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 (CO_{2} dissociation, NO, N_{2}H_{4}, O_{3} synthesis) and solid material transformations (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 coupling (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 temperature 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.

## I. INTRODUCTION

The advent of inexpensive sources of renewable energy boasts abundant electricity produced with minimal environmental impact. Processing techniques that are capable of promoting novel transformations of matter, which were previously prohibitive due to large electricity requirements, may now become environmentally tenable. Similar concepts were articulated in the nuclear age,^{1} but unfortunately, experience in the decades following the introduction of nuclear energy revealed adverse environmental consequences of that technology. In the age of abundant inexpensive renewable electricity, the time for electricity-intensive processing concepts may have finally arrived.

The basic idea explored here is the *processor*. The processor uses an energy input to operate on a mass flow to affect a desirable change in state. Schematics of different types of processors, classical as well as open driven system (ODS), are presented in Fig. 1. This perspective is focused on single-phase and multiphase reactions involving gasses. More specifically, the ODS processor under consideration here involves a low temperature plasma, which is a partially ionized gas that promotes chemical and material transformations. In the classical paradigm, the energy used to drive the reaction occurring in the processor is often heated, which has been used as a means by which to raise the temperature of the system. A critical assumption made in the analysis of classical processors, termed the *local thermal equilibrium* assumption, is that at the same location in space, all species and degrees of freedom have the same temperature. The local thermal equilibrium assumption allows kinetic reaction engineering analysis by transition 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 temperature by supplying heat to the system has two effects: (1) it allows thermally activated processes to occur at higher rates and (2) it changes the equilibrium speciation that the reaction will tend toward. In other words, in general, changing temperature changes the reaction rate and can also change the direction of the reaction.

In the new paradigm, the processor uses a work input to operate on the mass flow. Work and heat are different forms of energy. Work can be described in terms of a generalized force multiplied by a generalized displacement.^{2} The important aspect is that the force can be described as a partial derivative of energy with respect to the displacement while other parameters remain constant. For example, one can describe chemical work as d*W*_{chem} = *g*_{i}d*N*_{i}, where *g*_{i} is the chemical potential of species *i*, and d*N*_{i} is the change in the number of moles of species *i* contained within the system. Electricity is also a form of work. Heat, on the other hand, is a form of energy that is not described in terms of forces and displacements. Another important distinction between heat and work is that heat carries entropy, while work does not. This difference in the type of energy that is supplied to the classical processor compared to the ODS processor results in different thermodynamic limits for the changes in state that can be achieved.

The thermodynamic limits of processors for causing changes in state of the mass that flows through them can be established using the first (energy balance) and second (entropy balance) laws of thermodynamics. The ODS processor takes an electrical work input and rejects heat to the environment. The classical processor takes a heat input. Neglecting changes in kinetic and potential energy of the mass flow, and assuming steady state, the first law is quite similar for both types of processors, and can be written as

where $N\u0307in,i$ is the molar flow rate of species *i* at the inlet, $N\u0307out,i$ is the molar flow rate of species *i* at the outlet, $H\u0303in,i$ is the specific enthalpy of *i* at the inlet, $H\u0303out,i$ is the specific enthalpy of *i* at the outlet, $W\u0307e$ is the electrical work flow into the ODS processor, $Q\u0307out$ is the heat flow rejected to the environment by the ODS processor, and $Q\u0307in$ 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

where *T* is the temperature of the processor, $S\u0303in,i$ is the specific entropy of *i* at the inlet, $S\u0303out,i$ is the specific entropy of *i* at the outlet, and $S\u0307gen$ is the entropy generated by the process, which according to the second law of thermodynamics is positive semidefinite. If the processor is reversible, then $S\u0307gen=0$. In the reversible limit, the ODS processor is capable of reducing the specific entropy of the mass because no entropy is associated with work and it is rejecting heat to the environment. However, the classical processor must always increase the specific entropy of the mass that flows through it. Therefore, according to Eqs. (3) and (4), only the ODS processor is capable of both increasing specific enthalpy and decreasing specific entropy. In the reversible limit, at constant temperature and pressure, it is clear from Eqs. (1)–(4) that the classical processor cannot increase the total free energy associated with the mass that flows through it, but the ODS processor certainly can.

Reversibility is achieved when a process occurs in equilibrium with its surroundings. Thus, if the temperature of the ODS processor is approximately the same as the temperature of the surroundings, then the heat transfer $Q\u0307out$ is reversible. It is advantageous from this perspective for the ODS processor to operate at ambient temperature, since it would minimize the entropy generation due to heat transfer across a temperature gradient at the wall of the system. Therefore, low temperature plasmas are attractive as processors, since the heat transfer occurs across a small temperature gradient to the ambient. Having established the thermodynamic limits of both types of processors in Eqs. (1)–(4), the next step is to analyze how the direction of chemical reactions occurring within them can be predicted.

The idea is that the reaction occurring within the processor will proceed from the inlet state toward an equilibrium state whereat the entropy will be maximized. For the classical processor, one can readily derive the mole fractions of different chemical species that the system will tend toward using well-established approaches. The task is to derive the chemical reaction equilibrium criterion, which can then be used to predict the relative mole fractions of different species, given an initial composition at the inlet. Consider a closed multiphase system of interest, which is in contact with a pressure reservoir via an adiabatic plunger, and a heat reservoir via a rigid diathermal wall (Fig. 2). The system of interest contains only species that participate in some chemical reaction. This example will focus on a single chemical reaction, but it can be extended to multiple reactions.^{2} The three subsystems comprise an isolated system. Due to interaction with the reservoirs via the adiabatic plunger and diathermal wall, the system of interest is at constant temperature and pressure, and therefore, the Gibbs free energy of the system of interest is minimized at equilibrium. For illustration, consider the reaction of hydrogen with carbon to form acetylene (Fig. 2). The gas phase is *α*, while the solid phase is *β*. The change in total entropy of the system of interest as a function of the change in internal energy of *α* phase, change in volume of *α* phase, and extent of reaction is

where $T\alpha $ and $T\beta $ are the temperatures of the phases, $P\alpha $ and $P\beta $ are the pressures in the phases, *g*_{i} is the chemical potential of species *i,* and $\xi $ is the extent of reaction. The symbol $\delta $ denotes a small variation of the variable it operates on. In Eq. (5), the surface energy of the *β* phase has been neglected. At equilibrium, the total entropy of the system is maximized and thus, for small variations $\delta S=0$. The variations $\delta U\alpha $, $\delta V\alpha $, and $\delta \xi $ are independent. Therefore, for $\delta S=0$ at equilibrium, the coefficients of $\delta U\alpha $, $\delta V\alpha $ and $\delta \xi $ must be zero. Thus, at equilibrium, the temperatures must be equal: $T\alpha =T\beta $, which is the criterion for thermal equilibrium. If the temperatures are equal, then the pressures must also be equal to make the coefficient of the volume term vanish. Likewise for the third term, and thus, we arrive at the condition for chemical reaction equilibrium in the classical processor: $\u2211i\nu igi=gC2H2\u2212gH2\u22122gC=0$, where $\nu i$ is the stoichiometric coefficient of *i* in the reaction. From the chemical reaction equilibrium criterion, the mole fractions of C_{2}H_{2}, H_{2}, and C in the product mixture can be calculated from a material balance and tabulated information using the well-known procedure involving standard state reference information and species fugacity. In other words, the chemical reaction equilibrium criterion allows for prediction of the product mixture that the classical processor will tend toward as it proceeds toward equilibrium, given a certain amount of carbon and hydrogen loaded into the vessel. In general, for other types of reactions such as the homogeneous dissociation reactions (e.g., CO_{2} → CO + 1/2O_{2}), one must also assume thermal equilibrium in order to derive the condition for chemical reaction equilibrium, which is required to predict the equilibrium composition. Without thermal equilibrium, prediction of the outcome of the chemical reaction would not have been possible using the preceding approach. Can such an approach also be applied to chemical reactions occurring in low temperature plasmas?

