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http://www.uio.no/studier/emner/matnat/kjemi/KJM5120/v09/undervisningsmateriale/Defects-and-transport-2009-Ch5-Diffusion.pdf This online pdf file trys to derive Fick's 1st law. In the deriving process(page 5.3), it is says that the drift velocity of a particle $v_i$ is proportional to the driving force acting on particle $i$, $F_i$, that is, $$v_i=B_iF_i$$, where $B_i$ is a constant.

My questions is 1. What does this drift velocity mean here? Is it just the total velocity of particle $i$?

  1. According to the Newton's second law, $v_i(t)=v_0(t)+Ft/m$. So why $v_i=B_iF_i$ here? Does it because $\sum{v_0(t)}=0$ as the system is macroscopic ally still so the total initial velocity should be zero. But in this case, $B_i$ should be a value that depends on time $t$? But according to the textbook, $B_i$ seems to be a constant that describes the mobility of particle i?

Can anybody give me some help on it? Thanks!

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2 Answers 2

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You are correct in assuming that $v_i$ refers to the total velocity of particle $i$. Note that, since this is an ideal diffusion problem, the direction of the velocity is specified - it points along the concentration gradient.

The issue here is that the author uses the term "driving force" rather colloquially, to refer to something that is not, in fact, a force. He states that the concentration gradient "may be considered to be" (rather than "is") a "driving force." In fact, what you are attempting to justify (i.e. the fact that diffusion velocity is proportional to concentration gradient) is taken as an assumption in Fick's Law. For this reason, I don't think Newtonian mechanics really applies here.

My best advice is to either a.) take the same assumptions as Fick's Law, and proceed, or b.) learn the necessary statistical mechanics to eliminate or justify the assumption made in Fick's Law.

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  • $\begingroup$ Thanks, I got your points! Fick's 1st law is just a empirical, true derivation needs statistical mechanics, here $v_i=B_iF_i$ is just the same assumption as the assumption Fick's law used. So rather that derivation of Fick's 1st law, the author just give another type of writting of fick's 1st law $\endgroup$
    – FaDA
    Dec 20, 2016 at 10:47
  • $\begingroup$ Yep, that's exactly it. Glad I could help! $\endgroup$ Dec 20, 2016 at 22:34
  • $\begingroup$ Also for ions in electric fields etc, the drift velocity is proportional to the force, the mobility being the constant of proportionality. @FaDA $\endgroup$
    – user137289
    Dec 21, 2016 at 0:11
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  1. No. Drift velocity does not refer to the total velocity. It refers to a non-zero time-averaged velocity.

In the case of electrons in metals, the total velocity is the random thermal motion of around 400m/s, which changes direction rapidly due to collisions with impurities, dislocations or phonons. If there is no electric field the drift velocity is zero. If there is an electric field the electrons have a small systematic motion of about 1mm/s in the direction of the field, on top of the random motion. This is the drift velocity.

  1. The time $t$ in your formula is actually the mean time between collisions. Since the random velocity is very much greater than the drift velocity, the mean time between collisions does not increase as electric field increases, it is approximately constant. Therefore the drift velocity is proportional to the driving force $E$.
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