I am using mixing models to calculate effective parameters (such as permittivity or permeability) of mixtures, i.e. inclusions with a given volume fraction in a host medium. There are several mixing formulas that can provide values of the effective parameter of the mixture, such as Maxwell-Garnett or Bruggeman.
Explanations can be found here :http://en.wikipedia.org/wiki/Effective_medium_approximations
Bruggeman's model treats the host medium and the inclusion similarly, and it can handle any number of different inclusions. But, there is a limitation due to the mathematical formulation.
Suppose the simple case of only one inclusion in a host medium, the equation used to find the effective parameter of the mixture is a second order polynomial. If the discriminant is negative (which sometimes happens) then the solution is complex. But this may be non physical depending on the application case (conductivity calculation, permittivity is neither the inclusion nor the host have losses, etc) where we know that a real value is requested.
For a higher number of inclusions, this may happen more often.
So, what is the solution in this case ? How to use Buggeman's model to get a real value for the effective parameter in the case the equation's solutions are complex ?
I am not sure to understand the problem. I have personally never used the pure Bruggeman model because of its limitations, but I used the GEM model instead, which is an extension of Bruggeman model (see attached paper, but I have others). With GEM model, there is no need to know whether the system is summetric or asymmetric, and it works perfectly well. I never observed the problem you describe by using GEM formulation.
Alain
Article Electrical conductivity of carbonaceous powders
I have discussed briefly the Clausius Massotti, Maxwell Garnet and Bruggeman effective medium approximations in the context of interpreting ellipsometric measurements in Chap 9 of my book "Surfaces, Interfaces and Thin Films for Microelectronics", Wiley, 2008 by Eugene A. Irene.
Dear Fabrice, when you take the Bruggeman equation for spherical inclusions of the same size, I can hardly imagine that you get complex solutions. I did some calculations with binary mixtures and I found always positive discriminantes, given that the conductivities or permittivities of both inclusions were taken positive.
Discriminant=(3*phi*(s_1-s_2)+2*s_2-s_1)^2+8*s_1*s_2
In this equation is "phi" the volume fraction of the phase 1. Further, s_1 and s_2 are the conductivities of phase 1 and 2 respectively.
Thank you Henk
Actually, I am dealing with unusual values of the permittivities, which sometimes lead to negative discriminants. I'll have a look to the formula you provide very soon, to check mine...
When simulating the complex behavior I used complex number calculus and Newton-Raphson method. At high frequencies I used any formula valid for the real permittivities as a starting value and, while gradually changed the frequency I always used the converged value otained for a slightly higher frequency. The convergence occurred within a few steps.
Dear Sirs,
I do have a question about this problem: I need to evaluate the equivalent dielectric of a mixture of water and air (let's say air = 10, 20, 30, 40 and 50% in volume, with pressure of 1, 2, 3, 4 or 5 bar in order to keep the air in).
I let my water pass through 2 cylinders that behave like a cylindric capacitor. How can I estimate the dielectric of the mixture?
Thanks for your help,
Samuele
Is it a mist (fog) or a gaseous mixture? What is the temperature? You can use the mixtrue formula for composites only if you have droplets in air or vice versa. If you have a gas mixture, you should use the classical mixture formula for dipolar liquids or gases.
I have air droplets inside water. And the temperature is about 20 °C. Can I use this formula?
In my opinion yes. At least you should try it. You should use a permittivity value of 1 for air and you should calculate the volume fraction of air.
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