The Thermodynamics of Deformed Metamorphic Rocks: a Review
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Journal of Structural Geology 33 (2011) 758e818 Contents lists available at ScienceDirect Journal of Structural Geology journal homepage: www.elsevier.com/locate/jsg Review The thermodynamics of deformed metamorphic rocks: A review Bruce E. Hobbs a,b,*, Alison Ord a, Klaus Regenauer-Lieb a,b a School of Earth and Environment, The University of Western Australia, Perth 6009, Australia b CSIRO, Western Australia, Perth 6102, Australia article info abstract Article history: The deformation of rocks is a disequilibrium and strongly non-linear phenomenon with a number of Received 7 September 2010 interacting chemical, thermal and microstructural processes operating simultaneously. We review Received in revised form progress in this area over the past 30 years. Deforming-chemically reacting systems are dissipative 22 January 2011 systems and hence are characterised by highly ordered structures that develop through cooperative Accepted 27 January 2011 processes once parameters such as critical strains, strain-rates, fluid infiltration rates, damage densities Available online 9 February 2011 or temperatures are attained. Such criticality is the hallmark of deformed rocks at all length scales and is This paper is dedicated to the memory and the basis for a diverse range of structures such as foliations and lineations produced by metamorphic work of Bruno Sander whose classical paper differentiation, rotation recrystallisation, folding, boudinage and micro to regional scale fracture systems. Uber Zusammenhange zwischen Teilbewe- Criticality is identified with classical criticality and not self-organised criticality. The first and second laws gung und Gefuge in Gesteinen was published of thermodynamics are used to show that such structural diversity arises from reaction-diffusion- ’ 100 years ago. Sander s work, often deformation equations. Criticality of the system is associated with the stored energy becoming non- forgotten and sometimes incorrectly convex and structures arise in order to minimise this non-convex energy. These structures are scale maligned, stands today as a fundamental contribution to Structural Geology. His ideas invariant and hence are characterised by fractal and minimal surface geometries. Thermodynamics is on the control of rock fabric by the move- a powerful discipline to integrate seemingly unrelated processes in structural geology and produce an ment picture or kinematic history are rein- integrated approach to the subject that crosses all length scales. forced by the thermodynamic approach to Ó 2011 Elsevier Ltd. All rights reserved. Structural Geology reviewed in this paper. Keywords: Generalised thermodynamics Coupled processes Reaction-diffusion Folding Boudinage Metamorphic differentiation Microstructure Critical systems Prologue developed particularly by De Groot (1952) and Prigogine (1955) with specific application to chemical systems, including diffusion, The last review of the application of thermodynamics to the and based largely on earlier work by Onsager (1931a,b). There was deformation of rocks was the paper: Non-hydrostatic Thermody- very little attention paid to deforming systems in that body of work namics and Its Geologic Applications by Paterson (1973) who although De Groot and Mazur (1962) had introduced the concept of emphasised that he was only considering systems at equilibrium. In internal variables to define the state of a deforming system. Land- order to appreciate developments since then one should under- mark papers were published by Biot (1955, 1958), Coleman and Noll stand the historical context in which that paper was written. The (1963), Coleman and Gurtin (1967) and Rice (1971, 1975). These extension of thermodynamics to systems not at equilibrium had papers were seminal in that they expressed the second law of been considered by many before Paterson’s paper and had been thermodynamics in terms of entropy production rate and in a way that the evolution of a general system, not at equilibrium, could be tracked with time; they introduced and clarified the concept and * Corresponding author. use of internal variables rather than the classical state variables of E-mail address: [email protected] (B.E. Hobbs). equilibrium thermodynamics. Importantly, these papers addressed 0191-8141/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsg.2011.01.013 B.E. Hobbs et al. / Journal of Structural Geology 33 (2011) 758e818 759 the mechanics of deforming systems as well as chemical aspects. in environments that cover temperatures ranging from atmo- The publication of many of these papers and the works by Truesdell spheric temperature to the solidus of many rocks, pressures and Noll (1965) and Truesdell (1966, 1969) was