The interesting thing about low temperature plasmas is that the local thermal equilibrium assumption cannot be made, and therefore most of the familiar results from equilibrium thermodynamics, such as Henry's law, Raoult's law, osmotic pressure, and chemical reaction equilibrium, cannot be applied in their commonly used form. In low temperature plasma, different species have temperatures that differ by more than an order of magnitude at the same location in space. Additional variables are required to specify the state of the system. Specifically, in addition to the gas temperature *T*_{g} and total pressure *P*, one must also specify or measure the electron temperature *T*_{e} and positive ion density *n*_{i}. The plasma is typically assumed to be overall charge neutral, and so it may be assumed that the positive ion density is approximately equal to the electron density. The free electrons are selectively heated to temperatures orders of magnitude higher than the neutral gas and ion temperatures.^{3} For the systems of interest here, neutral gas temperature is typically on the order 10^{2} K, while the electron temperature is on the order 10^{4} K at the same location in space. For example, in Figs. 3(a) and 3(b) are visible and infrared images of a flow-through 13.56 MHz argon discharge operating at a pressure of 2 mbar and an applied power of 20 W. In the hot zone, the gas temperature was measured by a fluorescence decay probe, with appropriate heat transfer corrections, to be 413 K; while the fused silica tube was measured to be 387 K by infrared emission. In contrast to the gas temperature, the electron temperature was measured to be 20 000 K by a double Langmuir probe^{4–6} positioned at the same location and the ion density was 1 × 10^{18} m^{−3}. Clearly, local thermal equilibrium cannot be assumed in that system, since the electrons are 2 orders of magnitude higher temperature than the neutral species at the same spatial location. For another example of why the local thermal equilibrium assumption cannot be made, consider small nanoparticles suspended in a low temperature plasma. The surfaces of the nanoparticles are under continuous ion and electron bombardment, and to maintain charge neutrality in the plasma, nanoparticles become negatively charged.^{7} Positive ions are attracted to the negatively charged particles, and when they recombine on the surface, a large amount of energy is released. The consequence is that at steady state, particles can be 100s of K higher temperature than the surrounding gas,^{8–11} and again the local thermal equilibrium assumption cannot be made (Fig. 4). Processes occurring in low temperature plasma must obey the macroscopic energy and entropy balances for an ODS in Eqs. (1) and (3); however, since the local thermal equilibrium assumption cannot be made, they are not limited by the familiar results of equilibrium thermodynamics. In other words, at a given neutral gas temperature and pressure, product mixtures or material phases may be observed that are highly unexpected from the perspective of classical processors that are governed by local thermal equilibrium.

There is no established means, based upon equilibrium thermodynamics, to calculate the equilibrium constant for a chemical reaction occurring in a low temperature plasma, which is a direct consequence of the fact that the local thermal equilibrium assumption cannot be made. To deal with this complication, the modeling approach that has been most widely employed to predict the outcomes of low temperature plasma processes has been kinetic modeling,^{12} which has been used successfully by computational and theoretical researchers.^{13–19} However, proposed kinetic reaction mechanisms are sets of equations intended to describe a physiochemical process occurring in reality, they are not reality itself, and as such, proposed models may or may not comply with the laws of thermodynamics. For classical processors, the usual way to ensure the thermodynamic admissibility of a proposed reaction mechanism is to compare the steady state result to the chemical reaction equilibrium criterion from thermodynamics that was derived above.^{20} However, since there is no established means by which to predict chemical reaction equilibrium in low temperature plasmas using thermodynamics, such comparisons cannot be made at present for kinetics models of low temperature plasmas. Therefore, it is not straightforward to ensure the thermodynamic admissibility of a proposed chemical kinetics or dynamical model of a plasma process. Furthermore, many empirical parameters such as collision cross-sections that are required by kinetics models remain unknown, and the computational approach remains formidable for many experimental researchers. A method to predict chemical reaction equilibrium based upon thermodynamics would be tremendously valuable. Unfortunately, the key concepts that are required to derive a condition for chemical reaction equilibrium in low temperature plasmas have not hitherto been elucidated.

This work begins a line of inquiry focused on using the framework of thermodynamics to predict the outcomes of physiochemical reactions in low temperature plasmas. Examples from the literature will be provided for gas-phase chemical reactions and material conversions that exhibit outcomes which are highly unexpected from the perspective of thermodynamics governed by local thermal equilibrium. Two very different methods of using thermodynamics for the theoretical analysis of low temperature plasma processes will be presented. The first method is very simple, but somewhat surprisingly appears to account for experimental outcomes in some instances. *Can a low temperature plasma process be described, using the familiar chemical reaction equilibrium criterion that requires thermal equilibrium, as evolving toward a state at an effective temperature?* In other words, if a system has a given total pressure *P*, overall elemental composition, gas temperature *T*_{g}, and electron temperature *T*_{e}, can the behavior be described as evolving toward a thermal equilibrium state at some intermediate temperature *T*_{eff}, where *T*_{g} <*T*_{eff} < *T*_{e}? The concept of an effective temperature has been used to understand ionization processes in nonequilibrium plasmas,^{21} but to the author's knowledge, it has not been applied to chemical reaction equilibrium. It will be shown that in some cases the idea of evolution toward an equilibrium state at an effective temperature is adequate to describe experimental observations for gas-phase reactions; while in other situations, it clearly is not. The effective temperature hypothesis is useful because it is simple, but it lacks generality. The second method is a general means by which to ensure that a dynamical model of a low temperature plasma complies with the laws of thermodynamics, which is termed the general equation for nonequilibrium reversible–irreversible coupling (GENERIC), which was developed by Oettinger.^{22} GENERIC does not require knowledge of the equilibrium state that the system is tending toward to assess a given kinetic model. The GENERIC framework is used here to prove the thermodynamic admissibility (i.e., compliance with laws of thermodynamics) of the concept of *superlocal equilibrium* by analyzing a commonly used multifluid hydrodynamic model. The concept of superlocal equilibrium is expected to be of utility in future work focused on deriving conditions for chemical reaction equilibrium in low temperature plasmas.

