Dyre, “Modified entropy scaling of the transport properties of the Lennard-Jones fluid,” J. Phys. Chem. B 123, 6345–6363 (2019). 8

work page 2019

[4] [44]

Practical formula for the shear viscosity of Yukawa fluids,

S. Khrapak, “Practical formula for the shear viscosity of Yukawa fluids,” AIP Adv. 8, 105226 (2018)

2018

[5] [48]

Universal scaling of transverse sound speed and its iso- morphic property in Yukawa fluids,

N. Yu, D. Huang, S. Lu, S. Khrapak, and Y. Feng, “Universal scaling of transverse sound speed and its iso- morphic property in Yukawa fluids,” Phys. Rev. E 109, 035202 (2024)

work page 2024

[6] [1]

Shear relaxation times of simple fluids,

R. D. Mountain and R. Zwanzig, “Shear relaxation times of simple fluids,” J. Chem. Phys. 44, 2777 (1966)

1966

[7] [2]

On the relation between self-diffusion and viscosity of liquids,

R. Zwanzig, “On the relation between self-diffusion and viscosity of liquids,” J. Chem. Phys. 79, 4507–4508 (1983)

1983

[8] [3]

Excess entropy and Stokes-Einstein relation in simple fluids,

S. A. Khrapak and A. G. Khrapak, “Excess entropy and Stokes-Einstein relation in simple fluids,” Phys. Rev. E 104, 044110 (2021)

2021