Frequency-dependent complex viscosity obtained for a liquid two-dimensional dusty plasma experiment

Strongly coupled plasmas in a liquid phase can be characterized by a complex viscosity $\eta \left(\omega \right)$, which is a function of frequency. Data from a single experiment with dusty plasma were analyzed to compare $\eta \left(\omega \right)$ obtained by two fundamentally distinct methods. In a nonequilibrium method, a pair of counterpropagating laser beams, separated by a gap, applied a sinusoidal shear to a two-dimensional liquid, and $\eta \left(\omega \right)$ was determined using the constitutive relation. In an equilibrium method, there was no externally applied shear, so $\eta \left(\omega \right)$ could be calculated with a generalized Green-Kubo relation. The results for these two methods are compared for the real and imaginary parts of $\eta \left(\omega \right)$. For both parts, it is confirmed that the two methods yield results that agree qualitatively in their trends with frequency, with the real part diminishing with $\omega$ and the imaginary part increasing with $\omega$, as expected for viscoelastic liquids. Quantitatively, the values of $\eta \left(\omega \right)$ obtained by the two methods differ slightly. For the experiment that we analyze, values for the real and imaginary parts of $\eta \left(\omega \right)$ are substantially greater than those reported in an earlier experiment, which we attribute to shear thinning effects in the earlier experiment. The experiment we analyze was designed to minimize shear thinning, unlike the earlier experiment.

Figure 1

Images from the top-view camera for liquid runs (a) with and (b) without an externally applied shear. The camera's field of view of $24\times 32\mathrm{mm}$, shown here, was centered in the dust particle cloud, which had a diameter of 64 mm. In (a), the broad arrows indicate the position and width of the two shear laser beams, which could be sinusoidally modulated at an externally controlled frequency $\omega$. These beams were the source for the externally driven shear that made possible the use of the nonequilibrium method of analysis. The 2-mm gap between the shear beams was also the region of interest (ROI) for analyzing the hydrodynamic flow at the applied frequency $\omega$. In (b), the analysis region for the equilibrium method was the inner rectangle defined by a gray border that has a thickness equal to $r_{\text{cut}}$. No shear beams were used for the runs analyzed with the equilibrium method. The position and thickness of the arrows in (a), as well as the width of the gray border in panel (b), are drawn to scale

Figure 2

Comparison of nonequilibrium experimental results. Data from the earlier experiment of Hartmann et al. [24] are plotted as solid symbols. Data points plotted as open triangles are from our analysis of the authors' recent experiment of Ref. [43]; each data point is the mean of the results for ten runs, with an error bar indicating the standard deviation of the mean. For normalization, viscosity is divided by $\rho a_{\text{ws}}^2{\omega }_{\text{pd}}$ and frequency is divided by the nominal 2D dust plasma frequency ${\omega }_{\text{pd}}$. While data from the two experiments generally have the same trends, there is a significant quantitative difference: Values for both ${\eta }^{\prime }\left(\omega \right)$ and ${\eta }^{\prime \prime }\left(\omega \right)$ are smaller by at least a factor of 1.4 in the earlier experiment of Hartmann et al., which may be attributed to shear thinning in the earlier experiment.

Figure 3

Frequency-dependent complex viscosity: (a) real component ${\eta }^{\prime }\left(\omega \right)$ and (b) imaginary component ${\eta }^{\prime \prime }\left(\omega \right)$. Circles represent equilibrium-method results obtained using Eqs. (15) and (16). Scatter of the four data points, each for one run, reflects the uncertainty. Triangles show nonequilibrium-method results, using Eqs. (12) and (13), as in Fig. 2. The static viscosity ${\eta }^0$ is indicated by the data points at $\omega =0$. These results are our analysis of data from the experiment of Ref. [43].