## II. GASSES

The reactions of simple gas molecules provide excellent examples for exploring the concept of an effective temperature in low temperature plasmas. Several selected examples will be discussed, namely, CO_{2} dissociation, NO production, as well as N_{2}H_{4} and O_{3} syntheses. There are other examples that could have been chosen, and the reader is referred to the monograph of Fridman for additional information,^{23} which the author consulted many times during the preparation of this article. CO_{2} dissociation has become topically relevant in the context of climate change driven by the greenhouse effect, as well as in the context of Mars exploration. Mars has a 95% CO_{2} atmosphere at a pressure of approximately 6 mbar. The CO_{2} could be a source of breathable oxygen for astronauts.^{24} Furthermore, CO_{2} dissociation is an excellent example of a system in which the concept of an effective temperature is largely adequate for understanding experimental outcomes of low temperature plasma processes. The reaction of N_{2} with O_{2} to form NO has been studied in the context of artificial fixation of atmospheric nitrogen, but more importantly for the present discussion, the reaction is a good example of the limits of the concept of an effective temperature for describing the outcome of a low temperature plasma process. Finally, two examples will be discussed in which the concept of an effective temperature is wholly inadequate for describing the outcome of low temperature plasma processes: the reaction of O_{2} to form O_{3}, and the reaction of N_{2} and H_{2} to make N_{2}H_{4}. Ozone production is important for air and water purification (e.g., the author's office building has an ozone generator in the air circulation system), while hydrazine is an important chemical intermediate and also a rocket fuel. The examples were chosen since they are relatively simple, involve nearly ideal gases, and are well-studied by the low temperature plasma community. For each of the reactions, the equilibrium speciation will be plotted as a function of temperature by assuming thermal equilibrium; for pressures similar to published experimental studies. The maximum possible yield of products at thermal equilibrium will then be compared to actual experimental outcomes. If there exists some temperature at which the measured experimental product speciation would be expected from an equilibrium analysis, then the concept of an effective temperature holds. If, on the other hand, the experimental product yield is greater than the theoretical maximum expected from thermal equilibrium analysis, then the concept of an effective temperature will be called into question.

### A. CO_{2} dissociation

The dissociation of carbon dioxide in plasmas is well-studied, with literature going back decades and several reviews have been published.^{23,25,26} There has recently been a resurgence of interest in the topic. Recent research has focused on the external energy efficiency of the reaction to form carbon monoxide: CO_{2} → CO + ½O_{2}, which is highly endothermic with Δ*H*_{rxn} = 2.9 eV. The reaction is limited by the production of carbon monoxide and mono-oxygen: CO_{2} → CO + O (Δ*H*_{rxn} = 5.5 eV).^{23} In thermal equilibrium, the highly endothermic nature of the reaction allows it to proceed only at high temperature. For a pressure of 10 mbar, the equilibrium speciation is plotted as a function of temperature in Fig. 5(a), and the equilibrium fraction of carbon present as CO is plotted in Fig. 5(b). The curves were generated using the chemical equilibrium with applications (CEA) code developed by National Aeronautics and Space Administration (NASA).^{27} The behavior is similar at other pressures, but shifted horizontally to account for the fact that CO_{2} dissociation produces additional gas molecules. The general trend is that as temperature increases, molecules become atomized.

If the system were in thermal equilibrium at an electron temperature of 20 000 K, then full dissociation would occur [Fig. 5(a)]. That observation leads to the expectation that plasmas tend to dissociate molecules. However, the fractional ionization is typically small in a low temperature plasma, approximately 3 × 10^{−5} in Fig. 3 for example, and thus hot electrons are relatively dilute. On the other hand, if the system was in thermal equilibrium at the neutral gas temperature, which is typically less than 1000 K (Fig. 3), no dissociation would occur. Experimental observations are in between these extremes. The focus here will be on select experimental reports of CO_{2} dissociation in low temperature plasmas containing information relevant to this perspective.

The idea that the system can be described as evolving toward an equilibrium state at an effective temperature requires that the output from the reactor is independent of the speciation of the input, provided that the molar flow rates of carbon and oxygen are the same for different feed configurations and the reaction time is sufficiently long. From that point of view, the output from a plasma process should be nominally the same if either CO_{2} or CO + ½O_{2} were fed into the reactor at the same total carbon to oxygen ratio. Brown and Bell performed exactly that experiment using radiofrequency (RF) electrodeless discharges, similar to the reactor illustrated in Fig. 3, at pressures of several millibar.^{28} Interestingly, they observed that at high powers the CO_{2} fraction in the reactor effluent was nominally the same for feeds comprised of pure CO_{2}, or feeds comprised of CO + ½O_{2} (Fig. 6). The products of CO_{2} dissociation were CO and O_{2}. The gas temperatures, estimated using a shielded thermocouple, were less than 1000 K. However, the effective temperature at which the equilibrium CO_{2} mole fraction matched the experimental output was found to be approximately 2500 K. The electron temperature was reported to be approximately 4 eV (46 000 K).^{29} The effective temperature of 2500 K is between the gas temperature of 1000 K and the electron temperature of 46 000 K. The idea of a system evolving toward an equilibrium state defined by an effective temperature in between the gas and electron temperatures appears to be adequate for describing the process.

Kinetics control how quickly a system reaches the equilibrium state. If the idea of a low temperature CO_{2} plasma proceeding toward an equilibrium state at an effective temperature is applicable, then that equilibrium state should be reached at long times and be relatively stable. Experimentally, such behavior can be probed by monitoring the reaction as a function of time in a closed system, or alternatively, as a function of residence time in an open system. The composition should tend toward a stable state at long times. Unfortunately, published work in which authors have studied conversion as a function of time is relatively uncommon. The kinetics will of course depend on the process details, e.g., plasma source, applied power, gas composition, and pressure; but out of necessity, results from slightly different configurations will be presented below to illustrate the steady state in low temperature CO_{2} plasmas.

Published experimental work suggests that the CO_{2} dissociation reaction in low temperature plasma initiates after approximately 10^{−5 }s and reaches a steady state after 10^{2} s. The kinetics obviously will depend on the plasma parameters, pressure, etc. Experimental reports that cover the entire 7 orders of magnitude in time are not forthcoming. Three different reports that probe different time scales, which unfortunately involve different experimental conditions, will be presented. The dominant products of CO_{2} dissociation in these studies were CO and O_{2}, and thus, only the CO_{2} conversion is plotted in Fig. 7. Taylan and Berberoglu have studied dissociation in the residence time range from 1 to 100 *μ*s using microhollow cathode discharges in mixtures of CO_{2} and argon at atmospheric pressure.^{30} They found significant conversion after approximately 10 *μ*s, which increased to approximately 14% after 128 *μ*s, which was the longest time explored. The results suggested that if the residence time would have been further increased, larger conversions could have been achieved (Fig. 7, curve a). Mori *et al.* have studied CO_{2} dissociation in He mixtures using a capillary plasma reactor at reduced pressures of approximately 40 mbar and residence times in the range from 0.1 to 2 s.^{31} They found that the conversion increased with increasing residence time and applied current to the plasma (Fig. 7, curves b–e). The increase in CO_{2} conversion with residence time exhibited a sublinear dependence that appeared to be tending toward saturation at a steady-state value. Williams and Smith have performed experiments in which they analyzed the dissociation as a function of time in sealed discharge tubes containing CO_{2}, N_{2}, and He.^{32} The pressure was 26 mbar, and the RF plasma was generated using an electrodeless configuration at a frequency of 29 MHz. The time range was 10–1000 s. Already at the initial time point taken at 48 s, the conversion was approximately 69% (Fig. 7, curve f), indicating that the majority of the conversion took place on a shorter time scale, as expected. The interesting observation is that the conversion increased slightly with time, but after approximately 120 s, the system reached a stable steady state, consistent with the idea that an equilibrium had been reached. This state was stable from 120 s until 600 s, at which time the experiment was terminated. The kinetics are consistent with the system tending toward an equilibrium state, which can be described by an effective temperature of approximately 2500 K. Thus, for CO_{2} in low temperature plasma, the answer to the question appears to be yes, one can imagine the system as evolving toward an equilibrium state at an effective temperature that is intermediate between the gas and electron temperatures. Below it will be demonstrated that other systems cannot be understood using the concept of an effective temperature, which raises a question. What determines if a chemical reaction occurring in a low temperature plasma can be understood as proceeding toward a thermal equilibrium state at an effective temperature? What determines the effective temperature?

