Influence of Noise on Force Measurements

Abstract

We demonstrate how the ineluctable presence of thermal noise alters the measurement of forces acting on microscopic and nanoscopic objects. We quantify this effect exemplarily for a Brownian particle near a wall subjected to gravitational and electrostatic forces. Our results demonstrate that the force measurement process is prone to artifacts if the noise is not correctly taken into account.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 is attracting an increasing amount of attention. Examples range from the elastic properties of biomolecules [1] to Casimir forces [2]. Instrumental in such trend has been the invention of new methods to measure ultra-small forces in a range down to few femtonewtons [3,4]. Apart from the technological challenge intrinsic to measuring such minute forces, it is important to realize that the general concept of how forces are measured in macroscopic systems cannot be simply scaled down to nanoscopic objects, mainly due to the presence of thermal noise affecting the motion of small objects (Brownian motion). As will be demonstrated below, force measurements in the presence of thermal noise are prone to artifacts, unless the noise is properly taken into account. However, despite the great number of experiments measuring forces in microscopic systems, the role of noise has not been adequately addressed yet.In this Letter, we measure the forces acting on a Brownian particle in front of a wall: using two widely employed force-measurement methods, we obtain strongly contrasting forces, which deviate in their magnitude and even their sign. We track this disagreement down to a well-known mathematical controversy about the interpretation of stochastic differential equations in the presence of a diffusion gradient and we demonstrate how experimental data should be analyzed in order to obtain the correct underlying forces and to avoid artefacts.When a microscopic body is suspended in a liquid, viscous forces prevail by several orders of magnitude over inertial effects [5]. The presence of a constant external force F in such an overdamped system results in a constant particle drift velocity v d = F/γ, where γ is the particle's friction coefficient. Since v d = dz/dt can be retrieved from the measured particle displacement dz within time dt, the force can accordingly be measured as F = γv d . In the case of large forces this obviously leads to a univocal result. However, when the force exerted on the object is comparable or even smaller than the random forces due to the permanent collisions with the surrounding liquid molecules, a different value of the force is measured in each experiment. Typically one can average over many independent measurementswhere we definev d = 1 dt dz j and j denotes the j-th experimental value.Indeed Eq. (1) is key in non-equilibrium force measurements, e.g. studying the relaxation of biomolecules [6][7][8], collo...

Hydrodynamics and Brownian motions of a spheroid near a rigid wall

“…42 The existence of this drift velocity must be accounted for when measuring the effect of external potentials on the Brownian motion. 43 Indeed, recent experiments showed that neglecting the drift velocity leads to an erroneous qualitative evaluation of the conservative forces acting on Brownian spheres. 44 We would like to point out that the presence of a "drift" velocity does not mean that there is a net particle flux in the velocity direction.…”

Comment on “Influence of Noise on Force Measurements”

“…In a recent Letter [1], Volpe et al describe experiments on a colloidal particle near a wall in the presence of a gravitational field for which they study the influence of noise on the measurement of force. Their central result is a striking discrepancy between the forces derived from experimental drift measurements via their Eq.…”

“…(1) of [1] because the nonlinearity of the diffusion coefficient enters the drift. Hence, it is impossible to derive the "force" F (z) from a measurement of the drift velocity, as in Eq.…”

Anomalous diffusion in stochastic systems with nonhomogeneously distributed traps

“…However, the other interpretations may also be important. For example, the experimental analysis of the colloidal particles diffusion near the boundary favours AII (λ I = 1) [44], as we have already mentioned. We performed the numerical calculations for AII, similar to those for SI, and found the same diffusion properties in respect both to the exponent and the proportionality coefficient.…”

“…Emergence of the multiplicative noise near a boundary has been experimentally demonstrated for some physical systems. For example, a description of the colloidal particles diffusion in terms of a constant diffusion coefficient appears possible only if a particle remains far from any boundary [44]. Moreover, presence of the multiplicative noise in a description of particles near a wall is necessary to reach a proper thermal equilibrium [45].…”