Application of elements of quantum mechanics in analysing AC conductivity and determining the dimensions of water nanodrops in the composite of cellulose and mineral oil
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Abstract
The paper presents a model of hopping conductivity at direct and alternating current, developed on the basis of the quantum phenomenon of electron tunnelling between neighbouring potential wells. The tunnelling takes place between neutral potential wells, resulting in an electrical dipole and additional thermally activated polarisation. An important parameter of the model is the time of existence of the dipole (relaxation time). After this time, the electron with the probability p can hop to the third well, which causes a direct current flow, or return with the probability (1 − p) to the first well, thus causing the flow of high frequency current. The model shows that for direct or low frequency current, the current density and conductivity do not depend on the frequency. In the high frequency region, the current density does not depend on the frequency either. Lowfrequency conductivity is 2p times smaller than highfrequency conductivity. In the transitional region there is a frequency dependence of the current density. A formula was derived for the dependence of the relaxation time on the distance between neighbouring potential wells and temperature. The dependence of the hopping probability on activation energy, potential dipole energy and temperature was determined. A model of hopping conductivity at direct and alternating current was used to analyse the experimental frequency dependence of conductivity and permittivity of the composite of electrotechnical pressboard, mineral oil and water nanoparticles. It was found that there is a high compatibility of the experimental course with the model. It was also found that in the composite of electrotechnical pressboard, mineral oil and water nanodrops there are at least two mechanisms of changes in low frequency and high frequency conductivity. For the lowfrequency stage, the value of the probability of p_{1} ≈ 0.2 and the relaxation time τ_{1} ≈ 0.02 s. The probability of hops for the highfrequency growth stage is p_{2} ≈ 0.0005 and the value of relaxation time τ_{2} ≈ 2 × 10^{−7} s. The occurrence of two episodes of increase in conductivity is related to the presence in the composite of electrotechnical pressboard, mineral oil and water nanoparticles of differences in distances between neighbouring nanoparticles. For a large number of nanoparticles randomly distributed in the percolation channel, the distribution of the probability of the occurrence of the distance is in a good approximation normal. This means that in the percolation channel there are pairs of neighbouring wells, the distances between which are both much smaller than the average and much larger. A pair of wells, the distance between which is smaller than medium, participates in the conductivity on the highfrequency stage with a short relaxation time. On the other hand, a pair of neighbouring wells for which the distance is greater than average causes a high resistance to DC or lowfrequency current flow as well as a higher relaxation time. On the basis of the dependences designated in laboratory tests of lowfrequency conductivity (10^{−4} Hz) from the moisture content, using the elements of quantum mechanics, the dimensions of water nanoparticles in electrotechnical pressboard impregnated with mineral insulating oil were determined. It was found that nanoparticles contain on average about 200 molecules of water, and their diameters are about 2.24 nm. This approach allowed to calculate the dielectric permittivity of cellulose and to determine that water nanoparticles are located inside the cellulose fibres.
Graphical abstract
Keywords
Electrotechnical pressboard Mineral oil Moisture content Water nanoparticle Quantum mechanics AC conductivityIntroduction
For about a hundred years the production of highvoltagepower transformers has relied on insulation in the form of cellulose paper, impregnated by insulating oil. Paper–oil insulation, under longterm exploitation, undergoes aging phenomena, which are accompanied by cellulose depolymerisation processes, (Gilbert et al. 2010; Jalbert et al. 2015) take place, as a result of which the molecules of water precipitate directly in the cellulose fibers. However, the main source of moisture in the cellulose is moisture that penetrates into the transformer and dissolves in the transformer oil. The water is then supplied by cellulose oil and absorbed by it, this happens because the solubility of water in cellulose is more than a 1000 times higher than in transformer oil (Oommen 1983). These factors cause that during longterm operation the moisture content in the cellulosemineral oil composite gradually increases from the initial level of about 0.8% by weight up to even more than 5% by weight. The growth of moisture content in paper–oil insulation significantly reduces its quality and the reliability work power transformers.
It follows that the level of moisture content in the fixed insulation of power transformers should be specified as accurately as possible. To determine the moisture content in paper–oil insulation in power transformers nondestructive electrical methods are used. These methods can be divided into: measurements in the Frequency Domain Spectroscopy (FDS) (Ekanayake et al. 2006; Jaya et al. 2013; Zhang et al. 2014; Walczak et al. 2006) and time domain measurements: Return Voltage Measurement (RVM) (Dey et al. 2010; Saha 2003) and Polarisation Depolarisation Current (PDC) (Wolny and Kędzia 2010; Saha and Purkait 2004).
