Elsevier Editorial System(tm) for Journal of Structural Geology Manuscript Draft Manuscript Number: SG-D-17-00310R1 Title: A review of numerical modelling of the dynamics of microstructural development in rocks and ice: Past, present and future Article Type: SI:JSG - 40th Anniversary Keywords: review, evolution of microstructures, numerical modelling, dynamic recrystallization, process interaction Corresponding Author: Dr. Sandra Piazolo, Ph.D. Corresponding Author's Institution: University of Leeds First Author: Sandra Piazolo, Ph.D. Order of Authors: Sandra Piazolo, Ph.D.; Paul D Bons; Albert Griera; Maria-Gema Llorens; Enrique Gomez-Rivas; Daniel Koehn; John Wheeler; Robyn Gardner; Jose R. A. Godinho; Lynn Evans; Ricardo A Lebensohn; Mark W Jessell Manuscript Region of Origin: Abstract: This review provides an overview of the emergence and current status of numerical modelling of microstructures, a powerful tool for predicting the dynamic behaviour of rocks and ice at the microscale with consequence for the evolution of these materials at a larger scale. We emphasize the general philosophy behind such numerical models and their application to important geological phenomena such as dynamic recrystallization and strain localization. We focus in particular on the dynamics that emerge when multiple processes, which may either be enhancing or competing with each other, are simultaneously active. Here, the ability to track the evolving microstructure is a particular advantage of numerical modelling. We highlight advances through time and provide glimpses into future opportunities and challenges.
Cover Letter
Revision of Ms. Ref. No.: SG-D-17-00310 Title: A review of numerical modelling of the dynamics of microstructural development in rocks and ice: Past, present and future Journal of Structural Geology Dear Prof. Passchier, Dear Cees,
Thank you for handling our contribution and the opportunity to revise our manuscript: A review of numerical modelling of the dynamics of microstructural development in rocks and ice: Past, present and future. We have now carried out the revisions in the light of the useful and constructive comments by the reviewer. We took great care in addressing all the comments. Reviewer #1 suggested to add a table with the models cited and processes modelled into the main text. We hope that you agree with the reviewer and us that this table that we now supply will be very useful to the community and therefore warrants to be placed in the main body of the manuscript. Below we have outlined our responses and reasoning behind those. We hope you agree with us that the manuscript is now acceptable for publication in JSG.
Kind regards, Sandra Piazolo (corresponding author, on behalf of all authors)
Comments to review: Reviewers' comments are given in black font below, with our replies and changes made shown in red font. A copy of the revised manuscript with all changes tracked is also included in this resubmission.
Reviewer #1: This review paper on the topic of microstructural numerical modeling is a valuable addition to the JSG volume. Modeling is an extremely useful component of research, and one that has probably been neglected in microstructural work more than in other fields. As such, any pathway - such as this paper - to expand the reach of modeling will serve the community well. This manuscript does a nice job of presenting the historical development and range of applications in current modeling approaches. The vignettes, such as diffusion creep, stylolite formation, and dissolution, provide readers with good fuel for consideration in their own research programs. The introduction is particularly compelling and well structured. This manuscript could be published as is. The graphics and writing are clear and raise no concerns about the validity of the work. We are glad to see the positive overall assessment of our manuscript Nevertheless, I have a few suggestions the authors may want to consider that I think would increase the impact of this contribution and be more accessible to a wider audience of new and old modeling
practitioners. 1. I would find a table of the different models cited, links to access the code, if available, and the different processes included in each of the models a useful reference. In fact we as the authors did start with such a compilation when initially writing this contribution but realized that this would significantly increase the length of this short review if it were included in the printed version, hence decided against such a table. We now, however, as suggested by the reviewer, provide the table, but restrict ourselves to the cited models stating this explicitly. We hope you agree that this table is valuable enough to be in the main text. 2. The discussion on lines 50-55 seems to stop a bit short of the reality of our current understanding of microstructural processes. I think some additional thoughts about how non-linear feedbacks and potentially chaotic systems operate in the microstructural domain would strengthen the argument for modeling. We have now included a couple of extra sentences discussing non-linear feedbacks and their potential importance. 3. For several sections, particularly "2.2. Multiprocess modelling - approaches", and "2.3. The numerical representation of a microstructure", a more thorough description of the pros and cons of different approaches (e.g., operator splitting, different ways to represent the microstructure) would be welcome. We agree that this would be desirable, however due to the space limitations a thorough discussion is not possible. However, we have now added several sentences to identify the main pros and cons of the different approaches. 4. In a similar vein, overall, the text leans much more in the descriptive direction (making it very readable), than in the technical. I suggest considering adding some more technical discussion about the challenges of the modeling and the areas that we can consider robust. For the audience this is intended for, we would like to keep this contribution mainly on the descriptive side, however we take the point that some of the technical challenges and achievements need to be highlighted more. Hence we added for each section another sentence specifically about technical issues/ advances and robust approaches. 5. Another topic that could be expanded builds off Lines 100-103, which mention the role of the microscale stress and strain fields. The non-evolutionary models mentioned in section 5.5 contribute to that, but this is an area that is ripe for more investigation (e.g., Powell, R., Evans, K. A., Green, E. C. R. and White, R. W., On equilibrium in non-hydrostatic metamorphic systems. J Metamorph Geol. Accepted Author Manuscript. doi:10.1111/jmg.12298; Jessell et al., 2005). We agree that this area ripe for more investigation, especially in the dynamic field. However, Powell et al. 2018, as well as being critical of co-author Wheeler, do not provide a way in which to understand
diffusion creep as coupled to other chemical processes. So instead we cite Wheeler 2018, who does. We have now added “…local variations in stress and strain play a significant role in the evolution of microstructures and material behaviour, with their effects on chemistry being particularly intriguing (Wheeler 2018)”. Wheeler, J., 2018. The effects of stress on reactions in the Earth: sometimes rather mean, usually normal, always important. Journal of Metamorphic Geology 36, 439-461. (https://doi.org/10.1111/jmg.12299) 6. Section 6 (line 518) provides a nice look back, and I would welcome more forward looking thoughts on the most pressing problems or the problems that have the best potential for near-term solutions. This group of authors has a unique and exceptional perspective on the field, and this seems a good opportunity to share with the community more details of what they see as the future. In section 5 we already discuss some of the future challenges, hence in section 6, for sake of brevity, we only highlight what we considered the main challenge – linking chemistry and deformation in a robust manner
These above suggestions are merely suggestions - the authors should feel free to take them or leave them. I have a few minor additional comments: Line 117: Perhaps begin the paragraph with the sentence that starts on line 121, to provide more of a segue. Done, we have restructured the paragraph accordingly. Line 297-298: "In this case dissolution and/or precipitation are driven by changes in elastic and surface energies as well as normal stress at the interface." This probably needs some citation, as I wouldn't say that there is universal agreement about the drivers of dissolution and precipitation (e.g., Kristiansen, Kai, Markus Valtiner, George W. Greene, James R. Boles, and Jacob N. Israelachvili. "Pressure solutionThe importance of the electrochemical surface potentials." Geochimica et Cosmochimica Acta 75, no. 22 (2011): 6882-6892.) This leads us to a change in Section 3.2 the general description of diffusion along stressed boundaries to: “The mechanism involves diffusion driven by gradients in normal stress along boundaries, and in simple models no other driving forces need to be included”. This is then contrasted to the mechanisms discussed in the next section 3.3 “In this case dissolution and/or precipitation are driven not just by normal stress gradients at the interface (c.f. section 3.2) but also by changes in elastic and surface energies. In addition to these driving forces, the detailed electrochemistry of interfaces has a kinetic effect on rates (Gratier et al. 2013) ”.
We decided to cite Gratier et al. here, as their work is clearly relevant and in line with results from coauthor J. Wheeler (Sheldon et al. 2003, & 2004) Sheldon, H.A., Wheeler, J., Worden, R.H., Cheadle, M.J., 2003. An analysis of the roles of stress, temperature and pH in chemical compaction of sandstones. Journal of Sedimentary Research 73, 64-71. Sheldon, H.A., Wheeler, J., Worden, R.H., Cheadle, M.J., 2004. An analysis of the roles of stress, temperature and pH in chemical compaction of sandstones: Reply. Journal of Sedimentary Research 74, 449-450. Line 488: The main description paper for the elastic code used by Naus-Thijssen is: Vel, S.S., Cook, A., Johnson, S.E., and Gerbi, C., 2016, Computational Homogenization and Micromechanical Analysis of Textured Polycrystalline Materials, Computer Methods in Applied Mechanics and Engineering, v. 310, p. 749-779, doi: 10.1016/j.cma.2016.07.037. Apologies, we have now added this reference.
Highlights (for review)
Highlights:
Review of numerical modelling of dynamic interaction of processes at the microscale Assessment of the effect of competing processes allows in-depth understanding Modelling provides insights into link between processes and material properties Modelling of the interplay between physical and chemical processes is needed Future developments promise enhanced understanding and predictive powers
*Manuscript Click here to view linked References
1
A review of numerical modelling of the dynamics of microstructural development in
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rocks and ice: Past, present and future
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S. Piazolo1, P. D. Bons2, A. Griera3, M.-G. Llorens2, E. Gomez-Rivas4,5, D. Koehn6, J.
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Wheeler7, R. Gardner8, J. R. A. Godinho9,, L. Evans8,10, R. A. Lebensohn11, M. W.
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Jessell12
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1
School of Earth and Environment, University of Leeds, Leeds LS2 9JT, United Kingdom
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2
Department of Geosciences, Eberhard Karls University Tübingen, Wilhelmstr. 56, 72074
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Tübingen, Germany
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(Cerdanyola del Vallès), Spain
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4
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080280 Barcelona, Spain
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5
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United Kingdom
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6
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Glasgow G12 8QQ, UK
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7
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Herdman Building, Liverpool University, Liverpool L69 3GP, U.K.
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8
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Systems/GEMOC, Department of Earth and Planetary Sciences, Macquarie University, NSW
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2109, Australia
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9
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M13 9PL, Manchester, UK.
Departament de Geologia, Universitat Autònoma de Barcelona, 08193 Bellaterra
Departament de Mineralogia, Petrologia i Geologia Aplicada, Universitat de Barcelona,
School of Geosciences, King’s College, University of Aberdeen, Aberdeen AB24 3UE,
School of Geographical and Earth Sciences, Gregory Building, University of Glasgow,
Dept. Earth, Ocean and Ecological Sciences, School of Environmental Sciences, Jane
Australian Research Council Centre of Excellence for Core to Crust Fluid
Henry Moseley X-ray Imaging Facility, School of Materials, The University of Manchester,
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Vic 3800, Australia
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11
Los Alamos National Laboratory, MST8, MS G755, Los Alamos, NM, 87545, USA
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Centre for Exploration Targeting, School of Earth Sciences, The University of Western
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Australia, 35 Stirling Hwy, Crawley, 6009, Australia
School of Earth, Atmosphere and Environmental Sciences, Monash University, Clayton,
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Corresponding author: Sandra Piazolo (
[email protected])
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Abstract
34
This review provides an overview of the emergence and current status of numerical
35
modelling of microstructures, a powerful tool for predicting the dynamic behaviour of rocks
36
and ice at the microscale with consequence for the evolution of these materials at a larger
37
scale. We emphasize the general philosophy behind such numerical models and their
38
application to important geological phenomena such as dynamic recrystallization and strain
39
localization. We focus in particular on the dynamics that emerge when multiple processes,
40
which may either be enhancing or competing with each other, are simultaneously active.
41
Here, the ability to track the evolving microstructure is a particular advantage of numerical
42
modelling. We highlight advances through time and provide glimpses into future
43
opportunities and challenges.
44 45
Keywords:
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recrystallization, process interaction
review,
47 48
1.
Introduction
evolution
of
microstructures,
numerical
modelling,
dynamic
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Structural geology studies how rocks deform under applied stress or strain. These studies are
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applied from the sub-grain scale to that of mountain belts and tectonic plates. To know what
51
has happened to a volume of rock millions of years ago, or to predict what would happen to a
52
rock volume under certain (future) conditions, we need to know the link between (i) the
53
boundary conditions (e.g. applied stress or strain, pressure-temperature conditions), (ii) the
54
intrinsic material properties (e.g. Young's modulus, crystal symmetry, slip systems) and (iii)
55
the processes that may be activated (e.g. dislocation creep, pressure solution, mineral phase
56
changes). These three together determine the evolution of the state of the rock (e.g.
57
developing cleavage, folds, lineation, stylolites) and its bulk material properties (e.g.
58
viscosity). The microstructure is thus a state variable of a rock, describing the state it
59
achieved as a result of the interplay between various processes and boundary conditions.
60
Importantly, the microstructure is not a passive log of events, but plays an active and central
61
role in the evolution of a rock throughout its history (Gottstein, 2004). In this context non-
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linear feedback between a specific microstructural configuration and local changes in
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intrinsic and extrinsic parameters may be important but difficult to predict. Hence, the
64
microstructure is the link that couples the material properties, boundary conditions and
65
processes that together control the behaviour and evolution of a rock. Consequently, the
66
analysis and correct interpretation of microstructures is crucial in gaining an understanding of
67
how rocks, including ice, deform on Earth and other planets. Here we define microstructure
68
as the full spatial, compositional and orientational arrangement of all entities in a rock,
69
typically on the scale of a thin section to hand specimen (roughly µm to cm) (Hobbs et al.,
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1976). These entities include minerals, grain and subgrain boundaries, crystal lattice
71
orientations, and chemical composition from the nano- to micro-scale.
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While the rock's temperature or elastic strain are ephemeral, the microstructure may be
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preserved for the geologist to interpret millions to billions of years later. Therefore,
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microstructures are one of the prime forensic tools to unravel the history of a rock that allows
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us to deduce the succession of strain rates, stresses, diagenetic and metamorphic conditions
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and the rock's rheology during deformation (e.g. Hobbs et al., 1976; Vernon, 2004; Passchier
77
and Trouw, 2005).
