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REVIEW 4 major objections 5 minor 22 references

Delayed Arm Retraction Controls the Nonlinear Oscillatory Response of Long-Chain-Branched Polymer Melts

T0 review · 4 major / 5 minor · reviewed 2026-07-13 · grok-4.5

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Pith's one-line read The NLI peak in branched polymer melts comes from delayed arm retraction that later erases backbone orientation.

desk verdict Plausible molecular story for LCB NLI peaks, but the architecture-to-peak link rides on an ad hoc Padé Ha rather than a derived LAOS convolution. read the letter →

arxiv 2607.09561 v1 pith:XGC4B6RH submitted 2026-07-10 cond-mat.soft cond-mat.dis-nncond-mat.mtrl-sciphysics.chem-ph

Dario Nichetti , Alessio Zaccone This is my paper · ORCID
classification cond-mat.softcond-mat.dis-nncond-mat.mtrl-sciphysics.chem-ph
keywords long-chainbranchingNonlinearityIndextubetheoryarmretractionLAOSFourierrheologypolymermeltsbranch-pointforcebalance
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

Long-chain-branched polymer melts show a Nonlinearity Index that rises, peaks, and then falls under large-amplitude oscillatory shear, unlike the more monotonic rise of linear chains. This paper claims that the peak is not a simple saturation of elasticity but a molecular competition: the backbone first builds nonlinear orientation almost as a linear chain would, while branch points act as temporary anchors; later, long arms retract and release the stored branch-point tension, erasing orientational memory and cutting higher-harmonic content. A single architecture parameter Ba = fb Za/Zbb sets how strong and how early that delayed relaxation is, so sparse branching can still push NLI above one before the fall, while dense branching yields lower, broader peaks. The result is a tube-based reading of nonlinear Fourier rheology that treats the peak height and width as direct reporters of branch architecture rather than chemistry-specific quirks.

What carries the argument

The delayed arm-retraction factor Ha(θ) (a compact Padé-type crossover that multiplies the linear-backbone NLI form) together with the architecture parameter Ba = fb Za/Zbb that sets the strength and onset of branch-mediated memory loss.

What would settle it

Measure NLI versus reduced strain for a series of model polymers that systematically vary only Za/Zbb (or effective branch functionality) while holding backbone entanglement number fixed; the observed peak height and post-peak decay should collapse onto the predicted Ba dependence, and fail if the algebraic Ha form or the fixed exponent product is wrong.

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Extended reading notes

Core claim

The characteristic NLI maximum of long-chain-branched melts arises because the backbone first develops nonlinear orientation as in the corresponding linear polymer, after which delayed arm retraction relaxes branch-point tension and progressively erases backbone orientational memory; the branched response is therefore NLIbr(θ) = NLI(1)(θ) Ha(θ), with Ha a delayed arm-retraction survival factor controlled by the architecture parameter Ba = fb Za/Zbb.

Load-bearing premise

The arm-retraction memory is replaced by a simple algebraic crossover whose decay exponent is fixed by hand to match a tube-dilation power law, rather than being derived from a full convolution of the arm-survival probability under oscillatory shear.

Editorial extensions

If this is right

  • Sparse and dense long-chain branching fall into two distinct nonlinear regimes, with sparse branching able to exceed NLI = 1 before the post-peak decay.
  • Peak height and width become molecular readouts of arm-to-backbone entanglement ratio, independent of the entanglement molecular weight Me.
  • The same delayed-retraction competition explains why branched NLI curves initially track linear ones and only later fall.
  • Nonlinear Fourier rheology can be inverted for an effective branching descriptor Ba without needing full constitutive simulation.

