Disorder-induced heating as a mechanism for fast neutral gas heating in atmospheric pressure plasmas

Recent findings suggest that ions are strongly correlated in atmospheric pressure plasmas if the ionization fraction is sufficiently high ( ≳10−5 ). A consequence is that ionization causes disorder-induced heating (DIH), which triggers a significant rise in ion temperature on a picosecond timescale. This is followed by a rise in the neutral gas temperature on a longer timescale of up to nanoseconds due to ion–neutral temperature relaxation. The sequence of DIH and ion–neutral temperature relaxation suggests a new mechanism for ultrafast neutral gas heating. Previous work considered only the case of an instantaneous ionization pulse, whereas the ionization pulse extends over nanoseconds in many experiments. Here, molecular dynamics simulations are used to analyze the evolution of ion and neutral gas temperatures for a gradual ionization over several nanoseconds. The results are compared with published experimental results from a nanosecond pulsed discharge, showing good agreement with a measurement of fast neutral gas heating.


Introduction
Fast gas heating refers to a rapid increase in gas temperature commonly observed in non-equilibrium low-temperature plasmas at or above atmospheric pressure [1][2][3][4][5][6][7][8].It is commonly thought to stem from the electron kinetic energy via the relaxation of electronically excited states of atoms and molecules * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.[1,2], or from the dissociation of molecules [9,10].In plasma assisted combustion, the abrupt temperature increase induces an acceleration of combustion chemistry, thus shortening the ignition delay time [2].Therefore, control over gas heating is pivotal for successful ignition [4].In plasma aerodynamics, the primary effect of a plasma discharge on flow is often linked to gas heating in the discharge area, underscoring the significance of an accurate description of fast gas heating [11,12].Employing various discharges for plasma flow control, the most efficient actions generally involve the rapid release of thermal energy at specific points within the discharge area [1,2].Each of these examples highlight the need to understand the mechanisms underlying fast gas heating and the timescales on which they operate.
Gas heating mechanisms in low-temperature plasmas have been studied in the past [1,[3][4][5][6][7][8][9][10][13][14][15][16][17].In air, electronically excited states contribute through the spontaneous dissociation or quenching of electronically excited nitrogen molecules, as identified by Popov in 2011 [1].In particular, dissociation reactions due to electron impact of O 2 and N 2 molecules, and processes of quenching of electronically excited N 2 molecules by oxygen and excited O atoms by nitrogen, can be noted as significant [1,10].Vibrationally excited states also contribute to gas heating due to non-thermal electron impact excitation storing energy in the vibrational modes of nitrogen molecules, which is then released through vibrationtranslation and vibration-vibration relaxations [6,[15][16][17].Energy exchange through electron-neutral elastic collisions further enhances fast gas heating in plasma discharges once the ionization fraction is high enough [3].All of these atomic physics mechanisms are described by a firm theoretical and experimental basis [1,2].Here, we describe a fundamentally different physical mechanism that should also be considered.Following previous work [18], we demonstrate that disorderinduced heating (DIH) of ions followed by ion-neutral temperature relaxation significantly increases the neutral gas temperature when a large ionization degree of the plasma is reached, independently of the ionization rate.Specifically, the neutral gas temperature increase obtained from molecular dynamics (MD) simulations with an ionization ramp over a timescale of several nanoseconds shows good agreement with a measurement of fast gas heating in a nanosecond pulse discharge in atmospheric air by Lo et al [4].This suggests that DIH is a mechanism for fast gas heating that should be considered alongside the previously studied atomic physics mechanisms.
DIH occurs if the ion-ion interactions are strongly correlated following ionization [18][19][20][21].The ion-ion correlation strength can be quantified by the Coulomb coupling parameter where Z is the ion charge state, T i is the ion temperature and a ii = (3/4π n i ) 1/3 is the mean ion separation.If Γ ii ≫ 1 after ionization, ions will reconfigure to a correlated lower potential energy state [22] at a timescale of the ion plasma period, which corresponds to picoseconds in atmospheric pressure plasmas with ion densities ⩾10 23 m −3 .This causes the ion temperature to rise.Following this, ion-neutral temperature relaxation through collisions occurs, causing the neutral gas temperature to rise on a nanosecond timescale.This heating effect is pronounced at large ion densities, elevating the neutral gas temperature by hundreds to thousands of Kelvin at ionization fractions larger than 10 −2 at atmospheric pressure.DIH and the subsequent ion-neutral 'equilibrium' temperature have previously been studied assuming instant ionization of a gas [18].This work also provided a model to determine the temperature at which ions and neutrals equilibrate after DIH.This concept is grounded in energy conservation arguments, as expressed by where T n (t = 0) is the neutral gas temperature before plasma formation, x i = n i /n is the ionization fraction after plasma formation, and T max i is the maximum ion temperature obtained from This model demonstrated good agreement with MD simulations of discharges at atmospheric pressure after an instantaneous ionization [18].DIH was found to be significant if the ionization fraction exceeds x i ≳ 10 −2 for a gas initially at atmospheric pressure.
Here, we extend this study to the more common case of an ionization ramp that is slow compared to the picosecond timescale of DIH following an instantaneous ionization pulse.Results show that since DIH is a consequence of conservation of energy, from a thermodynamics standpoint, the total increase in temperature does not depend on the ionization rate.Instead, it relies solely on the initial and 'final' potential configurations of the system, which are determined from the ionization fraction and pressure of the discharge.These findings offer a refined understanding of the underlying mechanisms, showing that when ionization occurs on a nanosecond timescale or slower, the evolution of the neutral gas temperature due to DIH closely follows the ion density profile.
In order to validate these predictions, we compare MD simulation results for a gradual ionization ramp with experimental measurements of neutral gas temperature in an atmospheric pressure air nanosecond pulsed discharge from Lo et al [4].The good agreement between the simulated evolution of the neutral gas temperature and experimental measurements suggests that DIH followed by ion-neutral temperature relaxation has a significant influence on neutral gas heating in these experiments.A conclusion is that DIH should be considered alongside atomic reaction-based mechanisms for fast gas heating.This novel gas heating mechanism stands out as the fastest reported to date.