78 79
One major problem in microstructural studies is that we normally only have a static record as
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post-mortem "images" (grain arrangement in thin section, chemical and orientation
81
characteristics derived using electron microscopy, 3D datasets from computer tomography,
82
neutron diffraction and synchrotron beam analysis, etc.). To interpret microstructures, we
83
need to know how they form and change. Means (1977) wrote: "a more valuable kind of
84
kinematic investigation is that in which the time sequence of incremental strains or
85
incremental displacements, the kinematic history, is correlated with progressive structural
86
change. This promises to reveal more about the physics of rock deformation and to provide
87
more sensitive structural methods for reconstructing tectonic history". Such investigations
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first became possible using in-situ analogue experiments that were introduced to geology by
89
influential scientists such as Cloos (1955) and Ramberg (1981), and at the microstructural
90
scale by Means (1977). In in-situ analogue experiments by Means (1977), inspired by those
91
of McCrone and Chen (1949), a thin-section sized sample of crystalline analogue material is
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sandwiched and deformed between glass plates to observe the changing microstructure under
93
a microscope. These experiments were followed by a large number of studies that
94
investigated deformation related features such as dynamic recrystallization and crystal
95
plasticity (e.g. Means, 1977; 1980; Means and Ree, 1988; Park and Means, 1996; Urai et al.,
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1980; Urai and Humphreys, 1981; Wilson, 1986; Jessell, 1986; Ree, 1991; Bons and Jessell,
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1999; Ree and Park, 1997; Herwegh and Handy, 1996, 1998; Nama et al., 1999; Wilson et
98
al., 2014), melt or fluid-bearing microstructures (e.g. Urai, 1983; Rosenberg and Handy,
99
2000, 2001;
Rosenberg, 2001; Schenk and Urai, 2004, 2005; Walte et al., 2005), the
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development of vein and fringe microstructures (e.g. Hilgers et al., 1997; Koehn et al., 2003)
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and the formation of -clasts (ten Brink et al., 1995). Many of these experiments made clear
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that when a polycrystalline material deforms, multiple processes act upon the microstructure
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and govern its dynamic behaviour throughout its deformation and post-deformation history
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(Fig. 1a; see also movies at http://www.tectonique.net/MeansCD/). In addition, pre-existing
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heterogeneities and/or those developing during deformation can have a significant effect on
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the dynamics of the system since local variations in stress and strain play a significant role in
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the evolution of microstructures and material behaviour, with their effects on chemistry being
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particularly intriguing (Wheeler, 2018).
109
Although changes in microstructural characteristics (e.g. grain size) can be quantified in such
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experiments, the underlying principles of the active processes still need to be deduced.
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Therefore, an additional tool is needed. Here numerical models simulating microstructural
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development based on several concurrent processes become important. Such numerical
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simulations share the advantage of in-situ experiments that the full microstructural
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development can be traced in form of time-series, with the opportunity to systematically
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study the effect of different processes and/or pre-existing heterogeneities on the
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microstructural development. In addition, numerical simulations are neither constrained by
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time nor specific boundary conditions. Once calibrated against laboratory experiments (e.g.
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Piazolo et al., 2004) or analytical solutions, they can be applied to a wide range of conditions
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or materials including those not attainable or suitable for laboratory experiments. In many
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cases, the developed microphysical behaviour may be applied to problems on the continental
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scale.
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In this contribution, we aim to give an overview of the concepts behind numerical modelling
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of microstructures, i.e. microdynamic modelling, and the achievements that have been made
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in this field. This is followed by a selection of examples of the current state of the art. Finally,
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future possibilities and directions are briefly discussed, including some work in progress. We
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focus on those studies that (i) are applied to geological materials, (ii) involve several
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processes and (iii) allow prediction and visualization of the development of the
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microstructure. Hence, we exclude studies where microstructures are not spatially mapped
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along with models in which grains are modelled as if embedded in a medium with averaged
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properties. Table 1 summarizes the microdynamic numerical models cited, the numerical
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method used and processes included in each model.
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2. Numerical simulation of microstructures
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2. 1. Emergence and philosophy
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Soon after the first in-situ experiments, computers had advanced to a stage allowing the first
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numerical simulations of microstructural development, with a full "image" of the
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microstructure, to appear in geology (Etchecopar, 1977; Jessell, 1988a,b; Jessell and Lister,
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1990). When using such models we accept the fact that the scientific description of the
139
phenomena studied does not fully capture reality. This is not a shortcoming but a strength, as
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general system behaviour and interrelationships can be studied systematically and
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transferred to geological questions. In a numerical model the behaviour of a system emerges
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from the piece-meal enumeration of the behaviour of each of its component parts. As a
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consequence, numerical models are powerful and necessary when the bulk behaviour is
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influenced by the local interaction of the components. In contrast, global, averaged or mean
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field solutions cannot fully encompass the effect of processes that interact with each other
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on the small scale. In nature, patterns emerge (e.g. orientation and arrangement of high-
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strain zones, foliation, fractures) which may be used to deduce conditions of their formation.
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Numerical models are powerful tools to investigate this link between patterns observed in
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nature and processes that are responsible for their development.
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2.2. Multiprocess modelling – approaches
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As shown convincingly by the aforementioned experiments, several processes act
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concurrently on a microstructure and control the dynamics of the microstructural evolution
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and material properties (e.g. Means, 1977, 1980). While there are a large number of models
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that focus on one process alone, there are fewer systems that aim to model the effect of
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multiple interacting processes. Here, we focus on the latter. Solving all necessary equations
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simultaneously is only practical when the number of interacting processes is small.
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Alternatively, the set of equations, hence processes, needs to be reduced to a manageable
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number necessitating a user driven selection which may result in the omission of important
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feedbacks. Operator splitting provides an alternative approach, whereby each process acts
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sequentially on a microstructure that was incrementally changed by all operating processes in
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the previous step. In this case, the number of equations to be solved is not limited; hence
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there is no need to artificially reduce the number of modelled processes. Clearly, length scale
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and time steps have to be considered carefully for this approach to be valid. The operator
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splitting approach is fundamental to the numerical platform Elle (Jessell et al., 2001, Bons et
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al., 2008, Piazolo et al., 2010; www.elle.ws) which is, within the realm of microdynamic
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modelling, the most used numerical platform in geology. However, operator splitting has also
168
been employed in other numerical schemes (e.g. Cross et al., 2015).
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2.3. The numerical representation of a microstructure
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It is necessary to describe numerically the microstructure so that in the model the processes
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can act upon it spatially. Up to now, for rocks and ice, microstructural models have been
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mostly restricted to two-dimensions. Two basic approaches can be taken: (i) a lattice data
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structure or (ii) an element data structure. In the lattice data structure, the microstructure is
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mapped onto an irregular or regular lattice, like the pixels in a digital image or gridded
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analysis points obtained during chemical mapping and orientation mapping of geological
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samples. Potts, cellular automata, lattice-spring, particle-in-cell, phase field and
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micromechanical Fast Fourier Transform (FFT) based models utilize the lattice data structure
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(Fig. 2; see also chapter 2 of Bons et al., 2008 for a detailed review of different numerical
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methods). Numerically the lattice data structure is easy to use and calculations are
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straightforward. In the element data structure, the microstructure is described by discrete
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elements which can be points, line segments, or polygons (Fig. 2). Finite-element models and
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boundary models typically use elements to describe a system. Element data structure is well
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suited for processes acting upon a surface, e.g. grain boundary migration, but is numerically
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more “expensive” due to necessary topology checks. Both elements and lattice points can
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have, in addition to their location in xy space, specific properties or store specific state
187
variables, such as mineral phase, chemical composition, crystallographic orientation, and
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stress and strain (rate) values, respectively. The numerical platform Elle (Jessell et al., 2001)
189
combines these two approaches: a microstructure is represented by boundary nodes
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connected by straight segments (Fig.2). A polygon enclosed by boundary nodes represents a
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grain or subgrain. Property variations within the individual polygons are defined at interior
192
points where information is recorded and tracked. Processes acting on the microstructure can
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move boundaries and/or interior points and/or change the properties at those points, e.g.
194
composition, crystallographic orientation or dislocation density. The system has now matured
195
to a stage that it can robustly be used for a large variety of models relevant to geology.
196 197
3. Examples of modelling several coupled processes
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3.1.
199
3.1.1. A historical perspective
200
Dynamic recrystallization is the response of a crystalline aggregate to lower its free energy by
201
formation and movement of sub- and grain boundaries (Means, 1983). It occurs by a number
202
of concurrent processes that act upon the polycrystalline material. For decades, geologists
203
have used microstructural characteristics developed during dynamic recrystallization to infer
204
dominance of processes and linked boundary conditions (e.g. Urai et al., 1986; Hirth and
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Tullis, 1992; Stipp et al., 2002). Following the first attempts of such a model (Etchecopar,
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1977), Jessell (1988a,b) and Jessell and Lister (1990) developed a numerical model
207
incorporating rotation of the crystal lattice and dynamic recrystallization for quartz. The data
208
structure used was a 100x100 hexagonal lattice structure where each lattice point represented
209
a grain or subgrain (Fig. 3a inset). Calculation of the crystal lattice rotation in Jessell and
210
Lister (1990) uses the Taylor-Bishop-Hill calculation method (Taylor, 1938; Bishop and Hill,
211
1951a, 1951b; Lister and Paterson, 1979) and the critical resolved shear stress of the different
212
slip systems in quartz. This work reproduced the general microstructures seen in a mylonite.
213
A major step forward was that the simulations showed that dynamic recrystallization can
214
significantly modify crystallographic preferred orientation (CPO) development. Piazolo et al.
215
(2002) advanced Jessell’s model by utilizing the more robust Elle data structure to describe
216
the local deformation and the dynamics of the grain boundary network (Jessell et al., 2001).
217
The model now included a finite element solution for incompressible, linear or nonlinear
218
viscous flow (BASIL; Barr and Houseman 1992, 1996) to compute the local velocity field,
219
the front tracking method for the motion of grain boundaries by moving nodes and segments
220
(e.g. Bons and Urai, 1992), the Taylor-Bishop-Hill formulation (Lister and Paterson, 1979)
221
and additional features such as crystal lattice rotation, formation of subgrains,
222
recrystallization by nucleation, grain boundary migration, recovery, work hardening and
Dynamic recrystallization
223
tracking of dislocation densities. This list of additional features shows the advance made in
224
approximately a decade in reproducing natural microstructures (Fig. 3b).
225
So far, the rheological anisotropy of minerals deforming by dislocation creep was not taken
226
into account in the calculation of the stress-strain rate field and lattice rotation. About another
227
decade after Piazolo et al. (2002), the viscoplastic implementation of the full-field crystal
228
plasticity micromechanical code based on Fast Fourier Transforms (VPFFT; Lebensohn,
229
2001) was introduced in geological microstructural modelling (Griera et al., 2011, 2013,
230
2015). The VPFFT is a spectral method that specifically assumes that deformation is
231
achieved by dislocation glide on crystallographic slip planes (each with their own critical
232
resolved shear stress) whose orientations are mapped on a regular grid (for details see Griera
233
et al. (2013), Montagnat et al. (2011, 2014) and Llorens et al. (2016a)). The simulations of
234
Griera et al. (2011, 2013) and Ran et al. (2018) illustrate that heterogeneous stress within and
235
between grains that emerge from inhomogeneous slip has a significant effect on the rotation
236
rate of porphyroclasts and -blasts and the strain (rate) field around these objects.
237
More recently, Llorens et al. (2016a; 2016b; 2017) and Steinbach et al. (2016; 2017) coupled
238
dynamic recrystallization (DRX) by grain boundary migration (GBM), intracrystalline
239
recovery and polygonisation with VPFFT crystal plasticity models, and applied them to study
240
the deformation of polar ice. Llorens et al. (2016a) presents a comprehensive description of
241
the method. They found that DRX produces large and equidimensional grains, but only
242
marginally affects the development of the c-axes CPO. However, DRX can alter the activity
243
of slip systems and does modify the distribution of a-axes. In simple shear, the strong
244
intrinsic anisotropic of ice crystals is transferred from the crystal to the polycrystal scale,
245
leading to strain localisation bands that can be masked by GBM (Llorens et al., 2016b).
246
Llorens et al. (2017) compared the dynamics of pure polar ice polycrystalline aggregates in
247
pure and simple shear deformation. It was found that, due to the vorticity of deformation, it is
248
expected that ice is effectively weaker in the lower parts of ice sheets (where simple shear
249
dominates) than in the upper parts (where ice is mostly deformed coaxially). The method was
250
also applied to reproduce the development of tilted-lattice (kink) bands found in ice cores
251
(Jansen et al., 2016) and crenulation cleavage during folding (Ran et al., 2018) as a result of
252
mechanical anisotropy. Steinbach et al. (2016) included air inclusions as a second phase to
253
simulate DRX in porous firn and found that DRX can occur despite the low strain and stress in
254
firn.
255
Apart from polar ice, the VPFFT-Elle model has also been used to analyse subgrain rotation
256
recrystallization of halite polycrystals in simple shear (Gomez-Rivas et al., 2017) by coupling
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VPFFT with the Elle routines that simulate intracrystalline recovery (Borthwick et al, 2013)
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and subgrain rotation. They found that recovery does not affect CPOs, but strongly decreases
259
grain size reduction. These authors also evaluated the use of mean subgrain misorientations
260
as a strain gauge.