### B. NO synthesis

In low temperature plasmas containing N_{2} and O_{2}, the idea of a system evolving toward an equilibrium state at an effective temperature starts to break down. Specifically, it will be seen that the amount of a product experimentally observed in the reactor effluent, specifically nitrogen monoxide (NO), can be greater than the maximum fraction expected from equilibrium analyses at *any temperature*, given the feed composition and total reactor pressure.

The synthesis of NO in N_{2}-O_{2} plasmas, which has a long history going back more than 100 years, has been comprehensively reviewed elsewhere.^{23,33} The reaction N_{2} + O_{2} → 2NO is endothermic (Δ*H *=* *1 eV per molecule) and only proceeds in thermal equilibrium at high temperature. The plasma-activated reaction was first reported over 200 years ago by Sir Humphrey Davy.^{33} Industrial scale production by the Birkeland-Eyde process was demonstrated in 1903.^{33} Early processes relied on thermal plasmas and high quenching rates to achieve product distributions that had equilibrium or near equilibrium speciation. Later work focused on low temperature nonequilibrium plasmas, which can be used to selectively excite molecular vibrational modes to increase the overall energy efficiency of the process.^{33}

In systems containing nitrogen and oxygen, nitrogen monoxide forms at temperatures intermediate between the diatomic and monoatomic states of the gases. The equilibrium speciation in a system with equal number of N_{2} and O_{2} molecules is plotted as a function of temperature in Fig. 8(a) for a pressure of 30 mbar. The fraction of nitrogen present as NO is plotted in Fig. 8(b). The plots were generated using the NASA CEA code. The maximum fraction of NO is approximately 4.4% at a temperature slightly higher than 3000 K. Below this temperature, the gasses are present as N_{2} and O_{2}; while for very high temperatures, the speciation is dominated by N and O [Fig. 8(a)]. Therefore, if the system can be thought of as evolving toward an equilibrium state at some effective temperature, then the fraction of nitrogen present as NO has a theoretical maximum value of a few percent that it cannot exceed.

In low temperature plasmas, the fraction of nitrogen monoxide in the reactor effluent can exceed the thermodynamic maximum predicted from thermal equilibrium analysis by a significant margin. Most work in the nitric oxide system was published in the 1970s and early 1980s, with key papers appearing in French and Soviet journals. These papers, many of which are not in English, have been described by Fridman.^{23} Some authors report having significantly exceeded the maximum equilibrium fraction of NO. Alekseev *et al.* examined NO production using plasma-beam discharges in air at pressures on the order of 10^{−2} mbar.^{34} Plotted in Fig. 9(a) is NO production as a function of power supplied to the plasma for a pressure of 6.6 × 10^{−2} mbar, reproduced from Ref. 34 At the conditions used in that work, the theoretical maximum value that the mole fraction of NO *x*_{NO}/($xN2+xNO+xO2$) can obtain at equilibrium is 1.1%, which is expected at a temperature of approximately 2300 K. The maximum value that Alekseev *et al.* observed was close to 20%, more than a factor of 10 greater than the theoretical maximum at thermal equilibrium. Rapakoulias *et al.* have studied nitrogen fixation in N_{2}-O_{2} plasmas in the pressure range from 5 to 40 mbar using RF inductive plasmas at an applied frequency of 40 MHz.^{35} They examined the effect of nitrogen-to-oxygen ratio in the feed gas, as well as the presence of WO_{3} and MoO_{3} catalysts.^{35} Interestingly, both catalysts significantly increased the fraction of fixed nitrogen. Even without the catalyst, the fraction of nitrogen fixed by the reaction [Fig. 9(b)] exceeded the theoretical maximum (Fig. 8). Moreover, the authors demonstrated that the product distribution did not depend on the residence time, indicating that a stable steady state had been reached. These results demonstrate that the idea of the system evolving toward an equilibrium state at an effective temperature cannot explain experimentally observed product distributions, since there is no temperature at the system pressure for which such large amounts of NO are expected.

### C. N_{2}H_{4} and O_{3} synthesis

Now two examples are presented that demonstrate the inability of the thermal equilibrium assumption to explain, much less predict, the outcomes of low temperature plasma processes. Specifically, the synthesis of N_{2}H_{4} from N_{2} and H_{2} and the synthesis of O_{3} from O_{2} will be discussed. The reduction of nitrogen by hydrogen to form ammonia via the reaction N_{2} + 3H_{2} → 2NH_{3} is one of the most important reactions for human society—it is a critical source of agricultural fertilizer. The equilibrium composition of a system that initially contains N_{2} and H_{2} at a pressure of 1000 mbar is presented in Fig. 10(a). Nitrogen reduction by hydrogen to form ammonia is thermodynamically favorable at low temperatures and high pressures [Fig. 10(a)]. On the other hand, hydrazine (N_{2}H_{4}), which is a partially reduced form of nitrogen, is thermodynamically unstable. In systems containing N_{2} and H_{2} at commonly encountered pressures, there is negligible hydrazine present at equilibrium [Fig. 10(a)], although it may form as a kinetic intermediate between N_{2} and NH_{3} under conditions where ammonia is stable. Another example is ozone. Ozone (O_{3}) is a compound that is not present in a significant fraction at equilibrium for a system that initially contains pure O_{2}. The equilibrium composition of a system containing initially pure O_{2} at a total pressure of 1000 mbar is plotted in Fig. 10(b). The maximum mole fraction for O_{3} is 1 part per million, which occurs at approximately 3500 K. Interestingly, both compounds can be synthesized in large mole fractions, from elemental precursor gasses (N_{2} + H_{2} or O_{2}) by low temperature plasma processes.

