Effect of constant collision mean free time on the boundary layer of the active collisional warm plasma

The plasma boundary layer is analyzed for a plasma in contact with a conducting plain surface where the ion temperature is comparable with the electron temperature and the plasma pressure is sufficiently high. The variations of electrical potential from the plasma-presheath boundary to the wall is studied using the fluidal formalism of plasma in three approaches; plasma and sheath asymptotic solutions and full solution. In the full solution approach, fluidal equations lead to a singularity when the ion velocity reaches the ion thermal speed. It is shown that removing the singularity causes a well-defined eigenvalue problem and leads to smooth solutions for the model equations. Some of the applicable aspects such as the floating velocity and density of ions, the floating electrical potential and an estimation of the floating thickness of the boundary layer are obtained. The dependency of these quantities on the ionization degree, the ion temperature and ion-neutral collision is examined too.


Mathematical modeling of plasma
The positive column of warm plasmas containing neutral atoms, electrons and singly-charged positive ions is studied near the wall region by assuming the plane geometry of conducting walls. Ions and electrons are created in the plasma through single-stage collisions between neutral atoms and energetic electrons. Then, they stream to the wall, recombine there and appear as the neutral atoms in the plasma again.
The conducting walls are charged and their electric potential decreases with respect to the plasma center. The arisen electric force, together with the thermal, collisional and inertial forces restricts the ions and electrons movement towards the wall. The steady state continuity and momentum transfer equations for ions and electrons, with along the Poisson equation comprise the governing equations as follows where n i , v i and m i are the ion density, velocity and mass respectively, P i and P e are the ion and electron stress tensors respectively, and E = −∇V is the electric field with V as the electric potential. Also, n e is the electron density, ν I stands for the ionization frequency, τ ec is the mean free time of the ion-neutral elastic collision, e is the elementary charge and ǫ 0 is the vacuum permittivity. The inertial and collisional forces have been neglected in Eq. (4) since electrons are much lighter than ions.
It is assumed that there is homogeneity parallel to the wall, so the space variations are found just in direction perpendicular to the wall ( x-axis). Out of symmetrical configuration of plasma around the central plane x = 0 , www.nature.com/scientificreports/ the studying of plasma will be done in the half space x ≥ 0 . A schematic diagram of the simulation zones in the plasma boundary layer is shown in Fig. 1.
The general stress force −∇ · P is reduced to −dP/dx = −k B T(dn/dx) for the both of ions and electrons in the simplest case, i.e. isotropic Maxwellian plasmas with the ion and electron isothermal flows 33 . Here, T could be replaced by the absolute temperature of ions T i or electrons T e . Also, the collision mean free time is truly given by τ ec = 1/n n σ s v s in which n n is the number density of inert neutral atoms and σ s is the momentum transfer cross section for the ions moving with the ion acoustic speed c s = √ k B T e /m i 26 . With these assumptions, it is straightforward to rewrite (1)-(4) in the normalized form as Using Eqs. (5) and (6) results in; where In Eqs. (5)- (8) and (11), the following dimensionless parameters and variables have been utilized; in which, the dimensionless parameters τ , α , and ǫ are the temperature, collision and smallness parameters, respectively, D = ǫ 0 k B T e /n 0 e 2 defines the Debye length at the plasma center, and F = dφ/dr is the normalized electric field. Then, one can easily conclude that N = exp (−φ) from Eq. (8).
Applying ǫ → 0 in the Poisson's equation, Eq. (7) results to I = N = exp (−φ) , that is called the plasma approximation. In the plasma approximation, there is no sheath in front of the wall. However, it is sometimes necessary to introduce the sheath region including a positive space charge even in the asymptotic limit ǫ → 0 . In order to study the sheath region, it is needed to use the new space coordinate ξ = (x − x w )/ D = (r − r w )/ǫ in which, x w (or r w ) is the location of the wall selected as the origin of the new coordinate. Equations (5)-(7) in the new coordinate space are transformed to www.nature.com/scientificreports/ knowing that N = exp (−φ) . Although, there is an explicit difference between them under the asymptotic condition ǫ → 0 , there is no difference between the set Eqs. (5)- (7) and (12)- (14) for ǫ > 0.

Asymptotic limit of plasma
The plasma asymptotic limit means applying the condition ǫ → 0 in the plasma approach equations. In this limiting case, Eqs. (5)-(7) turns to where the subscription "p" refers to the plasma asymptotic limit. Equations (15) and (16) can be solved analytically to get for 0 ≤ U p ≤ √ 1 + τ . However these universal functions are not practical and one needs to find the numerical full solutions in more applicable situations for ǫ > 0 , yet they represent some primary knowledge about the collision and temperature effects in the real plasmas. These functions are transferred to in the collisionless warm plasmas (α = 0) 29 , and are simplified to in the collisionless cold plasmas (τ = 0 and α = 0) 2 .
Apparently there is a singular point in Eqs. (15) and (16) which is introduced by U b = √ 1 + τ = U B and is called the Bohm criterion. This singularity point represents the plasma-sheath boundary where the breakdown of the quasi-neutrality is commenced. The other variables at the plasma-sheath boundary can be found using Eq. (18) as follows; www.nature.com/scientificreports/ By the way, these marginal variables in the collisionless cold plasmas are reduced to the well known quantities

