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Glass Thermodynamics: Clausius Theorem and a New Tensorial Definition of Temperature

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Glass Thermodynamics: Clausius Theorem and a New Tensorial Definition of Temperature Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B, February 2021, 62 (1), 8–18 Glass thermodynamics: Clausius theorem and a new tensorial definition of temperature Akira Takada,a,b,* Reinhard Conradtc & Pascal Richetd a University College London, Gower Street, London W1E6BT, UK b Ehime University, 10-13 Dogo-Himata, Matsuyama, Ehime 790-8577, Japan c UniglassAC GmbH, Nizzaallee 75, D-52072, Aachen, Germany d Institut de Physique du Globe de Paris, 1 rue Jussieu, 75005 Paris, France Manuscript received 23 July 2020 Accepted 23 September 2020 This paper aims at unifying the statistical mechanics and thermodynamics of glass through a new extended definition of temperature. Along with new insights into the Clausius theorem, this definition has been derived from a three-level model system. As previously shown in our study of a two-level system, introduction of an additional parameter, i.e. an internal temperature, is useful to describe the relation between energy and entropy changes in non-equilibrium states. Because a further extension is indispensable to deal with more realistic glass models, we define here temperature as a tensor whose diagonal components, the statistical temperatures, constitute a bridge between thermodynamic variables and a partition function in the framework of non-equilibrium statistical mechanics. On the other hand, the non-diagonal components, named thermodynamic temperatures, are used to extend the Clausius theorem within the framework of non-equilibrium thermodynamics. The new formalism is applied to a three-level model as the simplest example of multi- level system. It succeeds in describing the relation between energy changes and entropy in non-equilibrium states. In addition, this study sheds a new light on the interpretation of the second law of thermodynamics. As a result, Clausius’ definition of entropy, and its famous inequality for non-equilibrium cases can be merged into a single equality without the introduction of any additional terms. In addition, this study introduces a more general definition of the zeroth law of thermodynamics for non-equilibrium conditions, and extends the concept of fictive temperature as an order parameter to distinguish non-equilibrium glasses. 1. Introduction Much effort has thus been made to extend such a Historically, thermodynamics and statistical mechan- framework to non-equilibrium states, e.g. Ref. 11. As ics have been the two theoretical frameworks relevant an example, the standard formalism of spin glasses to the investigation of the thermal properties of gases, has made it possible to estimate the regions of exist- liquids and solids (i.e. crystals and glasses). Within ence of spin-glass phases and the hierarchy of spin the former, one correlates energy with temperature, clusters.(12–14) But the applicability of the second and pressure, volume, entropy and other thermodynamic third laws of thermodynamics to non-equilibrium variables of the macroscopic world.(1–3) For glasses, systems in general remains debated so that an issue the relation between elementary structural entities as important as the definition of partition functions and thermodynamics can be studied in this way,(4,5) for non-equilibrium states has not been settled yet. for example by calorimetric methods with which Within the framework of statistical thermodynamics, thermodynamic variables can be determined even for two main approaches have been followed to solve non-equilibrium states.(6,7) In statistical mechanics,(8,9) these problems. In the first approach, one introduces Boltzmann established his famous link between en- the concept of ensemble and investigates all possible tropy and the number of microstates whereas Gibbs microstates, the replicas of the spin-glass community. introduced the concept of ensemble, which made it In the second approach, one investigates the temporal possible to calculate partition functions and then all response of correlation function by using the fluctua- thermodynamics variables. tion–dissipation (FD) theorem. Because no decisive As a fundamental problem, however, the conven- results have been obtained in both cases and none tional framework of both thermodynamics and statis- seems easily applicable to structural glasses, the tical mechanics is valid for equilibrium states only.