interact with the liquid droplet and accumulate as a layer on the surface of droplet due to their higher thermal flux than ions [39, 45]. The APP is positively charged as compared to the deposition of negative charge on the surface of droplet due to the accumulation of electrons. The negative potential on the surface of the droplet is screened because Debye shielding confines the potential variations in this particular small region, called as a thin layer. The formation of a thin layer is termed as a sheath, and it forms a potential barrier to extinguish further spread of negative charge. The distribution of potential is progressed around the droplet when the dimensionless radius of discharge plasma channel alters from 0.01 to 0.1 as shown in **Figure 2(a)**. The sharply rising edge in electric potential around the droplet exhibits that the floating potential is immediately strengthened after entrance into atmospheric pressure plasma because of charging. Eventually, this develops a balance of electronic and ionic fluxes around the droplet, and this is verified by the presence of constant flux in the inner region, when the normalized radius (*ρ*) varies from 0.01 to 0.1 approximately as appeared in **Figure 2(b)**. It exhibits that the dynamic behavior of discharge participating species is evolved in the mentioned domain of normalized radii of droplet from 0.01 to 10. Therefore, it is evident from **Figure 2** that the characteristics of charge carriers are changed in the presence of nebulizer droplets and effectively perform an important role for the charging of the droplet. Based on this analysis, it is concluded that system of fluid model equations becomes simplified in the inner region due to the development of constant flux.

To distinguish the attributes of charge carriers around and on the surface of the droplet, the distributions of drift velocity and electric field are compared as shown in **Figure 3**. Considering the flux balance conditions around the droplet, the strength of magnitudes of drift velocity and electric field is altered from higher to lower values. As the intense charged layer is formed at the surface of the droplet, which is responsible for the organization of distorted electric field at the surface of the droplet. It is apparent from **Figure 3(a)** that the lowering in the drift velocity and electric field is linearly varying from *ρ* = 0.01 to 1, while a sharp fall appears at higher normalized radii *ρ*. This shows that the magnitudes of drift velocity and electric field are prominently reduced to very small values away from the surface of the droplet. This situation is entirely transformed at the surface of droplet, when the velocity of ionic species and strength of electric field are certainly boosted to the higher values as a function of droplet radius as exhibited in **Figure 3(b)**. The comparison of both situations reveals that the tremendous modification occurs at the surface of droplet

**Figure 3.** *(a, b) Spatial distributions of normalized drift velocity and electric field around and at the surface of droplet in APP.*

due to the formation of sheath including the accumulation of electrons. This can be verified from the sharp and steady change in the drift velocity and electric field as viewed from **Figure 3**. The sharp distortion in electric field from 0.0001 to 0.001 exhibits that the sheath developed around the droplet due to the deposition of negative charge on the surface of the droplet.

**Figure 4(a)** shows that the normalized magnitude of the droplet floating potential at the surface continuously changes over time due to the strong perturbations caused near the surface by ions and electrons. It is also evident from the mathematical expressions in Eqs (1) and (2) that the spread of perturbation is uniformly homogeneous around the droplet. To explore the interaction of droplets during the transport in the plasma channel, the surface charge on the droplet is evaluated from the floating potential (*ψ*4*πε*0*rd*) [40] as well as from the Rayleigh method (8*π* ffiffiffiffiffiffiffiffiffiffiffi *ε*0*γr*3 *d* q ) [41]. Based on **Figure 4(b)**, it can be demonstrated that the surface charge of the droplet is appropriately less than the charge determined by Rayleigh's criterion. In comparison with droplet charging, droplets remain intact during residence time and are not decomposed into smaller particles. Therefore, the scenario of droplet charging can be supported. The characteristics of floating potential are distinguished

*Interaction and Transport of Liquid Droplets in Atmospheric Pressure Plasmas (APPs) DOI: http://dx.doi.org/10.5772/intechopen.105010*

from **Figures 2(a)** and **4(a)** showing the strong behavior around and on the surface of the droplet. This shows that the evaporation is not a very important mechanism in certain domain of droplet radii (>5 μm), and the droplets survive successfully during the passage in the plasma channel, which act as the seed for nonuniform surface depositions. Therefore, the behavior of droplets in helium APP can be described by the spatio-temporal characteristics using the nonequilibrium Langumir-Knudsen law [42] and normalized one-dimensional normalized fluid model [37] in spherical coordinates.

## **3. Multidimensional coupled fluid-droplet model for PlasmaStream system**

Now the multidimensional modeling approach is preferred to explain and interpret the liquid-plasma interaction, while the multiphase flow can be described by the Eulerian–Lagrangian numerical scheme. In this approach, the Lagrangian methods are used to investigate droplet dynamics, and Eulerian methods are used to investigate the kinetics of the He-N2 gas mixture, in addition to major collision interactions initiated by chemical reactions. As a result of adding the coupling terms to the conservation equations of mass, momentum, and energy of the gas and liquid phases [43, 44], the fluid model of the plasma gas and stochastic liquid droplet model of the liquid precursor droplets were integrated including mutual interactions. The following set of equations is used for the modeling of flowing gas mixture as:

$$\frac{\partial \rho\_{\text{g}}}{\partial t\_{1}} + \rho\_{\text{g}} \overrightarrow{\nabla}.\overrightarrow{u}(r, z, t) = \text{S} \tag{3}$$

$$\frac{\partial \left(\rho\_{\mathcal{g}} \overrightarrow{u}\left(r, z, t\right)\right)}{\partial t\_1} = \overrightarrow{\nabla}P + \overrightarrow{\nabla}.\overrightarrow{\Gamma}\_{\mu}\nabla\overrightarrow{u}\left(r, z, t\right) + \text{S}\_{\mathsf{M}}\tag{4}$$

$$\frac{\partial \left(\rho\_{\text{g}} h\right)}{\partial t\_{1}} = \overrightarrow{\nabla} \cdot \frac{k}{C\_{\text{p}}} \overrightarrow{\nabla} h + \overrightarrow{j} \, \overrightarrow{E} + \text{S}\_{\text{E}} \tag{5}$$

$$
\rho\_{\text{g}} \frac{\partial Y}{\partial t\_1} = \overrightarrow{\nabla} . D\_{\text{AB}} \nabla Y + \text{S} \tag{6}
$$

**Table 1** lists the terminology and description of variables for the above set of equations. Eqs (3–6) represent the fluid model equations of mass density, momentum, and energy of the gas mixture, while the models are coupled through source terms (S, SM, and SE). In this case, Implicit Continuous Eulerian (ICE) method is used because this has already been tested for the numerical solution of gas flows of any Mach number [33]. This has also considered as the effective for the calculation of terms containing momentum exchange between two phases. For the presence of species in APP, the kinetics behavior of discharge species (e�, He+ , He2 + , N2 + , He\* , and He2 \* ) are determined by solving the generalized continuity equation as written as follows:

$$\frac{\partial n\_{\rm sp}(r, z, t)}{\partial t\_2} + \overrightarrow{\nabla} \cdot \left( \overrightarrow{\Gamma}\_{\rm sp}(r, z, t) + n\_{\rm sp} \overrightarrow{u}(r, z, t) \right) = \mathbf{S}\_{\rm sp} \tag{7}$$

where the index **sp** corresponds to the distinctive species (electrons, ions, neutrals, and radicals), *Γ* ! spð Þ¼ *r*, *z*, *t* sgn qsp *μ*sp*E* ! *n*sp � *D*sp∇ ! *n*sp indicates the flux of discharge

*Interaction and Transport of Liquid Droplets in Atmospheric Pressure Plasmas (APPs) DOI: http://dx.doi.org/10.5772/intechopen.105010*

