Availability and reactivity of N₂(v) for NH₃ synthesis by plasma catalysis

An RF-driven atmospheric pressure plasma jet (inner diameter = 2 mm, outer diameter = 3 mm) coupled with a packed bed of metal wools downstream of the plasma was used in this study. The plasma jet was driven at 12.8 MHz with two duty cycle modulations, one at 20 kHz with a 50% duty cycle and one at 50 Hz with a 50% duty cycle to allow for species densities background subtraction in the MBMS. The volumetric gas flow rate was kept at 16.67 cm³ s⁻¹, referenced to standard temperature and pressure (1 bar, 298 K) throughout this study. The Ar/N₂/H₂ flow was passed through an O₂/H₂O trap (Agilent, OT3-4) to remove impurities before entering the plasma jet. More details about the setup and operation of the jet are described in prior work [15].

The reactor setup is displayed schematically in figure 1. The distance from the end of the cylindrical ground electrode to the nozzle (end of the quartz tube) was varied from 5 to 15 mm. The sampling orifice of the MBMS was placed adjacent (<0.2 mm) to the nozzle to sample the gas composition at the effluent of the reactor. When catalyst was used, the distance between the end of the ground electrode and the catalyst bed was kept constant at 5 mm, such that MBMS measurements made 5 mm from the ground to the nozzle are representative of species densities at the inlet of the catalyst bed. The length of the catalyst bed was varied from 2–10 mm with 0.0024 g mm⁻¹ Fe, 0.0036 g mm⁻¹ Ni, or 0.0033 g mm⁻¹ Ag wool loaded as catalyst.

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Fe (99.5% purity by EDS), Ni (99.6% purity by EDS), and Ag (97.1% purity by EDS) metal wools (EA consumables) are used as catalysts. Scanning electron microscopy was used to determine that the diameter of the metal wool fibers was ∼70 μm (as displayed in our prior work) [15]. Surface areas of the fibers were calculated based on the assumption that fibers were cylinders with infinite length.

The experimental MBMS setup and methods for detection, background subtraction, ionizer electron energy calibration, and determination of response factors for N, H, and NH₃ are described in detail in previous work [15]. MBMS signal was quantified when the signal reached steady-state within the 50 Hz modulation cycle, and densities of species were not observed to change over longer (seconds-minutes) timescales with or without the 50 Hz modulation. Error bars associated with reported species densities or MBMS signals represent 95% confidence intervals that arise from the acquisition of MBMS signal.

Densities of N₂(v) predicted by kinetic models are validated by the procedure outlined by Jiang et al [16] and are summarized below. MS signal is measured for m/z = 28 (N₂) for MS ionizer energies between 12.1 and 15.6 eV, values at or below the threshold energy for ground-state N₂ ionization. Background-subtracted signals $\left( {S_{{\text{N}}_2^ + }^{{\text{plasma on}}}\left( {{E_{{\text{el}}}}} \right) - S_{{\text{N}}_2^ + }^{{\text{plasma off}}}\left( {{E_{{\text{el}}}}} \right)} \right)$ were normalized to the signal measured at 15.6 eV for each operating condition as described in equation (1) to yield the normalized signal at the given electron energy (${S_{{\text{exp, norm}}}}\left( {{E_{{\text{el}}}}} \right)$):

$$\begin{align}{S_{{\text{exp, norm}}}}\left( {{E_{{\text{el}}}}} \right) = \frac{{S_{{\text{N}}_2^ + }^{{\text{plasma on}}}\left( {{E_{{\text{el}}}}} \right) - S_{{\text{N}}_2^ + }^{{\text{plasma off}}}\left( {{E_{{\text{el}}}}} \right){ }}}{{S_{{\text{N}}_2^ + }^{{\text{plasma on}}}\left( {15.6{\text{ eV}}} \right) - S_{{\text{N}}_2^ + }^{{\text{plasma off}}}\left( {15.6{\text{ eV}}} \right){ }}}.\end{align} \tag{ 1 }$$

Background subtraction was enabled by modulating the plasma at a frequency of 50 Hz, which allowed for the measurement of the plasma-on signal and the background signal with a multichannel scaler in a single measurement. Experimentally measured signal is compared to densities of individual vibrational levels predicted by the state-to-state kinetic model, based on the proportionality of mass spectrometer signal to the summation of ionization cross sections of individual vibrational levels at the given electron energies (${\sigma _{iz,v = i}}\left( {{E_{{\text{el}}}}} \right)$) multiplied by the density of the corresponding individual vibrational level $\left( {{n_{{{\text{N}}_2}\left( {v = i} \right)}}} \right)$ as shown in equation (2):

$$\begin{equation}S_{{\text{N}}_2^ + }^{{\text{plasma on}}}\left( {{E_{{\text{el}}}}} \right) \propto \mathop {{{\mathop \sum \nolimits}}}\limits_{i = 0}^{i = n} {\sigma _{iz,v = i}}\left( {{E_{{\text{el}}}}} \right){n_{{{\text{N}}_2}\left( {v = i} \right)}}.\end{equation} \tag{ 2 }$$

