Chapter 6 – Forces  217

restoring optical force on a trapped particle along with its viscous drag coefficient γ due to the

viscosity of the surrounding water solvent, as well as random thermally driven fluctuations in

force (the Langevin force, denoted as a random functional of time F(t)):

(6.7)

kx

v

F t

+

=

( )

γ

where x is the lateral displacement of the optically trapped bead relative to the trap

center and v is its speed (Figure 6.2c), and 〈

〉

F(t) when averaged over large times is zero.

The inertial term in the Langevin equation, which would normally feature, is substantially

smaller than the other two drag and optical spring force terms due to the relatively small

mass of the bead involved and can be neglected. The motion regime in which optically

trapped particles operate can be characterized by a very small Reynolds number, with the

solution to Equation 6.7 being under typical conditions equivalent to over-​damped simple

harmonic motion. The Reynolds number Re is the measure of ratio of the inertial to drag

forces:

(6.8)

R

vl

e ^{= ρ}

η

where

ρ is the density of the fluid (this case water) of viscosity η (specifically termed the

“dynamic viscosity” or “absolute viscosity” to distinguish it from the “kinematic vis­

cosity,” which is defined as η/​ρ)

l is a characteristic length scale of the particle (usually the diameter of a trapped bead)

The viscous drag coefficient on a bead of radius r can be approximated from Stokes law as

6πrη, which indicates that its speed v, in the presence of no other external forces, is given by

(6.9)

v

kv

r

= 6π η

Note that this can still be applied to nonspherical particles in which r then becomes the

effective Stokes radius. Thus, the maximum speed is given when the displacement between

bead and trap centers is a maximum, and since the physical size of the trap in the lateral

plane has a diameter of ~λ, this implies that the maximum x is ±λ/​2. A reasonably stiff optical

trap has a stiffness of ~10⁻⁴ N m⁻¹ (or ~0.1 pN nm⁻¹, using the units that are commonly

employed by users of optical tweezers). The speed v of a trapped bead is usually no more

than a few times its own diameter per second, which indicates typical Re values of ~10⁻⁸. As

a comparison, the values associated with the motility of small cells such as bacteria are ~10⁻⁵.

This means that there is no significant gliding motion as such (in either swimming cells or

optically trapped beads). Instead, once an external force is no longer applied to the particle,

barring random thermal fluctuations from the surrounding water, the particles come to a

halt. To arrive at the same sort of Reynolds number for this non-​gliding condition of cells

for, for example, a human swimming, they would need to be swimming in a fluid that had a

viscosity of molasses (or treacle, for readers in the United Kingdom).

Equation 6.7 describes motion in a parabolic-​shaped energy potential function (if k is

independent of x, the integral of the trapping force kx implies trapping potential of kx²/​

2). The position of the trapped bead in this potential can be characterized by the power

spectral density P(ν) as a function of frequency ν of a Lorentzian shape (see Wang, 1945)

given by

(6.10)

P

k T

v

v

v

B

( ) =

+

(

)

2

3

2

0

2

π