The surface dielectric barrier discharge in this work is simplified as a two-dimensional (2D) model, as depicted in Figure 1. The simulation domain is 8 mm (length) × 3 mm (height) in Cartesian x–y coordinates. The length of the ambient air and dielectric barrier are 8 and 7 mm, respectively. The thickness of the gas gap is 1.5 mm above the dielectric barrier and 3 mm above the rest. The grounded electrode was placed on the right side of the dielectric barrier, and the single exposed electrode with a high voltage is on the upper left surface of the dielectric barrier. The barrier dielectric is given a space-varying permittivity ε r from 7.5 to 10.5, changed in linear or non-linear forms. The continuity equation for both charged and neutral species is Eq. 1. The drift-diffusion approximation for charged species is calculated by Eqs. 2, 3, and Eq. 4 is used to obtain diffusion flux for neutral species, where e, −, +, and m represent electrons, negative ions, positive ions, and neutral species. n is the number density, cm⁻³, and j is the flux, cm⁻² s⁻¹. ∂ n j ∂ t + ∇ ⋅ j j = S j , (1) j e , − = − μ e , − E n e , − − D e , − ∇ n e , − , (2) j + = μ + E n + − D + ∇ n + , (3) j m = − D m ∇ n m . (4)

In the equations above, E is the electric field vector, µ is mobility, D is the diffusion coefficient, and S _j is the source of species of species j (j = e, −, +), which can be derived from the chemical reactions. The working gas is simplified as 80% N₂ and 20% O₂. The species and reactions in the model are given in Table 1. The rate coefficients of electron impact reactions are obtained by solving Boltzmann’s equation using BOLSIG+ ¹ . Cross sections are taken from the Morgan, TRINITI, and Phelps databases ^{2 – 4} . What’s more, the local field approximation is applied when calculating parameters, and the electron mean energy and electron transport coefficient are given as a function of reduced electric field. Transport coefficients for ions and neutrals were estimated following Ref. [19].

The electric potential is found by: ∇ ⋅ ( ε r ∇ φ ) = − ρ / ε 0 , (5) where, φ and ρ are the electric potential and space charge density, respectively, ε 0 is the vacuum permittivity. ε r is the relative permittivity of the dielectric barrier. The main plasma parameters, including dielectric surface charge, are calculated in this work. ∂ σ s ∂ t = − e [ ( ∑ j + − ∑ j − − j e ) ⋅ n ] , (6) j e ⋅ n = ( − μ e E n e − γ ∑ j + ) ⋅ n , (7) where n is the unit vector perpendicular to the dielectric surface, γ is the secondary emission coefficient, γ = 0.01 . The secondary emission electron energy is set at 2.5 eV, which is in the same as the surface reactions.

In a uniform electric field, the initial charged particle density is uniformly distributed. In a non-uniform distorted electric field, it is generally considered that the initial charged particle density obeys a Gaussian distribution. For the non-uniform field discharge of needle plate structure, it is often assumed that the initial charged particles (electrons and positive ions) follow the Gaussian distribution and that the negative ion density is zero [20–22]. The initial electron distribution is given by: N 0 _ e = N m a x e x p ( − ( x − x 0 ) 2 2 σ 2 − ( y − y 0 ) 2 2 σ 2 ) , (8) where N _max = 10¹⁶ m⁻³, (x₀, y₀) = (0, 2 mm), σ = 62.5 µm. (x₀, y₀) are the spatial coordinates of the needle electrode head. The negative voltage applied is shown in Figure 2.

Considering the technical process and reliability of graded material in 3D printing, several dielectric permittivity distributions are proposed in the present work, as a function of x along the length of the dielectric barrier direction, as shown in Figure 3. Both ideal permittivity and practical permittivity distributions are depicted as in Figure 3, and the approximated function of ideal permittivity is ε p = 7.5 + 0.43 x . Obviously, non-linear distribution as a function of length along x direction is introduced, and to be a comparison of uniform distribution of dielectric permittivity, the case in the condition of ε r = 7.5 is studied as well.

Boundary conditions are given in Table 2.

