Cooling history

Figure 1. Bow development for two cooling histories of the composite panel. At the figure, the bow (m) is plotted versus cooling time (sec) for two cooling regimes. (Reprinted with permission from ref. 7. Copyright 1986 SPI, Inc.)...

The most often encountered thermal history by geologists is continuous cooling from a high temperature to room temperature (such as cooling of volcanic rocks, plutonic rocks, and metamorphic rocks). One of the many ways to approximate the cooling history is as follows ... 

The P-T-t paths of metamorphic rocks may be much more complicated. They are usually heated first to a peak temperature and then cooled to room temperature. There are a variety of metamorphic rocks. Figure l-18c shows a hypothetical cooling history of an ultra-high-pressure metamorphic rock, which was subducted to great depth and then returned to the surface. Before subduction, the premetamorphic rock could be a basalt or a sedimentary rock. From 0 to 2 Myr, the slab is modeled as being subducted at 0.08 m/yr and an angle of 45°. At 2 Myr,... 

Figure 1-20 Explanation of closure time, closure age, and closure temperature (Tc). (a) The cooling history, (b) " Ar accumulation history. The long dashed line shows the accumulation of " °Ar if there were no Ar loss since formation. The thin solid curve shows a real accumulation history. The short dashed line shows how the age is obtained from the present-day " °Ar/" °K ratio. For a mineral grain cooling down from 1200 K, when the temperature is between 1200 and 1037 K (at 5 Myr), all °Ar is lost once produced. Then from 1037 to 571K (at 30 Myr), there is partial loss, and the loss becomes smaller and smaller. Below 571 K, essentially all newly produced " Ar is retained. When °Ar/ °K ratio is determined, one calculates the age based on the present-day °Ar/" °K ratio and the age corresponds to the time of closure ta about 20 Myr). That is, the age is 130 Ma, although the mineral formed at 150 Ma. The temperature at t— 20 Myr is the closure temperature ( 704 K). Adapted from Dodson (1973).

Figure 1-21 Cooling history of two granitoid samples (78-419 and 78-592) from the Separation Point Batholith of New Zealand. Data are from Harrison and McDougal (1980). The emplacement age of the batholith is > 115 Ma. The curves are plotted to guide the eyes. It can be seen that cooling rate was high initially and decreases gradually as the temperature approaches surface temperature, as expected. The temperature versus time relation is roughly exponential.

The two parameters, Tf and Tb, are very useful in inferring cooling rates and it is helpful to give their expression under a more general cooling history. The expressions of Tf and Tb can be obtained using the definition of mean times (Equation 1-60) ... 

Although Xc characterizes the cooling history, sometimes one would like to know the cooling rate q rather than Xc. For asymptotic cooling, the cooling rate q is... 

Because the diffusion properties differ for different minerals, by dating several minerals in a single rock, one would obtain different apparent ages. The curve of Tc versus represents the cooling history (Figure 1-21). 

In addition to the fraction of mass loss as a function of a = (jDdt) /a, it is of interest to examine how the fraction of mass loss change with time and temperature during cooling to understand the concept of closure. Figure 5-13 shows how the remaining fraction in the phase (1 — T) depends on time and temperature for a specific cooling history. In this example, the whole history of the mineral is 100 Myr. There was mass loss in the first 5 M5n", or at T > 850 K. As the system is cooled below 850 K, no more mass loss occurred. Hence, the system became closed at the temperature of about 850 K. 

Figure 5-14 Diffusive loss of Ar that was initially in hornblende during cooling after complete cooling down t=rx>) for asymptotic cooling history with (a) a fixed cooling timescale but varying the initial temperature and (b) a fixed initial temperature but varying the cooling rate.

Figure 5-19 Numerically calculated " °Ar concentration profile (solid curves) compared with pure growth profile (dashed lines) at two times during a single cooling history. Same input parameters as Figure 5-18.

Box 5.1 Derivation of Equation 5-1 25 for the special case of first-order reversible reactions with f = 2 b, and an asymptotic cooling history with Too = 0 K. 

From the assumed cooling history of T= 1500/(1 + t/lO ) with tin years, the input cooling rate q at Tae is ... 

Cooling history of anhydrous glasses based on heat capacity measurements... 

For a given cooling history, the diffusivity depends on time, and the mean diffusion distance may be estimated by (JD df). Thus, if we know the dependence... 

Figure 5-25 (a) Diffusion profile across a diffusion couple for a given cooling history. This profile is an error function even if temperature is variable as long as D is not composition dependent, (b) Diffusion profile across a miscibility gap for a given cooling history. Because the interface concentration changes with time, each half of the profile is not necessarily an error function. 

The above results may be applied to infer the critical cooling rate for the concentration of the core to be affected by diffusion. It is necessary to define precisely what is meant when we say "the center is affected by diffusion." If we use center concentration of Ci + 0.01 (Cmid - Ci) as the criterion for center concentration to be affected by diffusion, then it would occur at z = 2(fDdt) /a = 0.3947. For an asymptotic cooling history, this means that 2 Dox) la = 0.3947, or Doxla = 0.0389. Combining with Equation 5-137 that J = RTq/ xE), we obtain the critical q ... 

Sometimes, the profile is so short that it cannot be resolved by the measurement technique. Such information may also be applied to constrain cooling rate. For example, if the spatial resolution of the measurement is I, the absence of a profile (i.e., a step-function profile) means that jDdtasymptotic cooling history, then Dqt < f, leading to... 

Next we turn to the inference of cooling history. The length of the concentration profile in each phase is a rough indication of (jDdf) = (Dot), where Do is calculated using Tq estimated from the thermometry calculation. If can be estimated, then x, Xc and cooling rate q may be estimated. However, because the interface concentration varies with time (due to the dependence of the equilibrium constants between the two phases, and a, on temperature), the concentration profile in each phase is not a simple error function, and often may not have an analytical solution. Suppose the surface concentration is a linear function of time, the diffusion profile would be an integrated error function i erfc[x/(4/Ddf) ] (Appendix A3.2.3b). Then the mid-concentration distance would occur at... 

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