Robust approximation rules for critical electric field of dielectric gas mixtures

A semi-analytic method for quickly approximating the density-reduced critical electric field for arbitrary mixtures of gases is proposed and validated. Determination of this critical electric field is crucial for designing and testing alternatives to SF6 for insulating high voltage electrical equipment. We outline the theoretical basis of the approximation formula from electron fluid conservation equations, and demonstrate how for binary mixtures the critical electric field can be computed from the transport data of electrons in the pure gases. We demonstrate validity of the method in mixtures of N2 and O2, and SF6 and O2. We conclude with an application of the method to approximate the critical electric field for mixtures of SF6 and HFO1234ze(E), which is a high interest mixture being actively studied for high voltage insulation applications.


Introduction
Presently, most high voltage electrical infrastructure on Earth relies on sulphur hexafluoride (SF 6 ).This inert gas is utilised in equipment such as circuit breakers, transformers, and transmission infrastructure due its excellent electric insulation and arcextinguishing capacity [1].Identified as a greenhouse gas in the Kyoto protocol [2], SF 6 has since been labeled by the Intergovernmental Panel on Climate Change as the most potent greenhouse gas with a global warming potential (GWP) approximately 23,500 times that of carbon dioxide (CO 2 ), and with an anticipated lifetime of up to 3200 years.
The bulk of SF 6 emissions come from the electric power industry, often during maintenance of electrical equipment or simply due to aging or faulty storage vessels and electrical equipment.Recently, some success has been reported in legislatively constraining SF 6 emissions [3,4].However, ultimately the global SF 6 gas content is still increasing year-on-year and growth in newly installed gas-insulated equipment is overpowering industry regulation and financial penalty policies in reducing SF 6 emissions.[4].
To date, a number of replacements to SF 6 have been deployed in the power industry, such as dry air, N 2 , CO 2 , or various admixtures of natural and proprietary synthetic gases [1].Despite some adoption of replacements with lower GWP, the widespread global use of SF 6 prevails in many industrial applications where it has not yet been bettered on an engineering basis for insulation and/or current quenching requirements.In addition, the logistical and/or financial burdens to replace SF 6 based equipment are often significant.
Driving the effort towards replacing SF 6 are many hours and millions of dollars spent per year in academic and commercial research performing dielectric breakdown experiments [5,6] or swarm experiments [7,8] to characterise insulating gases.There are many complexities and challenges in adequately performing and interpreting measurements of such experiments in their varying geometries (e.g.needle-plane, parallel plate, coaxial) and varying operating conditions (e.g.low or high pressure/temperature, gas purity, pulsed Townsend or steady state Townsend in the case of swarm experiments).As a result, theory and simulation are vital methods helping to complement experiments.
The power and flexibility of theory and simulation applied to highly unconstrained design problems can enable a quick and affordable scan of very large parameter spaces of material choices or design configurations.This computational sieve approach to rapidly explore, trial, and assess potential scenarios then informs the design of targeted and efficient experiments and solution prototypes.For example, insight through large simulations of breakdown scenarios in often complex geometries [9] or detailed kinetic calculations of electron energy distribution functions (EEDFs) in gas mixtures over density or pressure windows [10] are now capable with advances in computing.That being said, these non-trivial computational tasks can also present their own challenges and barriers to entry, such as advanced computational domain knowledge or access to sufficient computing power or time.
In addition to brute force computational methods, analytic or simplified semianalytic methods can be employed to provide very simple and rapid estimations of design parameters.In particular, simple design formulas can be accessed by a wide user base, allowing experiment scenarios to be quickly explored.In Section 2 we present a semianalytic method to rapidly predict the critical reduced electric field, (E/n 0 ) c , of gas mixture combinations based entirely on calculated electron transport properties of the constituent pure gases only.We first demonstrate the validity of this model in Section 3 by benchmarking the predictive capability with relatively simple mixtures of well studied gases, such as N 2 -O 2 , and SF 6 -O 2 .Finally, in Section 4 we demonstrate the ability of the model to correctly predict the positive synergy in (E/n 0 ) c for mixtures of SF 6 and HFO-1234ze(E).This molecule (1E)-1,3,3,3-Tetrafluoroprop-1-ene (C 3 H 2 F 4 ), often termed HFO-1234ze(E) or R-1234ze, is a hydrofluoroolefin (HFO) used as a refrigerant and is studied as a potential SF 6 replacement [11,8].

