Observation of a transition to a localized ultrasonic phase in soft matter

The study investigates Anderson localization of classical (ultrasonic) waves in three-dimensional soft matter. The research question is whether a soft elastic medium, doped with resonant encapsulated microbubbles (EMBs), can exhibit a transition from extended (diffusive) transport to a localized phase, and how this transition manifests in measurable quantities such as attenuation, mean free path, and late-time transmission. The context is that soft matter offers large contrast in fast/slow wave speeds and potentially strong, broadband scattering with low dissipation, which are favorable for observing localization. The purpose is to experimentally identify mobility edges, quantify critical parameters (e.g., critical density and critical exponent), and validate self-consistent localization theory (SCT) in this platform. The importance lies in demonstrating broadband control of ultrasonic localization in soft matter, enabling tunable acoustic/elastic materials and providing a testbed to compare localization theories with experiments.

The work builds on foundational theories of Anderson localization and mobility edges in disordered systems (Anderson 1958; Abrahams et al. 1979; reviews by Evers & Mirlin 2008). For classical waves, predictions exist for localization and mobility edges in scalar/acoustic systems (Kirkpatrick 1985; Sheng & Zhang 1986; Condat & Kirkpatrick 1987). Prior experiments have observed localization in optical and elastic systems, notably ultrasound in mesoglasses (Hu et al. 2008; Cobus et al. 2018) and critical regimes for light (Störzer et al. 2011; Sperling et al. 2013). The Ioffe-Regel criterion (kℓ ~ 1) is a qualitative guide but may not be quantitatively predictive in 3D for resonant scatterers (Skipetrov & Sokolov 2018). Theoretical treatments indicate anomalies in diffusion near resonance and the role of internal resonances in slowing energy velocity. Numerical studies predict critical densities p_c ~ 0.08–0.1 k^3 for resonant point scatterers and cold atoms, which the present work compares against experimentally. The study also references EMB acoustic models for resonance and damping (Chen et al. 2009; Khismatullin 2004), and self-consistent theories for dynamics of localization (Skipetrov & van Tiggelen 2004, 2006; Kroha et al. 1993).

Experimental platform: A soft gel (Carbopol ETD 2050, 0.2 wt%) doped with encapsulated microbubbles (EMBs) is cast within a 4 mm-thick Uralite 3140 polymer shell to form a water-immersible sample with a 10 mm-thick doped gel layer (L_G = 10 mm, L_u = 4 mm). The polymer shell is impedance-matched to water and provides structural support. EMBs are Expancel microspheres with gas cores (isobutane/isopentane) and polymer shells primarily polyacrylonitrile (PAN). SEM confirms near-spherical geometry and shell thickness on the order of ~100–220 nm (measured collapsed thickness 350–580 nm; corrected ~219 nm). Doping volume fractions studied include φ ≈ 1.0%, 1.2 ± 0.9%, 2.7 ± 0.5%, and 9.9 ± 1.8%. EMB diameter distribution spans ~1–140 µm with many between 60–95 µm. Predicted EMB resonance frequencies f0 vs diameter overlap the experimental band (≈400–800 kHz). The undoped gel exhibits longitudinal phase velocity v_t ≈ 1498 m/s; at 700 kHz λ ≈ 2140 µm ≫ D, ensuring radial EMB oscillations near resonance. Acoustic radiation dominates damping in the EMB scattering model. Measurements: Normal-incidence through-transmission and reflection in water using a 0.5 MHz piston transducer (source) and a Reson TC 4035 hydrophone. Excitation uses a Gaussian-derivative narrow impulse (Δt < 10 µs) from an arbitrary waveform generator, with bandpass filtering and RF amplification; received signals are preamplified and digitized. For each dataset, 1000 acquisitions are averaged; background subtraction applied. Water reference (no sample) is used to reference sound level (SL = 20 log10(P_t/P_ref)). Temperature 296–298 K. Independent speckles are sampled by translating the sample transversely in >λ steps (~3 mm at 500 kHz); hydrophone aperture is smaller than speckle area to ensure independence. Signal processing and derived quantities: Time-windowing separates the coherent (ballistic) and incoherent (multiply scattered) fields. Frequency-dependent attenuation α(f) and scattering mean free path ℓ_s are extracted from the coherent field using the relation for the average Green’s function ⟨G(x,ω;x0,ω)⟩ = G0 exp(−(x−x0)/(2ℓ_s)), yielding normalized intensity I/I0 = exp(−(x−x0)/ℓ_s). For a slab of thickness L_G, α and ℓ_s are related via I/I0 = e^{−(α/k_s)} and ℓ_s = (2α)^{-1} (as used by the authors). Mobility edges are identified as attenuation peaks (α maxima) accompanied by sharp anomalies (minima/discontinuities) in ℓ_s/λ. The phase velocity and phase change Δθ are obtained to assess transport regimes. Late-time transport analysis: The incoherent field is digitally bandpass-filtered to frequency windows targeting ranges near and between mobility edges. The normalized transmitted intensity peak envelope I/I0(t) is computed (I = P^2/(2ρ_G v_1) with ρ_G and v_1 the gel density and longitudinal velocity). Diffusive regime fits use the exponential decay I/I0 = exp(−t/τ_D) (lowest diffusion eigenvalue), from which τ_D, D* = L^2/(π^2 τ_D), and diffusion length ℓ_D = √(D* τ_D) are obtained. Deviations from diffusion and localization dynamics are fitted with the self-consistent theory (SCT) of localization for open 3D media, where late-time transmission follows T(t) ~ exp(−η t / τ_p + 1), with free-fit parameters: bare diffusion coefficient D_e, localization length ξ, and absorption time τ_a. Fits are performed for multiple φ and frequency windows bracketed by mobility edges. Internal reflections are estimated via reflectivity R from reflected SL spectra and the ratio z0/L_G; absorption effects are bounded by τ_a extracted from SCT fits to ensure they do not mimic localization signatures. Controls and calibrations: Finite-thickness transmission corrections are applied (−0.64 dB undoped, −0.36 dB doped). Undoped sample attenuation α ≈ 15 Np/m at 700 kHz confirms low intrinsic loss. Reflection spectra show absence of resonant reflection for f > 400 kHz, supporting attribution of α peaks to mobility edges rather than Fabry–Perot modes.