#### **Table 1.**

*Variables in fluid-droplet model (2D and 3D).*

species including drift and diffusion components, μsp and Dsp correspond to mobility and diffusion coefficient of discharge species, respectively, while the Einstein relation (Dsp = μspkBTsp/qsp) is used for the calculation of numeric values. The discharge species density is denoted by *n*sp, and *S*sp exhibits to the source term for the construction and destruction of discharge species based on the chemical reactions as investigated in [45]. The electron mean energy plays a vital role in the transport coefficient and reaction rate coefficient of electron impact reactions that are updated at every time step in the plasma model indicating a strong coupling between the two-phase flow. The electron mean energy (*ε*) is determined by using the electron energy density balance equation as:

$$\frac{\partial n\_{\varepsilon}(r,z,t)}{\partial t\_{2}} + \overrightarrow{\nabla} \cdot \left( \overrightarrow{\Gamma}\_{\varepsilon}(r,z,t) + \varepsilon n\_{\varepsilon} \overrightarrow{u}(r,z,t) \right) = \mathbb{S}\_{\varepsilon} \tag{8}$$

The electron energy density is calculated using *nε*ð Þ¼ *r*, *z*, *t ne*ð Þ *r*, *z*, *t* ̄*ε*ð Þ *r*, *z*, *t* , where *n*e exhibits the electron density and *Γ* ! *ε* corresponds to the flux. Considering all of the terms in [46] together, the impact of joule heating, electron energy losses, and implicit correction terms is denoted by the energy source term (*S*ε). To solve the particle balance equation and electron energy density equation numerically, the alternating

direction implicit (ADI) solver is used [47]. For the estimation of the electric field (E! ), Poisson's equation is coupled with the plasma fluid model, and it can be expressed as:

$$\overrightarrow{\nabla} . \left( \varepsilon\_0 \overrightarrow{E} (r, z, t) \right) = -\sum\_{\text{sp}} q\_{\text{sp}} n\_{\text{sp}} (r, z, t) \tag{9}$$

where *ε*0 is the permittivity of the free space, and the space charge density is the product of species charge (*q*sp) and density (*n*sp). From the gradient of the electric potential (�*∂V=∂r*, � *∂V=∂z*) in two dimensions, we calculate the electric field strengths along r and z. Here, the numerical solution to Poisson's equation is efficiently achieved by using the successive over relaxation (SOR) method as applied in [48]. The local field approximation is used to calculate the effective electric field for the transport of ions in discharge plasma, in which the transport and reaction rate

coefficients are tied to the electric field [49]. In this numerical model, secondary emission of electrons is not involved because ions have small energies compared to electrons, according to [50].

### **3.1 Geometry of a plasma chamber and experimental chamber**

A range of siloxane monomers including hexamethyldisiloxane (HMDSO), *n*-hexane, tetraethyl orthosilicate (TEOS), and water were used to deposition coating on a PlasmaStream system using the laser diffraction particle size analysis technique as shown in **Figure 5(a)**. For plasma generation, a 7.5-cm-long polytetrafluoroethylene (PTFE) tube with a 1.5-cm inner diameter is used. **Figure 5(b)** shows the schematic geometry for the simulation region used in modeling the experiment, in which an oscillating sinusoidal potential (*V*) is imposed to a thin metal electrode at a specific frequency (*f*), and the substrate is grounded. Experimental investigations are performed on silicon substrates with resistivities ranging from 0 to 100 V cm1 (450 and 300 mm thick), for the deposition of coatings [19]. Assuming axisymmetry at the inlet boundary, we introduce the laminar profile of gas flow velocity. **Figure 5(b)** shows a pin metal electrode separated by two gas flows (Q1 and Q2 l min1 ) on the top side of the chamber (AB). The nebulizer droplets are introduced from the central location (B) with a flow rate Q3 μl min1 and are mixed with APP. At the central channel of APP, the droplets of different types of precursors which are served as discrete parcels are propelled along a stream of He-N2 (99% He and 1% N2) carrier gas until they reach the gaseous state (**Table 2**).

#### **Figure 5.**

*(a) Symbolic diagram of experimental chamber of PlasmaStream™ atmospheric pressure plasma jet deposition system with laser source [19] and (b) schematic geometry for the fluid-droplet model.*

*Interaction and Transport of Liquid Droplets in Atmospheric Pressure Plasmas (APPs) DOI: http://dx.doi.org/10.5772/intechopen.105010*

**Table 2.**

*Fluid-droplet model (2D) using initial and boundary conditions.*

#### **3.2 Stochastic liquid droplet model**

For a single liquid droplet falling vertically under the combined influence of a number of forces, the following equations described in [17] estimate the transient position of the droplet, its velocity, radius, and temperature.

$$\frac{d\overrightarrow{\dot{x}\_p}}{dt\_1} = \overrightarrow{\dot{u}\_p} \tag{10}$$

$$\frac{\mathbf{d}\overrightarrow{u}\_p}{\mathbf{d}t\_1} = \overrightarrow{F} \tag{11}$$

$$\frac{\mathrm{d}r\_p}{\mathrm{d}t\_1} = \left(\frac{\lambda\_\mathrm{g}}{\rho\_\mathrm{l}c\_{\mathrm{p}\_\mathrm{g}}}\right) \left(\frac{\mathrm{Nu}\_\mathrm{g}}{2r\_\mathrm{p}}\right) \left(\frac{Y\_{V\_\mathrm{s}} - Y\_V}{1 - Y\_{V\_\mathrm{s}}}\right) \tag{12}$$

$$\frac{\mathbf{d}T\_p}{\mathbf{d}t\_1} = \left( \left( \frac{3\lambda \mathbf{l} N \mathbf{u}\_1}{2\rho\_l c\_l r\_\mathbf{p}^2} \right) + 3 \frac{\mathbf{d}r\_p / \mathbf{d}t\_1}{r\_\mathbf{p}} \right) \left( T\_\mathbf{s} - T\_\mathbf{p} \right) \tag{13}$$

where the index *p* corresponds to a particular parcel and the details of variables in Eqs (10–13) are available in **Table 1**. In this model, an x-squared distribution (*R r*ð Þ¼ d ð Þ 1*=r*d *e* �*r*d *r*d ) of droplets in various parcels is introduced by taking the rectangular computational mesh, where *rd* and *r*d are the radius and mean radius of droplets. The index *i* and *j* of each cell increases along the radial and axial directions. The accuracy of the numerical simulations is obtained by mesh independence, and the mesh size is implemented as (*r*, *z*) = (20, 30) in these simulations, which are similar as used in [17, 21]. In the above equations, the dynamic nature of droplets is similar in each parcel, but collisions occur between different parcels. The formula for the evaluation of the collision frequency (νc) is mentioned as follows:

$$
\mu\_{\rm c} = n\_{\rm p\_2} \pi (r\_{\rm col} + r\_{\rm con})^2 \Big| \overrightarrow{u}\_{\rm r} \Big| / V\_{\rm ijk} \tag{14}
$$

