The influence of temperature on the electrical resistivity of the cellular polypropylene and the effect of activation energy

Determination of electrical resistivity in the dependence of temperature

Determination of dependence of surface electrical resistance on temperature

The schematic diagram of experimental measurements for the determination of the surface electrical resistivity is presented in Figure 1. It is important for the guard electrode to have a positive potential in order to gather electrons on it.

The surface electrical resistivity, according to the proposed scheme, can be determined by the relation(Vila & Sessler 2001):

${\rho }_s=\frac{2\pi }{ln\frac{D_2}{D_1}}·R_S$

(1)

where D ₂and D ₁ are, respectively, the inner diameter of the guard electrode and the central electrode diameter, while R _s is the surface resistance experimentally determined by the electrometer.

The maximal error of the methodology and apparatuses used is 6%.

Determination of dependence of volume electrical resistance to temperature

The schematic diagram of experimental measurements for the determination of the volume electrical resistivity is presented in Figure 2. In this case it is important for the electrode under the surface of the model to have a positive potential in order to gather electrons on it.

Calculation of the volume electrical resistivity (ρ _v ) from the measured volume resistance (R _v ), (${\rho }_v=\frac{S{R}_v}{d}$), involves, among other things, the real surface S $=\frac{\pi D_1^2}{4}$ of measurement electrode, but, since the model material is homogeneous and isotropic, fringes of the current lines in the area of the edges, may effectively increase the dimensions of the electrode diameter (1993; Vila et al. 2005) by the amount Bg , where,

$B=1-\frac{2\delta }{g};\delta =\left(\raisebox{1ex}{$2d$}\!\left/ \!\raisebox{-1ex}{$\pi $}\right.\right)lncosh\left(\raisebox{1ex}{$\mathit{\pi g}$}\!\left/ \!\raisebox{-1ex}{$4d$}\right.\right)$

while d and g are, respectively, the model thickness and the distance between central and guard electrode. Therefore, the exact “effective” area (S _eff ) of contact between the central electrode and the model is:

$S_{\mathit{eff}}=\frac{\pi }{4}{D}_1^2\left(1+\frac{\mathit{gB}}{D_1}\right)$

(2)

where, D ₁ is the central electrode diameter. Thus, the volume electrical resistivity is determined by the relation:

${\rho }_v=\frac{\pi }{4d}{D}_1^2{\left(1+\frac{\mathit{gB}}{D_1}\right)}^2{R}_v$

(3)

But, in our experimental conditions $\frac{\mathit{gB}}{D_1}$ << 1; thus we can write:

${\rho }_v=\frac{\pi }{4d}{D}_1^2\left(1+\frac{2\mathit{gB}}{D_1}\right)R_v$

(4)

The volume resistivity R _v is measured with the same apparatus, as well as the surface resistance.

The maximal error of the methodology and apparatus used is 6.4%.

Surface electrical resistivity

are presented in Figure 3.

The exponential approximation process can be divided into two intervals. The first involves the temperature range 293 K − 378 K, where a ₁ = 188.67Ω and b ₁ = − 1212 K, while the second involves the temperature range 383K − 453 K, where a ₂ = 0, 27Ω and b ₂ = 1285 ± 61 K.

Volume electrical resistivity

are presented in Figure 4.

The exponential approximation process can be divided into three intervals. The first involves the temperature range 293 K − 373 K, where α ₁ = 11, 24Ωm and β ₁ = 63 ± 16 K. The second one involves the temperature range 383 K − 413 K, where α ₂ = 2, 43×10^− 7 Ωm and β ₂ = 6901 ± 1473 K. The third, involves the temperature range 418 K − 448 K, where α ₃ = 7, 29×10^− 18 Ωm and β ₃ = 17738 ± 3768 K.

Interpretation of experimental results

The summation in the equation above is performed for all types of charge carriers.