Large yields of hydrazine from low temperature N_{2}-H_{2} plasmas have been reported. The most striking examples were published in Soviet journals and were written in Russian, and so again Fridman's translation is used.^{23} Hydrazine yield as a function of pressure for electron-beam supported N_{2}-H_{2} discharges is plotted in Fig. 11 for different gas feeding strategies.^{36} Significant N_{2}H_{4} fractions on the order of 10% were observed in the pressure range from 100 to 1000 mbar. To the author's knowledge, work has not been carried out to determine if such large N_{2}H_{4} fractions on the order of 10% are independent of the feed composition. In other words, if NH_{3} were fed into the reactor instead of N_{2} and H_{2}, but at the same nitrogen to hydrogen ratio, then would N_{2}H_{4} be observed in the same amount? Unfortunately, without the answer to that question, it is unclear if N_{2}H_{4} is simply a kinetic intermediate in the overall reaction of N_{2} + 3H_{2} → NH_{3,} which can be observed by appropriately manipulating the space time; or if the reported hydrazine fraction is observed at steady state, independent of the feed composition. To answer that question, one would need to carry out an experiment similar to the one illustrated in Fig. 6. Keep plasma parameters constant and examine the hydrazine fraction at long times for a feed of N_{2} + 3H_{2} and a feed of NH_{3} at the same total nitrogen and hydrogen ratio. Nevertheless, it is clear that the low temperature plasma process is capable of producing large mole fractions of a material that is not expected at thermal equilibrium.

Ozone can be produced at mole fractions exceeding 10% from pure O_{2} at atmospheric pressure using low temperature plasma, approximately 5 orders of magnitude higher than the maximum theoretical concentration expected from thermal equilibrium [Fig. 10(b)]. High concentration ozone produced by low temperature plasma has been used in a wide range of applications, for example, advanced oxidation processes for destruction of organic contaminants^{37} and pollution,^{38} and also as a gaseous precursor in chemical vapor deposition processes for thin film fabrication.^{39} Higher ozone mole fractions can be reached in pure O_{2} compared to air because the production of nitrogen monoxide (vide supra) suppresses the production of O_{3}.^{23} Reproduced from the work of Eliasson *et al.*,^{40} the fraction of ozone produced in an oxygen dielectric barrier discharge is plotted in Fig. 12 as a function of specific energy applied to the plasma and the neutral gas temperature. As the specific energy applied to the plasma increases, the fraction of O_{3} increases and then eventually saturates at a constant value (Fig. 12). Interestingly, the maximum value at which the ozone concentration saturates depends on the neutral temperature. As neutral temperature increases, the maximum ozone concentration decreases. Figure 12 is a clear example of the importance of neutral gas temperature in producing highly nonequilibrium material configurations. As neutral temperature increases, the rate at which species relax toward their thermal equilibrium configuration increases, which decreases the amount of a nonequilibrium substance such as ozone. From Fig. 12, it is clear that the concept of an effective temperature is inadequate to describe the system because the fraction of ozone produced is 5 orders of magnitude higher than the theoretical maximum at equilibrium. Furthermore, since the system is already in its thermal equilibrium state before the reaction (pure O_{2} at 278 K), the high fraction of O_{3} cannot be explained as a kinetic intermediate between the initial composition and the equilibrium composition. It is now undeniable that plasmas can increase the specific free energy of mass flows and cause changes in chemical state that move away from thermal equilibrium. Plasma processing can produce nonequilibrium materials.

## III. SOLID MATERIALS

Continuing on the idea of producing nonequilibrium materials using low temperature plasma, in this section, situations are described wherein endergonic transformations of solid materials are promoted by low temperature plasma, and a clear example is also presented of the synthesis of a material with nonequilibrium chemical composition.

There is tremendous interest in identifying transformations promoted by low temperature plasma that increase the free energy of the material being processed (i.e., endergonic transformations). Such transformations could be useful for energy storage and energetic material applications, as well as activating inert materials to increase their reactivity. Beyond those applications, there is an opportunity to discover fundamentally new configurations of matter that possess novel properties. Despite the allure, clear examples of endergonic transformations of solid phase materials promoted by low temperature plasma are scarce in the literature.

One of the most well-known situations in which plasmas can increase the free energy of a solid is surface activation. One processing configuration, illustrated in Fig. 13(a), is to bring a low temperature plasma jet into contact with a macroscopic solid using an impinging flow. After plasma treatment, the surface energy can be measured by optical contact angle analysis using a set of probe liquids and an appropriate model.^{41,42} For example, plotted in Fig. 13(b) is the surface energy of a polyether ether ketone sample before and after treatment with a low temperature RF plasma jet generated using a mixture of air and argon at a pressure of 1 mbar for 2 min. The contact angles formed between liquid droplets and the polymer samples were measured via the sessile drop method using pure water, diidomethane, and ethylene glycol [Fig. 13(b)]. The surface energy was calculated using the Kitazaki-Hata model.^{43} Plasma treatment increased the total surface energy from 54 to 153 mJ m^{−2}, which was mainly caused by an increase in the polar and hydrogen bonding components [Fig. 13(c)]. Similar results have been reported by other researchers.^{44,45} Activation of polymeric surfaces by low temperature plasmas has been used to increase the bond strength in assemblies constructed using adhesives^{42,46–53} and also autoadhered assemblies.^{45,54} Beyond polymers, plasma activation has also been used for bonding semiconductors to one another, for example, wafer bonding in silicon on insulator device manufacture.^{55,56} Moving forward, the increase in surface energy could have profound consequences for the behavior of nanoparticles processed by low temperature plasma. When size is in the nanometer range, the surface energy increases the chemical potential of the particle phase significantly,^{57} which can have many effects that could be beneficial, for example, larger critical nucleation size,^{58} changes in phase behavior,^{57,59} and shifts in chemical reaction equilibrium.^{60} Low temperature plasma activation of nanoparticle surfaces to increase the surface energy, and thereby increase the chemical potential of the solid phase, is a means by which to increase the free energy of the material without significantly altering its atomic structure or composition.

An example of an endergonic transformation of nanoparticles in low temperature plasma has been recently published.^{61} In that work, polydispersed aerosols comprised of bismuth (Bi) suspended in argon were made to pass through a low temperature plasma (Fig. 14). The plasma transformed the size distribution of the aerosol to make it more monodispersed. Such behavior is unexpected, since known aerosol growth mechanisms either preserve the width of the size distribution (i.e., condensation) or cause the distribution to become wider (i.e., coagulation).^{58,62} The specific entropy of an aerosol increases as the size distribution becomes more polydispersed.^{63,64} Thus, coagulation tends to increase specific entropy as the aerosol ages, which is reasonable if there is no work input into the system. The interesting thing about the result in Fig. 14 is that the plasma has acted to decrease the width of the size distribution, and therefore decrease the specific entropy of the aerosol. The narrowing of the size distribution is most convincingly represented in Fig. 15, where the mass distribution before and after plasma treatment clearly shows an increase in the mass contained within the narrow size range of the monodispersed peak. If enthalpy contributions to the free energy are neglected, which is reasonable considering the particles are relatively large (low surface area to volume ratio) and the composition is not changing (pure Bi), then the plasma has acted to increase the specific free energy of the aerosol by decreasing the specific entropy while the enthalpy remains nominally constant. The reader is referred to the original publication for additional experimental and mechanistic details.^{61}