In the above Eq. (14), the collector and contributor droplets are shown by *r*col and *r*con, while the relative velocity between parcels is denoted by *u* ! r ¼ *u* ! p1 � *u* ! p2 . The number of droplets in the associated parcel is exhibited by *n*p2 , and the volume of the cell containing both parcels is represented by *V*ijk. *P*0 ¼ *e*�*n* is employed to calculate the probability of absent collision. Here, *n* ¼ *νc*Δ*t*1 is the mean number of parcels and Δ*t*1 is used the computational time step in the case of Lagrangian approach. The collision and critical impact parameters (*b*, *b*cr) are described below to express the collisions between the droplets by the following relations as:

$$\begin{aligned} b &= (r\_{\text{col}} + r\_{\text{con}}) \sqrt{Y} \\ b\_{\text{cr}} &= (r\_{\text{col}} + r\_{\text{con}}) \sqrt{E\_{\text{coal}}} \end{aligned} \tag{15}$$

where *E*coal is the coalescence efficiency and defined as *Ecoal* ¼ min 1ð Þ *:*0, 2*:*4*ζ γ*ð Þ*=We* . The complex function *ξ* (*γ*), is approximated by the polynomial for simplicity [17] as *ξ* (*γ*) = *γ* 3 *–*2.4 *γ* 2 *- γ* and *γ* = *r*con/*r*col. The parameter (*Y*) is determined by a random number that exists within the interval (0, 1). In case the condition (*b* ≥ *b*cr) is true, then the result of a collision is grazing, which arises within a brief period following an injection of droplet pulse. Collisions caused by grazing occur when the droplets in the parcels and collectors collide so that they preserve their size radii and temperatures, but this collision also changes their velocities. The droplets in the plasma chamber coalesce when (*b* < *b*cr) occurs during downward displacement, leading to the formation of a new droplet with a higher radius. Nevertheless, the main step is to combine stochastic parcel method and ICE technique, in which gas flow rates and droplet velocities are integrated. For the discretization of mathematical equations, it is recommended to use first-order accuracy of finite difference method (FDM). A detailed description of the discretization procedure for these models is given in [17, 46]. Accordingly, **Tables 3–5** contain the description of the fluid-droplet model, including parameters and constants from an online database [51].

#### **Table 3.**

*Fluid-droplet model initialization.*

#### **Table 4.**

*Parameters used in fluid-droplet model (2D).*

#### **Table 5.**

*Density of electrons and vapor species using three-dimensional fluid-droplet model (3D).*

#### **3.3 Initial and boundary conditions**

Assumptions are made regarding laminar profiles of the axial, radial, and swirl velocities (*u*, *v*, *w*) in the PlasmaStream model. For grounded electrodes and solid metal substrates, the flux of electrons, heavy charged particles, neutral gas species, and electron energy density are calculated by considering the modified boundary conditions during the alternate cycles of sinusoidal voltage as mentioned in [52, 53]. The expression *v*th,sp ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8*kBT*sp*=πm*sp p is used for the evaluation of thermal velocity. The freeware Boltzmann solver (BOLSIG+) [54] was employed for the estimation of electron mobilities as well as reaction rate coefficients of excitation and ionization processes. Moreover, two different time steps are implemented to control the liquid phase of droplets (Δ*t*1) and plasma in the fluid-droplet model (Δ*t*2). Plasma equations can be numerically solved with a small time step constraint imposed on the plasma model. Courant–Friedrichs-Levy (CFL) condition is met when time step is selected for the case plasma model using an inequality, *Δt*2 < *ε*0*=* P spj*q*spj*μ*sp*n*sp � � [37] as compared to the liquid droplet phase. On the other hand, the coupling source terms are used to

solve the numerical solution of the liquid and gas phases by using the average values of electric field. For the gas and precursor droplets, we take into account small values of swirl velocities, since they have no significant effect on the numerical solution of twodimensional equations. **Table 3** summarizes the operating conditions for the fluiddroplet model as described in the experiment.

## **4. Dynamics of interaction in two-phase flow**

Utilizing a coupled fluid-droplet model, this section investigates the dynamics of interactions between different precursor droplets and APP. By displaying a quantitative understanding of droplet transit in the plasma chamber, we are able to explore the temporal mean profiles of droplet radii and temperatures, droplet count and size distributions, and vapor species density. To establish the authenticity of modeling outcomes, we performed a comparison of simulation results with experimentally measured size of HMDSO droplets using the laser diffraction particle size analysis technique in which the plasma discharge tube was positioned between the laser source and the capture lens as rationalized in detail in [19]. In order to evaluate the effectiveness of the results derived using the fluid-droplet model, the effect of evaporation is compared by taking four liquid precursors into account: HMDSO, n-hexane, TEOS, and water. The time for the evaporation of the entire pulse of droplets is dependent on many factors, while we address an impact of precursor and gas flow rates to describe the characteristics of APP. A similar profile of gas velocity is assumed as provided by the experimental measurements for the numerical simulations [19]. The initial injection velocity of droplets is distinctive in various sections of this chapter.

### **4.1 Behavior of two-phase flow**

Immediately after the liquid droplets interact with discharge plasma, the evaporation of the droplets begins.  The effect of interaction between two-phase flow is investigated at different time instants, such as 3, 6, and 9 ms as shown in **Figure 6**. Because the volatile HMDSO liquid droplets are exposed to atmospheric pressure plasmas within discharge plasma pulses, the vaporization rate of liquid droplets increases rapidly from 0 to 3 ms, which can be proved by a sharp peak in vapor species density along the pulse of droplets. In discharge plasma, the impact of droplet vaporization increases the lifetime of remaining droplets. The lifetime of the survived droplets in discharge plasma is increased by the impact of vaporization of droplets that further decelerates the evaporation of the remaining droplets in the plasma chamber. This effect expands spatially with time because the vapor species density (*ρ*g*Y*) increases along the evaporating pulse of droplets as shown in **Figure 6(a–c)**. Due to the higher temperatures of the plasma gas mixture in comparison with the liquid droplets, vaporization of droplets dominates in nonequilibrium APP. Based on the increase in vapor species density, the droplets vaporize fairly rapidly between 0 and 6 ms; however, this process impels to slow down with a reduction in the number of droplets in the APP at 9 ms, as discussed in [43].

*Interaction and Transport of Liquid Droplets in Atmospheric Pressure Plasmas (APPs) DOI: http://dx.doi.org/10.5772/intechopen.105010*

**Figure 6.**

*Top row (a, b, c) shows vapor species density and bottom row (a*0 *, b*0 *, c*0 *) exhibits temperature of the gas at 3, 6, and 9 ms using HMDSO precursor flow rate = 100 μl min*�*1 , gas flow rate = 5 l min*�*1 , f = 20 kHz, and V0 =* �*13.5 kV.*

A strong gradient along the pulse of evaporating droplets is evident in the spatial distributions of gas temperature (*T*g) as illustrated in **Figure 6(a**0 **–c**0 **)**. Vapor species density is produced as an additional species that is incorporated after droplets transfer into vapor after phase shifting. Due to evaporation, the temperature of the gas surrounding the HMDSO droplets has dropped sharply. As can be seen from the spatial distributions of gas temperatures, the area around the droplets pulse has been significantly affected by the evaporation as they fall downward. In the plasma chamber, as indicated clearly at three distinct points in time (3, 6, and 9 ms), the temperature of gas falls as the droplets evaporate, but the impact of the vaporization increases. Consequently, this can result in a lowered gas temperature which is useful and effective for the sensitive biomedical and industrial applications in different operating conditions of gas flow and precursor flow rates. Observations of gas temperature profiles indicate that distortions, caused by droplet dispersal, contribute significantly to the formation of a cold plasma at atmospheric pressure. Therefore, the diminish in gas temperature along with the increase in species density endorses the turbulent nature of the liquid precursor of the HMDSO in He-N2 atmospheric pressure plasma as shown in **Figure 6**.

## **4.2 Temporal evolution of a pulse of HMDSO droplets**

We analyzed the temporal mean profiles of radius, axis velocity, and temperature of HMDSO droplets in APP in order to understand the role of evaporation in APP. **Figure 7** shows the mean density of the vapor species during the entire lifetime of a pulse of droplets in the plasma chamber. A higher temperature of the APP enables the droplets to gain energy, resulting in evaporation around the surface of the droplets. While both the number of droplets and the temperature of the parcels are unchanged at the start, their radii and temperatures are reduced due to strong evaporation in domain I. Droplets in parcels contract in size due to evaporation, and some of them acquire nanometer-sized domain radii. For the various kinds of precursors in gas mixtures, the precedent for vaporization of droplets has been examined [17], and a similar benchmark is employed when trying to explain the evaporation behavior of droplets in APP. During interaction, a parcel is removed from the parcel distribution if its entire bundle of droplets evaporates. Droplet counts in parcels evaporate continuously, and its overall effect can be observed in the distribution of their mean radii. Due to drop-down of average temperature, the temperature of each droplet amplifies after gaining energy from the discharge plasma; however, due to the strong vaporization of droplets at the start, the rise in temperature of droplets diminishes in each parcel. Thus, the mean temperature of the remaining parcels actively curtails during the initial period of 1 ms. This is illustrated by a red dotted line in **Figure 7** which indicates that the mean concentration of vapor species increases moderately. This occurs because of evaporation of droplets in the domain I rather than being converted to vapor.

