Prostor

Abstract

Zero-dimensional reaction kinetic modeling in Ar is performed in order to evaluate a method for determination of the reduced electric field from intensity ratios of Ar excited states. Particular sets of Ar[2p] excited states are identified and selected based on their ability to reflect the value of the reduced electric field in the ratio of their populations. Balance equations for the selected species are combined and source terms are filtered and subsequently simplified via sensitivity analysis in order to be represented in a form that is convenient for use in the evaluation of measured line intensities. Various sources of uncertainties in the values of reaction rates and limitations of the proposed method with regards to time scales of electric field variations are discussed.

Reaction Kinetic Model (RKM):

M Stankov, M M Becker, T Hoder, D Loffhagen: “Extended reaction kinetics model for non-thermal argon plasmas and its test against experimental data” 2022 Plasma Sources Sci. Technol. 31 125002, DOI 10.1088/1361-6595/ac9332

• 23 species (with 408 processes):

  • \({\tt Ar[1p0]}\)

  • \({\tt Ar[1s5], Ar[1s4], Ar[1s3], Ar[1s2]}\)

  • \({\tt Ar[2p10], Ar[2p9], Ar[2p8], Ar[2p7], Ar[2p6], Ar[2p5], Ar[2p4], Ar[2p3], Ar[2p2], Ar[2p1] }\)

  • \({\tt Ar^*[hl], Ar_2^*[^3S_u^+,v=0], Ar_2^*[^1S_u^+,v=0], Ar_2^*[^3S_u^+,v>>0], Ar_2^*[^1S_u^+,v>>0] }\)

  • \({\tt Ar_2^+, Ar^+, e}\)

In this work: zero-dimensional modelling is performed, all spatial gradients are considered to be zero:

• Equations:

  • Balance equations: \[ \frac{\partial n_j(t)}{\partial t} = S_j(t). \]

  • Electron energy density: \[ \frac{\partial w_{\rm e}(t)}{\partial t} = - {\rm e}_0 \Gamma_{{\rm e}}(t)E(t)+P_{{\rm e}}(t), \] \[ \Gamma_{{\rm e}}(t) = - \frac{{\rm e}_0}{m_{\rm e}\nu_{\rm e}} E(t) n_{{\rm e}}(t). \]

• Parameters: \(T_{gas}=300\,{\rm K}\), \(N_{gas} = 2.44\times10^{25}\,{\rm m}^{-3}\), \(E(t)\) per case.

• Initial conditions: \(n_j=0\), but \(\bigl[{{\tt e}}\bigr]=2\bigl[{{\tt Ar^+}}\bigr]=2\bigl[{{\tt Ar_2^+}}\bigr]=10^8\hbox{--}10^{14}\,{\rm m}^{-3}\).

Balance equations together with equation for electron energy density are integrated in time for a given function of \(E(t)\).

Example: \(\bigl[{{\tt Ar[2p3]}}\bigr]\) and \(\bigl[{{\tt Ar[2p5]}}\bigr]\)

Governing equations

\[\begin{align*} \frac{{\mathrm d}\bigl[{{\tt Ar[2p3]}}\bigr]}{{\mathrm d}t} = & \bigl[{{\tt e}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{12} \\ &-\bigl[{{\tt Ar[2p3]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] \left(k_{298}+\ldots+k_{307}\right) \\ &-\bigl[{{\tt Ar[2p3]}}\bigr] \left( \nu_{383}+\nu_{384}+\nu_{385}\right) \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p4]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{308} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p2]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{293} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar^{*}[hl]}}\bigr] \nu_{404} \\ &\quad\quad\quad\quad\quad\quad\quad\quad+\hbox{ (neglected 31 source terms for {\tt Ar[2p3]})} \end{align*}\]

\[\begin{align*} \frac{{\mathrm d}\bigl[{{\tt Ar[2p5]}}\bigr]}{{\mathrm d}t} = & \bigl[{{\tt e}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{10} \\ &-\bigl[{{\tt Ar[2p5]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] \left(k_{318}+\ldots+k_{320}\right) \\ &-\bigl[{{\tt Ar[2p5]}}\bigr] \left( \nu_{377}+\nu_{378}\right) \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p3]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{299} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p4]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{309} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar^{*}[hl]}}\bigr] \nu_{402} \\ &\quad\quad\quad\quad\quad\quad\quad\quad+\hbox{ (neglected 31 source terms for {\tt Ar[2p5]})} \end{align*}\]

