DOI: 10.5281/zenodo.10829760
title: Architecture Controls Phonon Propagation in All-Solid Brush Colloid Metamaterials - datasets

description: Data sets to figures in the publication https://doi.org/10.1002/smll.202304157

Fig1 - a) Experimental dispersion plot for close-packed PS particles (diameter d = 307 nm) infiltrated in PDMS (red filled circles), and silica (SiO2)-PS GNP assembly (square symbols, d = 214 nm) with the empty symbols denoting the dispersionless, highly localized, rotational mode originating from dipole torsional modes of the individual particles; the wavenumber is normalized with respect to qBZ along ΓM direction. Calculated band structure along the [111] fcc high-symmetry direction for the PS opal infiltrated in (fluid) PDMS b) and for the DP980 colloidal crystal c) assuming respectively PBCs and IBCs (kT = 0.021 GPa nm−1). Solid lines: longitudinal bands in (b) and non-degenerate bands including inactive bands in (c); dotted lines: quasi-flat (highly localized) band originating from dipole torsional modes (see main text). Shaded regions denote hybridization gaps of dipole-resonance origin (LHG for longitudinal modes; HG for all modes). The horizontal red arrow in (b) indicates the position of the quadrupolar resonant frequency of the individual PS sphere in PDMS. Note that only non-degenerate bands that correspond to longitudinal phonons are shown (we have omitted transverse phonon modes since they were not observed experimentally in Figure 1a).

 

 

Fig 2- Top panel: experimental BLS spectra of three PS tethered SiO2 nanoparticle GNP films with different grafting densities, DP1300 (σ = 0.53nm−2 in (a)), DP530 (σ = 0.27nm−2 in (b)) and DP1170 (σ = 0.08 nm−2 in (c)) at a wave vector q (arrows in (d–f)) where a hybridization gap (HG, patterned areas in (d–f)) opens in the dispersion diagrams in (d–f)). The spectra are recorded with VV (black) and VH (grey) polarizations. The isotropic spectra obtained from the subtraction of the VH (depolarized) from the experimental polarized (VV) spectra are represented by Lorentzian lines (red). Bottom panel: experimental dispersion relations of the three systems in (a–c) and their optical images as insets in (d–f). The frequency is obtained from the isotropic spectra recorded at different q's. The direction of q is selected in the transmission and reflection (grey-shaded area) geometries and the magnitude of q is tuned by changing the scattering angle. The HGs denoted by patterned area are clearly observed in each system. The open circles represent the localized mode with q-independent frequency denoted by arrows. The effective medium acoustic modes are represented by red lines in the low-q regime. Note the dip in the BLS intensity at the frequency inside the gap (minimum DOS)

 

Fig3 - Dispersion relation of a) DP1300 and b) DP1170 swollen with 20 wt.% DMP (solid symbols). For comparison, the phonon dispersion in the pristine GNP films (open symbols) is shown in (a,b). The black and red lines denote the low-frequency acoustic regime of DP1170 and plasticized DP1170, while the blue and black dashed lines are to guide the eyes. The vertical arrows indicate the position of the HG indicated by the hatched and shaded areas, whereas the horizontal lines with arrows indicate the frequency fLO of the flat mode. Insets: Experimental VV (blue) and VH (grey) for the two plasticized samples and optical images of DP1300 with 20% DMP in (a). The isotropic spectra obtained from the subtraction of the VH (depolarized) from the polarized (VV) spectra are represented by Lorentzian lines (red) as in Figure 2a,c.

 

 

Fig4-Theoretical band diagram of the a–c) sparsely DP1170 (d = 140 nm) and d–f) densely grafted DP1300 (d  =  225 nm) considering PBCs with bulk PS sound velocities ( m s−1, m s−1) along [111] (left column, plots (a,d)), and, IBCs along [111] (middle column, plots (b,e)) and [112] (right column, plots (c,f)); the parameters used for the calculations are: kL = 1.16 GPa nm−1, kT = 0.20 GPa nm−1) with higher than bulk PS sound velocities ( , ), for DP1170 (plots b,c) and kL =  0.615 GPa nm−1, kT = 0.030 GPa nm−1 with bulk PS sound velocities for DP1300 (plots (e,f)). Solid and open circles indicate the experimental points. Hatched regions denote hybridization gaps (LHG for longitudinal modes; HG for all modes). Along the high symmetry line ΓL of the fcc Brillouin zone (BZ), dark/light solid and dotted blue lines denote non-degenerate (longitudinal, i.e., of Λ1 symmetry), doubly-degenerated (transverse, i.e., of Λ3 symmetry) and deaf (i.e., of Λ2 symmetry) computed bands, respectively. Along [112] that includes the low symmetry line ΓM of the fcc BZ all bands are non-degenerate of mixed character. The position of the flat band of dipole torsional origin is indicated by a red arrow.

 

 

Fig5 - a) Evolution of the effective medium slope for the different colloidal SiO2-PS GNP assemblies (filled symbols, left axis) and of the enhanced transverse velocity ratio for PS (open symbols, right axis) as a function of the interparticle distance, d = dcal (Table 1), showing a non-linear decay with increasing PS filling fraction (dashed curve is a guide to the eye). b) Redshifted variation of the localized-mode frequency, fLO, for the GNP colloids with increasing distance d. Blue dotted line denotes the flat mode frequency for a fcc crystal calculated along ΓL (taken at the middle of the BZ) assuming PBCs and bulk velocities for PS; solid gray line: interpolated curve for the various samples. c) Power-law variation of the localized-mode frequency with the tangential stiffness kT. d) The tangential stiffness kT as a function of the crowding parameter for the DP1170, DP530 and DP1300 with decreasing σ. In (b,c), all scales are logarithmic, symbols are color-indexed with the grafting-chain density value of the corresponding labeled samples.