The analysis of dielectric relaxation processes in the field of frequency and time is based on models, each of which has a more or less reliable physical explanation.
It appears from formulas (1, 2) that in the case of ionic conductivity there should be a linear dependence of conductivity on the content of moisture in the pressboard. Based on the analysis of experimental results (Zukowski et al. 2014), it was found that the dependence of direct current conductivity on the moisture content is much stronger than the linear one (exponential), and the DC conductivity takes place via the quantum phenomenon of electron tunneling between water molecules.
Until recently, the basis for the analysis of AC test results was the Debye dielectric relaxation model (Jonscher and Andrzej 1983), which, as it turned out, has several features that do not have experimental confirmation. Therefore, in the area of pulsation values ω = 2πf lower than the inverse relaxation time τ (ω < 1/τ), a quadratic dependence of conductivity on frequency takes place in this model. In moistened pressboard impregnated by isolation oil, this dependence is weaker than the square one (Walczak et al. 2006). Next disadvantage of the Debye model is the reduction permittivity in the square from the frequency. Measurements of permittivity of impregnated electrotechnical pressboard with a specific moisture content prove that these dependencies are weaker (Jonscher and Andrzej 1983). To justify the occurrence of weaker frequency dependencies of conductivity and dielectric permittivity than square ones, a number of empirical models have been developed (Cole and Cole 1941, 1942; Havriliak and Havriliak 1997).
The most important unfavorable conclusion resulting from the above models is the argument that when the frequency of the electric field tends to zero (to constant voltage), conductivity also tends to zero. This is contradictory to before determined measurement results, which in the ultralow frequency range have a certain conductivity value, close to the DC conductivity.
In works (Zukowski et al. 2014, 2015a) based on laboratory tests observed that the DC conductivity in the composite of cellulose, mineral oil and water nanoparticles takes place by hopping (tunneling) of electrons between the potential wells produced by water nanodrops.
In order to analyze obtained on the basis of laboratory tests the AC current of the composite of cellulose, mineral oil and water nanoparticles, the following model of hopping conductivity at alternating and direct current will be used. This model was initially created to describe the conductivity in the semiconductor material of gallium arsenide, illuminated with high doses of highenergy hydrogen ions (Żukowski et al. 2007) and was then extended in a series of works for the case of metaldielectric nanocomposites see for example (Kołtunowicz et al. 2013).

In the material there are nearneutral potential wells, located in sufficient vicinity, in which electrons are located.

External constant or alternating electric field forcing current is weak and does not change the probability of P(T) electrons hopping from one neutral potential well to another, but only leads to asymmetry of their hops, which can be taken into account with the Debye factor (Mott et al. 1979):
After the hop of the electron from one neutral well of potential to another, it stays in it for the life of the dipole (time of relaxation) τ. At that time, the first well has a positive charge, while the second one has a negative charge. This means the occurrence in the material containing inert wells the potential of an additional thermally activated polarization caused by the hopping exchange of charges.
As can be seen from further analysis, this coefficient is a function of frequency.
In Zukowski et al. (1997) it was shown that in systems of wells interacting with each other, time τ is not the same for all wells. Its value should depend on the distance between wells, which are distributed randomly in the material. The authors assumed that the distribution of τ times may be normal or similar to it.
As can be seen from Fig. 4, in the low frequency area σ_{L}(f) = const. The increase in frequency causes an increase in the conductivity, allowing for the fact that the slope of the growth section the value α_{max}(f) in formula (15) decreases with the increase in the probability p of a hop (Fig. 5), while in the high frequency region σ_{H}(f) = const.
As can be seen from Fig. 5, for small values of probability p there are values of the frequency coefficient α_{max} close to 2. This corresponds to the situation when the hop takes place between wells with neutral charge. As a result of such a hop, the wells obtain opposite charges (an electric dipole is created), and the electric field of the dipole forces the electron to return to the first well. The low values of α correspond to the large probability of the pjump. When the probability value rises to p = 0.5 then the current density ceases to depend on the frequency of the alternating current and the value of α is equal to 0.
The experimental verification of the model, made for semiconductors containing potential wells and metallic phase nanoparticle composites in dielectric matrices [see, for example Kołtunowicz et al. (2013) and Svito et al. (2014)], showed both qualitative and quantitative compatibility of the measured results with the model of stepped transmission of electrons on a constant current and alternating.
The purpose of the work was to determine the formulas for the key parameters of the hopping conductivity model at direct and alternating current—the probability of leaks and relaxation time, to analyze the AC conductivity of the composite of cellulose, mineral oil and water nanoparticles based on the hopping conductivity model at direct and alternating current as well as to determine on the basis of AC conductivity measurements the dimensions of water nanodrops by using elements of quantum mechanics.
