Page 112 - Optofluidics Fundamentals, Devices, and Applications

Optofluidic Trapping and Transport Using Planar Photonic Devices 93

for two important cases; the first being for spherical objects moving
through a stagnant fluid in an infinite domain. In such a case Eq. (5-2a)
reduces to the expression shown (often referred to as the Stokes drag
equation):
F =−6πμaU (5-3)
D
where U is the velocity of the particle relative to the bulk flow and a
is the particle radius.
The negative sign in the equation refers to the fact that the force
acts opposite the direction of the particle velocity. This equation is
only accurate when a particle is far from any no-slip boundaries (such
as walls). It can be shown that a modification of the Stokes drag equa-
tion can be made to approximate the drag for a particle moving near
an even solid surface. This equation (which is a form of Faxen’s law
[62,66]) is given as

−6πμaU
F = (5-4)
D ⎡ 3 ⎛ a⎞ 4 5 ⎤
1 ⎢ − 9 ⎛ ⎞ a + 1 ⎛ ⎞ a − 45 ⎜ ⎟ − 1 ⎛ ⎞ a ⎥
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎝ ⎝
⎝ ⎠ ⎥
⎢ ⎣ 16 ⎝ ⎠ h 8 ⎝ ⎠ h 256 h⎠ 16 h ⎦
where h is the distance between the particle center and the wall
surface.
5-4-3 Electromagnetic Forces on a Particle
As previously discussed, optical forces acting on particles can be
separated into two main categories. The optical trapping force acts
to pull a particle along the gradient of the electric field toward the
region of highest optical intensity. The radiation pressure forces
are due to the scattering and absorption of photons on the particle,
which push particles in the direction of optical intensity. As
described by Mishchenko et al. [67], this is an orthogonal decom-
position of the total force that is more generally described by the
surface integral of the time-averaged Maxwell stress tensor, T ,
M
as shown in Eq. (5-6a).
1
⋅
⋅
*
*
)
T = DE + HB − ( D E + H B I
*
*
M (5-6a)
2
where E = electric field
B = magnetic flux field
D = electric displacement
H = magnetic field
∗
∗
E and B = complex conjugates
I = isotropic tensor.