Time-Synchronized Microwave Cavity Resonance Spectroscopy and Laser Light Extinction Measurements as a Diagnostic for Dust Particle Size and Dust Density in a Low-Pressure Radio-Frequency Driven Nanodusty Plasma

Abstract

In a typical laboratory nanodusty plasma, nanometer-sized solid dust particles can be generated from the polymerization of reactive plasma species. The interplay between the plasma and the dust gives rise to behavior that is vastly different from that of pristine plasmas. Two of the key parameters in nanodusty plasma physics are, among other things, the dust particle size and the dust density. In this work, we introduce a novel method for the determination of these two quantities from the measurement of the free electron density using microwave cavity resonance spectroscopy and laser light extinction measurements. When comparing these two measurements to theory, one can determine the best-fitting dust particle size and dust density. Generally, cyclic behavior of the dust particle size and dust density was observed, of which the trends were relatively insensitive to varying the most stringent input assumptions. Finally, this method has been used to explore the behavior of the dust particle size and dust density for varying plasma powers.

1. Introduction

A dusty plasma comprises a partially ionized gaseous medium containing electrically charged solid dust particles, of which the size can typically range between a few nanometers and several micrometers [1,2]. Typical examples of dusty plasmas can be found both in astrophysics [3,4] and in industrial applications [5]. In the field of astrophysics, one can find dusty plasmas in the rings of Saturn [6,7,8], around Jupiter [9,10], and in the form of accretion disks [11]. In industrial applications, dust particles in plasma applications are often seen as a form of contamination [12]. For example, in photolithography machines, the presence of small dust particles in vulnerable places in the machine can lead to a drastic decrease in the microchip quality [13]. Therefore, the physics behind the release of (nano) particles from surfaces is a widely researched topic [14,15].

These applications formed the spark for investigating dusty plasmas in laboratory environments [1,16]. As dust particles in a plasma behave as electrostatically floating bodies, they will attain charge from the plasma. Generally, dust particles tend to charge negatively since the electron mobility is much higher than the ion mobility [2], which implies that the electron current towards the particle initially outweighs the ion current until an equilibrium between these currents is reached for a certain negative particle surface potential. This interplay between the dust particles and the plasma gives rise to a broad range of physical phenomena; strongly coupled (plasma-dust and dust-dust) interactions manifest in the form of crystallization [17], phase transitions [18], chain formation [19,20], or self-excited dust density waves [21,22,23,24,25,26]. In all of these phenomena, the dust charge plays a major role. The charge of micrometer sized dust particles has been measured using particle resonance methods [27,28] or by applying external electric fields [29,30]. For nanodusty plasmas, the dust charge density can for instance be estimated using laser-induced photodetachment (LIP) [31].

Apart from the charge, the dust particle size is a relevant parameter, especially in the case of nanodusty plasmas. In this type of dusty plasmas, dust particles are chemically grown in the plasma by means of admixing a reactive precursor gas with a noble background gas. For this, organic precursors (such as acetylene (C₂H₂) [32] or methane (CH₄) [33,34]), silicon-based [35], or organosilicon precursor gases [36,37] are often used. Such precursor gases easily dissociate into fragments, after which these fragments efficiently polymerize into larger species. In a nanodusty plasma, generally the growth can qualitatively be described in three distinct phases [38,39]. During the (i) nucleation phase, the plasma chemistry dictates the growth of protoparticles of only a few nanometers in size, which can be charged either positively, negatively, or neutrally. Once a critical density of protoparticles has been reached, the (ii) coagulation phase begins, during which the particles will rapidly coalesce. This phase is suddenly terminated by Coulombic repulsion, as the particles attain a permanently negative charge with increasing size [38,40]. From this point on, dust particle growth happens by means of (iii) accretion or surface growth, which is characterized by the collection of positive ions and radicals from the plasma. As the dust particles grow in size, the magnitude of the (outward pointing) ion drag force increases faster than the (inward pointing) electrostatic force, which causes the growing dust particles to drift outwards. Once the center of the discharge is sufficiently dust-free, a subsequent growth cycle can start.

In the field, much work has been devoted to modeling, ranging from detailed chemistries [41,42,43,44,45] to (discrete) sectional models for particle growth [46,47,48]. From an experimental point of view, several optical diagnostic techniques have been developed for monitoring the dust particle size, such as Mie ellipsometry [49,50,51,52,53,54], light extinction spectroscopy (LES) [55,56], time-resolved laser-induced incandescence (TIRE-LII) [57,58,59,60,61], and laser-induced particle explosive evaporation (LIPEE) [62,63].

In this work, we introduce a new method for in situ particle sizing, based on time-synchronized measurements of the free electron density in the plasma and laser light extinction by the dust cloud confined in it. Using these two quantities allows resolving both the dust particle size and the dust density. The experimental implementation of this method is relatively straightforward and, more importantly, the measurements are non-intrusive.

