Energy dissipation of a Brownian particle in a viscoelastic fluid

We evaluate the energy dissipation rate of an optically driven Brownian particle in a polymer solution utilizing the generalized version of Harada and Sasa’s equality [Phys. Rev. Lett. 95, 130602 (2005)] by Deutsch and Narayan [Phys. Rev. E 74, 026112 (2006)]. The irreversible work of a small system is estimated from readily obtainable quantities. By adopting the time-dependent memory function obtained by microrheology measurement, directly obtained works are in excellent agreement with those calculated from the generalized fluctuation dissipation theorem for nonequilibrium steady states. This result implies that the colloidal particle in a polymer solution can be described by the generalized Langevin equation.

Figure 1

Measurement of response function. (a) We shifted the tweezer position with a distance of $d=72.2\mathrm{nm}$ at $t=0$ and observed the relaxation processes to the new tweezer position. We repeated this and obtained an ensemble average. (b) Similarly in nonequilibrium steady states (NESSs), we added a bias $d$ and observed the relaxation processes.

Figure 2

Histogram of particle positions trapped in a single fixed tweezer and the corresponding potential profile estimated using the Boltzmann distribution. The number of the total count was $1015777$. The dashed line is the fitting curve of the harmonic function $U\left(x\right)=(1∕2)k{x}^2$ with a spring constant of $k=3373\pm 8k_{B}T∕\mathrm{\mu }{\mathrm{m}}^2=13.72\pm 0.03\mathrm{pN}∕\mathrm{\mu }\mathrm{m}$.

Figure 3

(a) The correlation function of the particle position. 51 trajectories of $69.1\mathrm{s}$ were averaged. The dashed line is a fitting curve $p∕({\omega }^{\varphi }+q)$ with $p=4.10\times {10}^{-16}$, $q=1.96$, $\varphi =1.41$. The rising of the spectrum in the high frequency region $(>1000\mathrm{rad}∕\mathrm{s})$ is due to the systematic error and aliasing effect. The data were smoothed with an exponentially increasing window. (b) Relaxation curve in equilibrium state (EQ). We transferred the tweezer position a small amount $(d=72.2\mathrm{nm})$ and observed the relaxation process to the new tweezer position. 1040 of $5\mathrm{s}$ trajectories were averaged. The dashed line is the inverse Fourier-Cos transform of the fitted line in (a), which corresponds to $1-k{C}_x\left(t\right)$. The shape of the relaxation curve can be well fitted by a stretched exponential function (data not shown), $⟨x\left(t\right){⟩}_{\epsilon }∕d=1-\exp [-(t∕\tau {)}^{\beta }]$ with $\beta =0.494\pm 0.002$ and $\tau =0.169\pm 0.001\mathrm{s}$.

Figure 4

The complex friction coefficient $\tilde{\gamma }\left(\omega \right)\equiv {\tilde{\gamma }}^{\prime }\left(\omega \right)-i{\tilde{\gamma }}^{″}\left(\omega \right)$. The steep drop of values in the high frequency region is due to our insufficient bandwidth for ${\tilde{C}}_x\left(\omega \right)$. The dashed line is the friction coefficient for pure water $9.273\times {10}^{-9}\mathrm{kg}∕\mathrm{s}$ at $20°\mathrm{C}$ for a particle with a diameter of around $1\mathrm{\mu }\mathrm{m}$. The dotted line is a fitting curve for ${\tilde{\gamma }}^{\prime }\left(\omega \right)$ in a frequency region from 10 to $300\mathrm{rad}∕\mathrm{s}$, which had a slope of $-0.454\pm 0.006$.

Figure 5

Typical trajectories for EQ and NESSs with switching rates of $\lambda =4$ and $12\mathrm{events}∕\mathrm{s}$. Black lines are trajectories of the particle. Gray lines are trajectories of the tweezer.

Figure 6

The correlation function of the velocity $\tilde{C}\left(\omega \right)$. We see that $\tilde{C}\left(\omega \right)$ in NESSs have higher spectrum than that in EQ. In the high frequency region, they converged to the same curve. We averaged 51 trajectories of $69.1\mathrm{s}$ each for each curve. The data were smoothened with an exponentially increasing window.

Figure 7

Relaxation curves in equilibrium state (EQ) and nonequilibrium states (NESSs). The dashed line is the same as that in Fig. 3b. We transferred the tweezer position a small amount $(d=72.2\mathrm{nm})$ and observed the relaxation process to the new tweezer position. 1040 of $5\mathrm{s}$ trajectories were averaged for each curve. We found that relaxation curves for EQ and NESSs coincided well.

Figure 8

(Color online) $\tilde{I}\left(\omega \right)$ (black) vs $\tilde{C}\left(\omega \right)-2k_{\mathrm{B}}T{\tilde{R}}^{\prime }\left(\omega \right)$ (blue). We plotted $\tilde{C}\left(\omega \right)-{\tilde{C}}_{\mathrm{EQ}}\left(\omega \right)$ instead of $\tilde{C}\left(\omega \right)-2k_{\mathrm{B}}T{\tilde{R}}^{\prime }\left(\omega \right)$ (see the main text). We found a significant deviation in the shape of them. The left and right vertical axes are for $\tilde{I}\left(\omega \right)$ and $\tilde{C}\left(\omega \right)-2k_{\mathrm{B}}T{\tilde{R}}^{\prime }\left(\omega \right)$, respectively.

Figure 9

(Color online) Both sides of the local equality [Eq. (4)] at each frequency. $\tilde{I}\left(\omega \right)$ (black) vs ${\tilde{\gamma }}^{\prime }\left(\omega \right)$ $\left[\tilde{C}\right(\omega )-2k_{\mathrm{B}}T{\tilde{R}}^{\prime }(\omega \left)\right]$ (blue).

Figure 10

The left-hand side (squares) and right-hand side (circles) of Deutsch and Narayan’s equality [Eq. (2)] were plotted against the switching rate. We have integrated both sides of the equality up to a frequency of $400\mathrm{rad}∕\mathrm{s}$ to avoid artifactual noise at high frequency (see the main text). For the right-hand side, we used ${\tilde{C}}_{\mathrm{EQ}}\left(\omega \right)$ instead of $2k_{\mathrm{B}}T{\tilde{R}}^{\prime }\left(\omega \right)$. The error bars were smaller than the size of the marks.