Signs of dynamic nonlinearity of the coal dielectrics depolarization upon excitation in electric and magnetic field

The authors conducted a series of experiments on samples treated immediately before measurements with a constant magnetic field of a neodymium magnet and/or an alternating electric field. The experiment has shown complex dynamics of electric potential with alternating polarity relative to the base exponential depolarization curve. The variety of responses to different types of excitation, and the irregular and unpredictable nature of fluctuations suggest that these are varieties of the same phenomenon – the nonlinear dynamics of ferroelectric domains. The applied methods of statistical processing of the cumulative data array suggest that the invariance of the relaxation dynamics of the accumulated electric potential after excitation can be characterized by three related nonlinear differential equations of the first order. The impact of an alternating electric field may consist in bringing the dipoles of ferroelectric domains in the molecular coal structure into a state of unstable equilibrium with their slow return to their original position. The impact of a constant magnetic field is determined by the interaction of the external magnetic field with the magnetization induced due to time-varying electric polarization. In any case, the observed fluctuations are due to changes in the domain structure topology of the condenser with coal matter.


Introduction
Coal matter refers to a polycomponent, multiphase, and metastable carbon material of natural origin.Structural variability at the atomic-molecular level is greatly reduced during carbonization.The impact of molecular metastability on the polarization of bituminous coals is manifested in low-energy movementstranslational movement and rotational orientation of dipoles of low-molecular compounds.Molecular mobility can be both individual and group in nature and lead to the formation of a dynamic domain structure.
Traditionally, spontaneous and electrically reversible polarization in an external electric field has been studied in inorganic ferroelectrics such as barium titanate.However, there are examples of organic ferroelectric molecular systems that are efficient, scalable, and cheap to operate.These materials are based on non-covalent molecules formed by two or more components, in which ferroelectricity arises either from molecular displacements or from the collective transfer of electrons or protons [1].
In such systems, individual molecules self-organize into ordered dynamic lattices under the impact of electric fields.The source of bistable polarization in organic ferroelectrics is either an intrinsic molecular dipole or an induced dipole as a result of supramolecular interactions between molecules.In all cases, this dipole must be located inside a non-centrosymmetric ordered structure [2].
In contrast to the generally accepted concept, where perfect single crystals are the key to superior ferroelectric properties, an alternative and easier to implement approach is possible."Ideal" ferroelectric behavior can be achieved in systems consisting of crystallites or nanoparticles electrostatically isolated from each other with a narrow size dispersion.Polarization switching in polycrystalline and disordered ferroelectrics is mainly dispersive in nature with a wide distribution of characteristic switching times described by the Preisach model [3].
Switching kinetics in ferroelectrics is usually described by nucleation and growth mechanisms.Dynamics is an important component of ferroelectricity since the correlated movement of dipoles is necessary to achieve a macro-effect.The behavior of the system is closely related to relaxation phenomena due to domain switching.The research on the nonlinear dynamics of cyclic oscillations on a ferroelectric capacitor in a resonant circuit showed the presence of characteristic bifurcations and the chaotic behavior of the system [4].
At the nanoscale scale, nonlinear dynamics are characteristic of ferroelectric domain switching induced by the tip of a scanning probe microscope, which exhibits intermittency, quasi-periodicity, and chaos.These effects are due to the interaction between polarization switching and screening charge dynamics [5].
In previous research, we studied the impact of environmental temperature and humidity on the parameters of natural self-potential and relaxation dynamics of charged coal samples of different ranks [6].The purpose of these studies is to consider the impact of external electric and magnetic fields on the slow relaxation dynamics of the electric charge in the studied coals, which are active dielectrics with ferroelectric properties.