Reading between the lines

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

  • If Ha is architecture-controlled, industrial LAOS screening of polyolefins or elastomers could rank long-chain branching density from a single NLI peak measurement.
  • The same force-balance-plus-delayed-retraction logic should appear in other branched topologies (combs, pom-poms) once the appropriate Ba is defined.
  • A first-principles LAOS convolution of the Milner–McLeish survival probability would either validate or replace the algebraic Ha form and fix the post-peak exponent without hand-tuning.
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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

4 major / 5 minor

Summary. The manuscript develops a minimal molecular tube model for the Nonlinearity Index (NLI) of long-chain-branched entangled melts under large-amplitude oscillatory shear. It extends a prior nonlinear tube-orientation description of linear polymers by introducing a branch-point force-balance factor ϕb = Zbb/(Zbb + fb Za) that reduces backbone orientation and a delayed arm-retraction survival factor Ha(θ) controlled by the architecture parameter Ba = fb Za/Zbb. The backbone first builds nonlinear orientation as in the linear case; subsequent arm retraction (assisted by CCR and tube dilation) erases orientational memory, producing an NLI maximum followed by post-peak decay. The model recovers the linear limit (Za = 0), distinguishes sparse versus dense branching regimes, and interprets the NLI peak height/width as a molecular indicator of branching architecture.

Significance. If the competition picture holds, the work supplies a transparent molecular interpretation of a distinctive experimental signature in nonlinear Fourier rheology and a direct architecture-to-NLI link via Ba. Strengths include recovery of the linear-polymer limit, a clear sparse/dense distinction, and an explicit connection between the NLI maximum and delayed arm retraction rather than simple orientational saturation. The framework is falsifiable in principle through systematic variation of arm length and branch density. These features make the paper potentially useful for interpreting LAOS data on industrial branched polymers, even though the quantitative architecture dependence currently rests on a compact algebraic approximation rather than a first-principles LAOS convolution.

major comments (4)
  1. [Maximum NLI and delayed arm-retraction crossover] Section “Maximum NLI and delayed arm-retraction crossover,” Eqs. (32)–(39): the load-bearing branch-memory factor is introduced as a Padé-type algebraic crossover Ha(θ) = [1 + Ba ((θ − θm)/θa)^p ]^(−q) with the product pq fixed by hand to 4/5 “to recover the tube-dilation asymptote.” The manuscript itself states that the true Milner–McLeish survival is the stretched exponential Ψa = exp[−(t/τa)^(1/2)] and that Ha is only a “compact crossover approximation.” No first-principles convolution of that survival probability under oscillatory shear is performed, nor is the fixed exponent product derived from the force balance. Because peak height, width and post-peak decay are controlled by Ha and Ba, the claimed quantitative molecular link from architecture to NLI shape is not yet secured; a derivation or numerical evaluation of the memory integral under LAOS is needed to justify the functional
  2. [Maximum NLI and delayed arm-retraction crossover] Eq. (30) and surrounding text: the backbone response NLI^(1)(θ) is taken directly from Eq. (83) of the authors’ prior preprint (Ref. 22) and is simply multiplied by Ha. The central claim therefore inherits any limitations of that unpublished linear-tube construction (including the values of Nmax and m). The manuscript should either re-derive the essential features of NLI^(1) within the present branched setting or demonstrate that the qualitative maximum survives under reasonable variations of the linear backbone form.
  3. [Comparison with experimental NLI curves] Comparison with experimental NLI curves and Fig. 1: the experimental support is a qualitative overlay of previously published NLI curves (Refs. 18–19) for chemically dissimilar materials. No quantitative least-squares comparison of predicted peak height or post-peak slope versus measured Ba (or Ma/Mbb) is shown, and free parameters (θm, θa, p, q, Nmax, m) remain unconstrained. Without at least one architecture series in which Ba is independently known, the claim that Ba “governs the height and width of the nonlinear peak” remains schematic.
  4. [Backbone stress and branch-point force balance] Eqs. (12)–(15) and the force-balance paragraph: the architecture factor ϕb is obtained from a minimal scalar entropic force balance that neglects hierarchical branch-point motion, dynamic tube dilation during the cycle, and stretch–orientation coupling already present in the stretch equation (19). While the limiting cases (Za → 0 and fb Za ≫ Zbb) are sensible, the intermediate quantitative reduction of |S_xy^(bb)| may be inaccurate for multi-arm or polydisperse architectures; a short sensitivity analysis or comparison with an existing hierarchical tube model would strengthen the claim that ϕb is the dominant architectural correction.
minor comments (5)
  1. [Maximum NLI and delayed arm-retraction crossover] Typographical error in the section heading “Maxmimum NLI and delayed arm-retraction crossover” (missing ‘i’).
  2. [CCR, stretch, and arm-retraction memory] Eq. (22) and the subsequent sentence contain a duplicated “is is”.
  3. [Predicted architecture dependence] Figure 2 is described as a “theoretical prediction” but is purely schematic; axis scales and parameter values used to generate the curves should be stated so that the figure can be reproduced.
  4. [References] References 22 and 23 are listed as “2026” preprints without arXiv identifiers or DOIs; full bibliographic details (or permanent links) should be supplied for reproducibility.
  5. [Nonlinear harmonic measure] The definition of NLI in Eq. (5) carries a conventional minus sign for strain-softening; a brief remark that the same construction applies to strain-hardening materials (with sign change) would avoid confusion for readers working with other chemistries.