The probability of hops and the lifetime of the dipole (relaxation time)
The hopping probability p, entering formula (10) and the following ones, is one of the key parameters of the model of the swift exchange of charges at direct and alternating current. Below, the formula will be derived for its value, taking into account changes in potential energy during hops, shown in Figs. 1, 2 and 3.
It follows from formula (27) that as the temperature rises, the frequency courses of conductivity, capacitance, loss tangent and frequency coefficient α(f) move parallel to each other into the area of higher frequencies of the forcing alternating current. This phenomenon can be taken into account by selecting the extremum on the experimental course and multiplying the measurement frequencies by the τ_{max} value characteristic of the extreme. In this way Figs. 4 and 5 were made. This allows to recalculate the conductivity, capacity, loss angle tangent and frequency coefficient α, obtained at any temperature, to the reference temperature of 20 °C in electrotechnology and 0 °C in physics. The advantages of this method of converting experimental runs into the reference temperature were used in Żukowski et al. (2015b).
The application of the model of direct and alternating current hopping conductivity for the analysis of the frequency dependencies of the conductivity of the composite of electrotechnical pressboard, mineral oil and water nanoparticles
By substituting the frequency value f_{s1} = 1 Hz, at which an increase in conductivity on the experimental course (Fig. 6) stops, we obtain τ_{1} ≈ 0.02 s.
It follows from formula (29) that with similar values of the potential well concentration at the first and second stage of conductivity, the dielectric permittivity should differ as many times as the relaxation time at the second stage is smaller than at the first stage, i.e. around 10^{5} times. This means that after the increase of the conductivity at the first stage and the transition to the second stage, the permeability of the composite electrotechnical pressboard, mineral oil and water nanoparticles should decrease to the value characteristic of the nonmoisture composites. As can be seen from Fig. 6, after the end of the first stage of growth, there is indeed a decrease values determined experimentally in dielectric permittivity to about 5.
By substituting for formula (30) an experimentally determined value of the directional coefficient—formula (5), it can be calculated that the distances between neighbouring potential wells that determine the conductivity on a high frequency conductivity rise are about 1.88 nm shorter than for the low frequency stage. Taking into account that the hop lengths have a probability distribution close to normal, and the average value is about 10 nm there is a high probability of the distance between potential wells differing by about 1.88 nm.
Analysis of the experimental frequency dependencies of conductivity, permittivity and frequency coefficient α(f) for the composite electrotechnical pressboard—mineral oil–water nanoparticles showed that there is a high compatibility of these runs with the model of hopping conductivity at direct and alternating current of the composite of electrotechnical pressboard, mineral oil and water nanoparticles. This model is based on the quantum phenomenon of electron tunneling between potential wells. Using this fact, elements of quantum mechanics will be used below to determine the dimensions of water nanoparticles in electrotechnical pressboard impregnated with mineral insulating oil based on the analysis of the experimental dependences of conductivity on the water content.
Determining the dimensions of nanodrops
In work (Zukowski et al. 2014) based on laboratory tests it is clearly demonstrated that the direct current conductivity in moist impregnated isolation oil is determined by the moisture content and is carried out by electron tunneling between the nearest potential wells created by water molecules. This is due to the exponential dependence of the DC conductivity on the moisture content.
As shown in paper (Shklovskii and Efros 1984), in the case of electron tunneling between the closest potential wells there is a very strong (exponential) dependence of conductivity on the concentration of wells of potentials N_{o}. In this case, dependence (5) can be used to analyze the dependence of conductivity on temperature and concentration of potential wells, provided that the activation energy does not depend on the concentration of the potential well. As was found in Żukowski et al. (2015b), the AC activation energy does not depend on the moisture content in the range of 1% by weight to 4% by weight and its value is ΔW ≈ (1.0582 ± 0.02224) eV. To apply formula (5) in the case of AC conductivity, one should select on the frequency dependencies for different moisture contents the frequency at which the conductivity depends only on the moisture content.
On the basis of the conductivity analysis, depending on the frequency, temperature and moisture content, the frequency and temperature appropriate for the application of formula (5) on the dependence of the hopping conductivity on the distance between potential wells are 10^{−4} Hz and 353 K.
As can be seen from Fig. 9, the relationship is a decreasing linear function, which is consistent with the formula for jump conductivity (5), a constant, within the uncertainty of measurement, directional coefficient β/R_{0} ≈ (− 6.1667 ± 0.23); the uncertainty of its determination is ± 3.63%. The R^{2} determination coefficient for the least squares approximation is close to one and equals R^{2} = 0.9672, which proves the high accuracy of the approximation.
