A review of nanoparticle decharging in atmospheric pressure plasma afterglows

Abstract

Plasma afterglows interacting with dust grains present a dynamic environment in which negatively charged dust grains leaving the plasma bulk experience an environment with plasma conditions transient in space and time. This review focuses on the impact of atmospheric pressure on the physics concerning the interaction between dust grains and the plasma afterglow. The four stage model commonly applied to low pressure dusty plasma afterglows provides a guiding framework for the analysis describing the phases of electron temperature relaxation, ambipolar diffusion, ambipolar-to-free diffusion transition and free diffusion. This work is completed by a non-exhaustive overview of research gaps and opportunities in the young and vibrant field of atmospheric pressure dusty plasma afterglows.

1 Introduction

Dust grains immersed in a plasma environment contribute strongly to the collective behavior by virtue of the electric charge obtained through interactions with the electrons, ions and electric fields. For many applications, from nanoparticle production [1, 2] to contamination control [3, 4], charged dust grains eventually leave the active plasma environment for subsequent processing. As a consequence, charged dust grains experience the transition from the active plasma region, through the ion-rich plasma afterglow with a net positive space charge, into an equilibrium environment containing neutral gas and long-lived radicals.

Early observations [5–7] of dust with residual charges in decaying plasmas at low pressure triggered investigations of dust (de)charging in temporal and spatial afterglow plasmas [8–18]. By contrast, the interaction of nanoparticles with atmospheric pressure afterglow plasmas constitutes a relatively unexplored field compared to low pressure dusty plasma afterglows. Nevertheless, the synthesis of nanocrystals at atmospheric pressure provides a low cost method to produce and deposit nanoparticles [19–22] with a specific structure [23, 24] and optical properties [25, 26], while the deposition of thin films using atmospheric pressure plasmas represents a cost effective alternative to vacuum processes [27–30] and provides the potential to include nanoparticles [20]. With the advent of these upcoming nanotechnologies, the demand for sensing abilities of such nanoparticles (or ultrafine particles) is also increasingly drawing attention [31–34]. Further progress of these technologies is limited by a lack of understanding of the interaction between nanoparticles and afterglow plasmas at atmospheric pressure.

Modeling efforts and experiments on atmospheric pressure afterglow plasmas interacting with dust grains have shown significant progress towards a more detailed understanding of these environments [35–41]. The models and experiments focused on the transition between the bulk plasma, where dust grains are negatively charged, towards an equilibrium gas in which dust grains can be left with residual positive charges. Such charging effects are heavily affected by deviations from bulk plasma properties such as quasi-neutrality, ambipolar diffusion and electron temperatures exceeding those of the gaseous and ion species. In turn, the charge state of nanoparticles strongly influences the aggregation of plasma species and coagulation with other nanoparticles [37]. Consequently, the spatial afterglow of such flow-through plasmas determines largely the resulting size distribution and material structure of nanoparticles synthesized using atmospheric pressure plasma reactors.

This review collects the existing knowledge on atmospheric pressure dusty plasma afterglows following the established model for low pressure dusty plasma afterglows [7, 15, 17, 42–45], and finalizes with a discussion of open gaps in this young and vibrant field of dusty plasma research.

2 Physics of dusty plasma afterglows

In the plasma afterglow, four stages can be identified during which the dust grain charge diminishes due to the decaying plasma conditions [15, 17]: (I) electron temperature relaxation, (II) ambipolar diffusion, (III) ambipolar-to-free diffusion transition, (IV) free diffusion, which are followed by an equilibrium with residual dust charge. In this work, there is no explicit definition of short τ⁰ and long τ^∞ time scales as used in the definitions by Ivlev et al [7] and Couëdel et al [15, 42–44]. Instead, the time scales denoted in this work are calculated for specific plasma conditions that approximate an early (close to the short time scales) or late (before reaching equilibrium where t → ∞) afterglow plasma at atmospheric pressure.

The physics of dust grains interacting with atmospheric pressure plasma afterglows is mostly impacted by the smaller mean free paths (i.e. much higher collision frequencies) compared to vacuum conditions. The mean free path l_sn of a charged or neutral species due to collisions with the neutral gas species is defined asfor species s, which could be electrons, ions or radicals, for example, where n_n denotes the neutral gas density and σ_sn the collision cross section. As a consequence, the transfer of energy and momentum occurs much more frequently and limits the mobility of the species involved in, for instance, electron temperature relaxation and the currents imposed on dust grains.