Asymptotic Limit of Sheath
The sheath asymptotic limit means applying the condition ǫ → 0 in the sheath approach equations. Using this condition in Eqs. (12)- (14) turns them to where subscription 's' refers to the sheath asymptotic solution. Combining Eqs. (22) and (23) results in with ψ s = φ s − φ b as the relative electric potential. Equation (25) in the cold plasmas reduces to the well-known relation 29 In order to find the sheath approach solutions, it is needed to solve the second-order differential equation (24) from the plasma-sheath boundary to the wall. To find the essential boundary conditions, it is necessary to expand I s and N s in powers of ψ s at the plasma-sheath border (where ψ s → 0 ) as follows Using above approximations in (24) results in for τ > 0 and for τ = 0 . The physical solutions for these two approximate equations are By applying Eqs. (31) and (32) in Eqs. (29) and (30) respectively, the unknown coefficients A and A ′ can be easily find as www.nature.com/scientificreports/ At a point sufficiently far from the wall, hence, sufficiently near to the plasma-sheath boundary, for example at ξ = −100 , ψ s can be determined from (31) and (32) in the warm and cold plasmas respectively. By knowing ψ s and using Eqs. (25) and (22), the values of I s and U s can be found at that point as the boundary conditions.
On the other hand, since the solution ψ s = ψ s (ξ ) is translationally invariant in ξ , it is needed to finish the numerical calculations at a proper point. Floating point, where the floating condition n if v xf = n ef v th /4 is fulfilled, is selected as the suitable ending point. In the floating condition, subscription 'f' denotes to the floating point and v th = √ 8k B T e /πm e is the electron thermal speed 31 . By using (22) at the floating point, the floating condition in the normalized form is is the relative electric potential at the floating point. Solving the sheath equations is trivial now.

Full solution of the eigenvalue problem
Having examined the both asymptotic limits in the plasma and sheath approaches, we have the full solution of plasma equations in both the body of plasma and near the wall in different approaches. In this section, we discuss about joining these two asymptotic solutions in the same approach smoothly. In other word, we seek to solve the first-order differential equations (7)-(10) for finite ǫ by integrating from the plasma center outwards.
It is clear from Eqs. (9) and (10) that derivatives of I and U are undetermined at the regular singularity τ is the normalized ion thermal speed, subscription 'r' refers to regular and the prime denotes differentiation with respect to the spatial coordinate r . To remove this ambiguity, the Hopital's rule obviously can be used here to get It should be noted that Eqs. (5) and (6)   www.nature.com/scientificreports/ is the only reasonable solution of Eq. (40) complying the condition U ′ r > 0 . Since a and b are two positive constants, the solution is reliable just for c < 0 . This condition for having physical solutions confines the amounts of α , ǫ and τ and connects them together. For some proper amounts of α , ǫ and τ , it is easy to determine U ′ r and I ′ r by utilizing Eqs. (41) and (37) respectively.

Methods and boundary conditions
By removing the ambiguity in U ′ r and I ′ r at the regular point U r , the set of differential equations (7)-(10) together with the appropriate boundary conditions should be solved as an eigenvalue problem for r w or the other variables at the wall. In order to find the boundary conditions at the plasma center, the power series expansion of the variables are used. Since I and φ are even functions and U is an odd function, they can be approximated by and therefore in which, the six unknown factors i 0 , i 2 , ϕ 2 , ϕ 4 , u 1 and u 3 should be specified. Using these power series in Eqs. (7)-(10) gives rise to and By using Eq. (47) iteratively (with i 0 = 1 as initial value) and finding the main factor i 0 , the other coefficients will be explicitly determined 30 .
On the other side, having been defined in "Asymptotic limit of plasma" section, the floating condition is used to end the calculations. This condition in the normalized shape is with β = √ m i /2πm e . Now, set of the equations in r (7)-(10) are integrated by the 6th order Runge Kutta Fehlberg method from the plasma center with the appropriate boundary conditions (42)-(46) until the boundary condition (48) is satisfied. Here, computations have been done for the electropositive gas helium with β ≈ 34.18.