Circularity Check

4 steps flagged · score 6.0 of 10

Load-bearing backbone NLI form and experimental NLI peak are imported from the authors' own prior preprints/papers; the new Ha factor multiplies that imported form, so peak shape is partly forced by prior definitions plus free crossover parameters.

  1. self citation load bearing [Maximum NLI and delayed arm-retraction crossover, Eq. (30)]
    "We use the nonlinear crossover form introduced in Ref. 22 for the backbone response of entangled polymers, NLI(1)(θ)=Nmax θ^m/(1+θ^m), θ=γ0/γc. This expression is Eq. (83) in Ref. 22 Here Nmax is the nonlinear saturation level reached by the corresponding backbone response, while m is the effective crossover exponent controlling the growth of higher harmonics. Equation (30) is not an independent constitutive equation but an analytical representation of the nonlinear backbone response obtained previously."

    The backbone nonlinear response that is multiplied by Ha to produce the claimed NLI maximum is not derived in this paper; it is imported as Eq. (83) of the authors' own prior preprint. The peak shape therefore inherits the functional form and parameters of that prior self-definition rather than being obtained from a fresh tube calculation.

  2. self citation load bearing [Comparison with experimental NLI curves; Fig. 1 caption and surrounding text]
    "The experimental behavior motivating the model is shown in Fig. 1, which compiles representative NLI data for both linear and long-chain-branched polymers obtained in previous studies by Nichetti and co-workers.18,19 ... Within the present theory, the initial increase reflects nonlinear buildup of backbone orientation, whereas the post-peak decay arises from arm retraction assisted by convective constraint release and tube dilation."

    The qualitative phenomenon the theory is said to explain (NLI maximum then decay only in LCB materials) is taken from the same authors' prior experimental papers. The theory is then fitted/compared to that self-compiled dataset, so the 'explanation' of the peak is partly a re-description of the authors' own earlier observations rather than an independent external benchmark.

2 more flagged steps
  1. ansatz smuggled in via citation [Maximum NLI... Eqs. (32)–(39); discussion of Ha]
    "The algebraic form adopted here, Eq. (32), is motivated by the asymptotic arm-retraction survival probability of Milner and McLeish,3,4,7 but is written as a simple Padé-type crossover function suitable for analytical treatment. ... Choosing pq=4/5 recovers the tube-dilation asymptote used in nonlinear tube theories. Thus Eq. (32) should be interpreted as a compact crossover approximation to the arm-retraction survival memory, constrained by the correct small-deformation limit and by the large-strain tube-dilation decay."

    The paper acknowledges that the true Milner–McLeish survival is a stretched exponential, yet replaces it by a free Padé Ha with product pq fixed by hand to 4/5. No first-principles convolution of Ψa under LAOS is performed. The quantitative dependence of peak height/width and post-peak decay on Ba is therefore controlled by an ansatz whose exponents are chosen to match a desired asymptote, not derived from the oscillatory force balance.

  2. self citation load bearing [Introduction and Discussion (nonaffine framing)]
    "This interpretation is consistent with the broader picture based on nonaffine elasticity20,21 developed in Refs. 22,23 In that framework, the NLI is interpreted as a Fourier-resolved measure of dynamic nonaffinity... The present work extends that viewpoint to long-chain-branched polymers by identifying arm retraction and branch-point relaxation as architecture-specific nonaffine relaxation mechanisms."

