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REVIEW 2 major objections 5 minor 45 references

Temperature dependence of nonlinear elastic moduli of polystyrene

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Pith's one-line read The temperature susceptibility of the nonlinear elastic moduli l and m of polystyrene is two orders of magnitude larger than that of the linear moduli, while the n modulus is essentially temperature independent.

desk verdict First temperature-dependent Murnaghan moduli for PS, though the headline effect may be partly biased by pressure–temperature coupling in the setup. read the letter →

arxiv 2502.01176 v1 pith:FU56IDCY submitted 2025-02-03 cond-mat.soft cond-mat.mtrl-sci

Andrey V. Belashov , Anna A. Zhikhoreva , Yaroslav M. Beltukov , Irina V. Semenova This is my paper · ORCID
classification cond-mat.softcond-mat.mtrl-sci PACS 43.35.Zc62.20.de81.05.Lg
keywords temperaturedependencenonlinearelasticmoduliMurnaghanpolystyreneacousto-elasticeffectultrasonicwavesglassypolymertime-temperaturesuperposition
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

This paper measures how the nonlinear elastic moduli of polystyrene change with temperature, using ultrasonic wave velocities under applied pressure. It reports that the Murnaghan moduli l and m change with temperature about a hundred times faster than the linear Lamé moduli λ and µ in the 25–65 °C range, while the third nonlinear modulus n barely changes. The authors argue this matters because nonlinear elastic response governs how polymers behave under dynamic loads, and its temperature sensitivity is poorly known. If the measurements are right, temperature corrections to polymer nonlinearity are far larger than previously assumed, and the n modulus offers a convenient temperature-insensitive reference.

What carries the argument

The central object is the set of Murnaghan third-order elastic moduli l, m, n, which extend the linear Lamé description to leading nonlinear terms. They are extracted from the acousto-elastic effect: the slopes αx, αy, αz of the effective elastic moduli M_j(P,T) = ρ $V_j^{2}$ versus applied pressure for longitudinal and two shear wave polarizations. Equations (2)–(4) convert these slopes and the temperature-dependent Lamé moduli into l, m, n. The argument rests on the linearity of these pressure slopes and on the assumption that the measured wave velocities at each nominal temperature represent the sample's true state.

What would settle it

A direct check would be to repeat the pressure-velocity sweeps at 55 and 65 °C with the sample held in a temperature bath or with local temperature sensors along the ultrasonic path, keeping heterogeneity below about 0.5 °C; if the extracted b_l and b_m change by more than their reported ±0.06 and ±0.04 GPa/°C uncertainties, the two-orders claim is compromised.

Watch

Extended reading notes

Core claim

Using the acousto-elastic effect, the authors measured velocities of longitudinal and shear ultrasonic waves in polystyrene under static pressures up to 16 MPa at temperatures from 25 to 65 °C and at four frequencies between 0.7 and 3 MHz. From the pressure slopes of the effective moduli they extracted the three Murnaghan third-order moduli l, m, and n via equations (2)–(4). The temperature slopes of the frequency-averaged moduli are b_l = −0.44 ± 0.06 GPa/°C, b_m = −0.12 ± 0.04 GPa/°C, and b_n = 0.00 ± 0.03 GPa/°C, compared with −5.0 ± 0.5 MPa/°C and −2.2 ± 0.5 MPa/°C for λ and µ. Thus l and m are roughly two orders of magnitude more temperature-sensitive than the linear moduli, while n shows no resolvable temperature dependence. The temperature susceptibility of l and m is essentially independent of frequency in the studied range.

Load-bearing premise

The 5 °C spatial temperature variation inside the sample at the high end of the range is not propagated into the slopes, so the nominal temperature may differ from the actual temperature along the ultrasonic path by several degrees; if that bias is systematic, the reported temperature susceptibilities could be distorted.

Editorial extensions

If this is right

  • Temperature corrections to nonlinear elastic response of polystyrene (and likely similar glassy polymers) are dominated by the l and m terms; ignoring them in dynamic-load modeling could misestimate stress by a growing margin as temperature rises.
  • The near-zero temperature slope of n means shear-shear nonlinear coupling is stable across the studied range, potentially usable as a temperature-insensitive baseline in ultrasonic nonlinearity measurements.
  • Since the temperature susceptibility of l and m is frequency-independent while their absolute values depend strongly on frequency, models can separate a frequency-dependent modulus magnitude from a temperature-shift contribution.
  • The observed behavior is consistent with pressure shifting relaxation processes differently at different temperatures, and if the time-temperature superposition interpretation holds, measurements at higher temperatures can extend the effective frequency range of nonlinear modulus data.

