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Shear-stress fluctuations in self-assembled transient elastic networks

J. P. Wittmer, I. Kriuchevskyi, A. Cavallo, H. Xu, and J. Baschnagel
Phys. Rev. E 93, 062611 – Published 20 June 2016
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  • INTRODUCTION
  • ALGORITHM AND TECHNICAL DETAILS
  • COMPUTATIONAL RESULTS
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Focusing on shear-stress fluctuations, we investigate numerically a simple generic model for self-assembled transient networks formed by repulsive beads reversibly bridged by ideal springs. With Δt being the sampling time and t☆(f)∼1/f the Maxwell relaxation time (set by the spring recombination frequency f), the dimensionless parameter Δx=Δt/t☆(f) is systematically scanned from the liquid limit (Δx≫1) to the solid limit (Δx≪1) where the network topology is quenched and an ensemble average over m-independent configurations is required. Generalizing previous work on permanent networks, it is shown that the shear-stress relaxation modulus G(t) may be efficiently determined for all Δx using the simple-average expression G(t)=μA−h(t) with μA=G(0) characterizing the canonical-affine shear transformation of the system at t=0 and h(t) the (rescaled) mean-square displacement of the instantaneous shear stress as a function of time t. This relation is compared to the standard expression G(t)=c̃(t) using the (rescaled) shear-stress autocorrelation function c̃(t). Lower bounds for the m configurations required by both relations are given.

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    • Received 25 February 2016

    DOI:https://doi.org/10.1103/PhysRevE.93.062611

    ©2016 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    ElasticityRheology
    Polymers & Soft Matter

    Authors & Affiliations

    J. P. Wittmer1,*, I. Kriuchevskyi1, A. Cavallo1, H. Xu2, and J. Baschnagel1

    • 1Institut Charles Sadron, Université de Strasbourg & CNRS, 23 rue du Loess, 67034 Strasbourg Cedex, France
    • 2LCP-A2MC, Institut Jean Barriol, Université de Lorraine & CNRS, 1 bd Arago, 57078 Metz Cedex 03, France

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

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    Issue

    Vol. 93, Iss. 6 — June 2016

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    Images

    • Figure 1
      Figure 1

      Addressed problem: (a) Shear-stress relaxation modulus G(t) after a tiny step strain δγ is imposed at t=0 (bold lines). (b) Permanent elastic network formed by beads connected by ideal harmonic springs (thin solid lines) without recombinations (f=0). (c) Self-assembled transient elastic network created by reversibly breaking and recombining springs with an attempt frequency f>0 per spring subject to a Metropolis criterion. The spring s thus connects the beads i and j on the left and the beads i and k on the right.

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    • Figure 2
      Figure 2

      Some technical details: (a) Model Hamiltonian with the bold line indicating the purely repulsive interaction between “harmonics spheres” [34] and the thin line the ideal spring between connected beads, (b) distribution p(r) of spring lengths r showing a maximum around the minimum of the spring potential at Rsp=2 and (c) distribution p(nsp) of the number of springs nsp being connected to a bead showing a maximum at nsp≈8. Only a negligible number of beads is not connected (nsp=0) or are dangling ends (nsp=1). In the current work nsp≤12 is imposed. The distributions shown in panels (b) and (c) are identical for all attempt frequencies f due to detailed balance.

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    • Figure 3
      Figure 3

      Snapshot of small square subvolume of linear length 10 containing 103 beads (disks) connected by 407 springs (straight lines). The width of the spring lines is proportional to the energy of the spring potential, Eq. (10). Short springs with r<Rsp=2 repel the beads (green lines), whereas longer springs (red lines) keep them together.

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    • Figure 4
      Figure 4

      Pair correlations for several attempt frequencies f. Inset: Radial pair correlation distribution function g(r) with r being the distance between two beads [2]. Main panel: Total coherent structure function S(q) with q being the length of the wave vector.

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    • Figure 5
      Figure 5

      Various “static” and “quasistatic” properties vs. Δx(f)≡Δt/t☆(f) with Δt=105 and t☆(f)=16/f. The data indicated for the smallest Δx correspond to f=0. The prediction Eq. (6) is indicated by the solid and the dash-dotted lines. Note that μ̃F≈μA and μ☆≈GF for all Δx.

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    • Figure 6
      Figure 6

      Shear-stress MSD h(t) shown for different frequencies f. Main panel: The MSD increases as h(t)∼t2 for small times t≪tA (thin solid line), shows an intermediate plateau with h(t)≈μF(f=0) for tA≪t≪t☆(f) (bold solid line) and approaches μA (dashed line) for even larger times t≫t☆(f). Inset: Comparison of y(x)=(μA−h(t))/G☆ with x=t/t☆(f), t☆(f)=16/f, and G☆=Geq(f=0) with the shear-stress response modulus G(t)/G☆ for f=0.01 and f=10−6 obtained from the shear-stress increment δτ̂(t) after applying a step-strain increment δγ=0.01 at t=0.

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    • Figure 7
      Figure 7

      Rescaled shear-stress ACF c̃(t)/G☆ vs. dimensionless time x=t/t☆(f) for a broad range of f. Also indicated are the similarly rescaled relaxation moduli G(t) obtained for f=0.01 and f=10−6 by applying a step strain δγ=0.01. The scaling clearly fails for small f (small Δx).

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    • Figure 8
      Figure 8

      Error bars δo/m for o=μA, μ̃F, and GF=μA−μF as a function of Δx=Δt/t☆(f). The error bars for μA are several orders of magnitude smaller than those for μ̃F. The deviations from μA−μ̃F=0 observed in Fig. 7 for small f are thus due to the fluctuations of μ̃F.

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    • Figure 9
      Figure 9

      Standard deviations δ[μA−h(t)] (filled symbols) and δc̃(t) (open symbols) for several attempt frequencies f as indicated. While δc̃(t) becomes similar to this bound for Δx≪1, δ[μA−h(t)] is orders of magnitude smaller in the same limit. The thin horizontal lines indicate δGF(f) for f=0 (bottom), f=10−5, f=0.01, and 10−3 (top); δ[μA−h(t)] is seen to approach this limit for t→Δt.

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    • Figure 10
      Figure 10

      GF(Δt) for subtrajectories of length Δt≤ttraj as a function of Δx≡Δt/t☆(f)∼Δt for different f. The data scales for Δt≫1. The existence of an additional time scale is visible for small Δt≪1. The thin solid line indicates Eq. (6).

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    • Figure 11
      Figure 11

      Comparison of G(t)=μA−h(t) and GF(Δt)≡μA−μF(Δt) for one example in the liquid limit (f=0.01, Δx=62.5). Confirming Eq. (4), GF(Δt) is equivalent to the weighted integral over G(t) indicated by the dotted line. Note that G(t≈Δt)≈GF(Δt) in the three time regimes where the response modulus has a plateau (shoulder). GF(Δt) is delayed with respect to G(t) due to the strong weight of small times to the integral Eq. (F1). The thin solid line indicates Eq. (6).

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