    The interpretive claim that NLI measures dynamic nonaffinity, and that the post-peak decay is loss of coherent backbone orientation, rests on the authors' own prior nonaffine papers (including the same Ref. 22 that supplies NLI(1)). The molecular story is therefore framed inside a self-citation chain rather than independently established.

full rationale

The paper's central claim is that the NLI maximum in LCB melts arises from competition between backbone nonlinear orientation and delayed arm retraction, written NLIbr(θ)=NLI(1)(θ)Ha(θ) with Ha controlled by Ba=fb Za/Zbb. That competition is a genuine architectural extension and is not purely definitional. However, the backbone piece NLI(1) is not re-derived here: it is taken verbatim as Eq. (83) of the authors' own prior preprint (Ref. 22), and the experimental motivation (Fig. 1) is compiled from the same group's NLI papers (Refs. 18–19). The nonaffine reading is likewise self-cited (Refs. 20–23). The new content is the multiplicative Ha Padé with free p,q,θa,θm and with the product pq fixed by hand to 4/5 to recover a tube-dilation asymptote. Because the peak shape is the product of an imported backbone form and a flexible algebraic survival factor whose parameters are not fixed by a first-principles LAOS convolution of Milner–McLeish, the quantitative architecture-to-peak mapping is partly forced by prior self-definitions plus free crossover choices. This is partial circularity (score 6), not total: the sparse/dense distinction and the force-balance ϕb remain independent content.

Assumptions & free parameters 7 free parameters · 6 assumptions · 3 invented entities

The central NLI-peak claim rests on tube theory and the authors’ prior linear NLI form, plus several minimal/ad hoc modeling choices (force-balance ϕb, Padé Ha, fixed pq=4/5, orientation-dominated regime). Free crossover parameters and self-imported NLI(1) carry much of the curve shape; independent evidence for the new entities is qualitative only.

free parameters (7)
  • Ba = fb Za/Zbb (branching strength in Ha)
    Treated as the principal control of peak height/width; can be molecular in principle but is used as a free architecture knob in schematic predictions without independent measured values for the plotted curves.
  • θm, θa (reduced-strain thresholds for arm-retraction onset and scale)
    Enter Ha(θ) and set where the peak and decay occur; not derived from first principles for LAOS, chosen to organize the crossover.
  • p, q with pq=4/5
    Exponents in the Padé arm-memory factor; product fixed by hand to recover a tube-dilation power-law asymptote rather than fitted from LAOS data or derived microscopically under oscillation.
  • m (crossover exponent in NLI(1))
    Imported from prior linear-polymer NLI form; controls growth of higher harmonics before arm retraction.
  • Nmax (backbone nonlinear saturation level)
    Sets the height scale of NLI(1); for linear chains claimed as 3 from prior constraint-tube theory, but effective peak in branched systems is reduced by Ha and ϕb.
  • Cc (prefactor in onset strain γc)
    Operational constant in γlin_c = Cc Z^{-1/2}; depends on definition of linear-regime limit.
  • βCCR, βs (CCR and stretch-relaxation coefficients)
    Standard phenomenological CCR strengths in effective disengagement and Rouse rates; not fixed by the paper.
assumptions (6)
  • domain assumption Milner–McLeish arm-retraction survival Ψa(t)=exp[−(t/τa)^{1/2}] with τa∼τe Za^{3/2} exp(3Za/2) for entangled arms.
    Used as the microscopic motivation for delayed arm memory (Eqs. 22–23, 34); standard branched-tube theory, not re-derived here.
  • domain assumption Nonlinear backbone NLI of entangled polymers is given by NLI(1)(θ)=Nmax θ^m/(1+θ^m) from the authors’ prior nonaffine tube-orientation theory.
    Eq. 30 explicitly identified as Eq. 83 of Ref. 22; entire branched construction multiplies this form by Ha.
  • ad hoc to paper Minimal entropic force balance at the T-branch yields ϕb=Zbb/(Zbb+fb Za) and S_xy^(bb)≃ϕb S_xy^lin.
    Eqs. 12–14; presented as simplest normalized force balance recovering linear and arm-dominated limits, not derived from a full network free energy.
  • domain assumption At large amplitude, stretch saturates so elastic stress is orientation-controlled: τE≃Ge S_xy^(bb).
    Eqs. 10–11; standard high-Wi tube modeling choice that drops stretch from the leading NLI argument.
  • ad hoc to paper Padé-type Ha(θ) is an adequate crossover approximation to Milner–McLeish survival under LAOS, with large-strain asymptote fixed by pq=4/5.
    Eqs. 32–39; algebraic form chosen for analytical treatment rather than computed from the oscillatory convolution of Ψa.
  • domain assumption CCR rate 1/τCCR=βCCR|γ̇| and product memory Mbr of reptation/CCR with arm survival control backbone orientation under LAOS.
    Eqs. 16–18, 24–25; standard nonlinear tube ingredients combined with arm survival.
invented entities (3)
  • Architecture factor ϕb = Zbb/(Zbb+fb Za)
    purpose: Reduces attainable backbone orientation and shifts onset strain via branch-point force transmission.
    New compact scalar for this paper’s force-balance picture; independent evidence is only limiting-case consistency, not separate measurement.
  • Delayed arm-retraction survival factor Ha(θ)
    purpose: Multiplies linear-backbone NLI to produce peak and post-peak decay once arm retraction activates.
    Core new dynamical object; form is phenomenological Padé motivated by Milner–McLeish, not independently measured.
  • Branching parameter Ba = fb Za/Zbb as NLI-peak controller
    purpose: Single architecture descriptor governing height and width of the nonlinear peak and sparse vs dense regimes.
    Elevated from a ratio of entanglement numbers to the principal molecular fingerprint of NLI curves; no independent calibration against known model-branch architectures is shown.