Theory
The macroscopic properties of a gas, like breakdown voltage, are often used to guide design of electrical equipment.Fundamentally, the macroscopic behaviours are underpinned by the microscopic interactions between electrons and gas molecules.Pulsed Townsend (PT) and steady-state Townsend (SST) swarm experiments have been used to study electron transport properties in insulating gases, such as drift velocity, diffusion coefficients, and importantly ionisation and attachment rates [1].One quantity of immediate relevance to the community is the critical electric field, where the ionisation rate equals the electron attachment rate.Beyond this field the rate of ionisation will exceed electron attachment and so it marks the transition of a gas from being a free electron sink to being a source, precipitating the dielectric breakdown of the medium.
Primary methods of approximating swarm properties in gas mixtures have largely been based on projecting functional dependence onto the reduced electric field, E/n 0 , which we will abbreviate in equations as Ě = E/n 0 for brevity.One then approximates quantities of a gas mixture as linear combinations of pure gas quantities evaluated at a common value of E/n 0 , weighted by gas density fractions of the mixture.For example, for simple gas mixtures, many have employed Blanc's law [12] to approximate drift velocity where W m is the electron drift velocity in the mixture, W A and W B are drift velocities in gas A and gas B, and where n A and n B are present gas densities of constituent gases A and B evaluated at a common value of E/n 0 .In a similar fashion, a so-called Wieland approximation has been used to approximate ionisation, k i,m , or attachment, k a,m , rates of gas mixtures [13] where k m is a rate in the mixture and k A and k B are the rates in pure gases A and B evaluated at a common value of E/n 0 .Implicit in its derivation, Blanc's Law [12] does not inherently include inelastic collisions.Further assumptions inherent in the Blanc and Wieland approximations that limit their validity are that (i) electron impact cross sections can be added in simple linear combinations at a given value of E/n 0 , and (ii) the steady state EEDF is the same for each component gas, and the combination mixture, at a given value of E/n 0 .This can be a reasonably good approximation for some certain conditions, however in general this assumption of the EEDF can be overly restrictive.
Previous studies have highlighted shortcomings in the Wieland approximation [14,15], with one notable limitation being an inability to predict positive synergism in gas mixtures -where a higher critical electric field of the mixture, above that of either constituent gases, is obtained.To remedy shortcomings, extensions on this style of approximation, allowing for a non-linear perturbation around the basic linear combination, have been explored [7,8].Indeed, van Brunt [15] proposed a novel alternative parameterisation of swarm properties via consideration of electron temperatures, similar to what is presented in this current study.However, while this parameterisation identifies the crucial role of mean electron energy, it relies on multiple free parameters that require specification for each gas being considered.
To better account for the dependence of energy transfer on inelastic collisions as E/n 0 increases an alternative approach to Blanc's law based on electron mean energy was developed [16].This alternative approach for evaluating collision and transport data of electrons in gas mixtures assumes equivalence of a shared common electron mean energy (CME), ⟨ε⟩, instead of a common reduced electric field, E/n 0 [16].This assumption therefore allows for arbitrary EEDFs in either constituent gas or the mixture itself, with the restriction that the integrated moment of mv 2 /2 over each EEDF must be equivalent.The CME approach has been well adopted by many areas of the community, however use of Blanc's Law still persists in some studies.
Of particular note to this study is the prior demonstration [14] of using the CME approximation for estimation of non-conservative collision rates, ν * , in mixtures of molecular gases this assumption was shown to be a much better method of projecting complex, nonlinear electron kinetic effects onto a single macroscopic swarm variable.In arriving at the semi-analytic model presented in this study, we will also adopt the approximation for estimation of non-conservative collision rates in equation ( 4), and leverage the CME approach to determine E/n 0 in the mixture conditions; this will enable estimation of the critical reduced electric field.As a foundation for our approach, we refer to the momentum and energy conservation equations for electron transport, that can be derived by taking integral moments of the Boltzmann equation over trial functions for electron momentum, mv, and kinetic energy, mv 2 /2 [17,18,19,20].As per the CME approximation idea [16], it has been previously demonstrated [16,21] that in the steady state limit the electron momentum balance equations of both pure gases A and B, and an arbitrary mixture of the two, yield the following expression for the electron drift velocity in the mixture While from the steady state energy balance equations of both pure gases A and B, and an arbitrary mixture of the two, a second expression for the electron drift velocity in the mixture can be found With ( 5) and ( 6) both explicitly referring to the value of E/n 0 in the mixture, we can combine ( 5) and ( 6) to obtain either an expression for the drift velocity if one desires or an expression for the reduced electric field in the mixture where ĚA = ĚA (⟨ϵ⟩), ĚB = ĚB (⟨ϵ⟩), W A = W A (⟨ϵ⟩), and W B = W B (⟨ϵ⟩) are all functions of the common mean electron energy being considered, ⟨ϵ⟩.