2.1 Stage I: electron temperature relaxation

During the first phase (I), the electron temperature drops quickly due to the absence of an active ionization source. This temperature drop is caused by frequent collisions between energetic electrons and neutral gas particles at room temperature, which occurs on a time scale typically about τ_T ∼10^–4 s under low pressure conditions (given for p = 0.4 mbar [15]). The charge fluctuation time scale τ_Q ∼ 10^–6 − 10^–5 s, as given by Couëdel et al [15] for the same conditions, is much smaller than τ_T. This means that the dust charge in the low pressure afterglow remains in quasi-steady state with the plasma conditions. Hence, under vacuum conditions, the relaxation of the electron temperature occurs sufficiently slow with respect to the (nearly temperature-independent) charging time scale so that the magnitude of the dust charge decreases steadily with the decaying plasma conditions [15].

Under atmospheric conditions, this situation differs significantly due to the electron-neutral collisionality. This effect has been accounted for by various models [37, 46], where a global model for the temporal afterglow was used due to the lack of a spatial afterglow model for the electron temperature. Given the scaling of the electron temperature relaxation timescale [15, 47],where σ_en denotes the electron-neutral collision cross section and the electron thermal velocity, k_B the Boltzmann constant, T_e the electron temperature, and m_e the electron (rest) mass. As can be seen from Eq. 2, τ_T is reduced by the order of magnitude difference in neutral gas pressure for low temperature plasmas. As observed from experiments, the timescale of decay of the effective collision frequency (equivalent to the electron-neutral collision frequency) was on the order of ∼1μs for a helium plasma afterglow entering an air environment [48], and on the order of ∼ 0.1μs for electrons in an argon-oxygen afterglow [49]. Consequently, the electron temperature relaxation time scale is much shorter than that can be expected at low pressure.

The difference in electron temperature relaxation time scale between low and atmospheric pressure conditions implies that the dust charging time scale also needs to be assessed with respect to an enhanced pressure. The ion current experiences collisions much more frequently at atmospheric pressure. To assess the contribution of the collisionless , collision-enhanced and hydrodynamic ion currents, the (capture radius) Knudsen number is used aswith α = 1.22 accounting for the Maxwellian ion energy distribution and R₀ the capture radius, both as defined by Equation [5, 6] in Gatti and Kortshagen [50]. This Knudsen number is at most 0.02 for dust grain radii a_d = 10–300 nm, argon ions in argon at atmospheric pressure σ_in ≈ 5.8 × 10^–19 m⁻² [51], n_n = 2.6 × 10²⁵ m⁻³, plasma density of n_e = n_i = 10¹⁸ m⁻³, and otherwise similar conditions used by Couëdel et al [15]. Hence, this means that only the hydrodynamic ion current contributes significantly to the total ion current. Consequently, the charge fluctuation timescale can be determined using the collisionless electron current (i.e. ) and the hydrodynamic ion current by linearization of the current balance,where and are based on the definitions by Gatti and Kortshagen [50], ɛ₀ denotes the vacuum permittivity, and V the dust grain potential. This results in a charge fluctuation time scale at atmospheric pressure expressed as follows,where e denotes the elementary charge, μ_i = eD_i/T_i the ion mobility, D_i the ion diffusion coefficient, and T_i the ion temperature. From Eq. 5, it can be evaluated that 89 ns for n_e = n_i = 10¹⁸ m⁻³, a_d = 100 nm, T_e = 2 eV, T_i = 0.025 eV, and equilibrium dust floating potential V_d = −0.14 V. Clearly, the charge fluctuation time scale τ_Q is close to the order of the electron temperature relaxation time scale τ_T ∼100 ns, which is vastly different from the ordering at low pressure.

2.2 Stage II: ambipolar diffusion

The second stage concerns the decay of the plasma density, from (the end of) the bulk plasma, until the point at which free diffusion is achieved. Plasma loss is due to ambipolar diffusion onto the reactor walls, recombination at the surface of dust grains and volume recombination of charged species.