In domain II, the droplets start enhancing in temperature as they attain partial steady states in the plasma chamber, as indicated by a black dashed line. Droplet-plasma interactions ultimately result in vapor species being transferred into discharge plasma, and this in turn drives a greater density of He-N2 gas mixture. As a result, when intensive evaporation of droplets occurs, their momentary temperature quenches while in the case of reduced evaporation, it starts to increase, as demonstrated by the green dashed line in **Figure 7** for domains I and II. It is seen at 8 ms as highlighted in the

#### **Figure 7.**

*Mean distributions of droplet radius (solid line), droplet axial velocity (dashed line), droplet temperature (dashdotted line), and mean vapor species density (dotted) of parcels using HMDSO precursors at flow rate = 100 μl min1 , gas flow rate = 5 l min1 , f = 20 kHz, and* V*0 = 13.5 kV.*

*Interaction and Transport of Liquid Droplets in Atmospheric Pressure Plasmas (APPs) DOI: http://dx.doi.org/10.5772/intechopen.105010*

domain II with an arrow that the evaporation accelerates sharply because the number of liquid droplets becomes very small at this stage. Droplets are distributed in a new manner after parcels of droplets are eliminated by evaporation, as indicated by the small spikes in the profile of mean radius of droplets. According to **Figure 7**, it is obvious that the density of the mean vapor species grows until it reaches a stationary state just before the phase shift of the entire pulse of droplets as indicated by the dotted red line.

#### *4.2.1 Effect of HMDSO,* n*-hexane,TEOS, and water*

Through the incorporation of a relevant precursor, the characteristics of surface deposition coating can be adjusted, resulting in a discharge plasma that was altered fundamentally by evaporation and finite-shared interactions between the droplets. We consider three kinds of liquid precursor droplets, for example, HMDSO, *n*-hexane, TEOS, and water to analyze the properties of APP and the size radii of droplets exists in the range from 4 to 24 μm. As a result of strong intermolecular interactions within their molecules, the HMDSO liquid precursor evaporates completely within 4 ms, while the *n*-hexane, TEOS, and water droplets survive for longer. It suggests that HMDSO is more reactive and volatile than *n*-hexane, TEOS, and water under similar operating conditions when interacting with He-N2 discharge plasma. By avoiding the areas near the cathodes and grounded electrodes at the maximum value of discharge current density after 2 ms, the spatial profiles of vapor species and electron density are exhibited in the center of the plasma chamber, as shown in **Figure 8**. In the analysis of the electronic and vapor species distributions, the central section of the plasma chamber is used due to dominant effect of evaporation in APP rather than the area near the electrodes since it has higher density peaks. As demonstrated in **Table 4**, all of these precursors have varying physical and chemical characteristics.

#### **Figure 8.**

*Top row showed the spatial profiles of vapor species density and bottom row illustrated electron density, using (a, a*0 *) HMDSO, (b, b*0 *)* n*-hexane (c, c*0 *) TEOS, and (d, d*0 *) water at precursor flow rate = 100 μl min*�*1 , gas flow rate = 5 l min*�*1 , f = 20 kHz, and* V*0 =* �*13.5 kV.*

**Figure 8(a–d)** reveals that HMDSO precursor has higher vapor species density than the three other precursors, including *n*-hexane, TEOS, and water because of the high level of evaporation. Several factors influence the rate of evaporation when droplets interact with discharge plasma, such as surface tension, dynamic viscosity, mass density, and boiling point of the precursor. Droplets'surface tension provides a strong resistive force against evaporation, and since HMDSO has a very small surface tension in contrast to other precursors, this implies rapid evaporation of the droplets (as shown in **Table 4**). The temperature of gas mixtures amplifies due to higher value of electron mean energy (*ε* � 0.4 eV) around the pulse of droplets than other discharge species in APP. A convection effect occurs during the active phase of the discharge current pulse in which the precursor droplets can vaporize as a result of the energy gain in the He-N2 gas mixture. This happens due to the impact of electric field and neutral gas density which is evolved by the exchange of energies in two-phase flow through coupling source terms as represented in the group of Eqs (3–6). Since water droplets have a high mass density value, they are pulled downward by gravity to a much greater extent than liquid precursor droplets. According to **Figure 8(a–d)**, the evaporating pulse of water droplets is more likely to pull downward than other precursors. HMDSO and *n*-hexane liquid precursors do not exhibit significant spatial differences along the pulse due to their small differences in mass density. Hence, the vaporization of droplets causes changes in the density of the vapor species and neutral gas, influencing the discharge properties directly.

**Table 6** shows that the numerical values of vapor density for different species diminish from HMDSO to water. Due to a high rate of evaporation, the gradient of electron density along the pulse of droplets in HMDSO and *n*-hexane precursors is relatively larger than that of TEOS and water precursors as shown in **Figure 8(a**0 **–d**0 **)**. This develops conductive pathways and can be grasped by HMDSO or *n*-hexane as active channels, rather than TEOS or water. Moreover, water (H2O) is regarded as the best example of hydrogen bonding because it is polar molecule [55]. As well as having intensive intermolecular forces between their molecules, water droplets have a higher surface tension and density. These factors all contribute to the increased lifetime of water droplets in APP and a weak rate of evaporation of water relative to HMDSO, *n*hexane, and TEOS precursors. So, there is a higher probability of smashing of water droplets on the substrate surface due to survival in the plasma and the survived droplets responsible to build a nonuniform surface deposition. Based on the numerical values of vapor species and electron densities for these precursors, it can be seen that the trend is reducing as exhibited in **Figure 8**. Compared to other precursors, the electron density gradient in water droplets is sluggish, which provides an explanation for small vaporization of water droplets in APP. In turn, the simulations contrast emphasizes the importance of tailoring plasma deposition parameters for different liquid monomers in order to obtain a uniform coating.

#### **Table 6.**

*Maximum values of vapor species density and electron density.*

*Interaction and Transport of Liquid Droplets in Atmospheric Pressure Plasmas (APPs) DOI: http://dx.doi.org/10.5772/intechopen.105010*

#### *4.2.2 Impact of precursor flow rates*

In assessing the impact of different flow rates to achieve the desired features of the surface coating deposition, the initial size distribution of the HMDSO droplets is pivotal. In discharge plasma, droplets are introduced with similar size distributions in the domain (4 μm ≤ *r*d ≤ 18 μm) at the start. Using the laser diffraction particle size analysis technique as employed by atmospheric pressure plasma jet deposition, similar initial size distributions of droplets in the domain (4 μm ≤ *r*d ≤ 18 μm) are introduced in the discharge plasma. According to numerical simulations [56], there is only a small amount of mutual interaction between the droplets at low precursor flow rates (<100 μl min1 ). However, these interactions are important for altering the structure of discharge plasma at higher precursor flow rates. To demonstrate the significance of collective collision interactions between droplets and explore the effects on the atmospheric pressure plasma distribution, the fluid-droplet model is investigated at 200 and 500 μl min1 under identical numerical framework.