- Observation of Anderson-localized ultrasonic phase in soft matter: Localized phases spanning up to 246 kHz are observed in EMB-doped gels, bounded by mobility edges identified as attenuation peaks in α(f) and sharp minima/discontinuities in ℓ_s/λ. - Mobility edges and disorder control: Attenuation resonances (>200 Np/m) appear above ~400 kHz and shift to lower frequencies with increasing φ, evidencing a critical scatterer density. For φ = 1.2%, a low-frequency mobility edge occurs at f_c ≈ 527 kHz; a higher-frequency edge near ~773 kHz. For φ = 9.9%, a broad SL minimum centered at ~636 kHz (613–782 kHz) corresponds to a mobility edge with ρ_c ≈ 0.09 k^3; additional low-frequency features near ~350 kHz suggest another edge. - Mean free path criterion and critical density: The transition to localization requires both ℓ_s/λ ≤ 1 and reaching a critical density p = p_c. Within the localized phase ℓ_s/λ lies between 0.4 and 1.0. Example: φ = 2.7% at f = 526 kHz (center of phase), ℓ_s = 1.0 mm (ℓ_s/λ = 0.65), about 9.2× the EMB diameter at that resonance. Estimated critical density at the low-frequency edge (φ = 1.2%, f = 527 kHz): ρ_c ≈ 2.2 × 10^9 EMBs/m^3 ≈ 0.09 k^3, in agreement with numerical predictions (0.08–0.1 k^3) and larger than the Ioffe–Regel estimate (~1.0 × 10^9 m^−3). Total density example: ρ_tot ~ 5.6 × 10^10 EMBs/m^3. - Critical behavior of ℓ_s: Near mobility edges, the mean free path exhibits an anomalous power-law decrease with frequency: ℓ_s = a (f − f_c)^γ + b. Extracted critical exponents: γ = 1.08 ± 0.05 at f_c = 527 kHz (φ = 1.2%); γ = 1.01 ± 0.11 at the higher-frequency edge (~773 kHz). This provides an experimental signature of the phase transition (interpreted as a renormalization of ℓ_s in a weakly dissipative medium). - k_s ℓ_s values: At low-frequency mobility edges: k_s ℓ_s ≈ 3.2 ± 0.1 (φ = 1.2%) and 3.6 ± 0.1 (φ = 2.7%). Within localized phases (between edges) minima: k_s ℓ_s ≈ 2.4 ± 0.1 (φ = 1.2%) and 2.9 ± 0.1 (φ = 2.7%), comparable to mesoglass studies. - Transport regime change above mobility edge: For φ = 2.7%, a discontinuity in ℓ_s/λ at 618 kHz corresponds to a higher-frequency mobility edge; above this, ℓ_s/λ saturates ~1.4 and v_1 drops by ~2.5× (measured v_1 ≈ 334 m/s), consistent with a different extended-state regime influenced by EMB gas properties and wavelength-scale effects. - Diffusive regime (below mobility edge): For φ = 1.2% and 400–550 kHz, late-time intensity follows diffusion with τ_D = 8.9 ± 0.5 µs, D* = 1.14 ± 0.07 m^2/s, yielding ℓ_D ≈ 3 mm ≪ L_G. - Localization dynamics at/near mobility edge: For φ = 2.7% (340–465 kHz), late-times show deviation from diffusion; SCT fits yield ξ ≈ 9.62 mm and bare D_e ≈ 1.54 m^2/s, with reduced absorption time at the edge (τ_a ~ 2 τ_D), indicating enhanced attenuation of incoherent waves due to diffusivity fluctuations approaching the edge. - Localized phase (between mobility edges): Late-time transmitted intensity exhibits pronounced non-exponential decay consistent with SCT. Example SCT fit parameters: • φ ≈ 1.0%, 650–750 kHz: D_e ≈ 33.60 m^2/s, ξ ≈ 2.26 mm, τ_a ≈ 116 µs. • φ = 1.2%, 620–720 kHz: D_e ≈ 33.35 m^2/s, ξ ≈ 1.94 mm, τ_a ≈ 199 µs. • φ = 2.7%, 490–562 kHz: D_e ≈ 38.70 m^2/s, ξ ≈ 2.02 mm, τ_a ≈ 100 µs. ξ decreases with increasing φ and is on the order of the wavelength (e.g., ξ/λ ≈ 1.1 at φ = 2.7%, f = 490 kHz). Absorption lengths ≥ 62 mm and τ_a ≥ 100 µs indicate negligible absorption within the localized phase and that late-time deviations from diffusion are not due to absorption. - Intrinsic losses and controls: Undoped sample α ≈ 15 Np/m at 700 kHz, about 59× smaller than the maximum α measured for φ = 1.2% (881 Np/m), confirming low baseline dissipation. Reflection spectra show no resonant reflection above 400 kHz, supporting interpretation of α peaks as mobility edges.

