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 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 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 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. 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 _ ~ _ ~ þ _ _ ; ODS 0 Nin;i Hin;i N out;i Hout;i W e Qout i¼1 i¼1 (1) Xn Xn : ¼ _ ~ _ ~ 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, Q_ is the heat flow rejected to the envi- out _ 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 Sin;i Nout;i 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 Sin;i Nout;i 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. Therefore, classical processor takes a heat input. Neglecting changes in low temperature plasmas are attractive as processors, since kinetic and potential energy of the mass flow, and assuming the heat transfer occurs across a small temperature gradient steady state, the first law is quite similar for both types of pro- to the ambient. Having established the thermodynamic limits cessors, and can be written as of both types of processors in Eqs. (1)–(4), the next step is to

J. Vac. Sci. Technol. B, Vol. 36, No. 4, Jul/Aug 2018 048501-3 Elijah Thimsen: Beyond equilibrium thermodynamics 048501-3 analyze how the direction of chemical reactions occurring dVa, and dn are independent. Therefore, for dS ¼ 0 at equi- within them can be predicted. librium, the coefficients of dUa, dVa and dn must be zero. The idea is that the reaction occurring within the processor Thus, at equilibrium, the temperatures must be equal: will proceed from the inlet state toward an equilibrium state Ta ¼ Tb, which is the criterion for thermal equilibrium. If whereat the entropy will be maximized. For the classical pro- the temperatures are equal, then the pressures must also be cessor, one can readily derive the mole fractions of different equal to make the coefficient of the volume term vanish. chemical species that the system will tend toward using well- Likewise for the third term, and thus, we arrive at the condi- established approaches. The task is to derive the chemical tion forP chemical reaction equilibrium in the classical pro- ¼ ¼ reaction equilibrium criterion, which can then be used to pre- cessor: i igi gC2H2 gH2 2gC 0, where i is the dict the relative mole fractions of different species, given an stoichiometric coefficient of i in the reaction. From the chemi- initial composition at the inlet. Consider a closed multiphase cal reaction equilibrium criterion, the mole fractions of C2H2, system of interest, which is in contact with a pressure reser- H2, and C in the product mixture can be calculated from a voir via an adiabatic plunger, and a heat reservoir via a rigid material balance and tabulated information using the well- diathermal wall (Fig. 2). The system of interest contains only known procedure involving standard state reference informa- species that participate in some chemical reaction. This tion and species fugacity. In other words, the chemical reac- example will focus on a single chemical reaction, but it can tion equilibrium criterion allows for prediction of the product be extended to multiple reactions.2 The three subsystems mixture that the classical processor will tend toward as it pro- comprise an isolated system. Due to interaction with the res- ceeds toward equilibrium, given a certain amount of carbon ervoirs via the adiabatic plunger and diathermal wall, the sys- and hydrogen loaded into the vessel. In general, for other tem of interest is at constant temperature and pressure, and types of reactions such as the homogeneous dissociation reac- ! þ therefore, the Gibbs free energy of the system of interest is tions (e.g., CO2 CO 1/2O2), one must also assume ther- minimized at equilibrium. For illustration, consider the reac- mal equilibrium in order to derive the condition for chemical tion of hydrogen with carbon to form acetylene (Fig. 2). The reaction equilibrium, which is required to predict the equilib- gas phase is a, while the solid phase is b. The change in total rium composition. Without thermal equilibrium, prediction of entropy of the system of interest as a function of the change the outcome of the chemical reaction would not have been in internal energy of a phase, change in volume of a phase, possible using the preceding approach. Can such an approach and extent of reaction is also be applied to chemical reactions occurring in low temperature plasmas? 1 1 The interesting thing about low temperature plasmas is that dS ¼ 0 ¼ dSa þ dSb ¼ dUa Ta Tb the local thermal equilibrium assumption cannot be made, and therefore most of the familiar results from equilibrium thermo- Pa Pb gC2H2 gH2 2gC þ dVa dn; (5) dynamics, such as Henry’s law, Raoult’s law, osmotic pressure, Ta Tb Ta Ta Tb and chemical reaction equilibrium, cannot be applied in their where Ta and Tb are the temperatures of the phases, Pa and commonly used form. In low temperature plasma, different species have temperatures that differ by more than an order of Pb are the pressures in the phases, gi is the chemical potential of species i, and n is the extent of reaction. The symbol d magnitude at the same location in space. Additional variables denotes a small variation of the variable it operates on. In are required to specify the state of the system. Specifically, in Eq. (5), the surface energy of the b phase has been neglected. addition to the gas temperature Tg and total pressure P, one At equilibrium, the total entropy of the system is maximized must also specify or measure the electron temperature Te and and thus, for small variations dS ¼ 0. The variations dUa, positive ion density ni. 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 102 K, while the electron temperature is on the order 104 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 probe4–6 positioned at the same location and the ion density 18 3 FIG. 2. (Color online) Classical chemical reaction equilibrium. A closed sys- was 1 10 m . Clearly, local thermal equilibrium cannot tem in which hydrogen reacts with carbon to form acetylene. be assumed in that system, since the electrons are 2 orders of

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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. FIG. 3. (Color online) Hot electrons in a cool background gas in low temperature plasma. (a) Digital photograph of a capacitive-coupled, ﬂow-through, Unfortunately, the key concepts that are required to derive a radiofrequency (13.56 MHz), low temperature argon plasma at an applied condition for chemical reaction equilibrium in low temperature power set point of 20 W and pressure of 2.0 mbar. The tube is fused silica. plasmas have not hitherto been elucidated. The powered electrode was the aluminum ring on the left, while the elec- This work begins a line of inquiry focused on using trode on the right was grounded. (b) Thermal image of the reactor at the same conditions as (a). The emissivity used in the thermal image was that of the framework of thermodynamics to predict the outcomes fused silica. Te is the electron temperature, Tg is the gas temperature, and of physiochemical reactions in low temperature plasmas. nion is the ion density. Data courtesy of N. B. Uner. Examples from the literature will be provided for gas-phase chemical reactions and material conversions that exhibit out- magnitude higher temperature than the neutral species at the comes which are highly unexpected from the perspective same spatial location. For another example of why the local of thermodynamics governed by local thermal equilibrium. thermal equilibrium assumption cannot be made, consider Two very different methods of using thermodynamics for small nanoparticles suspended in a low temperature plasma. the theoretical analysis of low temperature plasma processes The surfaces of the nanoparticles are under continuous ion and will be presented. The ﬁrst method is very simple, but some- electron bombardment, and to maintain charge neutrality in the what surprisingly appears to account for experimental out- 7 plasma, nanoparticles become negatively charged. Positive comes in some instances. Can a low temperature plasma ions are attracted to the negatively charged particles, and when process be described, using the familiar chemical reaction they recombine on the surface, a large amount of energy is equilibrium criterion that requires thermal equilibrium, as released. The consequence is that at steady state, particles can evolving toward a state at an effective temperature? In other be 100s of K higher temperature than the surrounding gas,8–11 words, if a system has a given total pressure P, overall ele- and again the local thermal equilibrium assumption cannot be mental composition, gas temperature Tg, and electron tem- made (Fig. 4). Processes occurring in low temperature plasma perature Te, can the behavior be described as evolving must obey the macroscopic energy and entropy balances for an toward a thermal equilibrium state at some intermediate tem- ODS in Eqs. (1) and (3); however, since the local thermal equi- perature Teff, where Tg

