Abstract

Introduction

All growth criteria considered below are based on two main assumptions. Firstly, the growth is stochastic in time. Secondly, the probability of streamer growth is proportional to some function of local electric field r(E) depending on dielectric properties of a substance. Thus, the generation of new bonds is governed by some stochastic growth criteria in each time step.

The electric field potential was obtained by solving the Poisson equation. Charge transport along the streamer branches was computed according to Ohm's law for each conductive element.

"Physical" Time and Growth Criteria

In single-element models, we actually have to take both the new bond and the "physical" time interval in a stochastic way.

Niemeyer--Pietronero--Wiesmann Model

p(E_i ) = r(E_i ) / $\sum _{i=1}^n$ r(E_i ).

For the first time this model allowed one to simulate the formation of branching streamer structures having fractal dimension.

Field Fluctuation Model

E_i > E_* - d_i ,             (1)

a new region of a conductive phase arises. A random value d is assumed to take into account inhomogeneities in dielectrics, thermal and other fluctuations, including local microfields acting on the molecules.

The following probability distribution for fluctuations d is used

j(d) = exp(-d / g) / g ,             (2)

that is d = - g ln(x). Hereinafter x will be a random number that is uniformly distributed in the interval from 0 to 1. For a given distribution function, r(E) has the form

r(E) = A e^E/g ,             (3)

where

A = $\frac ${1}{t} exp(-E_* / g) ,

where A and g are constants that depend on dielectric properties of liquid. The quantity of E_* should be always chosen greater then E. After this, t should be accordingly changed to ensure A=const.

To each random value d_i corresponds the time of appearance of the i-th bond

t_i = - $\frac ${ln(1-exp(-d_i / g))} {r(E_i)}.             (4)

The distribution of times t_i is

j (t_i ) = r(E_i ) exp(-r(E_i ) t_i ).             (5)

Thus, condition (1) is exactly equivalent to the inequality t_i < t.

Biller's Model

Scaled Stochastic Time Models

p(E_i ,t) = $\int $\limits_{t}^{$\infty $} r(E_i ) exp(-r(E_i ) t) dt = exp(-r(E_i ) t).

Let there be n dielectric sites near the structure. Then the probability that no bonds arise in time t is equal to

P(t) = $\prod $\limits_{i=1}^{n} p(E_i ,t) = exp(-S_nt),

where S_n = S_i=1ⁿ r(E_i ) . The corresponding distribution function of the time of appearance of the first new bond is f(t) = S_nexp(-S_nt) . It has been proposed [6] to use the random value t_S = - ln(x) / S_i=1ⁿ r(E_i ) --- the "scaled stochastic time" --- as the "physical" time interval.

At the same time, a new bond in the single-element model can be selected in several ways:

• for instance, by adding, as in [5], the bond that has minimum delay time
  t_i = - ln(x_i ) / r(E_i ) to the conductive structure (see in [6]);
• using the NPW criterion;
• using the FFC. A bond that has the shortest of the generation times t_i calculated from Eq. (4) is chosen.

New Multi-Element Models

• The times t_i = - ln(x_i ) / r(E_i ) is computed for all candidate bonds as in Biller's model. All the bonds having t_i < t arise.
• Several time steps t_Sj = -ln(x_j ) / S_i=1ⁿ r(E_i ) are sequentially performed as in SSTM as long as S_j=1^m t_Sj < t.

Discussion

The remaining growth criteria with the "physical" time that were considered above (FFC, SSTM, Biller's, and modifications of them) use the same distribution (5) of the stochastic time and, therefore, describe correctly, in particular, the statistical time lag of breakdown. Thus, these models are essentially equivalent, provided that the function r(E) = A e^-E/g is the same.

The basic difference of single-element models from multi-element models is that they consider the competition between candidate bonds in different ways. The competition is neglected in multi-element models, and the first bond that appears is assumed to suppress the growth of other bonds in single-element models. All multi-element models allow one to introduce a constant time interval for all steps of the growth. This fact is important for gas-dynamics flow simulation.