261
3.1.2. Testing the effect of different combinations of recrystallization processes
262
We show results from three combinations of processes that occur during dynamic
263
recrystallization. These processes act on the same initial 10x10 cm aggregate of ice Ih
264
crystals (Fig. 4a). The numerical approach in these simulations is based on the VPFFT
265
micromechanical model in combination with several Elle processes (Llorens et al., 2016a,
266
2016b, 2017; Steinbach et al., 2016; 2017). The deformation-induced lattice rotation and the
267
estimation of geometrically necessary dislocation densities calculated from the stress and
268
velocity field provided by the VPFFT algorithm are used to simulate recrystallization by
269
intra-crystalline recovery, grain boundary migration and nucleation. Grain boundary
270
migration (GBM) is simulated based on the algorithm by Becker et al. (2008). Here, a front-
271
tracking approach is used in which the movement of boundaries is mapped through time.
272
Recovery reduces the intra-granular stored energy in a deformed crystal (Borthwick et al.,
273
2013). Nucleation creates new grain boundaries with areas of misorientation values above a
274
pre-defined threshold (Piazolo et al., 2002; Llorens et al., 2017; Steinbach et al., 2017).
275
Dextral simple shear deformation is applied in strain increments of ∆ = 0.04. Three different
276
combinations of recrystallization processes are tested: (1) GBM only, (2) GBM and recovery
277
or (3) GBM, recovery and nucleation (Fig. 4a). Simulations including VPFFT viscoplastic
278
deformation show a similar evolution of c-axis orientation, regardless of the dynamic
279
recrystallization processes included (Fig. 4a). The lack of influence of nucleation on the CPO
280
is due to the fact that new grains are modelled to have lattice orientations close to those of
281
their parent grains. However, inclusion of the nucleation process result in grain size reduction
282
(Fig. 4b). Results illustrate how different combinations of microdynamic processes affect the
283
microstructural characteristics (e.g. CPO, grain size, shape and orientation) in different ways
284
where no single microstructural parameter can grasp the full dynamics of polycrystalline
285
deformation.
286
3.2. Diffusion creep
287
Diffusion creep is a physico-chemical process that operates in large regions of the Earth – in
288
the upper Earth as “pressure solution” where diffusion is enhanced along fluid films and
289
elsewhere at low strain rates, maybe in large parts of the lower mantle (Karato, 1992). The
290
mechanism involves diffusion driven by gradients in normal stress along boundaries. In
291
simple models no other driving forces need to be included. It is implicit in our understanding
292
of this mechanism that grain boundary sliding occurs in conjunction with diffusion. Ford et
293
al. (2002) devised a numerical model to understand the evolution of grain shapes and CPO in
294
grain boundary diffusion creep. The mathematical framework allows for the consideration of
295
diffusion and sliding together. Operator splitting is not required and the evolution at each
296
timestep is based on a single matrix inversion operation. The model couples the
297
microstructure to an evolving system of local stresses (Fig. 5). The model predicts that grain
298
shapes become somewhat elongate, in accordance with experiments on calcite (Schmid et al.,
299
1987) and observations on an albite-bearing mylonite (Jiang et al., 2000) (Fig. 5). It also
300
predicts that grain rotations are not chaotic and that CPO may be present to high strains
301
(Wheeler, 2009), as seen in, for example, experiments on olivine-orthopyroxene (Sundberg
302
and Cooper, 2008) and predicted by numerical simulations of Bons and den Brok (1999). A
303
second phase, if insoluble, can be included and such a model was used by Berton et al. (2006,
304
2011) to prove the hypothesis that a different grain boundary diffusion coefficient along the
305
two-phase boundaries could explain fibre growth in pressure shadows. Not all model
306
predictions are in agreement with other investigations – for example some materials
307
deformed by diffusion creep e.g. olivine show quite equant grains (Karato et al., 1986). In
308
such experiments grain growth occurs but this process was not included in the numerical
309
model, hence results differ. In a later section we address this issue, but this example
310
illustrates that, in common with other numerical models, the applicability reflects the
311
processes included.
312
3.3. Stress driven dissolution, growth and dynamic roughening
313
Another example where stress leads to micro- and macrostructures is the roughening of grain
314
boundaries in the presence of a fluid. In this case dissolution and/or mineral growth are
315
driven not just by normal stress gradients at the interface (c.f. section 3.2) but also by changes
316
in elastic and surface energies. In addition to these driving forces, the detailed
317
electrochemistry of interfaces has a kinetic effect on rates (Gratier et al., 2013). In the
318
example presented here, the process is modelled in Elle by coupling a lattice spring code that
319
calculates the strain with a background fluid that dissolves lattice elements. This process can
320
produce transient patterns of interface geometry at the grain boundaries (Koehn et. al., 2006).
321
In addition, the process itself produces rough interfaces that can be seen on the larger scale as
322
stylolites (Koehn et al., 2007; Ebner et al., 2009). The scaling behaviour of stylolites is
323
important for stress inversion and compaction on the basin scale (Koehn et al., 2012; 2016)
324
and the shape of stylolites is partly rooted in the microstructure of the host-rock (Fig. 6).
325
Slower dissolving material on various scales, from small grains to fossils and layers, can
326
influence the dissolution process and thus the development of patterns, whereas deformation
327
leads to changes in long-range interactions (Koehn et al., 2016). Similarly, pinning behaviour
328
in the presence of a fluid is seen in grain growth where grain boundary pinning results in
329
restriction of grain growth. Pinning particles can also be moved around so that growing
330
grains become clean. Such a process can lead to layered rocks, for example, zebra dolomites
331
(Kelka et al., 2015).
332
4. Numerical simulations of microdynamic processes: Examples of ongoing work
333
4.1.
334
Even now, most of the numerical models that incorporated more than one process are
335
restricted to deformation processes alone. Although in diffusion creep there is an intrinsic
336
chemical aspect, the model described in 3.2 involves just a single phase of fixed composition.
337
There are some models that incorporate the growth of new mineral phases and investigate the
338
rheological effect of such growth (e.g. Groome et al., 2006; Smith et al., 2015), however the
339
location of new phase growth and/or growth rate is predefined. While Park et al. (2004)
340
showed convincingly the effect of microstructure on the diffusion pathways and chemical
341
patterns in garnet and biotite, there are now many opportunities available in a numerical
342
system such as Elle (Jessell et al., 2001) to couple local chemistry and microstructural
343
evolution. Below we present preliminary results from two current projects linking chemical
344
changes and a dynamically evolving grain network.
345
4.1.1. Grain boundary diffusion creep and grain growth
346
Here we present the first results using a model combining, using operator splitting,
347
deformation by diffusion creep (Ford et al., 2002; Wheeler, 2009) and surface energy driven
Linking chemistry and microstructural evolution
348
grain boundary migration (GBM), i.e. grain growth (modified after Becker et al., 2008). Four
349
different scenarios are shown: diffusion creep only, grain boundary migration only and two
350
combinations with medium and high grain boundary migration GBM rates (Fig. 7a). The
351
starting material is an almost regular hexagonal grain mesh with equant grains in which triple
352
junctions have had small random perturbations imposed (Fig. 7a). The perturbations are
353
required to avoid mathematical problems that arise in perfectly regular hexagonal networks
354
(Wheeler, 2010). Topology checks such as neighbour switching routines are performed after
355
each step in all simulations, ensuring that no topology problems arise. The microstructural
356
development is markedly different if grain growth is coupled with diffusion creep (Fig. 7a)
357
where with increasing GBM rate the aspect ratios are generally reduced relative to that of
358
comparable experiments modelling diffusion creep only. This is provisionally in accord with
359
the hypothesis that there will be a ‘steady state’ grain elongation developed which is a
360
function of the relative magnitudes of strain rate and grain growth kinetic parameters
361
(Wheeler, 2009) but more simulations are required (with less regular starting
362
microstructures). In contrast, for grain growth only, the microstructure does not change as the
363
initial configuration consists of stable near 120° triple junctions (Fig. 7a). Grain numbers (i.e.
364
grain size) do not change in any of the simulations.
365
4.1.2. Trace element partitioning between fluid and solid coupled with recrystallization
366
Field and experimental studies illustrate the importance of the time and length scales on the
367
evolution of grain networks during trace element diffusion (e.g. Ashley et al., 2014 and
368
references therein). Therefore, understanding the influence of recrystallization on trace
369
element distribution is essential for correctly interpreting patterns of trace element
370
distribution (e.g. Wark and Watson, 2006). However, to our knowledge there are currently no
371
numerical approaches able to simulate diffusion coupled with microstructure evolution.
372
Recently, a new Elle finite difference process that solves Fick’s second law to simulate trace
373
element diffusion in a polycrystalline medium has been developed. The polycrystal is defined
374
by n-phases where grain boundaries and grains are differentiated (Fig. 7b). The trace element
375
partitioning coefficient between different phases is also considered. The approach uses an
376
element data structure representing grain boundaries and a lattice data structure of a regular
377
grid of unconnected lattice point to track diffusion along the grain boundaries and the
378
physical and chemical properties within grains, respectively. To ensure elemental exchange
379
between grain boundaries and grain interiors, a grain interior-grain boundary partition
380
coefficient is implemented that takes the proximity to grain boundaries into account. This
381
allows the simulation of coupled bulk and grain boundary diffusion. This approach can be
382
fully coupled with other processes of the numerical platform Elle and therefore allows
383
simulating simultaneous diffusion, deformation and static or dynamic recrystallization. The
384
use of this approach is demonstrated through the modelling of diffusion of a tracer with
385
fractionation during static grain growth of a single-phase aggregate (Fig. 7b) (after that
386
presented by Jessell et al., 2001). Patterns of chemical concentration qualitatively resemble
387
those of Ti-distribution observed in recrystallized quartz in shear zones and can help to
388
understand the redistribution of Ti in quartz during dynamic recrystallization (e.g. Grujic et
389
al., 2011), among many other geological problems.
390
4.2.
391
We restricted this review to two-dimensional numerical approaches, as very few models have
392
been published that are in three dimensions and also specific to geological microstructures.
393
An exception is the phase field approach by Ankit et al. (2015) who investigate the growth of
394
crystals in an open fracture. However, within the material science community numerical
395
models exist that investigate the behaviour of a microstructure in response to single processes
396
in three dimensions (e.g. three dimensional crystal plasticity modelling platform DAMASK
397
(Roters et al., 2012; Eisenlohr et al., 2013), which includes the option of solving the
Modelling in three dimensions
398
micromechanical problem using a 3D implementation of the FFT-based formulation
399
(presently implemented in Elle in its 2-D version) and grain growth (e.g. Krill and Chen,
400
2002; Kim et al., 2006). With the technical advances made within the geological and material
401
science community it is now in principle possible to extend currently available 3D models to
402
geological questions and/or multi-process scenarios.
403
4.2.1. Dissolution of reactive surfaces in three dimensions
404
Mineral dissolution controls important processes in geoscience such as serpentinization (e.g.
405
Seyfried et al., 2007), retrogression and replacement reactions (Putnis and Austrheim, 2012),
406
deformation by dissolution-precipitation creep (e.g. Rutter, 1976) and also affects processes
407
relevant to society such as the stability of spent nuclear fuel and mine waste. Recent work has
408
shown that dissolution rates are linked to the evolving surface structure, and thus are time-
409
dependent (Godinho et al., 2012). Different surfaces have different structures in three
410
dimensions and the presence of etch pits play a major role in dissolution (Pluemper et al.,
411
2012). Hence, for accurate representation of dissolution behaviour, modelling in three
412
dimensions is important. Here, we present a numerical model that simulates the dissolution
413
process as a potential tool to quantify the links between dissolution rates, reactive surface
414
area and topography over periods of time beyond reasonable for a laboratory experiment. The
415
program uses empirical equations that relate the dissolution rate of a point of the surface with
416
its crystallographic orientation (Godinho et al., 2012) to simulate changes of topography
417
during dissolution, which ultimately results in the variation of the overall dissolution rate
418
(Godinho et al., 2012, 2014). The initial surface is composed of a group of nodes with a xy
419
position and a set height (z). For each lattice point a local surface orientation is calculated
420
from the inclination of the segment node and its neighbours. This orientation together with
421
the crystallographic orientation of the grain the surface belongs to is then used to calculate a
422
dissolution rate (surface reactivity). Based on this the displacement of the node using the
423
equations published in Godinho et al. (2012) is calculated. The model allows the graphical
424
display of the three-dimensional topographic development of the surface, tracking of the
425
variation of the surface area and calculation of the overall dissolution rate at each stage of the
426
simulation. Results obtained with the 3D simulation at three consecutive stages of dissolution
427
(150 hrs, 180 hrs, 276 hrs dissolution duration) show that numerical results are in accordance
428
to experimental results (Fig. 7c).
429 430
5. Numerical simulations of microstructures: Possibilities and challenges
431
5.1. Linking laboratory and natural data with numerical models
432
Over the last two decades numerical capabilities have advanced markedly and models have
433
come of age. Consequently, a large range of models and process combinations is now
434
available that can be utilized to gain further insight into the link between processes, material
435
properties and boundary conditions. This also includes polymineralic systems making the
436
models more appropriate to use to investigate processes occurring within polyphase rocks
437
(e.g. Roessiger et al., 2014; Steinbach et al, 2016). New avenues of effective verification
438
against laboratory and natural data are now opening up, due to the development of analytical
439
tools allowing rapid complete microstructural and microchemical analysis providing dataset
440
similar to those possible in numerical models (e.g. Steinbach et al, 2017, Piazolo et al., 2016);
441
many of the analytical data sets have a lattice data structure.
442
5.2. Linking chemistry and microstructural evolution
443
Advances in numerical methods, theoretical treatment of thermodynamic data, analytical
444
tools and theoretical and experimental insights into the coupling between chemical and
445
physical process that emerged over the last decade, call for a major effort in this area of
446
research. Examples of areas of opportunities include investigation of the (1) significance of
447
local stress versus bulk stress on mineral reactions and reaction rates (e.g. Wheeler, 2014), (2)
448
characteristics of replacement microstructures and their potential significance for
449
microstructural interpretation (e.g. Putnis, 2009, Spruzeniece et al., 2017), (3) mobility of
450
trace elements enhanced by deformation that change significantly the local elemental
451
distributions (e.g. Reddy et al., 2007, Piazolo et al., 2012, 2016) and (4) coupling between
452
reactive fluid-solid systems and hydrodynamics (Kelka et al., 2017). Increasing computer
453
processing speeds will aid the running of such models. However, the technical challenges
454
include the harmonisation of fundamentally different numerical approaches (crystal plasticity
455
versus diffusion creep, for example) and the resolution of fundamental mathematical
456
problems, e.g. the current lack of an internally consistent model for multiphase diffusion
457
creep, highlighted by Ford and Wheeler (2004).
