Rheological Laws: Atoms on the Move

Rheological Laws: Atoms on the Move

The Geodynamics 101 series serves to showcase the diversity of research topics and methods in the geodynamics community in an understandable manner. We welcome all researchers – PhD students to Professors – to introduce their area of expertise in a lighthearted, entertaining manner and touch upon some of the outstanding questions and problems related to their fields. For our first ‘101’ for 2018, we have an entry by postdoctoral researcher Elvira Mulyukova from Yale University about rheology and deformation occurring on atomic scales … it’s a fun and informative read indeed! Do you want to talk about your research? Contact us!

Elvira Mulyukova, Yale University

Most of us have an intuitive understanding that different materials resist being moved, or deformed, to different degrees. Splashing around in the mud is more energy-consuming (and fun, but never mind that) than in the water, and splashing around in the block of concrete is energy-intensive bordering on deadly. What are the physical reasons for these differences?

For Earth materials (rocks), the answer lies in the restless nature of their atoms: the little buggers constantly try to sneak out of their crystal lattice sites and relocate. Some are more successful at it than others, making those materials more easily deformable. A lattice site is really just other atoms surrounding the one that is trying to escape. You see, atoms are like a bunch of introverts: each is trying to escape from its neighbours, but doesn’t want to get near them. The ones that do escape have to overcome a temporary discomfort (or an increase in their potential energy, for those physically inclined) of getting close to their neighbours. This requires energy. The closer the neighbours – the more energy it takes to get past them. When you exert force on a material, you force some of the neighbours to be further away from our potential atomic fugitive, making it more likely for the atom to sneak in the direction of those neighbours. The fun part (well, fun for nerds like me) is that it doesn’t happen to just one atom, but to a whole bunch of them, wherever the stress field induced by the applied force is felt. A bunch of atoms escaping in some preferential direction is what we observe as material deformation. The more energy you need to supply to induce the mass migration of atoms – the stronger the material. But it’s really a question of how much energy the atom has to begin with, and how much energy is overall needed to barge through its detested neighbuors. For example, when you crank up the temperature, atoms wiggle more energetically and don’t need as much energy supplied from external forcing in order to escape – thus the material gets weaker.

“A lattice site is really just other atoms surrounding the one that is trying to escape. You see, atoms are like a bunch of introverts: each is trying to escape from its neighbours, but doesn’t want to get near them.” cartoon by Elvira Mulyukova

One more thing. Where are the atoms escaping to? Well, there happen to be sanctuaries within the crystal lattice – namely, crystalline defects such as vacancies (aka point defects, where an atom is missing from the otherwise ordered lattice), dislocations (where a whole row of atoms is missing), grain boundaries (where one crystal lattice borders another, which is tilted relative to it) and other crystalline imperfections. These regions are sanctuaries because the lattice is more disordered there, which allows for larger distances in-between the neighbors. When occupying a regular lattice site – the atom is sort of trapped by the crystalline order. Think of the lattice as an oppressive regime, and the crystalline defects as liberal countries that are welcoming refugees. I don’t know, is this not the place for political metaphors? *…whistling and looking away…*

Ok, enough anthropomorphisms, let’s get to the physics. If this is the last sentence you’ll read in this blog entry, let it be this: rocks are made up of atoms that are arranged into crystal lattices (i.e., ordered rows and columns of atoms), which are further organized into crystal grains (adjacent crystals tilted relative to each other); applying force to a material encourages atoms to move in a preferential direction of the largest atomic spacing, as determined by the direction of the applied force; the ability of the lattice sites to keep their atoms in place (call it a potential energy barrier) determines how easily a material deforms. Ok, so it was more like three sentences, but now you know why we need to get to the atomic intricacies of the matter to understand materials macroscopic behaviour.

Alright, so we’re applying a force (or stress, which is simply force per area) to a material and watch it deform (a zen-inducing activity in its own right). We say that a material behaves like a fluid when its response to the applied stress (and not just any stress, but differential stress) is to acquire a strain rate (i.e., to progressively shorten or elongate in one direction or the other at some rate). On geological time scales, rocks behave like fluids, and their continuous deformation (mass migration of atoms within a crystal lattice) under stress is called creep.

