## 3 Methods
### Theory  
#### 3.2.1 Microwave Cavity Resonance Spectroscopy  
The quasi-neutrality condition in the plasma bulk is used to relate the change of the free electron density to the dust charge density, i.e., the total amount of charge captured by the dust grains. The quasi-neutrality condition states that the total amount of positive charge should be balanced by the total amount of negative charge, as:  
\[n_{\mathrm{i+}}=n_{\mathrm{e}}+n_{\mathrm{i-}}+n_{\mathrm{d}}|Z_{\mathrm{d}}|, \tag{4}\]  
where \(n_{\mathrm{i+}}\) and \(n_{\mathrm{i-}}\) represent the positive and negative ion density, respectively, \(n_{\mathrm{d}}\) the dust density, and \(Z_{\mathrm{d}}\) the amount of elementary charges on the dust grains. Here, it is assumed that the ion densities do not change during the experiment so that \(n_{\mathrm{i+}}(t)=n_{\mathrm{i+,initial}}\) and \(n_{\mathrm{i-}}(t)=n_{\mathrm{i-,initial}}\), and that \(n_{\mathrm{d,initial}}=0\) as dust grains are not formed yet. Using these assumptions, the change of electron density \(\Delta n_{\mathrm{e}}\) can be derived from Equation (4):  
\[\Delta n_{\mathrm{e}}(t) =n_{\mathrm{e,initial}}-n_{\mathrm{e}}(t)\] \[=(n_{\mathrm{i+,initial}}-n_{\mathrm{i-,initial}})-(n_{\mathrm{i+ }}(t)-n_{\mathrm{i-}}(t)-n_{\mathrm{d}}(t)Z_{\mathrm{d}}(t)) \tag{5}\] \[=n_{\mathrm{d}}(t)Z_{\mathrm{d}}(t).\]  
Figure 3: The free electron density \(\hbar_{\mathrm{e}}\) in panel (**a**) and the transmittance \(\hat{T}\) signal in panel (**b**) from the moment of plasma ignition until the end of the second growth cycle. (**a**) The free electron density has been obtained using microwave cavity resonance spectroscopy (MCRS), which clearly shows a repetitive growth pattern after the initial cycle. (**b**) The transmittance signal shows similar repetitive features, but also sharp minima and maxima that are indicative of additional growth features. The data in this Figure has been published in [PERSON] et al. (2022) 2022 J. Phys. D. Appl. Phys. 55 395203 ([[https://doi.org/10.1088/1361-6463/ac802a](https://doi.org/10.1088/1361-6463/ac802a)]([https://doi.org/10.1088/1361-6463/ac802a](https://doi.org/10.1088/1361-6463/ac802a))) licensed under a Creative Commons Attribution (CC BY) license and used with the permission of the authors. The equilibrium charge number \(Z_{\rm d}\) of dust particles with a radius \(a\) is governed by the balance of currents towards a dust grain:  
\[I_{\rm i+}=I_{\rm i-}+I_{\rm e}, \tag{6}\]  
where the positive ion current \(I_{\rm i}\) (i.e., attracted towards the particle) is defined as:  
\[I_{\rm i+}=e\pi a^{2}n_{\rm i}\sqrt{\frac{8k_{\rm B}T_{\rm i}}{\pi m_{\rm i}}} \left(1-\frac{eV_{\rm d}}{k_{\rm B}T_{\rm i}}\right). \tag{7}\]  
For electrons and negative ions (i.e., repelled from the particle), the current is defined as  
\[I_{\rm e,i-}=-e\pi a^{2}n_{\rm e,i-}\sqrt{\frac{8k_{\rm B}T_{\rm e,i-}}{\pi m_ {\rm e,i-}}}\exp\left(\frac{eV_{\rm d}}{k_{\rm B}T_{\rm e,i-}}\right), \tag{8}\]  
based on the orbital-motion-limited (OML) theory [68]. Taking a closer look at the OML expressions, it can be seen that the negative ion current is insignificant with respect to the electron current by evaluating the ratio between the currents:  
\[ \begin{split}\frac{I_{\rm i-}}{I_{\rm e}}&=\frac{- e\pi a^{2}n_{\rm i-}\sqrt{\frac{8k_{\rm B}T_{\rm i-}}{\pi m_{\rm i-}}}\exp\left( \frac{eV_{\rm d}}{k_{\rm B}T_{\rm i-}}\right)}{-e\pi a^{2}n_{\rm e}\sqrt{\frac {8k_{\rm B}T_{\rm e}}{\pi m_{\rm e}}}\exp\left(\frac{eV_{\rm d}^{\prime}}{k_{ \rm B}T_{\rm e}}\right)}\\ &=\frac{n_{\rm i-}}{n_{\rm e}}\frac{m_{\rm e}T_{\rm i-}}{m_{\rm i -}T_{\rm e}}\exp\left(\frac{eV_{\rm d}}{k_{\rm B}T_{\rm i-}}\right)\ll 1,\end{split}  \tag{9}\]  
where typically \(m_{\rm e}\ll m_{\rm i-}\), \(T_{\rm i-}\ll T_{\rm e}\), and \(\exp\left(\frac{eV_{\rm d}}{k_{\rm B}T_{\rm i-}}\right)\ll 1\) because \(eV_{\rm d}\sim 1\) eV and \(k_{\rm B}T_{\rm i-}\approx 0.026\) eV (at room temperature).