Research ArticleAPPLIED PHYSICS

Propagation and attenuation of mechanical signals in ultrasoft 2D solids

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Science Advances  11 Sep 2020:
Vol. 6, no. 37, eaba6601
DOI: 10.1126/sciadv.aba6601
  • Fig. 1 Structure of colloidal samples.

    Bright-field microscopy images of (A) 2D colloidal crystal and (B) 2D colloidal glass of silica particles used in the tweezing experiments. Scale bars, 40 μm.

  • Fig. 2 Generation and detection of mechanical waves.

    (A) Schematic overview of our experiment. (B) Raw experimental particle trajectories in the direction of the driven particle (gray). Colors correspond to the particles in (A). (C) Amplitude spectrum of the driven particle. The red line corresponds to the driving frequency. (D) Trajectories of (B) after Fourier filtering at the driving frequency. (E) Unfiltered root mean square displacement and (F) Fourier-filtered amplitude map of particles in a crystal excited at ω = 3.1 rad/s; the color scale in (E) and (F) represents the amplitude in micrometers. Scale bars, 40 μm. The red arrow indicates the oscillation direction of the probe particle.

  • Fig. 3 Comparison of experimental and theoretical wave patterns.

    (A) Experimental amplitude map of the parallel displacement. (B) Parallel displacement amplitude predicted by our model. (C) Experimental phase shift for the parallel displacement. (D) Phase shift of parallel displacement predicted by our model. (E) Experimental map of perpendicular displacement. (F) Perpendicular displacement amplitude predicted by our model. (G) Experimental phase shift for the perpendicular displacement. (H) Phase shift of perpendicular displacement predicted by our model. Amplitudes and phases have units micrometers and radians, respectively. Scale bars, 40 μm. The red double-headed arrow indicates the oscillation direction. In all cases, ω = 0.52 rad/s; for the model calculations, E = 2.5 · 10−6 N/m and ν = 0.7.

  • Fig. 4 Analysis of the damped waves.

    (A) Bin-averaged phase of the parallel displacement in the direction of the excitation for ω = 0.52 rad/s. Inset shows the phase velocity versus probed frequency; dashed line depicts slope 12; error bars depict a 95% confidence interval. (B) Superposition of parallel displacement amplitude for different frequencies along θ = 0 (blue) and θ = π/2 (red) versus normalized distance. (C) Superposition of parallel displacement phase for different frequencies along θ = 0 (blue) and θ = π/2 (red) versus normalized distance. Lines in (C) and (D) are fits to the theory (with ν = 0.7). (D) Elastic modulus as a function of frequency, obtained from the phase (open symbols) and amplitude (filled symbols) of both the parallel (red) and perpendicular (blue) displacement components; error bars depict 95% confidence intervals.

  • Fig. 5 Waves in colloidal glasses.

    Experimental maps of (A) parallel displacement amplitude, (B) parallel displacement phase, (C) perpendicular displacement amplitude, and (D) perpendicular displacement phase of 2D colloidal glasses excited at ω = 0.52 rad/s. Amplitudes and phases have units micrometers and radians, respectively. Scale bars, 20 μm. (E) Superposition of parallel displacement amplitude of colloidal glasses for different frequencies along θ = 0 (blue) and θ = π/2 (red) versus normalized distance, together with theoretical prediction (E = 1.9 × 10−7 N/m, ν = 0.60). (F) Bin-averaged phase-distance plot to determine phase velocity for ω= 0.52 rad/s in the parallel direction.

Supplementary Materials

  • Supplementary Materials

    Propagation and attenuation of mechanical signals in ultrasoft 2D solids

    Jan Maarten van Doorn, Ruben Higler, Ronald Wegh, Remco Fokkink, Alessio Zaccone, Joris Sprakel, Jasper van der Gucht

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