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low temperature plasma processes. The reaction of N2 with O2 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 O2 to form O3,andthereactionofN2 and H2 to make N2H4. 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

FIG. 4. Hot particles in a cool background gas in a low temperature plasma. would be expected from an equilibrium analysis, then the con- Particle temperature as a function of time for small nanoparticles suspended cept of an effective temperature holds. If, on the other hand, in a low temperature plasma. The particle diameter is dp. The gas tempera- the experimental product yield is greater than the theoretical ture was assumed to be 300 K, and the plasma density was 5 1016 m 3. Reprinted with permission from Mangolini and Kortshagen, Phys. Rev. B maximum expected from thermal equilibrium analysis, then 79, 026405 (2009). Copyright 2009, American Physical Society. the concept of an effective temperature will be called into question. was developed by Oettinger.22 GENERIC does not require knowledge of the equilibrium state that the system is tending A. CO2 dissociation toward to assess a given kinetic model. The GENERIC The dissociation of carbon dioxide in plasmas is well- framework is used here to prove the thermodynamic admis- studied, with literature going back decades and several reviews sibility (i.e., compliance with laws of thermodynamics) of have been published.23,25,26 There has recently been a resur- the concept of superlocal equilibrium by analyzing a com- gence of interest in the topic. Recent research has focused monly used multiﬂuid hydrodynamic model. The concept of on the external energy efﬁciency of the reaction to form carbon superlocal equilibrium is expected to be of utility in future 1 monoxide: CO2 ! CO þ =2O2, which is highly endothermic work focused on deriving conditions for chemical reaction with DHrxn ¼ 2.9 eV. The reaction is limited by the production equilibrium in low temperature plasmas. of carbon monoxide and mono-oxygen: CO2 ! CO þ O 23 (DHrxn ¼ 5.5 eV). In thermal equilibrium, the highly endo- II. GASSES thermic nature of the reaction allows it to proceed only at high The reactions of simple gas molecules provide excellent temperature. For a pressure of 10 mbar, the equilibrium specia- examples for exploring the concept of an effective tempera- tion is plotted as a function of temperature in Fig. 5(a),andthe ture in low temperature plasmas. Several selected examples equilibrium fraction of carbon present as CO is plotted in will be discussed, namely, CO2 dissociation, NO production, Fig. 5(b). The curves were generated using the chemical equi- as well as N2H4 and O3 syntheses. There are other examples librium with applications (CEA) code developed by National that could have been chosen, and the reader is referred to the Aeronautics and Space Administration (NASA).27 The behav- monograph of Fridman for additional information,23 which ior is similar at other pressures, but shifted horizontally to the author consulted many times during the preparation of account for the fact that CO2 dissociation produces additional this article. CO2 dissociation has become topically relevant in gas molecules. The general trend is that as temperature the context of climate change driven by the greenhouse effect, increases, molecules become atomized. as well as in the context of Mars exploration. Mars has a 95% If the system were in thermal equilibrium at an electron CO2 atmosphere at a pressure of approximately 6 mbar. The temperature of 20 000 K, then full dissociation would occur 24 CO2 could be a source of breathable oxygen for astronauts. [Fig. 5(a)]. That observation leads to the expectation that Furthermore, CO2 dissociation is an excellent example of a plasmas tend to dissociate molecules. However, the frac- system in which the concept of an effective temperature is tional ionization is typically small in a low temperature largely adequate for understanding experimental outcomes of plasma, approximately 3 105 in Fig. 3 for example, and

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1 thus hot electrons are relatively dilute. On the other hand, if comprised of pure CO2, or feeds comprised of CO þ =2O2 the system was in thermal equilibrium at the neutral gas (Fig. 6). The products of CO2 dissociation were CO and O2. temperature, which is typically less than 1000 K (Fig. 3), The gas temperatures, estimated using a shielded thermocou- no dissociation would occur. Experimental observations ple, were less than 1000 K. However, the effective tempera- are in between these extremes. The focus here will be on ture at which the equilibrium CO2 mole fraction matched the select experimental reports of CO2 dissociation in low tem- experimental output was found to be approximately 2500 K. perature plasmas containing information relevant to this The electron temperature was reported to be approximately perspective. 4eV (46000K).29 The effective temperature of 2500 K is The idea that the system can be described as evolving between the gas temperature of 1000 K and the electron tem- toward an equilibrium state at an effective temperature perature of 46 000 K. The idea of a system evolving toward requires that the output from the reactor is independent of the an equilibrium state defined by an effective temperature in speciation of the input, provided that the molar flow rates of between the gas and electron temperatures appears to be ade- carbon and oxygen are the same for different feed configura- quate for describing the process. tions and the reaction time is sufficiently long. From that Kinetics control how quickly a system reaches the equilib- point of view, the output from a plasma process should be rium state. If the idea of a low temperature CO2 plasma 1 nominally the same if either CO2 or CO þ =2O2 were fed proceeding toward an equilibrium state at an effective tempera- into the reactor at the same total carbon to oxygen ratio. ture is applicable, then that equilibrium state should be reached Brown and Bell performed exactly that experiment using at long times and be relatively stable. Experimentally, such radiofrequency (RF) electrodeless discharges, similar to the behavior can be probed by monitoring the reaction as a func- reactor illustrated in Fig. 3, at pressures of several millibar.28 tion of time in a closed system, or alternatively, as a function Interestingly, they observed that at high powers the CO2 frac- of residence time in an open system. The composition should tion in the reactor effluent was nominally the same for feeds 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 neces- sity, results from slightly different configurations will be presented below to illustrate the steady state in low temperature CO2 plasmas. Published experimental work suggests that the CO2 dissociation reaction in low temperature plasma initiates after approximately 105 s and reaches a steady state after 102 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 CO2 dissociation in these studies were CO and O2, and thus, only the CO2 conversion is plotted in Fig. 7. Taylan and Berberoglu have studied dissociation in the residence time range from 1 to 100 ls using microhollow cathode discharges in mixtures of 30 CO2 and argon at atmospheric pressure. They found significant conversion after approximately 10 ls, which increased to approximately 14% after 128 ls, 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 CO2 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

FIG. 5. (Color online) Calculated thermal equilibrium in a system containing residence time and applied current to the plasma (Fig. 7, CO2. Equilibrium speciation at a constant pressure of 10 mbar as a function curves b–e). The increase in CO2 conversion with residence of temperature. Initially, the system contains pure CO2. (a) Mole fractions time exhibited a sublinear dependence that appeared to be of different species and (b) fraction of carbon present as CO. Only the spe- tending toward saturation at a steady-state value. Williams cies present at signiﬁcant mole fractions are shown. Other species considered in the calculations, which were present at negligible fraction at and Smith have performed experiments in which they ana- equilibrium, were C2,C2O, C3,C3O2,C4,C5, and O3. lyzed the dissociation as a function of time in sealed

