Diffusion is one of the most emphasized topics in the materials science curricula. Usually, Fick’s first law serves as a sort of a master equation for diffusion:
(1)



Generalized flux and driving force
We first have to know a little about how fluxes and gradients are fundamentally described in thermodynamics. But wait, isn’t thermodynamics about equilibrium? Doesn’t “flux” already imply non-equilibrium, because true equilibrium requires no flux? Yes, transport like diffusion belongs to the realm of non-equilibrium thermodynamics! But we will only touch on this topic very lightly, just to find the fundamental driving force that makes mass flux happen.
Turns out, there is a certain near-equilibrium way for a system to reach the equilibrium state (state of maximum entropy), which is by increasing entropy in a certain gradual manner (detailed discussion in this post). This near-equilibrium pathway is equivalent to when a flux of particles is linearly proportional to the driving force that is causing the flux. How do we know if something qualifies as a driving force and corresponding flux? You can multiply the two and see if it equals to how fast entropy is produced. Let’s give it a try.
The mass transport we are interested is specified by particle flux [mol·cm-2·s-1]. Chemical potential gradients drive this flux, and non-equilibrium thermodynamics suggests the following flux-force pair:
(2)





Derivation of Fick’s first law
In the near-equilibrium regime, transport is linear to the driving force:
(3)







Now by comparing Eqs. (3) and (1), we notice that Fick’s first law is equivalent to saying that is the cause of
. The easiest example corresponding to this situation is when the mobile species of interest is some dilute neutral species (e.g. isotope tracer) that follows ideal-solution behavior inside a host lattice. In this special case, configurational entropy determines the chemical potential such as
, where
is the mole fraction of the species,
is the maximum number of sites,
is
of some reference state, and
is the gas constant. By taking a spatial derivative, we get
(4)


Diffusivity from Fick’s law adopted to general equations
Now that you have seen how Fick’s first law is for such a special case, you might wonder how we should treat more general cases. Since Fick’s first law became so famous, it effectively turned into the equation that defines diffusivity. In other words, for non-dilute or non-ideal cases, we still use Eq. (1) as a “reference” equation to define . Recall that the actual general equation is Eq. (3), and that Fick’s first law provided us the relation
. We can thus rewrite the general equation, adopting
as defined by Fick’s first law:
(5)

Let’s say Eq. (4) is modified by a thermodynamic factor to correct for non-ideal behavior:
(6)





Recap
The general driving force for particle flux is the chemical potential gradient . Fick’s first law is a special case when
originates from configurational entropy in the dilute limit (the
term). Fick’s first law defines the diffusion coefficient
, which is then used to describe more general situations.
Last modified: Sep. 5, 2020
Photo Credit: Sarah Richter
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