the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 02 May 2024
An optimization tool for identifying Multiple Diffusion Domain Model parameters
Abstract. The Multiple-Diffusion Domain (MDD) model empirically describes the diffusive behavior of noble gases in some terrestrial materials and has been commonly used to interpret 40Ar/39Ar stepwise degassing observations in K-feldspar. When applied in this manner, the MDD model can be used to test crustal exhumation scenarios by identifying the permissible thermal paths a rock sample could have undergone over geological time, assuming the diffusive properties of Ar within the mineral are accurately understood. More generally, the MDD model provides a framework for quantifying the temperature-dependent diffusivity of noble gasses in minerals. However, constraining MDD parameters that successfully predict the results of step-heating diffusion experiments is a complex task and the assumptions made by existing numerical methods used to quantify model parameters can bias the absolute temperatures permitted by thermal modeling. For example, the most commonly used method (Lovera et al., 1997) assumes that no domains lose more than 60 % of their gas during early heating steps. This assumption is unverifiable, and we show that Lovera et al.’s (1997) procedure may bias predicted temperatures towards lower values when it is violated. To address this potential bias and to provide greater accessibility to the MDD model, we present a new, open-source method for constraining MDD parameters from stepwise degassing experimental results, called the “MDD Tool Kit.” This software optimizes all MDD parameters simultaneously and removes any need for user-defined Ea or regression-fitting choices used by other tools. In doing so, this new method eliminates assumptions about the domain size distribution. To test the validity of our thermal predictions, we then use the MDD Tool Kit to interpret the 40Ar/39Ar results of Wong et al. (2023) from the Grayback Fault, AZ, USA. Although the resulting thermal histories are consistently ~ 60–75 °C higher than those found by Wong et al. 2023), they agree with independent observations from apatite fission track, zircon fission track, and (U-Th)/He (Howard and Foster, 1996).
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CC1: 'Comment on gchron-2024-11', Daniil Popov, 22 May 2024
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This is an unsolicited comment, so I feel justified to leave aside the tool itself and instead write a few words on the broader context of this work. As someone whose entire PhD thesis was concerned with the utility of the MDD theory, I find it a little bit disappointing that the latest work specifically cited as criticising the MDD theory is that of Parsons et al. (1999). In fact, it is quite difficult let go of emotions here after having to deal with the kinds of reviews that my work received. Notwithstanding, I will try my best in focusing only on science.
The authors of this manuscript write that Lovera et al. (1997) make some unverifiable assumption about the domain sizes right in the abstract and then reaffirm this statements in the conclusions. I must disagree with this statement because I think that Lovera et al. (1997) along with other MDD studies make the whole bunch of unverifiable assumptions, beside which any inappropriate choice of domain sizes pales into insignificance. I would particularly stress the following:
- The MDD theory assumes that the dominant mechanism of 40Ar redistribution during the geological past of alkali feldspar is volume diffusion. However, there is actually no unambiguous evidence to support this assumption (following my work on the Itrongay feldspar the work of Flude et al., 2014, cannot be taken as such). Meanwhile, there is plentiful evidence for fluid-induced dissolution-reprecipitation in virtually all alkali feldspar, which is thus a more viable mechanism for moving 40Ar. This has been discussed at length in the context of 40Ar/39Ar dating by I. Villa and I. Parsons with co-authors, and many papers have been published after 1999. I would only stress here that the main alteration mineral in alkali feldspar that typically goes into a mass spectrometer (let us give it a benefit of doubt…) is alkali feldspar of slightly different composition (read I. Villa’s papers with this in mind!), and that it is not possible to deduce the mechanism of 40Ar loss from step-heating data without mineralogical evidence as explained in Popov and Spikings (2020).
- The MDD theory assumes the presence of some intragrain domains that remain unchanged throughout laboratory outgassing. However, neither the inventors of the MDD theory nor their following provided a complete explanation of which specific parts of alkali feldspar grains represents those domains, and there is no good evidence to support the unchangingness of alkali feldspar microtexture during laboratory outgassing. Rather, the opposite is truth, and the existing mineralogical evidence on the behaviour of alkali feldspar during in vacuo outgassing suggests that it undergoes a range of modifications (Fitz Gerald et al., 2006; Parsons et al., 2010; Popov et al., 2020), of which I find heating-induced fracturing to be particularly important one in the context of diffusion studies. Contrary to the predictions of the MDD theory, very large fragments of alkali feldspar do not exhibit the kind of MDD-like behaviour that smaller fragments do, however their behaviour is consistent with prolonged heating-induced fracturing during laboratory outgassing.
- The MDD theory further assumes that the boundaries that separate the alleged intragrain domains provide infinitely fast pathways for Ar loss both during laboratory heating and the geological past. Leaving open cracks aside, the only conceivable candidate for this role are incoherent grain boundaries, so the assumption can be taken as relating to these. Although assuming that incoherent grain boundaries provide infinitely fast pathways for chemical and isotopic transport is frequent in geology, the validity of doing so is not obvious to say the least. For starters, there are examples for the Ar-in-alkali-feldspar system where this assumption comes into clear contradiction with empirical evidence, including laboratory outgassing results for the Shap feldspar (Popov et al., 2020) and 40Ar/39Ar dating results for authigenic feldspar (Mark et al 2008). The finiteness of grain boundary diffusivity is further illustrated by the experiments of Baxter et al. (2007), in which 37Ar and 4He contained in grain boundaries of polycrystalline diopside aggregates were not released until heating to temperatures approaching those needed for the release of these isotopes from diopside itself. Interestingly, a different view on the noble gas behaviour in grain boundaries can be found in studies modelling their diffusion in nuclear fuel rods: it is generally assumed that noble gases leaving individual UO2 crystals need to form an interconnected network of bubbles at the boundaries before they can move far from their source crystals (Pizzocri et al, 2020, Journal of Nuclear Materials; Kim et al., 2022, Materials Theory). The situation could even be even worse: He migration in grain boundaries in Fe is reportedly slower than within crystals (Deng et al., 2013, Journal of Nuclear Materials).