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Pith. "Pith review of Delayed Arm Retraction Controls the Nonlinear Oscillatory Response of Long-Chain-Branched Polymer Melts." pith.science (2026). https://pith.science/paper/XGC4B6RH

@misc{pith2026260709561,
  author       = {Pith},
  title        = {Pith review of: Delayed Arm Retraction Controls the Nonlinear Oscillatory Response of Long-Chain-Branched Polymer Melts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XGC4B6RH}},
  note         = {Machine review of arXiv:2607.09561}
}
read the original abstract

Long-chain branching profoundly modifies the nonlinear oscillatory response of entangled polymer melts by introducing arm-retraction pathways absent in linear polymers. We present a molecular tube theory that explains the characteristic maximum of the Nonlinearity Index (NLI) observed experimentally in long-chain-branched polymers. The theory extends the recently developed nonlinear tube-orientation description of linear polymers by incorporating branch-point force transmission and delayed arm retraction. The backbone initially develops nonlinear orientation as in the corresponding linear polymer, whereas long-arm retraction subsequently relaxes the stored branch-point tension and progressively erases backbone orientational memory. This competition produces a characteristic NLI maximum followed by a post-peak decay. The theory predicts two distinct nonlinear regimes corresponding to sparse and dense long-chain branching and introduces an architecture parameter governing the height and width of the nonlinear peak. The resulting framework provides a molecular interpretation of nonlinear Fourier rheology and directly links the nonlinear harmonic response to polymer architecture.

Figures

Figures reproduced from arXiv: 2607.09561 by the authors.

Figure 1
Figure 1. Experimental Nonlinearity Index as a function of reduced strain amplitude [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Theoretical prediction of the Nonlinearity Index obtained by combining the back [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

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Works this paper leans on

22 extracted references · 1 canonical work pages

  1. [1]

    and Edwards, S

    Doi, M. and Edwards, S. F. , title =

  2. [2]

    2023 , doi =

    Zaccone, Alessio , title =. 2023 , doi =

    2023

  3. [3]

    Journal of Non-Newtonian Fluid Mechanics , volume =

    Giuseppe Marrucci , title =. Journal of Non-Newtonian Fluid Mechanics , volume =

  4. [4]

