Petr Pilný

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Cube mapping


The following text explains the principle of determination of the activation energy and other parameters using the mapping for JMA model – more precisely the determination of the activation energy, pre-exponential factor and parameter of the model. The correct solution will be sought in three axes – one could say in virtual cube – hence the name of the modulus in the OriTas program (since the version 3.9.x.x).

How the scan of the whole cube looks like

Picture 1

How the scan of the whole area looks is evident since the first chapter describing the cube mapping. In fact, it is a curve (figure 1) covering an infinitive solutions that are able to fit the data with good R. If we take each wall of the cube an translate into them the obtained dependence E/lnA/n then we get:

  • dependence of E on lnA is straight line
  • dependence of E on n is a power
  • dependence of lnA on n is a power
Plot 1

From the given dependences the most easily (see figure 1,2,3) determined can be the equation of the line, the other has the problem in data fitting (see following text). Evaluation of only one curve does not provide reasonable result (combination of E, lnA, n) following the suitable parameters.

Plot 2

Infinitely solution for one heating rate, one solution for the two rates

Plot 3

The solution is to apply several heating rates when the dependence of E/lnA simplifies its solution into the searching of the point, where the (two) lines intersect (see figure 4). The solution is then one for each of the two heating rates – although it does not seem it has a big advantage. In the case that some heating rate does not match to other results it could be a problem of experimental data or even of experiment (wrongly prepared sample, complicated process, etc.)

How it works in OriTas

Plot 4

Now it is a time to look in detail on process of mapping creation in OriTas – it is necessary to pass the “whole” area o parameters. Right at the beginning it is necessary to mentioned that it is of course impossible. If each axis (each parameter) split into 1000 pieces then we have to calculate 1000x1000x1000 of simulations and verified their agreement with downloaded data, all this for several heating rates. The result is unacceptable several hours/days of calculation to obtained values, which can conveniently be obtained also by another method in much shorter time.

The solution is either reduction of the area or counting of only selected part of all possible combinations. Based on the random selection, OriTas calculates only the small percentage (typically 1-2 %) of all calculations, so that the calculation of total number in the miliones is proceeded. This number is still relatively high and for 8 heating rates takes the time of analysis in minutes or tens of minutes. In the future, there is a way to minimise the time into few seconds (see end of this text).

OriTas starts passing through the whole cube, point by point, for each combination of parameters performs simulation and each simulation is compared with the experimental data and obtains the value of R (it is not really correlation coefficient, but also reflects the quality of the fit in the range from 0 to 1; the marking is possibly a bit confusing). If there is a reasonable match to the original data, the result is stored for further processing. Due to the above described distribution of solution for one heating rate (E vs. lnA is a straight line) is not required perfect matching of the data – this reduces the number of solutions found necessary to obtained required parameters. In the case of mentioned dependence E/lnA (straight line) simply two combinations of parameters is sufficient.

When the dependences of E/lnA (and also E/n and lnA/n) are obtained for all heating rates, then is sufficient to solve a system of two equations for each of two heating rates (found the intersection of two lines). The solution is the value of activation energy and lnA – the final value of E and lnA is then the average of the all obtained values. The great advantage is that in the case some heating rate (measurement) will not match the others, it will appear as E value determined from the combination of this heating rate with others will not match – so we can easily eliminateerroneous measurements or look for the reason why it is out of the other measurements.

Dependences of E/n and lnA/n are the power law and their spacing and obtaining of the equation for parameter n determination is complicated. Moreover, thanks to the distribution of acquired solutions into the space not into the curve is complicated and imprecise. Much easier is obtaining of E and lnA values by performing one additional scan for each heating rate and all values of n in a certain interval (typically 0.5-5) and then chooses one of them as the most appropriate solution. Again, we get the value of parameter n for each heating rate, so we can easily track the dependence of n on the heating rate.

What about other models than JMA

The testing of other models have not been done yet, but the whole computation cycle is universally written and after minor modification can be apply for RO(n) model as well as for several basic models (R2,R3,D2, D3). Theoretically, it would also be possible to apply SB model (using dependence m/n). However, these tests will be made in the future and based on their outcome I will see what to do next.

Acceleration algorithm or less drunk coffee and more results

Finally, I only briefly mention the possibility to speed up the algorithm and the process for a total run time in seconds. If the calculation is optimized so that in case of finding a sufficient number of solution for obtaining E/lnA dependence continued calculation of other heating rate, there is a further saving of time required to scan the whole area. My goal is adjusting current algorithm to achieve total time under 3-5 minutes now.

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