Setup
MD simulations were run using the open-source software LAMMPS [23].The configuration employed builds upon that described in previous work [18], but with the assumption of gradual ionization.Since electrons in atmospheric pressure plasmas are typically much hotter than ions and exhibit weak coupling, they were treated as a non-interacting background species for modeling the dynamics of ions and neutrals and were thus excluded from the simulation [18].Partially ionized monatomic N gas was simulated considering short (neutralneutral), medium (ion-neutral) and long (ion-ion) range interactions using the Lennard Jones, charged induced dipole and Coulomb potentials respectively, where ϵ = 99.8kB , σ = 0.3667 nm, α R = 7.5 for N [24] and a 0 is the Bohr radius.The particle-particle particle-mesh (P3M) method was used to simulate ion-ion interactions [25].This algorithm consists of an Ewald summation where the potential is calculated through a direct sum for particles that are closer than a cutoff distance, and through the particle mesh method for particles that are further apart.In the particle mesh method, the charges of ions are interpolated onto a grid where the Poisson equation is solved in the Fourier space assuming periodic boundary conditions, the result is then interpolated back to the positions of each particle.A separation distance of r c = 10 a ii was selected for differentiating the short and longrange parts of the Ewald summation, and a cutoff distance of 5σ was applied for the Lennard-Jones potential.To prevent close interactions that require a short timestep due to the attractive charge-induced dipole potential, we used a repulsive core with radius r ϕ .Previous work showed that a choice of r ϕ ≈ 0.133 a in at atmospheric pressure blocks the three-body recombination that is unable to be simulated in a classical MD simulation.Three-body recombination is physically expected to influence plasma dynamics on a longer timescale than is considered here [18].A cutoff distance of 5a in , where a in represents the average ion-neutral spacing, was used for the ionneutral direct-force calculations.
To study the evolution of a non-equilibrium discharge with an arbitrary ionization rate, a neutral monatomic N gas at room temperature and atmospheric pressure was simulated until equilibrium was reached as delineated in [18].This stage of the simulation was run with a Nosé-Hoover thermostat (NVT stage) followed by an NVE simulation once room temperature was reached [25].Here, NVE refers to the constant total number of particles (N), Volume (V) and total energy (E), equivalent to the microcanonical ensemble.The total energy remained constant during each run of the NVE stage, with the only exception that every time neutrals were converted to ions, the potential energy was affected due to the change in the interaction potential.In contrast, NVT refers to the constant number of particles (N), volume (V) and temperature (T), equivalent to the canonical ensemble.In this thermostating stage, energy is not conserved and it is necessary prior to the NVE simulation to have the gas at the desired temperature at equilibrium.After the first NVE stage, a second NVE simulation was conducted, including neutral-neutral, ion-neutral, and ion-ion interactions with the potentials detailed in equations (4a)-(4c).In this stage, the ion density n i (t) was an input parameter.The timestep used was 5 × 10 −4 (ω max pi ) −1 , where ω max pi is the maximum plasma frequency corresponding to the maximum ion density n max i .The total number of particles was 50 000.The simulation domain was a three-dimensional box with periodic boundary conditions, and the volume was set to ensure that at the initial room temperature and with 50 000 atoms, the total gas density was approximately 2.5 × 10 25 m −3 , corresponding to atmospheric pressure.
In this study, we compare our MD simulations with a nanosecond pulsed discharge experiment conducted in atmospheric air, as reported by Lo et al [4].The experimental setup involved the generation of a discharge via a positive high-voltage pulse applied to pin-to-pin electrodes over a duration of 15 ns.Within this time frame, a streamer phase emerged, propagating between the electrodes, which subsequently transitioned into a spark phase characterized by low voltage and high current.Optical emission spectroscopy measurements indicated an ultra-fast gas heating, reaching temperatures up to 1200 K at 15 ns post-current rise [4].In addition, the electron density was measured at the end of the streamer phase to be 9.2 × 10 24 m −3 .Our analysis is particularly concentrated on this initial streamer phase.
To replicate these experimental conditions in the MD simulations, we employed the following method during the second NVE stage.A subset of neutral atoms was randomly ionized at intervals of every n t timesteps, resulting in the conversion of approximately 10 neutral atoms into ions per ionization event.The value of n t was dynamically adjusted to achieve a desired ion density profile, denoted as n i (t).It is crucial to note that we have intentionally omitted certain heating mechanisms commonly believed to contribute to gas heating in order to isolate the effect of DIH and assess its potential impact on the neutral gas temperature.Specifically, our simulations employed a monatomic nitrogen gas model, thereby neglecting dissociation processes.Furthermore, quenching of electronically excited species was not accounted for and no external electric field was included.In addition, no electron-ion recombination process was included due to the slower timescale when compared to the timescale of the streamer phase simulated.
Figure 1(a) illustrates three distinct ionization profiles, x i (t) = n i (t)/n, used in the simulations: an instant ionization profile, a linear ionization profile, and an ionization profile proportional to the deposited energy measured in the experiment reported by Lo et al [4].The maximum ion density simulated was 9.2 × 10 24 m −3 (x i ≈ 0.368), aligning with the maximum electron density documented in [4].