This work can be seen as a follow-up after our previous work [36]. The basis of the experimental setup and the primary data set is the same, but the focus is on the development of the fitting method. The newly developed method proves to expand the understanding of the measurements, which supports the choice of using the same data set for a direct comparison between the results. The structure of this work is as follows: In Section 2 the experimental setup for the measurement of the free electron density and the laser light extinction is explained in more detail. Section 3 continues by explaining how these types of measurements can be used to disambiguate the dust particle size and the dust density. Section 4 shows the typical results of these measurements and Section 5 summarizes the most important findings of this work.

2. Experiment

This section will describe the used experimental setup. Section 2.1 shows a general description of the equipment needed for plasma generation and gas handling. The section will continue with the discussion of the two key diagnostics for the developed fitting method: (i) microwave cavity resonance spectroscopy (discussed in Section 2.2) and (ii) laser light extinction (discussed in Section 2.3).

2.1. Gas Handling and Plasma Generation

Figure 1 shows a schematic top view of the used experimental setup. Inside a vacuum vessel, a plasma was generated inside a stainless steel cylindrical cavity with an inner radius of $R=0.033 \mathrm{m}$ and a height of $H=0.04 \mathrm{m}$. The base pressure of the setup was $1\times {10}^{-5} \mathrm{Pa}$. In the experiments, argon was used as a background gas and hexamethyldisiloxane (HMDSO, $\begin{array}{c}\hfill \left(\mathrm{CH}3\right)3-\mathrm{Si}-\mathrm{O}-\mathrm{Si}-\left(\mathrm{CH}3\right)3\end{array}$) was used as a reactive precursor gas, with partial pressures of $p_{\mathrm{Ar}}=5.7 \mathrm{Pa}$ and $p_{\mathrm{HMDSO}}=0.8 \mathrm{Pa}$, respectively. This leads to an HMDSO content of $12.3\%$ and a total pressure of $p=6.5 \mathrm{Pa}$. The plasma was generated using a sinusoidal voltage with a frequency of $f=13.56 \mathrm{M}\mathrm{H}\mathrm{z}$ applied to the top wall of the cavity, while the side and bottom are electrically grounded. During the experiments, the plasma power was measured to be $P=23 \mathrm{W}$ using an impedance meter (Impedans Octiv Poly).

2.2. Microwave Cavity Resonance Spectroscopy

The first main diagnostic of this work is microwave cavity resonance spectroscopy (MCRS), which was used to experimentally measure the free electron density in the plasma. This diagnostic technique is based on the determination of the shift in resonance frequency of a standing microwave mode in a cavity due to a change in the permittivity of the medium through which the microwaves propagate. In the past, this measurement has been used to measure the free electron density in a wide variety of plasma types [31,64,65,66]. The difference between the resonance frequency with plasma present $f_{\mathrm{p}}$ and the vacuum resonance frequency $f_1$ is therefore a direct measure of the volume-averaged free electron density ${\widehat{n}}_{\mathrm{e}}$. In the current set of experiments, microwaves were introduced using a vector network analyzer (VNA, Keysight E5072A) in the frequency range f = 5.0–5.2 GHz, closely around the resonance frequency of the ${\mathrm{TE}}_{011}$ mode in the cavity in vacuum (i.e., without plasma), which was determined to be $f_1=5.099 \mathrm{GHz}\pm 0.150 \mathrm{MHz}$. This mode was chosen since its mode pattern (a toroidal region with a maximum intensity at $r=R/2$, see the inset in Figure 1) best resembles the region where the dust particles are expected to be present. The amount of input microwave power into the cavity was fixed at $P_{\mathrm{microwave}}=1 \mathrm{m}\mathrm{W}$, whereas the amount of reflected power was monitored at a sample rate of $10 \mathrm{H}\mathrm{z}$ for the full frequency range. It can be noted that the microwave power is much less than the plasma power, which indicates that the measurements are non-invasive. This resulted in a reflected power spectrum as a function of the microwave frequency f, which represented an (inverted) Lorentzian-type of power response $P\left(f\right)$, of which the frequency corresponding to the minimum in reflected power reveals the resonance frequency at the moment of measurement. The resonance frequency was determined using a Matlab program that fitted the Lorentzian curve to the measured reflected power spectra. In this way, the shift in resonance frequency $\mathrm{\Delta }f=f_{\mathrm{p}}-f_1$ has been determined over the course of the evolution of the dusty plasma.