In the light of the cited evidence, I think that it is imperative to be able to mineralogically identify the specific diffusion domains in any given sample before applying the MDD theory. Equivalently, it is imperative to identify the boundaries that separate the domains from one another, and once this is done, it is possible to proceed to evaluating how likely it is that the 40Ar concentration in them was maintained at 0 during the sample’s geological past. Would the tool offered in this manuscript be of any use in such an endeavour? I don’t know. One could probably argue that it could generate plausible grain size distributions that could be then sought in the analysed sample while keeping in mind the aforementioned problems with the MDD theory. However, to me this looks pretty much like introducing yet another epicycle into a model that clearly fails at fitting the entirety of the available empirical evidence.
Does my assessment of the manuscript mean that I am against its publication in this journal? I actually have no strong opinion about that. Although I respect this journal at least to some degree, my overall disaffection with the publication business and the institute of peer review is such that I am quite indifferent about their fate in general, so long that enough people adhere to the DORA convention. Besides, I have not provided any critique to the tool itself and only pre-emptively commented on its potential use (I would only add that the ‘validation’ approach from section 6 does not validate anything as explained in Popov and Spikings, 2020). Therefore, it well may be that the manuscript has some value as do abstract mathematical theories, and if it ever gets published with the proud title of a 'peer-reviewed paper', it’s better be here with the associated warnings against its blind application to geological problems, ideally repeated in the main text.
Just don’t move to geological interpretations of MDD modelling results before the assumptions behind it have been validated by mineralogical evidence!
Daniil Popov
Citation: https://doi.org/10.5194/gchron-2024-11-CC1 -
RC1: 'Comment on gchron-2024-11', Anonymous Referee #1, 19 Jun 2024
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This manuscript outlines a new optimized method for constraining Multiple Diffusion Domain (MDD) model parameters from step-wise degassing experiments. The authors argue that the most commonly used method (Lovera et al. 1997) could be yielding too low temperature predictions due to the assumption that each domain loses no more than 60% of its gas during the lower temperature heating steps. The user can introduce bias when fitting a regression to the lower temperature steps when quantifying the activation energy, which in turn can yield lower temperature predicted thermal-histories. The open-source software presented here (MDD Tool Kit) does not need user inputs for regression fits, and so does not rely on any assumptions about domain size distribution. To test the software, the authors then modelled a previously published dataset (Wong et al., 2023) using the MDD Tool Kit. The resulting thermal histories are consistent with the geological and independent constraints, but are consistently higher temperature to those predicted by Wong et al. 2023.
I find that this manuscript is a good fit for GChron. It is well written with clear figures and tables. The authors lay out problems they see with the current most used method, and describe the model well. I have a few general and specific relatively minor comments.
General points:
- The authors mention the criticism of the MDD model – it would be valuable to include a brief section discussing both the criticisms and supporting evidence of the model.
- MDD theory has many potential problems – e.g., the assumption that phases remain constant during heating experiments. While it is out of the scope of this work to show evidence for or against various assumptions, the manuscript would benefit from a section including more detail on what the different assumptions actually are and to include references to previous studies on how these have been tested.
- Which choice of misfit method do you think is best to use? How would you advise a user of this tool kit to evaluate which is the best method to use for their data? In some cases, the results can be pretty significantly different (e.g., the high temperature portion of GR27’s cooling history).
- There are a lot of variables in this study, a table defining them could be helpful for readers to refer back to.
Specific points:
- Figure 1 A and B the x axis ticks are not the same in the two plots.
- Figures 3 and 5 are not directly referenced in the text.
- The x axis for Figure 4 cooling histories are not the same – it might be quicker for readers to evaluate the cooling history vs stratigraphic depth if these were the same.
- The synthetic data results and discussion section is a little unclear to me. You say the MDD tool kit did not perform as well for Experiment B, but it looks (in table 3) that experiment B’s results are closer to the setup values.
- I think the final column for the supplement tables (the label for whether a step should be included in the optimization) is not correct – legend says 2 = not used, but the table the value is 0.
- I like that you include a parameter range larger than realistic to avoid any potential bias.
- Did you choose 30,000 iterations when modelling the data from Wong et al. (2023) to allow for complete convergence – Figure 2 suggests this happens earlier at ~18,000 iterations. How would you advise users to choose the number of iterations?
- The cooling history for GR1 is initially pretty high temperature, especially when you compare it to the pre-existing BtAr date. The cooling history from Wong et al., 2023 would suggest that the sample experienced relatively monotonic cooling since BtAr closure. Is this realistic?
Citation: https://doi.org/10.5194/gchron-2024-11-RC1
Model code and software
MDD Tool Kit Andrew L. Gorin and Joshua M. Gorin https://github.com/dgorin1/mddtoolkit
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