Reading between the lines

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

  • If the two-orders ratio holds generally for glassy polymers, one would expect similar temperature susceptibility in other sub-Tg thermoplastics; a quick test is to repeat the same acousto-elastic measurement on PMMA or polycarbonate and compare b_l / b_λ.
  • The 5 °C spatial temperature spread reported at 65 °C could contribute systematic error; a direct check would be to measure the temperature profile along the ultrasonic path and propagate its uncertainty into the fitted slopes.
  • The frequency-independence of b_l and b_m combined with the strong frequency dependence of l and m suggests that relaxation processes shift in frequency with temperature; measuring over a wider frequency range or using broadband excitation could directly test the time-temperature superposition prediction.
  • The insensitivity of n to temperature and frequency may be characteristic of segmental-relaxation-driven coupling; this could be probed by testing polystyrene with different cross-linker content.
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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

2 major / 5 minor

Summary. The paper reports measurements of the Murnaghan third-order elastic moduli l, m, and n of a polystyrene-based glassy polymer as functions of temperature (25–65 °C) and ultrasonic frequency (0.7–3 MHz), using the acousto-elastic effect. Pressure slopes α_j of the effective moduli M_j(P,T) are extracted at each nominal temperature, and Eqs. (2)–(4) are used to obtain l, m, and n. The central claims are that the temperature susceptibilities of l and m are about two orders of magnitude larger than those of the linear Lamé moduli λ and μ (slopes b_l = -0.44 ± 0.06 GPa/°C, b_m = -0.12 ± 0.04 GPa/°C versus b_λ = -5.0 ± 0.5 MPa/°C, b_μ = -2.2 ± 0.5 MPa/°C), that n has no resolvable temperature dependence (b_n = 0.00 ± 0.03 GPa/°C), and that the temperature susceptibilities of l and m are nearly frequency-independent over the studied range, as demonstrated by the shifted-moduli analysis in Eqs. (7)–(9).

Significance. If the results are correct, they provide a rare quantitative data set on the temperature dependence of third-order elastic moduli of a glassy polymer and a striking contrast between linear and nonlinear thermal susceptibilities. The analysis chain is transparent: the moduli are obtained directly from measured acousto-elastic slopes with standard formulas, with no fitted theoretical model for the target quantities. The shifted-moduli consistency check is a useful way to display frequency independence of the temperature slopes. The paper also connects the observations to relaxation processes and time-temperature superposition, and it compares with prior results on vitreous silica and metal-matrix composites. The main risk is systematic temperature bias in the pressure slopes, which is not propagated into the reported uncertainties.

