^{1}, G. M. Wang

^{1}, D. R. M. Williams

^{2}, Stephen R. Williams

^{1}, Denis J. Evans

^{1}and E. M. Sevick

^{1,a)}

### Abstract

Physical systems often respond on a timescale which is longer than that of the measurement. This is particularly true in soft matter where direct experimental measurement, for example in force spectroscopy, drives the soft system out of equilibrium and provides a non-equilibrium measure. Here we demonstrate experimentally for the first time that equilibrium physical quantities (such as the mean square displacement) can be obtained from non-equilibrium measurements via umbrella sampling. Our model experimental system is a bead fluctuating in a time-varying optical trap. We also show this for simulated force spectroscopy on a complex soft molecule—a piston-rotaxane.

I. INTRODUCTION

II. EXPERIMENTAL FORCE SPECTROSCOPY OF A COLLOIDAL PARTICLE

III. SIMULATED FORCE SPECTROSCOPY OF A PISTON-ROTAXANE MOLECULE

### Key Topics

- Relaxation times
- 10.0
- Molecular spectroscopy
- 6.0
- Colloidal systems
- 5.0
- Optical spectroscopy
- 5.0
- Optical tweezers
- 5.0

##### F03H3/00

## Figures

(a) Schematic diagram of trap strength, *k*, versus *t*, illustrating the experimental protocol to generate non-equilibrium trajectories of a colloidal particle. The protocol involves cycling the trap strength between *k* _{0}, a weak trap where particle fluctuations are large, to *k* _{1}, a strong trap where fluctuations are reduced. (b) Experimental trace of particle's displacement from the trap centre versus time (red lines) against the time-dependent trapping strength, *k* (blue lines).

(a) Schematic diagram of trap strength, *k*, versus *t*, illustrating the experimental protocol to generate non-equilibrium trajectories of a colloidal particle. The protocol involves cycling the trap strength between *k* _{0}, a weak trap where particle fluctuations are large, to *k* _{1}, a strong trap where fluctuations are reduced. (b) Experimental trace of particle's displacement from the trap centre versus time (red lines) against the time-dependent trapping strength, *k* (blue lines).

Scaled MSD of an optically trapped particle versus time. The averages are constructed from a set of over 15 000 experimental trajectories where the trapping constant increases linearly in time, from *k* _{0} to *k* _{1} in *t* = 25 ms. The open symbols result from simple averages of particle positions along the non-equilibrium trajectories, or 〈*x* ^{2}〉_{ k(t), t } scaled by the expected equilibrium mean, *k* _{ B } *T*/*k*(*t*), where The filled symbols are similarly scaled averages constructed from umbrella sampling or 〈*x* ^{2}(*t*)〉_{ eq, k(t)}: that is, the mean is constructed from the same particle positions used in 〈*x* ^{2}〉_{ k(t), t }, but where the non-equilibrium bias of each measure, Eq. (2), is removed. In this and other figures, the error in the mean corresponds to the standard deviation in data.

Scaled MSD of an optically trapped particle versus time. The averages are constructed from a set of over 15 000 experimental trajectories where the trapping constant increases linearly in time, from *k* _{0} to *k* _{1} in *t* = 25 ms. The open symbols result from simple averages of particle positions along the non-equilibrium trajectories, or 〈*x* ^{2}〉_{ k(t), t } scaled by the expected equilibrium mean, *k* _{ B } *T*/*k*(*t*), where The filled symbols are similarly scaled averages constructed from umbrella sampling or 〈*x* ^{2}(*t*)〉_{ eq, k(t)}: that is, the mean is constructed from the same particle positions used in 〈*x* ^{2}〉_{ k(t), t }, but where the non-equilibrium bias of each measure, Eq. (2), is removed. In this and other figures, the error in the mean corresponds to the standard deviation in data.

Scaled MSD of an optically trapped particle at the completion of (a) confinement protocol and (b) release protocol at different protocol rates, Open circles are scaled non-equilibrium averages, 〈*x* ^{2}〉_{ k(t), t }, that depend strongly upon experimental protocol; filled circles are scaled equilibrium averages 〈*x* ^{2}〉_{ eq, k(t)}, constructed using non-equilibrium umbrella sampling. All quantities are scaled by the experimentally determined equilibrium mean square displacement, 〈*x* ^{2}〉_{ eq }, which by equipartition, 〈*x* ^{2}〉_{ eq } = *k* _{ B } *T*/*k*, is also used to determine the trap constant, *k*. These experimental averages are constructed from 7.8 − 16 × 10^{3} experimental trajectories with *k* _{0} of 1.3–1.5 pN/μm and *k* _{1} between 4.5–4.8 pN/μm.

Scaled MSD of an optically trapped particle at the completion of (a) confinement protocol and (b) release protocol at different protocol rates, Open circles are scaled non-equilibrium averages, 〈*x* ^{2}〉_{ k(t), t }, that depend strongly upon experimental protocol; filled circles are scaled equilibrium averages 〈*x* ^{2}〉_{ eq, k(t)}, constructed using non-equilibrium umbrella sampling. All quantities are scaled by the experimentally determined equilibrium mean square displacement, 〈*x* ^{2}〉_{ eq }, which by equipartition, 〈*x* ^{2}〉_{ eq } = *k* _{ B } *T*/*k*, is also used to determine the trap constant, *k*. These experimental averages are constructed from 7.8 − 16 × 10^{3} experimental trajectories with *k* _{0} of 1.3–1.5 pN/μm and *k* _{1} between 4.5–4.8 pN/μm.

Schematic of a piston-rotaxane molecule showing *N* = 2 free rings threaded onto an axle that is end-tethered to a surface. The top-most ring, referred to as a piston-ring, is optically trapped in force spectroscopy.

Schematic of a piston-rotaxane molecule showing *N* = 2 free rings threaded onto an axle that is end-tethered to a surface. The top-most ring, referred to as a piston-ring, is optically trapped in force spectroscopy.

Scaled average force on the piston ring at the completion of (a) compression and (b) expansion protocol at different rates, of an *N* = 2 piston rotaxane. Open circles are scaled non-equilibrium averages, filled circles are scaled equilibrium averages constructed from non-equilibrium umbrella sampling, All averages are scaled by the simulated equilibrium mean force, 〈*f*〉_{ eq } determined at the same fixed trap centre, *x* _{0}(*t*). The compression/expansion rate is given by the speed of the optical trap centre, where distance and time is in simulation units of ring thickness, *d* and the diffusion time for a ring to diffuse a distance equal to its thickness, or *d* ^{2}ξ/(*k* _{ B } *T*).

Scaled average force on the piston ring at the completion of (a) compression and (b) expansion protocol at different rates, of an *N* = 2 piston rotaxane. Open circles are scaled non-equilibrium averages, filled circles are scaled equilibrium averages constructed from non-equilibrium umbrella sampling, All averages are scaled by the simulated equilibrium mean force, 〈*f*〉_{ eq } determined at the same fixed trap centre, *x* _{0}(*t*). The compression/expansion rate is given by the speed of the optical trap centre, where distance and time is in simulation units of ring thickness, *d* and the diffusion time for a ring to diffuse a distance equal to its thickness, or *d* ^{2}ξ/(*k* _{ B } *T*).

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