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Flux equations question

  1. Nov 10, 2007 #1
    Dear all,

    this is my first post in PF, and I really don't know if the level of my question is below, above or just right for this place, so bear with me.

    I'm writing a paper on electromigration, and I am the kind of person that wants to understand well every aspect of the subject he is working on. One of the articles I use is Electromigration in metals" by P. S. Ho and T. Kwok. The first equation is:

    [tex]Ji = \sum^{n}_{j=1} {Lij Xj}[/tex]

    And equation 2:

    Ji = T[tex]^{-1}[/tex] [tex]\sum[/tex][tex]^{n}_{j=1}[/tex] Lij [tex]\nabla[/tex] ([tex]\mu[/tex]j + qj[tex]\phi[/tex])

    where Lij are phenomenological coefficients correlating the flux of the ith component, to the driving force Xj of the jth component. ([tex]\mu[/tex]j + qj[tex]\phi[/tex]) is the electrochemical potential of the jth component.

    My pressing and (to me at least) not-so-trivial question is: what exactly are these components? I can understand that in case of electromigration, the atoms of the lattice could be one, the electrons could be another, and the impurities could be yet more components. That would be fine, if not for the fact that the electrochemical potential of the atoms (of the lattice, for instance) is not the same everywhere along the conductor.
    But then again, this equation surely wasn't written so that one would sum across all atoms in the system.

    Could someone explain the meaning of "component" in this context?
    Last edited: Nov 11, 2007
  2. jcsd
  3. Nov 11, 2007 #2

    Claude Bile

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    It is not uncommon to heap a bunch of variables into a single coefficient (or matrix of coefficients) - this is frequently done in optics as well. The important detail is often which terms are coupled to one another, the actual strength of the coupling is usually incidental. Often too, such coefficients are more easily measured than calculated from first principles.

    The term "component" can therefore be read two ways - either from a purely mathematical standpoint, or as a label that umbrellas all the various physical factors that determine the value of the coefficient.

  4. Nov 11, 2007 #3
    I actually understand the coefficients, and even - believe it or not - understand the conditions under which they can be simplified using Onsager's relations. I still don't understand what a component is, though. I must admit your answer left me as puzzled as before.
  5. Nov 11, 2007 #4

    Claude Bile

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    My apologies, I mentally exchanged the terms coefficient and component, whoops.

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