1.

INTRODUCTION

In this proceeding, we present a simple algorithm to simulate a Brownian particle in an optical trap [1]. We provide an implementation of this algorithm using MatLab^®, because this language is widely employed in the science and engineering. All algorithms can also be translated straightforwardly in the freeware programming language SciLab [2].

The interest of these simulations is dual. On the one hand, it is interesting to simulate an optically trapped particle to gain insight into how an optical trap works; since optical traps have found a widespread set of applications in fields as diverse as, e.g., cooling of single atoms, colloidal physics and biophysics, it can be useful for many students to understand how these optical traps work in simple and inexpensive simulations. On the other hand, an optically trapped particle constitutes a simple example of a stochastic phenomenon whose evolution is determined by both stochastic (the Brownian motion) and deterministic (the optical forces) forcing terms and can, therefore, be used as a model system for more general stochastic phenomena.

2.

LANGEVIN EQUATION

An optically trapped particle can be described by the Langevin equation in one dimension [1, 3]:

where x(t) is the trajectory of the particle with respect to the trap center, m is the particle mass, γ is the friction exerted by the surrounding medium on the particle, k is the optical trap stiffness, k_BT is the thermal energy unit, k_B is the Boltzmann constant, T is the absolute temperature and W(t) is a Wiener process.

For microscopic particles immersed in a liquid, viscous forces are several orders of magnitude larger than inertial forces, i.e. the particle is in a low-Reynolds-number regime [4]. The Reynolds number is the ratio between inertial and viscous forces acting on an object moving in a fluid. If the object has a characteristic dimension L and is moving at velocity ν through a fluid with viscosity η and density ρ, its Reynold number is Re = Lνρ/η. In the low-Reynold-number regime, i.e for Re < 1, the viscosity dominates over inertia. Considering, for example, an E. coli bacterium in swimming in water [5], L ≈ 1μm, ν ≈ 30μm/s, η = 0.001P as and ρ = 1000kg/m3, so that Re = 3e-5 ⪡ 1. One of the most striking aspect of low Reynolds number phenomena is that the speed of an object is solely determined by the forces acting on it at the moment; a good introduction to how is life at low Reynold numbers can be found in Ref. [4]. In general, most optical manipulation experiments take place at low Reynolds numbers, the exceptions being experiments in fluids with very low viscosity such as air [6]. Therefore, in Eq. (1) we can drop the inertial term obtaining:

where D is the Stokes-Einstein diffusion coefficient.

Eq. (2) is one of the simplest examples of a stochastic differential equation [7, 8]. In general, stochastic differential equations are obtained from ordinary differential equations adding a noise term, i.e. the Wiener process W(t). W(t) is characterized by the following properties:

3.

FINITE-DIFFERENCE SOLUTION

In order to simulate Eq. (2) it is possible to employ a finite difference algorithm, where the continuous-time solution x(t) is approximated by a discrete time sequence x_i evaluated at times t_i = i Δt. This is in practice done by doing the following substitutions in Eq. (2) [1, 7]:

where w_i is a sequence of Gaussian random numbers with the following properties:

Notice how these properties mimick the ones of the Wiener process given above.

By making the aforementioned substitutions in Eq. (2) one obtains the following finite difference equation:

which can now be solved numerically.

4.

SIMULATION CODE

In this section we present the MatLab® code that implements Eq. (3) in one and 3 dimensions following the codes provided with Ref. [1].

4.1

1D Simulation Code

This function implements Eq. (3) in one dimension. Inputs: number of samples N, timestep Dt, initial position x1, particle radius R, temperature T, fluid viscosity eta, and trap stiffness k. Outputs: particle position x in meters and time t in seconds. Figure 1 shows the result of this simulation.

Figure 1.

Simulation of an optically trapped particle in three dimensions with the following parameters: N = 1e+4; Dt = 1e-3 [s]; x1 = 0 [m]; R = 1e-6 [m]; T = 300 [K]; eta = 0.001 [Pa s – water]; kx = 1e-6 [N/m][N/m].

4.2

3D Simulation Code

This function implements Eq. (3) in three dimensions. Inputs: number of samples N, timestep Dt, initial position x1 y1 z1, particle radius R, temperature T, fluid viscosity eta, and trap stiffness kx ky kz. Outputs: particle position x y z in meters and time t in seconds. Figure 2 shows the result of this simulation.

Figure 2.

Simulation of an optically trapped particle in three dimensions with the following parameters: N = 1e+4; Dt = 1e-3 [s]; x1 = 0 [m], y1 = 0 [m], z1 = 0 [m]; R = 1e-6 [m]; T = 300 [K]; eta = 0.001 [Pa s — water]; kx = 1e-6 [N/m], ky = 1e-6 [N/m], kz = 0.2e-6 [N/m].

5.

DIFFUSION GRADIENTS

In the presence of diffusion gradients, which arise naturally, e.g., when a Brownian particle is trapped near another particle or near a surface, some additional correction are needed in order to account for a spurious drift [9, 10, 11, 12]. This spurious drift results from the fact that the Wiener process is multiplied by a function of the particle position and is needed to correct an indetermination inherently occurring in stochastic differential equations with multiplicative noise. How to deal with these cases is explained in Refs. [13, 14]. Similar considerations hold also in the case of diffusion gradients induced by, e.g., chemical gradients or temperature gradients [15, 16, 17].

6.

FURTHER NUMERICAL EXPERIMENTS

Using the code provided in Section 4, it is possible to simulate an optically trapped particle under various conditions and verify, e.g., that the particle variance in the optical trap is inversely proportional to the trap stiffness [18]. Furthermore, it can be interesting to apply these simulation techniques to simulate and study the behavior of a Brownian particle in more complex force fields: for example, a Brownian particle in the presence of external forces or torques [19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. Also time-varying force fields can be considered, which leads to more complex phenomena, such as Kramers transitions [29], stochastic resonant damping [30] and stochastic resonance [31].