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FIG. 7. (Color online) CO2 dissociation in low temperature plasma tends toward a stable equilibrium state. (a) Dissociation in an Ar-CO2 microhollow cathode discharge at 1000 mbar. [Data reproduced with permission from Taylan and Berberoglu, Plasma Source. Sci. Technol. 24, 015006 (2015). Copyright 2015, IOP Publishing.] (b)–(e) Dissociation in a He-CO2 capillary discharge at a pressure of 40 mbar for different applied currents (b) 2.2 mA, (c) 4.0 mA, (d) 8.0 mA, and (e) 16 mA. [Data reproduced with permission from Mori et al., Plasma Source Sci. Technol. 15, 609 (2006). Copyright 2006, IOP Publishing.] (f) Dissociation in a sealed electrodeless RF plasma containing 13% CO2,9%N2 and 78% He at a pressure of 26 mbar. [Data reproduced with permission from Williams and Smith, J. Phys. FIG. 6. Steady state speciation independent of feed composition. Fraction of D 18, 335 (1985). Copyright 1985, IOP Publishing.] carbon present as CO2 in the effluent from an electrodeless RF plasma reactor as a function of applied power for different feed compositions. The experimental results are symbols, circles for pure CO2 feed and squares for B. NO synthesis 1 CO þ =2O2. The slight difference between the experimental curves at high powers is due to a portion of the flow bypassing the plasma. The dashed In low temperature plasmas containing N2 and O2, the curves are model results that correct for the flow bypass. Reprinted with per- idea of a system evolving toward an equilibrium state at an mission from Brown and Bell, Ind. Eng. Chem. Fundam. 13, 203 (1974). effective temperature starts to break down. Specifically, it Copyright 1974, American Chemical Society. will be seen that the amount of a product experimentally

32 observed in the reactor effluent, specifically nitrogen monox- discharge tubes containing CO2,N2, and He. The pressure ide (NO), can be greater than the maximum fraction expected was 26 mbar, and the RF plasma was generated using an from equilibrium analyses at any temperature, given the feed electrodeless configuration at a frequency of 29 MHz. The composition and total reactor pressure. time range was 10–1000 s. Already at the initial time point The synthesis of NO in N2-O2 plasmas, which has a long taken at 48 s, the conversion was approximately 69% (Fig. 7, history going back more than 100 years, has been comprehen- curve f), indicating that the majority of the conversion took 23,33 sively reviewed elsewhere. The reaction N2 þ O2 ! 2NO place on a shorter time scale, as expected. The interesting is endothermic (DH ¼ 1 eV per molecule) and only proceeds observation is that the conversion increased slightly with in thermal equilibrium at high temperature. The plasma- time, but after approximately 120 s, the system reached a sta- activated reaction was first reported over 200 years ago by ble steady state, consistent with the idea that an equilibrium Sir Humphrey Davy.33 Industrial scale production by the had been reached. This state was stable from 120 s until Birkeland-Eyde process was demonstrated in 1903.33 Early 600 s, at which time the experiment was terminated. The processes relied on thermal plasmas and high quenching rates kinetics are consistent with the system tending toward an to achieve product distributions that had equilibrium or near equilibrium state, which can be described by an effective equilibrium speciation. Later work focused on low tempera- temperature of approximately 2500 K. Thus, for CO2 in low ture nonequilibrium plasmas, which can be used to selec- temperature plasma, the answer to the question appears to tively excite molecular vibrational modes to increase the be yes, one can imagine the system as evolving toward an overall energy efficiency of the process.33 equilibrium state at an effective temperature that is interme- In systems containing nitrogen and oxygen, nitrogen diate between the gas and electron temperatures. Below it monoxide forms at temperatures intermediate between the will be demonstrated that other systems cannot be under- diatomic and monoatomic states of the gases. The equilib- stood using the concept of an effective temperature, which rium speciation in a system with equal number of N2 and O2 raises a question. What determines if a chemical reaction molecules is plotted as a function of temperature in Fig. 8(a) occurring in a low temperature plasma can be understood as for a pressure of 30 mbar. The fraction of nitrogen present as proceeding toward a thermal equilibrium state at an effective NO is plotted in Fig. 8(b). The plots were generated using temperature? What determines the effective temperature? the NASA CEA code. The maximum fraction of NO is

JVST B - Nanotechnology and Microelectronics: Materials, Processing, Measurement, and Phenomena 048501-8 Elijah Thimsen: Beyond equilibrium thermodynamics 048501-8 approximately 4.4% at a temperature slightly higher than 3000 K. Below this temperature, the gasses are present as N2 and O2; 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 per- cent 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 102 mbar.34 PlottedinFig.9(a) is NO production as a function of power supplied to the plasma for a pressure of 6.6 102 mbar, reproduced from Ref. 34 At the conditions used in that work, the theoretical maximum value þ þ that the mole fraction of NO xNO/(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 N2-O2 plasmas in the pressure range from 5 to 40 mbar using RF FIG. 8. (Color online) Equilibrium speciation at a constant pressure of 30 inductive plasmas at an applied frequency of 40 MHz.35 They mbar in a system with equal number of moles of nitrogen and oxygen. (a) examined the effect of nitrogen-to-oxygen ratio in the feed Speciation as a function of temperature. (b) Fraction of nitrogen present as 35 NO as a function of temperature. Only species present at significant mole gas, as well as the presence of WO3 and MoO3 catalysts. fractions are shown. Other species considered in the calculations, which Interestingly, both catalysts significantly increased the fraction were present at negligible fraction at equilibrium were NO2,NO3,N2O, of fixed nitrogen. Even without the catalyst, the fraction of N2O3,N2O4,N2O5,N3, and O3. nitrogen fixed by the reaction [Fig. 9(b)] exceeded the theoretical maximum (Fig. 8). Moreover, the authors demonstrated pressures [Fig. 10(a)]. On the other hand, hydrazine (N2H4), that the product distribution did not depend on the residence which is a partially reduced form of nitrogen, is thermody- time, indicating that a stable steady state had been reached. namically unstable. In systems containing N2 and H2 at com- These results demonstrate that the idea of the system evolving monly encountered pressures, there is negligible hydrazine toward an equilibrium state at an effective temperature cannot present at equilibrium [Fig. 10(a)], although it may form as a explain experimentally observed product distributions, since kinetic intermediate between N2 and NH3 under conditions there is no temperature at the system pressure for which such where ammonia is stable. Another example is ozone. Ozone large amounts of NO are expected. (O3) is a compound that is not present in a significant fraction at equilibrium for a system that initially contains pure O2. C. N2H4 and O3 synthesis The equilibrium composition of a system containing initially Now two examples are presented that demonstrate the pure O2 at a total pressure of 1000 mbar is plotted in Fig. inability of the thermal equilibrium assumption to explain, 10(b). The maximum mole fraction for O3 is 1 part per mil- much less predict, the outcomes of low temperature plasma lion, which occurs at approximately 3500 K. Interestingly, processes. Specifically, the synthesis of N2H4 from N2 and H2 both compounds can be synthesized in large mole fractions, þ and the synthesis of O3 from O2 will be discussed. The reduc- from elemental precursor gasses (N2 H2 or O2)bylowtem- tion of nitrogen by hydrogen to form ammonia via the reac- perature plasma processes. tion N2 þ 3H2 ! 2NH3 is one of the most important reactions Large yields of hydrazine from low temperature N2-H2 for human society—it is a critical source of agricultural fertil- plasmas have been reported. The most striking examples izer. The equilibrium composition of a system that initially were published in Soviet journals and were written in 23 contains N2 and H2 at a pressure of 1000 mbar is presented in Russian, and so again Fridman’s translation is used. Fig. 10(a). Nitrogen reduction by hydrogen to form ammonia Hydrazine yield as a function of pressure for electron-beam is thermodynamically favorable at low temperatures and high supported N2-H2 discharges is plotted in Fig. 11 for different

