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A link between anomalous viscous loss and the boson peak in soft jammed solids

Nature Physics volume 21, pages 262–268 (2025)Cite this article

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Abstract

Understanding the mechanical properties of soft jammed solids that consist of densely packed particles, such as foams and emulsions, requires insights into the microscopic origins of linear viscoelasticity—how a solid responds to an infinitesimal deformation. Here we perform microrheology experiments on concentrated emulsions and measure the storage and loss moduli for a wide range of frequencies. We applied a linear response formalism for microrheology to a soft sphere model that undergoes the jamming transition. We find that the theory quantitatively explains the experiments. Our analysis reveals that the anomalous viscous loss seen in emulsions results from the boson peak, which is a universal vibrational property of amorphous solids and reflects the marginal stability in soft jammed solids. We show that the anomalous viscous loss is universal in systems with various interparticle interactions as it stems from the universal boson peak; it even survives below the jamming density at which thermal fluctuation is pronounced and the dynamics becomes inherently nonlinear.

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Fig. 1: Link between vibration and linear viscoelasticity revealed in this work.
Fig. 2: Complex modulus of the concentrated emulsions obtained by the microrheology experiments.
Fig. 3: Direct comparison between theory and experiment.
Fig. 4: Scaling analysis of the experimentally measured complex modulus.
Fig. 5: Anomalous viscous loss in varieties of models.

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Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

Code availability

The computer codes used in this study are available from the corresponding authors upon reasonable request.

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Acknowledgements

We thank H. Mizuno for insightful discussions, and S. Inokuchi, K. Nishi and M. Annaka for their technical support. This work was supported by Hosokawa Powder Technology Foundation grant no. HPTF21509 (Y.H.); JST SPRING grant no. JPMJSP2108 (Y.H.); JST CREST grant no. JPMJCR24T2 (D.M.); JST ERATO grant no. JPMJER2401 (A.I.); and JSPS KAKENHI grant nos. JP20H01868 (A.I.), JP20H00128 (A.I), JP21H01048 (D.M.), JP22H04848 (D.M.), JP22K03552 (H.E.), JP23KJ0368 (Y.H.) and JP24H00192 (A.I.). This work was also supported by the JSPS Core-to-Core Program ‘Advanced core-to-core network for the physics of self-organizing active matter’ (grant no. JPJSCCA20230002; D.M.)

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Authors and Affiliations

  1. Graduate School of Arts and Science, The University of Tokyo, Tokyo, Japan

    Yusuke Hara & Atsushi Ikeda

  2. Department of Physics, Kyushu University, Fukuoka, Japan

    Ryosuke Matsuoka, Hiroyuki Ebata & Daisuke Mizuno

  3. Research Center for Complex Systems Biology, Universal Biology Institute, The University of Tokyo, Tokyo, Japan

    Atsushi Ikeda

Authors
  1. Yusuke Hara
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  2. Ryosuke Matsuoka
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Contributions

Y.H., D.M. and A.I. designed the research. Experimental data were collected by R.M. and D.M. Y.H. carried out the simulations. Y.H., R.M., H.E., D.M. and A.I. analysed the data, discussed the results and wrote the paper.

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Correspondence to Yusuke Hara, Daisuke Mizuno or Atsushi Ikeda.

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Nature Physics thanks Lars Pastewka and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Comparison between macrorheology and microrheology.

a. The macroscopic complex modulus of the soft sphere model calculated by the formula Eq.(32). The packing fraction is set to be ϕ = 0.66. b. The microscopic complex modulus of the soft sphere model calculated by the formula Eq.(19). The packing fraction is set to be ϕ = 0.66. Reproduced from Fig. 3c.

Extended Data Fig. 2 Scaling analysis of the vibrational density of states.

a. P(α−2)/(2α−2)D(ω) is plotted against ω/P1/2 for the soft sphere model, where P is the pressure of the packings. The results for P = 10−5 − 10−2 with N = 8000 are included. According to Eq. (4), the vDOS of the soft sphere model obeys the scaling relation ωeD(ω) = f(ω/ω*) with f(x) ∝ x2 at x < 1. Because the pressure follows the scaling relation \(P\propto {(\phi -{\phi }_{J})}^{\alpha -1}\), we can express the characteristic frequencies as ωe ∝ P(α−2)/(2α−2) and ω* ∝ P1/2. Therefore, the scaling relation can be rewritten as P(α−2)/(2α−2)D(ω) = g(ω/P1/2). The data collapse in the figure confirms this scaling relation of D(ω). b. Same as a but for \(\tilde{D}(\omega )\), the density of eigenfrequencies of \(\propto \tilde{{\bf{{\mathcal{M}}}}}\). The data collapse confirms that D(ω) and \(\tilde{D}(\omega )\) share the same scaling relation.

Extended Data Fig. 3 Scaling analysis of the loss modulus of the soft sphere model.

\({G}^{{\prime\prime} }/\left({\left(\phi -{\phi }_{J}\right)}^{(\alpha -2)/2}{\omega }^{1/2}\right)\) are plotted with respect to \(\omega /{(\phi -{\phi }_{J})}^{(\alpha -2)}\) for our numerical model. This is the same scaling analysis as in Fig. 4a, but for G″ of the numerical model at ϕ = 0.64, 0.65 and 0.66. The result shows that \({G}^{{\prime\prime} }/\left({\left(\phi -{\phi }_{J}\right)}^{(\alpha -2)/2}{\omega }^{1/2}\right)\) approaches to a constant with decreasing the frequency, except for the lowest frequencies. This confirms that our numerical model exhibits \({G}^{{\prime\prime} }\propto \sqrt{\omega }\).

Source data

Source Data Fig. 2

The (x, y) data for Fig. 2a,b.

Source Data Fig. 3

The (x, y) data for Fig. 3a–c.

Source Data Fig. 4

The (x, y) data for Fig. 4a–g.

Source Data Fig. 5

The (x, y) data for Fig. 5a–d.

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Hara, Y., Matsuoka, R., Ebata, H. et al. A link between anomalous viscous loss and the boson peak in soft jammed solids. Nat. Phys. 21, 262–268 (2025). https://doi.org/10.1038/s41567-024-02722-7

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