The time step is decreased because of the competition among the great number of branches during simulation of the streamer propagation by single-element models. It is reasonable to continue the computations using one of the multi-element models in this case.

The FFC criterion of conductive phase generation satisfies a specific form (3) of the function r(E), which is considered to describe dissociation and ionization in a strong electric field. All other models allow us to use any other dependences r(E) corresponding to various physical mechanisms of streamer inception and propagation. For example, one can use the well-known dependence r(E) = (E/E₀ )^h / t₀ or the tunnel effect r(E) = exp(-C/E ) / t₀ .

Computer Simulation

L(j) = - 4 p Q/h³ ,

$\frac{dQ}{dt}$ = - $\Div $(I_m) ,

I_m = - s_m s   D j_m / l_m ,

where L is the finite-difference Laplace operator, h is the lattice bond length, $\Div $(I_m ) = S_m=1^M I_m is the sum over all currents I_m flowing out of some vertex of the graph, M is the number of neighboring graph vertices, Dj_m is the electric potential difference along the bond, l_m is the bond length, s is its cross section, and s_m is its conductivity.

We used time-implicit finite-difference method for the equation of charge transport to obtain the equation for computation of electric potential

L(jⁿ⁺¹)h³ + 4 p t s   \Div(s_m   Dj_mⁿ⁺¹ / l_m) = - 4 p Qⁿ_i,j,k ,

where t is a step in time and n is its number. Charge transport is computed using new values of the electric-field potential jⁿ⁺¹

Qⁿ⁺¹_i,j,k - Qⁿ_i,j,k = s t S_m=1^M s_m Dj_mⁿ⁺¹ / l_m

to ensure charge conservation.

The calculation method developed is stable over a wide range of parameter 4 p s t s / h² up to 10⁵.

The simulation was carried out on lattices with sizes up to 90x90x90. The conductivity s was assumed to be constant and equal for all branches of the streamer structure. Streamer propagation velocity was ensured equal for all bonds including both types of diagonals, provided that the electric potential difference between their ends was the same. This statement was tested specially.

Results

For example, typical streamer structures obtained with multi-element FFC model (Fig. 1) and single-element Biller's model (Fig. 2) are shown. A denser streamer structure is formed when the applied voltage V increases, provided the conductivity is the same (Figs. 1,a--c and 2,a--c). Increase in s is equivalent to reduction of V (Figs. 1d and 2d). There is an area that is filled densely with streamer channels in the vicinity of the central electrode. These patterns coincide within stochastic variations. The time dependences of the growth velocity (Fig. 3a,b) and the total charge of structure (Fig. 3c,d) practically coincide for all the models with "physical" time provided that the function r(E) is the same. An increase in s causes a strong rise of the growth velocity (curves 4 in Fig. 3a,b) owing to the reduction of the charge relaxation time in the channels. The maximum electric field strength in front of the streamer tips is roughly constant during the growth and depends mainly on V and s .

Figures 4 and 5 show the graphs of the logarithm of the conductive bond density averaged over 15 streamer patterns versus the logarithm of the distance r from the centre. There is no fixed value of the fractal dimension for the streamer structure at finite s . The differential fractal dimension D_r introduced in [6] depends on r . D_r is close to the space dimension near the central electrode and reduces when the radius increases, particularly in the growth region. The area with
D_r � 3 expands as voltage increases (Fig. 4a,b). The electric field there is strong so that the streamer channels can propagate rapidly without charge relaxation in them.

 

Figure 4. Differential fractal dimensions of streamer structures for FFC model. V = 10(a) and 30(b) (s = 1).

Some discrepancies between single-element and multi-element models exist in the high-field region in the vicinity of the central electrode. There multi-element models give a higher density of streamer channels (Fig. 5b). These differences practically vanish in the case where the time step t in multi-element models is small, and, hence, the competition of neighboring bonds becomes insignificant.

 

Figure 5. Differential fractal dimensions of streamer structures. (a) Biller`s model, (b) FFC (�) and Biller's (+) models, V = 5, s = 1.

The same simulations were performed with SST models and new multi-element modification of Biller's model. These results agree with those described above.

Conclusion

Acknowledgements

References

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