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Beyond Equilibrium Thermodynamics In Beyond equilibrium thermodynamics in the low temperature plasma processor Elijah Thimsena) Interface Research Group, Department of Energy, Environmental and Chemical Engineering, Washington University in Saint Louis, One Brookings Drive, Box 1180, Saint Louis, Missouri 63130 (Received 15 January 2018; accepted 8 May 2018; published 27 June 2018) Low temperature plasmas are open driven thermodynamic systems capable of increasing the free energy of the mass that flows through them. An interesting thing about low temperature plasmas is that different species have different temperatures at the same location in space. Since thermal equilibrium cannot be assumed, many of the familiar results of equilibrium thermodynamics cannot be applied in their familiar form to predict, e.g., the direction of a chemical reaction. From the perspective of classical processing governed by thermal equilibrium, examples of highly unexpected gas-phase chemical reactions (CO2 dissociation, NO, N2H4,O3 synthesis) and solid material transfor- mations (surface activation, size-focusing, and hyperdoping) promoted by low temperature plasmas are presented. The lack of a known chemical reaction equilibrium criterion prevents assessment of predictive kinetics models of low temperature plasmas, to ensure that they comply with the laws of thermodynamics. There is a need for a general method to predict chemical reaction equilibrium in low temperature plasmas or an alternative method to establish the thermodynamic admissibility of a proposed kinetics model. Toward those ends, two ideas are explored in this work. The first idea is that chemical reactions in low temperature plasmas proceed toward a thermal equilibrium state at an effective temperature intermediate between the neutral gas temperature and the electron temperature. The effective temperature hypothesis is simple, and surprisingly is adequate for elucidation in some systems, but it lacks generality. The general equation for nonequilibrium reversible–irreversible cou- pling (GENERIC) is a general beyond equilibrium thermodynamics framework that can be used to rigorously establish the thermodynamic admissibility of a set of dynamic modeling equations, such as a kinetic model, without knowledge of the final state that the system is tending toward. The use of GENERIC is described by way of example using a two-temperature hydrodynamic model from the literature. The conclusion of the GENERIC analysis presented in this work is that the concept of superlocal equilibrium is thermodynamically admissible and may be applied to describe low temper- ature plasmas, provided that appropriate terms are included for exchange of internal energy and momentum between different species that may have different temperatures and bulk velocities at the same location in space. The concept of superlocal equilibrium is expected to be of utility in future work focused on deriving equilibrium criteria for low temperature plasmas. VC 2018 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1116/1.5022470 I. INTRODUCTION types of processors, classical as well as open driven system The advent of inexpensive sources of renewable energy (ODS), are presented in Fig. 1. This perspective is focused boasts abundant electricity produced with minimal environ- on single-phase and multiphase reactions involving gasses. mental impact. Processing techniques that are capable of More specifically, the ODS processor under consideration promoting novel transformations of matter, which were pre- here involves a low temperature plasma, which is a partially viously prohibitive due to large electricity requirements, ionized gas that promotes chemical and material transfor- may now become environmentally tenable. Similar concepts mations. In the classical paradigm, the energy used to drive were articulated in the nuclear age,1 but unfortunately, expe- the reaction occurring in the processor is often heated, rience in the decades following the introduction of nuclear which has been used as a means by which to raise the tem- energy revealed adverse environmental consequences of that perature of the system. A critical assumption made in the technology. In the age of abundant inexpensive renewable analysis of classical processors, termed the local thermal electricity, the time for electricity-intensive processing con- equilibrium assumption, is that at the same location in cepts may have finally arrived. space, all species and degrees of freedom have the same The basic idea explored here is the processor. The pro- temperature. The local thermal equilibrium assumption cessor uses an energy input to operate on a mass flow to allows kinetic reaction engineering analysis by transition affect a desirable change in state. Schematics of different state theory, for example, and also allows for prediction of the final state that reactions will tend toward at long times using equilibrium chemical thermodynamics. Raising the a)Electronic mail: [email protected] temperature by supplying heat to the system has two effects: 048501-1 J. Vac. Sci. Technol. B 36(4), Jul/Aug 2018 2166-2746/2018/36(4)/048501/19 VC Author(s) 2018. 