major comments (2)
  1. [Section 2, protocol description and Fig. 2] The pressure slopes α_j used in Eqs. (2)–(4) are not obtained under isothermal conditions. The protocol heats the sample from 25 °C to 65 °C at each fixed pressure, and the authors explicitly note that increasing pressure improves jaw contact and changes heat transfer, so the thermal state at different pressures is not identical. The reported surface-temperature heterogeneity reaches 5 °C at 65 °C (Fig. 1c), while the longitudinal velocity changes with temperature at roughly −2 m/s per °C (Table 1). Over the 16-MPa pressure span the genuine acousto-elastic velocity change is only of order a few m/s, so a pressure-induced effective-path-temperature shift of order 1 °C would bias α_j at the same level as the signal. This bias is not propagated into b_l and b_m or into the two-orders-of-magnitude claim. The authors should either quantify dT_eff/dP along the ultrasonic path from the thermal-imaging data, or provide a direct isothermal pressure-stepping validation at least at two temperatures.
  2. [Section 3.2, Fig. 5 and slopes b_n] The conclusion that n is temperature-independent within error rests on b_n = 0.00 ± 0.03 GPa/°C, but the individual n(T) points are not shown with uncertainties, and n is obtained from the small difference α_y − α_z (Eq. (4)), where the two shear sensitivities are nearly equal. The reported slope uncertainty does not include correlated errors in α_y and α_z or the temperature-heterogeneity effect described in the previous comment. Please provide a table of individual l, m, and n values with uncertainties at each temperature and frequency, and state how the slope uncertainties were computed, including whether correlations between the α_j were accounted for.
minor comments (5)
  1. [Section 3.2 and Fig. 5(a)] There is an apparent sign inconsistency: the text reports b_l = −0.44 ± 0.06 GPa/°C, while the label in Fig. 5(a) appears as '0.44 ± 0.06 GPa/°C'. In addition, the wording 'increased in their absolute values' should be reconciled with the sign of the plotted quantities, since the figures appear to show positive values of l and m that decrease with temperature.
  2. [Section 3.1] The sentence 'This behavior is in good agreement with data previously published elsewhere and can be explained by rising viscosity of the material at elevating temperatures' appears to say the opposite of the intended meaning; viscosity of polymers typically decreases with increasing temperature.
  3. [Section 2, Eq. (1)] The text states that 'the correct value of the velocity is V_j = V_i(P)(1 + ε_x)' with ε_x = νP/E, and then says this correction is 'taken into account further when using Eqs. (2)–(4)'. It is unclear whether Eq. (1) already includes this correction or whether it is applied only later; please clarify to avoid double-counting or omission, especially because E is temperature-dependent and could slightly affect the temperature slopes.
  4. [Figures 4 and 5] The individual data points in Figs. 4 and 5 are plotted without error bars. Adding error bars, at least at representative frequencies and temperatures, would help the reader assess the significance of the reported slopes and of the claimed frequency independence.
  5. [Section 4] The phrase 'time-temperature superposition principle is reliable only for thermodynamically simple systems' would be more standard as 'thermorheologically simple systems'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the nonlinear moduli are extracted directly from measured pressure-velocity slopes via standard acousto-elastic relations.

full rationale

The central derivation chain is not circular. The nonlinear Murnaghan moduli l, m, and n are computed from experimentally measured pressure dependences of ultrasonic velocities, using the explicit algebraic relations in Eqs. (2)-(4), with linear Lame moduli obtained from independent zero-pressure velocity measurements at each temperature. The reported temperature susceptibilities b_l, b_m, b_n are simply the slopes of these directly extracted moduli versus temperature; there is no fitted parameter that is then renamed as a prediction. The shifted-moduli analysis in Eqs. (7)-(9) is a consistency check rather than a derivation of the moduli, and it is not forced by construction because using the average slope to shift individual frequency curves could fail to collapse them if the per-frequency slopes actually differed. The methodological self-citations [42,43] supply measurement equations and experimental procedures, but these equations are standard acousto-elasticity results (related to Murnaghan and Thurston-Brugger theory) rather than an unverified premise loaded with the paper's own conclusions. The acknowledged temperature heterogeneity up to 5 degrees C, and the possibility that pressure changes heat transfer, are experimental limitations that could affect slope accuracy; however, they are not circularity. Thus the paper's temperature-dependence claims have independent empirical content and are not equivalent by construction to their inputs.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The experiment uses a standard acousto-elastic method whose theoretical content is contained in Murnaghan's theory and the authors' prior papers. The fitted values are the pressure slopes (alpha) and temperature slopes (b), which are the measured observables; no additional free constants are introduced. The main non-standard assumption is the treatment of the temperature heterogeneity and of the industrial copolymer as a homogeneous isotropic polystyrene.