Results
Figure 1(b) illustrates the evolution of ion and neutral gas temperatures for each of the ionization profiles.In the cases of gradual ionization, the ion temperature exhibits a peak on a ns timescale.This is associated with ionization events that cause DIH on a timescale that is shorter than the ion-neutral relaxation timescale.Following this initial surge, the ion temperature declines as the ion and neutral temperatures approach one another on a timescale of a few nanoseconds.At later times, the ion and neutral temperatures are approximately equal, and rise at the same rate.This incremental rise is attributed to DIH, occurring continuously with ion-neutral temperature relaxation as the neutral gas undergoes ionization during the simulation.Ionization proceeds at a much slower pace compared to DIH and the ensuing ion-neutral temperature relaxation.Therefore, both ions and neutrals elevate their temperature according to the same temperature profile, defined by the ionization fraction x i (t).The dynamics of ion and neutral temperatures contrasts with the instantaneous ionization case, assumed in [18], where for the same ionization fraction ions heat much faster, on a ps timescale to a maximum temperature of ∼2800 K.Then, ions cool through ion-neutral collisions, increasing the neutral gas temperature on a ∼ns timescale.
It is noteworthy that the moment the ionization ceases (at 15 ns for the gradual ionization profiles) the temperatures of the ion and neutral gases align, reaching a state referred to as 'equilibrium'.The 'equilibrium' temperature between ions and neutrals, discerned in figure 1(b), is measured at 1275 K and does not depend on the ionization dynamics, since all the simulated n i (t) profiles reach the same final temperature.This is in agreement with the model described in equation ( 2), which forecasts an equilibrium temperature of 1272 K. Beyond the 15 ns mark, the simulations persists without alterations in the ionization degree, and no further changes in the temperature are observed.It is noteworthy that although the three different MD simulations have different ionization timescales, they all converge to the same 'equilibrium' temperature.The congruence of the final temperature for all the simulations and its accord with equation (2) highlights that the total energy released by DIH is independent of the ionization dynamics, depending only on the initial conditions and final ionization state.