048501-1 048501-2 Elijah Thimsen: Beyond equilibrium thermodynamics 048501-2 Xn Xn : ¼ _ Á ~ À _ Á ~ þ _ À _ ; ODS 0 Nin;i Hin;i N out;i Hout;i W e Qout i¼1 i¼1 (1) Xn Xn : ¼ _ Á ~ À _ Á ~ þ _ ; Classical 0 N in;i Hin;i N out;i Hout;i Qin i¼1 i¼1 (2) _ where N in;i is the molar flow rate of species i at the inlet, _ ~ N out;i is the molar flow rate of species i at the outlet, Hin;i is ~ the specific enthalpy of i at the inlet, Hout;i is the specific _ enthalpy of i at the outlet, W e is the electrical work flow into _ the ODS processor, Qout is the heat flow rejected to the envi- _ ronment by the ODS processor, and Qin is the net heat flow into the classical processor. It is clear from Eqs. (1) and (2) that both types of processors are capable of increasing the specific enthalpy of the mass that flows through them. The difference emerges in the entropy balance. Again assuming steady state, the entropy balance for both processors is _ Xn Xn Qout _ ~ _ ~ _ ODS : 0 ¼ þ N in;i Á Sin;i À Nout;i Á Sout;i þ Sgen; T i¼1 i¼1 (3) FIG. 1. (Color online) Schematic of processors as open systems: (a) ODS and (b) classical. _ Xn Xn Qin _ ~ _ ~ Classical : 0 ¼ þ Nin;i Á Sin;i À Nout;i Á Sout;i T ¼ ¼ (1) it allows thermally activated processes to occur at higher i 1 i 1 þ _ ; rates and (2) it changes the equilibrium speciation that the Sgen (4) reaction will tend toward. In other words, in general, chang- ~ where T is the temperature of the processor, Sin;i is the spe- ing temperature changes the reaction rate and can also ~ cific entropy of i at the inlet, S ; is the specific entropy of i change the direction of the reaction. out i at the outlet, and S_ is the entropy generated by the pro- In the new paradigm, the processor uses a work input to gen cess, which according to the second law of thermodynamics operate on the mass flow. Work and heat are different forms is positive semidefinite. If the processor is reversible, then of energy. Work can be described in terms of a generalized S_ ¼ 0. In the reversible limit, the ODS processor is capa- force multiplied by a generalized displacement.2 The impor- gen ble of reducing the specific entropy of the mass because no tant aspect is that the force can be described as a partial deriva- entropy is associated with work and it is rejecting heat to the tive of energy with respect to the displacement while other environment. However, the classical processor must always parameters remain constant. For example, one can describe increase the specific entropy of the mass that flows through ¼ chemical work as dWchem gidNi,wheregi is the chemical it. Therefore, according to Eqs. (3) and (4), only the ODS potential of species i,anddNi is the change in the number of processor is capable of both increasing specific enthalpy and moles of species i contained within the system. Electricity is decreasing specific entropy. In the reversible limit, at con- also a form of work. Heat, on the other hand, is a form of stant temperature and pressure, it is clear from Eqs. (1)–(4) energy that is not described in terms of forces and displace- that the classical processor cannot increase the total free ments. Another important distinction between heat and work is energy associated with the mass that flows through it, but the that heat carries entropy, while work does not. This difference ODS processor certainly can. in the type of energy that is supplied to the classical processor Reversibility is achieved when a process occurs in equi- compared to the ODS processor results in different thermody- librium with its surroundings. Thus, if the temperature of the namic limits for the changes in state that can be achieved. ODS processor is approximately the same as the temperature The thermodynamic limits of processors for causing _ of the surroundings, then the heat transfer Qout is reversible. changes in state of the mass that flows through them can be It is advantageous from this perspective for the ODS proces- established using the first (energy balance) and second (entropy sor to operate at ambient temperature, since it would mini- balance) laws of thermodynamics. The ODS processor takes an mize the entropy generation due to heat transfer across a electrical work input and rejects heat to the environment. The temperature gradient at the wall of the system.
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