458
5.3. Linking brittle and ductile deformation: elasto-viscoplastic behaviour
459
When examining the rock record, it is clear that in many cases, brittle and ductile behaviour
460
often occurs at the same time within a rock (e.g. Bell and Etheridge, 1973, Hobbs et al.,
461
1976). With the potential significance of grain-scale brittle behaviour now measurable on
462
seismic signals (e.g. Fagereng et al., 2014) there is an increased need in developing numerical
463
techniques that allow us to model the dynamic link between brittle and ductile behaviour.
464
Such elasto-viscoplastic behaviour combines the elastic reversible fast deformation with a
465
viscous time dependent flow and a plastic behaviour. These behaviours can be included in
466
continuous or discontinuous models. For example, in the numerical platform Elle, a lattice
467
structure is used to deform the model elastically up to a critical stress where bonds fracture
468
(plastic behaviour) and the particles themselves deform as a function of stress and time
469
(viscous behaviour). In this case the viscous behaviour conserves the volume whereas shear
470
forces and differential stresses converge to zero (Sachau and Koehn, 2010, 2012; Arslan et
471
al., 2012; Koehn and Sachau, 2014). Alternatively, linking the elasto-visocplastic FFT based
472
(EVPFFT) (Lebensohn et al., 2012) with models such as Elle would allow calculation of the
473
Cauchy stresses that are the local driving force for grain scale damage processes.
474
One of the major challenges is the large range of time-scales in these processes, with
475
fracturing and fluid flow on fast to intermediate and viscous deformation and potentially
476
reactions on very long time scales. This complexity either requires the assumption that some
477
processes are instantaneous or it requires an “up-scaling in time” or non-linear time scales in
478
models.
479 480
5.4. Expansion of capability to three dimensions
481
With the advent of supercomputers and new numerical approaches now is the time to develop
482
techniques to investigate microstructural development in three dimensions. This is of
483
particular importance if material transport such as aqueous fluid and/or melt flow is to be
484
considered. This would enable, for example, modelling of strain fringes, shear veins and en-
485
echelon tension gashes. However, these require models that link brittle and ductile behaviour
486
along with modelling in three dimensions.
487
One of the biggest problems faced with three-dimensional models of microstructures are the
488
three-dimensional topology changes that are common in dynamic microstructural
489
development (e.g. Fig. 1-7). This is a major problem if an element data structure with
490
segments, i.e. grain boundaries is used. However, the phase field approach does not work
491
with such distinct boundaries, and is therefore well suited for 3D problems (e.g. Ankit et al.
492
2014). In addition, a three dimensional network of unconnected nodes, in which there is no
493
physical movement of boundaries but only changes in the properties of the 3D nodes (voxels)
494
(cf. Fig. 7c) may be a way forward (e.g. Sachau and Koehn, 2012).
495 496
5.5. Link between geophysical signals and microstructure
497
At a time where there is an ever-increasing amount of geophysical data being collected,
498
numerical simulations that are used to test the link between microstructural development and
499
geophysical signal will become increasingly important. Cyprych et al. (2017) showed that not
500
only crystallographic preferred orientation but also the spatial distribution of phases i.e. the
501
microstructure, has a major impact on seismic anisotropy. Therefore, a direct link between
502
the microdynamic models such as Elle allowing tracking of microstructural changes through
503
time and space and geophysical signal generation offers a wealth of new opportunities
504
including interpretation of strong reflectors in the lower crust and mantle. Current efforts by
505
Johnson, Gerbi and co-workers (Naus-Thijsen et al., 2011; Cook et al., 2013; Vel et al., 2016)
506
are in line with this direction.
507 508
5.6. Application of numerical models to polymineralic rock deformation and ice-related
509
questions
510
The dynamic behaviour of the Earth is strongly influenced by the deformation of
511
polymineralic rocks. At the same time Earth’s polar ice caps and glacial ice which often
512
include ice and a second phase (e.g. air inclusions, dust, entrained bedrock) is of major
513
importance to society, especially in view of changing climate (e.g. Petit et al., 1999; EPICA,
514
2004). Application of the current numerical capabilities to polycrystalline ice and
515
polymineralic rocks is therefore urgently needed. Over the last years, there has been an
516
increased effort in this direction (Roessiger et al., 2011, 2014; Piazolo et al., 2015; Llorens et
517
al., 2016a, 2016b, 2017; Jansen et al., 2016; Steinbach et al. 2016, 2017), which promises to
518
continue. Here, the development of the link between elemental mobility and microstructural
519
development is of major importance, as only with such models the can chemical signals of,
520
for example, ice cores be correctly interpreted.
521
522
5.7. Upscaling: Utilizing operator splitting and utilities developed for microdynamic
523
systems to larger-scale problems
524
One of the strengths of the numerical approach taken by the microstructural community has
525
been the close link between different processes and the ability of the models to take into
526
account the local differences in properties such as stress, strain and chemistry. The technique
527
of operator splitting has proven extremely powerful. Furthermore, the ability to model
528
anisotropic material behaviour utilizing for example VPFFT viscoplastic deformation
529
formulations (Lebensohn, 2001), has enabled realistic and dynamic models. Upscaling this
530
approach to investigate problems at a large scale e.g. folding (Llorens et al., 2013a, 2013b;
531
Bons et al., 2016; Ran et al., 2018) and shear deformation (Gardner et al., 2017) have shown
532
to be very beneficial. There is great scope to expand further on this in view of fluid flow,
533
mineralization and fault formation.
534 535
6. Numerical simulations of microstructures: Lessons learnt and future challenges
536
Numerical simulations of microstructural development have caught our imagination over the
537
last three decades. They have markedly advanced our ability to explain phenomena and
538
patterns we observe in nature and experiments by allowing us to test the link between
539
boundary conditions, material properties, processes, and microstructural development.
540
Importantly, models, especially those that couple several process and/or investigate pre-
541
existing heterogeneities can train the geologist to think of the dynamics of the system rather
542
than a linear development. For example, different patterns of strain localization observed in
543
nature can be explained by differences in the relative rates of interacting processes (e.g.
544
Jessell et al., 2005; Gardner et al., 2017). At the same time, specific indicative microstructural
545
parameters can be developed to help interpret natural microstructures (e.g. Piazolo et al.,
546
2002; Gomez-Rivas et al., 2017, Llorens et al., 2017; Steinbach et al., 2017).
547
However, including chemistry coupled to other processes remains a particular challenge.
548
Whilst, for example, trace-element diffusion can be enacted in parallel with other processes
549
(section 4.1.2), chemical transport of major elements and diffusion creep cannot yet be fully
550
integrated with many other processes. Indeed, when multiphase systems are considered, there
551
are unsolved problems with diffusion creep modelling even in the absence of other processes
552
(Ford and Wheeler, 2004). This challenge is closely linked to our current inability to
553
confidently model grain boundary sliding.
554
Nevertheless, the studies we describe have shown that numerical models are extremely
555
powerful in providing benchmark results to investigate what kind of microstructure may
556
develop under certain conditions. These models are sophisticated mind experiments that are
557
firmly based on physical and chemical laws for which the theory is well known individually
558
but their interaction is difficult to predict analytically.
559 560
Acknowledgements:
561
We would like to thank Win Means for opening up a whole new perspective on
562
microstructures with his inspirational in-situ experiments. The authors thank the DFG, ARC,
563
ESF, NSF, EU through Marie Curie Fellowship to SP, the Government of Catalonia's
564
Secretariat for Universities and Research for a Beatriu de Pinós fellowship to EGR (2016 BP
565
00208), and NERC (NERC grant NE/M000060/1) for support of the numerical endeavours.
566
The authors thank C. Gerbi for his helpful and constructive review as well as C. Passchier for
567
editorial handling of the manuscript.
568 569 570
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915
916
Table Captions
917
Table 1 List of numerical models of microstructural (mm to dm) development identifying
918
processes modelled, numerical method used and providing relevant references. This list is
919
restricted to geological applications and those referenced in the main text. Note processes are
920
categorized as "S" for static (material points do not move) and "D" for dynamic (material
921
points move). Furthermore, unless stated otherwise models are two-dimensional.
922
Abbreviations: TBH – Taylor Bishop Hill calculation method for crystal lattice rotation,
923
VPFFT – Viscoplastic Fast Fourier Transform based model, EVPFFT - Elasto-viscoplastic
924
Fast Fourier Transform based model, FEM Finite Element, reXX – recrystallization.
925 926
Figure Captions
927
Figure 1 Microstructural development during in-situ deformation of the rock analogue
928
octochloropropane within a circular shear zone (Bons and Jessell, 1999); dashed red lines
929
indicate the shear distribution between the two steps shown. Experiments run with top to the
930
right shear at an average strain rate of 4.6·10-4 s-1 where the strain rate near edge of the shear
931
zone ( 1.2·10-3 s-1) is 10 x higher than in the top half of the image ( 1.2·10-4 s-1). (a) t1 at a
932
bulk shear strain of 40; (b) t1+16 min.. Note multiple concurrent processes: grain boundary
933
migration, leading to dissection of grains (locations 1 & 2), subgrain rotation (black arrow),
934
nucleation (white arrow). The different shear rates lead to a different balance of
935
recrystallization processes and differences in microstructures. At the low shear rate grains are
936
equant, have straight sub-grain boundaries and basal planes at an angle to the NS and EW
937
polarisers. The high-strain rate zone shows serrate grain boundaries, an oblique grain-shape
938
foliation, basal grains approximately parallel to the shear-zone boundary, as well as shear
939
localisation on grain boundaries (location 3), indicative of grain boundary sliding/shearing.
940
Such "micro-shear zones" may now have been detected in polar ice sheets as well (Weikusat
941
et al., 2009).
942
Figure 2 Numerical representation of a microstructure. (a) Micrograph of quartzite; (b)
943
numerical representation combining an element data structure with nodes (black circles),
944
segments (black lines) and polygons (enclosed area) and a lattice data structure with
945
unconnected lattice points (open circles). This structure is used in the numerical platform Elle
946
(see text for details).
947
Figure 3 Numerical modelling of dynamic recrystallization – a historical perspective;
948
mineral modelled is quartz; simple shear (see text for details); (a) numerical microstructure
949
after = 1; different grey scales signify different crystallographic orientations; top inset shows
950
data structure of hexagonal lattice points (modified after Jessell and Lister, 1990); (b)
951
numerical microstructure after = 1; colours show crystallographic orientation, grain
952
boundaries are red, subgrain boundaries black; for data structure see inset (modified after
953
Piazolo et al., 2002).
954
Figure 4 Numerical modelling of dynamic recrystallization – testing the effect of process
955
combination on microstructural development; model parameters: mineral modelled - ice; time
956
step - 20 years, simple shear; ∆ - 0.04 per time step. FFT and GBM signify fast fourier
957
transform formulation for crystal plasticity and grain boundary migration, respectively; (a)
958
initial microstructure and results after=2.4 for different process combinations. Results are
959
shown as grain network with orientation related colour coding according to crystal
960
orientations relative to the shortening direction y (see legend) and in pole figures. In the latter
961
the colour bar indicates the multiples of uniform distribution; (b) grain area distribution
962
normalized to the initial average grain area for all models shown.
963
Figure 5 Results from diffusion creep modelling in pure shear; 2D microstructure as in the
964
starting frame of Fig 1b of Wheeler (2009). Shown is the oblique view of microstructure with
965
patterns of normal stress shown along grain boundaries in the 3rd dimension as “fences”. Red
966
arrows show stretching direction, hence stresses are tensile (shown as negative) on
967
boundaries at a high angle to stretching direction are tensile. Fences are colour coded
968
according to dissolution rate with blue low and red high. When there is no relative grain
969
rotation the fences have a single colour and the stress is parabolic. When there is relative
970
grain rotation the fences vary in colour and the stress is a cubic function of position.
971
Figure 6 Dynamic development of stylolite roughness in numerical simulations; (a) time
972
series (left to right) with the stylolite nucleating in the middle of a slow dissolving layer
973
(layer in green and stylolite in black colour). Once the stylolite has dissolve the layer on one
974
side the layer starts to pin and teeth develop. (b) Variation of the pinning strength of the layer
975
in three different simulations showing a strong dependency. The compaction (movement of
976
upper and lower walls) is shown in quite arrows. L in the picture on the right hand side is the
977
initial position of the layer and P shown as black arrows indicates the pinning of the layer
978
upwards and downwards during dissolution.
979
Figure 7 Example of new development in microdynamic numerical modelling; preliminary
980
results (see text for details). (a) Coupling of diffusion creep and surface energy driven grain
981
boundary migration (GBM); (left) starting microstructure; (middle) microstructure at stretch
982
2; (right) graph showing average aspect ratio versus stretch; note that the microstructure after
983
a significant period of exclusive grain boundary migration is the same as the starting
984
microstructure as no movement occurs as all triple junctions are 120°. Number of grains
985
stays constant for all simulations. (b) Evolution of chemical concentration of an arbitrary
986
element during surface energy driven GBM. The material is a single-phase polycrystalline
987
aggregate with different initial chemical content. This example assumes very low bulk
988
diffusion (Dbulk =1e-20 m2/s) and fast grain boundary diffusion (Dboundary =1e-8 m2/s), hence
989
grain boundary diffusion dominates. Colour code indicates chemical concentration and white
990
lines represent grain boundaries. (c) Dissolution of reactive surfaces in 3D using the example
991
of fluorite dissolution. Microstructures after three dissolution periods are show (150hrs, 180
992
hrs, 276 hrs); surfaces correspond to the {111} plane at the start of experiment/simulation;
993
colours identify different depths where blue signifies low and red high; images are 60 μm
994
width. Top panel row shows numerical results. Lower panel shows experimental results using
995
confocal microscopy images of a grain of a sintered CaF2 pellet at the same three dissolution
996
times as the numerical models. Note the formation of etch pits with similar triangular shape
997
and the faster/enhanced dissolution of the grain boundaries in both experiment and numerical
998
simulations.