Synthesis of materials that have a nonequilibrium chemical composition is a compelling research direction for low temperature plasma processing. The vast majority of nanocrystal materials that have been synthesized by low temperature plasmas are an equilibrium phase at some temperature and pressure, for example, Si,^{65,66} Ge,^{67} InP,^{68} Cu_{2}S,^{69,70} SnS,^{69} ZnS,^{69} Ni,^{71–73} SiC,^{74,75} and TiN.^{76} For those synthesis processes, the principle role of the plasma is to provide some desirable material property, for example, monodispersed size distribution, or high crystal quality with low defect density.^{77,78} There are reports of processes that produce kinetically stable materials that are not in their equilibrium atomic configuration for any temperature at the system pressure, for example, synthetic crystals of diamond phase carbon.^{79,80} However, if the decomposition of the gas-phase precursor at the reactor conditions is taken into account, then the overall process often results in a decrease in the specific free energy, despite the observation that a nonequilibrium material has been produced. For example, methane spontaneously decomposes at the substrate temperature and total pressure used to grow macroscopic crystals of diamond by microwave plasma-enhanced chemical vapor deposition,^{79} but the lowest chemical potential for carbon at those conditions is graphite phase. In other words, the gas-to-solid conversion decreases the specific free energy (i.e., CH_{4} decomposition), but the kinetics are controlled to arrest a nonequilibrium phase (i.e., diamond). The process is therefore exergonic. The material product can be considered as an intermediate in a process that has moved the feedstock closer to thermal equilibrium without reaching it, somewhat similar to the interpretation of the hydrazine result (vide supra). A more recent example is the synthesis of a kinetically arrested material with a nonequilibrium chemical composition—hyperdoped silicon nanocrystals.

The amount of boron dissolved in silicon nanocrystals synthesized by low temperature plasma has been reported to exceed the thermodynamic limit by more than an order of magnitude. The solubility of boron in silicon is plotted as a function of temperature in Fig. 16, which was reproduced from the data of Vick and Whittle.^{81} At thermal equilibrium, the maximum solubility of boron in silicon is approximately 0.5 at. %. Astonishingly, Zhou *et al.* have recently reported synthesis of crystalline silicon nanocrystals containing up to 31 at. % of boron using low temperature plasma.^{82} Doping of silicon nanocrystals synthesized in low temperature plasma has been recently reviewed by Ni *et al.*^{83} Here, the focus will be on two reports, by Pi *et al.*^{84} and Zhou *et al.*,^{82} which present the main experimental evidence in support of the claim that the maximum equilibrium boron solubility has been exceeded by a significant margin.

The synthesis of boron-doped Si nanocrystals has been described by Pi *et al.*^{84} The basic concept is to feed a silicon precursor (e.g., SiH_{4}) and a boron precursor (e.g., B_{2}H_{6}), both carried by an inert noble gas (e.g., Ar), into a tubular low temperature plasma reactor [Fig. 17(a)]. In the low temperature plasma, the precursors decompose to nucleate nanoparticles that subsequently grow.^{78} An ensemble of such nanoparticles can be collected as the product of the plasma reaction. In general, the product may contain nanoparticles with different overall composition, and different radial distribution of composition, which are illustrated schematically in Fig. 17(b). For example, nanoparticles may be comprised of: pure silicon, pure boron, silicon-rich core with boron-rich surface (B@Si), boron-rich core with silicon-rich surface (Si@B), or ideally, alloyed particles in which the boron is intimately mixed with the silicon at the atomic scale and the composition does not depend on radial position (B:Si). In other words, in the product collected from the reaction, in general there is a distribution in composition among the population and not all particles will contain the same fraction of boron, and in addition to that, each individual nanoparticle may have a composition that depends on position in the interior. Given these composition distributions, it is not obvious that the synthesis process illustrated in Fig. 17(a) will result in the desired material in which the boron and silicon are intimately mixed at the atomic length scale. Targeted material characterization experiments must be performed to establish that the product contains dissolved amount of boron above the solubility limit.

Etching the nanoparticle product using hydrofluoric acid (HF) can be used to assess the location of the boron within the individual nanoparticles.^{84} When exposed to air, silicon and boron will partially oxidize on the surface to form SiO_{2} and B_{2}O_{3}. Strong HF solutions will dissolve B, SiO_{2}, and B_{2}O_{3}, but will not dissolve a silicon matrix. Thus, if the nanoparticle product is etched using HF, dissolution may be assumed of any pure boron particles, boron surface coatings, or SiO_{2} native surface oxide. Thus, any remaining material is comprised of nanoparticles that have a silicon matrix or a silicon shell, and any boron present in the sample after etching is contained within that matrix or shell [Fig. 17(b)]. When Pi *et al.* etched their silicon doped nanocrystal product to remove the SiO_{2} and boron/boron oxide on the nanoparticle surfaces, they found that atomic concentration of boron increased, suggesting that the boron was predominantly present in the core of the nanocrystals.^{84} Recent work on the infrared absorption characteristics of these boron doped silicon nanocrystals found features consistent with a plasmonic response, suggesting that the boron was increasing the free charge carrier concentration. Such optical behavior is expected if the boron were acting as an electronic dopant, although the doping mechanism is unclear at present.^{85–88} Later characterization by x-ray photoelectron spectroscopy and high angle annular dark field scanning transmission electron microscopy (HAADF-STEM) also revealed that boron was enriched in the core of the nanocrystals compared to the surface.^{82} For example, presented in Fig. 18 is an HAADF-STEM elemental map of silicon nanocrystals hyperdoped with 31 at. % of boron.^{82} The crystal structure became increasingly strained and disordered as the amount of boron in the product increased. The authors reported that the size of the silicon nanocrystals, which was characterized by transmission electron microscopy (TEM) and x-ray diffraction, did not change significantly with increasing boron content from 0 to 31 at. %. At present, the observation of the insensitivity of size to the boron content is probably the strongest evidence that the boron is indeed forming an alloy with the silicon (i.e., B:Si) and not simply present as a core surrounded by some kind of silicon shell (i.e., Si@B). The author does note, however, that size is a weak function of the number of atoms in a particle, and the expected change in diameter for 31 at. % boron is probably less than the standard deviation of the size distribution. Nevertheless, the evidence presented is certainly consistent with hyperdoping of silicon with boron above the equilibrium solubility limit.

## IV. BEYOND EQUILIBRIUM THERMODYNAMICS

A number of examples have now been presented of low temperature plasmas moving systems away from equilibrium, or producing configurations of matter that are unexpected in a system governed by thermal equilibrium. Given that the local thermal equilibrium assumption cannot be applied to low temperature plasmas, how can expectations be formed theoretically about the direction of the reaction? The answer is clearly important for design of low temperature plasma processes. For now, kinetic models of plasma processes are probably the most reliable method to predict process outcomes. Since the criterion for chemical reaction equilibrium is currently unknown for low temperature plasmas, a different method must be used to ensure that a set of kinetic modeling equations complies with the laws of thermodynamics.

This section has two objectives. First, an example is provided for how to use the GENERIC framework to prove the thermodynamic admissibility of a set of dynamic modeling equations. The advantage of using GENERIC is that thermodynamic admissibility can be established without knowledge of the state that the system is tending toward. That feature is essential to ensure that a set of modeling equations for a low temperature plasma complies with the laws of thermodynamics, because methods of predicting, e.g., chemical reaction equilibrium have not been developed. The second objective of this section is to establish the concept of superlocal equilibrium, and prove that it complies with the laws of thermodynamics, provided appropriate exchange terms are used. The superlocal equilibrium concept is expected to be critical in future work that will focus on deriving equilibrium criteria for low temperature plasma, e.g., chemical reaction equilibrium.