The droplet size distributions at 0 ms are represented with black bars, whereas the density bars at 200 and 500 μl min1 are depicted with blue and red bars, respectively, as shown in **Figure 9(a)**. Vaporization of droplets is responsible for the shrinkage of their size domains, along with the events of mutual interactions involving grazing and coalescence, respectively, at 200 μl min1 at 1 ms. Based on the comparison of size distributions of droplets with blue and red bars, it is apparent that the collision events are substantially lower at 200 than 500 μl min1 . The droplet radii increment more rapidly as the coalescence of droplets continues to amplify at 500 μl min1 , and it clear from the size domain (2.5 μm ≤ *r*d ≤ 40 μm) of droplet radii relative to the initial size of droplets as shown in **Figure 9(a)** with black bars. Droplets scatter and settle during descent in the plasma chamber as they pass through these collision events gradually diminishing with time.

In the previous research work [56], they observed that the droplets with larger radii are pulled toward the substrate surface by gravity and electric potential that attract the droplets toward substrate surface. The evaporation of droplets continues to squeeze the radii of droplets in the plasma chamber as the droplets attain the steady state during downward drag at 500 μl min1 . In spite of mutual interactions making a significant contribution to droplet coalescence, the size domain of droplets has

#### **Figure 9.**

*(a) Black bars corresponds to 0 ms and blue as well as red at 1 ms. (b) Distribution of droplets radius at 12 ms using hexamethyldisiloxane using 200 and 500 μl min1 precursor flow rates, 5 l min1 gas flow rate, and 20 kHz as frequency (f) and* V*0 = 13.5 kV.*

become smaller, and it occurs within the range (100 nm ≤ *r*d ≤ 18 μm) at 12 ms in both cases as marked in **Figure 9(b)**. The smaller domain of droplet radii is caused by the ejection of higher radii of droplets from the plasma chamber indicating with blue bars at 200 μl min�1 with a smaller population as well as size domain in contrast to red bars at 500 μl min�1 as shown in **Figure 9(b)**. A large part of the droplets cross the plasma channel without evaporating completely and the remaining droplets found a decent opportunity to evaporate strongly in APP during downward drag. In the plasma chamber, droplets are found in the nanometer range prior to phase transition to vapors as evidenced by the multiple bar distributions.

## *4.2.3 Spatial distributions of electrons and N2 ions*

It has been observed that many nonthermal APP characteristics are associated with the presence of trace quantity of N2 impurities in the helium gas because these impurities trigger a reduction of the ignition potential by the action of Penning ionization [29, 45]. To determine whether Penning ionization dominates in forming charge carriers along the pulse of droplets, the spatial profiles of charge carriers can provide a better understanding at the peak value of discharge current density at 12 ms to highlight its impact. As compared to the plasma chamber, He+ and He2 + ions densities are greater near the momentary cathode electrode during the breakdown phase in contrast to N2 + ions around the pulse of droplets indicating coercive effect of Penning ionization.

**Figure 10(a, a**0 **)**showed the 3D spatial structure of electrons and N2 + ion species at the maximum value of discharge current density along the radial and axial axes at 200 and 500 μl min�1 , respectively. This happens by the evaporation of droplets around the pulse. As depicted in **Figure 10**, the mass flow of liquid precursor in discharge plasma at 500 μl min�1 is greater than 200 μl min�1 because of broader effective area of evaporation at 500 μl min�1 . The density of electrons surrounding the droplets is slightly greater than the density of N2 + ions, implying that electrons generate through multiple channels, such as direct ionization, Penning ionization, and stepwise ionization. Unlike N2 + ions, He and He2 + ions densities are smaller around the pulse of droplets in the plasma chamber which highlights an impact of Penning ionization. A uniform layer is formed on the substrate surface if the evaporation effect is spread evenly along the radial axis. There is a higher probability at 500 μl min�1 because of the formation of uniform sheet of electrons along the radial axis as opposed to 200 μl min�1 . According to the spatial distribution of charge carriers, a faster precursor flow rate results in greater density values at 500 μl min�1 , underlining why higher precursor flow rates result in higher density values. This discussion clarifies that different precursor flow rates in APP influence the kinetics of charge carriers.

#### *4.2.4 Impact of gas flow rates*

A classification of the behavior of complex interactions between liquid droplets and APP based on similar initial size distributions of droplets in the domain is made by considering different gas flow rates with the domain's initial size distributions (5 μm ≤ *r*d ≤ 24 μm). At the inlet boundary of the PlasmaStream system, the gas flow is laminar, but the time it takes before droplets acquire settling velocity differs with gas flow rates during the downward fall. In APP, the properties of droplets and charge carriers are dynamically modified at different gas flow rates, such as 5, 10, and 20 l min�1 that are exhibited with the spatial distribution of vapor species and electron densities as shown in **Figure 11**. Evaporation length is longer at 20 l min�1 than at 5

*Interaction and Transport of Liquid Droplets in Atmospheric Pressure Plasmas (APPs) DOI: http://dx.doi.org/10.5772/intechopen.105010*

#### **Figure 10.**

*3D profiles of electron density (a, b) and N2 + ion density (a*0 *, b*0 *) for HMDSO precursor droplets at flow rates of 200 and 500 μl min*�*1 , gas flow rate = 5 l min*�*1 , f = 20 kHz, and* V*0 =* �*13.5 kV.*

and 10 l min�1 gas flow rates. With greater gas flow rates, this amplification develops because evaporation and convection of gas mixtures occur in the plasma chamber. It takes around 2 ms for an entire pulse of droplets to evaporate at a gas flow rate of 20 l min�1 , which is smaller than the gas flow rates at 5 and 10 l min�1 . The spatial profiles of the densities of the vapor species in **Figure 11(a–c)** demonstrate how the rate of cooling shifts quickly toward the exit axis when the gas flow rate increases.

In order to determine the effect of different gas flow rates on electron distribution during droplet-plasma interactions, **Figure 11(a**0 **–c**0 **)** exhibits the distribution of electrons during the droplet-plasma interaction. In plasma channel, the evaporation rate is greater locally when droplets travel at a low gas flow rate, such as 5 l min�1 . When the surrounding region of droplets is saturated due to evaporation, the remaining droplets in the pulse evaporate at a lower rate due to the impact of this saturated environment. As time passes in discharge plasma, the effect of evaporation spreads across it, increasing the rate of evaporation again. This process is relatively slow at 5 l min�1 as compared to 10 l min�1 and 20 l min�1 . Since gas has a short residence time, its convection is agile at higher gas flow rates, causing a sharp evaporation of droplets. This means that there are smaller chances of saturation due to higher gas flow rates, such as 20 l min�1 which can be confirmed from the increment in the density of electrons from �3.50 � 1010 cm�3 to 4.50 � 1010 cm�3 , when the gas flow rate varies from 5 to 20 l min�1 . Due to the intense gravitational pull, evaporation of large radius drops in the

#### **Figure 11.**

*Distributions of (a, b, c) vapor species density and (a*0 *, b*0 *, c*0 *) electron density at precursor (HMDSO) flow rate at 100 μl min*�*1 using gas flow rates = 5 l min*�*1 , 10 l min*�*1 , and 20 l min*�*1 , f = 20 kHz, and* V*0 =* �*13.5 kV.*

plasma chamber occurs within a short period of time, which causes their journey to be accelerated. However, the small duration of evaporation at 20 l min�1 indicates the fast conversion of liquid material into vapor phase. According to these simulation outcomes of vapor species and electron densities, it is clearly observed that the characteristics of surface deposition can be readjusted by the control of the gas flow rates.