free parameters (4)
  • b_l = -0.44 ± 0.06 GPa/°C
    Slope of linear fit to l_av versus temperature (Fig. 5a); this is a central measured result, not a hidden assumption.
  • b_m = -0.12 ± 0.04 GPa/°C
    Slope of linear fit to m_av versus temperature (Fig. 5b).
  • b_n = 0.00 ± 0.03 GPa/°C
    Slope of linear fit to n_av versus temperature (Fig. 5c); consistent with zero.
  • alpha_x, alpha_y, alpha_z = not reported numerically
    Dimensionless pressure slopes of effective moduli Mj(P) for longitudinal (x) and two shear waves (y, z); extracted by linear regression at each temperature and frequency, then used in Eqs. (2)-(4) to compute l, m, and n.
assumptions (5)
  • standard math Murnaghan third-order elastic theory (Eqs. 2-4) correctly relates pressure slope coefficients to the moduli l, m, n.
    Invoked in Section 2; standard theory from Murnaghan [31] and prior paper [42].
  • domain assumption The sample is isotropic and homogeneous, and the vice pressure creates a uniform uniaxial stress state.
    Section 2 describes block bars of a commercial copolymer; no anisotropy, texture, or stress concentration is considered.
  • domain assumption The effective moduli Mj(P) are linear in pressure over 0-16 MPa, so the slopes alpha_j fully characterize the acousto-elastic response.
    Section 2: 'Fitting of linear dependencies of the effective moduli Mj(P,Ti) allowed us to determine the dimensionless slope coefficients alpha_j'.
  • domain assumption The sample remains in the glassy state over 25-65 C with no phase transition or relaxation that would make the moduli ill-defined.
    Section 2: glass transition about 100 C; softening occurs below; measurements reproducible below 75 C per Lamberson [44].
  • domain assumption The up-to-5 C temperature heterogeneity does not bias the assignment of a nominal temperature to the measured pressure-velocity slopes.
    Section 2 and Fig. 1(c): heterogeneity rises from 0.2 C to 5 C; its effect on the slopes is not quantified.

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Pith. "Pith review of Temperature dependence of nonlinear elastic moduli of polystyrene." pith.science (2026). https://pith.science/paper/FU56IDCY

@misc{pith2026250201176,
  author       = {Pith},
  title        = {Pith review of: Temperature dependence of nonlinear elastic moduli of polystyrene},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FU56IDCY}},
  note         = {Machine review of arXiv:2502.01176}
}
abstract

Nonlinear elastic properties of polymers and polymeric composites are essential for accurate prediction of their response to dynamic loads, which is crucial in a wide range of applications. These properties can be affected by strain rate, temperature, and pressure. The temperature susceptibility of nonlinear elastic moduli of polymers remains poorly understood. We have recently observed a significant frequency dependence of the nonlinear elastic (Murnaghan) moduli of polystyrene. In this paper we expand this analysis by the temperature dependence. The measurement methodology was based on the acousto-elastic effect, and involved analysis of the dependencies of velocities of longitudinal and shear single-frequency ultrasonic waves in the sample on the applied static pressure. Measurements were performed at different temperatures in the range of 25-65 {\deg}C and at different frequencies in the range of 0.75-3 MHz. The temperature susceptibility of the nonlinear moduli $l$ and $m$ was found to be two orders of magnitude larger than that of linear moduli $\lambda$ and $\mu$. At the same time, the observed variations of $n$ modulus with temperature were low and within the measurement tolerance. The observed tendencies can be explained by different influence of pressure on relaxation processes in the material at different temperatures.

Figures

Figures reproduced from arXiv: 2502.01176 by the authors.

Figure 1
Figure 1. (a) examples of spatial distributions of temperature on the sample surface at different mean temperatures (indicated on top of each image). (b) example of the thermal image of the sample and vise jaws, (c) dependence of temperature inhomogeneity in the sample on the mean temperature. Experiments on the determination of nonlinear elastic moduli at different sample temperatures involved measurements of the velocities … view at source ↗
Figure 2
Figure 2. (a) changes in temperature, pressure and phase of the longitudinal ultrasonic wave at the frequency of 1.5 MHz and the initial pressure on the sample (pressure at room temperature) of 7.5 MPa. (b) phase of the detected ultrasonic wave at different pressures and temperatures of the sample, (c) dependence of the velocity of longitudinal ultrasonic waves at 1.5 MHz on the static pressure applied on the sample and sampl… view at source ↗
Figure 3
Figure 3. Velocities of longitudinal and shear ultrasonic waves as a function of temperature at the frequency of 1.5 MHz (a,b) and as a function of frequency at the temperature of 25 ◦C (c,d) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Nonlinear moduli as a function of frequency for different temperatures. moduli shown in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: a–c) Averaged nonlinear moduli as a function of temperature. d–f) Shifted nonlinear moduli as a function of frequency for different temperatures. The temperature can shift the relaxation time as well as the mechanical coupling with the pressure. Qualitatively, the depe…

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.