The established theoretical approach for modeling low temperature plasmas has been kinetic modeling for the prediction of process outcomes. The equations that comprise such models must comply with the laws of thermodynamics. GENERIC is a beyond equilibrium thermodynamics framework that has been introduced by Oettinger in 1997, and has been described in detail in his book.^{22} The purpose is to rigorously ensure that a set of dynamical modeling equations, which are supplied into the GENERIC framework, comply with the laws of thermodynamics. The use of GENERIC will be illustrated by way of example using a two-temperature hydrodynamic model.

To use GENERIC, one must already have a set of equations that constitute a dynamical model. GENERIC is then used to analyze those equations. For example, to build a hydrodynamic model of a fluid, one can write expressions for mass, momentum, and energy conservation. GENERIC can then be used to ensure that the system of equations does not violate the first and second laws of thermodynamics. For a dynamical model to be thermodynamically admissible, it must satisfy the following equation:^{22}

where *x* is a set independent variables required for a complete description of the system (e.g., mass density, momentum density, and internal energy density), *E*(x) and *S*(x) are the total energy and total entropy as a function of the set of variables *x*, and *L*(*x*) and *M*(*x*) are the Poisson and friction matrices. The elements of the Poisson matrix *L* and friction matrix *M* are linear operators. The Poisson and friction matrices provide different contributions to the time evolution of the set of variables *x*, reversible changes due to energy (*L*), and irreversible changes due to entropy (*M*). For a model to be thermodynamically admissible, the matrices must meet several criteria. The Poisson matrix must satisfy the Jacobian identity, be antisymmetric, and satisfy the following degeneracy criterion: $L(x)\u22c5(\delta S(x)/\delta x)=0$. The criterion $L(x)\u22c5(\delta S(x)/\delta x)=0$ expresses that the entropy is not affected by the operator that generates the reversible dynamics. The friction matrix must also satisfy several criteria: *M* must be symmetric, positive semidefinite, and also in the absence of external fields, must satisfy the degeneracy criterion: $M(x)\u22c5(\delta E(x)/\delta x)=0$. The criterion $M(x)\u22c5(\delta E(x)/\delta x)=0$ expresses the conservation of energy in an isolated system, wherein work is transformed into internal energy. In the presence of external fields, the friction matrix *M* remains the same, but the right hand side of the degeneracy criterion may no longer be zero.^{22} For simplicity, in the present analysis, the effect of external fields will not be explicitly considered.

The use of GENERIC will be illustrated using a simple model for a low temperature plasma. One of the simplest models for a low temperature plasma is a two temperature system. Electrostatic effects are neglected. The system contains heavy ideal gas species (e.g., Ar) at a temperature *T*_{g} and partial pressure *P _{n}*, and light species (e.g., electrons) at a much higher temperature

*T*

_{e}and volumetric concentration

*n*

_{e}. The light species are theoretical particles that have the mass of an electron but no electrostatic charge, and are treated as ideal gas particles. The state variables

*T*

_{g},

*P*

_{n},

*T*

_{e}, and

*n*

_{e}are parameters that can be obtained from experimental measurements (Fig. 3) and can be controlled using external parameters such as applied power, vacuum pumping speed, and external cooling or heating. No external field is considered in the model, although the presence of such an external source of work at some point in the history of the system is implicit in the condition that

*T*

_{g}≠

*T*

_{e}at the same location in space. Such a system is described as being in superlocal equilibrium. The idea behind superlocal equilibrium is that at a given point in space, the temperature of all molecules of a given species is the same, e.g., all electrons have

*T*

_{e}and all Ar atoms have

*T*

_{g}, but the temperatures of different species may not be the same, i.e.,

*T*

_{g}≠

*T*

_{e}. The idea is illustrated in Fig. 19. Superlocal equilibrium has been used successfully in multifluid modeling of various low temperature plasma processes in the literature.

^{13,14}

The justification for neglecting electrostatic effects is the following. In low temperature plasmas, the electrons have very high kinetic energy, on the order of several electron-volts. The kinetic energy is sufficient to break many chemical bonds, for example by electron impact dissociation. In such a collision, the electron loses kinetic energy, however the expectation is that the electron is not captured by its collision partner. The main driver of chemical processes occurring in the bulk of the plasma is assumed to be the high kinetic energy of electrons, and the electrostatic charge of those electrons is assumed to play a less important role. The two temperature model, while quite simplistic, may be sufficient to capture the essential physics required to describe chemical processes occurring in the bulk of the plasma. The two temperature model described here was adapted from multifluid mass, momentum, and energy balance equations published in the literature.^{14} The equations explored in this work have been used to simulate kinetic and transport phenomena in low temperature plasmas in a wide variety of situations.^{13} Since the equations have been taken from elsewhere, they are not rigorously derived here. Instead, GENERIC is used as a test of whether the assumptions used in the derivation of those hydrodynamic equations comply with the laws of thermodynamics. The thermodynamic admissibility of the intrinsic dynamics of the system will be demonstrated in the absence of external interactions (e.g., external fields), although in principle those aspects could also be included.^{22}

There are three equations for each of the two species (i.e., argon and electrons): mass, momentum, and energy balances. The equations contain the usual terms from transport phenomena,^{89} and in addition, also have terms that account for momentum and energy exchange between the different species, which may be out of local equilibrium and have different velocities and temperatures. Several simplifying assumptions were made: (1) species are neither created nor destroyed (i.e., no source term in mass balance), (2) no electrostatic effects or external fields, and (3) no chemical reactions. The system contains only two species: argon atoms and theoretical particles that have the mass of an electron but no electrostatic charge. With these simplifying assumptions, the equations reveal the intrinsic or natural dynamics of the two-temperature system. The mass, momentum, and energy balances for species *i* are^{13}

where $\rho i$ is the mass density of species *i*, *t* is time, $M\u2192i$ is the momentum density of species *i*, *P*_{i} is partial pressure, $v\u2192i$ is velocity, *m*_{i} is the single particle mass, $\mu \xaf\xafi$ is the viscous stress tensor, $\nu ij$ is the frequency factor for collisions between species *i* and species *j*, $\epsilon i$ is the internal energy density of species *i*, *k*_{i} is the thermal conductivity, *T*_{i} is temperature, $C\u0303vi$ is the mole specific heat of *i* at constant volume, *N*_{A} is Avogadro's number and *k*_{B} is the Boltzmann constant. The mass balance [Eq. (7)], the first three terms on the right hand side of the momentum balance [Eq. (8)], and the first four terms of the energy balance [Eq. (9)], are well-known results of transport phenomena, and their description can be found in textbooks.^{89} More importantly for the present discussion, the thermodynamic admissibility of the standard terms in Eqs. (7)–(9) has been presented using GENERIC elsewhere,^{22} and so it is not necessary to further consider them here. The last terms of Eqs. (8) and (9), on the other hand, are something interesting. The goal here is to establish the thermodynamic admissibility of the term $\u2212\u2211j(1/mi+mj)\rho i\u2009\rho j\nu ij(v\xafi\u2212v\xafj)$ in Eq. (8), and $3kB\u2211j(1/(mi+mj)\u20092)\rho i\u2009\rho j\nu ij(Tj\u2212Ti)$ in Eq. (9) using GENERIC. These terms describe the exchange of momentum and the exchange of internal energy between different species that are traveling at different bulk velocities and have different temperatures at the same location in space. The terms are referred to as the momentum exchange and internal energy exchange terms. The exchange terms describe the interaction of species in a system governed by superlocal equilibrium. Both exchange terms represent dissipative processes that are only present if the system is out of local equilibrium, and thus, the terms tend to zero as local equilibrium is approached.