#### *4.2.5 Effect of gas flow rates on gas temperature and electron mean energy*

By applying similar conditions, the spatial patterns of gas temperature and electron mean energy are investigated at distinctive gas flow rates. While falling, the droplets reach a stationary state, indicating that the initial velocity is most important for mutual interactions, rather than after settling in the plasma chamber. In the process of transport of two-phase flow, the local mass flux ratio (*ρ*d*u*d/*ρ*gas*u*gas) alters until the droplets are all converted into vapor phase. This emerges as the reduction of gas temperature in the vicinity of droplets pulse by the evaporation process. **Figure 12(a–c)** reveals that as the rate of gas flow amplifies, the perturbation caused by the droplets in the mixture enhances. A change in the rate of gas flow can modify the interaction between the droplets and discharge plasma; however, the presence of impurities in the gas in two-phase flow can also play a significant role at atmospheric pressure. Impurities such as nitrogen (N2) and oxygen (O2) can

*Interaction and Transport of Liquid Droplets in Atmospheric Pressure Plasmas (APPs) DOI: http://dx.doi.org/10.5772/intechopen.105010*

#### **Figure 12.**

*Contour distributions of (a, b, c) temperature of gas and (a*0 *, b*0 *, c*0 *) electron mean energy using HMDSO droplets at different gas flow rates = 5*, *10, and 20 l min*�*1 , precursor flow rate = 100 μl min*�*1 , f = 20 kHz, and* V*0 =* �*13*.*5 kV in He-N2.*

enhance the rate of ionization because they can lead to an increase in the vaporization rate of droplets. The gas mixture ingests heat energy from the discharge plasma, and thereafter, it results as a substantial improvement in the evaporation of droplets in APP. As a result, the temperature of the gas falls along with the pulses of droplets from 5 to 20 l min�1 and the duration of evaporation is increased with an increment in the gas flow rate.

The temperature of electrons is calculated by the numerical solution of electron energy density equation considering the elastic and inelastic collision energy losses as well as joule heating as employed in [43, 45]. The maximum values of discharge current density during alternate cycles of discharge current pulses are observed near the thin cathode electrode and grounded substrate when the electron mean energy reaches the peak, although the prime focus lies in this case is to gain an understanding for the dynamic updates in the structure of electron mean energy along the pulse of droplets. The regions adjacent to the cathode and grounded substrate are excluded in this scenario in order to investigate the effects of evaporation on the electrons in the bulk of APP. The mean energy of electrons is reduced near the evaporating pulse of HMDSO droplets, as demonstrated by the spatial distributions in **Figure 12(a**0 **–c**0 **)**. Because, the spatial spread of electron mean energy is wider at 20 l min�1 than at 5 and 10 l min�1 , resulting a quicker cooling transfer downward in the plasma chamber at a higher gas flow rate. This demonstrates that the distribution of discharge plasma changes throughout droplet transit and evaporation. The spatial structures of gas

temperature and electron mean energy show that when the gas flow rate increases from 5 to 20 l min1 , the length of cooling increases in the plasma chamber.

## *4.2.6 Significance of He and He-N2 gases*

**Figure 13(a)** exhibited the average distribution of generation rates of stepwise ionization in the pure He and He-N2 gases. The main sources of ionization are direct and stepwise mechanisms in He gas, while Penning ionization is more important in the He-N2 gas mixture than other generation rates of ionization along the pulse of droplets. The stepwise ionization process is illustrated as a dotted line in this case and is responsible for ionization in He gas due to the abundance of a high density of metastables. The numeric value of the rate of stepwise ionization is 2.0 1014 cm3 s 1 in pure helium gas, and it curtails to 1.0 1012 cm3 s 1 in He-N2 gas mixture. **Figure 13(b)** shows the rate of net ionization in atmospheric pressure plasmas, which is the outcome of direct and Penning ionization processes. Through Penning ionization, He-N2 gas mixture experiences a faster rate of metastable destruction. It eliminates the vast majority of metastables, increasing the rate of ionization in large parts of APP. As a result, the density of metastables is reduced to a small amount in He-N2 gas. The net ionization rate in He-N2 discharge plasma is imperatively greater than pure He gas, while its numeric values 2.0 1015 cm3 s 1 in He-N2 diminishe to 1.0 1012 cm3 s 1 in pure He gas as shown in **Figure 13(a, b)**. The imbalance in the rate of net ionization is due to Penning ionization caused by the presence of trace quantity of nitrogen impurities, and this is highlighted in **Figure 13(b)** by a solid line. Droplets in the plasma chamber vaporize rapidly due to the impact of net ionization in He-N2 gas mixture, and therefore their lifetime is reduced. Based on the above outcomes, it is clearly observed that the rate of evaporation is amplified in He-N2 gas mixture as compared to pure helium gas, which is consistent with the previous [19] simulation modeling results.

To explore the behavior of plasma species, **Figure 14** showed the line-averaged distributions electrons, He2 + ions, N2 + ions, and metastables (He∗ ) species density over various cycles in the pure helium using solid black lines plus symbols, while showing the red and green lines plus symbols in He-N2 gas mixture. These distributions are obtained by neglecting the areas near the cathode and grounded substrate to explore the performance of droplet-plasma interaction during transport in the plasma

#### **Figure 13.**

*Distribution of averaged generation rates of (a) stepwise ionization and (b) net ionization in the pure He and He-N2 gases using gas flow rate = 10 l min1 and HMDSO precursor flow rate = 100 μl min1 , f = 20 kHz, and* V*0 = 13*.*5 kV.*

#### *Interaction and Transport of Liquid Droplets in Atmospheric Pressure Plasmas (APPs) DOI: http://dx.doi.org/10.5772/intechopen.105010*

chamber. The direct excitation is diagnosed as the central mechanism in the pure helium and He-N2 gas mixtures, but the relevant contribution of metastables becomes divergent in both gas mixtures. As can be seen from the line distribution of species density in **Figure 14(a, b)**, electrons and molecular helium ions are responsible for keeping quasi-neutrality in pure helium gas. On the other hand, the presence of nitrogen impurity molecules in He-N2 gas changes the situation, resulting in the existence of N2 + ions and electrons in APP, as shown in **Figure 14(a**0 **, d**0 **)**. This clearly shows that the Penning process is the primary ionization mechanism for supplying electrons along the pulse of droplets in He-N2 discharge plasma. The line-averaged behavior of electrons suggests that the ionization rate is amplified by the evaporation of droplets because of the high impact of Penning ionization than direct and stepwise ionization rates. The temporal distribution of metastables shows that they have a higher density in pure helium than in He-N2 gas mixture, as seen by the black line with hollow triangle and red line plus solid triangles in **Figure 14(c, c**0 **)**. This occurs due to higher destruction rate of metastables through Penning ionization as compared to stepwise ionization. Therefore, the above discussion showed that the major role of interaction between the droplets and plasma depends on the chemical reactions that are happening during two-phase flow.

Under comparable operating conditions, **Figure 15** shows a clear distinction in the spatial distributions of electron density in pure helium and He-N2 gas mixtures. At the highest value of discharge current density, the contour spatial distributions of electron density are exhibited in both gas mixtures. In the plasma chamber, the distributions of electron density are shown in **Figure 15(a, b)**, whereas the bottom row depicts the distribution of HMDSO droplets throughout the pulse as illustrated in **Figure 15(a**0 **, b**0 **)**, neglecting the thin cathode electrode and grounded substrate. The electron dynamics along the pulse of evaporating droplets are made clearer using this method. The electron density is significantly greater near the thin cathode electrode than along the pulse of droplets in the case of pure He, but its magnitude falls from 1.91 � 1011 to 1.22 � 1010 cm�3 as shown in **Figure 15(a, a**0 **)**. The similar trend in the contraction of electron density is observed in He-N2 gas mixture in which it alters from 3.15 � 1011 to 5.55 � 1010 cm�3 as displayed in **Figure 15(b, b**0 **)**. It is clear from the shift in the magnitudes of the electron density that its value (2.595 � 1011 cm�3 ) in He-N2 mixture

#### **Figure 14.**

*Comparison of line-averaged species density (electrons, He2 + ions, N2 + ions, and He ) in the pure helium and He-N2 gas mixtures using HMDSO droplets at precursor flow rate = 100 μl min*�*1 , gas flow rate = 10 l min*�*1 , f = 20 kHz, and* V*0 =* �*13*.*5 kV.*

is greater than the value (1.788 � 1011 cm�3 ) in the pure helium gas. This suggests that the rapid rate of evaporation is accelerated by the significant ionization effect in the He-N2 gas mixture, which accelerates the desolvation of droplets in the APP. As a result, this creates a feasible environment in discharge plasma to achieve the desired uniform deposition coating.