Population ratio

• All populations densities appear in ratios to other densities (relative measurements if spectral line intensities do suffice, are directly related to densities)

• Second order processes \({\tt A}^*+{\tt e}\rightarrow\ldots\) or \({\tt A}^*+{\tt B}^*\rightarrow\ldots\) appear to be of minor importance:

• Electron density cancels out

• Ratio of electron impact excitation rates \(\left(\frac{k_{10}}{k_{12}}\right)\) is a function of electron energy or \(E/N\):

\[\begin{align*} \left(\frac{k_{10}}{k_{12}}\right) = \frac{\bigl[{{\tt Ar[2p5]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]} \frac{\displaystyle \frac{{\mathrm d}\ln{\bigl[{{\tt Ar[2p5]}}\bigr]}}{{\mathrm d}t} +\left( \nu_{377...378}\right) +\bigl[{{\tt Ar[1p0]}}\bigr] \left( \left(k_{318...320}\right) -\frac{\bigl[{{\tt Ar[2p4]}}\bigr]}{\bigl[{{\tt Ar[2p5]}}\bigr]}k_{309} -\frac{\bigl[{{\tt Ar[2p3]}}\bigr]}{\bigl[{{\tt Ar[2p5]}}\bigr]}k_{299} \right) -\frac{\bigl[{{\tt Ar^{*}[hl]}}\bigr]}{\bigl[{{\tt Ar[2p5]}}\bigr]} \nu_{402} -\frac{\ldots}{\bigl[{{\tt Ar[2p5]}}\bigr]} } {\displaystyle \frac{{\mathrm d}\ln{\bigl[{{\tt Ar[2p3]}}\bigr]}}{{\mathrm d}t} + \left(\nu_{383...385}\right) + \bigl[{{\tt Ar[1p0]}}\bigr] \left( \left(k_{298...307}\right) - \frac{\bigl[{{\tt Ar[2p4]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]}k_{308} - \frac{\bigl[{{\tt Ar[2p2]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]}k_{293} \right) - \frac{\bigl[{{\tt Ar^{*}[hl]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]} \nu_{404} -\frac{...}{\bigl[{{\tt Ar[2p3]}}\bigr]} } \end{align*}\]

Identification of sensitive ratios

Combination of \(\bigl[{{\tt Ar[2pi]}}\bigr]\)/\(\bigl[{{\tt Ar[2pj]}}\bigr]\) for \(i,j\in[1,10]\), ratio: ‘row/col’:

• 2p1/2p3 proves too sensitive to \(\bigl[{{\tt Ar^[hl]}}\bigr]\)

• Ratios involving 2pj, for \(j>5\) are impractical for too many 2p states involved

• Lets have a look at: 2p2/2p3, 2p5/2p3, 2p2/2p3, 2p2/2p4 in details: we run simulation at constant \(E/N\) and:

  • evaluate how simplified governing equation reflect real \(E/N\)

  • evaluate iportance of individual terms (processes) (can we drop some?)

\(\bigl[{{\tt Ar[2p5]}}\bigr]/\bigl[{{\tt Ar[2p3]}}\bigr]\)

\[\begin{align*} \left(\frac{k_{10}}{k_{12}}\right) = \frac{\bigl[{{\tt Ar[2p5]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]} \frac{\displaystyle \frac{{\mathrm d}\ln{\bigl[{{\tt Ar[2p5]}}\bigr]}}{{\mathrm d}t} +\left( \nu_{377...378}\right) +\bigl[{{\tt Ar[1p0]}}\bigr] \left( \left(k_{318...320}\right) -\frac{\bigl[{{\tt Ar[2p4]}}\bigr]}{\bigl[{{\tt Ar[2p5]}}\bigr]}k_{309} -\frac{\bigl[{{\tt Ar[2p3]}}\bigr]}{\bigl[{{\tt Ar[2p5]}}\bigr]}k_{299} \right) -\frac{\bigl[{{\tt Ar^{*}[hl]}}\bigr]}{\bigl[{{\tt Ar[2p5]}}\bigr]} \nu_{402} -\frac{\ldots}{\bigl[{{\tt Ar[2p5]}}\bigr]} } {\displaystyle \frac{{\mathrm d}\ln{\bigl[{{\tt Ar[2p3]}}\bigr]}}{{\mathrm d}t} + \left(\nu_{383...385}\right) + \bigl[{{\tt Ar[1p0]}}\bigr] \left( \left(k_{298...307}\right) - \frac{\bigl[{{\tt Ar[2p4]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]}k_{308} - \frac{\bigl[{{\tt Ar[2p2]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]}k_{293} \right) - \frac{\bigl[{{\tt Ar^{*}[hl]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]} \nu_{404} -\frac{...}{\bigl[{{\tt Ar[2p3]}}\bigr]} } \end{align*}\]