*Manuscript mark-up changes Click here to view linked References
1
A review of numerical modelling of the dynamics of microstructural development in
2
rocks and ice: Past, present and future
3 4
S. Piazolo1, P. D. Bons2, A. Griera3, M.-G. Llorens2, E. Gomez-Rivas4,5, D.
5
Koehn5Koehn6, J. Wheeler6Wheeler7, R. Gardner7Gardner8, J. R. A.
6
Godinho8Godinho9,, L. Evans7,9Evans8,10, R. A. Lebensohn10Lebensohn11, M. W.
7
Jessell11Jessell12
8
1
School of Earth and Environment, University of Leeds, Leeds LS2 9JT, United Kingdom
9
2
Department of Geosciences, Eberhard Karls University Tübingen, Wilhelmstr. 56, 72074
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Tübingen, Germany
11
3
12
(Cerdanyola del Vallès), Spain
13
44
14
080280 Barcelona, Spain
15
5
16
United Kingdom
17
56
18
Glasgow G12 8QQ, UK
19
67
20
Herdman Building, Liverpool University, Liverpool L69 3GP, U.K.
21
78
22
Systems/GEMOC, Department of Earth and Planetary Sciences, Macquarie University, NSW
23
2109, Australia
24
89
25
Manchester, M13 9PL, Manchester, UK.
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Departament de Geologia, Universitat Autònoma de Barcelona, 08193 Bellaterra
Departament de Mineralogia, Petrologia i Geologia Aplicada, Universitat de Barcelona,
School of Geosciences, King’s College, University of Aberdeen, Aberdeen AB24 3UE,
School of Geographical and Earth Sciences, Gregory Building, University of Glasgow,
Dept. Earth, Ocean and Ecological Sciences, School of Environmental Sciences, Jane
Australian Research Council Centre of Excellence for Core to Crust Fluid
Henry Moseley X-ray Imaging Facility, School of Materials, The University of
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26
910
27
Vic 3800, Australia
28
1011
Los Alamos National Laboratory, MST8, MS G755, Los Alamos, NM, 87545, USA
29
1112
Centre for Exploration Targeting, School of Earth Sciences, The University of Western
30
Australia, 35 Stirling Hwy, Crawley, 6009, Australia
School of Earth, Atmosphere and Environmental Sciences, Monash University, Clayton,
31 32
Corresponding author: Sandra Piazolo (
[email protected])
33 34
Abstract
35
This review provides an overview of the emergence and current status of numerical
36
modelling of microstructures, a powerful tool for predicting the dynamic behaviour of rocks
37
and ice at the microscale with consequence for the evolution of these materials at a larger
38
scale. We emphasize the general philosophy behind such numerical models and their
39
application to important geological phenomena such as dynamic recrystallization and strain
40
localization. We focus in particular on the dynamics that emerge when multiple processes,
41
which may either be enhancing or competing with each other, are simultaneously active.
42
Here, the ability to track the evolving microstructure is a particular advantage of numerical
43
modelling. We highlight advances through time and provide glimpses into future
44
opportunities and challenges.
45 46
Keywords:
47
recrystallization, process interaction
review,
48 49
1.
Introduction
evolution
of
microstructures,
numerical
modelling,
dynamic
50
Structural geology studies how rocks deform under applied stress or strain. These studies are
51
applied from the sub-grain scale to that of mountain belts and tectonic plates. To know what
52
has happened to a volume of rock millions of years ago, or to predict what would happen to a
53
rock volume under certain (future) conditions, we need to know the link between (i) the
54
boundary conditions (e.g. applied stress or strain, pressure-temperature conditions), (ii) the
55
intrinsic material properties (e.g. Young's modulus, crystal symmetry, slip systems) and (iii)
56
the processes that may be activated (e.g. dislocation creep, pressure solution, mineral phase
57
changes). These three together determine the evolution of the state of the rock (e.g.
58
developing cleavage, folds, lineation, stylolites) and its bulk material properties (e.g.
59
viscosity). The microstructure is thus a state variable of a rock, describing the state it
60
achieved as a result of the interplay between various processes and boundary conditions.
61
Importantly, the microstructure is not a passive log of events, but plays an active and central
62
role in the evolution of a rock throughout its history (Gottstein, 2004). In this context non-
63
linear feedback between a specific microstructural configuration and local changes in
64
intrinsic and extrinsic parameters may be important but difficult to predict. Hence, the
65
microstructure is the link that couples the material properties, boundary conditions and
66
processes that together control the behaviour and evolution of a rock. Consequently, the
67
analysis and correct interpretation of microstructures is crucial in gaining an understanding of
68
how rocks, including ice, deform on Earth and other planets. Here we define microstructure
69
as the full spatial, compositional and orientational arrangement of all entities in a rock,
70
typically on the scale of a thin section to hand specimen (roughly µm to cm) (Hobbs et al.,
71
1976). These entities include minerals, grain and subgrain boundaries, crystal lattice
72
orientations, and chemical composition from the nano- to micro-scale.
73
While the rock's temperature or elastic strain are ephemeral, the microstructure may be
74
preserved for the geologist to interpret millions to billions of years later. Therefore,
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75
microstructures are one of the prime forensic tools to unravel the history of a rock that allows
76
us to deduce the succession of strain rates, stresses, diagenetic and metamorphic conditions
77
and the rock's rheology during deformation (e.g. Hobbs et al., 1976; Vernon, 2004; Passchier
78
and Trouw, 2005).
79 80
One major problem in microstructural studies is that we normally only have a static record as
81
post-mortem "images" (grain arrangement in thin section, chemical and orientation
82
characteristics derived using electron microscopy, 3D datasets from computer tomography,
83
neutron diffraction and synchrotron beam analysis, etc.). To interpret microstructures, we
84
need to know how they form and change. Means (1977) wrote: "a more valuable kind of
85
kinematic investigation is that in which the time sequence of incremental strains or
86
incremental displacements, the kinematic history, is correlated with progressive structural
87
change. This promises to reveal more about the physics of rock deformation and to provide
88
more sensitive structural methods for reconstructing tectonic history". Such investigations
89
first became possible using in-situ analogue experiments that were introduced to geology by
90
influential scientists such as Cloos (1955) and Ramberg (1981), and at the microstructural
91
scale by Means (1977). In in-situ analogue experiments by Means (1977), inspired by those
92
of McCrone and Chen (1949), a thin-section sized sample of crystalline analogue material is
93
sandwiched and deformed between glass plates to observe the changing microstructure under
94
a microscope. These experiments were followed by a large number of studies that
95
investigated deformation related features such as dynamic recrystallization and crystal
96
plasticity (e.g. Means, 1977; 1980; Means and Ree, 1988; Park and Means, 1996; Urai et al..,
97
1980; Urai and Humphreys, 1981; Wilson, 1986; Jessell, 1986; Ree, 1991; Bons and Jessell,
98
1999; Ree and Park, 1997; Herwegh and Handy, 1996, 1998; Nama et al., 1999; Wilson et
99
al., 2014) and), melt or fluid-bearing microstructures (e.g. Urai, 1983; Rosenberg and Handy,
100
2000, 2001; Rosenberg, 2001; Schenk and Urai, 2004, 2005; Walte et al., 2005) and), the
101
development of vein and fringe microstructures (e.g. Hilgers et al., 1997,; Koehn et al..,
102
2003) and the formation of -clasts (ten Brink et al., 1995). Many of these experiments made
103
clear that when a polycrystalline material deforms, multiple processes act upon the
104
microstructure and govern its dynamic behaviour throughout its deformation and post-
105
deformation history (Fig. 1a; see also movies at http://www.tectonique.net/MeansCD/). In
106
addition, pre-existing heterogeneities and/or those developing during deformation can have a
107
significant effect on the dynamics of the system since local variations in stress and strain play
108
a significant role in the evolution of microstructures and material behaviour. , with their
109
effects on chemistry being particularly intriguing (Wheeler, 2018).
110
Although changes in microstructural characteristics (e.g. grain size) can be quantified in such
111
experiments, the underlying principles of the active processes still need to be deduced.
112
Therefore, an additional tool is needed. Here numerical models simulating microstructural
113
development based on several concurrent processes become important. Such numerical
114
simulations share the advantage of in-situ experiments that the full microstructural
115
development can be traced in form of time-series, with the opportunity to systematically
116
study the effect of different processes and/or pre-existing heterogeneities on the
117
microstructural development. In addition, numerical simulations are neither constrained by
118
time nor specific boundary conditions. Once calibrated against laboratory experiments (e.g.
119
Piazolo et al.., 2004) or analytical solutions, they can be applied to a wide range of conditions
120
or materials including those not attainable or suitable for laboratory experiments. In many
121
cases, the developed microphysical behaviour may be applied to problems on the continental
122
scale.
123
In this contribution, we focus on those worksIn this contribution, we that (i) are applied to
124
geological materials, (ii) involve several processes and (iii) allow prediction and visualization
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125
of the development of the microstructure. Hence, we exclude studies where microstructures
126
are not spatially mapped along with models in which grains are modelled as if embedded in a
127
medium with averaged properties. We aim to give an overview of the concepts behind
128
numerical modelling of microstructures, i.e. microdynamic modelling, and the achievements
129
that have been made in this field. This is followed by a selection of examples of the current
130
state of the art. Finally, future possibilities and directions are briefly discussed, including
131
some work in progress. We focus on those studies that (i) are applied to geological materials,
132
(ii) involve several processes and (iii) allow prediction and visualization of the development
133
of the microstructure. Hence, we exclude studies where microstructures are not spatially
134
mapped along with models in which grains are modelled as if embedded in a medium with
135
averaged properties. Table 1 summarizes the microdynamic numerical models cited, the
136
numerical method used and processes included in each model.
137 138
2. Numerical simulation of microstructures
139
2. 1. Emergence and philosophy
140
Soon after the first in-situ experiments, computers had advanced to a stage allowing the first
141
numerical simulations of microstructural development, with a full "image" of the
142
microstructure, to appear in geology (Etchecopar, 1977; Jessell, 1988a,b; Jessell and Lister,
143
1990). When using such models we accept the fact that the scientific description of the
144
phenomena studied does not fully capture reality. This is not a shortcoming but a strength, as
145
general system behaviour and interrelationships can be studied systematically and
146
transferred to geological questions. In a numerical model the behaviour of a system emerges
147
from the piece-meal enumeration of the behaviour of each of its component parts. As a
148
consequence, numerical models are powerful and necessary when the bulk behaviour is
149
influenced by the local interaction of the components. In contrast, global, averaged or mean
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150
field solutions cannot fully encompass the effect of processes that interact with each other
151
on the small scale. In nature, patterns emerge (e.g. orientation and arrangement of high-
152
strain zones, foliation, fractures) which may be used to deduce conditions of their formation.
153
Numerical models are powerful tools to investigate this link between patterns observed in
154
nature and processes that are responsible for their development.
155 156
2.2. Multiprocess modelling – approaches
157
As shown convincingly by the aforementioned experiments, several processes act
158
concurrently on a microstructure and control the dynamics of the microstructural evolution
159
and material properties (e.g. Means, 1977, 1980). While there are a large number of models
160
that focus on one process alone, there are fewer systems that aim to model the effect of
161
multiple interacting processes. Here, we focus on the latter. Solving all necessary equations
162
simultaneously is only practical when the number of interacting processes is small.
163
Alternatively, the set of equations, hence processes, needs to be reduced to a manageable
164
number necessitating a user driven selection which may result in the omission of important
165
feedbacks. Operator splitting provides an alternative approach, whereby each process acts
166
sequentially on a microstructure that was incrementally changed by all operating processes in
167
the previous step. In this case, the number of equations to be solved is not limited; hence
168
there is no need to artificially reduce the number of modelled processes. Clearly, length scale
169
and time steps have to be considered carefully for this approach to be valid. The operator
170
splitting approach is fundamental to the numerical platform Elle (Jessell et al., 2001, Bons et
171
al., 2008, Piazolo et al., 2010; www.elle.ws) which is, within the realm of microdynamic
172
modelling, the most used numerical platform in geology. However, operator splitting has also
173
been employed in other numerical schemes (e.g. Cross et al., 2015).
174
175
2.3. The numerical representation of a microstructure
176
It is necessary to describe numerically the microstructure so that in the model the processes
177
can act upon it spatially. Up to now, for rocks and ice, microstructural models have been
178
mostly restricted to two-dimensions. Two basic approaches can be taken: (i) a lattice data
179
structure or (ii) an element data structure. In the lattice data structure, the microstructure is
180
mapped onto an irregular or regular lattice, like the pixels in a digital image or gridded
181
analysis points obtained during chemical mapping and orientation mapping of geological
182
samples. Potts, cellular automata, lattice-spring, particle-in-cell, phase field and
183
micromechanical Fast Fourier Transform (FFT) based models utilize the lattice data structure
184
(Fig. 2; see also chapter 2 of Bons et al., 2008 for a detailed review of different numerical
185
methods). Numerically the lattice data structure is easy to use and calculations are
186
straightforward. In the element data structure, the microstructure is described by discrete
187
elements which can be points, line segments, or polygons (Fig. 2). Finite-element models and
188
boundary models typically use elements to describe a system. Element data structure is well
189
suited for processes acting upon a surface, e.g. grain boundary migration, but is numerically
190
more “expensive” due to necessary topology checks. Both elements and lattice points can
191
have, in addition to their location in xy space, specific properties or store specific state
192
variables, such as mineral phase, chemical composition, crystallographic orientation, and
193
stress and strain (rate) values, respectively. The numerical platform Elle (Jessell et al.., 2001)
194
combines these two approaches: a microstructure is represented by boundary nodes
195
connected by straight segments (Fig.2). A polygon enclosed by boundary nodes represents a
196
grain or subgrain. Property variations within the individual polygons are defined at interior
197
points where information is recorded and tracked. Processes acting on the microstructure can
198
move boundaries and/or interior points and/or change the properties at those points, e.g.