Ionization cross sections are obtained from NIST [24]. Ionization cross sections for N₂(v) are assumed to be the same as ionization cross sections for ground state N₂ with the electron energy shifted by the vibrational energy as described in previous work [15]. Ionization cross sections were corrected to account for the finite electron energy width of the ionizer as described previously [15]. It has been shown that electronically excited N₂ does not contribute to the plasma-on signal for similar operating conditions as those investigated in this work [16]. For the case of the plasma-off signal, only ground state nitrogen contributes to the signal, as shown in equation (3):

$$\begin{equation}S_{{\text{N}}_2^ + }^{{\text{plasma off}}}\left( {{E_{{\text{el}}}}} \right) \propto {\sigma _{iz,v = 0}}\left( {{E_{{\text{el}}}}} \right){n_{{{\text{N}}_2}}}.\end{equation} \tag{ 3 }$$

Equations (2) and (3) can be combined and normalized to relate normalized, background subtracted signal with the densities of N₂(v) in the sampled gas, as shown in equation (4):

$$\begin{align}&\frac{{S_{{\text{N}}_2^ + }^{{\text{plasma on}}}\left( {{E_{{\text{el}}}}} \right) - S_{{\text{N}}_2^ + }^{{\text{plasma off}}}\left( {{E_{{\text{el}}}}} \right)}}{{S_{{\text{N}}_2^ + }^{{\text{plasma on}}}\left( {15.6\,{\text{eV}}} \right) - S_{{\text{N}}_2^ + }^{{\text{plasma off}}}\left( {15.6\,{\text{eV}}} \right)}}\,\nonumber\\&\quad = \frac{{\mathop \sum \nolimits_{i = 0}^{i = n} {\sigma _{iz,v = i}}\left( {{E_{{\text{el}}}}} \right){n_{{{\text{N}}_2}\left( {v = i} \right)}} - {\sigma _{iz,v = 0}}\left( {{E_{{\text{el}}}}} \right){n_{{{\text{N}}_2}}}}}{{\mathop \sum \nolimits_{i = 0}^{i = n} {\sigma _{iz,v = i}}\left( {15.6\,{\text{eV}}} \right){n_{{{\text{N}}_2}\left( {v = i} \right)}} - {\sigma _{iz,v = 0}}\left( {15.6\,{\text{eV}}} \right){n_{{{\text{N}}_2}}}}}.\end{align} \tag{ 4 }$$

Normalized, background subtracted mass spectrometer signal can be predicted at a given ionizer electron energy using the obtained densities of N₂(v) from the state-to-state kinetic model as shown in equation (5) to obtain a kinetic-model predicted MS signal (${S_{{\text{model, norm}}}}\left( {{E_{{\text{el}}}}} \right)$):

$$\begin{align}&{S_{{\text{model, norm}}}}\left( {{E_{{\text{el}}}}} \right)\nonumber\\&\quad = \frac{{\mathop \sum \nolimits_{i = 0}^{i = n} {\sigma _{iz,v = i}}\left( {{E_{{\text{el}}}}} \right){n_{{{\text{N}}_2}\left( {v = i} \right),{\text{model}}}} - {\sigma _{iz,v = 0}}\left( {{E_{{\text{el}}}}} \right){n_{{{\text{N}}_2}}}}}{{\mathop \sum \nolimits_{i = 0}^{i = n} {\sigma _{iz,v = i}}\left( {15.6\,{\text{eV}}} \right){n_{{{\text{N}}_2}\left( {v = i} \right),{\text{model}}}} - {\sigma _{iz,v = 0}}\left( {15.6\,{\text{eV}}} \right){n_{{{\text{N}}_2}}}}}.\end{align} \tag{ 5 }$$

The experimental signal is normalized per equation (1) and compared to the signal predicted by the state-to-state kinetic model in equation (5) to assess the accuracy of the VDF predicted by the kinetic model.

The N₂ VDF for different experimental conditions is determined with a quasi-zero-dimensional model that simultaneously solves the electron energy equation, the heavy species (e.g. Ar, N₂, H₂, N, H, etc) energy equation, and the equations for individual species balances using the ANSYS Chemkin-Pro software. The model incorporates electron impact reactions [5, 6], including vibrational excitation, electronic excitation, dissociation, and ionization; state-specific vibration–vibration (VV) energy transfer for N₂–N₂ [25] and vibration–translation (VT) relaxation for N₂–Ar [26, 27], N₂–N₂ [26, 27], N₂–N [28, 29], N₂–H₂ [30], and N₂–H [30]; chemical reactions among radical species [30–32]; energy transfer involving excited electronic states of Ar and N₂ [30, 32–39]; and ion recombination reactions [30, 32, 37, 40] to describe the evolution of species densities and the VDF. Rates for electron impact reactions at different electron temperatures were determined by solving the Boltzmann equation using Bolsig for the inlet composition of the plasma jet [41]. Functional forms of the rate constants used in the state-to-state kinetic model are described in the supporting information in section S1.

