Talk:Fick's laws of diffusion

Suggestions for improvement

1. The reasoning behind Fick's 1st Law is pretty evident; can someone provide an explanation or brief derivation of Fick's 2nd Law?

~Consider Fick's 1st law acting either side of a given point. Then, as I believe Darken put it: "What goes in but doesn't come out, stays there", hence Fick's 2nd! —Preceding unsigned comment added by 143.167.129.202 (talk) 10:14, 9 October 2007 (UTC)

2. I've seen another form of Fick's 2nd Law that goes:

$\frac{\partial \phi}{\partial t} = \frac{\partial}{\partial x} (D \frac{\partial \phi}{\partial x})$

What's the difference between this form and the form listed in the article?

3. Including the del-operator forms of the equations would be helpful.

• 1) Imagine a plane in space, and two small volumes A and B straddling the plane (one on each side). In a unit of time, a particle in volume A has the same probability to cross the plane into B as the particle in B has to cross into A. However, if there are more particles in box A than B, then on-net, more particles will flow from A to B.

Let P be the probability of any particle crossing the boundary, let NA and NB be the number of particles in volumes A and B respectively. Then the expected net number of particles to cross the boundary in a unit of time , Q, is the expected number of particles that leave A minus the expected number of particles that leave B:

$Q = P*NB - P*NA$

Dividing by the volume V of the box,

$q2 = -P\frac{NA-NB}{V}$

As the volume of the box goes to zero, NA/V-NB/V is spatial derivative of the concentration dc/dx. As the area of the boundary and the time interval correspondingly go to zero, q2 becomes the flux q. This results in Fick's Law:

$q = -P\frac{dc}{dx}$

This easily generalizes to higher dimensions since you could do the same thing for a plane in each of the x,y, and z directions:

$q = -P\frac{dc}{dx} - P\frac{dc}{dy} - P\frac{dc}{dz} = -P \nabla c.$

If, for some reason, the particles like to flow more in one direction than another (say they are molecules in a material that permits flow more easily in the z-direction than the x or y directions), then you would have a different probability in each direction:

$q = -P_x\frac{dc}{dx} - P_y\frac{dc}{dy} - P_z\frac{dc}{dz}$

If you hadn't oriented your coordinate system exactly along the high and low probability directions, then when you change coordinates you would get cross terms:

$q = -(P_{xx}\frac{dc}{dx} + P_{xy}\frac{dc}{dx} + P_{xz}\frac{dc}{dx} + P_{yx}\frac{dc}{dy} + P_{yy}\frac{dc}{dy} + P_{yz}\frac{dc}{dy} + P_{zx}\frac{dc}{dz} + P_{zy}\frac{dc}{dz} + P_{zz}\frac{dc}{dz})$

This can be expressed more concisely as

$q = -P \nabla c$

if we understand P to be a matrix (the diffusivity tensor).

• 2) This is a combination of Fick's law with the conservation of mass law, which says that the net flux through the boundary of any volume must equal the change in concentration in the volume. Mathematically,

$\frac{d}{dt}\int_V c dV = \int_{\partial V} q \cdot \vec{dA}$

Using the divergence theorem,

$\frac{d}{dt}\int_V c = \int_{V} \nabla \cdot q$

Since this holds for any arbitrary volume, we can take away the integral,

$\frac{dc}{dt} = \nabla \cdot q = -\nabla \cdot (P \nabla c)$

67.9.148.47 (talk) 07:37, 15 December 2008 (UTC)

4. There's an example of a 1D solution. It would be useful to include which equation does this solution solve (D constant). —Preceding unsigned comment added by 193.2.69.149 (talk) 14:15, 26 May 2009 (UTC)

Dimensions for D

Could someone explain how we obtain the dimensions of [length² time^-1] for the diffusion coefficient/diffusivity D?