J. Vac. Sci. Technol. B, Vol. 36, No. 4, Jul/Aug 2018 048501-9 Elijah Thimsen: Beyond equilibrium thermodynamics 048501-9

FIG. 10. (Color online) Equilibrium composition in systems initially containing N2-H2 and O2 showing that both N2H4 and O3 are present at a negligible mole fraction. (a) Equilibrium composition as a function of temperature for a system at a constant pressure of 100 mbar that initially contains only FIG. 9. (Color online) Nitrogen ﬁxation as oxides by low temperature plasma N2:H2 at a ratio of 1:3. The following species were considered in the calcu- in amounts greater than the maximum predicted by thermal equilibrium. (a) 9 Nitrogen monoxide as a function of applied power for a low temperature air lation, but were found to be present at a mole fraction less than 10 for the plasma (Ref. 34). [From Alekseev et al., International Symposium on assigned temperatures: H, N, NH, NH2,N2H2,N2H4,N3, and N3H. (b) Plasma Chemistry. Copyright 1979 by International Union of Pure and System at a pressure of 1000 mbar that initially contains pure O2. Applied Chemistry. Data reproduced with permission of International Union of Pure and Applied Chemistry.] (b) Conversion of N in N2-O2 plasmas as a 6. Keep plasma parameters constant and examine the hydra- function of the nitrogen to oxygen ratio in the feed gas reproduced with per- zine fraction at long times for a feed of N þ 3H and a feed 15 2 2 mission from Rapakoulias et al., Rev. Phys. Appl. , 1261 (1980). of NH at the same total nitrogen and hydrogen ratio. Copyright 1980, EDP Sciences. The curves in (b) are for a different cata- 3 lysts, or lack thereof, in the discharge zone. Nevertheless, it is clear that the low temperature plasma pro-

36 cess is capable of producing large mole fractions of a mate- gas feeding strategies. Signiﬁcant N2H4 fractions on the rial that is not expected at thermal equilibrium. order of 10% were observed in the pressure range from 100 Ozone can be produced at mole fractions exceeding 10% to 1000 mbar. To the author’s knowledge, work has not been from pure O2 at atmospheric pressure using low temperature carried out to determine if such large N2H4 fractions on the plasma, approximately 5 orders of magnitude higher than the order of 10% are independent of the feed composition. In maximum theoretical concentration expected from thermal other words, if NH3 were fed into the reactor instead of N2 equilibrium [Fig. 10(b)]. High concentration ozone produced and H2, but at the same nitrogen to hydrogen ratio, then by low temperature plasma has been used in a wide range would N2H4 be observed in the same amount? Unfortunately, of applications, for example, advanced oxidation processes 37 38 without the answer to that question, it is unclear if N2H4 is for destruction of organic contaminants and pollution, and simply a kinetic intermediate in the overall reaction of also as a gaseous precursor in chemical vapor deposition pro- 39 N2 þ 3H2 ! NH3, which can be observed by appropriately cesses for thin ﬁlm fabrication. Higher ozone mole fractions manipulating the space time; or if the reported hydrazine can be reached in pure O2 compared to air because the produc- fraction is observed at steady state, independent of the feed tion of nitrogen monoxide (vide supra) suppresses the produc- 23 40 composition. To answer that question, one would need to tion of O3. Reproduced from the work of Eliasson et al., the carry out an experiment similar to the one illustrated in Fig. fraction of ozone produced in an oxygen dielectric barrier

JVST B - Nanotechnology and Microelectronics: Materials, Processing, Measurement, and Phenomena 048501-10 Elijah Thimsen: Beyond equilibrium thermodynamics 048501-10

FIG. 11. (Color online) Hydrazine yield as a function of pressure for electron-beam supported discharges: (squares) discharge in N2-H2 mixture, FIG. 12. (Color online) Ozone fraction in the effluent from a dielectric barrier and (circles) discharge in nitrogen followed by mixing with hydrogen. From discharge that was fed pure O2. Specific energy is the power applied to the Fridman, Plasma Chemistry. Copyright 2008 by Cambridge University plasma divided by the molar flow rate of O2. The three curves correspond to Press. The plot is reproduced with permission of Cambridge University different neutral temperatures. [Data reproduced with permission from Press. The original data were reported by Ref. 36. Eliasson et al.,J.Phys.D20, 1421 (1987). Copyright 1987, IOP Publishing.] discharge is plotted in Fig. 12 as a function of specific energy energy storage and energetic material applications, as well as applied to the plasma and the neutral gas temperature. As the activating inert materials to increase their reactivity. Beyond specific energy applied to the plasma increases, the fraction of those applications, there is an opportunity to discover funda- O3 increases and then eventually saturates at a constant value mentally new configurations of matter that possess novel (Fig. 12). Interestingly, the maximum value at which the ozone properties. Despite the allure, clear examples of endergonic concentration saturates depends on the neutral temperature. As transformations of solid phase materials promoted by low tem- neutral temperature increases, the maximum ozone concentra- perature plasma are scarce in the literature. tion decreases. Figure 12 is a clear example of the importance One of the most well-known situations in which plasmas of neutral gas temperature in producing highly nonequilibrium can increase the free energy of a solid is surface activation. material configurations. As neutral temperature increases, the One processing configuration, illustrated in Fig. 13(a),isto rate at which species relax toward their thermal equilibrium bring a low temperature plasma jet into contact with a mac- configuration increases, which decreases the amount of a non- roscopic solid using an impinging flow. After plasma treat- equilibrium substance such as ozone. From Fig. 12,itisclear ment, the surface energy can be measured by optical contact that the concept of an effective temperature is inadequate angle analysis using a set of probe liquids and an appropriate to describe the system because the fraction of ozone produced model.41,42 For example, plotted in Fig. 13(b) is the surface is 5 orders of magnitude higher than the theoretical maximum energy of a polyether ether ketone sample before and after at equilibrium. Furthermore, since the system is already in treatment with a low temperature RF plasma jet generated its thermal equilibrium state before the reaction (pure O2 at using a mixture of air and argon at a pressure of 1 mbar for 278 K), the high fraction of O3 cannot be explained as a kinetic 2 min. The contact angles formed between liquid droplets intermediate between the initial composition and the equilib- and the polymer samples were measured via the sessile drop rium composition. It is now undeniable that plasmas can method using pure water, diidomethane, and ethylene glycol increase the specific free energy of mass flows and cause [Fig. 13(b)]. The surface energy was calculated using the changes in chemical state that move away from thermal equilib- Kitazaki-Hata model.43 Plasma treatment increased the total rium. Plasma processing can produce nonequilibrium materials. surface energy from 54 to 153 mJ m2, which was mainly caused by an increase in the polar and hydrogen bonding III. SOLID MATERIALS components [Fig. 13(c)]. Similar results have been reported Continuing on the idea of producing nonequilibrium by other researchers.44,45 Activation of polymeric surfaces materials using low temperature plasma, in this section, sit- by low temperature plasmas has been used to increase the uations are described wherein endergonic transformations of bond strength in assemblies constructed using adhe- solid materials are promoted by low temperature plasma, sives42,46–53 and also autoadhered assemblies.45,54 Beyond and a clear example is also presented of the synthesis of a polymers, plasma activation has also been used for bonding material with nonequilibrium chemical composition. semiconductors to one another, for example, wafer bonding There is tremendous interest in identifying transformations in silicon on insulator device manufacture.55,56 Moving for- promoted by low temperature plasma that increase the free ward, the increase in surface energy could have profound energy of the material being processed (i.e., endergonic consequences for the behavior of nanoparticles processed by transformations). Such transformations could be useful for low temperature plasma. When size is in the nanometer