The spatial distribution of electron density with dielectric barrier permittivity ε r = 7.5 and ε p are shown in Figure 4. A streamer starts after the applied voltage rises to about −10 kV, at about 10 ns. Within the first 10 ns, the discharge transfers from a corona-like to a streamer-like discharge, mainly due to the upper side of the applied voltage. The negative discharge streamer propagates along the surface of the dielectric barrier, but the streamer starting positions with uniform permittivity ε r = 7.5 seem to be farther than those with graded permittivity ε p . This also means that the streamer starts at an earlier time on the graded dielectric barrier in this case. Streamer approximately driven by high electric field, both of the direction and magnitude. The magnitude of electric field strength near the surface of the dielectric barrier has not changed greatly due to the graded permittivity. As a comparison, the direction of electric field strength is much different in graded dielectric than in the uniform distribution of permittivity, as is depicted in Figure 5B. Near the surface inside the dielectric with permittivity ε p 1 and ε p 2 , there is ε p 1 E p 1 ⋅ s i n ( δ 1 ) = ε p 2 E p 2 ⋅ s i n ( δ 2 ) , where δ 1 ( δ 2 ) are the angles between E p 1 ( E p 2 ) and x direction. According to the given graded permittivity in Figure 3, the graded permittivity increases with the length of the dielectric barrier in the x-direction, thus, ε p 2 > ε p 1 . As a consequence, δ 2 is smaller than δ 1 , which means the direction of electric field strength is concentrated on the streamer propagation direction, introducing functionally graded material. Besides, compared with Figure 5A, it shows a little rise in electric field magnitude in Figure 5B. At 25 ns, before the falloff of the negative voltage pulse, the streamer reaches 2.3 mm with uniform permittivity and almost 3 mm with graded permittivity. This may provide a longer propagation of streamer controlling without changing the electric field magnitude greatly.

Figure 6 shows the spatial distribution of surface charge density with uniform and graded permittivity. The charge density along the dielectric barrier has a wider distribution range under the condition of graded permittivity. The E y component is shown in Figure 7, as the propagation of the streamer changes the sign and the surface charge decreases in accumulation when the E y component falls to zero. The slower-drifting ions, including, N 4 + , N 2 + significantly change the charge polarity on the surface layer of the dielectric barrier. The charges on the surface generate a strong electric field, leading to the new ionization of the gas. The charge density keeps increasing during the falloff stage of applied voltage and corresponds to the propagation of the streamer. It can be found that surface charge density is much higher with the graded permittivity applied. As depicted in Figures 5B, 7, the usage of graded permittivity dielectric changes the electric field component softly, with almost no distortion of it. But the surface charge density is considerably increased; this may benefit some applications such as plasma catalysis [23], which create more intense reaction rates and reactive molecules.

What’s more, surface charge density increases with the graded permittivity, which is inflected by the length along the dielectric. With the development of discharge, the streamer slides over the surface of the dielectric barrier, driven by the increasing electric field caused by the graded permittivity in the form of step-rise. The graded permittivity applied in this case, as presented, is more obviously in a changing electric field. The tiny adjustment by the graded permittivity dielectric barrier may result in continuously extended discharge propagation and more surface charge density (Figure 6).

To understand the influence of graded permittivity on some reactive molecules, the spatiotemporal evolution of N 4 + , N 2 + ion densities is shown in Figures 8, 9, respectively. It can be found that positive ions N 2 + are always produced in regions of high electric fields. This is mainly due to the possible ionization. From Figures 8A, B, it shows a higher ionization rate where N 2 + is relatively concentrated with the application of graded permittivity dielectric. Compared with the distribution of N 2 + , the density of reactive species N 4 + shows a relatively uniform distribution along the streamer. N 4 + is produced from the recombination and association reactions in which the species N 2 + will be consumed, since N 2 + is produced mainly in the head of the streamer. In the aspect of the early stage during the upper side of voltage, the distance between the concentration of N 2 + ( N 4 + ) and the high voltage electrode is shorter with the graded dielectric barrier, and the density is higher. This indicates the production rates of two species are faster, the usage of graded permittivity dielectric may speed up the reaction rates. In the late stage of discharge development, a reversal density of N 2 + and N 4 + appears near the negative high voltage electrode, mainly due to the drift process.