The resistance to deformation is termed viscosity (let’s denote it µ), which basically tells you how much strain rate (˙e) you get for a given applied differential stress (τ). Buckle up, here comes the math. For a given dimension (say x, and for the record – I’ll only be dealing with one dimension here to keep the math symbols simple, but bear in mind that µ, ˙e and τ are all tensors, so you’d normally either have a separate set of equations for each dimension, or some cleverly indexed symbols in a single set of equations), you have:

So if I’m holding a chunk of peridotite with a viscosity of 1020 Pa s (that’s units of Pascal-seconds, and that’s a typical upper mantle viscosity) and squeezing it in horizontal direction with a stress of 108 Pa (typical tectonic stress), it’ll shorten at a rate of 5 · 10-13 s-1 (typical tectonic rates). A lower viscosity would give me a higher strain rate, or, equivalently, with a lower viscosity I could obtain the same strain rate by applying a smaller stress. If at this point you’re not thinking “oh, cool, so what determines the viscosity then?,” I failed massively at motivating the subject of this blog entry. So I’m just gonna go ahead and assume that you are thinking that. Right, so what controls the viscosity? We already mentioned temperature (let’s call it T), and this one is a beast of an effect. Viscosity depends on temperature exponentially, which is another way of saying that viscosity depends on temperature hellavulot. To throw more math at you, here is what this dependence looks like:

where R = 8.3144598 J/K/mol (that’s Joule per Kelvin per mol) is the gas constant and E is the activation energy. Activation energy is the amount of energy that an atom needs to have in order to even start thinking about escaping from its lattice site, which of course depends on the potential energy barrier set up by its neighbours. Let’s say your activation energy is E = 530·103 J mol-1 . If I raised your temperature from 900 to 1000 K (that’s Kelvin, and those are typical mid-lithospheric temperatures), your viscosity would drop by a factor of ∼ 1000. That’s a three orders of magnitude drop.

Like I said, helluvalot. If instead you had a lower activation energy, say E = 300 · 103 J mol-1 , the same temperature experiment would bring your viscosity down by a factor of ∼ 50, which is less dramatic, but still significant. It’s like running through peanut butter versus running through chocolate syrup (running through peanut butter is a little harder… I clearly need to work on my intuition-enhancing examples). Notice, however, that while the temperature dependence is stronger for materials with higher activation energies, it is more energy-consuming to get the creep going in those materials in the first place, since atoms have to overcome higher energy barriers. There’s more to the story. Viscosity also depends on pressure (call it P), which has a say in both the energy barrier the atoms have to overcome in order to escape their neighbours, as well as how many lattice defects (called sanctuaries earlier) the atoms have available to escape to. The higher the pressure, the higher the energy barrier and the fewer lattice sanctuaries to resort to, thus the higher the viscosity. Throwing in the pressure effect, viscosity goes as:

The exact dependence of viscosity on pressure is determined by V – the activation volume.

Alright, we’re finally getting to my favourite part – the atoms’ choice of sanctuary sites. If the atomic mass migration happens mainly via point defects, i.e., by atoms hopping from one single lattice vacancy to another, the deformation regime is called diffusion creep. As atoms hop away, vacancies accumulate in regions of compressive stress, and fewer vacancies remain in regions of tensional stress. Such redistribution of vacancies can come about by atoms migrating through the bulk of a crystal (i.e., the interior of a grain, which is really just a crystal that is tilted relative to its surrounding crystals), or atoms migrating along the boundary of a crystal (i.e., a grain boundary). In both cases, the rate at which atoms and vacancies get redistributed depends on grain size (let’s denote it r). The larger the grains – the more distance an atom has to cover to get from the part of the grain that is being compressed to the part that is under tension. More math is due. Here is what the viscosity of a material deforming by diffusion creep looks like:

Exponent m depends on whether the atoms are barging through the bulk of a grain (m = 2), or along the grain boundaries (m = 3). What’s that new symbol B in the denominator, you ask? That’s creep compliance (in this case – diffusion creep compliance), and you two have already met, sort of. Creep compliance specifies how a given creep mechanism depends on pressure and temperature:

For diffusion creep of upper mantle rocks, I typically use m = 3, B0 ∼ 13 µmm MPa-1 s-1 (which is just a material-specific prefactor), Ediff = 300 · 103 J mol-1 and Vdiff = 5 cm3 mol-1 from Karato and Wu (1993). Sometimes I go bananas and set Vdiff = 0 cm3 mol-1 , blatantly ignoring pressure dependence of viscosity, which is ok as long as I’m looking at relatively modest depth-ranges, like a few tens of kilometers.