Comparison with experiment
Figure 2 presents a comparison between the neutral gas temperature obtained from MD and the published measurements from Lo et al [4].MD results are shown for both of the gradual ionization profiles depicted in figure 1(b).The final 'equilibrium' temperature from the MD simulations aligns well with the experimental results.The simulations also agree with the experimental measurements of the temperature evolution, extending from the initiation of the discharge up to 15 ns.Moreover, the temperature determined using an ionization profile founded on the deposited energy profile measured experimentally, demonstrates good agreement with the measured temperature [4].
It is important to highlight that the MD simulations did not incorporate heating mechanisms related to plasma chemistry, such as spontaneous dissociation or quenching of electronically excited molecules, nor did they include the external electric field, elastic electron-neutral collisions or electron-ion recombination.Furthermore, the simulation domain consisted of a small periodic box with a total number of atoms and ions of 50 000, and thus excluded geometric effects.The simulation setup rests exclusively on thermodynamic arguments and it still is able to recover the experimentally observed temperature profile, including only interactions between ion and neutral atoms through the potentials described in equations (4a)-(4c).Conversely, Lo et al [4] attribute the detected gas heating in the streamer phase to the well-recognized quenching of electronically excited species by oxygen molecules.However, the commonly assumed mechanism is more likely to have occurred over 10's of ns and was thus a significant mechanism for spark-timescale temperature increases but not the streamer initiation-timescale heating.This suggests that DIH and ionneutral temperature relaxation might constitute an essential ultra-fast heating mechanism, one that has been overlooked in the current analysis of cold atmospheric pressure plasmas.
In Lo et al [4], the discharge transitioned into a spark after 15 ns, with the gas heating persisting over an extended timescale.This concurs with the timescale argument employed here and in our previous work [18], suggesting that while alternative heating mechanisms commonly found in partially ionized plasmas still occur, DIH can significantly influence the Evolution of the neutral gas temperature from molecular dynamics, using the gradual ionization fraction profiles shown in figure 1(a) and from a nanosecond pulsed discharge in atmospheric air [4].
ion and neutral gas temperatures on a much shorter timescale and should be incorporated in future analyses.
While only a monatomic gas was simulated in the streamer phase it is expected that molecules play an important role in this stage of the discharge.In this context, it was recently found that the presence of molecules leads to a somewhat lower 'equilibrium' temperature due to the larger number of degrees of freedom in the plasma [26].Hence, if molecules were included here, the amount of potential energy released during DIH would be distributed among translational and rotational degrees of freedom leading to a smaller neutral gas temperature.
It is noteworthy that DIH produces a considerable increase in the ion temperature on a short, picosecond, timescale.This rapid increase in the temperature leads to a considerable increase in the gas pressure.In a real plasma, this increase in pressure would lead to an expansion of the plasma.However, we would expect this to happen on a much longer timescale than DIH or the ionization stage of this discharge.

Conclusions
This work carried out MD simulations of the neutral gas temperature evolution in a partially ionized atmospheric pressure plasma, offering a comparison with experimental data from a nanosecond discharge in atmospheric air [4].The simulations modeled an ionization profile that matched the measured energy deposition profile and ranged from an ionization fraction of x i = 0-0.368,corresponding to the experimentally measured ion density of 9.2 × 10 24 m −3 , after 15 ns.The simulation results were found to align well with the final neutral gas temperature observed in the experiment.They also captured the temporal evolution of the gas temperature through the discharge pulse.The main conclusion is that DIH of ions followed by ion-neutral temperature relaxation is a viable mechanism for ultrafast gas heating in these experiments.The findings also reinforce that the total energy released by DIH is independent of the ionization dynamics or the gas composition.No additional gas heating mechanisms beyond DIH were included.Hence, DIH and the subsequent ion-neutral temperature relaxation should be accounted for in the analysis of fast gas heating in atmospheric pressure plasmas.

Figure 1 .
Figure 1.(a) Ionization fraction profiles used in the MD simulations.(b) Evolution of ion and neutral temperatures from MD simulations for each ionization fraction profile shown in (a).Continuous lines represent the ion temperature while dashed lines represent the neutral gas temperature.A vertical dashed line marks the instant at which ionization stops for the cases of gradual ionization.

Figure 2 .
Figure 2. Evolution of the neutral gas temperature from molecular dynamics, using the gradual ionization fraction profiles shown in figure1(a) and from a nanosecond pulsed discharge in atmospheric air[4].