Mechanics and statistics of the worm-like chain

Andrew Marantan, L. Mahadevan

The worm-like chain model is a simple continuum model for the statistical mechanics of a flexible polymer subject to an external force. We offer a tutorial introduction to it using three approaches. First, we use a mesoscopic view, treating a long polymer (in two dimensions) as though it were made of many groups of correlated links or “clinks,” allowing us to calculate its average extension as a function of the external force via scaling arguments. We then provide a standard statistical mechanics approach, obtaining the average extension by two different means: the equipartition theorem and the partition function. Finally, we work in a probabilistic framework, taking advantage of the Gaussian properties of the chain in the large-force limit to improve upon the previous calculations of the average extension.

DOI

An optical levitation system for a physics teaching laboratory

Oscar Isaksson, Magnus Karlsteen, Mats Rostedt and Dag Hanstorp

We describe an experimental system based on optical levitation of an oil droplet. When combined with an applied electric field and a source of ionizing radiation, the setup permits the investigation of physical phenomena such as radiation pressure, light diffraction, the motion of a charged particle in an oscillating electric field, and the interaction of ionizing radiation with matter. The trapping occurs by creating an equilibrium between a radiation pressure force and the force of gravity. We have found that an oil droplet can be trapped for at least nine hours. The system can be used to measure the size and total electric charge on the trapped droplet. The intensity of the light from the trapping laser that is scattered by the droplet is sufficient to allow the droplet to be easily seen with the naked eye, covered by laser alignment goggles. When oscillating under the influence of an ac electric field, the motion of the droplet can be described as that of a driven, damped harmonic oscillator. The magnitude and polarity of the charge can be altered by exposing the droplet to ionizing radiation from a low-activity radioactive source. Our goal was to design a hands-on setup that allows undergraduate and graduate students to observe and better understand fundamental physical processes.

DOI

Simulation of a Brownian particle in an optical trap

Giorgio Volpe and Giovanni Volpe
An optically trapped Brownian particle is a sensitive probe of molecular and nanoscopic forces. An understanding of its motion, which is caused by the interplay of random and deterministic contributions, can lead to greater physical insight into the behavior of stochastic phenomena. The modeling of realistic stochastic processes typically requires advanced mathematical tools. We discuss a finite difference algorithm to compute the motion of an optically trapped particle and the numerical treatment of the white noise term. We then treat the transition from the ballistic to the diffusive regime due to the presence of inertial effects on short time scales and examine the effect of an optical trap on the motion of the particle. We also outline how to use simulations of optically trapped Brownian particles to gain understanding of nanoscale force and torque measurements, and of more complex phenomena, such as Kramers transitions, stochastic resonant damping, and stochastic resonance.
DOI

Orthonormal basis for nonparaxial focused fields in two dimensions, and its application to modeling scattering and optical manipulation of objects

Krista Lombardo and Miguel A. Alonso

A new family of rigorous solutions to the two-dimensional Helmholtz equation is discussed. These solutions form a complete orthonormal basis in terms of which any focused field can be expressed. The directionality of each element of this basis is regulated by a parameter which is related to an imaginary displacement of the focus. The forces and torques that these fields and any linear combination of them exert on a homogeneous circular scatterer can be calculated analytically, regardless of the relative position between the scatterer and the focus.

DOI

Optical tweezers for undergraduates: Theoretical analysis and experiments

M. S. Rocha

A theoretical treatment of optical tweezers is presented at a level suitable for undergraduates. We explore the Rayleigh and the geometrical optics regimes with an emphasis on the latter. We discuss a model for the geometrical optics regime, including spherical aberration effects, and show that the model can easily be implemented numerically. A comparison of the model with experimental data yields excellent agreement between theory and experiment. We also briefly discuss a theory of optical tweezers valid for microspheres of any size.

DOI