According to simulation outcomes, it has been rarely identified the decomposition of HMDSO precursor droplets at small precursor flow rates (10–100 μl min�1 ), which can be confirmed by considering the criterion of Rayleigh limit. In APP, the magnitude of surface charge on the droplets in the mentioned size domain of radii is less than Rayleigh limit as discussed in [24]. The area of ionization around the pulse of droplets is large in He-N2 gas, while it squeezes in pure helium gas combination, according to the spatial profiles of electron density. This means that the composition

*Profiles of electron density in the plasma chamber (a, b), electron density around the pulse of HMDSO droplets (a*0 *, b*0 *) in the pure He and He-N2 gases at gas flow rate = 10 l min*�*1 , HMDSO precursor flow rate = 100 μl min*�*1 , f = 20 kHz, and* V*0 =* �*13*.*5 kV.*

### *Interaction and Transport of Liquid Droplets in Atmospheric Pressure Plasmas (APPs) DOI: http://dx.doi.org/10.5772/intechopen.105010*

of the operating gas mixture affects droplet evaporation. If the gas mixture contains small amount of impurities that are capable of modifying the properties of discharge plasma, and ultimately, the evaporation and mutual interactions of droplets are effected. In APP, the droplets acted as the perturbing agent at the beginning; however, they became part of the operating gas mixture after phase transformation later as vapors. The entire set of interactions is glued through the coupling source terms providing a clear description of heat transfer in two-phase flow. The characteristics of cold plasma can be manipulated by the control of precursor and gas flow rates, because the rate of evaporation inflates with an increment of precursor and gas flow rates. This kind of cold APP is identified as suitable for the medical and industrial applications [6, 9, 25, 26]. In case of incomplete evaporation of droplets, they are not considered as appropriate for the coating deposition applications due to the presence of droplets in discharge plasma. As can be seen from the above contrast, N2 impurities are extremely helpful in enhancing ionization activities during transport of carrier of gas in the plasma chamber, which therefore enhances vaporization of droplets, resulting in a homogenous plasma environment appropriate for surface coatings.

#### *4.2.7 Comparison of model and experimental observations*

To verify the numerical simulation results, it is critical to do a comparison with experimental measurements using a separate setup of laser diffraction particle size analysis technique with the PlasmaStream atmospheric pressure jet deposition system. At location B in the PlasmaStream system in **Figure 5(b)**, a pulse of HMDSO precursor droplets is injected within the size range of 1 μm ≤ *r*d ≤ 6 μm in APP. At small precursor flow rates (≤100 μl min1 ) [56], the mutual interactions between HMDSO droplets, such as grazing and coalescence, are minimal. The fluid-droplet model developed in this chapter's simulation study had similar experimental settings as those listed in **Table 3**, and the initial injection velocity of droplets was assumed to be the same as 1.5 103 cm s1 in this case. The radii of droplets are reduced steadily by the main contribution of evaporation on the surface during downward fall in the plasma chamber.

The initial distribution of droplets exists in the range (≥1 μm) before introducing into plasma chamber, and the size radii of droplets continuously contract due to evaporation as clearly observed by the presence of droplets within nanometer range (100 nm ≤ *r*d ≤ 900 nm) as shown in **Figure 16(a)** at 7 ms. This demonstrates that the HMDSO liquid precursor in He-N2 discharge plasma is highly volatile. The experimental size distributions are recorded in two different experiments in the plasma chamber as shown in different colors in **Figure 16(b)**, but the size domain of droplet radii is similar in both cases. It is evident from the size domain of droplets as mentioned in dotted curly brackets in **Figure 16(a, b)** that the entire bunch of droplets lies in the domain (500 nm ≤ *r*d ≤ 5 μm) at 7 ms, which exhibits a good agreement between the coupled fluid-droplet model and experimental size distributions. In the experimental measurements, the split at 0.5 μm is a result of the limiting resolution of the laser diffraction imaging lens arrangement. However, as seen in the size distribution of droplets, the fluid-droplet model also offers information regarding the occurrence of minimum feasible radii of droplets in the nanoscale range as highlighted in **Figure 16(a)**. The foregoing comparison clearly demonstrates that under similar operational constraints, the numerical model and experimental measurements are synchronized nicely. As a result, the similarity of results increases confidence that the numerical simulations utilizing the 2D-coupled fluid-droplet model can accurately describe the complicated interaction between two-phase flow.

**Figure 16.**

*Comparison of (a) fluid-droplet model and (b) experimental observations of size distribution of droplets using HMDSO precursor droplets at flow rate = 100 μl min*�*1 and gas flow rate = 5 l min*�*1 , f = 20 kHz, and* V*0 =* �*13.5 kV.*

#### *4.2.8 3D profiles of droplet-plasma interaction*

The dynamic characteristics of evaporation utilizing TEOS and HMDSO precursors under the same limitations are described in this section using three-dimensional profiles species density in APP. The validity and legitimacy of multidimensional (2D and 3D) numerical modeling outcomes have already been described in [43] by comparing with experimental measurements. **Figure 17** shows the iso-contours of electrons (red) and vapor species density (green) at three consecutive time instants (*t*1 = 0.6, *t*2 = 1.6, and *t*3 = 2.6 ms) during the evaporating pulse of two distinct precursor droplets (TEOS and HMDSO). In the case of both precursors, the iso-contours of vapor species density revealed that their volumes grow with time; however, in the discharge plasma, TEOS expands in volume faster due to higher evaporation of droplets than HMDSO. Because the mass density of TEOS and HMDSO droplets differs, the bunch of TEOS droplets experiences a stronger gravitational attraction than the HMDSO droplets, and eventually, it manifests itself as a disparity in their volumetric spread as clearly contrasted in the top and bottom rows of **Figure 17**. As shown by the blue dashed line, the affected volume along the pulse of droplets extends and pulls downward. On the other hand, the rate of vaporization of HMDSO droplets is greater than TEOS in He-N2 discharge plasma because of its less surface tension and boiling point as compared to TEOS. This is supported by **Table 5** that showed greater values of vapor species density for HMDSO than TEOS, while these features are in accordance with the previous results discussed in the previous subsections using two-dimensional-coupled fluid-droplet model.