At sufficiently high stresses, a whole row of atoms can become mobilized and move through the crystal, instead of the meagre one-by-one atomic hopping between the vacancies. This mode of deformation is called dislocation creep. Dislocations are really just a larger scale glitch in the structure of atoms (compared to vacancies). They are linear lattice defects, where a whole row of atoms can be out of order, displaced or missing. It requires more energy to displace a dislocation, because you are displacing more than one atom, but once it’s on the move, it accommodates strain much more efficiently than in the each-atom-for-itself diffusion creep regime. As the material creeps, dislocations get born (nucleated), get displaced and get dead (annihilated). Dislocations don’t care about grain size. What they do care about is stress. Stress determines the rate at which dislocations appear, move and disappear. I know you saw it coming, more math, here is the dislocation creep viscosity:

Exponent n dictates the stress dependence of viscosity. Stress dependence of dislocation creep viscosity is a real pain, making the whole thing nonlinear and difficult to use in a geodynamical model. Not impossible, but rage-inducingly difficult. Say you’re trying to increase the strain rate by some amount, so you increase the stress, but then the viscosity drops, and suddenly you have a monster of a strain rate you never asked for. Ok, maybe it’s not quite this bad, but it’s not as good as if the viscosity just stayed constant. You wouldn’t be able to have strain localization, form tectonic plate boundaries and develop life on Earth then, but maaaan would you be cracking geodynamic problems like they were peanuts! I’m derailing. Just like all the other creeps, dislocation creep has its own compliance, A, that governs its dependence on pressure and temperature:

For dislocation creep of upper mantle rocks, I typically use n = 3, A0 = 1.1 · 105 MPa-n s -1 (which is just a material-specific prefactor), Edisl = 530 · 103 J mol -1 and Vdiff = 20 cm3 mol-1 , similar to Karato and Wu (1993). Just like for diffusion creep, I sometimes just set Vdisl = 0 cm3 mol-1 to keep things simple.

We’re almost done. Allow me one last remark. A rock has an insane amount of atoms, crystal grains and defects, all subject to local and far-field conditions (stress, temperature, pressure, deformation history, etc). A typical rock is therefore heterogeneous on the atomic (nano), granular (micro) and outcrop (meter) scales. Thus, within one and the same rock, deformation will likely be accommodated by more than just one mechanism. With that in mind, and sticking to just two deformation mechanisms described above, we can mix it all together to get:

This is known as composite rheology, where we assumed that the strain rates accommodated by diffusion and dislocation creep can be simply summed up, like so:

Alright. If you got down to here, I salute you! Next time you’re squeezing a peridotite, or splashing in the mud, or running through peanut butter – give a shout out to those little atoms that enable you to do such madness. And if you want to get to the physics of it all, you can find some good introductory texts in Karato (2008); Turcotte and Schubert (2002).


S. Karato. Deformation of earth materials: an introduction to the rheology of solid earth. Cambridge Univ Pr, 2008. 

S. Karato and P. Wu. Rheology of the upper mantle: a synthesis. Science, 260(5109):771–778, 1993. 

D.L. Turcotte and G. Schubert. Geodynamics. Cambridge Univ Pr, 2002.

The jelly sandwich lithosphere: elastic bread, the jelly, and gummy bears

The jelly sandwich lithosphere: elastic bread, the jelly, and gummy bears

The Geodynamics 101 series serves to showcase the diversity of research topics and methods in the geodynamics community in an understandable manner. We welcome all researchers – PhD students to Professors – to introduce their area of expertise in a lighthearted, entertaining manner and touch upon some of the outstanding questions and problems related to their fields. This month Vojtěch Patočka from the Charles University in Prague, Czech Republic, discusses the rheology of the lithosphere and its food analogies. Do you want to talk about your research? Contact us!

It is becoming increasingly obvious that geodynamics and cooking are closely related, especially to the participants of a recent workshop in the Netherlands. Food analogies are helping students to get a physical grasp of continuum mechanics and to forget their lunches at university canteens. Here we take a basic look at a field where talking about fine cuisine has long been established: the rheology of the lithosphere. But first we step back a little.

Elastic solids deform when a force is applied and return to their original shape when the force is removed. Viscous fluids eventually take the shape of a container that is applied to them: they spontaneously flow to a state of zero shear stress. One can hardly imagine more different materials than these two, and yet the Earth’s mantle is sometimes modelled as an elastic shell and sometimes considered to be a viscous fluid.