The experiments directly address whether soft matter with resonant microbubble scatterers can host a three-dimensional Anderson-localized ultrasonic phase and how to identify the transition. The identification of mobility edges via attenuation peaks and concurrent sharp minima/discontinuities in ℓ_s/λ, together with late-time deviations from diffusion consistent with SCT, demonstrate a localized phase bounded by mobility edges. The observation that ℓ_s exhibits a power-law decrease near the edge with a critical exponent ≈1 provides a clear and practical experimental signature of the transition in this weakly dissipative system, even though standard theory does not predict localization signatures in the mean free path away from edges. The extracted critical density ρ_c ≈ 0.09 k^3 agrees with numerical predictions for resonant point scatterers, highlighting the role of resonance-driven disorder and effective-medium spatial dispersion. The study also reveals a distinct high-frequency extended regime with reduced phase velocity, likely governed by EMB gas dynamics and wavelength-scale effects. Overall, the results validate that soft matter platforms allow broadband control over disorder strength (via frequency and φ), enable tuning of localization length ξ (down to order λ), and support late-time transport consistent with localization theory. These findings are relevant for designing materials with tunable acoustic and elastic properties and for benchmarking localization theories under weak dissipation and finite-size constraints.

This work demonstrates broadband control and experimental observation of Anderson localization of ultrasound in soft matter doped with resonant encapsulated microbubbles. Localized phases up to 246 kHz wide are realized, with mobility edges identified by attenuation peaks and sharp anomalies in ℓ_s/λ. The transition is accompanied by an anomalous, power-law reduction of the mean free path with critical exponent near unity. Within the localized phase, ℓ_s/λ ranges from 0.4 to 1.0, late-time transmission follows self-consistent localization theory, and ξ decreases with increasing φ to values comparable to the wavelength. A critical density ρ_c ≈ 0.09 k^3 is measured, in agreement with theoretical predictions for resonant scatterers, and a distinct extended-state regime appears at higher frequencies. Future work should develop theoretical frameworks explaining the anomalous mean-free-path behavior at mobility edges in systems with many internal resonances, refine understanding of effective-medium spatial dispersion, and explore application-driven tuning of localization (e.g., via EMB design, shell mechanics, and gas composition) for acoustic metamaterials, thermal transport control, and phonon engineering.

- Mean free path anomalies at mobility edges are not fully explained by existing field-theoretic models; further theory is needed to account for many internal resonances and near-vanishing ℓ_s at edges. - Measurements are performed in a weakly dissipative but not perfectly lossless medium; although τ_a ≥ 100 µs indicates negligible absorption within localized phases, dissipation could subtly affect critical parameters compared to idealized models. - Finite sample thickness (L_G = 10 mm) imposes finite-size effects; near the mobility edge ξ can approach L_G, limiting observable loop sizes and potentially biasing extracted ξ. - Identification of mobility edges relies on attenuation peaks and ℓ_s/λ anomalies coupled with SCT fits; while corroborated by late-time behavior, this approach depends on model assumptions and frequency window selections. - High-frequency extended regime interpretation (e.g., reduced v_1 ≈ 334 m/s) involves contributions from EMB gas properties and wavelength-scale effects; detailed microscale modeling was beyond scope. - EMB size and volume fraction distributions introduce topological and resonance disorder; while characterized, sample-to-sample variability and polydispersity may influence precise mobility edge positions and ξ. - The Ioffe–Regel condition (k_s ℓ_s ~ 1) is only qualitatively applicable for resonant scatterers; quantitative thresholds may vary across systems, limiting generalizability.