Collisional processes between N₂(v) and NH_x species were not considered in this model, since these species densities are orders of magnitude lower than the species that most contribute to VT relaxation, and their VT rate coefficients are not expected to be larger than the VT rate coefficients of other species by orders of magnitude (see section 3.1). Vibrational excitation of H₂ was not considered in the kinetic model because its inclusion is not expected to significantly alter N₂(v) and H densities. H₂(v) densities are expected to be low, since rates of VT relaxation of H₂(v) by H are >1000× faster than VV energy transfer from N₂ to H₂ due to the differences in their single vibrational quantum energies [30, 42]. Because of the fast VT relaxation of H₂(v), vibrationally enhanced H₂ dissociation is estimated to be negligible in comparison to dissociation of H₂ by electron impact or dissociation of H₂ from quenching of excited states of N₂ or Ar [36, 43, 44]. Rate constants for VV energy transfer from N₂ to H₂ are lower than VV energy transfer from H₂ to N₂ because H₂ has a higher single vibrational quantum energy than N₂ [22]. As a result, VV energy transfer tends to depopulate H₂(v) and populate N₂(v), so it is not expected that VV energy transfer from N₂ to H₂ will depopulate N₂(v) more than VT relaxation.

Inputs into the model include the inlet gas composition, inlet gas and electron temperature, flow rate, reactor geometry, radial heat transfer coefficient, and power deposition profile (displayed in section S2 of the supporting information). The plasma volume is considered to be the volume between the needle electrode and the ground electrode plus the region of emission beyond the electrodes. The total power deposited into the plasma is equal to the experimentally measured power. The volumetric power density input from the beginning of the plasma to the end of the ground electrode is constant, and then drops linearly from the end of the ground electrode to the end of the visible emission of the plasma jet (figure S1(a)) [16]. The power deposition profile in the model neglects the 20 kHz modulation cycle used in experiments but operates with the same average power as in experiments, since models of RF atmospheric pressure plasma jets demonstrate that the 20 kHz modulation cycle does not significantly alter species densities measured experimentally or predicted by the model but does significantly increase the computation time [45].

The output of the model includes species densities, gas temperature, and electron temperature as a function of the gas residence time or axial position. These outputs are displayed in the supporting information (section S2). A small but finite power deposition in the afterglow is necessary to enable Chemkin to solve the electron energy equation in the afterglow. Simulations that compare species densities and temperature outputs with increasing power deposition in the afterglow show that these outputs do not significantly change as the power deposition in the afterglow varies, indicating that the power deposition does not significantly impact the results and conclusions of this study (figure S3). The heat transfer coefficient is estimated to match the gas temperature predicted by the model with the experimentally measured gas temperature at the 3.5 cm axial position, although simulations with different heat transfer coefficients show that the variation of this parameter does not significantly alter species densities predicted by the model (figure S4).

Loss pathways for N₂(v) in the catalyst bed were analyzed through inclusion of surface-mediated vibrational relaxation into a subset of the previously described gas-phase reaction set that considers N₂ VV and VT processes. Inlet N₂(v) densities were taken from the plasma kinetic model at the axial position of the start of the catalyst bed and densities of other species were taken from MBMS measurements. The gas temperature was determined from thermocouple measurements with continuous operation of the plasma, and the surface temperature was estimated to be the average gas temperature when a 50 Hz modulation cycle is included, as described in previous work [15]. Decay of N and H in the catalyst bed was fitted to an exponential function that passes through the densities of N and H at the inlet and the detection limits of these species (1.5 × 10¹³ cm⁻³ and 5 × 10¹⁴ cm⁻³, respectively) at 2 mm of catalyst bed, although this assumption does not significantly influence the N₂ VDF because neither of these species are the dominant quenchers of N₂(v) at the investigated operating conditions (see section 3.1).

Vibrational relaxation on the surface is modeled according to the approach of Marinov et al [20], which uses collision theory to describe collisions between N₂(v) and the surface and loss coefficients (${\gamma _{{\text{loss}}}}$) to express the probability of vibrational relaxation when N₂(v) collides with the surface. Loss coefficients are assumed to scale linearly with vibrational level (v). The rationale behind this assumption is discussed by Marinov et al [20] We use a value of ${\gamma _{{\text{loss}},v = 1}}{ }$ ≈ 0.001–0.0026 as measured by Black et al [19] over stainless steel and Cu. Equation (6) displays the rate constant (${k_{v \to v - 1,{\text{surface}}}}$) for vibrational relaxation of N₂(v) on a surface, which is the thermal flux to the surface ($\frac{1}{4}{v_{{\text{th}},{{\text{N}}_2}}}$) multiplied by the loss probability:

$$\begin{equation}{k_{v \to v - 1,{\text{surface}}}} = \frac{1}{4}{v_{{\text{th}},{{\text{N}}_2}}}\left( {v{\gamma _{{\text{loss}},v = 1}}} \right).\end{equation} \tag{ 6 }$$

