High-bandwidth viscoelastic properties of aging colloidal glasses and gels

We report measurements of the frequency-dependent shear moduli of aging colloidal systems that evolve from a purely low-viscosity liquid to a predominantly elastic glass or gel. Using microrheology, we measure the local complex shear modulus $G^{*}\left(\omega \right)$ over a very wide range of frequencies (from $1\mathrm{Hz}\text{to}100\mathrm{kHz}$). The combined use of one- and two-particle microrheology allows us to differentiate between colloidal glasses and gels—the glass is homogenous, whereas the colloidal gel shows a considerable degree of heterogeneity on length scales larger than $0.5\mu \mathrm{m}$. Despite this characteristic difference, both systems exhibit similar rheological behaviors which evolve in time with aging, showing a crossover from a single-power-law frequency dependence of the viscoelastic modulus to a sum of two power laws. The crossover occurs at a time $t_0$, which defines a mechanical transition point. We found that the data acquired during the aging of different samples can be collapsed onto a single master curve by scaling the aging time with $t_0$. This raises questions about the prior interpretation of two power laws in terms of a superposition of an elastic network embedded in a viscoelastic background.

Figure 1

Light scattering-intensity reduced with respect to scattered intensity from toluene for two samples: (a) $3.2\mathrm{wt}\%$ Laponite and (b) $0.8\mathrm{wt}\%$ Laponite, $6\mathrm{mM}$ NaCl. The data points are taken at late stages of aging when the scattering intensity has stabilized.

Figure 2

Normalized displacement power spectral densities $⟨{\mid x\left(\omega \right)\mid }^2⟩∕2D_0$ of silica probe particles in a glassy sample ($3.2\mathrm{wt}\%$, bead diameter $1.16\mu \mathrm{m}$) and a gel-like sample ($0.8\mathrm{wt}\%$, $6\mathrm{mM}$ NaCl, bead diameter $0.5\mu \mathrm{m}$) in the $x$ direction with increasing age after preparing the sample. Aging times are given in the legend. The solid squares show the PSD of a bead in pure water for comparison. All experiments were done at $21°\mathrm{C}$.

Figure 3

Glass data: the symbols show the shear moduli $G^{\prime }\left(\omega \right)$ and $G^{″}\left(\omega \right)$ (absolute magnitude) as a function of frequency measured using $1.16\text{−}\mu \mathrm{m}$ silica probe particles in a $3.2\mathrm{wt}\%$ Laponite solution in pure water with increasing aging time after preparing the sample. Aging times are given in the legend. The lines show the fits of $G^{\prime }\left(\omega \right)$ and $G^{″}\left(\omega \right)$ according to $C_1{(-i\omega )}^a+C_2{(-i\omega )}^b$ in which $C_2=0$ for aging times $t_a<120\min$.

Figure 4

Gel data: the symbols show the shear moduli $G^{\prime }\left(\omega \right)$ and $G^{″}\left(\omega \right)$ (absolute magnitude) as a function of frequency measured using a $0.5\text{−}\mu \mathrm{m}$ silica probe particle in a $0.8\mathrm{wt}\%$ Laponite solution in $6\mathrm{mM}$ NaCl water with increasing aging time after preparing the sample. Aging times are given in the legend. The lines show the fits of $G^{\prime }\left(\omega \right)$ and $G^{″}\left(\omega \right)$ according to $C_1{(-i\omega )}^a+C_2{(-i\omega )}^b$ in which $C_2=0$ for aging times $t_a<100\min$.

Figure 5

Glass data: the shear moduli $G^{\prime }\left(\omega \right)$ and $G^{″}\left(\omega \right)$ (magnitude) at two different stages of aging in a $3.2\mathrm{wt}\%$ Laponite solution derived from one-particle (lines) and two-particle (symbols) MR using $1.16\text{−}\mu \mathrm{m}$ silica probe particles. The distance between the two particles was $6\mu \mathrm{m}$. The aging times are shown in the figure. Note that in the late stages of aging the material becomes too stiff to obtain a good cross-correlation signal between the two beads over the background noise.