met with some ranging from atmospheric to about two GPa, mass transport both controversy largely because Truesdell in particular (and in a not in the solid state and in fluids such as aqueous fluids and melts, very subtle manner) pointed to serious flaws in the works of De thermal transport, both by advection and conduction, and chem- Groot and Prigogine. The situation was not helped by the emphasis ical reactions. A thermodynamic approach emphasises that such of the Truesdell School on mathematical formalism and rigour. By processes do not operate independently of each other but are the time Paterson’s paper appeared a large amount of confusion strongly coupled through the second law of thermodynamics. existed in the non-equilibrium literature (and even in the equilib- Each process dissipates energy, that is, it produces entropy, and so rium areas that Paterson chose to address) and to a large extent this the coupling is dictated by the ways in which the entropy confusion continues in the literature. The essential areas of production rate is partitioned between the processes. By entropy contention were: (i) The “Curie Principle or Theorem” that claimed production rate we mean the amount of heat produced at each quantities of different tensorial character cannot interact. The claim point in the system per unit time divided by the current temper- involved restrictions placed on the entropy production rate for atureatthatpoint(Truesdell, 1966). If s_ is the specific entropy coupled processes arising from the “theorem”. (ii) The claim that production rate at a particular point then the specifictotaldissi- processes “close” to equilibrium are linear whereas those “far” from pation function is definedatthatpointasF ¼ Ts_ where T is the equilibrium are non-linear. (iii) The claim that systems evolve so absolute temperature; the dissipation function, F, is a scalar and that the entropy production rate is a minimum. It turns out in has the units Joules per kilogram per second; the overdot repre- hindsight (and we address these issues in this paper) that all three sents differentiation with respect to time. The total dissipation claims are without merit but in 1973 (and even today in some rate at each point, F, is the sum of the individual dissipation rates minds) the controversy was rife. Paterson chose not to address arising from each dissipative process operating at that point. The these issues. Since then the work of many, including in particular dissipation rates that concern us are those arising from plastic Ziegler (1963, 1983), Lavenda (1978), Coussy (1995, 2004, 2010), deformation, Fplastic, mass transfer (by diffusion or byadvection in Houlsby and Puzrin (2006a) and Ross (2008), has clarified points of a moving fluid), Fmass transfer, chemical reactions, Fchemical,and contention to the extent that the application of thermodynamics to thermal transport, Fthermal transport,andso deforming systems is now routine in many disciplines and some progress can be made in applying the concepts of thermodynamics F ¼ Ts_ ¼ Fplastic þFmasstransfer þFchemical þFthermaltransport 0 (1) to deforming-chemically reacting geological systems. That is the subject of this paper. where the inequality is a direct expression of the second law of thermodynamics for a system not at equilibrium; the equality holds 1. Introduction for equilibrium. Eq. (1) is the fundamental equation that enables various processes to be coupled in a thermodynamically admissible Over the past 30 years there have been dramatic developments manner (such that the laws of thermodynamics are obeyed); when in structural geology, metamorphic petrology, physical metallurgy, the individual dissipation functions are expressed in an explicit continuum mechanics, physical chemistry and thermodynamics manner Eq. (1) is often called the ClausiuseDuhem inequality that are relevant to processes that operate within the Earth but (Truesdell and Noll, 1965). If necessary additional dissipation func- these developments have evolved largely independently of each tions could be added to Eq. (1) to represent other dissipative other and many have not yet been incorporated into main-stream processes such as fracturing, grain-size reduction or microstructural structural geology. On the other hand many contributions made evolution. Eq. (1) is true independently of the material properties of within structural geology over the past 30 years take on new the material; it is true for homogeneous and inhomogeneous significance when viewed in the context of new knowledge. This materials and for isotropic and anisotropic materials. Complicated paper attempts to integrate some of these developments into interactions can be incorporated into Eq. (1) by introducing rela- a common framework that couples together the various processes tions (determined