Two different classes of hairy self-suspended
nanoparticles in
the melt state, polymer-grafted nanoparticles (GNPs) and star polymers,
are shown to display universal dynamic behavior across a broad range
of parameter space. Linear viscoelastic measurements on well-characterized
silica-poly(methyl acrylate) GNPs with a fixed core radius (R
core) and grafting density (or number of arms f) but varying arm degree of polymerization (N
arm) show two distinctly different regimes of response.
The colloidal Regime I with a small N
arm (large core volume fraction) is characterized by predominant low-frequency
solidlike colloidal plateau and ultraslow relaxation, while the polymeric
Regime II with a large N
arm (small core
volume fractions) has a response dominated by the starlike relaxation
of partially interpenetrated arms. The transition between the two
regimes is marked by a crossover where both polymeric and colloidal
modes are discerned albeit without a distinct colloidal plateau. Similarly,
polybutadiene multiarm stars also exhibit the colloidal response of
Regime I at very large f and small N
arm. The star arm retraction model and a simple scaling
model of nanoparticle escape from the cage of neighbors by overcoming
a hopping potential barrier due to their elastic deformation quantitatively
describe the linear response of the polymeric and colloidal regimes,
respectively, in all these cases. The dynamic behavior of hairy nanoparticles
of different chemistry and molecular characteristics, investigated
here and reported in the literature, can be mapped onto a universal
dynamic diagram of f/[R
core
3/ν0)1/4] as a function of
(N
armν0
f)/(R
core
3), where ν0 is the monomeric volume. In this diagram, the two regimes
are separated by a line where the hopping potential ΔU
hop is equal to the thermal energy, k
B
T. ΔU
hop can be expressed as a function of the overcrowding
parameter x (i.e., the ratio of f to the maximum number of unperturbed chains with N
arm that can fill the volume occupied by the polymeric
corona); hence, this crossing is shown to occur when x = 1. For x > 1, we have colloidal Regime I with
an overcrowded volume, stretched arms, and ΔU
hop > k
B
T, while polymeric Regime II is linked to x <
1. This single-material parameter x can provide the
needed design principle to tailor the dynamics of this class of soft
materials across a wide range of applications from membranes for gas
separation to energy storage.