199
composition, crystallographic orientation or dislocation density. The system has now matured
200
to a stage that it can robustly be used for a large variety of models relevant to geology.
201 202
3. Examples of modelling several coupled processes
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203
3.1.
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204
3.1.1. A historical perspective
205
Dynamic recrystallization is the response of a crystalline aggregate to lower its free energy by
206
formation and movement of sub- and grain boundaries (Means, 1983). It occurs by a number
207
of concurrent processes that act upon the polycrystalline material. For decades, geologists
208
have used microstructural characteristics developed during dynamic recrystallization to infer
209
dominance of processes and linked boundary conditions (e.g. Urai et al., 1986; Hirth and
210
Tullis, 1992; Stipp et al., 2002). Following the first attempts of such a model (Etchecopar,
211
1977), Jessell (1988a,b) and Jessell and Lister (1990) developed a numerical model
212
incorporating rotation of the crystal lattice and dynamic recrystallization for quartz. The data
213
structure used was a 100x100 hexagonal lattice structure where each lattice point represented
214
a grain or subgrain (Fig. 3a inset). Calculation of the crystal lattice rotation in Jessell and
215
Lister (1990) uses the Taylor-Bishop-Hill calculation method (Taylor, 1938; Bishop and Hill,
216
1951a, 1951b; Lister and Paterson, 1979) and the critical resolved shear stress of the different
217
slip systems in quartz. This work reproduced the general microstructures seen in a mylonite.
218
A major step forward was that the simulations showed that dynamic recrystallization can
219
significantly modify crystallographic preferred orientation (CPO) development. Piazolo et al.
220
(2002) advanced Jessell’s model by utilizing the more robust Elle data structure to describe
221
the local deformation and the dynamics of the grain boundary network (Jessell et al., 2001).
222
The model now included a finite element solution for incompressible, linear or nonlinear
223
viscous flow (BASIL; Barr and Houseman 1992, 1996) to compute the local velocity field,
Dynamic recrystallization
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224
the front tracking method for the motion of grain boundaries by moving nodes and segments
225
(e.g. Bons and Urai, 1992), the Taylor-Bishop-Hill formulation (Lister and Paterson, 1979)
226
and additional features such as crystal lattice rotation, formation of subgrains,
227
recrystallization by nucleation, grain boundary migration, recovery, work hardening and
228
tracking of dislocation densities. This list of additional features shows the advance made in
229
approximately a decade in reproducing natural microstructures (Fig. 3b).
230
So far, the rheological anisotropy of minerals deforming by dislocation creep was not taken
231
into account in the calculation of the stress-strain rate field and lattice rotation. About another
232
decade after Piazolo et al. (2002), the viscoplastic implementation of the full-field crystal
233
plasticity micromechanical code based on Fast Fourier Transforms (VPFFT; Lebensohn,
234
2001) was introduced in geological microstructural modelling (Griera et al., 2011, 2013,
235
2015). The VPFFT is a spectral method that specifically assumes that deformation is
236
achieved by dislocation glide on crystallographic slip planes (each with their own critical
237
resolved shear stress) whose orientations are mapped on a regular grid (for details see Griera
238
et al. (2013), Montagnat et al. (2011, 2014) and Llorens et al. (2016a)). The simulations of
239
Griera et al. (2011, 2013) and Ran et al. (2018) illustrate that heterogeneous stress within and
240
between grains that emerge from inhomogeneous slip has a significant effect on the rotation
241
rate of porphyroblastporphyroclasts and -clastblasts and the strain (rate) field around these
242
objects.
243
More recently, Llorens et al. (2016a; 2016b; 2017) and Steinbach et al. (2016; 2017) coupled
244
dynamic recrystallization (DRX) by grain boundary migration (GBM), intracrystalline
245
recovery and polygonisation with VPFFT crystal plasticity models, and applied them to study
246
the deformation of polar ice. Llorens et al. (2016a) presents a comprehensive description of
247
the method. They found that dynamic recrystallizationDRX produces large and
248
equidimensional grains, but only marginally affects the development of the c-axes CPO.
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249
However, DRX can alter the activity of slip systems and does modify the distribution of a-
250
axes. In simple shear, the strong intrinsic anisotropic of ice crystals is transferred from the
251
crystal to the polycrystal scale, leading to strain localisation bands that can be masked by
252
GBM (Llorens et al., 2016b). Llorens et al. (2017) compared the dynamics of pure polar ice
253
polycrystalline aggregates in pure and simple shear deformation. It was found that, due to the
254
vorticity of deformation, it is expected that ice is effectively weaker in the lower parts of the
255
ice sheets (where simple shear dominates) than in the upper parts (where ice is mostly
256
deformed coaxially). In Jansen et al. (2016) thisThe method was also applied to reproduce the
257
development of tilted-lattice (kink) bands found in ice cores (Jansen et al., 2016) and
258
crenulation cleavage during folding (Ran et al., 2018) as a result of mechanical anisotropy.
259
Steinbach et al. (2016) included air inclusions as a second phase to simulate DRX in porous
260
firn and found that DRX can occur despite the low strain and stress in firn.
261
Apart from polar ice, the VPFFT-Elle model has also been used to analyse subgrain rotation
262
recrystallization of halite polycrystals in simple shear (Gomez-Rivas et al., 2017) by coupling
263
VPFFT with the Elle routines that simulate intracrystalline recovery (Borthwick et al, 2013)
264
and subgrain rotation. They found that recovery does not affect CPOs, but strongly decreases
265
grain size reduction. These authors also evaluated the use of mean subgrain misorientations
266
as a strain gauge.
267
3.1.2. Testing the effect of different combinations of recrystallization processes
268
We show results from three combinations of processes that occur during dynamic
269
recrystallization. These processes act on the same initial 10x10 cm aggregate of ice Ih
270
crystals (Fig. 4a). The numerical approach in these simulations is based on the VPFFT
271
micromechanical model in combination with several Elle processes (Llorens et al., 2016a,
272
2016b, 2017; Steinbach et al., 2016; 2017). The deformation-induced lattice rotation and the
273
estimation of geometrically necessary dislocation densities calculated from the stress and
274
velocity field provided by the VPFFT algorithm are used to simulate recrystallization by
275
intra-crystalline recovery, grain boundary migration and nucleation. Grain boundary
276
migration (GBM) is simulated using a front-tracking approach based on the algorithm by
277
Becker et al. (2008). Here, a front-tracking approach is used in which the movement of
278
boundaries is mapped through time. Recovery reduces the intra-granular stored energy in a
279
deformed crystal (Borthwick et al., 2013). Nucleation creates new grain boundaries with
280
areas of misorientation values above a pre-defined threshold (Piazolo et al.., 2002; Llorens et
281
al., 2017; Steinbach et al., 2017). Dextral simple shear deformation is applied in strain
282
increments of ∆ = 0.04. Three different combinations of recrystallization processes are
283
tested; namely: (1) GBM only, (2) GBM and recovery or (3) GBM, recovery and nucleation
284
(Fig. 4a). Simulations including VPFFT viscoplastic deformation show a similar evolution of
285
c-axis orientation, regardless of the dynamic recrystallization processes included (Fig. 4a).
286
The lack of influence of nucleation on the CPO is due to the fact that new grains are modelled
287
to have lattice orientations close to those of their parent grains. However, inclusion of the
288
nucleation process result in grain size reduction (Fig. 4b). Results illustrate how different
289
combinations of microdynamic processes affect the microstructural characteristics (e.g. CPO,
290
grain size, shape and orientation) in different ways where no single microstructural parameter
291
can grasp the full dynamics of polycrystalline deformation.
292
3.2. Diffusion creep
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293
Diffusion creep is a physiophysico-chemical process that operates in large regions of the
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294
Earth – in the upper Earth as “pressure solution” where diffusion is enhanced along fluid
295
films and elsewhere at low strain rates, maybe in large parts of the lower mantle (Karato,
296
1992). The mechanism involves diffusion driven by gradients in normal stress along
297
boundaries. In simple models no other driving forces need to be included. It is implicit in our
298
understanding of this mechanism that grain boundary sliding occurs in conjunction with
299
diffusion. Ford et al. (2002) devised a numerical model to understand the evolution of grain
300
shapes and CPO in grain boundary diffusion creep. The mathematical framework allows for
301
the consideration of diffusion and sliding together. Operator splitting is not required and the
302
evolution at each timestep is based on a single matrix inversion operation. The model couples
303
the microstructure to an evolving system of local stresses (Fig. 5). The model predicts that
304
grain shapes become somewhat elongate, in accordance with experiments on calcite (Schmid
305
et al., 1987) and observations on an albite-bearing mylonite (Jiang et al., 2000) (Fig. 5). It
306
also predicts that grain rotations are not chaotic and that CPO may be present to high strains
307
(Wheeler, 2009), as seen in, for example, experiments on olivine-orthopyroxene (Sundberg
308
and Cooper, 2008).) and predicted by numerical simulations of Bons and den Brok (1999). A
309
second phase, if insoluble, can be included and such a model was used by Berton et al. (2006,
310
2011) to prove the hypothesis that a different grain boundary diffusion coefficient along the
311
two-phase boundaries could explain fibre growth in pressure shadows. Not all model
312
predictions are in agreement with other investigations – for example some materials
313
deformed by diffusion creep e.g. olivine show quite equant grains (Karato et al., 1986). In
314
such experiments grain growth occurs but this process was not included in the numerical
315
model, hence results differ. In a later section we address this issue, but this example
316
illustrates that, in common with other numerical models, the applicability reflects the
317
processes included.
318 319
3.3. Stress driven dissolution, growth and dynamic roughening
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320
Another example where stress leads to micro- and macrostructures is the roughening of grain
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321
boundaries in the presence of a fluid. In this case dissolution and/or precipitationmineral
322
growth are driven not just by normal stress gradients at the interface (c.f. section 3.2) but also
323
by changes in elastic and surface energies as well as normal stress at the interface.. In
324
thisaddition to these driving forces, the detailed electrochemistry of interfaces has a kinetic
325
effect on rates (Gratier et al., 2013). In the example presented here, the process is modelled in
326
Elle by coupling a lattice spring code that calculates the strain with a background fluid that
327
dissolves lattice elements. This process can produce transient patterns of interface geometry
328
at the grain boundaries (Koehn et. al., 2006). In addition, the process itself produces rough
329
interfaces that can be seen on the larger scale as stylolites (Koehn et al., 2007; Ebner et al.,
330
2009). The scaling behaviour of stylolites is important for stress inversion and compaction on
331
the basin scale (Koehn et al., 2012; 2016) and the shape of stylolites is partly rooted in the
332
microstructure of the host-rock (Fig. 6). Slower dissolving material on various scales, from
333
small grains to fossils and layers, can influence the dissolution process and thus the
334
development of patterns, whereas deformation leads to changes in long-range interactions
335
(Koehn et al., 2016). SimilarSimilarly, pinning behaviour in the presence of a fluid is seen in
336
grain growth where grain boundary pinning results in restriction of grain growth. Pinning
337
particles can also be moved around so that growing grains become clean. Such a process can
338
lead to layered rocks, for example, the Zebrazebra dolomites (Kelka et al., 2015).
339
4. Numerical simulations of microdynamic processes: Examples of ongoing work
340
4.1.
341
Even now, most of the numerical models that incorporated more than one process are
342
restricted to deformation processes alone. Although in diffusion creep there is an intrinsic
343
chemical aspect, the model described in 3.2 involves just a single phase of fixed composition.
344
There are some models that incorporate the growth of new mineral phases and investigate the
345
rheological effect of such growth (e.g. Groome et al., 2006; Smith et al., 2015), however the
346
location of new phase growth and/or growth rate is predefined. While Park et al. (2004)
347
showed convincingly the effect of microstructure on the diffusion pathways and chemical
348
patterns in garnet and biotite, there are now many opportunities available in a numerical
Linking chemistry and microstructural evolution
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349
system such as Elle (Jessell et al., 2001) to couple local chemistry and microstructural
350
evolution. Below we present preliminary results from two current projects linking chemical
351
changes and a dynamically evolving grain network.
352
4.1.1. Grain boundary diffusion creep and grain growth
353
Here we present the first results using a model combining, using operator splitting,
354
deformation by diffusion creep (Ford et al., 2002,; Wheeler, 2009) and surface energy driven
355
grain boundary migration (GBM), i.e. grain growth (modified after Becker et al., 2008). Four
356
different scenarios are shown: diffusion creep only, grain boundary migration only and two
357
combinations with medium and high grain boundary migration GBM rates (Fig. 7a). The
358
starting material is an almost regular hexagonal grain mesh with equant grains in which triple
359
junctions have had small random perturbations imposed (Fig. 7a). The perturbations are
360
required to avoid mathematical problems that arise in perfectly regular hexagonal networks
361
(Wheeler, 2010). Topology checks such as neighbour switching routines are performed after
362
each step in all simulations, ensuring that no topology problems arise. The microstructural
363
development is markedly different if grain growth is coupled with diffusion creep (Fig. 7a)
364
where with increasing GBM rate the aspect ratios are generally reduced relative to that of
365
comparable experiments modelling diffusion creep only. This is provisionally in accord with
366
the hypothesis that there will be a ‘steady state’ grain elongation developed which is a
367
function of the relative magnitudes of strain rate and grain growth kinetic parameters
368
(Wheeler, 2009) but more simulations are required (with less regular starting
369
microstructures). In contrast, for grain growth only, the microstructure does not change as the
370
initial configuration consists of stable near 120° triple junctions (Fig. 7a). Grain numbers (i.e.
