# Ultrasmall Force Measurement

The concept of force plays a central role in our understanding of nature

The concept of force plays a central role in our understanding of nature. Due to the ongoing trend towards miniaturization, the investigation of forces relevant at microscopic and nanoscopic length scales therefore is an important field of research. Over the past decades several methods to measure ultra-small forces with resolution in the sub-pico Newton range have been developed. Total Internal Reflection Microscopy (TIRM) is an extremely sensitive non-invasive technique to measure the interaction potentials between a micron- or submicron sized particle and a wall with femto Newton resolution. The equilibrium distribution of the particle-wall separation distance z is sampled monitoring the intensity I scattered by the Brownian particle under evanescent illumination. Central to the data analysis is the knowledge of the relation between I and the corresponding z, which typically must be known a priori. For purely dielectric particles, an exponential dependence can be assumed, however, severe deviations may occur when e.g. surfaces with metallic coatings are considered.

Apart from the technological challenge intrinsic to dealing with forces in the order of femto Newton, it is important to realize that the general concepts we apply in our everyday life cannot be simply scaled down to microscopic objects, mainly due to the presence of thermal noise affecting the motion of small objects (Brownian motion).

For microscopic objects immersed in a liquid, viscous forces prevail by several orders of magnitude over inertial effects, i.e. they are overdamped. This implicates that a constant force F leads to a constant terminal drift velocity v, so that v = F/γ with γ the friction coefficient. However, things are complicated by the presence of Brownian noise. Indeed, the presence of a spatially varying Brownian noise leads to the presence of spurious forces, i.e. forces that exist only due to and in the presence of thermal noise. If overlooked, this leads to erroneous forces, which may severely affect the physical interpretations of experimental data.

The fact that standard TIRM relies on the a priori knowledge of I(z) poses considerable constraints to the experimental conditions where TIRM can be applied. Besides metallic surfaces which can be easily functionalized with SAMs and hence are interesting for biological purposes, this also affects measurements with high penetration depths which offer the possibility to look further into the medium. Thus a method to in situ determine the correct I(z) without any assumptions on the shape is highly desirable.

Indeed this is possible by utilizing additional information about the system:

- Between the particle and the wall there is a hydrodynamic interaction leading to a distant dependent friction and hence a reduced diffusivity if the particle approaches the wall. The resulting distance dependence of the diffusion coefficient D(z) is well described theoretically by the “Brenner-formula” which has been established in 1961 and has been proven to be correct several times.
- As Brownian motion is a gaussian stochastic process, the propagator p(Δz;z0, Δt) giving the probability of a Brownian increment Δz within a time step Δt, starting at a certain position z0 is perfectly symmetric for sufficiently small time steps Δt.

Given the trajectory z(t) of the particle, it is possible to calculate p(Δz;z0, Δt) and D(z) experimentally.

The algorithm’s basic idea can be summarized as follows:

- Guess an I(z) and determine the putative trajectory z(t).
- Calculate p(Δz;z0, Δt) and D(z) and check if the conditions resulting from 1. and 2. are satisfied.
- Try with modified I(z) until the conditions are fulfilled.

The resulting I(z) can be used to calculate high precision trajectories z(t) on metallic surfaces, high penetration depths and s-polarization to determine the forces between particle and wall or to study Brownian dynamics. Furthermore this technique can be used to double-check standard TIRM measurements or even to automatize the whole TIRM data evaluation process, maybe providing the missing link to a widespread use of TIRM in other fields like biology where autonomous evaluation procedures are highly appreciated.

### Further information

*Novel perspectives for the application of total internal reflection microscopy [101]G. Volpe, T. Brettschneider, L. Helden, C. Bechinger, Opt. Express 17, 23975 (2009)download*

*Influence of noise on force measurements [105]*

G. Volpe, L. Helden, T. Brettschneider, J. Wehr, and C. Bechinger, Phys. Rev. Lett. 104, 170602 (2010)

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