Page 115 - Optofluidics Fundamentals, Devices, and Applications

96 Cha pte r F i v e

particle moving along a waveguide, we shift reference frames by
keeping the particle still and applying boundary conditions that sim-
ulate the flow moving by it. In the low-Reynolds number regime the
drag force is linearly proportional to the flow velocity, so the simplest
method is to apply a “slip” velocity boundary condition on all sur-
faces of some low flow speed (say 1 μm/s) opposite the direction one
expects the particle to move. By integrating Eq. (5-2) over the particle
surface a drag force is obtained, which can be scaled to the particle
speed by simply dividing it by the imposed slip velocity. For cases
where an imposed flow is incident on the particle, such as in [Ref. 69],
it is appropriate to use standard microfluidic boundary conditions
(see Erickson [71] for details).
Figure 5-7 shows the results of finite element-based computations
performed on dielectric spheres of various sizes.
From Fig. 5-7b it is apparent that the propulsive force follows an
approximate squared relation with particle size (F ∝ a ) and from
2
scat
the experiments described earlier we know that F is also propor-
EM
tional to the optical intensity. As such it is reasonable to qualitatively
approximate F = C a I , where C is a constant and an unknown
2
EM, scat 1 o 1
function of the physical parameters of the system and I is the optical
o
intensity. Using Faxen’s law approximation F = 6πηaU /g(a/h),
flow o
where g(a/h) is the denominator in Eq. (5-4). If the distance between
the bottom of the particle and the top of the waveguide is small com-
pared to the particle diameter, a/h ~1 then g(a/h) is a constant and we
can derive the following approximate equation descriptive of the
transport velocity:
2
Ca I aI
U = 1 = C o (5-10)
o 2 η
6πη /( /
ag a h)
where C is a different constant comprising of the same system phys-
2
ical parameters as C . We note from Eq. (5-10) that (based on these
1
computations) the transport velocity in a ≥ λ appears to contain only
a linear dependence on particle size (as opposed to the much stronger
dependence in the Rayleigh regime).
5-4-5 Comments on the Influence of Brownian Motion
and Trapping Stability
The transport path taken by a particle with a diameter greater than
approximately 1 μm is largely deterministic and well described by
Eq. (5-10). Below 1 μm, however, they begin to be significantly affected
by Brownian motion and the hydrodynamic solution becomes pre-
dictive of the average motion of the transported particle (about
which it will diffuse), rather than its actual pathline. The underlying
cause of Brownian motion is the random fluctuating collisions of sol-
vent molecules impacting larger microscopic particles. Brownian motion