The rate of N₂(v) loss by surface-mediated vibrational relaxation (${r_{v \to v - 1,{\text{surface}}}}$) is written as shown in equation (7):

$$\begin{equation}{r_{v \to v - 1,{\text{surface}}}} = \chi {k_{v \to v - 1,{\text{surface}}}}{n_{{{\text{N}}_2}\left( v \right)}}\frac{{SA}}{V}.\end{equation} \tag{ 7 }$$

$\frac{{SA}}{V}$ represents the surface area of the fibers per volume of reactor, and ${n_{{{\text{N}}_2}\left( v \right)}}$ is the number density of vibrationally excited N₂ of level v. The $\chi$ term in equation (7) captures the effect that external mass transfer has on rates of quenching [46]. $\chi$ is defined in equation (8), with the mass transfer coefficient for N₂ (${k_g}$) calculated by methods described in previous work [15]:

$$\begin{equation}\chi = \frac{{{k_g}}}{{{k_g} + {k_{v \to v - 1,{\text{surface}}}}}}.\end{equation} \tag{ 8 }$$

NH₃ is proposed to be formed from N₂(v) by dissociative adsorption of N₂(v) over transition metal surfaces to form N adatoms, which then undergo hydrogenation and desorption to form NH₃ [8]. Rate constants for dissociative adsorption of N₂(v) are calculated according to the approach used by Mehta et al [8] The rate constant for dissociative adsorption consists of the pre-exponential factor (A) multiplied by an exponential factor that accounts for an activation barrier modified by the vibrational energy of N₂(v), which is shown in equation (9):

$$\begin{equation}{k_{{\text{ads}},v}} = A{e^{\left( { - \frac{{{E_{\text{a}}} - \alpha {E_{\text{v}}}}}{{{\text{RT}}}}H\left( {{E_{\text{a}}} - \alpha {E_{\text{v}}}} \right)} \right)}}.\end{equation} \tag{ 9 }$$

The exponential factor changes as a function of vibrational level through subtraction of the activation energy for ground-state N₂ dissociative adsorption (E_a) by the vibrational energy (E_v) of the corresponding vibrational level multiplied by a Fridman–Macheret parameter (α). DFT-derived values for N₂(v) dissociative adsorption activation energies are computed using the scaling relations reported by Mehta et al [8] using dissociative adsorption energies for N₂ on terrace sites reported in the CatApp database [47], since terrace sites should be the most abundant sites on the metal wools with 70 μm diameter used in experiments. The Fridman–Macheret parameter, α, describes how effectively vibrational energy can be used to overcome activation barriers. $E_{\text{a}}^{\text{f}}$ is defined as the activation barrier for the forward reaction and $E_{\text{a}}^{\text{r}}$ is the activation barrier for the reverse reaction, which together define α in equation (10):

$$\begin{equation}\alpha = \frac{{E_{\text{a}}^{\text{f}}}}{{E_{\text{a}}^{\text{f}} + E_{\text{a}}^{\text{r}}}}.\end{equation} \tag{ 10 }$$

When $\alpha {E_{\text{v}}}$ exceeds ${E_{\text{a}}}$, the exponent is set to zero through use of a Heaviside function (H). Equation (11) describes the rate of N₂(v) loss to dissociative adsorption (${r_{{\text{ads}},v}}$), with $\chi$ taking the same form as in equation (8) with ${k_{v \to v - 1,{\text{surface}}}}$ replaced by ${k_{{\text{ads}},v}}$:

$$\begin{equation}{r_{{\text{ads}},v}} = \chi {k_{{\text{ads}},v}}{n_{{{\text{N}}_{\text{2}}}\left( v \right)}}{\theta _{**}}\frac{z}{2}\frac{{SA}}{V}.\end{equation} \tag{ 11 }$$

In equation (11), the $\frac{z}{2}$ term accounts for coordination effects associated with N₂ adsorption on a surface. It is assumed that z = 6, which corresponds to dimer adsorption on the (111) plane of a face-centered cubic lattice. Assumptions about the values for the pre-exponential factor (A) and ${\theta _{**}}$ (fractional coverage of sites available for dimer adsorption) will be described in more detail in section 3.3. Values of $\frac{{SA}}{V}$, ${\gamma _{{\text{loss}},v = 1}}$, ${E_{\text{a}}}$, and $\alpha$ used to determine reaction rates for the different metals are reported in table S.7.

