Page 113 - Optofluidics Fundamentals, Devices, and Applications

94 Cha pte r F i v e

We note that the use of the time-independent Maxwell stress ten-
sor is justified here since the transport processes of interest occur on
timescales much longer than the optical period (for more informa-
tion interested readers are directed to a review article that discusses
electromagnetic stress tensors [68]). When expanded out, Eq. (5-6a)
becomes

⎛ 1 ⎞
⋅
*
* *
BH
*
*
⎜ DE + B H − 2 (DE * + ⋅ * ) DE + B H * y DE + BH * z ⎟
x
x
y
x
x
x
x
x
z
x
⎜ 1 ⎟
*
*
DE +B
T = ⎜ DE * + B H * DE * + B H * − ( ⋅ * ⋅H ) ) DE + B H * ⎟
M ⎜ y x y x y y y y 2 y z y z ⎟
⎜ 1 * ⎟
⎜ ⎝ DE + B H * x DE + B H * y DE + B H − 2 ( ⋅ * + ⋅ ) ⎟ ⎠
H
DE
*
BH
*
*
*
z
z
z
z
y
z
z
z
z
x
(5-6b)
where the subscripts x, y, and z signify the coordinate directions. By
integrating the time-independent Maxwell stress tensor on a surface
enclosing the particle of interest, we can determine the total electro-
magnetic force acting on the system, F , given by
EM
F = T ( ∫ n ⋅ ) dS (5-7)
EM M
s
where n is the unit vector normal to the particle surface. As we [69]
and others [52] have shown, the E and H fields can be computed
either through a full solution to Maxwell’s equations or by solving
the time harmonic wave equation via the finite element method and
the integration of Eq. (5-7) carried out numerically. For further infor-
mation on how to carry out these computations, readers are referred
to Refs. 52 and 69.
5-4-4 Solutions in Different Transport Regimes
The set of equations in the preceding section represent a relatively
basic, but general, model for optofluidic transport, ignoring such
effects as heating, surface friction, and electrical double layer
repulsion. Despite this the basic model has proven to be relatively
predictive of observed experimental behaviors [49]. In this section
we discuss how to implement these models for two transport regimes
of interest: (1) when the transported particle radius, a, is much smaller
than the wavelength of light, λ, and (2) when the particle radius is
approximately the same or much larger than λ.
Transport in the Development in the a << k Regime
This first regime is referred to as the Rayleigh regime and is defined
by the assumption that the electromagnetic field is uniform as it
impinges on the particle (hence the limitation that a << λ). For this case