Particles undergoing dielectrophoresis move in a fluid, therefore their velocities will be significantly slowed by the resistance of drag forces. The net force on a particle undergoing traveling-wave dielectrophoresis as a result of the electrode configuration presented in figure 2.5 is
(4.1) where represents opposing drag forces and is the sum of the noise components described earlier. In order to describe the separation of particles, it is best to analyze the system
in terms of particle velocities rather than forces since we are interested in controlling the motion of particles. But in order to determine velocity, we have to understand the nature of the drag forces that will be exerted on the particles.
In fluid mechanics, the Reynolds number is the ratio used to measure the relative importance of inertial forces to viscous forces and is defined as [52]
(4.2)
where ρm is the fluid density, νm is the velocity of the fluid, L is the characteristic length of the system and η is the dynamic viscosity of the fluid medium. The medium used in dielectrophoresis is typically of low viscosity like water, with a value of .
At microfludic scales, the characteristic length is small, making the Reynolds number low; therefore micro and nanometer scale particles undergoing dielectrophoresis typically experience laminar Stokes flows in which inertia is negligible [52].
In general, the motion of an object in a fluid can be described by a set of partial differential equations known as the Navier-Stokes equations [53]. For the case of a small Reynolds number particle undergoing laminar flow in an incompressible, Newtonian fluid, these differential equations reduce to a simple closed form [52]. Micro and nanoscale particles undergoing dielectrophoresis meet the previously mentioned criteria (e.g., water or an electrolytic solution), therefore the drag force exerted on moving particles can be modeled using Stoke’s law:
( ) (4.3)
where is the velocity of the particle and the friction factor term is γ = 6πηrp.
Micro and nanoscale particles have extremely small mass (mp), therefore their acceleration time constant, τ = mp/γ is also very small and they reach steady state very quickly.
For example, an average sized yeast cell weighs [54], which results in a time constant of τ = 238 ns. The amount of dielectrophoretic force that can be generated using moderate voltages typically causes particles to move very slowly, at speeds on the order of μm/s. Therefore, the time period during which the particle is accelerating is negligible, and while the particle is moving, it can be assumed that it is not accelerating. Since the particle is not accelerating, there is no net force on the particle ( ) Therefore using equations 4.1 and 4.3, a particle undergoing motion can always be assumed to be traveling at its terminal velocity:
⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ (4.4)
For our multiplexing method, it is desirable have control over both the x and y components of velocity. Assuming for a moment that noise is negligible and the fluid in the medium is not moving, using expressions 2.11 and 4.4 we can further express the x-component of particle velocity at a given frequency in terms of the key parameters of interest: TWDEP force magnitude, the sign of the electric field phase gradient along the x-axis and the sign of the imaginary part of the Clausius-Mossotti factor
( ) ( ) [ { ( )}] | ( )|
(4.5)
Likewise, the y-component of particle velocity will be a function the DEP force magnitude, the sign of the electric field magnitude gradient along the y-axis and the sign of the real part of the Clausius-Mossotti factor:
( ) ( ) [ { ( )}] | ( ) |
At this point we define a new variable , which is representative of the controllable sign of the force components. For the x-component of velocity, this directional variable has a value of
( ) [ { ( )}] (4.7)
The sign of the Clausius-Mossotti factor is fixed for a particle at a given frequency and the phase gradient can be externally used to control its sign. For the y-component of velocity, since in the configuration of figure 2.5 the electrodes are at the bottom of the chamber, the electric field intensity decreases as the height along the Y-axis increases. Therefore the sign of the electric field intensity gradient will always be negative and the overall sign of the y-component of velocity will be opposite the sign of the real part of the Clausius-Mossotti factor making
[ { ( )}] (4.8)
The sign of can only be controlled via selection of frequency as phase has no bearing on its value. In terms of these new variables, the controllable, frequency dependent velocity components can be rewritten as
( ) | ( )| (4.9)
( ) | ( )| (4.10)
In addition to being able to control the directions of these velocity components, for our time-multiplexing dielectrophoresis method we wish to also be able to control the average value of the velocities over time. In our methodology, we require a supporting hardware platform (described in chapter 5) with the ability to change the field configuration at any given time, where the field configurations are defined in terms of their frequency and phase gradient . For a field configuration of , applied for time interval of that is then switched to a field
configuration of , for duration of , the time-averaged net differential velocity components particle becomes
̅ 〈 ( , )〉 ( ) | ( )| ( ) | ( )| (4.11) ̅ 〈 ( , )〉 ( ) | ( )| ( ) | ( )| (4.12)
The term ( ) can also be referred to as the duty cycle, D of the system, which will vary from 0 ≤ D ≤ 1. Substituting in the variable for the duty cycle, the time-averaged velocity components that define the motion of a particle, in a time-multiplexing dielectrophoresis field are
̅ | ( )| ( ) | ( )| (4.13)
̅ | ( )| ( ) | ( )| (4.14)
In the next section we explain how these differential velocity components generated by time- multiplexing dielectrophoresis fields can be exploited to separate particles.
4.2 SEPARATING PARTICLES VIA TIME-MULTIPLEXED