Quantization of Acoustic Modes in Dumbbell Nanoparticles

The vibrational eigenmodes of dumbbell-shaped polystyrene nanoparticles are recorded by Brillouin light spectroscopy (BLS), and the full experimental spectra are calculated theoretically. Different from spheres with a degeneracy of ($2l+1$), with $l$ being the angular momentum quantum number, the eigenmodes of dumbbells are either singly or doubly degenerate owing to their axial symmetry. The BLS spectrum reveals a new, low-frequency peak, which is attributed to the out-of-phase vibration of the two lobes of the dumbbell. The quantization of acoustic modes in these molecule-shaped dumbbell particles evolves from the primary colloidal spheres as the separation between the two lobes increases.

Figure 1

Vibrational eigenmodes of polystyrene spherical nanoparticles. (a) Experimental (black line) and calculated (red line) BLS spectra of SP190. The blue lines represent the calculated intensities of the individual eigenmodes broadened to Gaussian profiles by the instrumental width (0.35 GHz). Six spheroidal eigenmodes are indicated by black arrows. Four torsional eigenmodes are indicated by gray arrows. Left inset: a reduced spectrum, $I{f}^2$. Right inset: a scanning electron microscopy (SEM) image of the nanoparticles (scale bar: 500 nm). The relatively low signal-to-noise ratio is due to the short accumulation time. (b) Illustration of the mode patterns (cross-sectional view, for $m=0$) of the six spheroidal eigenmodes in (a). The cyan, dashed circles represent nanoparticles at rest. The black, solid lines represent the nanoparticles surface in the specific vibrational eigenmodes. The gray, dashed lines indicate displacements in internal volumes. The orange arrows indicate local deformations. $t$ and ($t+T_{n,l}/2$) represent two chosen time instants separated by half of the vibrational period $T_{n,l}$ of the $(n,l)$ mode.

Figure 2

Vibrational eigenmodes of polystyrene dumbbell-shaped nanoparticles. (a) Experimental (black line) and theoretical (red line) BLS spectra of DB232-151-214. The blue lines represent the theoretical intensities of the individual eigenmodes broadened by the instrumental width (0.35 GHz) and the dashed line denotes a linear background added to the calculated total spectrum to account for the Rayleigh wing in the experimental spectrum. Characteristic eigenmodes in the five peaks ($A$ to $E$) are indicated by black arrows. The subscript “2” means the mode is doubly degenerate. Six pure torsional eigenmodes are indicated by gray arrows. Left inset: a reduced spectrum, $I{f}^2$ (the abrupt intensity drop at around 1.6 GHz is due to the shutter). Right inset: an SEM image of the nanoparticles (scale bar: 500 nm). (b) Illustration of the mode patterns (cross-sectional view) of the six singly degenerate spheroidal eigenmodes in (a). See the Fig. 1 caption for more details. $T_{\mathrm{DB},i}$ represents the vibrational period of the corresponding eigenmode.

Figure 3

Vibrational eigenmodes and BLS spectra of asymmetric dumbbell-shaped nanoparticles. (a) Eigenmodes of DB232-$d_{12}\text{−}214$ at varying $d_{12}$’s. The red lines denote doubly degenerate modes. The blue, solid (dashed) lines represent singly degenerate modes with nonzero (zero) Brillouin intensities. The eigenmodes contributing to peaks $A$ and $B$ in Fig. 2 are circled. The black, filled (open) triangles denote spheroidal (torsional) eigenmodes of SP232. From bottom to top: $T(1,2)$, $S(1,2)$, $S(2,1)$, $T(1,3)$, and $S(1,3)$. The green, filled (open) triangles denote spheroidal (torsional) eigenmodes of SP214. From bottom to top: $T(1,2)$, $S(1,2)$, $S(2,1)$, and $T(1,3)$. Inset: a schematic of the DB. (b) Calculated BLS spectra of DB232-$d_{12}\text{−}214$ at five $d_{12}$’s. The gray arrows in (b3)–(b5) indicate the lowest-frequency mode. The black arrow in (b3) indicates the new, intermediate-frequency peak at $d_{12}=100\mathrm{nm}$. See the caption of Fig. 2 for more details. Insets: schematics of DBs.