3.1. Photoionization in DP

Ferland's Cloudy codes are one of the most promising tools to study the photoionization in a stellar environment (Ferland et al. Reference Ferland, Korista, Verner, Ferguson, Kingdon and Verner1998; Salz et al. Reference Salz, Banerjee, Mignone, Schneider, Czesla and Schmitt2015). To illustrate the photon-induced ionization, we define ionization factor ${\chi (H)} = N(HII)/{n_H}$; where N(HII) and ${n_H}$ are the density of H⁺ ions and total hydrogen, respectively. The plot for ${\chi (H)}$ as a function of depth is depicted in figure 2(a–d). For higher hydrogen density values, a declining trend of ${\chi (H)}$ is recorded, which may be attributed due to reducing intensity and red-shifting of the photons as a consequence of multiple collisions with the atoms, molecules or dust. A similar trend is also recorded for electron density (n_e), as it mostly depends on the ${\chi (H)}$ (figure 3a–d). Note that for photoionization, the energy of the incident photon $(h\nu )$ must be greater than the threshold energy $({\phi _{\textrm{th}}})$ of the system. The kinetic energy of the photoionized electrons can be expressed as (Güémez & Fiolhais Reference Güémez and Fiolhais2018)

(3.1)\begin{equation}{T_p} = h\nu - {\phi _{\textrm{th}}},\end{equation}

where, ${T_p}$ is the kinetic energy of the photoionized electrons.

Figure 2. The ionization profile of hydrogen for different hydrogen density conditions.

Figure 3. Variation of electron density with depth of the DP system. Note that a declining trend of electron density is witnessed with depth for ${n_H} = {10^2}-{10^4}\;\textrm{c}{\textrm{m}^{ - 3}}$; the electron density remains almost unchanged for ${n_H} = 10\;\textrm{c}{\textrm{m}^{ - 3}}$.

3.2. Dusty plasma thermal properties effected by the electrostatics of the dust/grains

The Cloudy codes were also used to probe the thermal properties of the DP. The temperature profile in DP is presented in figure 4(a–d) for different hydrogen density 10–10⁴ cm⁻³. The DP with lower hydrogen density $(n_H=10\;\textrm{cm}^{-3})$ shows a monotonic decreasing trend of temperature with depth of DP (figure 4a). Apart from the inverse square fall of radiation intensity, it also red shifted due to multiple collisions between photon and DP constituent particles; which is speculated for a monotonic lowering trend of temperature in the case ${n_H} = 10\;\textrm{c}{\textrm{m}^{ - 3}}$. Note that similar trends have been witnessed in the existing literature (Liu & Danziger Reference Liu and Danziger1993). As can be seen from figure 4(b–d), the temperature profile exhibits three different regions, for DP with higher ${n_H}(\mathrm{\ > }10\;\textrm{c}{\textrm{m}^{ - 3}})$. In the first region, the temperature drops monotonically, a bump can be witnessed in the second region, a sharp drop in temperature can be recorded in the third region. It is well established that the electron temperature is co-related to its velocity as ${v_e} = \sqrt {{k_B}{T_e}/{m_e}}$. Note that in a PN DP environment, the velocity of the electron depends on three factors: first, the gradient dependent flow $(\textrm{diffusion}\;\textrm{velocity} \propto \Delta n)$, i.e. the diffusion; second, electrostatic drift due to the presence of the ionized PAH grains which is present in the DP; and finally, due to the photo-energy absorption process, as mentioned in Section (3.1). Considering all these factors, the net kinetic energy of electrons can be expressed as

(3.2)\begin{gather}{T_{\textrm{net}}} = {T_p} + {T_d} + \beta {d_g}{\phi _{\textrm{el}}} - \varGamma ,\end{gather}

(3.3)\begin{gather}= > {T_{\textrm{net}}} = h\nu - {\phi _{\textrm{th}}} + \alpha \Delta n_e^2 + \beta {d_g}{\phi _{\textrm{el}}} - \gamma n\Delta r.\end{gather}

Here, ${d_g}$ is the dust-to-gas ratio, the kinetic energy due to diffusion and electrostatic dust potential are termed as ${T_d} = \alpha \Delta n_e^2$ and ${\phi _{\textrm{el}}}$ respectively. Here α, β and γ are constants. The energy loss due to inelastic collisions or scattering is termed as $\varGamma = \gamma n\Delta r$, which is speculated to be increased with depth (Δr) of the DP medium, along with the number density of the system owing to multiple collisions. Considering, α = 0.00002, β = 10⁵ and γ = 0.36 × 10⁻¹⁸, the plots for T _net in accordance to (3.3) have been presented in figure S2(a,b) (supplementary material), which is similar to the temperature profile (figure 4a,d) for ${n_H} = 10\;\textrm{c}{\textrm{m}^{ - 3}}$ and 10⁴ cm⁻³, respectively. The same is also true for other hydrogen density conditions (not shown). Further details can be found in the supplementary material.