371
grain size) do not change in any of the simulations.
372 373
4.1.2. Trace element partitioning between fluid and solid coupled with recrystallization
374
Field and experimental studies illustrate the importance of the time and length scales on the
375
evolution of grain networks during trace element diffusion (e.g. Ashley et al., 2014 and
376
references therein). Therefore, understanding the influence of recrystallization on trace
377
element distribution is essential for correctly interpreting patterns of trace element
378
distribution (e.g. Wark and Watson, 2006). However, to our knowledge there are currently no
379
numerical approaches able to simulate diffusion coupled with microstructure evolution.
380
Recently, a new Elle finite difference process that solves Fick’s second law to simulate trace
381
element diffusion in a polycrystalline medium has been developed. The polycrystal is defined
382
by n-phases where grain boundaries and grains are differentiated (Fig. 7b). The trace element
383
partitioning coefficient between different phases is also considered. The approach uses an
384
element data structure representing grain boundaries and a lattice data structure of a regular
385
grid of unconnected lattice point to track diffusion along the grain boundaries and the
386
physical and chemical properties within grains, respectively. To ensure elemental exchange
387
between grain boundaries and grain interiors, a grain interior-grain boundary partition
388
coefficient is implemented that takes the proximity to grain boundaries into account. This
389
allows the simulation of coupled bulk and grain boundary diffusion. This approach can be
390
fully coupled with other processes of the numerical platform Elle and therefore allows
391
simulating simultaneous diffusion, deformation and static or dynamic recrystallization. The
392
use of this approach is demonstrated through the modelling of diffusion of a tracer with
393
fractionation during static grain growth of a single-phase aggregate (Fig. 7b) (after that
394
presented by Jessell et al., 2001). Patterns of chemical concentration qualitatively resemble
395
those of Ti-distribution observed in recrystallized quartz in shear zones and can help to
396
understand the redistribution of Ti in quartz during dynamic recrystallization (e.g. Grujic et
397
al. 2011).., 2011), among many other geological problems.
398
4.2.
399
We restricted this review to two -dimensional numerical approaches, as very few models
400
have been published that are in three dimensions and also specific to geological
401
microstructures. An exception is the phase field approach by Ankit et al. (2015) who
402
investigate the growth of crystals in an open fracture. However, within the material science
403
community numerical models exist that investigate the behaviour of a microstructure in
404
response to single processes in three dimensions (e.g. three dimensional crystal plasticity
405
modelling platform DAMASK (Roters et al., 2012; Eisenlohr et al., 2013), which includes the
406
option of solving the micromechanical problem using a 3D implementation of the FFT-based
407
formulation (presently implemented in Elle in its 2-D version) and grain growth (e.g. Krill
408
and Chen, 2002; Kim et al., 2006; Zöllner, 2011). With the technical advances made within
409
the geological and material science community it is now in principle possible to extend
410
currently available 3D models to geological questions and/or multi-process scenarios.
Modelling in three dimensions
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411 412
4.2.1. Dissolution of reactive surfaces in three dimensions
413
Mineral dissolution controls important processes in geoscience such as serpentinization (e.g.
414
Seyfried et al., 2007), retrogression and replacement reactions (Putnis and Austrheim, 2012),
415
deformation by dissolution-precipitation creep (e.g. Rutter, 1976) and also affects processes
416
relevant to society such as the stability of spent nuclear fuel and mine waste. Recent work has
417
shown that dissolution rates are linked to the evolving surface structure, and thus are time-
418
dependent (Godinho et al., 2012). Different surfaces have different structures in three
419
dimensions and the presence of etch pits play a major role in dissolution (Pluemper et al..,
420
2012). Hence, for accurate representation of dissolution behaviour, modelling in three
421
dimensions is important. Here, we present a numerical model that simulates the dissolution
422
process as a potential tool to quantify the links between dissolution rates, reactive surface
423
area and topography over periods of time beyond reasonable for a laboratory experiment. The
424
program uses empirical equations that relate the dissolution rate of a point of the surface with
425
its crystallographic orientation (Godinho et al., 2012) to simulate changes of topography
426
during dissolution, which ultimately results in the variation of the overall dissolution rate
427
(Godinho et al., 2012, Godinho et al., 2014). The initial surface is composed of a group of
428
nodes with a xy position and a set height (z). For each lattice point a local surface orientation
429
is calculated from the inclination of the segment node and its neighbours. This orientation
430
together with the crystallographic orientation of the grain the surface belongs to is then used
431
to calculate a dissolution rate (surface reactivity). Based on this the displacement of the node
432
using the equations published in Godinho et al. (2012) is calculated. The model allows the
433
graphical display of the three-dimensional topographic development of the surface, tracking
434
of the variation of the surface area and calculation of the overall dissolution rate at each stage
435
of the simulation. Results obtained with the 3D simulation at three consecutive stages of
436
dissolution (150 hrs, 180 hrs, 276 hrs dissolution duration) show that numerical results are in
437
accordance to experimental results (Fig. 7c).
438 439
5. Numerical simulations of microstructures: Possibilities and challenges
440
5.1. Linking laboratory and natural data with numerical models
441
Over the last two decades numerical capabilities have advanced markedly and models have
442
come of age. Consequently, a large range of models and process combinations is now
443
available that can be utilized to gain further insight into the link between processes, material
444
properties and boundary conditions. This also includes polymineralic systems making the
445
models more appropriate to use to investigate processes occurring within
446
polymineralicpolyphase rocks (e.g. Roessiger et al., 2014; Steinbach et al, 2016). New
447
avenues of effective verification against laboratory and natural data are now opening up, due
448
to the development of analytical tools allowing rapid complete microstructural and
449
microchemical analysis providing dataset similar to those possible in numerical models; (e.g.
450
Steinbach et al, 2017, Piazolo et al., 2016); many of the analytical data sets have a lattice data
451
structure.
452 453
5.2. Linking chemistry and microstructural evolution
454
Advances in numerical methods, theoretical treatment of thermodynamic data, analytical
455
tools and theoretical and experimental insights into the coupling between chemical and
456
physical process that emerged over the last decade, call for a major effort in this area of
457
research. Examples of areas of opportunities include investigation of the (1) significance of
458
local stress versus bulk stress on mineral reactions and reaction rates (e.g. Wheeler, 2014;
459
Hobbs and Ord, 2014), (2) characteristics of replacement microstructures and their potential
460
significance for microstructural interpretation (e.g. Putnis, 2009, SpuzienceSpruzeniece et al.,
461
2017), (3) mobility of trace elements enhanced by deformation that change significantly the
462
local elemental distributions (e.g. Reddy et al., 2007, Piazolo et al., 2012, 2016) and (4)
463
coupling between reactive fluid-solid systems and hydrodynamics (Kelka et al., 2017).
464
Increasing computer processing speeds will aid the running of such models. However, the
465
technical challenges include the harmonisation of fundamentally different numerical
466
approaches (crystal plasticity versus diffusion creep, for example) and the resolution of
467
fundamental mathematical problems, e.g. the current lack of an internally consistent model
468
for multiphase diffusion creep, highlighted by Ford and Wheeler (2004).
469
5.3. Linking brittle and ductile deformation: elasto-viscoplastic behaviour
470
When examining the rock record, it is clear that in many cases, brittle and ductile behaviour
471
often occurs at the same time within a rock (e.g. Bell and Etheridge, 1973, Hobbs et al.,
472
1976). With the potential significance of grain-scale brittle behaviour now measurable on
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473
seismic signals (e.g. Fagereng et al., 2014) there is an increased need in developing numerical
474
techniques that allow us to model the dynamic link between brittle and ductile behaviour.
475
Such elasto-viscoplastic behaviour combines the elastic reversible fast deformation with a
476
viscous time dependent flow and a plastic behaviour. These behaviours can be included in
477
continuous or discontinuous models. For example, in the numerical platform Elle, a lattice
478
structure is used to deform the model elastically up to a critical stress where bonds fracture
479
(plastic behaviour) and the particles themselves deform as a function of stress and time
480
(viscous behaviour). In this case the viscous behaviour conserves the volume whereas shear
481
forces and differential stresses converge to zero (Sachau and Koehn, 2010, 2012; Arslan et
482
al., 2012; Koehn and Sachau, 2014). Alternatively, linking the elasto-visocplastic FFT based
483
(EVPFFT) (Lebensohn et al., 2012) with models such as Elle would allow calculation of the
484
Cauchy stresses that are the local driving force for grain scale damage processes.
485
One of the major challenges is the large range of time-scales in these processes, with
486
fracturing and fluid flow on fast to intermediate and viscous deformation and potentially
487
reactions on very long time scales. This complexity either requires the assumption that some
488
processes are instantaneous or it requires an “up-scaling in time” or non-linear time scales in
489
models.
490 491
5.4. Expansion of capability to three dimensions
492
With the advent of supercomputers and new numerical approaches now is the time to develop
493
techniques to investigate microstructural development in three dimensions. This is of
494
particular importance if material transport such as aqueous fluid and/or melt flow is to be
495
considered. This would enable, for example, modelling of strain fringes, shear veins and en-
496
echelon extensiontension gashes. However, these require models that link brittle and ductile
497
behaviour along with modelling in three dimensions.
498
One of the biggest problems faced with three-dimensional models of microstructures are the
499
three-dimensional topology changes that are common in dynamic microstructural
500
development (e.g. Fig. 1-7). This is a major problem if an element data structure with
501
segments, i.e. grain boundaries is used. However, the phase field approach does not work
502
with such distinct boundaries, and is therefore it is well suited for 3D problems (e.g. Ankit et
503
al. 2014). In addition, a three dimensional network of unconnected nodes, in which there is
504
no physical movement of boundaries but only changes in the properties of the 3D nodes
505
(voxels) (cf. Fig. 7c) may be a way forward (e.g. Sachau and Koehn, 2012).
506 507
5.5. Link between geophysical signals and microstructure
508
At a time where there is an ever-increasing amount of geophysical data being collected,
509
numerical simulations that are used to test the link between microstructural development and
510
geophysical signal will become increasingly important. CrypchCyprych et al. (2017) showed
511
that not only crystallographic preferred orientation but also the spatial distribution of phases
512
i.e. the microstructure, has a major impact on seismic anisotropy. Therefore, a direct link
513
between the microdynamic models such as Elle allowing tracking of microstructural changes
514
through time and space and geophysical signal generation offers a wealth of new
515
opportunities including interpretation of strong reflectors in the lower crust and mantle.
516
Current efforts by Johnson, Gerbi and co-workers (Cook et al., 2013; Naus-Thijsen et al.,
517
2011; Cook et al., 2013; Vel et al., 2016) are in line with this direction.
518 519
5.6. Application of numerical models to polymineralic rock deformation and ice-related
520
questions
521
The dynamic behaviour of the Earth is strongly influenced by the deformation of
522
polymineralic rocks. At the same time Earth’s polar ice caps and glacial ice which often
523
include ice and a second phase (e.g. air inclusions, dust, entrained bedrock) is of major
524
importance to society, especially in view of changing climate (e.g. Petit et al., 1999; EPICA,
525
2004). Application of the current numerical capabilities to polycrystalline ice and
526
polymineralic rocks is therefore urgently needed. Over the last years, there has been an
527
increased effort in this direction (Roessiger et al., 2011, 2014; Piazolo et al., 2015; Llorens et
528
al., 2016a, 2016b, 2017; Jansen et al., 2016; Steinbach et al. 2016, 2017; Roessiger et al.,
529
2011, 2014), which promises to continue. Here, the development of the link between
530
elemental mobility and microstructural development is of major importance, as only with
531
such models the can chemical signals of, for example, ice cores be correctly interpreted.
532 533
5.7. Upscaling: Utilizing operator splitting and utilities developed for microdynamic
534
systems to larger -scale problems
535
One of the strengths of the numerical approach taken by the microstructural community has
536
been the close link between different processes and the ability of the models to take into
537
account the local differences in properties such as stress, strain and chemistry. The technique
538
of operator splitting has proven extremely powerful. Furthermore, the ability to model
539
anisotropic material behaviour utilizing for example VPFFT viscoplastic deformation
540
formulations (Lebensohn, 2001), has enabled realistic and dynamic models. Upscaling this
541
approach to investigate problems at a large scale e.g. folding (Llorens et al., 2013a, 2013b;
542
Bons et al.., 2016; Ran et al., 2018) and shear deformation (Gardner et al., 2017) have shown
543
to be very beneficial. There is great scope to expand further on this in view of large scale
544
shear deformation, fluid flow, mineralization and fault formation.
545 546
6. Numerical simulations of microstructures: Lessons learnt and future challenges
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547
Numerical simulations of microstructural development have caught our imagination over the
548
last three decades. They have markedly advanced our ability to explain phenomena and
549
patterns we observe in nature and experiments by allowing us to test the link between
550
boundary conditions, material properties, processes, and microstructural development.
551
Importantly, models, especially those that couple several process and/or investigate pre-
552
existing heterogeneities can train the geologist to think of the dynamics of the system rather
553
than a linear development. For example, different patterns of strain localization observed in
554
nature can be explained by differences in the relative rates of interacting processes (e.g.
555
Jessell et al.., 2005; Gardner et al.., 2017). At the same time, specific indicative
556
microstructural parameters can be developed to help interpret natural microstructures (e.g.
557
Piazolo et al., 2002; Gomez-Rivas et al., 2017, Llorens et al., 2017; Steinbach et al., 2017).
558
Such
559
However, including chemistry coupled to other processes remains a particular challenge.