N₂ VDFs predicted by the state-to-state gas-phase kinetic model for different inlet compositions at the 3.5 cm axial position, which corresponds to the location of the front of the catalyst bed when a catalyst bed is included in experiments, are displayed in figure 2(a). Over 40% of N₂ is vibrationally excited for all of the investigated operating conditions. The VDFs differ from Treanor or Boltzmann distributions (figure S5), which highlights the necessity of kinetic modeling to accurately determine the VDF and investigate N₂(v)-enabled reactions. A comparison between the normalized signal measured by MBMS with the kinetic model-predicted mass spectrometer signal at different electron energies is displayed in figure 2(b) for the different process conditions. The mass spectrometer signal predicted by the state-to-state kinetic model (equation (5)) displays quantitative agreement with the experimentally measured signal (equation (1)), showing that the VDFs predicted by the kinetic model are representative of the distribution of N₂(v) available to react in the catalyst bed.

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Figures 2(c) and (d) display experimentally measured N and H densities at the same axial position as the N₂(v) measurements for comparison. Figure S2 compares experimentally measured values for N, H, and NH₃ densities with those predicted by the state-to-state kinetic model. The kinetic model predicts H densities accurately within ∼2×, overpredicts N densities by 2.4–6.4×, and underpredicts NH₃ densities by around an order of magnitude. Possible explanations for these discrepancies are discussed in the supporting information in section S2, and the implications that these discrepancies have on N₂(v) densities predicted by the kinetic model are discussed below.

A comparison between N₂(v), N, and H densities in figure 2 shows that the combined N₂(v> 0) densities are at least 10× greater than H densities and 100× greater than N densities, with densities of N₂(v) becoming comparable to N densities around v = 9, 12, and 8 for 0.5% N₂ and 0.5% H₂ in Ar; 5% N₂ and 0.5% H₂ in Ar; and 0.5% N₂ and 2.5% H₂ in Ar conditions, respectively. As N₂ concentration is increased from 0.5% to 5% with a constant 0.5% H₂ concentration, the corresponding densities of N₂(v) increase by around a factor of ten, while the N density only increases by a factor of two. As H₂ concentration is increased from 0.5% to 2.5% with a constant 0.5% N₂ concentration, the densities of N₂(v= 0−5) vary by less than 10% but densities of N₂(v> 8) are at least an order of magnitude lower because the higher H and H₂ densities result in increased rates of VT relaxation for the higher vibrational levels [30]. The H density increases by a factor of 3.5 as the H₂ concentration is increased from 0.5% to 2.5% at constant 0.5% N₂ because more H₂ is available to react in gas-phase dissociation processes. N₂(A) densities predicted by the kinetic model at the inlet of the catalyst bed are ∼10¹¹ cm⁻³, which is orders of magnitude lower than N₂(v> 0) densities and the N densities at the inlet of the catalyst bed, and is orders of magnitude lower than the observed NH₃ formation in the catalyst bed, even though the densities of N₂(A) reported by the model in the afterglow may be overestimated because of afterglow power deposition required by Chemkin.

To determine how inaccuracies in N, H, and NH₃ densities predicted by the state-to-state kinetic model may influence the VDF due to changes in VT rates, a sensitivity analysis was performed in which this model was run with VT rate constants for N, H, or H₂ changed to be the same as VT rate constants for Ar (or N₂, as it is assumed in the model that ${k_{{\text{VT,Ar}}}}$ = ${k_{{\text{VT,}}{{\text{N}}_{\text{2}}}}}{ }$ [14]), since Ar is the species with the lowest VT rate constants. These results are displayed in figure 3 and show that the inclusion of N₂(v) VT relaxation by H₂ with the VT rate constants for N and H the same as Ar produces a VDF that is closest to the VDF produced with the inclusion of the species-specific VT rate constants for all species. VT relaxation of N₂(v) by N or H contributes comparatively less than VT relaxation of N₂(v) by H₂ to the VDF. The fact that H₂ has the strongest influence on the VDF explains why the kinetic model can accurately predict the VDF even if there are discrepancies between N, H, and NH₃ densities that are experimentally measured and predicted by the model. It also affirms the observation that increasing the H₂ density in the discharge results in a loss of N₂(v) densities in the tail of the distribution. This is detrimental to NH₃ formation from N₂(v), since high levels of N₂(v) are proposed to be the most reactive for dissociative adsorption, and thus NH₃ formation, over catalytic surfaces [8].

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MBMS measurements and state-to-state kinetic modeling simulations were performed to investigate consumption pathways of N₂(v) in the packed bed. Figure 4(a) displays densities of N₂(v) determined by the kinetic model at 35 and 45 mm (locations of inlet and outlet of catalyst bed when present) with an empty tube (only gas-phase reactions considered). Small changes in N₂(v) densities from 3.5 to 4.5 cm are predicted by the model due to VT relaxation in the gas phase, which is quantitatively consistent with MBMS measurements that show small changes in the N₂(v) signal at 14.6 and 15.1 eV in figure 4(b).

