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Diffusion -continued

Diffusion -continued. Determining D, Arhenius plots, Intro to closure temperatures. Recall the general solution for an infinite sheet immersed into two half spaces. What controls the speed of decay of C = f(t)?. Answer: The width of the sheet - h D - the most important.

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Diffusion -continued

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  1. Diffusion -continued Determining D, Arhenius plots, Intro to closure temperatures

  2. Recall the general solution for an infinite sheet immersed into two half spaces

  3. What controls the speed of decay of C = f(t)? Answer: The width of the sheet - h D - the most important

  4. Is the diffusion coefficient constant? If not, what can cause it to change? Temperature ( the most important factor) Material properties (also very important) Pressure, oxygen fugacity - less relevant

  5. Physics of diffusion • So far diffusion has been described only mathematically • What causes transfer of atoms in a crystalline lattice? • What are the mechanisms of transfer?

  6. Mechanisms

  7. Diffusion coefficient • As the T increases, the probability of an atom having sufficient local thermal energy to jump from its original position to an adjacent one by the mechanisms above. • It follows that diffusion is a thermally activated process and that the change in D is primarily determined by T.

  8. Formal expression

  9. Arrhenius distribution • In words: as the temperature is lowered, a threshold is reached at which the number of vacancies is primarily impurities-derived, and not thermal. • Below that temperature, there is low probability of atoms moving freely in and out of the system.

  10. Plotting Arrhenius distributions If one plots Log D vs. 1/T, D forms a linear array.

  11. Other factors Diffusivity decreases with pressure, but the effect is small compared to temperature

  12. How does one calibrate a diffusion curve? • Get several data points, D and T • Plot them on an Arrhenius curve (log D vs 1/T) • Fit them like an isochron and recover the “relevant” diffusion parameters D0 and E.

  13. How do we get D? • Imagine solving an equation like for a given temperature (T of an experiment): For a grain that was doped with an artificial or spiked isotope of the element of interest. Use C, C0, H, and t all of which are under control. Ultimately, get (D,T) pairs.

  14. Example- Pb diffusion

  15. Example 1 Ar diffusion in Kspar

  16. Example 2 Ar in mica- note the grain size effect

  17. Example 3 Various elements in zircon

  18. He diffusivity in monazite standard 554 (from the Catalina Mts)

  19. Other effects LogD Water pressure on U diffusion in zircon

  20. Back to diffusion profiles Need to quantify a large vs small relaxation of the C function and thus the likelihood of exchange of an element and its isotopes with surrounding medium.

  21. The concept of closure temperature • A simplified version of just that, that relies on a number of assumptions; has been defined by Dodson (1973), reason why this mathematical treatment is sometimes referred to as Dodson theory; • A closure temperature for a diffusing species can be solved for numerically and is a unique number for an element undergoing volume diffusion in and our of a mineral;

  22. More.. • More modern approaches to closure temperature have showed that different parts of a crystal close at different T, and that one can effectively defined a closure T profile for certain materials (Dodson, 1986; Ganguly, several recent papers); • One key aspect of closure T concept is to know its limitations and simplifying assumptions.

  23. From Hodges, 2005

  24. Assumptions At a peak temperature To , the mineral does not retain any daughter products; Tc is therefore independent of To which is only true for relatively fast diffusing species; slow diffusing species require a totally different treatment and they are essentially leaky chronometers The geometry of the diffusing species can be approx as a sphere, cylinder or plane sheet The phase is surrounded by an infinite medium that has much greater diffusivity than the phase; The cooling path is a very specific one:

  25. Cooling rate dT/dt is proportional to 1/T The only way one can make this tractable mathematically

  26. Formulation E- activation energy, R- gas ct, A, geometric parameter, constant, D0-preexponential constant, R, grain size, dT/dt = cooling rate. Tc = closure temp

  27. Geometric constant A A= 55 -sphere A=27 cylinder A=8.7 plane sheet

  28. What is important in this formula? • The equation is iterative in Tc, needs to be solved numerically, usually in two iterations; • It depends on grain size, shape, and cooling rate - I.e. if a rock cools fast it’s minerals have different closure temps compared to a slower cooling path ..

  29. Example • Consider a biotite that cooled at 5 C/Ma, through closure temp for argon • E= 47 kcal/mol, and D0 = 2 cm2/sec, r= 0.3 cm • Assume an cylinder geometry (A=27), one can solve for Tc of Ar in biotite = 300 0C • We can use that closure temp for Ar-Ar geochronology = the Ar-Ar system will start ticking at around 300 C (and below) if all the assumptions above are true…

  30. How do these parameters influence the result? • Larger grain size- higher Tc • Faster cooling rate, higher Tc

  31. Example • Ar in hb

  32. Example 2 • Pb in various

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