Approximation rule for critical reduced electric field
By using the earlier approximation for a mixture ionisation or attachment collision frequency in equation ( 4), and the expression in equation ( 8) for equivalent reduced electric field in the mixture, we can now state the procedure to extract a value of critical electrical field as: (i) Using a platform of choice (e.g.Boltzmann equation solver, Monte Carlo simulation) compute steady state electron swarm parameters for pure gases A and B.
(iii) Using the available swarm data and desired density fraction given by ( 2), evaluate ionisation and attachment coefficients of a mixture as a function of mean energy (iv) Using the available data, evaluate (8) to obtain the reduced electric field, Ěm (⟨ϵ⟩), in the mixture as a function of mean energy.
(v) Evaluate the net effective ionisation coefficient which can now be expressed, or plotted for convenience, using Ěm (⟨ϵ⟩) as a function k eff,m (E/n 0 ) to determine the zero-crossing and thus the critical reduced electric field of the gas mixture.
So, with zero knowledge of the electron transport in the mixture, we have detailed an approximation of the critical reduced electric field of a gas mixture.This uses only the electron swarm data computed for pure gases, which is a standard exercise being made evermore accessible by multiple freely available software for download, or even codes evaluated online via a web browser [22,23,24,25].To validate the use of this presented method, we now present benchmarking results of approximating the critical reduced electric field for mixtures of relatively well-studied molecular gases.

Benchmarking and model validation
To validate the proposed method to rapidly approximate the critical reduced electric field, we compare to the calculated critical reduced electric field determined for various mixture combinations.The calculated value is determined by first solving the Boltzmann kinetic equation for the EEDF in a mixture gas using a well-benchmarked multi-term solution technique and software framework [19,26,27].One can then compute the effective net electron creation rate, k eff,m , as the difference of the integrated ionisation and attachment rates k i,m − k a,m .The critical reduced electrical field is naturally determined when k eff,m = 0.
We first present results for a simplified case of N 2 and O 2 mixtures, since only the O 2 target will contribute to electron attachment processes.We then follow with benchmarking of SF 6 and O 2 mixtures, where the addition of SF 6 provides a large attachment cross-section to rigorously test the proposed approximation.For the calculations detailed in this study electron scattering cross sections for SF 6 , O 2 , and N 2 were obtained from the Biagi database † of the LXCat online database [28].

