Aging in dense suspensions of soft thermosensitive microgel particles studied with particle-tracking microrheology

Using particle tracking microrheology, we studied the glass transition in dense suspensions of thermosensitive microgel particles. These suspensions can be tuned reversibly between the glass state at low temperature and the liquid state at high temperature. In the glass state, the ensemble averaged mean squared displacements (MSDs) of added fluorescent tracer particles depend on the age of the suspension. We also determine the local viscoelastic moduli, $G^{\prime }$ and $G^{″}$, from the MSDs using the Generalized Stokes-Einstein Relation and compare them to the bulk moduli, measured using conventional rheometry. With particle tracking, one probes the viscoelastic moduli in a lower frequency range than with macrorheology, which makes it possible to determine the mean relaxation time that is inaccessible with macrorheology. In the glass state, the mean relaxation time increases linearly with the age of the sample and the short time particle displacement distributions are non-Gaussian, indicating inhomogeneity of the system. The observed difference between conventional and microrheology is explained quantitatively assuming that the tracer particles are surrounded by a viscoelastic liquid shell, different from the bulk.

Figure 1

(Color online) The radius of gyration $R_g$ of a polyNipam-polyNipmam microgel particle as a function of the temperature $T$ measured with static light scattering. The line is drawn to guide the eye.

Figure 2

(Color online) The mean squared displacement $⟨\Delta r^2⟩$ of the probe particles measured for $c=4\mathrm{\%}w/w$ at different temperatures (from top to bottom $32°\text{C}$, $31°\text{C}$, $30°\text{C}$, and $27°\text{C}$, with $\varphi =1.18$, 1.28, 1.37, and 1.56, respectively) and $t_w=300\text{s}$. The open squares indicate the detection limit of our CSLM setup.

Figure 3

(Color online) The MSD $⟨\Delta r^2⟩$ as a function of time at $30°\text{C}$ (○, $\varphi =1.37$) and $31°\text{C}$ (▽, $\varphi =1.28$) measured after $t_w=300\text{s}$ (open symbols) and $t_w=3000\text{s}$ (filled symbols).

Figure 4

(Color online) Panel a: the MSD ($c=4\mathrm{\%}w/w$ and $T=27°\text{C}$, $\varphi =1.56$) measured at several waiting times [300 s (filled ○), 800 s (filled △), 1300 s (filled ▽), 1800 s (filled ◇), 3000 s (filled ◻), 3500 s (open ◻), 4000 s (open ▽), and 4500 s (open ◇)]. Panel b: the same data but the MSDs and $t-t_w$ have been normalized with ${⟨\Delta r^2⟩}_p$ and the crossover time $\tau$, respectively. Inset: $\tau$ as a function of the waiting time $t_w$ in units of 1000 s.

Figure 5

(Color online) Panel a: the displacement probability $\left[P\left(\Delta y\right)\right]$ at $31°\text{C}$ (◇, $\varphi =1.28$) and $32°\text{C}$ (△, $\varphi =1.18$) taken at $t_w=300\text{s}$ and $t-t_w=10\text{s}$ and compared to Gaussian fits (lines). Panel b: $P\left(\Delta y\right)$ at $27°\text{C}$ (○, $\varphi =1.56$) and $30°\text{C}$ (△, $\varphi =1.37$) taken at $t_w=300\text{s}$ and $t-t_w=10\text{s}$ and compared to a double Gaussian (lines). The dotted lines indicate the mobile and immobile populations. Panel c: $\left[P\left(\Delta y\right)\right]$ at $27°\text{C}$ $\left(\varphi =1.56\right)$ taken at $t_w=300\text{s}$ and $t-t_w=10\text{s}$ (closed symbols) and $t-t_w=100\text{s}$ (open symbols). At $t-t_w=10\text{s}$, $P\left(\Delta y\right)$ is well described with a double Gaussian while at $t-t_w=100\text{s}$, can be described with one Gaussian.

Figure 6

(Color online). The non-Gaussian parameter ${\alpha }_2$ $\left(t_w=300\text{s}\right)$ at several temperatures: $27°\text{C}$ (◻, $\varphi =1.56$), $30°\text{C}$ (○, $\varphi =1.37$), $31°\text{C}$ (△, $\varphi =1.28$), and $32°\text{C}$ $\left(◁,\varphi =1.18\right)$.

Figure 7

(Color online) The normalized $G^{\prime }$ (open symbols) and the $G^{″}$ (filled symbols) of a $4\mathrm{\%}w/w$ suspension ($T=27°\text{C}$, $\varphi =1.56$) at several waiting times [300 s (○), 800 s (△), 1300 s (▽), 1800 s (◇), 3000 s (◻), 3500 s (○), 4000 s (△), and 4500 s (▽)] calculated from Fig. 4, using Mason’s approximation of the Generalized Stokes-Einstein Relation.

Figure 8

(Color online) Elastic (filled symbols) and loss modulus (open symbols) measured at $32°\text{C}$ (a) and $27°\text{C}$ (b) corresponding to $\varphi =1.18$ and 1.56, respectively. The waiting times are: 3 s (○), 30 s (◻), 300 s (▽), 1000 s (△), and 3000 s $(◁)$. The lines represent the SGR predictions. At $32°\text{C}$, the moduli have been plotted vs $\omega$ and at $27°\text{C}$ vs $\omega t$, to obtain a collapse to master curves. (Panel b, for $\omega t>200$, see text.)

Figure 9

(Color online) Comparison between the macro-(○) and microrheology (△) at $T=27°\text{C}$ $\left(\varphi =1.56\right)$ and $t_w=1300\text{s}$. The open (filled) symbols represent the measured $G^{\prime }$ $\left(G^{″}\right)$ values. The blue dashed curves are the results of the calculation, described in Sec. 4B, using the analytic representation of the macro results (full red lines) as input.

Figure 10

Cell model to calculate the drag force $F_d$ on the probe particle with radius $a$, due to an oscillating velocity field $U$. The surrounding shell, with viscosity ${\eta }_{\text{shell}}^{\ast }$, has a radius $b$ $\left(\beta =b/a\right)$.