At t1, the conducting channel is forming an intensive ionization along the pulse of both precursor droplets, as shown in **Figure 17(a, a**0 **)**. At *t*2, **Figure 17(b, b**0 **)** exhibits how the overlapped density layers of electrons (red) and vapor species density (blue) are expanding, with a growth of vapor species continuing until the full pulse of droplets dissolves in the discharge plasma. In terms of electron kinetics, they are present throughout the plasma chamber at the highest value of negative discharge current density, as illustrated in **Figure 17** by the red isocontours. This demonstrates a progressive increase in electron density in the plasma chamber. The numerical values listed in **Table 5** indicate that the electron

*Interaction and Transport of Liquid Droplets in Atmospheric Pressure Plasmas (APPs) DOI: http://dx.doi.org/10.5772/intechopen.105010*

#### **Figure 17.**

*Spatio-temporal iso-contours of electrons (red) and vapor species density (green) using TEOS (top row, a, b, c) and HMDSO (bottom row, a*0 *, b*0 *, c*0 *) at t1, t2, and t3 ms at precursor flow rate = 100 μl min*�*1 , gas flow rate = 5 l min*�*1 , f = 20 kHz, and V0 =* �*13.5 kV [44].*

density around the pulse of HMDSO droplets is higher than TEOS. HMDSO has a smaller effect on the volume of vapor species compared with TESO due to the robust evaporation rate in APP. Additionally, the numerical simulation outcomes highlighted that the duration of evaporation for the entire pulse of TEOS droplets is almost twice as long as its counterpart for HMDSO droplets. Thus, TEOS droplets scatter more than HMDSO droplets causing the volume of vapor species to increase. It is evident that these observations differ sharply from those of the top and bottom rows as shown in **Figure 17**. Following the discussion above, it is evident that liquid droplets respond very differently to discharge plasma, depending on the type of precursor. As a result, the numerical modeling outcomes are deemed to be adequate for describing a good understanding of complex interaction of droplets with plasma in APP. These quasi-volumetric characteristics suggest that the internal structure of discharge plasma in two-phase flow is highly complicated, which may be further rationalized by considering a complex chemistry set comprising chemical reactions between vapors and discharge species.

## **5. Conclusion**

In this chapter, we discussed how droplet-plasma interaction is initiated in the atmospheric pressure plasmas. For describing the properties of plasma and explain droplet transport and interaction with the plasma, the one-dimensional normalized fluid model is developed considering Boltzmann distribution of electrons around the droplets. Based on the simulation outcomes, it is concluded that the evaporation and charging of droplets are considered as the most dominant mechanisms within the plasma channel. After developing and testing multidimensional fluid models (2D and 3D) in two-phase flow under distinct precursor and gas flow rates, we examined the implications of complicated interaction between the droplets and plasmas. The characteristics of droplet-plasma interaction manifest that the evaporation of droplets is noticed as the prime mechanism, which has been verified by using different types of liquid precursors (HMDSO, n-hexane, TEOS, and water). A deeper observation at the mean profiles of droplet radii showed that the mutual interactions between the droplets are a major factor to modify the structure of discharge plasma in the limit of higher precursor flow rates (≥200 μl min1 ). Additionally, the spatial and temporal distributions of droplets and APP are explored to understand the interactive behavior of various precursors, showing that evaporation is remarkably intensive in the case of HMDSO and n-hexane compared to TEOS and water. In the plasma chamber, the convection of cooling is much shifted downward at 20 l min1 compared with smaller gas flow rates, justifying the significance of the evaporation process at different gas flow rates (5, 10, and 20 l min1 ). To establish the validity of the numerical simulation results, we compared the outcomes from the 2D fluid-droplet model with experimental observations. Finally, a 3D view of droplet-plasma interaction is presented using spatio-temporal iso-contours of electrons and vapor species density using HMDSO and TEOS to show the volumetric spread in APP.

## **Acknowledgements**

This work is supported by Science Foundation Ireland under grant 08/SRC/I1411.

## **Author details**

Muhammad M. Iqbal1 \* and Mark M. Turner2 \*

1 CCT College Dublin, Dublin, Ireland

2 School of Physical Sciences, Dublin City University, Dublin, Ireland

\*Address all correspondence to<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Interaction and Transport of Liquid Droplets in Atmospheric Pressure Plasmas (APPs) DOI: http://dx.doi.org/10.5772/intechopen.105010*

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Section 4


*4.3. Finite-Time Convergence*

The finite-time convergence of the interval width [*x*(*k*)] is one of the primary issues concerning the tight initialization and stability of the interval estimator. Therefore, this section proves that [*x*(*k*)] converges to a known upper-bounded value in finite time provided by the uncertain quantities. Hence, we introduce the following Lemma to compute the upper bound on [*x*(*k*)].

**Lemma 2.** *The following inequality determines the upper bound on the width of the interval vector provided by (9) and (24):*

$$\mathcal{IG}\left[\mathbf{x}(k)\right] \le \left\|\boldsymbol{\phi}(k)E\_{n-1}\right\|\_{\infty} \mathcal{O}\left[\mathbb{G}\right] + \left\|\boldsymbol{\phi}(k)\right\|\_{\infty} \mathcal{O}\left[w\right] + \left\|\left|\boldsymbol{\varphi}\_{v}(k)\right\|\_{\infty} \mathcal{O}\left[v\_{n}\right], \quad \forall k \ge 3. \tag{28}$$

**Proof.** Firstly, the recursive system (24) is employed as an interval predictor during the initialization phase for *k* = 0, 1, 2 to provide tight bounds on the interval vector of the SEIR model (4). As a result, the upper limit on the width of interval vector provided by (24) is given by

$$\mathcal{U}\left[\mathbf{x}(k)\right] \le \left\| \prod\_{\ell=1}^{k} A(k-\ell) \right\|\_{\infty} \mathcal{U}\left[\mathbf{x}(0)\right] + \mathcal{U}\left[\eth(k-1)\right].\tag{29}$$

Secondly, the proposed interval state estimator (9) is used for *k* ≥ 3. Then, Equation (23) implies

$$\left|\mathbb{U}\left[\mathbf{x}(k)\right]\right| \le \left\|\left[\phi(k)E\_{n-1}\right]\right\|\_{\infty} \left\|\mathbb{U}\left[\mathbb{S}\right] + \left\|\left[\phi(k)\right]\right\|\_{\infty} \left\|\mathbb{U}\left[w\_{n-1}\right] + \left\|\left[\phi\_{v}(k)\right]\right\|\_{\infty} \left\|\mathbb{U}\left[v\_{n}\right] \right\|\right.\tag{30}$$

whereas

$$\begin{split} & \| \| \boldsymbol{\phi}(k) \boldsymbol{E}\_{n-1} \| \_{\infty} \boldsymbol{\eth} \left[ \boldsymbol{\eth} \right] + \| \boldsymbol{\phi}(k) \| \_{\infty} \boldsymbol{\eth} \left[ \boldsymbol{w}\_{n-1} \right] + \| \boldsymbol{\phi}\_{\boldsymbol{\nu}}(k) \| \_{\infty} \boldsymbol{\eth} \left[ \boldsymbol{v}\_{\boldsymbol{\nu}} \right] \\ & \leq \| \boldsymbol{\phi}(k) \boldsymbol{E}\_{n-1} \| \_{\infty} \boldsymbol{\eth} \left[ \boldsymbol{\eth} \right] + \| \boldsymbol{\phi}(k) \| \_{\infty} \boldsymbol{\eth} \left[ \boldsymbol{w} \right] + \| \boldsymbol{\phi}\_{\boldsymbol{\nu}}(k) \| \_{\infty} \boldsymbol{\eth} \left[ \boldsymbol{v}\_{\boldsymbol{\nu}} \right]. \end{split} \tag{31}$$

Based on (30) and (31), one can easily determine that

$$\mathbb{G}\left[\mathbf{x}(k)\right] \le \left\|\boldsymbol{\phi}(k)E\_{n-1}\right\|\_{\infty} \mathbb{G}\left[\mathbb{G}\right] + \left\|\boldsymbol{\phi}(k)\right\|\_{\infty} \mathbb{G}\left[w\right] + \left\|\boldsymbol{\varphi}\_{v}(k)\right\|\_{\infty} \mathbb{G}\left[v\_{n}\right].\tag{32}$$

This completes the proof of Lemma 2.

**Remark 1.** *It is worth noting that we do not need observer gain to design the interval estimator. Therefore, the impact of gain that leads to pessimistic state enclosures for the traditional-type interval observer design method [27,28] is avoided. However, it is more demanding in term of computation time. In addition, unlike Kalman filter-type state estimators, the exact values of exogenous signals are not necessarily known and hence represent an advantage while dealing with practical applications.*