560
Whilst, for example, trace-element diffusion can be enacted in parallel with other processes
561
(section 4.1.2), chemical transport of major elements and diffusion creep cannot yet be fully
562
integrated with many other processes. Indeed, when multiphase systems are considered, there
563
are unsolved problems with diffusion creep modelling even in the absence of other processes
564
(Ford and Wheeler, 2004). This challenge is closely linked to our current inability to
565
confidently model grain boundary sliding.
566
Nevertheless, the studies we describe have shown that numerical models are extremely
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567
powerful in providing benchmark results to investigate what kind of microstructure may
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568
develop under certain conditions. These models are sophisticated mind experiments that are
569
firmly based on physical and chemical laws for which the theory is well known individually
570
but their interaction is difficult to predict analytically.
571
572
Acknowledgements:
573
We would like to thank Win Means for opening up a whole new perspective on
574
microstructures with his inspirational in-situ experiments. The authors thank the DFG, ARC,
575
ESF, NSF, EU through Marie Curie Fellowship to SP and NERC for support of the numerical
576
endeavours., the Government of Catalonia's Secretariat for Universities and Research for a
577
Beatriu de Pinós fellowship to EGR (2016 BP 00208), and NERC (NERC grant
578
NE/M000060/1) for support of the numerical endeavours. The authors thank C. Gerbi for his
579
helpful and constructive review as well as C. Passchier for editorial handling of the
580
manuscript. Formatted: Font: Bold
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Formatted: English (United Kingdom)
Formatted: English (United Kingdom)
934
935
Table Captions
936
Table 1 List of numerical models of microstructural (mm to dm) development identifying
937
processes modelled, numerical method used and providing relevant references. This list is
938
restricted to geological applications and those referenced in the main text. Note processes are
939
categorized as "S" for static (material points do not move) and "D" for dynamic (material
940
points move). Furthermore, unless stated otherwise models are two-dimensional.
941
Abbreviations: TBH – Taylor Bishop Hill calculation method for crystal lattice rotation,
942
VPFFT – Viscoplastic Fast Fourier Transform based model, EVPFFT - Elasto-viscoplastic
943
Fast Fourier Transform based model, FEM Finite Element, reXX – recrystallization.
944 945
Figure Captions
946
Figure 1 Microstructural development during in-situ deformation of the rock analogue
947
deformationoctochloropropane within a circular shear zone (Bons and Jessell, 1999); dashed
948
red lines indicate the shear distribution between the two steps shown; octachloropropane,.
949
Experiments run with top to the right shear, at an average strain rate isof 4.6·10-4 s-1 where
950
the strain rate near edge of the shear zone ( 1.2·10-3 s-1) is 10 x higher than in the top half of
951
the image ( 1.2·10-4 s-1). (a) at t1 at a bulk shear strain of 40; (b) t1+16 min; note.. Note
952
multiple concurrent processes: grain boundary migration, leading to dissection of grains
953
(locations 1 & 2), subgrain rotation (black arrow), nucleation (white arrow). The different
954
shear rates lead to a different balance of recrystallization processes and differences in
955
microstructures. At the low shear rate grains are equant, have straight sub-grain boundaries
956
and basal planes at an angle to the NS and EW polarisers. The high-strain rate zone shows
957
serrate grain boundaries, an oblique grain-shape foliation, basal grains approximately parallel
958
to the shear-zone boundary, as well as shear localisation on grain boundaries (location 3),
Formatted: Space After: 0 pt
959
indicative of grain boundary sliding/shearing. Such "micro-shear zones" may now have been
960
detected in polar ice sheets as well (Weikusat et al., 2009).
961
Figure 2 Numerical representation of a microstructure;. (a) micrographMicrograph of
962
quartzite; (b) numerical representation combining an element data structure with nodes (black
963
circles), segments (black lines) and polygons (enclosed area) and a lattice data structure with
964
unconnected lattice points (open circles). This structure is used in the numerical platform Elle
965
(see text for details).
966
Figure 3 Numerical modelling of dynamic recrystallization – a historical perspective;
967
mineral modelled is quartz; simple shear (see text for details); (a) numerical microstructure
968
after = 1; different grey scales signify different crystallographic orientations; top inset shows
969
data structure of hexagonal lattice points (modified after Jessell and Lister, 1990); (b)
970
numerical microstructure after = 1; colours show crystallographic orientation, grain
971
boundaries are red, subgrain boundaries black; for data structure see inset (modified after
972
Piazolo et al., 2002).
973
Figure 4 Numerical modelling of dynamic recrystallization – testing the effect of process
974
combination on microstructural development; model parameters: mineral modelled is- ice;
975
intrinsic grain boundary mobility M0 = 1·10-10 m2kg-1s-1 (see Llorens et al., 2017 for
976
details); time step =- 20 years, simple shear; ∆ =- 0.04 per time step. FFT and GBM signify
977
fast fourier transform formulation for crystal plasticity and grain boundary migration,
978
respectively; (a) initial microstructure and results after=2.4 for different process
979
combinations. Results are shown as grain network with orientation related colour coding
980
according to crystal orientations relative to the shortening direction y (see legend) and in pole
981
figures, where. In the latter the colour bar indicates the multiples of uniform distribution; see
982
text for further details; (a) initial microstructure (left) and numerical results and results
Formatted: Font: Not Bold, English (United Kingdom)
Formatted: Font: Not Bold, English (United States)
983
after=2.4; (b) grain area distribution normalized to the initial average grain area for all
984
models shown.
985
Figure 5 Results from diffusion creep modelling in pure shear; 2D microstructure as in the
986
starting frame of Fig 1b of Wheeler (2009);). Shown is the oblique view of microstructure
987
with patterns of normal stress shown along grain boundaries in the 3rd dimension as “fences”;
988
red”. Red arrows show stretching direction, hence stresses are tensile (shown as negative) on
989
boundaries at a high angle to stretching direction are tensile. Fences are colour coded
990
according to dissolution rate with blue low and red high. When there is no relative grain
991
rotation the fences have a single colour and the stress is parabolic. When there is relative
992
grain rotation the fences vary in colour and the stress is a cubic function of position.
993
Figure 6 Dynamic development of stylolite roughness in numerical simulations; (a) time
994
series (left to right) with the stylolite nucleating in the middle of a slow dissolving layer
995
(layer in green and stylolite in black colour). Once the stylolite has dissolve the layer on one
996
side the layer starts to pin and teeth develop. (b) Variation of the pinning strength of the layer
997
in three different simulations showing a strong dependency. The compaction (movement of
998
upper and lower walls) is shown in quite arrows. L in the picture on the right hand side is the
999
initial position of the layer and P shown as black arrows indicates the pinning of the layer
1000
upwards and downwards during dissolution.
1001
Figure 7 Example of new development in microdynamic numerical modelling; preliminary
1002
results (see text for details). (a) Coupling of iffusiondiffusion creep and surface energy driven
1003
grain boundary migration (GBM); (left) starting microstructure; (middle) microstructure at
1004
stretch 2; (right) graph showing average aspect ratio versus stretch; note that the
1005
microstructure after a significant period of exclusivelyexclusive grain boundary migration is
1006
the same as the starting microstructure as no movement occurs as all triple junctions are 120°;
1007
number°. Number of grains stays constant for all simulations. (b) Evolution of chemical
1008
concentration of an arbitrary element during surface energy driven GBM. The material is a
1009
single-phase polycrystalline aggregate with different initial chemical content. This example
1010
assumes very low bulk diffusion (Dbulk =1e-20 m2/s) and fast grain boundary diffusion
1011
(Dboundary =1e-8 m2/s), hence grain boundary diffusion dominates; colour. Colour code
1012
indicates chemical concentration; and white lines displayingrepresent grain boundaries. (c)
1013
Dissolution of reactive surfaces in 3D; material using the example of fluorite; microstructures
1014
dissolution. Microstructures after three dissolution durationsperiods are show (150hrs, 180
1015
hrs, 276 hrs); surfaces correspond to the {111} plane at the start of experiment/simulation;
1016
colours identify different depths where blue signifies low and red high; images are 60 μm
1017
width; note the formation of etch pits with similar triangular shape and the faster/enhanced
1018
dissolution of the grain boundaries in both experiment and numerical simulation; (top. Top
1019
panel row) shows numerical results; (lower. Lower panel row)shows experimental results
1020
using confocal microscopy images of a grain of a sintered CaF2 pellet at the same three
1021
dissolution times.
1022 1023
as the numerical models. Note the formation of etch pits with similar triangular shape and
1024
the faster/enhanced dissolution of the grain boundaries in both experiment and numerical
1025
simulations.
Formatted: Space After: 0 pt, Don't adjust space between Latin and Asian text, Don't adjust space between Asian text and numbers
Formatted: English (Australia)
a
t1
b
t1+16 min
strain rate increase
Figure1
Figure 2
unconnected polygon node
fsp
qtz
a
b
segment
node
Figure 3
grain
a
γ=1
b
γ=1
Figure 4
initial
FFT+GBM
γ = 2.4 {0001}
(a) c
a
{1120}
FFT+GBM+recovery +nucleation
FFT+GBM+recovery
{1010}
γ = 2.4 {0001}
{1120}
{1010}
γ = 2.4 {0001}
{1120}
b
8 {0001} {1120} {1010}
(b) 30
log (area/initial average grain area)
y {0001}
{1120} {1010}
frequency
0
FFT+GBM FFT+GBM+recovery
20
FFT+GBM+recovery +nucleation 10
0
-1.5
-1
-0.5
0
0.5
1
1.5
{1010}
Figure 5
0.04 0.02
Stress
0 -0.02 -0.04 -0.06
5
-0.08
4 3 0
2 1
1 2
0 3
-1 4
-2 5 -3
6 -4
Figure 6
t1
nucleation t 2 compaction
roughening in time compaction
t3 t4 teeth growth compaction
p L p
layer dissolved
small parts pinning increase in pinning strength
layer pinning
Figure 7
a
diffusion creep average aspect ratio
3
starting geometry
en masse neighbour switch diffusion creep
diffusion creep + low GBM rate 2
diffusion creep + low GBM rate diffusion creep + high GBM rate
diffusion creep + high GBM rate 1 1
2
3 Stretch
4
b
progression in time 10
c etch pits
150 hrs
enhanced dissolution
180 hrs
chemical concentration
100
grain boundary
276 hrs
1.62 micron
enhanced dissolution relative height etch pits - 1.54 micron
Table 1
(micro) structure/process General papers – review, overview, state of the art Single processes Crystal lattice rotation for single phase
Numerical Method
References Jessell et al., 2001; Jessell and Bons, 2002; Bons et al., 2008; Piazolo et al., 2010
D D D
slip, rotation, translation TBH VPFFT
D
VPFFT – 3D
Etchecopar, 1977 Lister and Paterson, 1979 Lebensohn et al., 2001; Montagnat et al. ,2011, 2014 Roters et al., 2012; Eisenlohr et al., 2013
Grain rotations for two phases
D
VPFFT
Elasto-viscoplastic behaviour
D D
Diffusion creep for single phase
D
EVPFFT lattice spring model + viscous deformation front tracking
Diffusion creep for two phases
D
front tracking
Dissolution and/or precipitation
D
lattice spring model
Dissolution at reactive surfaces
D
front tracking front tracking - 3D
Grain growth – isotropic surface energy – isotropic surface energy
S SD
Roessiger et al. 2011, 2014 Bons & Urai, 1992
S S SD S SD
front tracking front tracking (triple points only) front tracking front tracking Potts – 2 & 3D front tracking phase field – 3D
D S D
front tracking front tracking FEM – thin sheet model
Hilgers et al. ,1997; Koehn et al., 2003 Becker et al., 2008; Roessiger et al., 2011 Barr and Houseman, 1992, 1996
S D
finite difference VPFFT
Park et al., 2004 Griera et al., 2011, 2015; Ran et al., 2018
S
Potts-like
Borthwick et al., 2013
D
FEM + front tracking
Groome et al., 2006; Smith et al., 2015
D
TBH+Potts-like
Jessell, 1988a, b; Jessell & Lister, 1990
S D
front tracking TBH+FEM+front tracking
Roessiger et al., 2011 Piazolo et al., 2002
D
VPFFT+ front tracking
D
FEM+front tracking
Jansen et al., 2016; Llorens et al., 2016a, b, 2017; Steinbach et al., 2016, 2017; Gomez-Rivas et al., 2017; Piazolo et al., this contribution Jessell et al., 2005; Gardner et al., 2017
D D S
FEM front tracking front tracking
Cross et al., 2015 Piazolo et al. this contribution Piazolo et al. this contribution
– anisotropic surface energy – growth with pinning – isotropic grain growth – two phases Growth of crystals into a crack or in strain fringes Strain-induced grain boundary migration Single phase ductile deformation without reXX + crystallography Grain boundary diffusion Strain/stress localisation Combination of Processes Intracrystalline recovery and subgrain rotation Ductile deformation and grain growth polyphase Dynamic reXX: - crystal plastic deformation, grain boundary migration, rotation recrystallization - grain growth and polygonisation - crystal plastic deformation, grain boundary migration, rotational recrystallization, nucleation, recovery - crystal plastic deformation, grain boundary migration, rotational recrystallization, recovery Strain/stress localization - linked to grain size variations & rheology - linked to grain size variations & rheology Diffusion and grain growth Trace element diffusion and reXX
Griera et al., 2011, 2013, 2015; Ran et al., 2018 Lebensohn et al., 2012 Sachau and Koehn, 2010, 2012; Arslan et al., 2012; Koehn and Sachau, 2014 Ford et al., 2002, 2004; Wheeler, 2009, 2010 Berton et al., 2006, 2011; Ford and Wheeler, 2004 Koehn et al., 2006, 2007, 2012, 2016; Ebner et al., 2009, Godinho et al., 2014 Piazolo et al. this contribution
Becker et al., 2008; Piazolo et al., 2016 Kelka et al., 2015 Kim et al. 2006; Krill and Chen, 2002 Roessiger et al., 2014 Ankit et al., 2015