The Slater perturbation theorem relates the shift in resonance frequency $\mathrm{\Delta }f$ to the change in permittivity, from which the volume-averaged free electron density ${\widehat{n}}_{\mathrm{e}}$ in the plasma can be derived as [67]:

$${\widehat{n}}_{\mathrm{e}}=\frac{8{\pi }^2{\epsilon }_0{m}_{\mathrm{e}}{f}_{\mathrm{p}}^2}{e^2{f}_1}\frac{\mathrm{\Delta }f}{\mathcal{V}},$$

(1)

with the vacuum permittivity denoted by ${\epsilon }_0$, the plasma frequency by $f_{\mathrm{p}}$, the elementary charge by e, the vacuum resonance frequency by $f_1$ (i.e., without plasma), and the electric- field-squared-weighted volume ratio by $\mathcal{V}$. The fact that the plasma volume $V_{\mathrm{p}}$ does not occupy the complete cavity volume $V_{\mathrm{c}}$ (due to the sheath regions at the plasma boundary) is accounted for by this electric-field-squared-weighted volume ratio $\mathcal{V}$:

$$\mathcal{V}=\frac{{\int \int \int }_{V_{\mathrm{p}}}{\left|\mathbf{E}\left(\mathbf{r}\right)\right|}^2{d}^3\mathbf{r}}{{\int \int \int }_{V_{\mathrm{c}}}{\left|\mathbf{E}\left(\mathbf{r}\right)\right|}^2{d}^3\mathbf{r}}.$$

(2)

The microwave electric field $\mathbf{E}\left(\mathbf{r}\right)$ is determined using COMSOL Multiphysics using the dimensions of the cavity used in the experiments, and zero potential boundary conditions at the walls of the cavity.

2.3. Laser Light Extinction

The second key diagnostic for this approach is laser light extinction. The experimental implementation is rather straightforward: A green laser ($\lambda =532 \mathrm{n}\mathrm{m}$) was sent radially through the center of the cavity via two slits. After passing through the cavity, its intensity was measured using a photodiode (Thorlabs DET10A). When a dust cloud is present in the cavity, part of the light will be scattered or absorbed by the dust particles, which decreases the intensity measured by the photodiode. The quantity of interest is the transmittance $\widehat{T}$:

$$\widehat{T}=\frac{I}{I_0},$$

(3)

i.e., the ratio between the light intensity with $\left(I\right)$ and without $\left(I_0\right)$ the presence of a dust cloud, respectively. The transmittance of the dust cloud was recorded as a function of time using an oscilloscope (Agilent DSO7054A).

3. Methods

The time-synchronized measurement of the free electron density in the plasma and laser light extinction by the dust cloud can be used to determine the dust particle size and dust density. Figure 2 provides a schematic overview of the fitting method, which will be discussed in more detail in the remainder of this section. In Section 3.1, the experimental part (depicted in green in Figure 2) is discussed: The raw data is shown and the measured quantities are derived. Section 3.2 follows by deriving the matching quantities from theory (depicted in blue in Figure 2). Section 3.3 continues with the fitting procedure for obtaining the dust particle size and density from the decrease in electron density and the transmittance, as shown by the central part in Figure 2.

3.1. Measurement Data

Figure 3 shows the experimental measurement data, i.e., the free electron density ${\widehat{n}}_{\mathrm{e}}$ measured using MCRS in panel (a) and the transmittance signal $\widehat{T}$ measured by laser light extinction in panel (b), from the moment of plasma ignition at $t=0 \mathrm{s}$ until the end of the second dust growth cycle at $t=250 \mathrm{s}$. In this section, we will discuss the general behavior of the signals over the course of time.

From the free electron density, it can be seen that the electron density is initially about ${\widehat{n}}_{\mathrm{e}}\approx 12\times {10}^{15} {\mathrm{m}}^{-3}$, but quickly decreases by a factor of 6 to ${\widehat{n}}_{\mathrm{e}}\approx 2\times {10}^{15} {\mathrm{m}}^{-3}$ about $20 \mathrm{s}$ after plasma ignition. For the current method, we are mainly interested in the decrease of the electron density with respect to the initial value, $\mathrm{\Delta }{\widehat{n}}_{\mathrm{e}}\left(t\right)={\widehat{n}}_{\mathrm{e},\mathrm{initial}}-{\widehat{n}}_{\mathrm{e}}\left(t\right)$ (see Figure 3a), since it can be attributed to the charging of the growing dust particles. Note that here the subscript `initial’ refers to the moment shortly after ignition of the plasma; i.e., when the electron and ion densities have built up, but when there are no dust particles yet. In practice, this corresponds to the values at the time of the first measurement point, which is typically measured $100 \mathrm{m}\mathrm{s}$ after switching on the plasma. At about $t=35 \mathrm{s}$ the transmittance starts to decay linearly. During this period, the electron density changes only slightly up to the point that the transmittance value is at its minimum. This moment in time, corresponding to a minimum in $\widehat{T}$, represents the formation of the void: a volume in the discharge center from which the dust grains are being removed when the outward-pointing ion drag force overcomes the inward-pointing electrostatic force. After this moment, both the electron density and transmittance start to increase again, signaling that the dust grains are effectively being transported toward the plasma edge and will eventually leave the plasma. It is interesting to note that both the electron density (around $t=110 \mathrm{s}$) and the transmittance (around $t=150 \mathrm{s}$) do not fully recover back to their initial values before the subsequent growth cycle starts. This indicates that a new growth cycle commences as soon as the center of the discharge is sufficiently dust-free, even though there is still a substantial amount of dust in the periphery.