J. Vac. Sci. Technol. B, Vol. 36, No. 4, Jul/Aug 2018 048501-11 Elijah Thimsen: Beyond equilibrium thermodynamics 048501-11 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 pres- 65,66 67 68 69,70 69 sure, for example, Si, Ge, InP, Cu2S, SnS, 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 FIG. 13. (Color online) Polymer surface activation using low temperature distribution, or high crystal quality with low defect den- plasma. (a) Flowing air-argon plasma impinging on a polymer sample. (b) sity.77,78 There are reports of processes that produce kineti- Example of an optical contact angle measurement by the sessile drop method. (c) Surface energy calculated using the Kitazaki-Hata model from contact cally stable materials that are not in their equilibrium atomic angle data measured using water, diidomethane, and ethylene glycol for a pol- configuration for any temperature at the system pressure, for yether ether ketone sample. The increase in surface energy is primarily attrib- example, synthetic crystals of diamond phase carbon.79,80 uted to increases in the polar and hydrogen bonding components, which is However, if the decomposition of the gas-phase precursor at consistent with x-ray photoelectron spectroscopy that revealed an increase in the oxygen content of the surface after plasma exposure (data not shown here). the reactor conditions is taken into account, then the overall process often results in a decrease in the specific free energy, grow macroscopic crystals of diamond by microwave plasma- despite the observation that a nonequilibrium material has enhanced chemical vapor deposition,79 but the lowest chemi- been produced. For example, methane spontaneously decom- cal potential for carbon at those conditions is graphite phase. poses at the substrate temperature and total pressure used to In other words, the gas-to-solid conversion decreases the

JVST B - Nanotechnology and Microelectronics: Materials, Processing, Measurement, and Phenomena 048501-12 Elijah Thimsen: Beyond equilibrium thermodynamics 048501-12

FIG. 14. (Color online) Size focusing of a metal aerosol by a low temperature plasma for two different mass concentrations. (a) Schematic of the experiment. A test aerosol of Bi particles was generated and passed through the low temperature plasma. Both the test aerosol and the plasma-treated aerosol were characterized by TEM, and the mass concentration was characterized by gravimetric ﬁlter measurements. For a test aerosol mass concentration of 0.54 mg m3, TEM images of (b) the test aerosol, (c) plasma treated aerosol, and (d) frequency distribution of size extracted from (b) and (c) using IMAGEJ. The parameter rg is the geometric standard deviation of lognormal ﬁts to the size distributions. A value of rg less than or equal to 1.1 is an accepted criterion for monodispersed. Panels (e)–(g) are the same characterizations as (b)–(d), but for a different mass concentration: 2.7 mg m3. Figure reprinted with permission from Uner and Thimsen, J. Phys. Chem. C 121, 12936 (2017). Copyright 2017, American Chemical Society.

speciﬁc free energy (i.e., CH4 decomposition), but the kinetics 0.5 at. %. Astonishingly, Zhou et al. have recently reported are controlled to arrest a nonequilibrium phase (i.e., diamond). synthesis of crystalline silicon nanocrystals containing up to The process is therefore exergonic. The material product can 31 at. % of boron using low temperature plasma.82 Doping be considered as an intermediate in a process that has moved of silicon nanocrystals synthesized in low temperature the feedstock closer to thermal equilibrium without reaching plasma has been recently reviewed by Ni et al.83 Here, the it, somewhat similar to the interpretation of the hydrazine focus will be on two reports, by Pi et al.84 and Zhou et al.,82 result (vide supra). A more recent example is the synthesis of which present the main experimental evidence in support of a kinetically arrested material with a nonequilibrium chemical the claim that the maximum equilibrium boron solubility has composition—hyperdoped silicon nanocrystals. been exceeded by a signiﬁcant margin. The amount of boron dissolved in silicon nanocrystals The synthesis of boron-doped Si nanocrystals has been synthesized by low temperature plasma has been reported to described by Pi et al.84 The basic concept is to feed a silicon exceed the thermodynamic limit by more than an order of precursor (e.g., SiH4) and a boron precursor (e.g., B2H6), both magnitude. The solubility of boron in silicon is plotted as a carried by an inert noble gas (e.g., Ar), into a tubular low tem- function of temperature in Fig. 16, which was reproduced perature plasma reactor [Fig. 17(a)]. In the low temperature from the data of Vick and Whittle.81 At thermal equilibrium, plasma, the precursors decompose to nucleate nanoparticles the maximum solubility of boron in silicon is approximately that subsequently grow.78 An ensemble of such nanoparticles

J. Vac. Sci. Technol. B, Vol. 36, No. 4, Jul/Aug 2018 048501-13 Elijah Thimsen: Beyond equilibrium thermodynamics 048501-13

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 SiO2 and B2O3. Strong HF solutions will dissolve B, SiO2,andB2O3, 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 SiO2 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. FIG. 15. (Color online) Plasma-treatment of a Bi aerosol increases the mass etched their silicon doped nanocrystal product to remove the contained within the narrow size range of the monodispersed peak. The mass distributions are for a test aerosol concentration of 2.7 mg m 3. The SiO2 and boron/boron oxide on the nanoparticle surfaces, they data incorporate the frequency distribution in Fig. 14(g) as well as mass con- found that atomic concentration of boron increased, suggesting centration determined by filter measurements. Figure reprinted with permis- that the boron was predominantly present in the core of the sion from Uner and Thimsen, J. Phys. Chem. C 121, 12936 (2017). 84 Copyright 2017, American Chemical Society. nanocrystals. Recent work on the infrared absorption charac- teristics of these boron doped silicon nanocrystals found fea- can be collected as the product of the plasma reaction. In gen- tures consistent with a plasmonic response, suggesting that the boron was increasing the free charge carrier concentration. eral, the product may contain nanoparticles with different Such optical behavior is expected if the boron were acting overall composition, and different radial distribution of com- as an electronic dopant, although the doping mechanism is position, which are illustrated schematically in Fig. 17(b).For unclear at present.85–88 Later characterization by x-ray photo- example, nanoparticles may be comprised of: pure silicon, electron spectroscopy and high angle annular dark field scan- pure boron, silicon-rich core with boron-rich surface (B@Si), ning transmission electron microscopy (HAADF-STEM) also boron-rich core with silicon-rich surface (Si@B), or ideally, revealed that boron was enriched in the core of the nanocrys- alloyed particles in which the boron is intimately mixed with tals compared to the surface.82 For example, presented in Fig. the silicon at the atomic scale and the composition does not 18 is an HAADF-STEM elemental map of silicon nanocrystals depend on radial position (B:Si). In other words, in the prod- hyperdoped with 31 at. % of boron.82 The crystal structure uct collected from the reaction, in general there is a distribu- became increasingly strained and disordered as the amount tion in composition among the population and not all particles of boron in the product increased. The authors reported that will contain the same fraction of boron, and in addition to the size of the silicon nanocrystals, which was characterized that, each individual nanoparticle may have a composition by transmission electron microscopy (TEM) and x-ray diffrac- that depends on position in the interior. Given these composition, did not change significantly with increasing boron content tion 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