The first step in a GENERIC analysis is to define the level of description, or more specifically, the set of variables *x*. Since Eqs. (7)–(9) are written in terms of mass, momentum and internal energy densities, these variables constitute a convenient *x.* For the two-temperature system containing only two species

where, for example, index 1 refers to the argon atoms and index 2 refers to the electrons.

The next step is to calculate the derivatives of total energy of the system and total entropy of the system with respect to the set of variables *x*. The total energy of the system contains kinetic energy associated with momentum and internal energy: $E=\u2211i=12\u222b\u222b\u222bdV((1/2)(M\u2192i2/\rho i)+\epsilon i)$. The total entropy can be found by simply integrating the entropy density $Si$, which depends on mass density and internal energy density: $S=\u2211i=12\u222b\u222b\u222bdV\u2009Si(\rho i,\epsilon i)$. The derivatives of energy and entropy with respect to the set of variables *x* in Eq. (6) are evaluated in the following way:

From thermodynamics, $Ti=\u2202\epsilon i\u2202S$ and $\u2202S\u2202\rho i=\u22121migiTi$ where *g*_{i} is the chemical potential of species *i*.^{2} Thus, it can be shown for the two-species system that

The three derivative column vectors in Eqs. (10) and (11) have been constructed using only the level of description, i.e., the set of variables that have been chosen to describe the system, and different contributions to the total energy and total entropy of the system. It is the matrices *L* and *M* that will be used to determine if the modeling equations, specifically the exchange terms in Eqs. (8) and (9), are thermodynamically admissible. Since the terms of interest correspond to dissipative effects that do not explicitly depend on external inputs, i.e., they are part of the intrinsic dynamics of the system, they must be accounted for by the friction matrix *M*. Further discussion of this point may be found in the literature.^{22} Recall that the friction matrix must meet several criteria for the equations to be thermodynamically admissible: (1) *M* must be symmetric, (2) *M* must be positive semidefinite, and (3) in the absence of external interactions, *M* must meet the degeneracy criterion: $M\u22c5(\delta E/\delta x)=0$. Furthermore, the product $M\u22c5(\delta S/\delta x)$ should reproduce the exchange terms of interest in the rows of the column vector associated with the momentum density and internal energy density of species 1 and 2. The task now is to construct an *M* matrix and verify that it complies with its constraints.

The construction of the *M* matrix can be simplified by noting that the total *M* matrix is the sum of contributions of rank one, each of which by itself is expected to be symmetric degenerate, and positive semidefinite^{90}

where *M*_{HD} is the friction matrix associated with the standard hydrodynamic terms in Eqs. (2)–(4), which has been comprehensively described elsewhere.^{22} *M*_{EX} and *M*_{MX} are the friction matrices associated with the internal energy exchange term and the momentum exchange term to be analyzed. To maintain symmetry, the first and fourth rows, and the first and fourth columns of *M*_{MX} and *M*_{EX} are expected to be zero to be consistent with the mass balance equation for both species. In other words, the energy and momentum exchange terms do not affect the mass balance if no source term is included in Eq. (7). The friction matrix for the internal energy exchange term is a dyadic product that can be written as

where $\beta \u2032=3kB(1/(m1+m2)2)\rho 1\u2009\rho 2\nu 12$. The friction matrix *M*_{EX} is symmetrical, positive semidefinite, and satisfies the degeneracy criterion $M\u22c5(\delta E/\delta x)=0$. Moreover, it reproduces the internal energy exchange term exactly

An issue emerges with the momentum exchange term. Despite the best efforts of the author (approximately 1 week of focused effort was spent on this by both the author and H. C. Oettinger), a friction matrix *M*_{MX} could not be identified that reproduces only the momentum exchange term in Eq. (8) and also meets all three criteria for a friction matrix. However, if one allows for an additional term in the energy equation (9) that increases the internal energy of both species due to the bulk velocity difference between them, then the momentum exchange term is thermodynamically admissible according to the following friction matrix, which is also a dyadic product.

where $\beta =(1/m1+m2)\rho 1\u2009\rho 2\nu 12$ and *T*_{12} is the reduced temperature of species 1 and 2: $T12=T1T2/(T1+T2)$. The friction matrix *M*_{MX} meets all three criteria. When multiplied by the entropy derivative [see Eq. (6)], *M*_{MX} reproduces the momentum exchange term and also produces a new term in the energy balance

Therefore, it is suggested that the inclusion of an additional term in the energy balance equation (9) is required for the superlocal equilibrium model to be thermodynamically admissible. That additional term is proposed to be: $\u2211j(1/2)(1/mi+mj)\rho i\u2009\rho j\nu ij(v\xafi\u2212v\xafj)\u20092$. The new term appears to be the discrete counterpart of the viscous dissipation term of hydrodynamics. With the inclusion of this term, the internal energy density equation has the following form:

The conclusion of this GENERIC analysis is that the concept of superlocal equilibrium is thermodynamically admissible if appropriate terms are included that describe the exchange of momentum and heat between species that have different momentum density and temperature at the same location in space. The task for future work will be to use the superlocal equilibrium concept to derive equilibrium criteria for low temperature plasma, e.g., chemical reaction equilibrium.

## V. SUMMARY AND CONCLUSIONS

In conclusion, macroscopic energy and entropy balances of the low temperature plasma processor reveal that in the reversible limit it is capable of increasing the free energy of the mass that flows through it. Different species have different temperatures at the same location in space in low temperature plasmas, and as such, the thermal equilibrium assumption cannot be applied. Many of the familiar results of equilibrium thermodynamics, such as the chemical reaction equilibrium criterion, cannot be applied in the familiar form. Several examples were given of gas and material transformations that are unexpected from the perspective of thermodynamics governed by thermal equilibrium. For the CO_{2} dissociation reaction, the concept of an effective temperature for the chemical reaction appears to be adequate for describing experimental outcomes. Are there other chemical systems wherein the concept of an effective temperature is also applicable? Unfortunately, the effective temperature concept is not general. GENERIC was described, by way of example, as a means to ensure the compliance of a dynamic model with the laws of thermodynamics. The GENERIC analysis provided here proves that the concept of superlocal equilibrium complies with the laws of thermodynamics if appropriate terms are included to describe the exchange of momentum and internal energy between species that may have different temperatures and bulk velocities at the same location in space.

## ACKNOWLEDGMENTS

Necip B. Uner provided the data presented in Fig. 3 and participated in useful discussions during the preparation of the manuscript. Hans Christian Oettinger generously helped the author use the GENERIC framework, and Qinyi Chen checked calculations. Harold E. “Trey” Oldham plasma treated the polymer sample and measured surface energy for the example used in Fig. 13. This work was partially supported by the National Science Foundation under Grant Agreement No. PHY-1702334.