Benchmark: N 2 and O 2 mixtures
Mixtures of molecular nitrogen and oxygen are often studied as a surrogate for air.As a result, there has been a wealth of scientific data and understanding produced for electron scattering cross-sections and subsequent transport in pure N 2 , O 2 , and indeed mixtures of the two.Therefore, it is a good candidate system for benchmarking and validation of theory and methods.
For validation in the case of N 2 and O 2 mixtures, in Figure 1 we present the results of approximating both the drift velocity, W , and net effective ionisation rate, k eff , as a function of E/n 0 ; for clarity of plot presentation we show comparisons for mixture fractions at 25%, 50%, and 75% ratios.The present study is chiefly concerned with the net effective ionisation rate, however since drift velocity is also a common transport quantity used in assessment of swarm analysis we include it in Figure 1 as an additional demonstration of the applicability of the approximation method presented in this work.For both transport quantities the approximation produces excellent qualitative agreement compared to the exact calculations.As a quantitative measure of the critical reduced electric field accuracy, Table 1 shows that, at worst, the agreement is about 5-6%, which was representative of the general case for other mixture ratios not shown in Figure 1.Mixtures of sulphur hexafluoride, SF 6 , and molecular oxygen, O 2 , have been well studied in the semiconductor etching industry.As it was for our previous benchmark case, choosing a well studied pair of molecules is a wise choice for benchmarking and validation of theory and methods.In particular, validation in the presence of a highly attaching gas like SF 6 accompanied by another attaching gas is worth demonstrating before trialing the methods presented in this study on more complex molecules such as HFO-1234ze(E).
For validation in the case of SF 6 and O 2 mixtures, in Figure 2 we present the results of approximating both the drift velocity, W , and net effective ionisation rate, k eff , as a function of E/n 0 ; for clarity of presentation we show comparisons for mixture fractions at 25%, 50%, and 75% ratios.Once again, Figure 2 demonstrates excellent qualitative agreement produced by the approximation, for both drift velocity and net creation rate.In these cases the transition from pure SF 6 to O 2 extremes is somewhat monotonic.In terms of the critical reduced electric field values, Table 2 values of relative error summarise that there is generally improved accuracy compared with the prior N 2 /O 2 benchmark case.As an additional point of interest, Figure 3 shows the EEDFs at (a) the critical reduced electric field point for pure SF 6 , O 2 , and a mixture of 50% SF 6 and 50% O 2 .Here, subfigure (a) shows that for the wide range of critical E/n 0 values there is some variation in EEDFs.The mixture EEDF generally resembles that for SF 6 , with some deviation around the distribution peak.Notably the EEDF in O 2 peaks at a lower energy.Subfigure (b) shows the same EEDF for the mixture as in (a), however for pure SF 6 and O 2 the EEDFs are shown corresponding to the same electron mean energy value at the mixture's critical E/n 0 , ⟨ϵ⟩ = 7.0 eV.This demonstrates that for the bulk of the distribution population, the EEDFs are indeed quite similar, particularly between approximately 0.1-20 eV.This demonstration highlights that abstracting the dynamics of the bulk population of EEDFs onto the common mean energy ⟨ϵ⟩, invoked in the derivation of our approximation method, is indeed a reasonable approach.When discussing the nature of EEDFs it is often instructive to compare against a Maxwellian distribution often used by the community.For this purpose, Figure 3(b) also allows comparison against a Maxwellian EEDF of the same ⟨ϵ⟩ as the calculated distributions.This highlights that while the bulk of these computed EEDFs reasonably follow the Maxwellian shape about the peak, the sharp roll-off to a depleted tail of higher energy electrons is not sufficiently captured by a Maxwellian.