FIG. 17. (Color online) Doping of silicon nanocrystals with boron using low temperature plasma synthesis. (a) A silicon precursor, boron precursor, and plasma gas are fed into the reactor, and a nanocrystal product is collected from the efﬂuent. The ensemble of nanocrystals in the product can have a variety of different B:Si ratios and spatial distribution of composition, which FIG. 16. (Color online) Solubility of boron in silicon at thermal equilibrium. are illustrated schematically in (b), ranging from pure Si, to pure B, boron Republished with permission from Vick and Whittle, J. Electrochem. Soc. rich surface with silicon rich core (B@Si), or vice-versa (Si@B), or ideally, 116, 1142 (1969). Copyright 1969, Clearance Center, Inc. intimately mixed boron-silicon alloy (B:Si).

JVST B - Nanotechnology and Microelectronics: Materials, Processing, Measurement, and Phenomena 048501-14 Elijah Thimsen: Beyond equilibrium thermodynamics 048501-14 from 0 to 31 at. %. At present, the observation of the insensi- The established theoretical approach for modeling low tem- tivity of size to the boron content is probably the strongest evi- perature plasmas has been kinetic modeling for the prediction dence that the boron is indeed forming an alloy with the of process outcomes. The equations that comprise such models silicon (i.e., B:Si) and not simply present as a core surrounded must comply with the laws of thermodynamics. GENERIC is by some kind of silicon shell (i.e., Si@B). The author does a beyond equilibrium thermodynamics framework that has note, however, that size is a weak function of the number of been introduced by Oettinger in 1997, and has been described atoms in a particle, and the expected change in diameter for 31 in detail in his book.22 The purpose is to rigorously ensure that at. % boron is probably less than the standard deviation of the a set of dynamical modeling equations, which are supplied size distribution. Nevertheless, the evidence presented is cer- into the GENERIC framework, comply with the laws of ther- tainly consistent with hyperdoping of silicon with boron above modynamics. The use of GENERIC will be illustrated by way the equilibrium solubility limit. of example using a two-temperature hydrodynamic model. To use GENERIC, one must already have a set of equa- IV. BEYOND EQUILIBRIUM THERMODYNAMICS tions that constitute a dynamical model. GENERIC is then A number of examples have now been presented of low used to analyze those equations. For example, to build a temperature plasmas moving systems away from equilibrium, hydrodynamic model of a fluid, one can write expressions or producing configurations of matter that are unexpected in for mass, momentum, and energy conservation. GENERIC a system governed by thermal equilibrium. Given that the can then be used to ensure that the system of equations does local thermal equilibrium assumption cannot be applied to not violate the first and second laws of thermodynamics. For a dynamical model to be thermodynamically admissible, it low temperature plasmas, how can expectations be formed 22 theoretically about the direction of the reaction? The answer must satisfy the following equation: is clearly important for design of low temperature plasma dx dEðxÞ dSðxÞ processes. For now, kinetic models of plasma processes are ¼ LðxÞ þ MðxÞ ; (6) dt dx dx probably the most reliable method to predict process outcomes. Since the criterion for chemical reaction equilibrium where x is a set independent variables required for a complete is currently unknown for low temperature plasmas, a different description of the system (e.g., mass density, momentum den- method must be used to ensure that a set of kinetic modeling sity, and internal energy density), E(x) and S(x) are the total equations complies with the laws of thermodynamics. energy and total entropy as a function of the set of variables This section has two objectives. First, an example is pro- x,andL(x)andM(x)arethePoissonandfrictionmatrices. vided for how to use the GENERIC framework to prove the The elements of the Poisson matrix L and friction matrix M thermodynamic admissibility of a set of dynamic modeling are linear operators. The Poisson and friction matrices provide equations. The advantage of using GENERIC is that thermo- different contributions to the time evolution of the set of vari- dynamic admissibility can be established without knowledge ables x, reversible changes due to energy (L), and irreversible of the state that the system is tending toward. That feature changes due to entropy (M). For a model to be thermodynam- is essential to ensure that a set of modeling equations for ically admissible, the matrices must meet several criteria. a low temperature plasma complies with the laws of thermody- The Poisson matrix must satisfy the Jacobian identity, be anti- namics, because methods of predicting, e.g., chemical reaction symmetric, and satisfy the following degeneracy criterion: equilibrium have not been developed. The second objective of LðxÞðdSðxÞ=dxÞ¼0. The criterion LðxÞðdSðxÞ=dxÞ¼0 this section is to establish the concept of superlocal equilib- expresses that the entropy is not affected by the operator that rium, and prove that it complies with the laws of thermody- generates the reversible dynamics. The friction matrix must namics, provided appropriate exchange terms are used. The also satisfy several criteria: M must be symmetric, positive superlocal equilibrium concept is expected to be critical in semidefinite, and also in the absence of external fields, must future work that will focus on deriving equilibrium criteria for satisfy the degeneracy criterion: MðxÞðdEðxÞ=dxÞ¼0. The low temperature plasma, e.g., chemical reaction equilibrium. criterion MðxÞðdEðxÞ=dxÞ¼0 expresses the conservation

FIG. 18. (Color online) Silicon nanocrystals hyperdoped with 31 at. % B. (a) HAADF-STEM image, (b) silicon elemental map and (c) boron elemental map. Reprinted with permission from Zhou et al., Part. Part. Syst. Charact. 32, 213 (2014). Copyright 2015, John Wiley & Sons, Inc.

