Force measurement in the presence of Brownian noise: Equilibrium-distribution method versus drift method

The study of microsystems and the development of nanotechnologies require alternative techniques to measure piconewton and femtonewton forces at microscopic and nanoscopic scales. Among the challenges is the need to deal with the ineluctable thermal noise, which, in the typical experimental situation of a spatial diffusion gradient, causes a spurious drift. This leads to a correction term when forces are estimated from drift measurements [G. Volpe, L. Helden, T. Brettschneider, J. Wehr, and C. Bechinger, Phys. Rev. Lett. 104, 170602 (2010)]. Here we provide a systematic study of such an effect by comparing the forces acting on various Brownian particles derived from equilibrium-distribution and drift measurements. We discuss the physical origin of the correction term, its dependence on wall distance and particle radius, and its relation to the convention used to solve the respective stochastic integrals. Such a correction term becomes more significant for smaller particles and is predicted to be on the order of several piconewtons for particles the size of a biomolecule.

Figure 1

Main techniques of measuring forces at the microscopic and nanoscopic scales, classified according to their force resolution and the working conditions—surface and bulk—for which they are best suited: atomic force microscopy (AFM) [12], photonic force microscopy (PFM) [13, 14, 15], and total internal reflection microscopy (TIRM) [16, 17, 18].

Figure 2

Schematic procedure for measuring a force $F(z)$ by the equilibrium-distribution method. From (a) the measured equilibrium probability distribution $p(z)$ of a particle one obtains (b) the potential-energy distribution $U(z)=-\ln p(z)$, which then gives (c) the force $F(z)=-\frac{\partial }{\partial z}U(z)$ [Eq. (1)].

Figure 3

Schematic view of the propagator $p(\Delta z;z_0,\Delta t)$, which gives the distribution of the particle position increments $\Delta z$ from its initial position $z_0$ after a time step $\Delta t$. For sufficiently small $\Delta t$ the distribution is gaussian with standard deviation $\sqrt{2{\mathit{Dz}}_0\Delta t}$ [Eq. (4)]. The force can be estimated from the measured drift according to Eq. (5).

Figure 4

In the presence of (a) a diffusion gradient, the propagator $p(\Delta z;z_0,\Delta t)$ depends on where $D$ is evaluated, i.e., at (b) $z_{\text{initial}}$, (c) $z_{\text{middle}}$, or (d) $z_{\text{final}}$. In the latter two cases this results in a shift of the position distribution mean—the spurious drift.

Figure 5

Total internal reflection microscopy. (a) Sketch of a typical TIRM geometry with a single colloidal particle in front of a planar transparent wall. The particle is illuminated by the evanescent field created by the total internal reflection of a laser beam at the glass-fluid interface while undergoing Brownian motion. The scattered intensity $I$ is measured using a photomultiplier (PMT). The forces acting on the particle are due to gravity $G_{\mathrm{eff}}$ and electrostatic interactions $F_{\mathrm{el}}$ and are indicated by opposing arrows. When (b) the relationship between the scattered intensity and distance $I(z)$ is known, (c) the intensity time series $I(t)$ can be converted into (d) the trajectory $z(t)$.

Figure 6

Influence of $\Delta t$. At the distance $z_0=220\hspace{0.5em}\mathrm{nm},(\mathrm{a})$ $\Delta t=2\hspace{0.5em}\mathrm{ms}$ leads to a Gaussian distribution of the position increments, but for (b) $\Delta t=20\hspace{0.5em}\mathrm{ms}$, a clear deviation from a Gaussian distribution appears. At $z=400\hspace{0.5em}\mathrm{nm}$, where the spatial variation of the drift is much smaller, (c) the distribution is again Gaussian for $\Delta t=2\hspace{0.5em}\mathrm{ms}$ and (d) remains almost Gaussian also for $\Delta t=20\hspace{0.5em}\mathrm{ms}$.

Figure 7

Comparison of measured (solid circles) and calculated (line) normalized vertical diffusion coefficient $D_{\perp }/D_{\mathrm{SE}}$ for an $R=400\hspace{0.5em}\mathrm{nm}$ particle as a function of the particle-wall separation $z$.

Figure 8

Force derived by the equilibrium-distribution method (squares) for the $400$-$\mathrm{nm}$-radius particle (see Table ) and theoretical expectation (line) according to Eq. (6). The inset shows the corresponding potential $U(z)$.

Figure 9

Force derived by the drift method for the very same particle as in Fig. 8. The force measured according to Eq. (5) with $\alpha =1$ (solid circles) coincides with the theory (line), while with $\alpha =0$ (circles) and $\alpha =0.5$ (triangles) there is a clear disagreement.

Figure 10

Comparison of forces obtained for the $655$-$\mathrm{nm}$-radius particle (see Table ). For the equilibrium-distribution method (solid squares) and drift method with correction term (open squares) the results are in agreement with the theoretical expectation (line) according to Eq. (6). Neglecting the correction term leads to a clear deviation (solid circles). These data are the same as those presented in Ref. [9].

Figure 11

Comparison of forces obtained for the $1180$-$\mathrm{nm}$-radius particle (see Table ). For the equilibrium-distribution method (solid squares) and the drift method with the correction term (open squares) the results are in agreement with the theoretical expectation (line) according to Eq. (6). Neglecting the correction term leads to a clear deviation (solid circles).

Figure 12

Distance dependence of the theoretically calculated spurious force $\alpha \gamma \frac{d}{\mathit{dz}}D$ with $\alpha =1$ for various particle radii $R$ (solid lines). Experimental data are shown for $R=400\hspace{0.5em}\mathrm{nm}$ (circles; cf. Fig. 9), $R=655\hspace{0.5em}\mathrm{nm}$ (squares; cf. Fig. 10), and $R=1180\hspace{0.5em}\mathrm{nm}$ (triangles; cf. Fig. 11).

Figure 13

Distance dependence of the spurious drift $\frac{d}{\mathit{dz}}D$ for various particle radii $R$ (solid lines). Experimental data are shown for $R=400\hspace{0.5em}\mathrm{nm}$ (circles; cf. Fig. 9), $R=655\hspace{0.5em}\mathrm{nm}$ (squares; cf. Fig. 10), and $R=1180\hspace{0.5em}\mathrm{nm}$ (triangles; cf. Fig. 11).