Of the three types of electrical conductivity observed in polymers, for the interpretation of experimental results we will consider the electrical conductivity which is conditioned by the ions movement. The reason for this is that in our case the fields applied are weak (E≈ 2kVcm ⁻¹) and they extend in short time intervals (1min) (Sessler et al. 1986; Sawwa et al. 1980). Also the temperature range where the experiments are made is generally more than 300K, so the ionic conduction effect is more probable, because a combination of Schottky and Pool-Frenkel effects are observed in lower temperatures (Nevin & Summe 1981).

where ∆U−the potential barrier between II and III equilibrium states; l ₂ − the distance between these states; f ₀− the frequency of vibrations in II and III states (Figure 5);

Knowing that the velocity of ions movement is v = χE, and the current density, j = nqv = nqχE = γE, derives γ = nqχ.

The first exponential term in equation (11) takes into account the “liberation” and ion movement under the influence of thermal movement. The second exponential term takes into account the influence of the field in the charge carriers concentration, while the multiplier $\mathit{sh}\left(\frac{q{l}_{2}E}{2\mathit{kT}}\right)$, takes into account the influence of the field in the mobility of ions.

which is in full accordance with the fitting formula of experimental results.

Exponential type functions and activation energy

Exponential approximations of experimental results are of two types, increasing and decreasing functions. The question is: How do we explain this?

Formula (17) shows clearly that the “key” to answering this question is the activation energy. It is usually identified as the energy barrier that must be surmounted to enable the occurrence of the bond redistribution steps required to convert reactants into products.

Variation of activation energy and electrical resistivity

In the graph of the dependence of the surface electrical resistivity on temperature (Figure 3), one can clearly distinguish two zones:

In the first zone, we notice an exponential increase. In this case W = − 10, 09kJmol ⁻¹ < 0 , thus the reaction is exothermal. The temperature increase leads to an intensification of thermal movement of ions. In exothermal reactions, ions have enough energy to surmount the potential barrier and undergo reaction. This means that the number of ions which contribute to conductivity is reduced, thus the electrical resistivity will be increased.

The second zone corresponds to an exponential decrease.

In this case W = 10, 68kJmol ⁻¹ > 0, thus the reaction is endothermic. In endothermic reactions, ions do not have enough energy to surmount the potential barrier and easily undergo the reaction. This means that a large number of ions contribute to conductivity thus the electrical resistivity will be decreased.

In the graph of the dependence of the volume electrical resistivity on temperature (Figure 4), we can distinguish three exponentially decreasing zones:

This means that reactions that occur are characterized by a positive activation energy W > 0 and with the increase of the temperature, the exponential $e^{\frac{W}{\mathit{RT}}}$ decreases. In the first zone (where W ₁ = 0, 52kJmol ⁻¹ > 0) ions do not have enough energy to undergo the reaction. Thus, the electrical resistivity decreases.

In the second zone W ₂ = 57, 40kJmol ⁻¹ > W ₁ the energy barrier that ions have to surmount to undergo the reaction, is higher.

So, the number of ions that contribute to conductivity is larger than in the first zone and therefore the electrical resistivity continues to decrease, but the decrease happens more rapidly than in the first zone.

In the third zone, W ₃ = 147, 54kJmol ⁻¹ > W ₂ > W ₁, the energy barrier is higher than in two other zones. This means that the number of ions that are able to undergo the reaction is very small. The conductivity increases. The electrical resistivity continues to decrease, but the decrease happens more rapidly than in the first and second zones.

Estimation of activation energy

The estimations show that in case of cellular polypropylene (VHD 50), the average activation energy, for the temperature range between 293 and 453 K, is W = 41, 2kJmol ⁻¹. This is in full accordance with the order of the values of activation energies, obtained by other researchers, for polypropylene: W = 38.94kJmol ⁻¹ (Eckstein et al. 1998), W = 40, 61 − 41, 87kJmol ⁻¹ (Pearson et al. 1988) and W = 41, 87kJmol ⁻¹ (Fujiyama et al. 2002). This fact clearly confirms the accuracy of experimental measurements.