Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                

Shear-stress fluctuations and relaxation in polymer glasses

I. Kriuchevskyi, J. P. Wittmer, H. Meyer, O. Benzerara, and J. Baschnagel
Phys. Rev. E 97, 012502 – Published 11 January 2018
PDFHTMLExport Citation
Abstract
Authors
Article Text
  • INTRODUCTION
  • ALGORITHMICAL DETAILS
  • EXPECTATION VALUES
  • STANDARD DEVIATIONS, DISTRIBUTIONS, AND…
  • SHEAR-STRESS RELAXATION MODULUS
  • Standard deviation δG(t)
  • SYSTEM-SIZE EFFECTS
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We investigate by means of molecular dynamics simulation a coarse-grained polymer glass model focusing on (quasistatic and dynamical) shear-stress fluctuations as a function of temperature T and sampling time Δt. The linear response is characterized using (ensemble-averaged) expectation values of the contributions (time averaged for each shear plane) to the stress-fluctuation relation μsf for the shear modulus and the shear-stress relaxation modulus G(t). Using 100 independent configurations, we pay attention to the respective standard deviations. While the ensemble-averaged modulus μsf(T) decreases continuously with increasing T for all Δt sampled, its standard deviation δμsf(T) is nonmonotonic with a striking peak at the glass transition. The question of whether the shear modulus is continuous or has a jump singularity at the glass transition is thus ill posed. Confirming the effective time-translational invariance of our systems, the Δt dependence of μsf and related quantities can be understood using a weighted integral over G(t).

    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    13 More
    • Received 31 October 2017

    DOI:https://doi.org/10.1103/PhysRevE.97.012502

    ©2018 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Elastic deformationShear deformationShear flowsViscoelasticity
    Polymers & Soft Matter

    Authors & Affiliations

    I. Kriuchevskyi1,2, J. P. Wittmer1,*, H. Meyer1, O. Benzerara1, and J. Baschnagel1

    • 1Institut Charles Sadron, Université de Strasbourg & CNRS, 23 rue du Loess, 67034 Strasbourg Cedex, France
    • 2LAMCOS, INSA, 27 av. Jean Capelle, 69621 Villeurbanne Cedex, France

    • *joachim.wittmer@ics-cnrs.unistra.fr

    Article Text (Subscription Required)

    Click to Expand

    References (Subscription Required)

    Click to Expand
    Issue

    Vol. 97, Iss. 1 — January 2018

    Reuse & Permissions
    Access Options
    Author publication services for translation and copyediting assistance advertisement

    Authorization Required

    Log In
    Other Options
    ×

    Images

    • Figure 1
      Figure 1

      Radial pair correlation function g(r) for one temperature above and one below the glass transition temperature Tg showing that our coarse-grained polymer model does not crystallize. Also indicated are the equilibrium bond length lbond=0.967, the position of the minimum of the LJ potential rmin=21/6, and the potential cutoff distance rcut=2.3.

      Reuse & Permissions
    • Figure 2
      Figure 2

      Average specific volume v per monomer as a function of T for Γ=2×10−5. The thin solid lines indicate linear fits to the glass and the liquid branches. Using this dilatometric criterion, one defines a glass transition temperature Tg≈0.38 from the intersection of both linear asymptotes.

      Reuse & Permissions
    • Figure 3
      Figure 3

      Shear modulus μsf(T)=〈μ¯sf(T)〉 for different sampling times Δt using a linear representation. The transition becomes more and more steplike with increasing Δt but remains continuous for all Δt sampled. Also included is the shear-stress relaxation modulus G(t) taken at a time t=104 (crosses). The vertical dashed line indicates the (Δt-independent) glass transition temperature [Eq. (4)], operationally defined using a dilatometric criterion during the continuous temperature quench. The thin solid line corresponds to a cusp singularity with an effective exponent α≈0.2. Inset: zoom for T around Tg for Δt=100000 emphasizing that the transition characterized by μsf(T) remains continuous.

      Reuse & Permissions
    • Figure 4
      Figure 4

      μA,μ0,μ1,μF, and μsf vs T using a half-logarithmic representation. Only data for Δt=Δtmax=105 are given. For large temperatures μsf≈μ1 and μF≈μ0≈μA. With decreasing temperature μsf increases rapidly around Tg, but remains continuous. μ0 and μ1 increase rapidly below T≈0.3 and μ1−μsf and μ0−μA become thus finite. Inset: μF(T) using linear coordinates emphasizing the maximum near Tg.