Ambipolar diffusion dominates free diffusion as long as the electron Debye length λ_De is much smaller than the reactor size Λ during the early afterglow. In low temperature plasmas, ambipolar diffusion is governed by the ion diffusivity D_i [52, 53], due to the high mobility of electrons μ_i ≪ μ_e, so thatand the ambipolar diffusion time scale τ_amb boils down to [54]:By evaluation of Eq. 7 for low and atmospheric pressure conditions, one arrives at ambipolar diffusion time scales of 2 × 10^–4 s and 2 × 10^–2 s for electrons and ions at room temperature. This shows that ambipolar diffusion is much more limited at atmospheric pressure assuming similar conditions for the reduced electric field and mobility, and that the ambipolar diffusion time scales linearly with neutral gas pressure.

The absorption of charged species at the surface of the dust grains provides the second plasma loss mechanism, if the dust density is sufficiently high [55, 56] by assuming here n_d = 10¹³ m⁻³. As the dust grains are negatively charged by the bulk plasma, the decrease of the ion density is governed by the flux of ions necessitated for recombination, Γ_i, lost at the surface of the dust grains, :Under low pressure conditions, the particle absorption time τ_A is thus determined by the OML ion current [15],where denotes the thermal ion velocity with m_i the ion mass. At atmospheric pressure, p = 1000 mbar, the hydrodynamic ion current is dominant so that the particle absorption time becomes:where λ_DL denotes the linearized Debye length [57]. Using Eq. 9, 10 for the particle absorption time scales, it can be seen that the orderingdiffers vastly from the ordering at atmospheric pressureThe outcome for low pressure agrees with findings under such conditions following the four stage model [42]. By contrast, the recombination of ions at the dust surface provides a significant sink compared to reactor wall losses at atmospheric pressure.

Third, ion-electron recombination plays a significant role at atmospheric pressure. For the purpose of comparison, the recombination time scale is based on collisional-radiative recombination for Ar⁺ with electrons [58]:Under vacuum, n_e = 10¹⁵ m⁻³ and T_e = T_i, the recombination time scale is in the order of seconds already in the early afterglow, while τ_CR ∼ 10^–5 s for n_e = 10¹⁸ m⁻³ and electrons at room temperature. Please note that also dissociative recombination of molecular ions and other three-body processes can become significant and that this analysis is purely illustrative for the comparison. Consequently, electron-ion recombination could play an important role for plasma loss in the afterglow, although its contribution diminishes quickly with decreasing plasma density.

In conclusion, the plasma density decays steadily due to the loss of electrons and ions to the reactor walls and the surface of dust grains, and by volume recombination at atmospheric pressure. Moreover, the dust density, plasma density, dust size and the gas composition can significantly influence the foregoing ordering of time scales and dominance of the associated processes. The steady decay of the plasma density amounts to two important implications. First, the electron Debye length increases during the plasma decay up to a point at which it becomes comparable to the reactor size, i.e. Λ/λ_De ∼ 1. Second, the electron density decay results in an increasing significance of the total dust grain charge to the quasi-neutrality condition, i.e., the Havnes parameter P_H = |q_d|n_d/(en_e) ∼ 1, where the threshold condition is strongly affected by the value of the dust density [53]. Ambipolar diffusion is halted when one of these conditions is met and consequently, charged species transport becomes governed by free diffusion.

2.3 Stage III: ambipolar-to-free diffusion transition

During the third phase (III), a transition from ambipolar to free diffusion occurs when the electron Debye length tends towards the size of the reactor. At this point, ambipolar diffusion is no longer dominantly driving the transport and the ratio of these processes is important to determine the ratio of electron and ion density during the afterglow process. Several models on dust decharging in the low pressure afterglow have accounted for this transition explicitly such as the model by Couëdel et al [42] and Denysenko et al [17]. In both low [42] and atmospheric pressure [41] cases, it has been noted that a minor deviation from ambipolar diffusion, early in the afterglow, can greatly affect the residual dust charge by broadening the dust charge distributions and resulting in positive residual charges.