3.2. 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:

$$n_{\mathrm{i}+}=n_{\mathrm{e}}+n_{\mathrm{i}}-+n_{\mathrm{d}}\left|Z_{\mathrm{d}}\right|,$$

(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 $n_{\mathrm{i}}+\left(t\right)=n_{\mathrm{i}}+,\mathrm{initial}$ and $n_{\mathrm{i}}-\left(t\right)=n_{\mathrm{i}}-,\mathrm{initial}$, and that $n_{\mathrm{d}},\mathrm{initial}=0$ as dust grains are not formed yet. Using these assumptions, the change of electron density $\mathrm{\Delta }{n}_{\mathrm{e}}$ can be derived from Equation (4):

$$\begin{array}{cc}\hfill \mathrm{\Delta }{n}_{\mathrm{e}}\left(t\right)& =n_{\mathrm{e}},\mathrm{initial}-n_{\mathrm{e}}\left(t\right)\hfill \\ \hfill & =\left(n_{\mathrm{i}}+,\mathrm{initial}-n_{\mathrm{i}}-,\mathrm{initial}\right)-\left(n_{\mathrm{i}}+\left(t\right)-n_{\mathrm{i}}-\left(t\right)-n_{\mathrm{d}}\left(t\right)Z_{\mathrm{d}}\left(t\right)\right)\hfill \\ \hfill & =n_{\mathrm{d}}\left(t\right)Z_{\mathrm{d}}\left(t\right).\hfill \end{array}$$

(5)

The equilibrium charge number $Z_{\mathrm{d}}$ of dust particles with a radius a is governed by the balance of currents towards a dust grain:

$$I_{\mathrm{i}+}=I_{\mathrm{i}-}+I_{\mathrm{e}},$$

(6)

where the positive ion current $I_{\mathrm{i}}$ (i.e., attracted towards the particle) is defined as:

$$I_{\mathrm{i}+}=e\pi a^2{n}_{\mathrm{i}}\sqrt{\frac{8k_{\mathrm{B}}{T}_{\mathrm{i}}}{\pi m_{\mathrm{i}}}}\left(1-\frac{e{V}_{\mathrm{d}}}{k_{\mathrm{B}}{T}_{\mathrm{i}}}\right).$$

(7)

For electrons and negative ions (i.e., repelled from the particle), the current is defined as

$$I_{\mathrm{e},\mathrm{i}-}=-e\pi a^2{n}_{\mathrm{e},\mathrm{i}-}\sqrt{\frac{8k_{\mathrm{B}}{T}_{\mathrm{e},\mathrm{i}-}}{\pi m_{\mathrm{e},\mathrm{i}-}}}exp\left(\frac{e{V}_{\mathrm{d}}}{k_{\mathrm{B}}{T}_{\mathrm{e},\mathrm{i}-}}\right),$$

(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:

$$\begin{array}{cc}\hfill \frac{I_{\mathrm{i}-}}{I_{\mathrm{e}}}& =\frac{-e\pi a^2{n}_{\mathrm{i}-}\sqrt{\frac{8k_{\mathrm{B}}{T}_{\mathrm{i}}-}{\pi m_{\mathrm{i}}-}}exp\left(\frac{e{V}_{\mathrm{d}}}{k_{\mathrm{B}}{T}_{\mathrm{i}-}}\right)}{-e\pi a^2{n}_{\mathrm{e}}\sqrt{\frac{8k_{\mathrm{B}}{T}_{\mathrm{e}}}{\pi m_{\mathrm{e}}}}exp\left(\frac{e{V}_{\mathrm{d}}}{k_{\mathrm{B}}{T}_{\mathrm{e}}}\right)}\hfill \\ \hfill & =\frac{n_{\mathrm{i}}-}{n_{\mathrm{e}}}\frac{m_{\mathrm{e}}{T}_{\mathrm{i}-}}{m_{\mathrm{i}-}{T}_{\mathrm{e}}}exp\left(\frac{e{V}_{\mathrm{d}}}{k_{\mathrm{B}}{T}_{\mathrm{i}-}}\right)\ll 1,\hfill \end{array}$$

