An introduction to diffusion

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## An introduction to diffusion

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**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.