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Vibrational relaxation of N₂(v) on the catalyst surface is included in the reaction set of the state-to-state kinetic model with ${\gamma _{{\text{loss}},v = 1}}$ = 0.0018 ± 0.0008, as described in section 2.5, to model loss of N₂(v) in the packed bed of catalyst. Figures 4(c), (e) and (g) display densities of N₂(v) determined by the kinetic model after 2 and 10 mm of the Fe, Ni, or Ag catalyst bed with the inclusion of vibrational relaxation on the surface in the reaction set. Kinetic model-predicted MS signals are in quantitative agreement with loss of MBMS signal as displayed in figures 4(d), (f) and (h). These results show that quenching of N₂(v) on the surface can account for the observed decrease in N₂(v) MBMS signal when catalyst is present without consideration of reactions that consume N₂(v) to form NH₃.

NH₃ is proposed to be formed from N₂(v) through dissociative adsorption on the catalyst surface. The rate of this reaction is described by equations (9)–(11). N₂(v) could contribute to NH₃ formation when it can undergo dissociative adsorption faster than vibrational relaxation on the surface, which occurs when $\frac{{{r_{{\text{ads}},v}}}}{{{r_{v \to v - 1,{\text{surf}}}}}} > 1$. Ignoring mass transfer effects, this criterion is expressed by equation (12):

$$\begin{equation}\frac{{{r_{{\text{ads}},v}}}}{{{r_{v \to v - 1,{\text{surf}}}}}} = { }\frac{{3A{e^{\left( { - \frac{{{E_{\text{a}}} - \alpha {E_{\text{v}}}}}{{RT}}H\left( {{E_{\text{a}}} - \alpha {E_{\text{v}}}} \right)} \right)}}{\theta _{**}}{n_{{{\text{N}}_2}\left( v \right)}}\frac{{SA}}{V}}}{{\frac{1}{4}{v_{{\text{th}},{{\text{N}}_2}}}\left( {v{\gamma _{{\text{loss}},v = 1}}} \right){n_{{{\text{N}}_2}\left( v \right)}}\frac{{SA}}{V}}} > 1.\end{equation} \tag{ 12 }$$

To determine which vibrational levels have $\frac{{{r_{{\text{ads}},v}}}}{{{r_{v \to v - 1,{\text{surf}}}}}} > 1$, values for A and ${\theta _{**}}$ must be known. A can be determined using transition state theory with knowledge of the transition state entropy of the adsorption process, and the determination of ${\theta _{**}}$ requires microkinetic modeling to determine steady-state coverages of adsorbates. An upper bound on $\frac{{{r_{{\text{ads}},v}}}}{{{r_{v \to v - 1,{\text{surf}}}}}}$ for vibrational level v, ${( {\frac{{{r_{{\text{ads}},v}}}}{{{r_{v \to v - 1,{\text{surf}}}}}}{ }} )_{{\text{max}}}}$, can be defined by setting $A = \frac{1}{4}{v_{{\text{th}},{{\text{N}}_2}}}$, which is an upper bound on the pre-exponential factor and assumes that the adsorbate retains two degrees of translational freedom on the catalytic surface, and by setting ${\theta _{**}} = 1$, which assumes that all surface sites are unoccupied. After cancellation of identical terms in the numerator and denominator, ${( {\frac{{{r_{{\text{ads}},v}}}}{{{r_{v \to v - 1,{\text{surf}}}}}}{ }} )_{{\text{max}}}}$ takes the form shown in equation (13):

$$\begin{equation}{\left( {\frac{{{r_{{\text{ads}},v}}}}{{{r_{v \to v - 1,{\text{surf}}}}}}{ }} \right)_{{\text{max}}}} = { }\frac{{3{e^{\left( { - \frac{{{E_{\text{a}}} - \alpha {E_{\text{v}}}}}{{{\text{RT}}}}H\left( {{E_{\text{a}}} - \alpha {E_{\text{v}}}} \right)} \right)}}}}{{v{\gamma _{{\text{loss}},v = 1}}}}.\end{equation} \tag{ 13 }$$

Vibrational levels for which ${( {\frac{{{r_{{\text{ads}},v}}}}{{{r_{v \to v - 1,{\text{surf}}}}}}{ }} )_{{\text{max}}}} > 1$ represent an upper bound on vibrational levels that can participate in NH₃-forming surface reactions faster than they lose their energy to vibrational relaxation on the surface. Figure 5 plots ${( {\frac{{{r_{{\text{ads}},v}}}}{{{r_{v \to v - 1,{\text{surf}}}}}}{ }} )_{{\text{max}}}}$ as a function of vibrational level at 400 K over Fe, Ni, and Ag terrace sites, and table 1 tabulates values of v, the percentage of N₂(v), and percentage of vibrational energy at the inlet of the catalyst bed for which ${( {\frac{{{r_{{\text{ads}},v}}}}{{{r_{v \to v - 1,{\text{surf}}}}}}{ }} )_{{\text{max}}}} > 1$.