You may be thinking:

Sure, it is a matter of what time scale is relevant for the process at hand

– just as J.C. Maxwell already thought: “Hence, a block of pitch may be so hard that you cannot make a dent in it by striking it with your knuckles; and yet it will in the course of time flatten itself by its weight, and glide downhill like a stream of water.” To visualise Maxwell’s dreams, the University of Queensland has been continuously running a pitch drop experiment for the past 90 years.

It is for the reason above that seismologists vibrate a Hookean Earth in their computers and geodynamicists play with viscous fluids. Are both of them right? Surprisingly, only the seismologists are. There is direct evidence that the outer parts of planetary mantles have an important elastic component even on geological time scales, which implies that treating the entire mantle as viscous is wrong. A textbook example can be found near the deepest part of the world’s oceans. Fig. 1, adopted from the bible by Turcotte & Schubert, shows how one can fit the bathymetric profile across the Mariana trench to the shape of a bent elastic plate. Note that the slab is subducting at the rate of a few centimeters per year, meaning that each segment of the slab is loaded for tens of millions of years before it disappears into the mantle and it still retains the elastic strain.

Figure 1: Comparison of a bathymetric profile across the Mariana trench (solid line) with the deflection of a thin elastic plate subject to end loading (dashed line). Distance xbx0 is the half-width of the forebulge from which the thickness of the plate can be inferred. Adopted from section 3 of Turcotte & Schubert, 2002.

Flexure studies and effective elastic thickness

Other similar examples, referred to as flexure studies, are summarised nicely by Watts et al., 2013. They include deflections under seamounts that litter oceanic floors and also some much longer lasting structures, such as continental foreland basins with the Ganges basin representing a particularly popular case. The free parameter that plays first fiddle in flexure studies is the thickness of elastic plate that matches observation. What does the resulting value, known as the effective elastic thickness, actually tell us?

I will use a dirty trick of bad journalism and quote an innocent geology blog slightly out of context:

The Indian crust is cold and rigid. Clever folk can do the maths on the shape of the crust as it bends down. This confirms that the pattern matches the model for rigid, elastic deformation. It also allows us to calculate the plate’s flexural rigidity, which is a measure of its strength. This means quantifying the rheology of real bits of the earth, which is a very useful trick

The math referred to involves purely elastic deformation and the flexural rigidity in its definition depends only on Young’s modulus, thickness, and Poisson ratio of the plate. Are the clever folk trying to fool us into believing that the lithosphere is an elastic plate? From lab measurements and geology we know for sure that it is not. Brittle failures are an abundant feature in both the crust and upper mantle, and various solid state creep mechanisms must be active in the deeper parts of the lithosphere.

The best fit for the Indo-Australian plate subducted below the Ganges basin is obtained with an elastic plate of circa 90 km thickness and for the Pacific plate at the Mariana trench it is circa 30 km. In both cases the plates are actually much thicker. What the computed values of 90 and 30 km tell us, is how much of elastic energy is present within the plate over the process of subduction, despite the brittle failures and despite the ductile creep activated in response to shear stresses within the plate.

To move away from sinking slabs, think about the story of a seamount that popped out on an ocean floor. Imagine a fresh, unstressed segment of oceanic plate that gets suddenly loaded by the uninvited underwater volcano. Small intra-plate fractures may immediately form, depending on the size of the load, and ductile creep will begin to continuously deform the plate’s deeper parts. The elastic energy present in the material upon loading gets partially released, either immediately through cracking or gradually via ductile creep. Measuring effective (or equivalent) elastic thickness merely tells us the amount of elastic energy left at the time of measurement.

This is all well known to the authors of flexure studies. In fact, they often re-draw the Christmas trees (see Fig. 2) and provide constraints on rheological zonation of the Earth. Pioneers in the field are E.B. Burov and A.B. Watts, whose papers are often accidentaly googled by chefs searching for latest trends in the dessert industry due to the extensive use of the words jelly sandwich and crème brûlée ([4] and [5], for breakfast see also [6]). The main point of the discussion is to determine the brittle/ductile transition and the active creep mechanism in a realistic, compositionally stratified lithosphere. This is usually complicated by the fact that lithologies measured in lab experiments strongly depend on composition, water content and temperature – and these are not well constrained in the real Earth.