The ratio of ambipolar and free diffusivity is often based on the dimensionless (squared) ratio of the reactor size Λ and the electron Debye length λ_De [59]. In early studies [59, 60], it was stated already that the transition does not proceed abruptly, but rather that deviations from ambipolar diffusion occur at ratios Λ/λ_De ∼100. Although such studies have been performed decades ago, Λ/λ_De ∼10 is also used to define the ambipolar-to-free diffusion transition in work on the afterglow [61–63]. Nevertheless, in different works [17, 42], the ambipolar-to-free diffusion transition is explicitly accounted for to provide estimates for the dust residual charge. As the authors point out, the accuracy of the diffusion transition is key to determine the dust decharging process, because the losses to dust grains may also affect the transition from ambipolar to free diffusion.

2.4 Stage IV: free diffusion

During the fourth phase (IV), free diffusion occurs when charge separation is allowed on reactor length scales, after which electrons and ions diffuse independently. This is formalized by the condition that the electron Debye length λ_De is similar to the order of the reactor length Λ, i.e. Λ/λ_De ∼ 1. From this moment, the electrons are quickly lost to the reactor walls and dust grains due to the mass difference (i.e. m_i ≫ m_e) at the same temperature. As a consequence, the dust grains are left in a positive space charge region which results in an increasingly less negative dust charge by absorption of ions. In the end, the charge distribution is severely changed compared to the previous phase to such extent that part of the dust grains can become positively charged [42].

The positive ion density is transported to the reactor walls by free diffusion, or absorbed at the surface of the dust grains. The time scale of free diffusion of the ions τ_free is governed by the diffusivity and the reactor length scale,for Λ = 1 mm and the ion diffusion coefficient,where the parameters were defined already for Eqs. 3–9 for argon ions diffusing in argon gas at atmospheric pressure. Because Λ = 100 mm typically at low pressure, and the pressure typically equals ∼0.1 mbar, the free diffusion time scale for the ions is the same as that at atmospheric pressure.

The loss of ions due to absorption by the dust grains can be evaluated using Fuchs’ theory of aerosol charging, although there is serious debate about the applicability of the theoretical assumptions underlying the model to small particles [40, 64]. Using Fuchs’ theory [65], the neutralization time scale of the dust grains due to ion absorption during the free diffusion regime can be assessed usingwhere β_i = 10^–11 m³ s⁻¹ obtained from Suresh et al [41] for n_i = 10¹⁴ m⁻³ at p = 1000 mbar and T_n = 300 K. By comparison of [14–16], it can be concluded that the ions are quickly lost by absorption to the dust grains down to an ion density of about n_i ∼10¹² m⁻³, after which free diffusion to the walls plays an equally significant role. Consequently, this means that the ion density present during this stage is critical in determining the residual dust charge.

The dust residual charge becomes frozen, when the plasma density is insufficient to provide any significant current. The frozen charge condition, as described by Chaubey et al [45] and Ivlev et al [7], states that the charge becomes frozen when τ_L ≲ τ_Q. At very low pressure, plasma loss occurs much faster than charge fluctuations, i.e. τ_L ≪ τ_Q, and the dust charge can be frozen already from the beginning of the afterglow. At atmospheric pressure, however, the plasma loss timescale can exceed the charge fluctuation timescale, which agrees with the scaling of the frozen charge condition with pressure as described by Ivlev et al [7].

In Figure 1, the timescales discussed in this work are depicted as a function of the particle radius a_d in the range of 1–1000 nm, evaluated for three different phases as indicated in the figure caption. The conditions related to phase 1 are similar to those used for evaluation of the electron temperature relaxation stage. It can be seen that the charge fluctuation timescale τ_Q ≲ τ_T in the early afterglow (phase 1 and 2), which means that the charge is in quasi-steady state with the plasma conditions for all sizes depicted. Because the electron temperature relaxation is much faster than the timescales of plasma losses, the phase 2 and 3 timescales are the target of further analysis. These conditions are based on the ambipolar diffusion stage (phase 2) and the free diffusion stage (phase 3). Recombination occurs much faster than the ambipolar and particle absorption losses as and , which implies that the plasma density will quickly decrease due to recombination in pure argon. Under conditions where recombination is negligible, the dominant loss process depends strongly on the particle radius due to the intersection point between and the ambipolar timescale around a_d ∼500 nm. This implies that particles with a larger size will absorb a significant part of the plasma ions, while the smaller particles do not have sufficient time to interact with the ions and thus have their charge frozen earlier in the afterglow. For phase 3, the neutralization timescale is smaller than the free diffusion timescale , which indicates that particle decharging also occurs throughout this phase. A detailed calculation of the neutralization timescale, through evaluation of β_i for different particle radii (see Eq. 16), is out of scope for this work, but could provide interesting insight in the relation to the other timescales.

