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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 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
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
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?
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.
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.
Plotting Arrhenius distributions If one plots Log D vs. 1/T, D forms a linear array.
Other factors Diffusivity decreases with pressure, but the effect is small compared to temperature
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.
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.
Example 1 Ar diffusion in Kspar
Example 2 Ar in mica- note the grain size effect
Example 3 Various elements in zircon
He diffusivity in monazite standard 554 (from the Catalina Mts)
Other effects LogD Water pressure on U diffusion in zircon
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.
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;
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.
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:
Cooling rate dT/dt is proportional to 1/T The only way one can make this tractable mathematically
Formulation E- activation energy, R- gas ct, A, geometric parameter, constant, D0-preexponential constant, R, grain size, dT/dt = cooling rate. Tc = closure temp
Geometric constant A A= 55 -sphere A=27 cylinder A=8.7 plane sheet
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 ..
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…
How do these parameters influence the result? • Larger grain size- higher Tc • Faster cooling rate, higher Tc
Example • Ar in hb
Example 2 • Pb in various