(10) purpose of the present study is to explore new ground with a third kind of approach. *Corresponding author. Email [email protected] The basic point here is to rely only on instantane- DOI: 10.13036/17533562.62.1.011 ous atomic coordinates, velocities and energies. As 8 Physics and Chemistry of Glasses: European Journal of Glass Science and Technology Part B Volume 62 Number 1 February 2021 A. TAKADA ET AL: GLASS THERMODYNAMICS: CLAUSIUS THEOREM AND A NEW TENSORIAL TEMPERATURE generally accepted, two extensive variables such as the system by internal or external effects. volume and the number of particles can be calculated The experimental or theoretical determination of as states variables at each instant without requiring this uncompensated heat, as it was called by Clausius, either the concept of ensemble or the FD theorem. has always been a matter of concern. In one of the It follows that entropy can also be calculated as most popular approaches, one introduces a general- an extensive variable without requiring replicas or ized thermodynamic force F and thermodynamic temporal responses. This statement directly leads flow X; to our new “spatial sampling” framework in which d S=FdX (4) all thermodynamic variables can be defined for each i microstate, contrary to the commonly held opinion For chemical reactions, the force F is de Donder’s affin- that they can be defined only for macrostates. Build- ity and the fluxX then denotes reaction coordinates.(21) ing on this starting point, we make in addition use In other situations, the combination of force F and of a tensorial form of temperature that originates in fluxX may represent a temperature gradient and the an analogy with the non-equilibrium deformation resulting heat flow, or a concentration gradient and of materials under force. Whereas a unique value a mass flow. This formulation has thus been applied of pressure specifies the mechanical state of a body to thermal conduction, diffusion, chemical reaction under an isotropic force, pressure must be expressed and many other non-equilibrium phenomena.(22) instead in a tensorial form under an anisotropic stress. For thermal conduction, the time change of entropy Consequently, the problem is to define a new diS/dτ is statistical temperature T for non-equilibrium states sta d S/dτ=F(dX/dτ) (5) by a combination of thermodynamic and statistical i mechanical methods. To solve it, we will first briefly F=−grad(lnT)/T (6) review the Clausius theorem, which underlies calo- rimetric entropy determinations, and its relationship dX/dτ=dQ/dτ (7) with the concept of fictive temperature. We will then turn to the concept of partition function, which is a where τ is time and Q is heat flow. An important as- core feature of statistical mechanics. As a starting sumption is that the formulation of Equations (4) or point toward more complex systems, we will then (5)–(7) relies on “local equilibrium.” For example, T extend our previous work(15–20) with a three-level in Equation (6) must be an equilibrium temperature model to illustrate how one can unify the two differ- with respect to spatial coordinates. For this reason, ent frameworks of thermodynamics, as represented applications of Equations (5)–(7) are limited to by the Clausius theorem, and statistical mechanics, as phenomena close to equilibrium or to steady-state represented by partition function, through a tensorial non-equilibrium as expressed by a linear response definition of temperature. function (i.e. a linear force–flow relation). Independently of any non-equilibrium considera- 2. Clausius theorem tions, the concept of “fictive temperature” is used to characterize the structural and thermodynamic As is well known, Clausius theorem is one expres- states of a glass.(23,24) Since the fictive temperature sion of the second law of thermodynamics. Along a differs from the temperature of a heat bath or that of reversible cycle, the environment, it represents an additional internal temperature characterizing the extent of departure dS=dQ/T (1) ext from equilibrium. From a practical point of view, where dS, dQ and Text are the entropy and energy structural relaxation models based on this concept changes and the external temperature (i.e. heat bath have long proven useful to analyze residual stresses or environmental temperature), respectively. On the and tempering processes. On the theoretical side, an other hand, in an irreversible cycle, investigation of a two-level system(15,16) demonstrated that such an internal temperature T unrelated to dS>dQ/T (2) int ext either Equation (2) or (3) can be uniquely defined The entropy appearing in Equation (2) is usually split so as to satisfy Clausius theorem even under non- into two contributions, equilibrium conditions; dS=deS+diS (3) dS=dQ/Tint (8) In the absence of mass transfer between the system We call Tint thermodynamic temperature because it is at and its environment, the first, ed S, is the entropy as- the roots of the fundamentally important thermody- sociated with the heat dQ transferred from or to the namic Equation (4). As already stated, however, this external heat bath under the reversible conditions of concept of a single additional temperature cannot be Equation
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