• Well assuming x is in [length], t is in [time] and $\phi$ on the both sides is in the same units, then we don't have other choices. abakharev 01:39, 23 March 2006 (UTC)

D is velocity of flow, dimension length¹time^-1, divided by gradient of concentration, dimension length^-1. --Vaughan Pratt (talk) 07:02, 25 July 2008 (UTC)

Article name

Shouldn't the article be called Fick's laws of diffusion ? MP (talk) 09:20, 25 September 2006 (UTC)

• I was just thinking the same, although it's only minor maybe we should change itAzo bob (talk) 10:38, 31 March 2008 (UTC)
• I agree as well. I think I know how to do it (it is actually MOVING a page that is done to accomplish this). I haven't done this before though, so I want to read up on it some more . Help:Moving a page --Lacomj (talk) 22:06, 17 December 2008 (UTC)
• Done. Lone Skeptic (talk) 03:15, 11 December 2009 (UTC)

variant phi characters

This is picky, but some sources distinguish between the two ways of drawing phi, i.e. ɸ and φ. It's the same character, but different equations pick one or the other for historical reasons. I suggest picking one and sticking with it. Ojcit 22:59, 2 October 2006 (UTC)

I prefer $\phi$, to ensure that the symbol in text is identical to the symbol in equations (typeset with latex). Berland 11:21, 29 January 2007 (UTC)

A Biological Perspective

How does it combine with Graham's law to show "the exchange rate of a gas across a fluid membrane?" Graham's law deals with different molar masses, does that mean you just have to worry about the different molar masses of the gas with the fluid?

In a shake flask of microorganisms the demand for oxygen exceeds the supply; if you measure the oxygen in the liquid phase when the microorganisms are trying to grow exponentially, the oxygen concentration is observed to be zero. Does Fick's Law term of (P2-P1) change the partial pressure of oxygen only so as to raise the oxygen concentration only, when the experimenter mixes in some extra oxygen into the gas phase? Richard8081 21:46, 14 August 2007 (UTC)

History Section

Can somebody please improve the history section? It's not particularly important but still, for people who need to know the background information, there's a lot left to be desired.--Gabycs 21:12, 14 March 2007 (UTC)

Vandalism removed

Removed something about his cod piece. —Preceding unsigned comment added by 80.202.72.196 (talk) 21:08, 15 October 2007 (UTC)

Fick's law in biology

For the transport of CO2 across cell walls of leaves Fick's law takes the form dq/dt = -(c2 - c1)/R where c2 < c1 and R is referred to as the resistance and is used for uniform flow, i.e., the flow is constant in time. It would appear that the area of the membrane and its thickness would have to be constant. dc/dt=d(q/V)/dt=(dq/dt)/V=-Δc/RAΔx=-(1/RA)Δc/Δx so Fick's law is of the form,

dc/dt = - K Δc/Δx

which is not the diffusion equation. The flow goes down a concentration gradient. Also by the ideal gas law P=cR'T so dq/dt = -Δc/R = -ΔP/(R*R'T) = - K'ΔP so we have,

dq/dt = - K' ΔP

It might be better to redo Fick's law to make the constants independent of the area and thickness of the membrane so that they are characteristic of the material of the membrane and not the dimensions.

I will restore some of the changes I made. —Preceding unsigned comment added by Jbergquist (talk • contribs) 11:19, 21 October 2007 (UTC)

I have been reviewing the derivation of the diffusion equations for temperature and concentration. Both use a constitutive equation and a continuity equation stated in vector form. For thermal conductivity heat transfer takes place with no net exchange of carriers which would involve convection. Diffusion takes place at constant temperature and pressure.

