How Fast Does a Charge Decay?

Associate Professor Neils Jonassen authored a bi-monthly static column that appeared in Compliance Engineering Magazine. The series explored charging, ionization, explosions, and other ESD related topics. The ESD Association, working with In Compliance Magazine is re-publishing this series as the articles offer timeless insight into the field of electrostatics.

Professor Jonassen was a member of the ESD Association from 1983-2006. He received the ESD Association Outstanding Contribution Award in 1989 and authored technical papers, books and technical reports. He is remembered for his contributions to the understanding of Electrostatic control, and in his memory we reprise “Mr. Static”.

~ The ESD Association

Reprinted with permission from: Compliance Engineering Magazine, Mr. Static Column Copyright © UBM Cannon

There’s a phrase that has been bothering me for years: “How do you remove static electricity?” At one level the question makes sense. Everybody involved in electrostatics understands what the inquirer is trying to ask. But at a physics level, as well as a linguistic one, the phrasing is more dubious. A better expression of the question would be “How do you neutralize the field from static charges?”

Why is this phraseology better? Well, first of all, the field from a charge (distribution) is a well-defined concept, which static electricity is not. And secondly, when you do neutralize a field (or “remove static electricity”), you very rarely remove anything from the charged body. (When you ground a negatively charged conductor with a metallic wire and avoid all kinds of discharges, you lead away the excess electrons. But that is the only case where charges can be removed.)

In order for neutralization to happen, the charged area has to be in contact with a medium containing charge carriers of the opposite polarity. A force from the field then acts upon these charge carriers, and, if they have some ability to move, they’ll eventually plate out on the charged area. The field from the plated-out carriers will superimpose the original field, resulting in a steadily decreasing “total” field. In other words the static charge is decaying.

So let’s change the question from how to remove static electricity to how fast does a charge decay.

Bulk and Surface Decay

It is easier to describe the decay if we consider separately bulk decay, where charges move through the interior of the material, and surface decay, where the movement of charges takes place primarily in a surface layer.

Bulk Resistivity. The rate at which neutralization takes place in a given field depends upon the conductivity γ of the medium. A field E will release a current with the density (current per unit area) j given by

j = γE    (1)

Equation 1 is often written

E = ρj    (2)

where

       (3)

is the bulk resistivity of the medium. Equations 1 and 2 are both versions of Ohm’s law (in differential form). The field from a given charge will always be proportional to the charge, but the factor of proportionality will depend upon the geometry and dielectric properties of the charged body and its surroundings.

Let’s look at a simple example. Figure 1 (situation 1) shows a piece of a material, with the resistivity ρ and the permittivity ε, resting on a grounded plane. A charge q is evenly distributed on the surface of the material. We assume that the distance to other grounded objects is much larger than the dimensions of the charged sample.

Figure 1: Bulk decay of charge, situation 1.

If the charge density is σ (C • m²), then the field strength in the material is

     (4)

According to Equation 2 this field will produce a (negative) current

But the current density j is also the rate at which the surface density decreases, that is

     (5)

The solution to Equation 5 is

      (6)

where σo is the initial value of the charge density. Thus it appears that the charge is being neutralized exponentially with the time constant

τ_o = ερ    (7)

Equation 7 is generally valid when the field from the charge to be neutralized extends exclusively through the medium with the resistivity ρ and the permittivity ε.

Consider a sample of Plexiglas with ρ ≈ 1013 Ω • m and ε ≈ 3 • 10¹¹ F • m¹ (ε_r ≈ 3.4). A charge on it will decay with a time constant of about 300 seconds. It should be appreciated that the rate of decay is determined not only by the resistivity, but also by the permittivity. So if we have a sample with the same resistivity as the Plexiglas, but with twice the permittivity, the rate of decay will be half that of the Plexiglas.

The situation is more complicated, however, if the field from the charge, or rather the flux, extends through several dielectrics with different resistivities and permittivities. Thus, a brief digression to discuss electrical flux is useful here.