J. Vac. Sci. Technol. B, Vol. 36, No. 4, Jul/Aug 2018 048501-15 Elijah Thimsen: Beyond equilibrium thermodynamics 048501-15 of energy in an isolated system, wherein work is transformed of external interactions (e.g., external fields), although in prin- into internal energy. In the presence of external fields, the ciple those aspects could also be included.22 friction matrix M remains the same, but the right hand side of There are three equations for each of the two species (i.e., the degeneracy criterion may no longer be zero.22 For sim- argon and electrons): mass, momentum, and energy balances. plicity, in the present analysis, the effect of external fields The equations contain the usual terms from transport phenom- will not be explicitly considered. ena,89 and in addition, also have terms that account for The use of GENERIC will be illustrated using a simple momentum and energy exchange between the different spe- model for a low temperature plasma. One of the simplest cies, which may be out of local equilibrium and have different models for a low temperature plasma is a two temperature velocities and temperatures. Several simplifying assumptions system. Electrostatic effects are neglected. The system con- were made: (1) species are neither created nor destroyed (i.e., tains heavy ideal gas species (e.g., Ar) at a temperature Tg no source term in mass balance), (2) no electrostatic effects or and partial pressure Pn, and light species (e.g., electrons) at a external fields, and (3) no chemical reactions. The system con- much higher temperature Te and volumetric concentration tains only two species: argon atoms and theoretical particles ne. The light species are theoretical particles that have the that have the mass of an electron but no electrostatic charge. mass of an electron but no electrostatic charge, and are With these simplifying assumptions, the equations reveal the treated as ideal gas particles. The state variables Tg, Pn, Te, intrinsic or natural dynamics of the two-temperature system. 13 and ne are parameters that can be obtained from experimen- The mass, momentum, and energy balances for species i are tal measurements (Fig. 3) and can be controlled using exter- dq i ¼rM~ ; (7) nal parameters such as applied power, vacuum pumping dt i speed, and external cooling or heating. No external field is dM~ considered in the model, although the presence of such an i ¼rP rðÞq ~v ~v rl dt i i i i i external source of work at some point in the history of the X 6¼ 1 system is implicit in the condition that Tg Te at the same q q ðÞ ; þ i j ij vi vj (8) location in space. Such a system is described as being in j mi mj superlocal equilibrium. The idea behind superlocal equilib- e ~ d i ¼r r r~ l r~ Cvi rðÞq ~ rium is that at a given point in space, the temperature of all ki Ti Pi vi i vi iviTi molecules of a given species is the same, e.g., all electrons dt miNA X 1 have Te and all Ar atoms have Tg, but the temperatures of þ 3k q q ðÞT T ; (9) 6¼ B ðÞþ 2 i j ij j i different species may not be the same, i.e., Tg Te. The idea j mi mj q ~ is illustrated in Fig. 19. Superlocal equilibrium has been where i is the mass density of species i, t is time, Mi is the used successfully in multifluid modeling of various low tem- momentum density of species i, Pi is partial pressure, ~vi is perature 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 FIG. 19. (Color online) Superlocal and local equilibrium. At a spatial location that is governed by superlocal equilibrium (left), all molecules of a given spe- the equations have been taken from elsewhere, they are not cies (e.g., Ar), represented by the numerical index, have the same bulk velocity rigorously derived here. Instead, GENERIC is used as a test signified by the arrows, and the same temperature signified by the shade of the of whether the assumptions used in the derivation of those circle fill (or color online). However, the bulk velocity and temperature of spe- hydrodynamic equations comply with the laws of thermody- cies 1 may differ from species 2. In a system governed by local equilibrium (right), at a given location in space, all species have the same bulk velocity and namics. The thermodynamic admissibility of the intrinsic temperature. Low temperature plasmas are proposed to be governed by superlo- dynamics of the system will be demonstrated in the absence cal equilibrium.

JVST B - Nanotechnology and Microelectronics: Materials, Processing, Measurement, and Phenomena 048501-16 Elijah Thimsen: Beyond equilibrium thermodynamics 048501-16

l velocity, mi is the single particle mass, i is the viscous stress entropy can be found by simply integrating the entropy den- tensor, ij is the frequency factor for collisions between species sity Si, whichP dependsÐÐÐ on mass density and internal energy e 2 i and species j, i is the internal energy density of species i, ki density: S ¼ ¼ dV Siðq ; eiÞ. The derivatives of ~ i 1 i is the thermal conductivity, Ti is temperature, Cvi is the mole energy and entropy with respect to the set of variables x in specific heat of i at constant volume, NA is Avogadro’s number Eq. (6) are evaluated in the following way: and k is the Boltzmann constant. The mass balance [Eq. (7)], 2 3 2 3 B d d the first three terms on the right hand side of the momentum 6 7 6 7 6 dq 7 6 dq 7 balance [Eq. (8)], and the first four terms of the energy balance 6 7 6 7 [Eq. (9)], are well-known results of transport phenomena, and dE 6 d 7 dS 6 d 7 ¼ 6 7 E and ¼ 6 7 S: 89 d 6 ~ 7 d 6 ~ 7 their description can be found in textbooks. More impor- x 6 dM 7 x 6 dM 7 tantly for the present discussion, the thermodynamic admissi- 4 d 5 4 d 5 bility of the standard terms in Eqs. (7)–(9) has been presented de de using GENERIC elsewhere,22 and so it is not necessary to @ei @S 1 gi further consider them here. The last terms of Eqs. (8) and From thermodynamics, Ti ¼ @ and @q ¼ where gi is S i mi Ti (9), on the other hand, are something interesting. The the chemical potential of species i.2 Thus, it can be shown goal here isP to establish the thermodynamic admissibility of for the two-species system that ð = þ Þq q ð Þ the term j 1 mi mj i j ij vi vj in Eq. (8),and 2 3 P 1 g 3k 1=ðm þ m Þ 2 q q ðT T Þ in Eq. (9) using 2 3 6 1 7 B j i j i j ij j i 1 6 m T 7 2 6 1 1 7 GENERIC. These terms describe the exchange of momentum 6 v1 7 6 7 6 2 7 6 0 7 and the exchange of internal energy between different species 6 v 7 6 7 6 1 7 6 1 7 that are traveling at different bulk velocities and have different d 6 7 d 6 7 E 6 1 7 S 6 T1 7 temperatures at the same location in space. The terms are ¼ 6 7 and ¼ 6 7: (11) dx 6 1 7 dx 6 1 g 7 referred to as the momentum exchange and internal energy 6 v2 7 6 2 7 6 2 2 7 6 7 exchange terms. The exchange terms describe the interaction 6 7 6 m2 T2 7 4 v2 5 6 7 of species in a system governed by superlocal equilibrium. 6 0 7 4 5 Both exchange terms represent dissipative processes that are 1 1 only present if the system is out of local equilibrium, and thus, T2 the terms tend to zero as local equilibrium is approached. ThefirststepinaGENERICanalysisistodefinethelevel The three derivative column vectors in Eqs. (10) and (11) have of description, or more specifically, the set of variables x.Since been constructed using only the level of description, i.e., the set of variables that have been chosen to describe the system, and Eqs. (7)–(9) are written in terms of mass, momentum and inter- different contributions to the total energy and total entropy of nal energy densities, these variables constitute a convenient x. the system. It is the matrices L and M that will be used to deter- For the two-temperature system containing only two species mine if the modeling equations, specifically the exchange terms 2 3 in Eqs. (8) and (9), are thermodynamically admissible. Since dq 6 1 7 the terms of interest correspond to dissipative effects that do 6 dt 7 6 7 not explicitly depend on external inputs, i.e., they are part of 2 3 6 ~ 7 q 6 dM1 7 the intrinsic dynamics of the system, they must be accounted 1 6 7 6 7 6 dt 7 for by the friction matrix M. Further discussion of this point 6 ~ 7 6 7 6 M1 7 6 e 7 may be found in the literature.22 Recall that the friction matrix 6 7 6 d 1 7 6 e 7 6 7 must meet several criteria for the equations to be thermody- 6 1 7 dx 6 dt 7 x ¼ 6 7 ! ¼ 6 7; (10) namically admissible: (1) M must be symmetric, (2) M must be 6 q 7 dt 6 dq 7 6 2 7 6 2 7 positive semidefinite, and (3) in the absence of external interac- 6 7 6 7 6 M~ 7 6 dt 7 ðd =d Þ¼ 4 2 5 6 7 tions, M must meet the degeneracy criterion: M E x 0. 6 dM~ 7 ðd =d Þ e 6 2 7 Furthermore, the product M S x should reproduce the 2 6 @ 7 6 t 7 exchange terms of interest in the rows of the column vector 4 5 associated with the momentum density and internal energy de2 dt density of species 1 and 2. The task now is to construct an M matrix and verify that it complies with its constraints. where, for example, index 1 refers to the argon atoms and The construction of the M matrix can be simplified by index 2 refers to the electrons. noting that the total M matrix is the sum of contributions of rank one, each of which by itself is expected to be symmetric The next step is to calculate the derivatives of total energy 90 of the system and total entropy of the system with respect to degenerate, and positive semidefinite the set of variables x. The total energy of the system contains M ¼ MHD þ MEX þ MMX; (12) kinetic energy associated with momentum and internal P ÐÐÐ where MHD is the friction matrix associated with the standard ¼ 2 ð = Þð ~2=q Þþe energy: E i¼1 dV 1 2 Mi i i . The total hydrodynamic terms in Eqs. (2)–(4), which has been

