An introduction to diffusion Thermochron, Fall 2005
TEXTS • Heat conduction - Carslaw and Jaegger • Diffusion - Crank, 1975
Diffusion is analogous to heat conduction • The rate of transfer of diffusing substance through unit area of a section is proportional to the concentration gradient measured normal to the section (Fick’s law, 1855).
Similar to heat transfer- same thing, really • Rate of transfer of heat per unit area is proportional to the thermal gradient (Fourier’s Law of heat conduction, 1822). • One deals with the diffusion of heat the other with the diffusion of mass.
Diffusion coefficient QUESTIONS?? What are the units for D? Why the negative sign for both forms of diffusion?
Answers • The minus sign reflects the fact that diffusion takes place in the direction opposite to increasing concentration (or heat); • Units are area/time, e.g. cm2/sec. • These simple formulations apply only to perfectly isotropic mediums.
Deriving the fundamental equation for diffusion in an isotropic medium Requires applying the law of conservation of energy to a volume, e.g. a parallelepiped of lengths 2x, 2y, and 2z. Ein+Egenerated=Estored+Eout
Some implications • The changes in concentration due to diffusion over a characteristic time interval t will propagate a distance on the order of: (Dt)1/2 • Similar equation for conduction • This simplification x= (Dt)1/2 gives us a mean to determine back of the envelope diffusion scales. • If the length scale is x (grain size), how long will it take for diffusion to operate at that length scale? Note: As you will see soon, diffusion is a thermally activate process and the assumption of constant D is a very simplifying one. However, D is typically constant over a given small range of temperatures.
Example • Upper mantle minerals have grain sizes of mm length scale. Let’s say 1 mm on average. • The diffusion coefficient for most elements in olivines and pyroxenes (at T= 1000 0C) is on the order of: 1 x 10-15 cm2/sec. • What is the time scale of diffusional equilibration of a mantle rock and what is the prospect of using these rocks for geochronology of mantle events?
Answers: • Takes less than 1 My (in this case 0.3 Ma) to erase any diffusional gradients; • Prospects of mantle rocks to preserve any kind of diffusional gradients and thus age info = not good to zero.
Same applies to heat conduction • Similar to x= (Dt)1/2 , one can apply x= (Kt)1/2 to heat conduction; • E.g. what is the time scale for a granite body of size x to cool? r cold 1 km body, 10 km, 100 km body hot Note the non-linearity
Does a magma body ten times larger cool ten time slower? 100 times slower !
Goals • Fundamentally, we are in pursuit of diffusion laws and the solutions to those laws because they constrain the effective temperatures below which the effective transport of elements and isotopes of interest stops. • That is when the isotopic clock starts and quantifying the conditions under which that happens is equally important as the decay process itself.
A road map for the next three lectures • Find simple analogies to exemplify diffusion of mass • Try to work out analytical solutions to super simplified cases (sphere, plane geometries) • Bring in the math baggage associated with this; • Apply these solutions to the classic formulation of closure temperature (Dodson, 1973) • Modern variations on the closure temperature theory
An example • Imaginary experiment: pool of water and drop a chlorine tablet. The pool is an infinite reservoir that has no chlorine to start with. • What will be the distribution of Cl as a function of x and t in the pool?????
Set initial and boundary conditions Ctablet=C0; Cpool=0 t=0 Differentiating, one obtains the following solution:
If the concentration is originally located in a point source and the medium is a plane: Which is the classic, analytical solution to a point source and works ok for your pool although the chlorine tablet is not a infinitely small and the pool is not a infinitely large sheet..
The problem is symmetric with respect to x; • The Pi-form of the constant is determined by the fact that the chlorine diffuses not just in the directions x and -x, but radially around the tablet.
Analogy • A garnet immersed in an infinitely large reservoir of biotite
Case 2: two long metal bars placed in contact end to end. • Another classic example in the diffusion book. • Has an analytical solution; • Exemplifies the general form of solutions to more complicated problems;
Solution: Consider that the half space is composed of an infinite number of point sources (our previous problem). Because of that, one can obtain an analytical solution by superposing the infinite # of point sources. Overall, in math, this sumation of linear solutions leads to an exponential distribution of the solution.
Looks like the solution is of the form of a well-known mathematical function called the error function (erf) Solutions to erf are available in tabulated form or as an excel add-in etc etc.
Solution: This is very similar to the heat conduction solution for equivalent boundary / initial conditions which the temperature at the interface stays ct as the average between the two t’s. applies to dike intrusions etc
Case 3: plane source of limited extent - • A particular case of 2 with the source is limited from -h<x>+h • Integration is thus from x- h to x+h instead of x to infinity;
Some concluding remarks from this intro lecture • Mass diffusion is (almost) identical in its treatment with heat conduction - aka heat diffusion • Simplifying calculations can give us order of mag info on length and time scales of diffusion • Analytical solutions are found for some of the simplest initial and boundary conditions • The most common type of solution of these problems involves the error function; • More complicated diffusion problems do not have analytical solutions; discussions so far focused on constant D and isotropic 1D examples.