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Copyright Brian J. Kirby. With questions, contact Prof. Kirby here. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.

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Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

5.3 Analytic representations of electrodynamic quantities: complex permittivity and conductivity [electrodynamics top]

Section G.3 in the appendices discusses the fact that, if the time dependence of our voltages are sinusoidal, the relations between the voltage/current, their integrals, and their derivatives take on a special form. We take advantage of this by using the analytic representation of the sinusoidal signals, in which case we exchange the real description of a physical quantity with its analytic representation, which is easier to work with but involves complex numbers. For hydraulic circuit analysis, this replaces hydraulic resistance and compliance with the complex hydraulic impedance, which obeys a simple Hagen-Poiseuille-type relation (microfluidics textbook nanofluidics textbook Brian Kirby Cornell = microfluidics textbook nanofluidics textbook Brian Kirby Cornellmicrofluidics textbook nanofluidics textbook Brian Kirby Cornell). This highlights that the relative response of a resistor (which impedes flow or current) and a capacitor (which stores volume or charge) differs by a factor of , which means that the flow stored by compliance is delayed by π∕2 as compared to that through a microchannel, and the ratio of the currents between the two is proportional to ω. This analysis is mathematically possible because the frequency spectra of real functions are symmetric around the origin, and was physically possible because the governing equations for circuits are linear and homogeneous. For the equations of electrodynamics, a similar approach is possible—this approach makes analysis of system responses to sinusoidal fields enormously simpler.

Just as we might have for a circuit, we can replace a real electric field microfluidics textbook nanofluidics textbook Brian Kirby Cornell = microfluidics textbook nanofluidics textbook Brian Kirby Cornell0cosωt with its analytic representation microfluidics textbook nanofluidics textbook Brian Kirby Cornell = microfluidics textbook nanofluidics textbook Brian Kirby Cornell0expjωt and replace the real equations of electrodynamics with equivalent analytic representations. We can furthermore drop the expjωt and write the result strictly in terms of the phasors. For example, we can (See Exercise 5.8) replace microfluidics textbook nanofluidics textbook Brian Kirby Cornell = εmicrofluidics textbook nanofluidics textbook Brian Kirby Cornell with
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.51)

where microfluidics textbook nanofluidics textbook Brian Kirby Cornell is the analytic representation of the electrical displacement. For electrodynamic systems, we combine conductivity and permittivity into a complex parameter microfluidics textbook nanofluidics textbook Brian Kirby Cornell or microfluidics textbook nanofluidics textbook Brian Kirby Cornell. In so doing, the constitutive relations for electric displacement (microfluidics textbook nanofluidics textbook Brian Kirby Cornell = εmicrofluidics textbook nanofluidics textbook Brian Kirby Cornell) and ohmic current density (microfluidics textbook nanofluidics textbook Brian Kirby Cornell= σmicrofluidics textbook nanofluidics textbook Brian Kirby Cornell) can be combined to form
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.52)

where the complex permittivity microfluidics textbook nanofluidics textbook Brian Kirby Cornell isdefined as
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.53)

Similarly, we can replace microfluidics textbook nanofluidics textbook Brian Kirby Cornell= σmicrofluidics textbook nanofluidics textbook Brian Kirby Cornell with
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.54)

where the complex conductivity microfluidics textbook nanofluidics textbook Brian Kirby Cornell isgiven by:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.55)

5.3.1 Complex description of dielectric loss

For determination of electric charge transport and solution of electrodynamic equations, the complex permittivity microfluidics textbook nanofluidics textbook Brian Kirby Cornell from Equation 5.53 correctly describes both displacement and ohmic current. For describing the conversion of electric field to heat, though, the conductivity σ describes on one of the loss mechanisms. The energy of an applied electric field can be directly absorbed by a medium, a process called dielectric loss. Thus conductivity leads to charge transport and, in turn, heat generation, while dielectric loss leads to heat generation directly. The dielectric loss for a mode i is proportional to χe,i(0)ωτi. We write the applied field E = E0cosωt using its analytic representation microfluidics textbook nanofluidics textbook Brian Kirby Cornell = E0expjωt, and group the polarization response and dielectric loss terms into a complex permittivity microfluidics textbook nanofluidics textbook Brian Kirby Cornell= ε+microfluidics textbook nanofluidics textbook Brian Kirby Cornellε′′. Here εdenotes the real part of the permittivity, i.e., the measureof the polarization response to the electric field:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.56)

and the imaginary component ε′′ denotes the(imaginary) dissipation of electrical (polarization) energy into internal energy of the molecules of the medium:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.57)

From these relations, we see that εis a measure of how much the medium polarizes in response to an electric field, while ε′′ is the product of the relative medium response times ωτi. The energy dissipation is low at high frequency, because the medium has no time to polarize and no energy is transferred to the medium from the field. The energy dissipation is low at low frequency, because, though the medium polarizes, the total energy stored per cycle is low. However, at frequencies near 1∕τi, the medium is polarized quite strongly (one half of the polarization at steady-state) and the number of cycles is high. Thus significant energy is pumped into the medium for each cycle, and there are many cycles per second. The total energy conversion to heat is thus proportional to σ∕ω+ε′′.

The real and imaginary components of the complexpermittivity are related by the Kramers-Krönig relations, discussed in Appendix G:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.58)

and
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.59)

While these relations are exact, they do not immediately lend themselves to an intuitive interpretation. For the sigmoidal functions that describe the permittivity, the following relation holds, which is exact at ω = 1∕τi and approximate for frequencies near ω = 1∕τi:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.60)

If plotted on a semilog axis, the dissipative response is approximately equal to the derivative of the reactive response. Thus energy dissipation coincides with those frequencies where the permittivity is decreasing with ω.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 5.11: Ohmic and displacement current. (a): ohmic current in an electrolyte describes the motion of free charges (ions) in response to electric field. The current is proportional to the electric field and the conductivity σ is the constitutive constant. (b): displacement current describes the motion of bound charges (partial charges of covalent bonds, atomic nuclei and electron orbitals, etc) in response to this electric field. The displacement current is proportional to the time derivative of the electric field and the electrical permittivity ε0 is the constitutive constant.


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Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

Copyright Brian J. Kirby. Please contact Prof. Kirby here with questions or corrections. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.


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