The biological usage is that of Fick's first law. It differs from conduction which the resistance causes Joule heating while the passage of a gas through a barrier is associated with the Joule-Thomson effect which causes cooling. The matter requires further study. As stated the "flow law" may require some modification such as an A² term as in Poiseuille's law which also involves a connection between a flow and pressure differences. --Jbergquist 00:26, 22 October 2007 (UTC)

The term "rate of diffusion" can be confusing. The reference cited uses flux = P A Δc where the flux would contain an extra A. P is the permeability of the membrane and is more easily determined than a conductivity K. One should probably use K if one wants to express the law in terms of a constant which is not dependent on the amount of material or its geometry. So one would arrive at R = d/KA² as the formula for the resistance. --Jbergquist 23:33, 22 October 2007 (UTC)

The reason why one might want to use permeability is because it follows the same rules of combination as conductance since P=KA/d. Hence doubling the thickness of a membrane halves the permeability and doubling the area would double the permeability. --Jbergquist 00:11, 23 October 2007 (UTC)

• I think I'd prefer to see this kind of information in a separate article, with a link from this general article. I.e., this is more of a special case. --Lacomj (talk) 22:19, 17 December 2008 (UTC)

Ideas for improvement

Here are a few ideas for improvement of this article that I'd be able to help with. I appreciate any comments. Some are more radical than others. --Lacomj (talk) 22:45, 17 December 2008 (UTC)

1. Discussion (brief) of generalized transport equations (i.e., compare with heat, momentum, charge transport).
2. Derivation of Fick's 2nd Law
3. Expand section on solutions to Fick's 2nd law. Erf solution is most common, but there are a few other important ones.
   1. An example solution to a 2D case analogous to the 1D case? Lack of clear example in the literature.71.37.148.130 (talk) 22:31, 16 October 2011 (UTC)maehlen
4. Brief discussion of more advanced topics including:
   1. multicomponent diffusion (not simply binary diffusion)
   2. Onsager's equations & thermodynamics
   3. Random walks, stoichastics, and Green's function representations
   4. Diffusion isotropy
5. Move the sections on the diffusivity (temperature and pressure dependence) to a new article on Diffusivity-- which can be expanded considerably to include:
   1. Intrinsic diffusivity
   2. Tracer diffusivity

An idea for improvement

Fickian diffusion is an ideal case and real cases are different. Why the real case is different from fickian diffusion can be discussed

• sounds like a good idea to me, although anomalous diffusion can get quite complex, so it'd need to be a fairly general statement. I did touch a bit on this in the revised history section, where I mention that there is such a thing as "non-Fickian" diffusion. It can be expended a bit perhaps, or maybe this should instead be the emphasis for a separate page? --Lacomj (talk) 00:24, 24 December 2008 (UTC)

– You don't need to turn to anomalous (eg single file) diffusion to find examples of non-Fickian behaviour. There are simple examples in ternary mixtures (see eg the books and publications of Krishna) where transport goes against the direction of the concentration gradient. Most would consider this to be non-Fickian behaviour, since Fick's law is really a phenomenological law relating diffusive flux to concentration gradients. At best, it shows that the Fickian diffusion coefficients are highly dependent on fluid composition (much more so than the Maxwell-Stefan diffusion coefficients). Snowjeep (talk) 17:00, 16 October 2009 (UTC)

Re-location of text on diffusion coefficient to mass diffusivity

I believe it might be appropriate to re-locate the sections that deal exclusively with the diffusion coefficient (=mass diffusivity) to its own article (mass diffusivity). Any opinions, better ideas? Cheers. Stan J. Klimas (talk) 22:43, 5 March 2009 (UTC)

Correction within notes necessary

Hello,

the notes are not correct. For Ficks original article it has to read: "Poggendorffs Annalen der Physik" short: "Pogg. Ann."

Unfortunately the notes are hidden in some "ref.list" for which I do not know to get access. Perhaps somebody elso knows and can do this correction?

there is also the lonk to the abstract: http://onlinelibrary.wiley.com/doi/10.1002/andp.18551700105/abstract. I will try to get it into the page.

Best regards, Denkgenau (talk) 06:48, 3 May 2011 (UTC)

I corrected this. Next time you could just edit the whole article and do a cntrl-f search for the ref definition. Thank you for your effort. --Rosentod (talk) 09:07, 6 May 2011 (UTC)