Ramsauer–Townsend resonances in precompressed granular chains: general impurity arrangements and disorder

A 2016 Physical Review E paper showed that a symmetric contiguous double impurity in a strongly precompressed granular chain produces a Ramsauer–Townsend (RT) type resonance — perfect transmission of a plane wave at a frequency tunable anywhere within the host chain's pass band — and stated in its conclusions that this paves the way for a systematic study of RT resonances in granular crystals: resonances for more general impurity arrangements (asymmetric pairs, multi-site clusters, ordered or disordered impurity distributions), and the extension from a finite disordered segment in a homogeneous host to fully disordered granular crystals where disorder-induced localization and superdiffusive transport compete with resonant transparency. The follow-up by the same group measured superdiffusive energy transport in fully disordered nonlinear chains (Nat. Commun. 2018) but did not address RT resonances; the closest adjacent works — quasiperiodic granular chains, 2D impact scattering by multiple point defects, RT-like resonance modes in weakly inhomogeneous force chains, and viscoelastic shock-impurity scattering — leave the resonance theory for general arrangements and its interplay with full disorder unresolved. The problem remains open.

Background

A granular chain is a one-dimensional array of elastically contacting spherical particles whose interactions obey Hertz's law: the contact force between neighbors scales as $A(\Delta+u_n-u_{n+1})^{3/2}$, where $\Delta$ is the static overlap produced by an applied precompression force and $u_n$ is the displacement of particle $n$ from its equilibrium position. Under strong static precompression ($\Delta$ large compared with the dynamic relative displacements), the Hertzian nonlinearity can be linearized, and the chain supports small-amplitude plane waves with an acoustic dispersion relation and a finite cutoff (upper band-edge) frequency — the transmission band of the host chain. The precompressed, linearized granular chain is thus an experimentally realizable macroscopic platform for lattice scattering theory.

When one or a few particles of different mass or radius ('impurities') replace particles of the host chain, an incident plane wave is partly reflected and partly transmitted. Martínez, Yasuda, Kim, Kevrekidis, Porter, and Yang [Phys. Rev. E 93, 052224 (2016)] derived closed-form reflection and transmission coefficients for a single impurity and for a symmetric contiguous double impurity (two identical impurities occupying adjacent sites). For a single impurity, the transmission coefficient $|T|^2$ decays monotonically with wavenumber $k$ and vanishes as $k\to\pi$. For the symmetric double impurity, they identified a resonance at a wavenumber $k_r$ at which transmission is perfect ($|T|^2=1$), in analogy with the Ramsauer–Townsend (RT) effect of quantum scattering — the transparency of certain low-energy particle beams to atoms — and showed that the RT resonance can be tuned to any frequency within the host pass band by choosing the impurity size ratio. They confirmed the predictions experimentally using laser-Doppler-vibrometer measurements of particle velocities, and demonstrated numerically that reflectionless modes (generalizations of the RT resonance) persist in chains containing many double impurities, including a finite disordered segment of such impurities embedded in an otherwise homogeneous host. Their analysis was restricted to the linearized, strongly precompressed regime, and quantitative theory–experiment discrepancies remain near the upper band edge, attributed in the paper to wave localization from 'beyond-cutoff' frequency components of the excitation, the intrinsic frequency response of the piezoelectric actuator, and possible particle misalignment.

The disordered-chain context is set by two numerical studies the source itself cites: Martínez, Kevrekidis, and Porter [Phys. Rev. E 93, 022902 (2016)] and Achilleos, Theocharis, and Skokos [Phys. Rev. E 93, 022903 (2016)], which reported superdiffusive energy transport (spreading faster than diffusive) and energy localization in one-dimensional disordered granular crystals — chains whose particles have randomly varying properties. In such fully disordered chains one expects Anderson localization, the exponential spatial localization of waves by disorder, to compete with any resonant transparency mechanism.

