Interpretation of experimental results
Charge transport in materials is conditioned by electrons, ions or colloidal charged particles. Electrical conductivity increases with the increasing of the concentrationn
_{
i
}, charge q
_{
i
} and mobility χ
_{
i
} of these charged particles:
\gamma ={\displaystyle \sum _{i}{n}_{i}}{q}_{i}{\chi}_{i}
(7)
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 PoolFrenkel effects are observed in lower temperatures (Nevin & Summe 1981).
In the: “bonded” state (not dissociated), the ion is in the I state with the lowest energy (Figure 5). As a result of thermal movement fluctuation, there is a possibility of “liberation” of a portion of the “bonded” ions and overcoming in their II state. Characteristic of the II state is the existence of vacancies in which the ion may be located. If the number of ions in theI state is n
_{0}, then, the number of ions in the II state, in conditions of an electric field applied, is (Sazhin 1970):
n={n}_{0}\frac{exp\left(\frac{\mathit{\Delta E}q{l}_{1}E}{\mathit{kT}}\right)}{\left[1+exp\left(\frac{\mathit{\Delta E}q{l}_{1}E}{\mathit{kT}}\right)\right]}
(8)
where, ∆E where is the difference in ion’s energy between I and II states, while l
_{1} is the distance between I and II state. For the ion mobility in moving from II state to III state and beyond, in the field direction the following expression can be used (Nevin & Summe 1981):
\chi =\frac{{f}_{0}{l}_{1}}{3E}exp\left(\frac{\mathit{\Delta U}}{\mathit{kT}}\right)\mathit{sh}\left(\frac{q{l}_{2}}{2\mathit{kT}}E\right)
(9)
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.
If the electrical conductivity is conditioned by the movement of a type of ion, then by neglecting the second term of (8), we obtain:
\gamma =\frac{{n}_{0}{f}_{0}{l}_{1}q}{3E}\phantom{\rule{0.3em}{0ex}}exp\phantom{\rule{0.2em}{0ex}}\left(\frac{\mathit{\Delta E}q{l}_{1}E}{\mathit{kT}}\right)\phantom{\rule{0.3em}{0ex}}exp\phantom{\rule{0.2em}{0ex}}\left(\frac{\mathit{\Delta U}}{\mathit{kT}}\right)\phantom{\rule{0.2em}{0ex}}\mathit{sh}\phantom{\rule{0.2em}{0ex}}\left(\frac{q{l}_{2}E}{2\mathit{kT}}\right)
(10)
or
\gamma =\frac{{n}_{0}{f}_{0}{l}_{1}q}{3E}\phantom{\rule{0.3em}{0ex}}exp\phantom{\rule{0.2em}{0ex}}\left(\frac{\mathit{\Delta E}+\mathit{\Delta U}}{\mathit{kT}}\right)\phantom{\rule{0.3em}{0ex}}exp\phantom{\rule{0.2em}{0ex}}\left(\frac{q{l}_{1}E}{\mathit{kT}}\right)\phantom{\rule{0.2em}{0ex}}\mathit{sh}\phantom{\rule{0.2em}{0ex}}\left(\frac{q{l}_{2}E}{2\mathit{kT}}\right)
(11)
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}\phantom{\rule{0.1em}{0ex}}\left(\frac{q{l}_{2}E}{2\mathit{kT}}\right), takes into account the influence of the field in the mobility of ions.