(9)

where typically $m_{\mathrm{e}}\ll m_{\mathrm{i}-}$, $T_{\mathrm{i}-}\ll T_{\mathrm{e}}$, and $exp\left(\frac{e{V}_{\mathrm{d}}}{k_{\mathrm{B}}{T}_{\mathrm{i}-}}\right)\ll 1$ because $e{V}_{\mathrm{d}}\sim 1 \mathrm{e}\mathrm{V}$ and $k_{\mathrm{B}}{T}_{\mathrm{i}-}\approx 0.026 \mathrm{e}\mathrm{V}$ (at room temperature). As a consequence, the dust floating potential $V_{\mathrm{d}}$ can be determined from Equation (6) in combination with Equations (7) and (8), neglecting the contribution of $I_{\mathrm{i}-}$. This relates to the equilibrium charge $Q_{\mathrm{d}}=e{Z}_{\mathrm{d}}$ following the spherical capacitor model:

$$Q_{\mathrm{d}}=e{Z}_{\mathrm{d}}=4\pi {\epsilon }_{0}a{V}_{\mathrm{d}}.$$

(10)

From Equation (5), it can be seen that the quantity $n_{\mathrm{d}}{Z}_{\mathrm{d}}$ can be related to the change in the electron density due to the formation and growth of the dust particles. In order to make the OML equations only a function of a and $n_{\mathrm{d}}$, one should make reasonable assumptions for the (negative) ion density and the electron temperature. From modeling efforts in the nucleation phase of nanodusty plasmas [69,70,71,71], the (total) positive ion density is expected to be (usually about one order of magnitude) higher than the electron density, due to the presence of negative ions. Therefore, in this work it is assumed that at plasma ignition the total positive ion density is five-fold the electron density, i.e., $n_{\mathrm{i}+,\mathrm{initial}}=5n_{\mathrm{e},\mathrm{initial}}$, so that, by quasi-neutrality and the initial absence of dust, $n_{\mathrm{i}-,\mathrm{initial}}=4n_{\mathrm{e},\mathrm{initial}}$ and $n_{\mathrm{d},\mathrm{initial}}=0$. For the electron temperature a constant value of $T_{\mathrm{e}}=3 \mathrm{e}\mathrm{V}$ is assumed. Section 4.2 provides more details about these assumptions and performs a sensitivity analysis.

3.2.2. Laser Light Extinction

Using the Lambert–Beer law [1], the transmittance can be related to properties of the dust particles:

$$T=exp\left(-Q_{\mathrm{ext}}\left(n,{\lambda }_{\mathrm{exc}},a\right)\pi a^2{n}_{\mathrm{d}}L\right).$$

(11)

In this equation, $Q_{\mathrm{ext}}\left(n,{\lambda }_{\mathrm{exc}},a\right)$ is the extinction coefficient, a is the radius of the dust, $n_{\mathrm{d}}$ is the dust density, and L is the extinction path length, which is taken to be equal to the internal diameter of the cavity, i.e., $L=2R=0.066 \mathrm{m}$. The extinction coefficient is determined using Mie theory and is a function of the complex refractive index, the excitation wavelength, and the dust particle radius. Here, the refractive index at ${\lambda }_{\mathrm{exc}}=532 \mathrm{n}\mathrm{m}$ of $n=1.45+0.00i$ was used, as determined from spectroscopic ellipsometry measurements on HMDSO thin films reported in literature [72]. Unfortunately, the optical characterization data of dust particles formed in a low-pressure Ar-HMDSO plasma is not available in current literature. It should therefore be noted that the processing conditions (i.e., pressure, power, and background gas) of the thin films are slightly different from the conditions in this work and that it is assumed that the optical properties of thin films are similar to those of a cloud of dust particles.

3.3. Optimization Procedure

This section will provide more details about the optimization procedure for obtaining the dust particle size and dust density from the measurements. As illustrated in Figure 2 and the previous sections, we have obtained two measured quantities $\left(\mathrm{\Delta }{\widehat{n}}_{\mathrm{e}},\widehat{T}\right)$, which can be directly compared to theoretically determined values $\left(n_{\mathrm{d}}{Z}_{\mathrm{d}},T\right)$ using Equations (5) and (11), respectively. For the sake of clarity, the experimentally measured quantities are denoted with a hat symbol, while the quantities derived from theory are not.

