Unusual High-Frequency Mechanical Properties of Polymer-Grafted Nanoparticle Melts

Brillouin light spectroscopy is used to measure the elastic moduli of spherical polymer-grafted nanoparticle (GNP) melts as a function of chain length at fixed grafting density ($0.47\mathrm{chains}/{\mathrm{nm}}^2$) and nanoparticle radius (8 nm). While the moduli follow a rule of mixtures (Wood’s law) for long chains, they display enhanced elasticity and anomalous dissipation for graft chains $<100\mathrm{kDa}$. GNP melts with long polymers at high $\sigma$ have a dry zone near the GNP core, surrounded by a region where the grafts can interpenetrate with chain fragments from adjacent GNPs. We propose that the departures from Wood’s law for short chains are due to the effectively larger silica volume fraction in the region where sound propagates—this is caused by the short, interpenetrated chain fragments being pushed out of the way. We thus conclude that transport mechanisms (of gas, ions, sound, thermal phonons) in GNP melts are radically different if interpenetrated chain segments can be “pushed out of the way” or not. This provides a facile new means for manipulating the properties of these materials.

Figure 1

(a) Schematic of polymer brush structure in GNP assemblies. (b) Transmission electron microscopy (TEM) image for GNP melts with molecular weight $M{W}_n=80\mathrm{kDa}$ and grafting density $\sigma \approx 0.47\mathrm{chains}/{\mathrm{nm}}^2$. (c) SAXS patterns for GNPs with increasing $M{W}_n$ (bottom—31 kDa; top—132 kDa, hollow symbols). The data were modeled as a product of a polydisperse hard sphere form factor (with known radius and dispersity) with a monodisperse Percus-Yevick hard sphere structure factor, with two parameters $R_{\mathrm{HS}}$ and ${\varphi }_{\mathrm{HS}}$, the hard sphere radius and volume fraction (solid lines). (d) Estimated GNP radius, $R_{\mathrm{HS}}$ (symbols) and theoretically predicted radius $R_p=R_c+h_{\mathrm{dry}}+h_{\mathrm{inter}}/2$ (solid line, see panel (a); [14]) plotted vs $M{W}_n$ of the polymer graft. The dashed line is $R_{\mathrm{dry}}=R_c+h_{\mathrm{dry}}$.

Figure 2

(a) Anti-Stokes Brillouin light scattering (BLS) spectra at two different scattering wave vectors $q$’s for a representative GNP melt with graft molecular weight, $M{W}_n=41\mathrm{kDa}$. The incident angle is $\alpha$, the phonon wave vector, $\mathbf{q}={\mathbf{k}}_{\mathbf{s}}-{\mathbf{k}}_{\mathbf{i}}$ is defined by the wave vectors of the scattered ($k_s$) and incident ($k_i$) light with wavelength $\lambda$ ($=532\mathrm{nm}$). The BLS spectra represented (red lines) by single Lorentzian line are recorded at two different $q$ directions: in-plane with $q=0.001{\mathrm{nm}}^{-1}$ (upper, transmission) and $q=0.0304{\mathrm{nm}}^{-1}$ (lower panel, reflection geometry). (b) The corresponding acoustic dispersion relation obtained from the BLS spectra in (a), and the shaded blue region indicates the spectra collected in the reflection geometry [lower panel in (a)]. For comparison, the acoustic dispersion relation of $M{W}_n=132\mathrm{kDa}$ is also shown. (c),(d) Experimental longitudinal $c_L$ (c) and transverse $c_T$ (d) sound velocities for different molecular weights at $\sigma \approx 0.47\mathrm{chains}/{\mathrm{nm}}^2$ for PMA grafted ${\mathrm{SiO}}_2$ GNP. The blue circle and red square points indicate the trend with ${\varphi }_{\mathrm{NP}}$ (top axis) and $M{W}_n$ (bottom axis), respectively. The blue dotted line indicate the Wood’s law representation of the experimental $c_L$ and $c_T$ vs ${\varphi }_{\mathrm{NP}}$ in (c),(d). Red dashed lines are guides to the eye.

Figure 3

(a) Bulk ($K$), Young’s ($E$), shear ($G$), and longitudinal ($M$) moduli of GNPs GNP’s computed from the sound velocities of Figs. 2 and 2. The dashed line is the theoretical prediction based on the effective medium Wood’s law and the shaded area is for the structural transition. (b) The modulus ratio for the dry region and bulk polymer, $M_{\mathrm{dry}}/M_P$, computed from the Wood’s law assuming either a three-layer (core, dry and interpenetrated regions) or closely packed hard sphere (core, dry region) contribution indicated by the gray and blue filled symbols, respectively. In the hard sphere model, the polymer grafts of the interpenetrated region occupy the interstitial (void) regions schematically indicated (in blue) in the inset. Modulus data for chain lengths which deviate from Wood’s law are shown in the graph. (c) Hypersonic sound absorption $\mathrm{\Gamma }/q^2$ in GNPs for different graft $M{W}_n$. The black dashed line denotes the sound absorption in the bulk PMA matrix, whereas the red dashed is guide to the eye.