J. Vac. Sci. Technol. B, Vol. 36, No. 4, Jul/Aug 2018 048501-17 Elijah Thimsen: Beyond equilibrium thermodynamics 048501-17 2 3 22 comprehensively described elsewhere. MEX and MMX are the 0 6 7 friction matrices associated with the internal energy exchange 6 7 6 0 7 term and the momentum exchange term to be analyzed. To 6 7 6 1 7 maintain symmetry, the first and fourth rows, and the first and 6 3k q q ðÞT T 7 d 6 B ðÞþ 2 1 2 12 2 1 7 fourth columns of MMX and MEX areexpectedtobezerotobe S 6 m1 m2 7 MEX ¼ 6 7: (14) consistent with the mass balance equation for both species. In dx 6 0 7 6 7 other words, the energy and momentum exchange terms do 6 7 6 0 7 not affect the mass balance if no source term is included in Eq. 6 7 4 1 5 (7). The friction matrix for the internal energy exchange term 3k q q ðÞT T B ðÞþ 2 1 2 12 1 2 is a dyadic product that can be written as m1 m2 2 3 000000 6 7 6 7 An issue emerges with the momentum exchange term. 6 0000007 6 7 Despite the best efforts of the author (approximately 1 week 6 001001 7 ¼ b0 6 7; of focused effort was spent on this by both the author and H. MEX T1T26 7 (13) 6 0000007 C. Oettinger), a friction matrix MMX could not be identified 6 7 4 0000005 that reproduces only the momentum exchange term in Eq. (8) and also meets all three criteria for a friction matrix. 00100 1 However, if one allows for an additional term in the energy equation (9) that increases the internal energy of both species b0 ¼ =ð þ Þ2 q q where 3kB 1 m1 m2 1 2 12. The friction matrix due to the bulk velocity difference between them, then the MEX is symmetrical, positive semidefinite, and satisfies the momentum exchange term is thermodynamically admissible degeneracy criterion M ðdE=dxÞ¼0. Moreover, it reprodu- according to the following friction matrix, which is also a ces the internal energy exchange term exactly dyadic product.

2 3 00 000 0 6 7 6 ðÞ ðÞ 7 6 042v2 v1 0 42v2 v1 7 6 7 b 6 02ðÞv v ðÞv v 2 02ðÞv v ðÞv v 2 7 ¼ 6 2 1 2 1 1 2 2 1 7; MMX T126 7 (15) 2 6 00 000 07 6 7 4 0 42ðv1 v2Þ 042ðv1 v2Þ 5 2 2 02ðÞv2 v1 ðÞv2 v1 02ðv1 v2Þðv2 v1Þ

b ¼ð = þ Þq q where 1 m1 m2 1 2 12 and T12 is the reduced tem- Therefore, it is suggested that the inclusion of an additional perature of species 1 and 2: T12 ¼ T1T2=ðT1 þ T2Þ. The fric- term in the energy balance equation (9) is required for the tion matrix MMX meets all three criteria. When multiplied by superlocal equilibrium model to be thermodynamically the entropy derivative [see Eq. (6)], MMX reproduces the Padmissible. That additional term is proposed to be: ð = Þð = þ Þq q ð Þ 2 momentum exchange term and also produces a new term in j 1 2 1 mi mj i j ij vi vj . The new term appears the energy balance to be the discrete counterpart of the viscous dissipation term of hydrodynamics. With the inclusion of this term, the inter- 2 3 nal energy density equation has the following form: 0 6 7 6 1 7 6 q q ðv v Þ 7 e ~ 6 þ 1 2 12 2 1 7 d i ¼r r r~ l r~ Cvi rðÞq ~ 6 m1 m2 7 ki Ti Pi vi i vi iviTi 6 7 dt miNA 6 1 1 q q ðÞ 2 7 X 6 1 2 12 v2 v1 7 1 dS 6 2 m þ m 7 þ3kB q q ijðÞTj Ti ¼ 6 1 2 7: ðÞm þ m 2 i j MMX d 6 7 (16) j i j x 6 0 7 X 6 1 1 7 þ 1 1 q q 2: 6 q q ð Þ 7 i j ijðÞvi vj (17) 6 1 2 12 v1 v2 7 2 m þ m 6 2 m þ m 7 j i j 6 1 2 7 4 5 1 1 q q ðÞ 2 þ 1 2 12 v2 v1 The conclusion of this GENERIC analysis is that the con- 2 m1 m2 cept of superlocal equilibrium is thermodynamically admissible

JVST B - Nanotechnology and Microelectronics: Materials, Processing, Measurement, and Phenomena 048501-18 Elijah Thimsen: Beyond equilibrium thermodynamics 048501-18 if appropriate terms are included that describe the exchange of 10N. J. Kramer, R. J. Anthony, M. Mamunuru, E. S. Aydil, and U. R. 47 momentum and heat between species that have different Kortshagen, J. Phys. D , 075202 (2014). 11J. E. Daugherty and D. B. Graves, J. Vac. Sci. Technol., A 11, 1126 momentum density and temperature at the same location in (1993). space. The task for future work will be to use the superlocal 12M. Capitelli, Fundamental Aspects of Plasma Chemical Physics Kinetics equilibrium concept to derive equilibrium criteria for low tem- (Springer, New York, 2016), p. 318. 13M. J. Kushner, J. Phys. D 42, 194013 (2009). perature plasma, e.g., chemical reaction equilibrium. 14R. Le Picard, A. H. Markosyan, D. H. Porter, S. L. Girshick, and M. J. Kushner, Plasma Chem. Plasma Process. 36, 941 (2016). V. 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JVST B - Nanotechnology and Microelectronics: Materials, Processing, Measurement, and Phenomena