The optimization procedure works as follows: (i) For a wide range of a and $n_{\mathrm{d}}$, the theoretical values of $\left(n_{\mathrm{d}}{Z}_{\mathrm{d}},T\right)$ are calculated. Then, (ii) the relative errors with respect to the measured values $\left(\mathrm{\Delta }{\widehat{n}}_{\mathrm{e}},\widehat{T}\right)$ are calculated using ${\chi }_{\mathrm{mcrs}}=\left|\frac{n_{\mathrm{d}}{Z}_{\mathrm{d}}-\mathrm{\Delta }{\widehat{n}}_{\mathrm{e}}}{\mathrm{\Delta }{\widehat{n}}_{\mathrm{e}}}\right|$ and ${\chi }_{\mathrm{ext}}=\left|\frac{T-\widehat{T}}{\widehat{T}}\right|$. Figure 4 shows a qualitative visualization of the relative errors for a single data point in the measured time trace. The red curve in the plot represents pairs of $\left(a,n_{\mathrm{d}}\right)$ for which the goodness of fit with respect to the MCRS measurements is high, i.e., the relative MCRS error ${\chi }_{\mathrm{mcrs}}$ is low. The blue curve corresponds to a region for which the goodness of fit with respect to the extinction measurements is high, i.e., the relative extinction error ${\chi }_{\mathrm{ext}}$ is low. Since both of the diagnostics are measured in a time-synchronized fashion and therefore probe the same sample of dust, the most probable $\left(a,n_{\mathrm{d}}\right)$ pair lies on the intersection of the two regions; where both ${\chi }_{\mathrm{mcrs}}$ and ${\chi }_{\mathrm{ext}}$ are low, which is indicated with the magenta region in Figure 4. In the example of Figure 4, this would result in a dust particle radius of $a=110 \mathrm{n}\mathrm{m}$ and a dust density of $n_{\mathrm{d}}=1.19\times {10}^{14} {\mathrm{m}}^{-3}$, as indicated by the black circle.

4. Results and Discussion

This section shows the typical results of the determination of the dust particle size and density from combined MCRS and extinction measurements. Section 4.1 shows the general results of a measurement and compares the findings to similar works. Section 4.2 continues with a sensitivity analysis for two quantities for which an assumption is necessary for the analysis and which are challenging to measure, namely the electron temperature and the negative ion density. Finally, in Section 4.3 the method is used to observe trends in the dust particle size and dust density when varying the plasma power.

4.1. General Results

In this section, the results of a typical time-resolved measurement of the dust particle size and dust density using the method proposed in this work are discussed. Using the developed optimization procedure explained in Section 3.3, the fitted dust particle radius a and the dust density $n_{\mathrm{d}}$ are shown in Figure 5. After igniting the plasma at $t=0 \mathrm{s}$, the dust particles will be generated through the polymerization of precursor species. The lower detection limit of the method is determined by the sensitivity of the extinction measurement; at early times, the dust particle size and/or density is too small to cause sufficient extinction of the laser light (i.e., $\widehat{T}\approx 1$). Therefore, the fitting procedure is started when $\widehat{T}\le 0.99$, i.e., from $t=34 \mathrm{s}$ onward, resulting in a detection limit for particles with a radius of $a_{\min }=59 \mathrm{n}\mathrm{m}$. As soon as the particles can generate enough extinction, a linear increase in the particle radius and a decrease in the dust density are observed. The growth rate in this linear regime is determined to be $1.30 \mathrm{n}\mathrm{m}/\mathrm{s}$, which lies in the same range as the experiments performed in $\begin{array}{c}\hfill \mathrm{C}2\mathrm{H}2\end{array}$ reported elsewhere [50,73].

At around $t=83 \mathrm{s}$, the dust density transitions into a sharper decrease. This coincides with the turning point of the transmittance (see panel (b) in Figure 3 and is indicated with the red vertical lines in Figure 5), which signifies the appearance of the void. It should be noted that this implies that with the presence of the void the path length L (in Equation (11)) is overestimated, which eventually underestimates the quantity $a^2{n}_{\mathrm{d}}$. From this moment on, the interpretation of the dust particle size and the dust density is complicated, since the path length is no longer fixed and since dust particles from two subsequent generations coexist, implying two different dust size distributions [51]. Therefore, the results shown in the following sections will be limited to the first growth cycle, and before the appearance of the void.

It is interesting to note that the mean dust charge $Q_{\mathrm{d}}$ also becomes available as soon as the dust particle radius and the dust density have been determined. Using Equations (5) and (10), the total (negative) dust particle charge is shown to become more negative over time, ranging from −50 elementary charges at $t=34 \mathrm{s}$ to −130 elementary charges at $t=83 \mathrm{s}$.

Several other researchers focused on the growth of dust particles in low-pressure nanodusty plasmas. In order to get a sense of whether the retrieved dust particle size is consistent with literature, an overview of the methods, precursor gases, and operating conditions have been provided in

Table 1. Overview of the characteristics of low-pressure dust growth as reported in the literature.

Author	Used Method	Precursor	Plasma Power (W)	(Total) Pressure (Pa)	Growth Time (s)	Dust Particle Size (nm)
Groth et al.  [50]	Ellipsometry	C2H2	8	24	85	200
Groth et al. [51]	Ellipsometry	C2H2	20	24	90	100
Dworschak et al. [73]	Ellipsometry + AFM	C2H2	8	25	130	150
Killer et al. [74]	Ellipsometry	C2H2	8	24	129	210
Tadsen et al. [23]	Ellipsometry	C2H2	8	11.5	85	184
Chutia et al. [75]	SEM	C2H2	1	20	600	300
Van de Wetering et al. [61]	LII	C2H2	7	10	${}^{\left(\mathrm{a}\right)}$	100–165 ${}^{\left(\mathrm{b}\right)}$
Mitic et al. [76]	White light scattering	C2H2	0.3	20	150	260
Eom et al. [57]	TiRe-LII	SiH4	5	2.7	1000	200

^(a): Continuous C₂H₂ flow, switched off after 35 min, ^(b): Depending on the refractive index data set.

