1. Diffusion (continued)
Lecture 17

2. Last time we introduced Fick’s First Law
Note minus sign. This means diffusion moves down the concentration gradient, i.e., from high to low concentration.

3. Deriving Fick’s Law
On a microscopic scale, the mechanism of diffusion is the random motion of atoms.
Consider two adjacent lattice planes in a crystal spaced a distance dx apart. The number of atoms (of interest) at the first plane is n1 and at the second is n2.
We assume that atoms can randomly jump to an adjacent plane and that this occurs with an average frequency ν (i.e., 1 jump of distance dx every 1/ν sec) and that a jump in any direction has equal probability.
At the first plane there will be νn1/6 atoms that jump to the second plane (there are 6 possible jump directions). At the second plane there will be νn2/6 atoms that jump to first plane. The net flux from the first plane to the second is then:

4. Deriving Fick’s Law
We’ll define concentration, c, as the number of atoms/unit volume n/x3, so:
Letting dc = -(c1 - c2) and multiplying by dx/dx
Letting then

5. Continuity Equation or: At fixed x and t
Imagine a volume and a flux of a species into that volume, Jx, at x and a flux out, Jx+dx, at x+dx.
Suppose nx atom/sec at x and nx+dx atoms/sec pass through the plane at x and x+dx, respectively.
The fluxes at the two planes are
Jx = nx/dx2-sec
and Jx+dx = nx+dx/dx2-sec.
Conservation of mass dictates that the increase in the number of atoms, dn, within the volume is just what goes in minus what comes out.
Over an increment of time dt this will be dn = (Jx - Jx+dx)dt. The concentration change is
or:
At fixed x and t

6. Deriving Fick’s Second Law
Continuity equation relates any change in flux along the gradient to change in concentration with time.
Since:
we have:

7. Fick’s Second Law
The second law says that the change in concentration with time at some point on a conc. gradient depends on the second derivative of the concentration gradient.
The straight line (∂2c/∂x2) is the steady state case. It is also the situation towards which a system will evolve over time.

8. Solutions to the Second Law
There is no single ‘solution’ to the second law, rather there are a variety of solutions that depend on the boundary conditions of a particular situation.
As a first example, consider a infinitesimally thin layer (e.g., Ir at K-T boundary) of material at x = 0 at t = 0. These, together with mass conservation, are our boundary conditions.
Mass conservation is

9. Thin Film Solution Boundary conditions were x = 0 at t = 0 and
Solution, from Crank (1975), fitting these conditions is:
If diffusion can occur only in one direction:

12. Adjacent Layers
Now suppose with have one layer of finite thickness with concentration co abutting a second with c=0.
Crank’s solution is to superimpose the solutions for an infinite number of thin flims, dξ, and to sum their contributions to the total concentration at Xp.
Solution is:

13. Adjacent Layers The solution to: is
where erf is the error function and has the approximate form:
also available as the ERF function in Excel.

14. Other examples
Book has several other examples such as diffusion of Ni in a spherical, zoned olivine crystal with an initially homogeneous conc. of 2000 ppm and a conc. fixed at the rim by equilibrium with a magma at 500 ppm.

15. Diffusion in Multicomponents Systems
So far, we have considered only what is called tracer or self diffusion - diffusion of a species who concentration is so small it has not affect on the system.
By definition, diffusion is a process in which there is no net transport across the boundary of interest (otherwise it is advection). Mass & charge balance require diffusion in one direction to be balanced by diffusion in another.
More generally, we can consider
“Chemical” diffusion
Multicomponent diffusion
Multicomponent chemical diffusion

16. Chemical Diffusion
“Chemical” diffusion refers to non-ideal situations where for one reason or another, we cannot treat our diffusing species as an inert particle, i.e., chemical interactions occur, we replace concentration with the chemical potential gradient.
where L is the chemical or phenomenological coefficient.
We have the additional task of deducing chemical potential from concentration, otherwise, treatment is similar.

17. Multicomponent Diffusion
This is where the concentration of an species is great enough that its diffusion depends on the diffusion of another species, e.g., Fe and Mg in ferromagnesian silicates (these elements tend to occupy the same site, so Mg can move in only if Fe moves out, etc).
Di,k is called the interdiffusion coefficient and computed as:
where n’s are the mole fractions.
The total solution is J = -DC (minus sign missing in book)
where J and C are (vertical) vectors of species and D is a matrix.

18. Multicomponent Chemical
Multicomponent chemical diffusion is a combination of the latter two, where we use interdiffusion coefficients and chemical potential.
Diffusion matrix is:
J = -LC
Bruce Watson’s experiments dissolving quartz sphere in molten basalt show an example where chemical diffusion must be used - note Na and K diffusing up a concentration gradient!

19. Mechanisms of Diffusion in Solids
Exchange: low probability
Interstitial: pretty much limited to He, etc.
Interstitiality: also low probability
Vacancy: most likely
Diffusion will then depend on both the probability of a jump and the probability of a vacancy.
Vacancies will increase with T:
where k is a rate constant and EH and activation energy for vacancy creation.

20. Diffusion Rates Probability of atom making a jump to vacancy is
where ℵ is number of attempts and EB is the activation or barrier energy.
Combining, total diffusion rate will be product of probability. of a vacancy times probability of a jump:
where m and n are constants
At higher T, thermal vacancies dominate and
Bottom line: like other things in kinetics, we expect diffusion rates to increase exponentially with temperature.

21. Diffusion Rates
In general, the temperature of the diffusion coefficient is written as:
In log form:
At lower temperature, where permanent vacancies dominate, we might see different behavior because:

22. Determining Diffusion Coefficients
In practice, one would do a experiments at a series of temperatures with a tracer, determine the profile, solve Fick’s second law for each T.
Plotting ln D vs 1/T yields Do and EB.
Diffusion coefficient is different for each species in each material.
Larger, more highly charged ions diffuse more slowly.

23. Reactions at Surfaces
Homogeneous reactions are those occurring within a single phase (e.g., an aqueous solution). Heterogeneous reactions are those occurring between phases, e.g., two solids or a solid and a liquid. Heterogeneous reactions necessarily occur across an interface.

24. Interfaces, Surfaces, and partial molar area
By definition, an interface is boundary between two condensed phases (solids and liquids).
A surface is the boundary between a condensed phase and a gas (or vacuum).
In practice, surface is often used in place of interface.
We previously defined partial molar parameters as the change in the parameter for an infinitesimal addition of a component, e.g., vi = (∂V/∂n)T,P,nj. We define the partial molar area of phase ϕ as:
where n is moles of substance.
Unlike other molar quantities, partial molar area is not an intrinsic property of the phase, but depends on shape, size, roughness, etc. For a perfect sphere: