In this section I discuss some of the physics that underlies these experiments. Initially, our goal was to utilise tilted Bessel beams as model thermodynamic systems. As we will see in section 4.9, the project evolved from a fundamental study of particle dynamics into the development of a new technique forstaticoptical sorting. I therefore stop short of deriving a complete thermodynamic description for our system, however it is useful to review some of the basics, particularly since some of the relevant papers were a key source of inspiration during the early stages of my research.
4.7.1
Brownian particle dynamics in a one-dimensional potential
Because the transverse profile of the Bessel beam is circularly symmetric, for the zeroth- order beam in this discussion we only need to consider the radial motion of the particle. For movement in one dimension,r, the equation of motion for the particle is given by the
4.7. Directed Brownian transport - theory 112
Langevin equation:
mr¨=−γr˙− d
drU(r) +ξ(t) (4.17) whereγdescribes the damping due to the viscosity of the surrounding fluid,ξis a term that represents the addition of Gaussian white noise to the system andU(r) describes the one- dimensional potential energy landscape. In the overdamped regime this equation reduces to
γr˙=−drdU(r) +ξ(t) (4.18) In their discussions of washboard potentials, both Marchesoni [36] and Tatarkova [27] present equations of this form, with the potential given asU(r) =−dcosr−F r. HereF is a constant which serves to provide the tilt to an otherwise symmetric potential structure.
For a particle in a potentialU(r), the time-evolution of the probability distribution of its position is described by the Fokker-Planck equation. I defineP(r, t|r0, t0) as the probability
that a particle will be found at positionrat timet if it was located initially atr0 at time t0. In the context of this work, the Fokker-Planck equation is given by [37]:
∂P ∂t = ∂(−∇U(r)P) ∂r +kBT ∂2P ∂r2 (4.19)
If the potentialU(r) has no time dependence,∂P/∂t= 0 and the solution forP is given by the Boltzmann distribution:
P(r) =P0exp −U(r) kBT (4.20)
If we have a system of two adjacent potential energy wells of depthU1andU2, for a particle
to make a transition from one well to the other it will have to overcome the potential energy barrier ∆Ui=Ui−Ub where Ub is the potential energy at the top of the inter-well barrier
(figure 4.16).
The probability of transition is proportional to theArrhenius factor: exp(−∆Ui/kBT).
Energy
x
0
U
bU
1U
2DU
2DU
1Figure 4.16: Diagram showing a configuration of two potential energy wells. To make the transition from well 2 to well 1, the particle must undergo a thermally activated transition over the inter-well barrier, which has a height ∆U2.
Kramers [38] solved the Focker-Planck equation for an overdamped particle, determining that the transition rate can be found through the expression
W =ωaωb 2πγ exp −∆Ui kBT (4.21)
whereωa describes the curvature of the potential around the stable equilibrium point at the
bottom of the well,ωb the curvature at the top of the barrier andγthe viscous damping due
to the surrounding fluid. If a detailed knowledge of the shape of the potential is available, equation 4.21 can be used to determine the Kramers’ transition rate exactly.
4.7.2
Kramers’ theory applied to optical manipulation
In the introduction to his 1940 paper, Kramers [38] states that he developed his theory to “elucidate some points in the theory of the velocity of chemical reactions.” In 1999, McCann
et al. published a paper in Nature [39] that quantified the thermally activated transitions of a Brownian particle between two closely-located optical traps, Their approach to the problem heavily influenced my own experiments, and I briefly review it here.
4.7. Directed Brownian transport - theory 114
was able to show that Kramers’ Theory applied to optically trapped particles. The sphere moved with Brownian motion and as such its thermally activated escape from the potential energy well associated with the optical trap is a stochastic process. To get a strong handle on the statistics, it is necessary to take large data sets. Manual data collection of large data sets can be tedious, however more importantly, since the transition event is random, the observer cannot predict when it will happen and may miss the event should their concentration temporarily wane.
Figure 4.17: Illustration taken from McCann et al. [39]. They used two closely spaced optical traps with an automated data collection system to study the inter-well transitions of a Brownian particle, verifying Kramers’ theory in the context of optical trapping.
From the excellent data obtained by McCann et al. [39], I could see that it is clearly beneficial to develop an automated measurement system for such an experiment. Their system used a pattern-matching algorithm to track particle locations to within 10nm. Since they were following the same particle continuously as it hopped back and forth between the two traps, they were able to leave the experiment running for long periods of time. In a single experiment they were able to measure up to 94,000 inter-well transitions. The collection of such large data sets can be extremely useful when studying stochastic processes, and I set about trying to develop an automated system to study the motion of particles in a Bessel beam.