Results and discussion
The general form of the plasma asymptotic solution and full solution for the normalized electrical potential φ is shown in Fig. 2 for τ = 0.5 and different values of α . As it is seen, the gradient of the electrical potential tends to infinity by approaching to the plasma-sheath border r b in the plasma asymptotic solution. This figure makes a comparison between the two solutions in the plasma approach for the same plasma parameters. By the results, the effect of the collision parameter is to decrease the boundary layer width (floating width) and to increase the electrical potential at the floating point (floating potential) for fixed values of τ and ǫ. Figure 3 gives the sheath asymptotic solution for the normalized electrical potential φ s for the same parameters as in Fig. 2 showing how the collision parameter increases the floating potential φ f . It is concluded that increasing the plasma pressure makes the floating electrical potential more negative, therefore reduces the floating current to the wall. By receding from the floating wall, the normalized electrical potential is reduced and saturated to the constant potential φ b at the plasma-sheath border which is an increasing function of the collision parameter in accordance with Eq. (21-a). Figures 2 and 3 demonstrate the same results in different approaches. www.nature.com/scientificreports/ Figure 4 represents the plasma asymptotic and full solutions of the normalized electrical potential distribution for α = 10 and different values of τ , in which the general form of the electrical potential variations in the both solutions is as in Fig. 2. It is seen that general effect of ion temperature τ is to enlarge the plasma boundary layer extension and to decrease the normalized floating potential φ f . Similar results can be seen in some recent works on the warm plasma with low ion temperature using approximate methods 27,28,39 . Figure 5 depicts the distribution of φ s and the dependence of φ f and φ b on τ in the sheath approach for the same parameters as in Fig. 3. It can be seen that both of the normalized electrical potential distribution and floating potential are decreasing function of the ion temperature τ . It also can be noted that the ion temperature has no effect on the electrical potential at the plasma-sheath border in accordance with Eq. (21-a) 28 . Figure 6 shows the full-solution for the normalized potential profile when the ions isothermally flow towards the wall for τ = 0.5 , α = 5 and some values for ionization rate ǫ . As can be observed, the effect of ǫ is to enlarge the floating width and to reduce the normalized floating potential φ f for a fixed value of τ and α . Indeed, increasing the ionization rate raises the ion density number or the positive space charge which results in growing the electric field and therefore the ion electric drag towards the wall in accordance with the motion of ion and  www.nature.com/scientificreports/ Poisson's equations. By increasing the ion velocity, the ion density and so the positive space charge will decrease according to continuity equation and this gives rise to decrease the electric field or the slope of electric potential. It can also be realized that the effects of ion temperature and ionization rate are qualitatively the same. Spatial distribution of the normalized positive space charge I − N in the full-solution approach is shown in Fig. 7 as a function of ionization rate ǫ . According to this figure, by receding from the plasma-sheath edge, the positive space charge in the boundary layer increases dramatically and after hitting to a peak begins to fall by approaching to the negative biased wall with φ w = 20 . Also, it can be seen that the positive space charge is an ascending and widening function of the ionization rate and it means that the slope of electric potential drops by growing the ionization rate according to the Poisson's equation. This result is in good consistency with Fig. 6 and as a result, one can say that the electric drag of ions towards the wall will be plummeted by the ionization rate 28 .
Tables 1, 2, 3 and 4 give the floating variables φ f , U f , r f and I f respectively, for ǫ = 0, 0.002, 0.02 , τ = 0.0, 0.1, 0.5, 1.0 and α = 0, 0.1, 1, 5, 10, 50, 100, 500 and 1000 . These tables quote that the floating ion density I f , floating ion velocity U f (ion kinetic energy) and floating width r f are decreasing functions, while the floating potential φ f is an increasing function of the collision parameter α (plasma pressure). In the special case ǫ = 0 , it    www.nature.com/scientificreports/ should be noted that I b /I f and therefore U f = √ 1 + τ I b /I f is independent of the collision parameter [according to (25) and (33)]. The ion temperature effects is totally the opposite of the collision effects and have already been studied in the collisionless plasmas in 28,29 . In order to complete the discussion, the effects of the ion temperature on the plasma variables in the collisional plasmas has been studied and the results have been summarized in Tables 1, 2, 3 and 4. The results show the weaker effects of ion temperature compared with the effects of ionization and collision parameters.  Table 3. Floating width r f (α, τ , ǫ).  Table 4. Floating ion density I f (α, τ , ǫ).

Summary and conclusions
This paper expands a numerical-analytical method to describe the boundary layer of collisional warm plasmas in touch with the conducting planar surface when the ion temperature is comparable to the electron temperature and the plasma pressure is sufficiently high such that the ion-neutral elastic collision in the presheath must be taken into account. Ions are created by electron impact ionization in the active plasma that allows the plasma boundary layer to be formed. Considering the effects of space-charge, the mathematical description of plasmas boundary layer by means of the hydrodynamic or fluidal equations leads to a singular point when the mean velocity of ions reaches the ion thermal speed U r = √ τ ( c th = √ k B T i /m i ). Starting with the ion continuity and momentum transfer equations including terms of collision, pressure and inertia, the positive column of thermal plasmas at high pressure is examined. By removing the singularity and finding smooth solutions, the essential smoothing conditions around the singular point within the boundary layer of plasma is investigated which leads to a well-defined eigenvalue problem.
The electrical potential distribution in the plasma boundary layer, has been shown in three approaches; plasma and sheath asymptotic solutions for ǫ = 0 and full solution for ǫ 0 . While the ion temperature and smallness parameter decrease the electric potential profile, the ion-neutral collision increases it in these three approaches. Also, the global floating variables including the ion velocity, ion density, electrical potential, boundary layer width, and how they are all affected by the plasma parameters (ion temperature, collision and smallness parameters) have been presented. All these consequences are much more pronounced with collision and smallness parameters than with the ion temperature.