Fluctuation-Dissipation Theorem in an Aging Colloidal Glass

We provide a direct experimental test of the fluctuation-dissipation theorem (FDT) in an aging colloidal glass. The use of combined active and passive microrheology allows us to independently measure both the correlation and response functions in this nonequilibrium situation. Contrary to previous reports, we find no deviations from the FDT over several decades in frequency (1 Hz–10 kHz) and for all aging times. In addition, we find two distinct viscoelastic contributions in the aging glass, including a nearly elastic response at low frequencies that grows during aging.

Figure 1

The displacement power spectral densities (PSD) as a function of frequency for $1.16\mu \mathrm{m}$ silica probe particles with increasing age after preparing the sample. Fluctuations were recorded for 45 seconds with the 830 nm laser focus and results averaged in $x,y$ directions for 1 bead several times. Aging times are given in the legend. The solid squares show the PSD of a bead in pure water for comparison. An acoustic noise signal around $f\approx 200\mathrm{Hz}$ is cut out from the curve at the latest stage of aging where the displacement signal was the lowest. All experiments were performed at $21°\mathrm{C}$.

Figure 2

Comparison of active and passive results: Real ${\alpha }^{\prime }(\omega )$ and imaginary ${\alpha }^{\prime \prime }(\omega )$ at $t_a=100$ and 300 min obtained from active (solid symbols) and passive (open symbols) microrheology performed on the same $1.16\mu \mathrm{m}$ diameter silica bead in the same sample. For the passive experiments, the imaginary parts of the response functions are obtained directly and real parts are calculated with a Kramers-Kronig integral. The lines show the fits to Eq. (2). At early stages of aging the data can be described with one power law, while at later stages, a superposition of two power laws is needed to describe the whole frequency range. The amplitude of oscillation for the active experiments was 77 nm.

Figure 3

The comparison of ${\alpha }^{\prime \prime }$ extracted from passive (open symbols) and active (solid symbols) measurements as a function of aging time. For the lowest frequency, we could not measure reliable data longer than 300 min, since the signal to noise to ratio became of the order of 1 with increasing aging time, as the material became stiff.

Figure 4

For $t_a<150\min $, the complex response function can be described by single power law of the form $1/(-ı\omega {)}^a$. After about 150 min, a slowly relaxing contribution emerges leading to a response function of the from in Eq. (2).