. The table indicates that under low-pressure conditions dust particles are produced with a typical radius of a = 100 nm to 300 nm, depending on the pressure, plasma power, and reactor geometry. Even though this range of dust particle sizes matches the retrieved particle size from the fitting procedure, a direct comparison is difficult. Therefore, in Appendix A.1, the method has been applied to MCRS and extinction data from previous work [31]. which could be directly compared to SEM images of the dust particles collected during the same measurement. In Appendix A, it is shown that the final retrieved particle size from the fitting method $\left(a_{\mathrm{fit}}=156 \mathrm{n}\mathrm{m}\right)$ is close to the mean dust particle size from the SEM image ($a_{\mathrm{SEM},\mathrm{mean}}=146 \mathrm{n}\mathrm{m}$ and a standard deviation of ${\sigma }_{\mathrm{SEM}}=7 \mathrm{n}\mathrm{m}$).

4.2. Sensitivity Analysis for the Electron Temperature and the Negative Ion Density

As mentioned in Section 3.2.1, the theoretical determination of the charge density $n_{\mathrm{d}}{Z}_{\mathrm{d}}$ and the transmittance T depends strongly on the assumed electron temperature through Equation (8) and the negative ion density through Equation (4). Both the electron temperature and negative ion density are difficult to determine experimentally or by modeling, and should therefore be assumed based on typical values found in the literature. In order to assess the sensitivity of the fitting procedure to these assumptions, the fitting procedure is also evaluated for different values for $T_{\mathrm{e}}$ and $n_{\mathrm{i}}-$.

Figure 6 shows the results of varying only the electron temperature $T_{\mathrm{e}}$, while still using the same data set. The figure depicts the obtained dust particle radius and dust density as a function of time for $T_{\mathrm{e}}=2 \mathrm{e}\mathrm{V}$, $T_{\mathrm{e}}=3 \mathrm{e}\mathrm{V}$ and $T_{\mathrm{e}}=4 \mathrm{e}\mathrm{V}$. Variations in the electron temperature lead to relatively small changes; the average dust particle size variation over the measured time range is $-7.0 \mathrm{n}\mathrm{m}$ and $+5.1 \mathrm{n}\mathrm{m}$, for $T_{\mathrm{e}}=2 \mathrm{e}\mathrm{V}$ and $T_{\mathrm{e}}=4 \mathrm{e}\mathrm{V}$, respectively, both with respect to the base case of $T_{\mathrm{e}}=3 \mathrm{e}\mathrm{V}$. For the dust density, the average variation is found to be $+0.50\times {10}^{14} {\mathrm{m}}^{-3}$ and $-0.26\times {10}^{14} {\mathrm{m}}^{-3}$, with respect to the default case.

This dependency can be explained as follows: From the OML theory, it can be derived that the dust floating potential $V_{\mathrm{d}}$ increases with the increasing electron temperature $T_{\mathrm{e}}$. Physically, this can be understood from the fact that a stronger potential barrier is necessary to repel more energetic electrons. For a constant or increasing a, the dust charge $Z_{\mathrm{d}}$ also increases with increasing $T_{\mathrm{e}}$, which implies a larger Coulomb repulsion among the dust particles. Via Equation (5), it can be seen that $n_{\mathrm{d}}$ must decrease for a (fixed) measured $\mathrm{\Delta }{\widehat{n}}_{\mathrm{e}}$. Using the Lambert–Beer law in Equation (11), it can be seen that the dust particle radius a indeed must increase with the decreasing dust density $n_{\mathrm{d}}$ for a fixed value of the measured transmittance.

Figure 7 shows the effect of varying the negative ion density $n_{\mathrm{i}}-$. Here, we see the dust particle radius and dust density as a function of time for $n_{\mathrm{i}}-=2n_{\mathrm{e}0}$, $n_{\mathrm{i}}-=4n_{\mathrm{e}0}$ and $n_{\mathrm{i}}-=8n_{\mathrm{e}0}$. The variations in dust particle radius are similar in magnitude to the deviations due to variations in the electron temperature; the average dust particle size variation over the measured time range is $+9.7 \mathrm{n}\mathrm{m}$ and $-11.4 \mathrm{n}\mathrm{m}$, for $n_{\mathrm{i}}-=2n_{\mathrm{e}0}$ and $n_{\mathrm{i}}-=8n_{\mathrm{e}0}$, respectively, both with respect to the default case of $n_{\mathrm{i}}-=4n_{\mathrm{e}0}$. Similarly, the average difference in the dust density is about $-0.44\times {10}^{14} {\mathrm{m}}^{-3}$ and $+0.95\times {10}^{14} {\mathrm{m}}^{-3}$, compared to the default case.