\(\bigl[{{\tt Ar[2p2]}}\bigr]/\bigl[{{\tt Ar[2p4]}}\bigr]\)

\[\begin{align*} \left(\frac{k_{13}}{k_{11}}\right) = \frac{\bigl[{{\tt Ar[2p2]}}\bigr]}{\bigl[{{\tt Ar[2p4]}}\bigr]} \frac{\displaystyle \frac{{\mathrm d}\ln{\bigl[{{\tt Ar[2p2]}}\bigr]}}{{\mathrm d}t} + \left(\nu_{386...389}\right) + \bigl[{{\tt Ar[1p0]}}\bigr] \left( k_{293...297} \right) - \frac{\bigl[{{\tt Ar^{*}[hl]}}\bigr]}{\bigl[{{\tt Ar[2p2]}}\bigr]} \nu_{405} -\frac{...}{\bigl[{{\tt Ar[2p2]}}\bigr]}} {\displaystyle\frac{{\mathrm d}\ln{\bigl[{{\tt Ar[2p4]}}\bigr]}}{{\mathrm d}t} +\left( \nu_{379...382}\right) +\bigl[{{\tt Ar[1p0]}}\bigr] \left( \left(k_{308...317}\right) -\frac{\bigl[{{\tt Ar[2p5]}}\bigr]}{\bigl[{{\tt Ar[2p4]}}\bigr]}k_{318} -\frac{\bigl[{{\tt Ar[2p3]}}\bigr]}{\bigl[{{\tt Ar[2p4]}}\bigr]}k_{298} \right) -\frac{\bigl[{{\tt Ar^{*}[hl]}}\bigr]}{\bigl[{{\tt Ar[2p4]}}\bigr]} \nu_{403} -\frac{\ldots}{\bigl[{{\tt Ar[2p4]}}\bigr]} } \end{align*}\]

\(\bigl[{{\tt Ar[2p2]}}\bigr]/\bigl[{{\tt Ar[2p3]}}\bigr]\)

\[\begin{align*} \left(\frac{k_{13}}{k_{12}}\right) = \frac{\bigl[{{\tt Ar[2p2]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]} \frac{\displaystyle \frac{{\mathrm d}\ln{\bigl[{{\tt Ar[2p2]}}\bigr]}}{{\mathrm d}t} + \left(\nu_{386...389}\right) + \bigl[{{\tt Ar[1p0]}}\bigr] \left( k_{293...297} \right) - \frac{\bigl[{{\tt Ar^{*}[hl]}}\bigr]}{\bigl[{{\tt Ar[2p2]}}\bigr]} \nu_{405} -\frac{...}{\bigl[{{\tt Ar[2p2]}}\bigr]}} {\displaystyle \frac{{\mathrm d}\ln{\bigl[{{\tt Ar[2p3]}}\bigr]}}{{\mathrm d}t} + \left(\nu_{383...385}\right) + \bigl[{{\tt Ar[1p0]}}\bigr] \left( \left(k_{298...307}\right) - \frac{\bigl[{{\tt Ar[2p4]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]}k_{308} - \frac{\bigl[{{\tt Ar[2p2]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]}k_{293} \right) - \frac{\bigl[{{\tt Ar^{*}[hl]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]} \nu_{404} -\frac{...}{\bigl[{{\tt Ar[2p3]}}\bigr]} } \end{align*}\]

Time variying fields?

Hopeful Expectation: Governing equations do contain time dependent terms, ratios do contain time derivatives. But let’s test it!

Acknowledgement

This contribution was supported by the Czech Science Foundation grant project number 21-16391S.