Problem Statement

The source paper's conclusions state that its results 'pave the way for a systematic study of the properties of Ramsauer–Townsend resonances in granular crystals'. The problem is to build that systematic theory of reflectionless transmission in strongly precompressed granular chains, beyond the symmetric contiguous double impurity:

1. General impurity arrangements. Establish when unit-transmission (RT) resonances occur for impurity configurations beyond the symmetric contiguous double impurity — asymmetric impurity pairs, multi-site impurity clusters, and graded, correlated, or statistically characterized impurity ensembles. Derive resonance conditions (existence regions, resonant wavenumber, tunability ranges) in closed form where tractable and map them numerically otherwise, determining how the resonance condition depends on the number, spacing, ordering, and relative properties of the impurities, and whether RT-like transparency can be engineered in more complex granular metamaterials (e.g., for embedding sensors or foreign objects with minimal wave interference).

2. Fully disordered crystals. Extend the analytic and experimental framework from a finite disordered scattering segment in a homogeneous host to fully disordered granular crystals, to determine how RT-like reflectionless mechanisms interact with disorder-induced localization and superdiffusive transport.

The common scientific objective uniting these components is one predictive scattering theory of reflectionless transmission in heterogeneous (ordered, correlated, and disordered) precompressed granular crystals: when, and under what conditions on impurity arrangement and disorder, does such a chain exhibit reflectionless transmission, and can that transparency be predicted, tuned, and experimentally validated? Results on individual arrangement classes (e.g., only asymmetric pairs, or only finite multi-site clusters) constitute meaningful partial progress toward this objective rather than separate unrelated deliverables.

References

1. Alejandro J. Martínez, Hiromi Yasuda, Eunho Kim, P. G. Kevrekidis, Mason A. Porter, Jinkyu Yang, 'Scattering of waves by impurities in precompressed granular chains', Physical Review E 93, 052224 (2016). DOI: 10.1103/physreve.93.052224; arXiv:1602.08651, https://arxiv.org/abs/1602.08651
2. Eunho Kim, Alejandro J. Martínez, Sean E. Phenisee, P. G. Kevrekidis, Mason A. Porter, Jinkyu Yang, 'Direct measurement of superdiffusive energy transport in disordered granular chains', Nature Communications 9, 640 (2018). DOI: 10.1038/s41467-018-03015-3, https://www.nature.com/articles/s41467-018-03015-3
3. Alejandro J. Martínez, P. G. Kevrekidis, Mason A. Porter, 'Superdiffusive transport and energy localization in disordered granular crystals', Physical Review E 93, 022902 (2016). DOI: 10.1103/physreve.93.022902; arXiv:1411.5746, https://arxiv.org/abs/1411.5746
4. V. Achilleos, G. Theocharis, Ch. Skokos, 'Energy transport in one-dimensional disordered granular solids', Physical Review E 93, 022903 (2016). DOI: 10.1103/physreve.93.022903; arXiv:1509.03551, https://arxiv.org/abs/1509.03551
5. C. Chong, Mason A. Porter, P. G. Kevrekidis, C. Daraio, 'Nonlinear coherent structures in granular crystals', Journal of Physics: Condensed Matter 29, 413003 (2017). DOI: 10.1088/1361-648X/aa7672; arXiv:1612.03977, https://arxiv.org/abs/1612.03977
6. Hrachya Kocharyan, Nikhil Karanjgaokar, 'Influence of interactions between multiple point defects on wave scattering in granular media', Granular Matter 24, 2 (2022). DOI: 10.1007/s10035-021-01177-4, https://link.springer.com/article/10.1007/s10035-021-01177-4
7. O. I. Gerasymov, A. Ya. Spivak, 'On the Wave Transmission in a Gently Perturbed Weakly Inhomogeneous Non-Linear Force Chain', Ukrainian Journal of Physics 65, 1008 (2020). DOI: 10.15407/ujpe65.11.1008, https://doi.org/10.15407/ujpe65.11.1008
8. Tao Liang, Wen-Shan Duan, 'Scattering of shock wave by impurities in a viscoelastic granular chain', The European Physical Journal E 42, 140 (2019). DOI: 10.1140/epje/i2019-11904-x, https://link.springer.com/article/10.1140/epje/i2019-11904-x
9. Alejandro J. Martínez, Mason A. Porter, P. G. Kevrekidis, 'Quasiperiodic granular chains and Hofstadter butterflies', Philosophical Transactions of the Royal Society A 376, 20170139 (2018). DOI: 10.1098/rsta.2017.0139; arXiv:1801.09860, https://arxiv.org/abs/1801.09860