It is very important to determine the fact that the activation energy of direct current conductivity ΔW does not depend on the moisture content. Activation energy is a function of two energy states of the jumping electron—the initial state of the valence electron energy in the neutral well from which the electron begins its jump, and the first free state on which the electron will be in the second well after the jump (Fig. 1).
The radius of electron location for the water nanoparticle based on the Bohr model for the hydrogen atom will be calculated below. The formula (5) for conductivity in the case of electron tunneling will be used for calculations. The formula (5) includes the radius of the location of the hopping valence electron R_{0} (socalled Bohr radius). For the calculation of the radius of the valence electron location of the water molecule, we will use an approach similar to the method of calculating the energy of ionization and the radius of the valence electron of an admixture of shallow phosphorus in semiconducting silicon described in Mott (1974).
Next, it is assumed that the valence electron is at the level of n = 1. The first ionization energy of the phosphorus atom is E_{1} = 10.49 eV (Buhl 1994). According to formula (34), the ionization energy of the phosphorus atom in the silicon atom should decrease ε^{2} = 156.25 times to the level of 0.067 eV. Comparing this result with the experienced ionization energy value of the phosphorus atom in 0.045 eV silicon (Buhl 1994), it can be concluded that the difference between experimental and calculated values is less than 1.5 times. This is a satisfactory outcome. As a result, shallow admixtures, such as phosphorus in silicon, are often called hydrogenlike admixtures. The difference between the experimental and theoretical values is probably connected with the fact that near the atom of the admixture of phosphorus, the polarization of silicon differs slightly from the state found in the ideal network that does not contain admixtures and defects. From formula (34) it follows that to obtain ionization energy 0.045 eV, the dielectric permeability close to the phosphorus atom should be about 10.5. This is a reduction of only 16% compared to pure silicon.
We shall build on the following assumptions.
The electron is tunneling from an inert molecule of water that is in a nanodrop. Tunneling takes place from the ground state to the first higher unoccupied state.
Due to the fact that the valence electron of a molecule of water is strongly shielded from the nucleus through lower electrons, its first ionisation energy is about 12.6 eV (Mallard and Linstrom 2000; Shirai et al. 2001; Zavilopulo et al. 2005; Kovtun 2015) and is very similar to the ionization energy of a hydrogen atom of 13,595 eV. The difference between these values is only 7.3%. Such close values of the ionization energy of the water molecule and the hydrogen atom mean that the structure of the energy states of the water molecule’s valence electron is similar to the structure of the hydrogen atom states.
Dielectric permittivity of pressboard impregnated with insulating oil ε_{r} ≠ 1. Thus, in formula (35) in the denominator there is a square of relative permittivity.
In formula (35), similarly as in the model of the hydrogenlike admixture of the phosphorus atom in silicon, we assume that the valence electron of the water molecule is in the ground state with the main quantum number j = 1. In contrast to the model of the hydrogenlike admixture, where the ionization energy is calculated, i.e. the electron’s hop into the state of k equal to infinity, in formula (35) in the case of tunneling for the first unoccupied state, one should write k = 2.
Substituting for formula (36) the value of AC activation energy, which is ΔW ≈ (1.05820 ± 0.2224) eV (Żukowski et al. 2015b), and known fixed values: m_{e}, e, ε_{0}, h, we obtain the relative permittivity of the insulating material around the water molecule, which the electron is tunneling. Its value is ε_{r} ≈ 5.62. This is a value similar to the dielectric permittivity of the pressboard in the frequency range above 1000 Hz. The obtained value of the relative permittivity of the material surrounding the molecule of water on which the electron is tunneling indicates that the molecule of water from which the electron tunnels is not in the insulating oil filling the spaces between the cellulose fibers. The dielectric permittivity value of the relative insulating oil is much lower and amounts to approximately 2.2 (CIGRE Working Group A2.35 2010). This means that the moisture nanoparticles are located inside the cellulose fibers, the dielectric permittivity of which is greater than the dielectric permittivity of the oil.
When a molecule of water from which an electron hops, is placed in pressboard impregnated with insulation oil with a relative permittivity ε_{r} ≈ 5.62, the Bohr radius, entering the formula for hopping conductivity (5) will increase (ε_{r})^{2} times, and its value will be R_{0} ≈ 1.657 nm.
Taking into account the fact that the water in the pressboard is in the form of nanodrops, the distance to which the electron tunnels is different than r_{1}. We will mark it r_{n}.
