Local mechanical properties of a hyperswollen lyotropic lamellar phase

The local and slow dynamics of a colloidal particle dispersed in a hyperswollen lyotropic lamellar phase was studied by passive and active microrheological methods. According to observation by a particle-tracking video microscopy, the motion of a particle follows a normal diffusion process in a lamellar phase with a large interlayer distance. However, trap-diffusion motion was also observed for a small interlayer distance. This characteristic motion was discussed by consideration of the time evolution of mean square displacement, the van Hove self-correlation function and the non-Gaussian parameter. These indicate that the dynamics of a particle becomes spatially heterogeneous in a lamellar phase whose interlayer distance is close to the size of the particle. By the measurement of local viscosity using optical tweezers, Newtonian behavior was observed at a high shear rate. In a lamellar phase with a small interlayer distance, non-Newtonian behavior was also observed at a low shear rate. The dependences of effective viscosity on the interlayer distance obtained by both methods are in agreement. The effective viscosity in the lamellar phase was found to be four to five times larger than that of water even for a large interlayer distance. Possible reasons for the increase in effective viscosity are discussed quantitatively.

Figure 1

(Color online) Schematic figure of a hyperswollen lyotropic lamellar phase. The thickness of the bilayers is $\delta$, and the radius of the colloidal particles dispersed between the bilayers is $a$. The spherical part (online green) of a surfactant molecule represents the hydrophilic part and the tail (online brown) represents the hydrophobic part. The interlayer distance $d$ can be controlled by varying the concentration of the surfactant and was set to be larger than $2a$. The membranes in a hyperswollen lyotropic lamellar phase undulate markedly owing to their thermal fluctuation.

Figure 2

(Color online) Examples of two-dimensional trajectories of a single colloidal particle in a hyperswollen lyotropic lamellar phase (left) and the time evolution of its $y$ position (right). (a) and (b): $d=150\text{nm}$. (c) and (d): $d=60\text{nm}$. The estimated potential depth of the trapping site enclosed by an oval is about $3.6k_{\text{B}}T$.

Figure 3

(Color online) Dependence of the mean-square displacement (MSD) of a single particle on time lag $\Delta t$ when $d=150$ and 60 nm. The data used to calculate MSD are the same as those used in Fig. 2. The solid lines are the best-fitted lines of $\text{MSD}=A{\left(\Delta t\right)}^{\alpha }$ for a small $\Delta t$. The best-fitted values of $\alpha$ are 1.0 for $d=150\text{nm}$ and 0.79 for $d=60\text{nm}$. The dotted line represents $\text{MSD}\propto \Delta t$.

Figure 4

Dependence of the non-Gaussian parameter ${\alpha }_2$ on time lag $\Delta t$ when $d=60\text{nm}$. The data used to evaluate ${\alpha }_2$ are the same as those used in Fig. 2. The time at which ${\alpha }_2$ is maximum (indicated by an arrow) corresponds to the crossover time from anomalous to normal diffusion in Fig. 3.

Figure 5

(Color online) Dependence of MSD on time lag $\Delta t$ for 100 particles when $d=60\text{nm}$. Particles are classified into two groups named A and B. In group A, MSD increases almost linearly increases with $\Delta t$. In group B, MSD tends to saturate at a certain value.

Figure 6

(Color online) Van Hove self-correlation function $P\left(\Delta x,\Delta t\right)$ at different time lags $\Delta t$. (a) $d=300\text{nm}$, (b) $d=60\text{nm}$. The thin lines are the best-fitted curves assuming a Gaussian distribution. The thick line in (b) is the best-fitted curve using Eq. (3).

Figure 7

(Color online) Dependence of the ensemble-averaged non-Gaussian parameter ${\alpha }_2$ on time lag $\Delta t$ for various interlayer distances $d$.

Figure 8

(Color online) Dependence of MSD averaged over about 100 mobile particles on time lag $\Delta t$. The slope of each line is $\alpha =1.0$. From the intercept of the lines, the self-diffusion coefficient is evaluated at for each interlayer distance $d$.

Figure 9

(Color online) Dependence of the effective viscosity $\eta$ on interlayer distance $d$. ◻: particle-tracking microscopy. ○: optical tweezers. The dotted line indicates $\eta$ for water at $25°\text{C}$.

Figure 10

(Color online) Dependence of the maximum velocity $v_{\text{c}}$ on the laser power $P_2$ (trapping force) when $d=75\text{nm}$. Newtonian behavior is exhibited at higher speeds, whereas, non-Newtonian (shear thinning) is exhibited at lower speeds. Yield stress inducing the motion of the particle and stick-slip motion were observed at low speeds and high powers.