The higher density and producing rates of N 2 + and N 4 + , caused by the introduction of graded permittivity, will benefit the plasma catalysis such as the application of synthesis of Ammonia [24]. Ni/ γ -Al ₂ O ₃ enhance plasma-promoted NH ₃ production and favors surface-adsorbed NH _x species. N 2 + and N 4 + on the surface of the catalyst Ni/ γ -Al ₂ O ₃ can be regarded as the catalytic efficiency of plasma. As the dominant species, including N 2 + and N 4 + , the introduction of graded permittivity to the solid catalyst will promote a faster synthesis rate of NH ₃ .

As for the aerodynamic applications, the momentum source due to the charged particles’ collisions with neutral gas molecules (body force) is given by [10] F = e ( n i − n − − n e ) E , (9) where E is the electric field vector, e represents elementary charge, and n _i , n ₋ and n _e are the positive ion density, negative ion density, and electron density, respectively. Figure 10 depicts the contours of a momentum source with uniform permittivity and one with graded permittivity. As is depicted in Figure 10A, the contours of the momentum source (volumetric force) show different signs inside and outside the streamer body, and are almost concentrated in the head of the streamer and a thin layer near the dielectric surface. In the thin layer near the surface, the sign of a body force is always in negative territory, and positive in the outer areas. This outcome is in accord with that in Ref. [10]. Compared with the gravity force in the atmospheric air, it is about 10 4 times greater. Thus, the body force in the layer, which is perpendicular to the contours, drags the streamer slide along the dielectric surface. The non-linear structure of the contours of the momentum source is always accompanied by a complex vortex, and as the propagation of the streamer, this kind of vortex moves along the dielectric surface barrier. The flow conditions will be significantly changed due to the existence of the vortex.

The numbers on the right side of the contours are the x-coordinates of the streamer head, as shown in Figure 10. It is obvious from Figure 10B that the contours cover a wider range on the dielectric surface and propagate at a faster speed. The graded permittivity introduced in the model shows an intense and speeded-up momentum exchange between charged ions, electrons, and neural molecules in the ambient air. Local vortex caused by body force is considered to be the possible way of cooling down of the temperature and a kind of diffusion effect to the streamer propagation. The longer propagation of vortex means that, with the same negative voltage pulse applied, the flow-field control along the dielectric surface seems to be more efficient. The “ionization wind” will slide faster and generate more thrust force on the surface of the functionally graded material dielectric barrier.

The 2-D model of different dielectric barrier layer thickness is shown in Figure 11, in order to investigate the influence of thickness of linear permittivity dielectric layer on SDBD, the thickness of graded permittivity dielectric barrier is reduced to 0.5 mm, and the rest of the dielectric barrier is in the uniform permittivity ε r = 7.5 . The 0.5 and 1.5 mm in the figure represent the thickness of the graded permittivity layer. The boundary conditions and negative voltage pulse applied to the model are the same as in Figure 1 and Figure 2. The graded permittivity applied here is still the ε p as in Figure 3.

The number on the right side of the streamer head is the x coordinate of it. Figure 12 depicts the spatial distribution of electron density with a layer of thin thickness. Compared with Figure 4B, it depicts a shorter distance of propagation in Figure 12, but somewhat at 30 ns (falloff of the negative voltage pulse), in the similar position of the streamer head. But the reversal of electron density distribution pointing at the high voltage electrode is suppressed, which can be seen in Figure 12. As indicated in Figure 5B, the graded permittivity dielectric affects the streamer propagation by enhancing the electric field distribution, and it mainly takes place along the surface of the dielectric barrier. As a consequence, with the same voltage applied, the electric field would not be dramatically changed in the E y component, which can also be identified from Figure 13. Figure 13 shows the E y component of different layer thicknesses with graded permittivity. The magnitude of E y component is a little higher with the thickness of 1.5 mm of graded permittivity dielectric barrier than with the thickness of 0.5 mm. Figure 14 shows the instantaneous distribution of surface charge density at 20, 25, and 30 ns with a layer of very thin thickness. The E y component is negative over the surface with a high enough charge, as shown in Figure 14. This shows that the thickness of the graded permittivity dielectric layer will not significantly change the surface charge density along the barrier surface, of which is correlated with the E y component.