Figure 6

The displacement PSDs of $0.5\text{−}\mu \mathrm{m}$ silica beads measured at different positions within an aged glass ($3.2\mathrm{wt}\%$ Laponite in pure water, $t_w\approx 5\mathrm{h}$) and within an aged colloidal gel ($0.8\mathrm{wt}\%$ Laponite in $6\mathrm{mM}$ NaCl solution, $t_a\approx 10\mathrm{h}$).

Figure 7

(Color) Local elastic modulus $G^{\prime }$ and loss modulus $G^{″}$ measured at different positions in an aged glassy sample of $3.2\mathrm{wt}\%$ Laponite in pure water $(t_a\approx 5\mathrm{h})$

Figure 8

Elastic and loss modulus as a function of aging time for a sample of $3.2\mathrm{wt}\%$ Laponite in pure water obtained from macrorheology and one-particle MR at $f=0.7\mathrm{Hz}$. The strain amplitude in the macrorheology measurements was 0.01.

Figure 9

(Color) Gel data: local elastic modulus $G^{\prime }$ and loss modulus $G^{″}$ measured at different positions in an aged gel sample of $0.8\mathrm{wt}\%$ Laponite in $6\mathrm{mM}$ NaCl solution $(t_a\approx 10\mathrm{h})$.

Figure 10

Gel data: the shear moduli $G^{\prime }\left(\omega \right)$ and $G^{″}\left(\omega \right)$ at different stages of aging derived from one- and two-particle MR of $0.5\text{−}\mu \mathrm{m}$ silica probe particles in a gel of $0.8\mathrm{wt}\%$ Laponite in $6\mathrm{mM}$ NaCl solution. The distance between the two beads was $4.66\mu \mathrm{m}$. For $t_a=129\min$ the shear moduli at two different positions are equal. At $t_a=205\min$ the shear moduli at the two positions are not equal. Furthermore, shear modulus at position 2 shows anisotropy. At a later time $t_a=510\min$ the shear moduli at the two positions are not equal, but the anisotropy observed earlier at position of bead 2 has disappeared.

Figure 11

Complex shear modulus at a late stage of aging $t_a\approx 8.5\mathrm{h}$ obtained from single-particle MR at two different positions of the sample, two-particle MR, and bulk rheology in a sample of $0.8\mathrm{wt}\%$ Laponite, $6\mathrm{mM}$ NaCl. The circles show $G^{\prime }$ and triangles show $G^{″}$ values.

Figure 12

The complex shear moduli of Laponite suspensions can be described as the sum of two power laws $C_1{(-i\omega )}^a+C_2{(-i\omega )}^b$ in which $C_2=0$ for waiting times $t_a<t_0$. The crossover times are $t_0=155,95,120,105,95\min$ for Laponite concentrations $2.8\mathrm{wt}\%$, $3\mathrm{wt}\%$, $p\mathrm{H}=10$, $3.2\mathrm{wt}\%$, $1.5\mathrm{wt}\%$, $5\mathrm{mM}$ NaCl and two different positions of $0.8\mathrm{wt}\%$, $6\mathrm{mM}$ NaCl, respectively. (a) The evolution of power-law exponents $a$ (solid symbols) and $b$ (open symbols) as a function of aging time for different concentrations of Laponite. (b) The exponents $a$ (solid symbols) and $b$ (open symbols) as a function of scaled aging time (c) The amplitude of viscoelastic contributions $C_1$ (solid symbols) and $C_2$ (open symbols) as a function of aging for different samples. (d) The same as panel (c) but plotted versus scaled aging time. The sample concentrations are shown in the legend.

Figure 13

(Color online) Bold symbols: the elastic modulus obtained from macrorheology at late stages of aging at the aging time that there is a crossover from a fast regime of aging to a slower regime as a function of concentration measured for different glassy samples and a gel sample of $0.8\mathrm{wt}\%$, with $6\mathrm{mM}$ salt at $f=0.05\mathrm{Hz}$. Open symbols: An estimate of the plateau value $G_P^{\prime }\propto kT∕D^3$ is shown for comparison.