By substituting in formula (43) formula (41), we obtain water nanoparticles that contain on average about 200 molecules of water. The diameter of such nanodrops is about 2.24 nm. When comparing the number of water molecules in nanodrops, determined on the basis of semiempirical calculations with the Xray of a doublenegative oxygen ion, amounting to about 220 and their diameters 2.32 nm (Żukowski et al. 2015a) with that obtained by elements of quantum mechanics (about 200), it can be seen that these last calculations give the number of water molecules in nanodrops about 0.91 times lower, their diameter being about 2.24 nm. This is consistent with the sufficient condition for the formation of nanoparticles (Pogrebnjak and Beresnev 2012; Pogrebnjak et al. 2018), which is that oil and water do not enter chemical reactions, they are not mutually soluble and the surface tension of water is much higher than that of insulating oil.
It seems that the calculation of nanodrop, made on the basis of laboratory tests lowfrequency conductivity with made using the basics of quantum mechanics are more reliable, because they allow to determine both the dimensions of the nanodrops and calculate the dielectric permittivity of the pressboard impregnated with insulating oil.
Semiempirical calculations and the use of elements of quantum mechanics made on the basis of laboratory tests of DC and AC conductivity of pressboard impregnated with insulating oil, clearly demonstrate that the moisture in pressboard is in the form of nanodrops. Therefore, such material should be treated as a composite of cellulose, mineral oil and water nanoparticles.
Conclusion
The paper presents a model of hopping conductivity at direct and alternating current, developed on the basis of the quantum phenomenon of electron tunneling between neighbouring potential wells. The tunneling takes place from one neutral potential well to another, resulting in an electrical dipole and additional thermally activated polarization. An important parameter of the model is the dipole lifetime τ. After this time, the electron can hop to the third well with the probability p, which causes a direct current flow, or return to the first well with the probability (1 − p), thus causing the flow of highfrequency current. The model shows that for direct or low frequency current, the current density and conductivity do not depend on the frequency. In the high frequency region, the current density does not depend on the frequency. Lowfrequency conductivity is 2p times smaller than highfrequency conductivity. In the transitional region there is a frequency dependence of the current density.
An analysis designated in laboratory tests of the frequency dependence of conductivity and permittivity for the composite of electrotechnical pressboard, mineral oil and water nanoparticles was made with the use of the direct and alternating current hopping conductivity model, which showed that there is a high correspondence between the waveforms and the model. It has been established that in the composite of electrotechnical pressboard, mineral oil and water nanocrusts there are at least two mechanisms of changes in conductivity. For the first stage, the value of the hopping probability p_{1} ≈ 0.2, and the value of the relaxation time τ_{1} ≈ 0.02 s. The probability of hops for the second stage of growth is p_{2} ≈ 0.0005 and the value of relaxation time τ_{2} ≈ 2×10^{−7} s. The occurrence of two sections of the conductivity increase is associated with differences in distances between neighbouring nanoparticles obtaining in the composite of electrotechnical pressboard, mineral oil and water nanodrops. In the percolation channel, connecting measuring electrodes for pressboard with a thickness of 1 mm, the number of nanoparticles and number the distance between the neighboring molecules them is in the order of 10^{5}. According to the central limit theorem, for such a large number of randomly distributed elements, the probability distribution of the occurrence of distance is approximately normal. This means that in the percolation channel there are pairs of neighbouring wells, the distances between which are both much smaller and much larger than the average. A pair of wells, the distance between which is smaller than average, participates in the permittivity with a short relaxation time. On the other hand, a pair of neighbouring wells for which the distance is greater than average causes a high resistance to DC or lowfrequency current as well as a higher relaxation time.
Using the elements of quantum mechanics on the basis in the laboratory tests of the dependences of lowfrequency conductivity from the moisture content was determined, the dimensions of water nanoparticles in electrotechnical pressboard impregnated with mineral insulating oil were determined. It was found that nanoparticles contain on average about 200 molecules of water, and their diameters are about 2.24 nm. This approach allowed to calculate the dielectric permittivity of cellulose, the value of which is ε_{r} ≈ 5.62. It is a value close to the static dielectric permittivity of the pressboard and is much higher than the permittivity of the insulating oil for which ε_{r} ≈ 2.2. The obtained value of the relative permittivity of the material surrounding the molecule of water to which the electron tunnels indicates that the water nanoparticles between which the electron is tunneling are inside the cellulose fibers and not in the insulating oil that fills the spaces between the cellulose fibers.
Notes
Funding
The funding was provided by Polish Ministry of Science and Higher Education [Grand No: Statute tasks8620/E361/S/2018 (S28/E/2018)].
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