      Reuse & Permissions
    • Figure 5
      Figure 5

      Temperature dependence of μA and μ0. Main panel: μA (open symbols) and μ0 (closed symbols) for two sampling times illustrating the Δt independence expected from the commutation of time and ensemble averages [Eq. (24)]. While μA(T) becomes (more or less) constant below Tg,μ0(T) is seen to increase strongly. The dashed-dotted and the solid lines indicate two linear fits with μA(T)=μA(Tg)[1−c(T/Tg−1)] with c=0.076 and 0.19 for, respectively, the low and high temperature regimes and Tg=0.38 for both. Inset: double-logarithmic representation of μ0/μA−1 vs T. The ratio decreases inversely with temperature for T≪Tg (solid line).

      Reuse & Permissions
    • Figure 6
      Figure 6

      Shear modulus μsf as a function of sampling time Δt for a broad range of T as indicated in the figure. μsf(Δt) decreases continuously with Δt. Note that a smaller temperature increment ΔT=0.01 is used around Tg (solid lines) where μsf(Δt;T) changes much more rapidly with T. The vertical lines mark the sampling times used in Fig. 3. The dashed-dotted lines are obtained using Eq. (1) by integrating the shear-stress relaxation modulus G(t).

      Reuse & Permissions
    • Figure 7
      Figure 7

      Standard deviation δμsf(T) for different sampling times Δt using a linear representation. The observed peak slightly below Tg becomes sharper with increasing Δt. The small filled symbols indicate the values predicted (Sec. 5e) according to the two-point approximation (39) from the standard deviation δG(t). While this allows to relate μsf(Δt) to δG(t) for T≪Tg and T≫Tg, it fails for large Δt around the glass transition.

      Reuse & Permissions
    • Figure 8
      Figure 8

      Distribution p(μ¯sf) of the time-averaged modulus μ¯sf: (a) p(μ¯sf) for T=0.35 and several Δt as indicated. The maximum μsf,max shifts to the left with increasing Δt and the histogram becomes more lopsided. (b) p(μ¯sf) for Δt=105 and a broad range of T. (c) μsf,med−μsf vs T for Δt=105.

      Reuse & Permissions
    • Figure 9
      Figure 9

      Standard deviations as a function of T: (a) δμA,δμ0,δμ1,δμF, and δμsf for Δt=105. δμA is found to be small and δμsf≈δμF for all T. δμ0 and δμ1 become rapidly similar below Tg and orders of magnitude larger than δμF confirming the presence of strong frozen shear stresses. (b) δμ1(T) for several Δt using the same symbols as in Fig. 7 for δμsf(Δt). While sampling time effects are seen to be irrelevant for T≪Tg, i.e., the frozen stresses cannot relax, they matter for temperatures around and above Tg.

      Reuse & Permissions
    • Figure 10
      Figure 10

      Correlation coefficient r01 as a function of T for different Δt showing that r01→0 for T≫Tg and r01≈1 for T≪Tg. The transition around T≈Tg depends again on Δt. Inset: T1/2(Δt) defined by r01(T1/2,Δt)=12 reveals a logarithmic decay of the correlations with increasing Δt.

      Reuse & Permissions
    • Figure 11
      Figure 11

      Stress relaxation modulus G(t) for a broad range of T using half-logarithmic coordinates. G(t) has been obtained by means of Eq. (20) using gliding averages, i.e., the statistics deteriorates for t→Δt and the data have been logarithmically averaged for clarity. The dashed vertical line marks the time used for G(t) in Fig. 3 and Table 2.

      Reuse & Permissions
    • Figure 12
      Figure 12

      Comparison of μsf(Δt),μ(Δt), and G(t) for T=0.1 and 0.5. μsf(Δt) is indicated by squares, μ(Δt) by stars, and G(t) by open circles if obtained using Eq. (20) and by pluses if obtained using Eq. (30) from the μsf(Δt) data. It is seen that μsf(Δt)≈μ(Δt) and that also both expressions for G(t) are essentially identical. (a) At sufficiently low temperatures μ(Δt)≈G(t=Δt) over a broad plateau. (b) At larger temperatures and longer times G(t) decays faster than its integral μ(Δt). The dotted lines indicate μ(Δt)∼1/Δtα and G(t)∼1/tα with α=12.