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Figure 6 compares the density of N₂(v) at the inlet of the catalyst bed for which ${( {\frac{{{r_{{\text{ads}},v}}}}{{{r_{v \to v - 1,{\text{surf}}}}}}{ }} )_{{\text{max}}}} > 1$, the N density at the same location, and the experimentally measured NH₃ formation (NH₃ density at 4.5 cm with catalyst minus NH₃ density at 3.5 cm) in the packed bed of Fe, Ni, or Ag. Measurements of NH₃ density at 3.5 cm and 4.5 cm without catalyst show that the change in NH₃ density from 3.5 to 4.5 cm is below the limit of uncertainty (1.0 × 10¹⁴ cm⁻³) for each process condition (figure S2), indicating that any changes in the NH₃ density with catalyst present can be attributed to reactions on the metal wools. Figure 5 and table 1 show that for Ni and Ag, ${( {\frac{{{r_{{\text{ads}},v}}}}{{{r_{v \to v - 1,{\text{surf}}}}}}{ }} )_{{\text{max}}}} > 1$ when v > 17 and v > 38, respectively. The density of N₂(v> 17) is at least 6 orders of magnitude lower than the N density and the observed NH₃ formation in the Ni catalyst bed (figure 6(b)). Similarly, figure 6(c) shows that the density of N₂(v > 38) is orders of magnitude lower than the N density and the observed NH₃ formation in the Ag catalyst bed. Thus, N₂(v) does not measurably contribute to NH₃ formation over Ni and Ag even though the densities of N₂(v> 0) exceed N densities at the inlet of the catalyst bed by 100× because most of the N₂(v) loses its energy to the catalyst faster than it can dissociate to form NH₃.

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Figure 5 and table 1 show that for Fe, the transition metal that binds nitrogen the most strongly and has the lowest activation barrier for dissociative adsorption of N₂ of the three investigated metals, ${( {\frac{{{r_{{\text{ads}},v}}}}{{{r_{v \to v - 1,{\text{surf}}}}}}{ }} )_{{\text{max}}}} > 1$ when v > 8. Figure 6(c) shows that the density of N₂(v> 8) is on the order of 10¹³–10¹⁵ cm⁻³ for the investigated operating conditions. This accounts for <2% of N₂(v> 0) and <7% of energy deposited into N₂(v) (table 1). N₂(v> 8) densities depend sensitively on the process conditions and can be either less than or greater than the N densities at the inlet of the catalyst bed, depending on the operating conditions (figure 6(a)). Qualitatively, this suggests that Fe could potentially dissociate N₂(v) in quantities that would contribute to NH₃ formation similar to the measured amounts. However, the data in figure 6(a) suggest that NH₃ does not form from N₂(v), since NH₃ formation does not correlate with N₂(v> 8) densities varying from 10¹³–10¹⁵ cm⁻³. NH₃ formation instead correlates with N consumption for the investigated operating conditions, as discussed in previous work [15].

A lack of NH₃ formation from N₂(v> 8) can be attributed to inaccuracies in the upper bound estimates on N₂(v) reactivity described in equation (13). DFT-based microkinetic modeling that not only considers kinetics for dissociative adsorption but also kinetics for other reactions that can happen on the transition metal surfaces suggests that vibrational excitation does not enhance rates of NH₃ formation over Fe [8, 10], likely because the dissociative adsorption step is not the rate limiting step in NH₃ formation over Fe. The overestimation of the pre-exponential factor for N₂(v) dissociative adsorption can also contribute to the overestimation of vibrational levels that can contribute to NH₃ formation reactions. Ultra-high vacuum studies of N₂ chemisorption on Fe(111) report that the pre-exponential factor for adsorption should be 0.03–0.003× lower than the upper-bound estimate used in equation (13) [48]. Reducing ${r_{{\text{ads}},v}}$ by a factor of 0.03× increases the minimum v for which $\frac{{{r_{{\text{ads}},v}}}}{{{r_{v \to v - 1,{\text{surf}}}}}} > 1$ from 9 to 12 over Fe, reducing the density of N₂(v) that can undergo dissociative adsorption faster than vibrational relaxation by an order of magnitude or more for the investigated operating conditions.

The use of equation (12) to assess the ability of N₂(v) to contribute to NH₃ formation can be applied to systems beyond this work with appropriate considerations. In packed bed DBDs, metal particles supported on a metal oxide are often used as catalysts. Metal particles with sizes below ∼10 nm have fractions of step sites >0.1 [49, 50]. These step sites have lower activation barriers for N₂ dissociative adsorption than terrace sites [51, 52]. As a result, a larger range of vibrational levels would be able to undergo dissociative adsorption faster than vibrational relaxation (figure S6). However, these metal nanoparticles are dispersed on a support material with a surface area much larger (i.e. ∼100×) than the surface area of the metal particles [53, 54]. N₂(v) undergoes vibrational relaxation on metal oxide supports with similar loss probabilities to metals [19, 20] but cannot participate in NH₃ formation reactions on the metal oxide supports, resulting in a larger fraction of N₂(v) directed away from possible NH₃-forming reactions.

