Page 117 - Optofluidics Fundamentals, Devices, and Applications

98 Cha pte r F i v e

transport can occur. The system we consider here is for a dielectric
particle trapped in the evanescent field of a waveguide, subject to a
microfluidic cross flow. Our approach, which is outlined in greater
detail in a recent paper [69], is to compute the work required to
remove a particle from a trapped state and determine a trapping sta-
bility parameter based on this. It can be easily shown that the down-
ward trapping force far exceeds that in the direction of flow, so our
stability condition is based on a particle being “swept” off the wave-
guide (since this is the most likely way it will be removed).
If we assume the spatial force variation in the flow direction is
like that of the evanescent field, then a similar relationship would
hold for a sufficiently wide waveguide:

F () = F exp(−γ ) x (5-11)
x
trap T0 f
where F represents the transverse trapping force at a point x = 0,
T0
which we designate as the trailing edge of the waveguide, and γ is
f
the trapping force decay rate. We will later show numerically that this
relationship holds for a particle moving laterally from a rectangular
waveguide. For a constant flow speed, we can assume that the drag
forces on the particle are relatively constant compared to the steep
decay of the transverse (x-direction) trapping force. Then, the net
force acting on a particle is
F ()x = F exp(−γ ) x + F (5-12)
net T0 f D
where F is the drag force exerted on the particle. The nature of this
D
equation is such that for any nonzero positive drag force, the net force
acting on the particle will at some point become a net positive force.
The crossing point can be analytically determined:

⎛F ⎞
−1
x = γ ln ⎜ T 0 ⎟ (5-13)
c f ⎝ F D ⎠
where x is the crossing point where the net force acting on the parti-
c
cle becomes a positive value. The work necessary to release a particle
from the trapping field can be shown to be

x c
x dx =
θ
θ
=
1
W trap ∫ F trap () F γ −1 [1 + (ln( ) − )] (5-14)
T0
f
0
where θ is the ratio of the drag force to the transverse trapping force
(F /F ). The equation is only valid for 0 ≤θ≤ 1, and it can be shown
D T0
that the limit of the nonlinear term goes to unity as θ goes to zero. We
can define the trapping stability of the particle by relating the work