      Reuse & Permissions
    • Figure 13
      Figure 13

      Generalized shear viscosity η with open symbols obtained using Eq. (3) and filled symbols using Eq. (37): (a) η(Δt;T) as a function of Δt for several T. As shown by the solid line, η(Δt) increases linearly in the solid limit. At higher temperatures η(Δt) increases less strongly and eventually levels off. The two dashed lines indicate the limits η(Δt)→η∞(T) for T=0.5 and 0.45. The vertical arrows mark for several temperatures the approximative position of the terminal relaxation time τ∞(T). (b) η(T;Δt) as a function of T for different Δt. The crosses represent η∞(T) from the inset of Fig. 17.

      Reuse & Permissions
    • Figure 14
      Figure 14

      Standard deviation δG(t) of the relaxation modulus G(t) presented in Fig. 11. All data have been obtained with gliding averages over time series of length Δt=105. Inset: double-logarithmic representation of δG(t) vs time t for several temperatures T. Main panel: δG as a function of temperature for several times t as indicated. δG is nonmonotonic with a strong maximum slightly below Tg.

      Reuse & Permissions
    • Figure 15
      Figure 15

      Test of system-size effects using the four chain numbers M=200, 384, 1536, and 3072: (a) Shear-stress relaxation modulus G(t) for five temperatures T. (b) μsf(T;Δt=105) for several M demonstrating the expected system-size independence of the continuous transition of the shear modulus for finite sampling times Δt. (c) Standard deviation δμsf(T) as a function of temperature T for Δt=105.

      Reuse & Permissions
    • Figure 16
      Figure 16

      μsf(Δt) and its contribution μ1(Δt) for the high temperature T=0.5 using logarithmic coordinates. The small filled circles represent uncorrected μsf(Δt) data where we do not take into account the impulsive truncation corrections for μA [43]. The uncorrected data saturate at a small, but finite value ΔμA≈0.28 (dashed horizontal line). If correctly shifted (open squares), μsf(Δt)=μ1(Δt) for all Δt and μsf(Δt)=η∞2/Δt for Δt≫τ∞ (solid line). The vertical arrow marks the terminal relaxation time τ∞=66 for T=0.5 set as reference for the rescaling of μ1(Δt) in Fig. 17.

      Reuse & Permissions
    • Figure 17
      Figure 17

      μ1(Δt) for higher temperatures where μA=μ0 holds: (a) Using double-logarithmic coordinates, the shear viscosity η∞(T) can be estimated for T≳0.38 by fitting μ1(Δt)=η∞2/Δt as indicated by the dashed lines. (b) η∞(T) increases over four orders of magnitude between T=0.55 and 0.38. The plus is a fair guess for T=0.37. For comparison, we indicate μ1(T;Δt)Δt/2 for Δt=100000,50000, and 10000 (open symbols). The bold line represents Eq. (B1). (c) Characterization of τ∞(T) by tracing y=μ1(Δt)τ∞/η∞ vs x=Δt/τ∞. (d) Terminal relaxation time τ∞(T) used for the rescaling of μ1(Δt).

      Reuse & Permissions
    • Figure 18
      Figure 18

      Sketch of properties of interest for the determination of the shear-stress relaxation modulus G(t) focusing on solids. Static properties are indicated by horizontal lines, G(t) by the bold solid line, c(t) by the thin dashed-dotted line, and h(t)=c(0)−c(t) by the bold dashed line. A canonical affine shear transformation at t=0 implies G(t=0)=μA while for large times G(t)→μsf. At variance to this, c(t) decays from c(t=0)=μ0 to c(t=Δt)=μ1. In general, μA≠μ0, hence, G(t)≠c(t).

      Reuse & Permissions
    • Figure 19
      Figure 19

      Ensemble-averaged ACF c(t) (open symbols) and associated standard deviation δc(t) (filled symbols) as a function of temperature T for several times t. Note that δc(t) is always larger c(t) for all T and t. The bold solid line indicates μ1(T) for Δt=105, the dashed line the corresponding standard deviation δμ1(T). Below Tg, one observes c(t)≈μ1(T) and δc(t)≈δμ1(T).

      Reuse & Permissions
    • Figure 20
      Figure 20

      Generalized shear-stress relaxation time τ(T;Δt) for several Δt with open symbols obtained using Eq. (E1) and filled symbols using Eq. (E3) taking advantage of the μsf(Δt) data. For both methods, the data are noisy and unreliable around Tg. This is at variance to the smooth τ∞(T) values (crosses) obtained in Fig. 17 from μ1(Δt).

      Reuse & Permissions
    ×

    Sign up to receive regular email alerts from Physical Review E

    Log In

    Cancel
    ×

    Search

    Article Lookup

    Paste a citation or DOI
    Enter a citation
    ×