The electrical flux or E-flux Φ_E through a surface S is defined as

ΦE = ∫ E • dS
s

If the surface S is a closed surface surrounding a charge q, then, assuming you have the same permittivity all over the surface S, the previous equation becomes

This is simply Gauss’ theorem, which enables calculation of the field from various charge distributions. Flux being a rather abstract concept, it can be helpful to envisage the situation as a charge “emitting” a certain number of field lines. The number of those field lines through a unit area (perpendicular to the field strength) is equal to the field strength. So the flux through a given area is, roughly speaking, the number of field lines through that area.

Now back to the more-complex situation. Figure 2 (situation 2) shows a sample with the thickness d, permittivity ε, and resistivity ρ, resting on a grounded plane, like that shown in Figure 1. But in this case another grounded plane is placed parallel to the sample at a distance x. Let us assume that the sample is Plexiglas, and that the space above the sample is vacuum (or air) with ε = ε_o and ρ ≈ ∞. The field (flux) from the charge is now shared between the Plexiglas and the air in such a way that the surface potential of a point on the charged surface is the same whether it’s calculated as the field strength in air multiplied by x or as the field strength in the dielectric multiplied by d. Thus the charge is expected to decay exponentially again, but now with a time constant τ given by

  (8)

Figure 2: Bulk decay of charge, situation 2.

For instance if we choose d = 0.01 m and x = 0.003 m (ε_r = 3.4, the relative permittivity of Plexiglas), we find that τ = 594 seconds. In other words, it takes about twice as long for the charge to decay, even from the same sample of material, simply because of the proximity of another grounded conductor.

The example shown in Figure 2 is a very simplified case, and often it will not be possible to predict the relevant value of the time constant for a given sample in a given geometrical environment.

Surface Resistivity. Special cases are the ones in which neutralization takes place in a shallow layer on the surface of the material. This could be a material treated with an antistatic agent or an insulative substrate onto which a conductive layer is evaporated. If such a layer is highly conductive as compared with the contacting materials, the neutralizing current will run only in this layer. However, part of the flux from the charge will run in the adjoining layers, and the “driving field,” that is, the field in the conductive layer, will depend upon the permittive properties of the adjoining insulators. Thus the rate of decay (and the time constant) will depend not only on the properties of the region where the decay takes place, but also on properties outside the region of decay. This is, in principle, the same problem shown in Figure 2. Usually the processes in thin layers are characterized by defining a surface resistivity ρs (in a way similar to the definition of bulk resistivity) by the equation

E_s = ρ_sj_s    (9)

This version of Ohm’s law states that a field Es along a surface with the surface resistivity ρs will cause a current with the linear current density j_s (current per unit length, A • m₁) in the layer given by equation (9).

Although in the matter of bulk resistivity it is possible in certain simplified cases (Figures 1 and 2) to derive a connection between the resistivity and the rate at which a charge is being neutralized, it is not nearly as simple in the matter of surface resistivity.

Figure 3 (situation 1) shows a piece of material A. At one end of A is a spot of negative charge and, at the other end, a grounded electrode B in direct contact with A. Between B and A is a field. Only that fraction of the flux that runs through the conductive layer will cause a current to neutralize the charge. There is no doubt that if the charge is, say, doubled, then the field strength will be doubled in every point, but the field distribution will be the same. And if the surface resistivity is doubled, the decay rate will be halved. With this geometry, it seems likely that we will have a time constant proportional to the surface resistivity. But, in contrast to simple situation 1 for bulk decay (Figure 1), we cannot theoretically predict–even if we measure the surface resistivity and know the permittivity of the conductive layer–the time constant for surface decay. This is because we don’t know how the flux is distributed between the conductive layer and the environment.

Figure 3: Surface decay of charge, situation 1.

Figure 4 (situation 2) shows a state similar to situation 2 for bulk decay (see Figure 2). Another grounded conductor C is in the neighborhood of the charged sample, but not in direct contact with it, so no neutralizing current will flow to C. And since the flux to B is now lower, so is the neutralizing current, and the time constant will have increased, even if the sample, the charge, and the grounding electrode arrangement is the same.

Figure 4: Surface decay of charge, situation 2.

This discussion has tacitly assumed that there is only one value for the resistivity (be it bulk or surface) independent of the field applied. Yet it is often found that the resistivity increases with decreasing field strength. Nevertheless, resistivities are usually determined at only one field strength (one voltage difference between a set of electrodes on the sample), and we have no way of knowing if this particular field strength is typical for the physical conditions during a decay process.

