# Electrical Properties

Several applications make the electrical properties of paper relevant. Static electricity causes problems in printing and the handling of dry paper. Important are also electrophotographic applications, such as laser printing and copying, and special applications, such as papers used in electrical insulation and capacitors. Already today and especially in the future, using paper as a base for printed electronics also makes this subject interesting. The various electrical properties of paper are much more complicated than, say, the thermal ones, since here the whole structure — pores, fibres, fillers and additives and coatings, and even the humidity — contributes. This first part introduces the most important theoretical considerations as background.

These above-mentioned matters are discussed more in detail in the following articles.

Theory

According to Ohm’s law, the current density, J, relates to electric field, E, by

(1)

where σ is electrical conductivity. In most materials, conductivity depends on the direction of the applied field. This makes σ a second rank tensor, which can be represented by a 3×3 matrix. In linear isotropic materials only, conductivity is a scalar quantity independent of the direction of electric field.

The inverse value of conductivity, resistivity, ρ = 1/σ, is often used. The resistivity of paper is usually 1010–1014 Ω·m. The high values mean that paper is an insulator. For comparison, the resistivities of copper and undoped silicon at room temperature are 2·10-4 and 106 Ω·m, respectively 1.

Usually, the volume (or bulk) resistivity and surface resistivity of paper are considered. Volume resistivity is measured in the z-direction through the thickness of paper. It has the ordinary units of Ω·m. Volume resistivity is a material parameter and should not depend on paper thickness. Surface resistivity is measured along the surface and has the units of Ω because one determines the current across a line on the surface 2,3. The current is not entirely limited to the surface layer unless its resistivity is much lower than the bulk resistivity. Sometimes lateral resistivity is calculated from volume resistivity measured in the plane of the sheet by dividing it with paper thickness 4.

The main problem with the resistivity of paper and other insulators is that the ordinary dc measurement gives ambiguous results. Among other things, the measured resistance is time-dependent. This is typical of a dielectric material. There are a few free charge carriers, and their activation energy is high.

The time-dependent current density in a dielectric material can be derived from Maxwell’s equations. It consists of an “Ohmic” current and a polarisation current,

(2)

where is the displacement or polarisation field that relates to the electric field via

(3)

Here ε(ω) is the frequency-dependent permittivity. It describes the polarisability of the material in an external electric field. Permittivity is usually given as the dielectric constant, εr = ε/ε0, where ε0 is the permittivity of vacuum. The effect of polarisation is to reduce the electric field inside the material. Larger permittivity gives stronger polarisation.

Figure 1 shows the polarisation mechanisms of a dielectric body 5. Positive charges of nonpolar molecules move in the direction of the applied field, and negative charges move in the opposite direction. Molecular forces oppose the charge displacement. If polar molecules exist, their permanent electric dipoles align themselves with the direction of the field. The total current density of Eq. 4 arises from the actual movement of charge carriers according to Ohm’s law and from the time-dependent changes in the electric displacement field. Figure 1. Polarisation mechanisms 5.

When the applied electric field varies with time, it is no longer reasonable to separate permittivity and conductivity. For this purpose, the expression of permittivity is the following complex quantity:

(4)

where ε´´(ω) is the dielectric loss term. In the complex permittivity, the dielectric constant is in the real part, εr = ε´ / ε0, and the ratio, tan δ ≡ ε´´/ ε´, is the loss tangent. If the material is not a perfect insulator, the apparent (measured) permittivity is the following 6:

(5)

From Eq. 5, one can see that the real part of permittivity is in the same phase with the applied field and does not cause any losses in an ac measurement. The imaginary part of permittivity and the conductivity term are out of phase and cause losses in an ac measurement — compare with an RC circuit and its impedance. However, only ε´´ is a true dielectric loss term. The second term is due to dc conduction. Figure 2 shows a schematic dielectric spectrum. The dc conductivity can be determined from the increase in the measured imaginary part of permittivity as ω → 0. Figure 2. Schematic dielectric spectrum with dc conduction losses 7.