    Philosophical Transactions of the Royal Society A , volume =

    Giuseppe Marrucci and Giovanni Ianniruberto , title =. Philosophical Transactions of the Royal Society A , volume =

  5. [5]

    Macaubas, P. H. P. and Demarquette, N. R. and Dealy, J. M. , title =. Rheologica Acta , volume =. 2005 , doi =

    2005

  6. [6]

    and Colby, R

    Rubinstein, M. and Colby, R. H. , title =

  7. [7]

    2026 , eprint =

    Nichetti, Dario and Zaccone, Alessio , title =. 2026 , eprint =

    2026

  8. [8]

    Approximate analytical description of the nonaffine response of amorphous solids , author =. Phys. Rev. B , volume =. 2011 , month =. doi:10.1103/PhysRevB.83.184205 , url =

    doi:10.1103/physrevb.83.184205 2011

Show all 22 references
  1. [9]

    S. T. Milner and T. C. B. McLeish , title =. Macromolecules , year =

  2. [10]

    McLeish, T. C. B. , title =. Advances in Physics , volume =. 2002 , doi =

    2002

  3. [11]

    , title =

    Wilhelm, M. , title =. Macromolecular Materials and Engineering , volume =. 2002 , doi =

    2002

  4. [12]

    and Wilhelm, M

    Hyun, K. and Wilhelm, M. and Klein, C. O. and Cho, K. S. and Nam, J. G. and Ahn, K. H. and Lee, S. J. and Ewoldt, R. H. and McKinley, G. H. , title =. Progress in Polymer Science , volume =. 2011 , doi =

    2011

  5. [13]

    Ewoldt, R. H. and Hosoi, A. E. and McKinley, G. H. , title =. Journal of Rheology , volume =. 2008 , doi =

    2008

  6. [14]

    Read, D. J. and McLeish, T. C. B. , title =. Macromolecules , volume =. 2001 , doi =

    2001

  7. [15]

    Read and Dietmar Auhl and Chinmay Das and Jaap den Doelder and Michael Kapnistos and Iakovos Vittorias and Tom C

    Daniel J. Read and Dietmar Auhl and Chinmay Das and Jaap den Doelder and Michael Kapnistos and Iakovos Vittorias and Tom C. B. McLeish , title =. Science , volume =. 2011 , doi =. https://www.science.org/doi/pdf/10.1126/science.1207060 , abstract =

    2011 doi

  8. [16]

    and Auhl, Dietmar and Das, Chinmay and den Doelder, Jaap and Kapnistos, Michalis and Vittorias, Ioannis and McLeish, Tom C

    Read, Daniel J. and Auhl, Dietmar and Das, Chinmay and den Doelder, Jaap and Kapnistos, Michalis and Vittorias, Ioannis and McLeish, Tom C. B. , title =. Journal of Rheology , volume =. 2012 , doi =

    2012

  9. [17]

    and Bach, Anders and Koutalas, Georgios and Vlassopoulos, Dimitris , title =

    Hassager, Ole and Rasmussen, Henrik K. and Bach, Anders and Koutalas, Georgios and Vlassopoulos, Dimitris , title =. Macromolecules , volume =. 2008 , doi =

    2008

  10. [18]

    A. E. Likhtman and T. C. B. McLeish , title =. Macromolecules , year =

  11. [19]

    Likhtman, A. E. and Graham, R. S. , title =. Journal of Non-Newtonian Fluid Mechanics , volume =. 2003 , doi =

    2003

  12. [20]

    Hoyle, D. M. and Auhl, D. and Harlen, O. G. and Barroso, V. C. and Wilhelm, M. and McLeish, T. C. B. , title =. Journal of Rheology , volume =. 2014 , doi =

    2014

  13. [21]

    Rubber Chemistry and Technology , volume =

    Scacchi, Michele and Nichetti, Dario , title =. Rubber Chemistry and Technology , volume =. 2025 , doi =

    2025

  14. [22]

    Academic Journal of Polymer Science , volume =

    Scacchi, Michele and Nichetti, Dario , title =. Academic Journal of Polymer Science , volume =. 2025 , doi =

    2025

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