FIGURE 1

3 Discussion

The foregoing analysis of the physics of atmospheric pressure dusty plasma afterglows is related to recent experimental and modelling work in this section to define open gaps in the field.

First, it is noted by various authors that a deviation from ambipolar diffusion early in the afterglow introduces growing differences between the electron and ion density, which greatly affect the final charge distribution of the nanoparticles [37, 41]. In the model by Chen et al [37], for example, the transition from ambipolar-to-free diffusion is simply put as two limiting cases due to the lack of a clear transition model, as stated by the authors. Consequently, an improved understanding of the transition from ambipolar-to-free diffusion could be gained by accurately modeling the ratio of the electron to ion density and the electric field, as both evolve dynamically in the spatial afterglow of flow-through plasmas.

Second, it is stated by multiple authors that accurate knowledge of the (absolute) ion and electron density in the afterglow is required to predict the dust charge distribution [41, 42]. It was found by Suresh et al through a comparison of their model to experiments by Sharma et al [38] that the fraction of charged particles increases with both electron and ion density present at the beginning of the spatial afterglow [41]. This implies that the electron density provides a tuning knob for the dust charge distribution, which shifts to positive residual mean charge for increasing electron density. Additionally, external electric fields have been used to manipulate the dust charge in the afterglow [11, 12, 14, 18, 35, 36, 45, 66]. In various experiments, it was found that a bias voltage can result in positively charged dust grains, or can be used to reduce the dust charge. However, it was noted that different mechanisms could be at play such as a decreased electron loss or thermionic emission from the wall [35]. Hence, the manipulation and measurement of the electron and ion density in future experiments would greatly assist the development of models and physical control of dust charge distributions.

Third, inconsistencies have been noted regarding the charge distribution of dust grains in relation to the dust grain size. In the afterglow, the characteristic charging time takes most time for the smallest dust grains due to the decreasing probability of a collision with charged species. For small nanoparticles with a_d ≲100 nm, the charge fluctuation time scale τ_Q typically exceeds the residence time τ_res in the afterglow meaning that the dust charge is not in a steady state [38]. However, this also suggests that dust particles below a critical size—for which τ_Q ∼ τ_res—should not exhibit a bipolar charge distribution, which is a conclusion opposite to their observations. For larger nanoparticles with a_d ≳100 nm, the dust grains became net positively charged for lower residence times (i.e. due to higher flow velocity) [36], which is in sharp contrast with the idea that shorter interaction time leads to charge distributions with mean negative charge. In conclusion, future endeavors could be targeting the transient dynamics of nanoparticles interacting with afterglow plasmas to capture the physics involved in the charging of small nanoparticles.

Fourth, there are several opportunities for future research that span interest across other fields. The presence of negative ions built from molecular species introduced by a precursor gas could have a major influence on the dust charging [38], but also on the depletion of electrons in the bulk and afterglow plasma [67]. The effect of neutral radicals and nanoparticles is not often mentioned in the foregoing literature, although neutral radicals contribute significantly to surface growth and an abundance of neutral nanoparticles in the bulk plasma already implies that coagulation becomes a significant mechanism for dust growth [68]. The relatively high fluxes compared to typical low pressure plasmas induce significant heating induced by recombination at the dust surface, which leads to increased hydrogen desorption and crystallization of the material [39].

Experimentalists, theoreticians and modellers clearly face many challenges in their effort to understanding dusty plasma afterglows. As the charge of dust particles in the plasma afterglow greatly impacts the material properties such as the crystallinity and size distribution, their combined effort will enable many applications from controlled synthesis of nanosized materials to contamination control at the nanometer scale.

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