Application to dielectric insulator design
In the search for alternatives to sulphur hexafluoride, SF 6 , industry and scientific communities are considering a wide range of options.One such option that has been heavily studied [29,11,5,30] is (1E)-1,3,3,3-Tetrafluoroprop-1-ene (C 3 H 2 F 4 ), often abbreviated to HFO-1234ze(E).This gas is one of an emerging generation of hydrofluoroolefins (HFOs) originally targeted for use as refrigerants in place of traditional hydrofluorocarbons.The industry use of HFOs is being spurred by their often very low ozone depeletion and global warming potentials.
Mixtures of sulphur hexafluoride, SF 6 , and HFO-1234ze(E), are also being studied to maximise insulating and arc quenching properties [8,31] as part of the large design and development research efforts to reduce SF 6 use in the high voltage electricity industry.For the HFO-1234ze(E) electron scattering cross-section data, a recently obtained crosssection set from a forthcoming study [32] applying neural network based swarm inversion techniques [33,34] is employed.
Here in Figure 4 we focus on probing mixtures of SF 6 and HFO-1234ze(E) for approximating the net effective ionisation rate, k eff , as a function of E/n 0 .For clarity of presentation we show comparisons for mixture fractions of 25%, 50%, and 75%.The results in Figure 4, and subsequent Figures 6 and 7, clearly demonstrate the well established non-monotonic behaviour of the critical field increasing with increasing SF 6 fraction but then reducing as the fraction of SF 6 approaches 100%.The ability of these mixtures to yield a critical reduced field greater than either case of pure gas has been termed as a positive synergy in the insulating gas literature.This positive synergy effect has been well studied and discussed [8,31,35] for insulating gases, and so being able to reproduce this important physics phenomenon from the present approximation formula, despite only using known data for pure gas cases, is an encouraging outcome of the present method.
Despite the added complication of the non-monotonic positive synergy phenomenon our present method again provides a good approximation to the critical reduced electric field for these mixtures in the regime of competing ionising and attaching processes.We can see from Table 3 there is excellent quantitative agreement between the approximated value for the critical reduced electric field for mixtures, with relative errors on the order of 1-2%.Comparing the EEDFs, in Figure 5(a), we can see quite different distributions for the case of pure SF 6 , HFO-1234ze(E), and a mixture of 50% SF 6 and 50% HFO-1234ze(E).At low energies the mixture EEDF resembles that of SF 6 , however beyond about 1 eV the mixture EEDF adopts qualitative features of the pure HFO-1234ze(E) distribution.To compare EEDFs corresponding to the same common electron mean energy, we refer to subfigure (b).Here, we see that when considering EEDFs yielding the same mean energy as the mixture critical point, ⟨ϵ⟩ = 5.2 eV, the distributions are once again very similar.Once again, this outcome validates the use of the underlying common mean energy assumption discussed in Section 2. We now demonstrate an example of high-throughput data generation and understanding that the present semi-analytic method offers, compared to having to calculate the exact EEDF via the Boltzmann equation for each system configuration.Scanning over a range of mixture combinations 0 ≤ x ≤ 1 and a range of applied E/n 0 values allows a surface of the net effective ionisation rate, k eff , to be constructed over the domain of possible parameter space, as shown in Figure 6.Considering the colour gradient changes projected to the base of the plot, one can clearly see the non-linear and non-monotonic features of the levels of k eff,m .For the possible mixture combinations in the case of k eff,m (E/n 0 ) = 0, black marker points are placed to highlight the critical field curve for the mixtures of SF 6 and HFO-1234ze(E).These points can be projected onto a more convenient 2D plot, like that commonly presented in literature, in Figure 7.For comparison the pulsed Townsend swarm measurements of Egüz et al. [7] have been plotted.Further, the critical electron mean energy, ⟨ϵ⟩ c , at a given critical electric field is plotted against the right-hand axis of Figure 7.In contrast to the non-monotonic nature of (E/n 0 ) c , the critical electron mean energy, ⟨ϵ⟩ c , plotted in Figure 7 demonstrates a monotonic increase with increasing fraction of SF 6 .By plotting and comparing (E/n 0 ) c and the equivalent critical ⟨ϵ⟩ c we highlight why indeed a mixture approximation of linear addition of constituent gas quantities with functional dependence on E/n 0 may fail.In contrast, the monotonic nature in ⟨ϵ⟩ c allows a linear combination of pure gas quantities with mean energy functional dependence to be used effectively.

