Electrokinetics and Electrohydrodynamics in Microsystems
Among the most promising techniques to handle small objects at the micrometer scale are those that employ electrical forces, which have the advantages of voltage-based control and dominance over other forces. The book provides a state-of-the-art knowledge
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CISM COURSES AND LECTURES
Series Editors: The Rectors Giulio Maier - Milan Franz G. Rammerstorfer - Wien Jean Salençon - Palaiseau
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(/(&752.,1(7,&6 $1' (/(&752+ 0, which is the case when the liquid is moved by the electric field. L The current j for given u and E0 can be determined from V = 0 E(x) dx. A certain charge injection law should be given in order to eliminate E0 . The expressions say that the charge density decreases from the emitter to the collector, while the field strength E increases. Injection of large amounts of charge has a limit, since the charge already present in the system repels the incoming charge. When the field at the emitter is too low, charge can no longer be removed from the emitter, and the pump reaches its space charge limit (Crowley et al., 1990). For space-charge-limited emission (SCLE), the electric field at the emitter is much smaller than the average field, E0
V /L, and this is the boundary condition that we are going to consider to close our problem (E0 ≈ 0). The result is an implicit equation for j ⎧ ⎫ 2 3/2 3 ⎬ εμ ⎨ 2jL u u uL V = + . (31) − − ⎩ ⎭ 3j εμ μ μ μ Once we know the electric current we can calculate the generated pressure as a function of V and u. The generated or maximum pressure for spacecharge-limited emission (SCLE) becomes ⎧ ⎫2 2 1/2 ⎨ 2jL 1 ε u⎬ u Δpmax = εEL2 = + − (32) 2 2 ⎩ εμ μ μ⎭
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Antonio Ramos
where EL = E(L) is the field at the collector. We can now obtain the liquid velocity u as a function of the external back pressure Δpout at different applied potentials V , taking into account that Δpmax is balanced by the pump internal hydrodynamic resistance plus the external back pressure: Δpmax = Rin Q + Δpout , where Rin = 128ηL/πD4 is the hydrodynamic resistance and Q = uπD2 /4 is the flow-rate. Let us analyse two limits: a) the liquid velocity is much smaller than the ion migration velocity (u μV /L); and b) the liquid velocity is much greater (u μV /L). Here V /L is an estimate of the electric field in the pump. When u μV /L, the current density is j ≈ ρμE and the final expressions for the current density, electric field, generated pressure and maximum flow-rate are 9 μεV 2 , (33) j= 8 L3
3V x E(x) = , (34) 2L L 9 εV 2 , 8 L2 9 εV 2 πD4 . = 8 L3 128η
Δpmax = Qmax
(35) (36)
When u μV /L, the current density is j ≈ ρu which implies that ρ is a constant of x and E is a linear function of x. The final expressions are E(x) =
2V x , L2
2uεV , L2 2εV 2 , Δpmax = L2 εV 2 πD4 . Qmax = 3 L 64η j=
(37) (38) (39) (40)
Here Δ
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