If, l
_{1} = l
_{2} = l (see Figure 5) we have:
\mathit{sh}\phantom{\rule{0.2em}{0ex}}\left(\frac{\mathit{qlE}}{2\mathit{kT}}\right)=\frac{1}{2}\left[\phantom{\rule{0.2em}{0ex}}exp\phantom{\rule{0.2em}{0ex}}\left(\frac{\mathit{qlE}}{2\mathit{kT}}\right)exp\phantom{\rule{0.2em}{0ex}}\left(\frac{\mathit{qlE}}{2\mathit{kT}}\right)\right]
(12)
and
exp\phantom{\rule{0.2em}{0ex}}\left(\frac{\mathit{qlE}}{\mathit{kT}}\right)\phantom{\rule{0.1em}{0ex}}\mathit{sh}\phantom{\rule{0.1em}{0ex}}\left(\frac{\mathit{qlE}}{2\mathit{kT}}\right)=\frac{1}{2}\left[\phantom{\rule{0.1em}{0ex}}exp\phantom{\rule{0.1em}{0ex}}\left(\frac{3\mathit{qlE}}{2\mathit{kT}}\right)\phantom{\rule{0.1em}{0ex}}exp\phantom{\rule{0.1em}{0ex}}\left(\frac{\mathit{qlE}}{2\mathit{kT}}\right)\right]
(13)
By neglecting the term exp\phantom{\rule{0.1em}{0ex}}\left(\frac{\mathit{qlE}}{2\mathit{kT}}\right), the relation (11) can be written:
\gamma =\frac{{n}_{0}{f}_{0}\mathit{lq}}{6E}\phantom{\rule{0.2em}{0ex}}exp\phantom{\rule{0.2em}{0ex}}\left(\frac{\mathit{\Delta E}+\mathit{\Delta U}}{\mathit{kT}}\right)\phantom{\rule{0.2em}{0ex}}exp\phantom{\rule{0.2em}{0ex}}\left(\frac{3\mathit{qlE}}{2\mathit{kT}}\right)
(14)
If we note {\gamma}_{0}=\frac{{n}_{0}{f}_{0}\mathit{lq}}{6E}\phantom{\rule{0.1em}{0ex}}exp\phantom{\rule{0.2em}{0ex}}\left(\frac{3\mathit{qlE}}{2\mathit{kT}}\right) and W
^{∗} = ΔE + ΔU, where W
^{*} is the activation energy for a molecule, we obtain:
\gamma ={\gamma}_{0}exp\phantom{\rule{0.1em}{0ex}}\left[\frac{{W}^{\ast}}{\mathit{kT}}\right]
(15)
while arguing for a mole, the above relation can be written:
\gamma ={\gamma}_{0}exp\phantom{\rule{0.1em}{0ex}}\left[\frac{W}{\mathit{RT}}\right]
(16)
where W is the activation energy for a mole. The inverse of the above formula ultimately determines the electrical resistivity dependence of the temperature:
\rho ={\rho}_{0}exp\phantom{\rule{0.1em}{0ex}}\left[\frac{W}{\mathit{RT}}\right]
(17)
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.
Its magnitude is identified as the difference between the energy of the molecules undergoing reaction and the overall average energy (Brown & Galweyl 1995). The activation energy may be positive, negative, or zero, depending on the complexity of the reaction being investigated. When the activation energy is negative, a standard interpretation of the observation is available: the reaction under investigation is multistep and involves at least one intermediate step (Turro et al. 1982). So, the type of reaction that clearly gives the change of activation energy sign is the multistep reaction. For a reaction with a “preequilibrium”, there are three activation energies to be taken into account; two of which refer to reversible steps of preequilibrium and one to the final step. The relative magnitudes of activation energies determine whether the total activation energy is positive or negative (Atkins 2006). We distinguish two types of reactions:

a)
Exothermic, associated with heat production and the activation energy, in this case, is negative (W < 0); b) endothermic, associated with heat absorption and activation energy, in this case, is positive (W > 0) .From the relation (16) we observe that with the increase of the temperature, for W < 0, the exponential {e}^{\frac{W}{\mathit{RT}}} increases, while for W > 0, the exponential {e}^{\frac{W}{\mathit{RT}}} decreases. This way the increasing or decreasing trend of exponential functions that give the temperature dependence of electrical resistivity, can be explained.From the graphs we see that, for the same model, there are different values of activation energy.
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
Although the fundamental object of our study is not the activation energy, we see important to stop on an estimation of it, comparing relations (5) or (6) of experimental approximations with relation (17) of theoretical interpretation:
\rho =A{e}^{\frac{B}{T}}={\rho}_{0}{e}^{\frac{W}{\mathit{RT}}}
(18)
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.