These trends can be explained as follows: For a higher negative ion density, the positive ion density also increases, which follows from the quasi-neutrality condition shortly after igniting the plasma. An increase in the positive ion density $n_{\mathrm{i}+}$ results in a larger ion current, which allows a higher electron current at the floating potential, as Equation (6) suggests. This implies that the floating potential becomes less negative for increasing the negative ion density, and that the dust charge $Z_{\mathrm{d}}$ decreases for the fixed dust particle radius a. For a fixed $\mathrm{\Delta }{\widehat{n}}_{\mathrm{e}}$, this results in a higher dust density $n_{\mathrm{d}}$. From the Lambert–Beer law, it can be seen that this implies a lower dust particle radius a, which is consistent with the effect of a decreasing dust charge $Z_{\mathrm{d}}$.

4.3. Influence of Changing Plasma Power on the Dust Particle Size and Dust Density

In the previous sections, we have introduced the method of determining the dust particle radius and the dust density from the measurement of the free electron density and the laser light extinction. In this section, we will continue by investigating the behavior of the dust particle radius and dust density as a function of plasma power. In these experiments, the partial pressures of argon and HMDSO are similar to those described in Section 3.1, while the power is varied from 8 W to 23 W. In line with the reasoning presented in Section 4.1 we selected the point in time right before the formation of the void, which coincides with the turning point in the transmittance (see Figure 3b). At this point, the dust particle radius and the dust density were recorded and plotted in Figure 8, both as a function of plasma power. In this Figure, the error bars arise from the propagation of errors in the measured electron density ${\widehat{n}}_{\mathrm{e}}$ and the transmittance $\widehat{T}$, which via the fitting procedure results in errors in the estimated a and $n_{\mathrm{d}}$. In order to guide the eye, polynomial fits are included through the data points.

Figure 8 shows that by increasing the plasma power, the determined dust particle radius decreases while the dust density increases. By increasing the plasma power, (i) the overall plasma densities increase, while (ii) the growth cycle time decreases. For both effects, we will hypothesize about what happens to the resulting dust particle size a during the accretion phase. An increase in the plasma densities will increase the ion/radical fluxes towards the particle, leading to faster particle growth. The decreasing growth cycle time is a direct result of a change in the force balance acting on the particles. By increasing the plasma power, both the densities and the electric field strength increase, which eventually causes the ion drag force to be able to overcome the electrostatic force at shorter timescales [36]. This leads to a shorter growth cycle, which translates to a shorter residence time of the dust particles in the plasma and eventually leads to smaller dust particles.

From the previous discussion, the two mentioned effects counteract each other. In our previous work [36] it was argued that using the transmittance $\widehat{T}$ enabled measuring only the combined quantity $a^2{n}_{\mathrm{d}}$, which made it difficult to differentiate between effects in the dust particle size and the dust density. However, using the newly developed method, Figure 8 shows that we are able to disentangle the behavior of the dust particle size and density evolution. This indicates that for higher plasma power, the effect of the shorter growth cycle leads to smaller dust particles, even though the fluxes towards the dust particles’ surfaces are larger.

5. Conclusions

In this work, we introduced a novel method for determining the dust particle size and dust density in a low-pressure nanodusty Ar-HMDSO plasma, derived from measurements of the free electron density and laser light extinction. Comparing the measured values of the decrease of the electron density and the transmittance to those predicted by OML theory and the Lambert–Beer law, respectively, the combined error can be minimized such that the best-fitting values of the dust particle size and dust density can be determined.

As expected from the raw data and literature, the dust growth in the first growth cycle is found to be linear, with a typical dust particle size of $\sim 140\mathrm{n}\mathrm{m}$ right before the appearance of the void. The dust density is usually in the order of $\sim {10}^{14}$${\mathrm{m}}^{-3}$. Furthermore, it was observed that the interpretation of the data can become obscured by the appearance of the void region and the growth of a successive generation of dust particles. Generally, the dust particle size and dust density show opposite trends: When the dust particle size increases, the dust density decreases. The fitting procedure relies on assumptions of the electron temperature and the negative ion density; two quantities that are difficult to estimate or experimentally determine. The sensitivity of the fitting procedure to these parameters was investigated, which yielded similar trends in both dust particle size and density, while only the absolute values shifted. Finally, the method was used to investigate the dependence of the dust particle size and dust density on the plasma power. Interestingly, a lower plasma power yielded larger dust particle sizes, which is a direct consequence of the slower dust growth cycle, i.e., the longer residence time in the plasma.

The proposed fitting procedure shows that the time-synchronized measurement of the free electron density of the plasma and the laser light extinction by the dust cloud can reveal information about the dust particle size and dust density. This novel, experimentally convenient and non-intrusive method provides more insight into the fundamentals of nanodusty plasma physics.