Elevated surface temperature can also increase $\frac{{{r_{{\text{ads}},v}}}}{{{r_{v \to v - 1,{\text{surf}}}}}}$, since the rate of dissociative adsorption has an exponential dependence on surface temperature. Figure S7 shows ${( {\frac{{{r_{{\text{ads}},v}}}}{{{r_{v \to v - 1,{\text{surf}}}}}}{ }} )_{{\text{max}}}}$ as a function of vibrational level at a surface temperature of 700 K. Even at 700 K, the lowest vibrational level where ${( {\frac{{{r_{{\text{ads}},v}}}}{{{r_{v \to v - 1,{\text{surf}}}}}}{ }} )_{{\text{max}}}} > 1$ for Ni is v = 17, which is only one vibrational level less than the lowest level at 400 K. For Fe, the lowest vibrational level where ${( {\frac{{{r_{{\text{ads}},v}}}}{{{r_{v \to v - 1,{\text{surf}}}}}}{ }} )_{{\text{max}}}} > 1$ is v = 5. N₂(v> 4) still only represents ∼10% of N₂(v>0) for the VDFs in figure 2(a), although the shapes of the VDF will change with temperature, since rates of VV and VT processes are temperature dependent, and rates of EV processes will change due to changes in electron temperature and density with temperature [55]. At temperatures near and above 700 K, NH₃ decomposition is catalyzed over transition metal surfaces [56] at rates which will limit the net NH₃ formation from N₂(v). Hence, while increasing the gas temperature would likely enhance dissociative adsorption of N₂(v), the beneficial effect for NH₃ formation will likely be limited in addition to requiring increased energy deposition.

Another method to maximize NH₃ formation from N₂(v) is to increase the densities of the vibrational levels of N₂(v) that can dissociate faster than relax. Figure 2(a) shows that by increasing N₂ from 0.5% to 5% with constant 0.5% H₂ produces a ∼10× increase in the N₂(v) density at each vibrational level. Increasing the fraction of N₂ further (for example, to 50%) would require the use of an alternative plasma source. DBDs can operate without a noble gas carrier but operate with E/N ∼100 Td, which favors energy deposition into electronic excitation or dissociation rather than vibrational excitation [11]. This results in relatively lower N₂(v) formation than N formation in the plasma compared to the investigated operating conditions using a RF source and consequently a lower relative contribution of N₂(v) to NH₃ formation compared to N. Microwave discharges or gliding arc discharges operate in regimes that favor vibrational excitation and can be sustained with molecular gases [12]; however, they also operate with gas temperatures >1000 K [57, 58], which can prevent net NH₃ formation due to the prominence of NH₃ decomposition above 700 K over catalytic surfaces [56]. For these discharges, the gas would have to be cooled to an appropriate temperature before reaching the surface over a timescale that does not result in significant loss of N₂(v) in the gas phase due to VT relaxation. Figure 2(a) also shows that increasing H₂ from 0.5% to 2.5% at constant 0.5% N₂ reduces N₂(v> 7) densities by at least an order of magnitude. Operating with lower % H₂ would increase densities of N₂(v) that are more reactive for dissociative adsorption; however, relatively lower H₂ densities could adversely affect rates of hydrogenation steps on the catalyst surface, and as a result reduce NH₃ formation rates and yields. Proximity of the plasma to the catalyst also dictates the densities of N₂(v) that can react to form NH₃. Figures S2(g)–(i) show that loss of N₂(v) from the end of the plasma to the beginning of the catalyst bed (∼300–650 μs) is most pronounced for the operating condition with the highest fraction of H₂, which exhibits a ⩾2× decrease in N₂(v) densities for v> 9 due to VT relaxation in the gas phase. Loss of N₂(v) in the gas phase would become more significant as the fraction of H₂ increases (for example, at or beyond 50%) and the transport timescale increases.

The choice of catalyst, plasma source, reactor design, gas/surface temperature, and gas composition dictate the production of N₂(v) in the plasma, loss of N₂(v) by relaxation in the gas phase and on surfaces, and utilization of N₂(v) to form NH₃ over the catalyst. While conditions that enhance the role of vibrational excitation in plasma-catalyzed NH₃ synthesis more than in this study might be found, the results in this work and literature precedent does not suggest that most, or even a significant fraction, of the energy directed towards vibrational excitation can be directed towards pathways that contribute to NH₃ formation, which limits the ability of N₂(v) to produce NH₃ by plasma catalysis with high energy efficiency.