Figure 2: The total force per unit width necessary to break or viscously deform a lithospheric section at a given strain rate. Plots like these are known as the Christmas tree plots, here adopted from Basin Analysis by P.A. Allen and J.R. Allen (without the food).

A geodynamical paradox

It is a paradox that in geodynamical modeling we often use the constraints from flexure studies and at the same time forget about elasticity, without which there would be no such studies at all. Recall that the primary result of shape fitting, for instance of the one depicted in Fig. 1, is that some elastic support is present. As I warned above, one can only hardly distinguish between a purely elastic plate of a given thickness and some other, thicker elasto-brittle or visco-elastic plate when looking at its surface flexure. However, there must be elasticity involved in some form: purely viscous or visco-plastic plates would not form the observed forebulge. Forebulges are related to the way elastic rods and plates transfer bending moment throughout the medium.

The tendency to disregard elasticity may be related to the way we use the word ‘rigid’ in the context of plate tectonics. In physics, rigid means ‘not deforming’. For plate tectonics to work the plates do not have to be rigid. They may not flow apart on geological time scales (they must remain plates), but some relatively small reversible deformation is well compliant with the concept. So the main point is: Be the lithosphere a sandwich or a crème brûlée, it is also flexible, even on geological time scales.

Batman and gummy bears

If you are in numerical modelling then there is good news for you. In the past two decades, several Robins, including myself, have enhanced the Batman codes to account for elasticity. Maybe all you have to do is to switch on the right button. The implementation of visco-elasticity is usually based on a method developed by L. Moresi, who also, according to a previous Geodynamics 101 blog post, coined the hero terminology I just borrowed. Visco-elasticity works in quite a simple way, just like gummy bears. I am currently running an experiment with them. They quickly bend and squeeze when tortured, their resistance governed by their shear modulus, and recover when let go. At the same time they can flow, the resistance being controlled by their viscosity. Let’s put a book on top of one and see if we can find its nose after a week…

The elasticity button

Don’t be afraid to push the elasticity button, if your code has it. Usually it won’t do anything dramatic to your simulations, but exceptions exist [7]. In thermal convection models without plate tectonics there is not much feedback between the lid and the underlying mantle, and so only the build-up of dynamic topography is affected [8]. In subduction modeling your slabs may obtain different dipping angles. In continental extensions the total amount of extension will become more important than the divergence rate [9]. And in the simulation of continental shortening mentioned above [7], the elastic energy accumulated in the entire model gets partially released upon the onset of a shear zone. In such cases, i.e. when large scale elastic strains suddenly influence a much smaller region, one can expect some earthquakes to shake the conventional view of elasticity in geodynamical modeling. And if you still do not care about elasticity but yet you made it all the way here, then you deserve a bonus: the convergence of Stokes solvers is way better for visco-elastic rheologies than for the viscous ones – if you are numerically troubled by large viscosity contrasts of your model, elasticity is the way to go for you.

(1) D.L. Turcotte and G. Schubert (2002), Geodynamics
(2) A.B. Watts, S.J. Zhong, and J. Hunter (2013), The Behavior of the Lithosphere on Seismic to Geologic Timescales, doi: 10.1146/annurev-earth-042711-105457
(3) P.A. Allen and J.R. Allen (2005), Basin analysis : principles and applications
(4) E.B. Burov and A.B. Watts (2006), The long-term strength of continental lithosphere: 'jelly sandwich' or 'creme brulée'?, doi: 10.1130/1052-5173(2006)016
(5) E.B. Burov (2009), Time to burn out creme brulee?, doi: 10.1016/j.tecto.2009.06.013
(6) E.H. Hartz, Y.Y. Podladchikov (2008), Toasting the jelly sandwich: The effect of shear heating on lithospheric geotherms and strength, doi: 10.1130/G24424A.1
(7) Y. Jaquet, T. Duretz, and S.M. Schmalholz (2016), Dramatic effect of elasticity on thermal softening and strain localization during lithospheric shortening, doi: 10.1093/gji/ggv464
(8) V. Patocka, O. Cadek, P.J. Tackley, and H. Cizkova (2017), Stress memory effect in viscoelastic stagnant lid convection, doi: 10.1093/gji/ggx102
(9) J.A. Olive, M.D. Behn, E. Mittelstaedt, G. Ito, and B.Z. Klein, The role of elasticity in simulating long-term tectonic extension, doi: 10.1093/gji/ggw044