Two-Particle Microrheology of Quasi-2D Viscous Systems

We study the spatially correlated motions of colloidal particles in a quasi-2D system (human serum albumin protein molecules at an air-water interface) for different surface viscosities ${\eta }_s$. We observe a transition in the behavior of the correlated motion, from 2D interface dominated at high ${\eta }_s$ to bulk fluid dependent at low ${\eta }_s$. The correlated motions can be scaled onto a master curve which captures the features of this transition. This master curve also characterizes the spatial dependence of the flow field of a viscous interface in response to a force. The scale factors used for the master curve allow for the calculation of the surface viscosity ${\eta }_s$ that can be compared to one-particle measurements.

Figure 1

(a),(b) Two-point correlation functions $D_{rr}(R,\tau )$ and $D_{\theta \theta }(R,\tau )$ for a sample with ${\eta }_{s,1p}=340\mathrm{nPa}\mathrm{s}\mathrm{m}$, with lag times $\tau =0.1$, 0.2, 0.4, and 0.8 s. The evenly spaced correlation functions imply that $D_{rr},D_{\theta \theta }\sim \tau$. Inset: One-particle MSD for the sample (diamonds), where the straight line is a fit giving ${\eta }_{s,1p}=340\mathrm{nPa}\mathrm{s}\mathrm{m}$, using Eq. (2).

Figure 2

Correlation functions $⟨D_{rr}/\tau ⟩$ (solid symbols) and $⟨D_{\theta \theta }/\tau ⟩$ (open symbols) as a function of particle separation $R$, for decreasing surface viscosities ${\eta }_s$. The samples are the following: (a) 2D dominated, ${\eta }_{s,1p}=340\mathrm{nPa}\mathrm{s}\mathrm{m}$ (diamonds); (b) crossover, ${\eta }_{s,1p}=72\mathrm{nPa}\mathrm{s}\mathrm{m}$ (circles); (c) 3D dependent, ${\eta }_{s,1p}=21.3\mathrm{nPa}\mathrm{s}\mathrm{m}$ (squares).

Figure 3

Master curve of scaled variables ${\overline{D}}_{rr}$ (solid symbols) and ${\overline{D}}_{\theta \theta }$ (open symbols) as a function of reduced particle separation $\beta$. Symbols are same as in Fig. 2, with two additional data sets [${\eta }_{s,1p}=2.1\mathrm{nPa}\mathrm{s}\mathrm{m}$ (stars) and ${\eta }_{s,1p}=1275\mathrm{nPa}\mathrm{s}\mathrm{m}$ (triangles)]. Solid lines represent theoretical fits to the data. Inset: One- and two-particle surface viscosities for the five samples shown in the master curve. The straight line has a slope of 1, indicating an equality between the two viscosities.

Figure 4

One-particle (solid symbols) and two-particle (open symbols) MSDs for four of the five samples shown in Fig. 3. Inset: radial scaling of $D_{rr}$ for the different samples at a lag time $\tau =0.53\mathrm{s}$.