Figure 4. (a–d) Electron temperature variation in DP for different hydrogen density condition. (e–h) The variation of average grain charge (in units of electronic charge per grain) is depicted for different hydrogen densities. In the figure, different colours correspond to grains of different radius (average); the increasing order of grain radius is shown with an arrowhead. Note that the average grain charge is the mean value of different charge states of a particular grain of shape and size in each zone.

The grain charging/discharging processes are quite common in a DP environment. The basics process of ionization/charge capture processes are depicted as

(3.4)\begin{gather}\textrm{Ionization}\;\textrm{of}\;\textrm{grain:}\;\textrm{Grain} + h\nu = \textrm{Grai}{\textrm{n}^ + } + {e^ - },\end{gather}

(3.5)\begin{gather}\textrm{Charge}\;\textrm{capture}\;\textrm{of}\;\textrm{grain:}\;\textrm{Grai}{\textrm{n}^ + } + {e^ - } = \textrm{Grain} + h\nu .\end{gather}

It is worth mentioning that the grain potential ${\phi _{\textrm{el}}}$ depends on the average charge state of the grain (Z), in addition to its size (a), and mathematically expressed as (Ferland Reference Ferland1996) ${\phi _{\textrm{el}}} = (Z + 1){e^2}/a$.

In a PN DP environment, the grains present in the DP exhibited ionization in the case of lower hydrogen density conditions $({n_H} = 10\;\textrm{c}{\textrm{m}^{ - 3}})$, whereas for ${n_H} > 10\;\textrm{c}{\textrm{m}^{ - 3}}$, it offered promising electron capturing ability (figure 4e–h). Due to a lesser interaction of photons with the DP in a lower hydrogen density regime, the transmitted radiation is more intense. As a consequence, photo-ionization effects of grains could be witnessed in the same regime. The maximum accumulated charge $({Z_{\textrm{max}}})$ due to ionization of grains having working potential (W) may be written as (Fortov et al. Reference Fortov, Nefedov, Vaulina, Petrov, Dranzhevski, Lipaev and Semenov2003)

(3.6)\begin{equation}{Z_{\textrm{max}}} = (h{\nu _{\textrm{max}}} - W){a_p}/{e^2}.\end{equation}

It is worth mentioning that the photoionized electron temperature does not depend on the radiation flux, but depends on the frequency of it. In accordance with equation (3.6), the positive charge accumulated in PAH grains is more for larger grain size particles, as can be witnessed from figure 4(e). To probe the stability of the grain distribution in space, we studied the variation of kinetic energy of the grains with respect to the charge variation on the grain surface. An increasing trend of grain temperature could be witnessed for higher charge accumulated cases, as depicted in figure S3 (supplementary material). For the ${n_H} = {10^4}\;\textrm{c}{\textrm{m}^{ - 3}}$ case, grain temperature varies from 80 K to 180 K. Notably, at a fixed charged state, larger grain exhibits a lower velocity.

On the other hand, in higher n_H cases, the radiation flux lowers after interaction with H-clouds and ionized them, which yield significant electron density in those cases. As the electron density is noticeably higher for ${n_H} > 10\;\textrm{c}{\textrm{m}^{ - 3}}$, which enriches the grain–electron capture probability. The electron densities for different hydrogen density conditions are shown in figure S4 (supplementary material). Upon accumulation of the electrons, the grains became negatively charged, as shown in figure 4(f–h). Note that ionization- and electron-capturing ability offer an increasing trend for grain particles, having larger radius (average).

The unusual tempearture profile can be attributed due to the electrostatic interactions between the charged PAH grains and the electrons. Due to the charging/discharging process of the PAH grains, it creates a local potential, i.e., suspected for accelerating the electrons, which results in an unusual temperature variation in DP. It is worth mentioning that, on neglecting the grain potential, the ${T_{\textrm{net}}}$ curve (as per equation (3.3)) varies significantly from the simulated expectations; which ensures the role of grains in the unusual temperature profile.

In the outer region of DP, discharge of grains could be noticed; which eventually withdraws the electrostatic interactions between the electron and grain, causing a noticeable fall in electron temperature.