Conclusion
In summary, we have presented the formulation and validation of a semi-analytic formula and method for approximating the critical reduced electric field for insulating gas mixtures.These findings demonstrate a rapid and accurate approximate approach to assessing the breakdown electric field properties of arbitrary insulating gas mixtures.Importantly, our results show immediate application and promise towards parameter space exploration and design studies of potential insulating gases to replace SF 6 in the high voltage electricity industry.While this study provides valuable insights into how industry standard electron transport data of individual pure gases can be combined to estimate mixture properties, further investigation is needed, for example, to develop improvements to the model to facilitate the use of experimentally measured swarm quantities as a function of E/n 0 .While high-throughput approximation and prototyping can be done using the presented method, the necessity of having calculated pure gas swarm data as a function of mean energy is a limitation.Future studies will explore this by studying functional dependencies of electron mean energy, characteristic energy, and quantities measured directly in swarm experiments.By continuing to investigate these aspects, we can expand the set of quality tools available to scientists and engineers in the design and development of effective, safe, and climate neutral insulating gas technologies for high voltage electricity industries.

Figure 1 .
Figure 1.Comparison of (top) flux drift velocity, W , and (bottom) net effective ionisation rate, k eff , for various mixtures of N 2 and O 2 as a function of E/n 0 .Exact values computed via multi-term solution of the EEDF from the Boltzmann equation, approximations determined from procedure proposed in this study.Horizontal axis in k eff plot drawn in thicker line to highlight where zero-crossing of curves denotes critical reduced electric field.

Figure 2 .
Figure 2. Comparison of (top) flux drift velocity, W , and (bottom) net effective ionisation rate, k eff , for various mixtures of SF 6 and O 2 as a function of E/n 0 .Exact values computed via multi-term solution of the EEDF from the Boltzmann equation, approximations determined from procedure proposed in this study.Horizontal axis in k eff plot drawn in thicker line to highlight where zero-crossing of curves denotes critical reduced electric field.

Figure 3 .
Figure 3.Comparison of EEDFs for SF 6 , O 2 , and a mixture of 50% SF 6 and 50% O 2 .(Left) EEDFs at each critical E/n 0 value for the three background gas cases.(Right) EEDFs for each background gas case associated with a mean electron energy ⟨ϵ⟩ = 7.0 eV corresponding to the mixture EEDF at the critical field threshold.

Figure 4 .
Figure 4. Comparison of net effective ionisation rate, k eff , for various mixtures of SF 6 and HFO-1234ze(E) as a function of E/n 0 .Exact values computed via multi-term solution of the EEDF from the Boltzmann equation, approximations determined from procedure proposed in this study.Horizontal axis in k eff plot drawn in thicker line to highlight where zero-crossing of curves denotes critical reduced electric field.

Figure 5 .
Figure 5.Comparison of EEDFs for SF 6 , HFO-1234ze(E), and a mixture of 50% SF 6 and 50% HFO-1234ze(E).(Left) EEDFs at each critical E/n 0 value for the three background gas cases.(Right) EEDFs for each background gas case associated with a mean electron energy ⟨ϵ⟩ = 5.2 eV corresponding to the mixture EEDF at the critical field threshold.

Figure 6 .
Figure 6.Surface of net effective ionisation rate, k eff , for mixtures of SF 6 and HFO-1234ze(E) as a function of E/n 0 and % fraction of SF 6 .Level curves projected on lower plane, with k eff = 0 points marked in black, .

Figure 7 .
Figure 7. (Left-hand axis) Approximate critical reduced electric field where attachment transitions to ionisation for increasing fractions of SF 6 in mixture with HFO-1234ze(E).Present approximation result shown in black markers, , swarm measurements of Egüz et al.[8] shown in red markers, .(Right-hand axis) Mean electron energy, ⟨ϵ⟩, at critical electric field plotted in triangle markers, , to demonstrate monotonic nature

Table 1 .
Errors in critical reduced electric field approximation for various mixtures of N 2 and O 2 .

Table 2 .
Errors in critical reduced electric field approximation for various mixtures of SF 6 and O 2 .

Table 3 .
Errors in critical reduced electric field approximation for various mixtures of SF 6 and HFO-1234ze(E).