Measurement of Decay Time

The previous considerations illustrate that only under very ideal conditions is it possible to calculate reasonably accurately from material parameters (resistivity and permittivity) how fast a charge on an insulator will decay. This is because of two main reasons:
The resistivity depends on the field strength from the decaying charge (and we rarely know this relationship), and even more importantly,
The driving field from a given charge depends on the permittive properties of the environment in a usually incalculable way.

So the obvious question is why not measure the decay time directly? If we are dealing with a highly resistive item, it is certainly possible to charge the material and measure how fast the field from the charge decays when the item is placed in a relevant environment. Usually we are interested in semiinsulative materials where the charges are neutralized in seconds or less. And the procedures of measurement have to allow for this.

Over the years several procedures have been developed, and, to be kind, none of them were very successful. A general shortcoming of all these methods is that they do not measure in situ. That is, the measurements are performed not on the material as it normally appears when it gets charged, but rather on sheet samples suspended in such a way as to facilitate charging as well as field measurement.

Probably the most commonly used method is Federal Test Method (FTM) Standard 101C, method 4046.1, where a sample is clamped between two electrodes (see Figure 5). A field meter is mounted pointing at the center of the sample midway between the electrodes. The sample is allegedly charged by the electrodes when they are connected to a voltage supply, and the charge decay is taken as the reading of the field meter after the electrodes are grounded. It seems difficult (at least to this author) to be sure that a reading of the field meter is a sign of an excess charge on the material, unless the material is truly conductive. Polarization may certainly show itself, at least with some materials, as an external field, and the rate of relaxation of polarization is not necessarily the same as that of a true excess charge.

Figure 5: FTM Standard 101C decay of field from charge.

Several other questions could be raised concerning this method. The most important one is that a decay time obtained by method 4046.1 for a sheet of a material of a given small size does not reveal much about how fast the field from a charge will be neutralized on a larger sample or item in another location.

In another method, the sample (again a suspended sheet) is charged by a corona discharge. The charger is then removed and replaced by a field meter. (Incidentally, we developed this method, which has the merit of placing a real charge on the surface of the material under investigation, at our laboratory as early as 1977, but ultimately abandoned it since our instrumentation was not fast enough.) Although it has been argued that the corona charging with air ions may be irregular, one could also argue that the charging experienced in everyday life is irregular too. So this should rather be deemed a virtue of the method. Still the main argument is that one does not measure the charge neutralization (decay) rate under circumstances that resemble normal use of the materials.
It should also be mentioned that it is not possible to distinguish between bulk and surface decay using either of these methods, or probably any other method for that matter. It may even be argued that the distinction does not make sense at all. Another objection to any principle, suggested or applied, for determination of charge decay time is that any method capable of detecting the presence and time variation of a charge on an object will occupy a certain fraction of the electrical flux from the charge, a fraction which, without the presence of the measuring equipment, might contribute to the rate of neutralization or decay of the charge. Thus the measured rate of decay will normally be different from (often larger than) the «natural,» undisturbed rate.

Conclusion

The considerations presented in this paper may make it seem as if we know nothing about the laws of decay of charges on insulators. This is not the case.

Although we can accurately predict the current I through a resistor with the resistance R from a voltage supply with output voltage V, we have to accept that static electricity (ESD, if you insist) is a little more complicated (and interesting). We also have to accept the fact that there’s no way you can predict the decay behavior of a manufactured item placed in an arbitrary environment by doing some laboratory measurements on a sample of the material of said item.

So what do we do when we have to choose between different materials? Well, we know that if we have two materials with different resistivities, bulk or surface, under similar circumstances the one with the lowest resistivity will mean the fastest decay time, although not in an unambiguous way. So the obvious advice is to choose the material with the lowest resistivity.

Niels Jonassen, MSc, DSc worked for 40 years at the Technical University of Denmark, where he conducted classes in electromagnetism, static and atmospheric electricity, airborne radioactivity, and indoor climate. After retiring, he divided his time among the laboratory, his home, and Thailand, writing on static electricity topics and pursuing cooking classes. Mr. Jonassen passed away in 2006.