Methods
A series of experiments were carried out on bituminous coals with a low content of mineral impurities.In all variants of the experiments, a constant ambient temperature of 27±3 °C and relative air humidity of 45±5% were maintained.Samples were used in the form of tablets with a diameter of 13 mm, manufactured by compaction method under a pressure of 10 tons/cm 2 of fraction powders < 10 microns, crushed in a molecular mill.
In the measuring apparatus for researching the dynamics of charge to a voltage of 9.75 V and discharge of the test capacitor with a coal sample, a precision operational amplifier INA116 from Texas Instruments was used with a high input resistance and low input capacitance connected according to a standard circuit.The amplified signal was digitized by a USB oscilloscope with a sampling period of t = 0.5 seconds.180-second time series were saved on a personal computer for further estimating the parameters of the nonlinear dynamic system in the Matlab program.
Immediately before the charge/discharge cycle, the samples were placed for 3 minutes in a constant magnetic field of a neodymium magnet with a pull-off force of 5.0 kg and/or an alternating lowfrequency electric field of 250 V/cm with an industrial frequency of 50 Hz without direct contact with the sample.

Experimental research and discussion of results
The applied treatment with constant magnetic and/or alternating electric fields in some samples leads to significant deviations from the exponential shape of the discharge curves (figure 1) peculiar to the relaxation of the induced electric potential.The level and nature of fluctuations are different for different samples.For example, the effect of magnetic field treatment for sample two is much stronger than for the first coal sample.
In all cases, the electrical potential has complex, unpredictable dynamics with varying polarity relative to the basic exponential trend of the depolarization curve.Fluctuations can fade over time and be stable throughout the entire recording period, have frequent sharp outbreaks that look like pulsed noise, and smooth long-term deviations from the control depolarization curve with a variable sign.In all experiments with a fixed effect, fluctuations range relative to the control depolarization curve of the untreated sample with different shapes, amplitudes, and durations.The resulting diversity of responses to different types of excitation, and the irregular and unpredictable nature of the fluctuations suggest that these are varieties of the same phenomenondeterministic chaos generated by the nonlinear dynamics of ferroelectric domains.
The duration of the recorded signals provides a data array sufficient to carry out the procedure for reconstructing the attractor using the Takens method [7] in a wide range of lag space.Using the phase portrait, one can estimate a number of system parameters: approximated entropy, correlation dimension, and the top Lyapunov exponent [8].
Entropy is used as a measure of the regularity and unpredictability of fluctuations in nonlinear time series.A regular time series has many repeating patterns, so the entropy value is small.A random signal is less predictable and has a high approximate entropy value.
The correlation dimension is directly proportional to the level of chaos in the system.It is a measure of the dimension of the phase space occupied by the signal.Its high value corresponds to a greater level of chaotic difficulty.
The top Lyapunov exponent characterizes the rate of divergence of infinitely close trajectories in phase space and, accordingly, the rate of loss of information about the initial conditions.A negative Lyapunov exponent indicates convergence of trajectories, while positive exponents exhibit divergence and chaos.
Since in all variants of the experiment, the correlation dimension varies within a small range of 2.72 -2.87 (table 1), the dimension of the phase space is the same and equal to three.The choice of lag 300 equal to a 30 second time interval is due to a compromise between the influence of random noise and the loss of information about the true dynamics of charge relaxation.
In the phase portrait of the differential depolarization time-domain signal x1(t) in the first case of excitation by a constant magnetic field, at least two attracting sets of unequal volume are observed (figure 2).The small value of the approximated entropy of 0.039 and the negative Lyapunov exponent "-0.49" indicate the quasi-periodicity of the time series and its movement towards a stable state of equilibrium.In the second case, with the same type of excitation, the cloud of displaying points occupies a larger volume (figure 3) compared to the first variant.The highest entropy of 0.346, as well as the generation of pulses of random polarity, amplitude, duration, and duty cycle, are signs of the intermittency mode particular to nonlinear dynamic systems.The negative Lyapunov exponent of "-0.49" shows a fairly rapid attenuation of bursts of interrupted equilibrium.The phase portrait of the differential depolarization signal of coal excited by a low-frequency alternating electric field is a strange attractor in the form of an open curve of a complex curved shape (figure 4).The volume of phase space elements has greatly compressed, which is a sign of a dissipative system.The lowest entropy value of 0.004 exhibits a high level of signal regularity, and the small and negative Lyapunov exponent of "-0.02" predicts a slow return of the system to a stable equilibrium state.
After excitation by a combination of two types of fields, the phase volume is compressed to 4-5 compactly located limited areas of phase space with a non-uniform probability density (figure 5).The dynamical system by jumps evolves from one attracting set of phase space to the next.The type of phase portrait and entropy of 0.052 characterize the presence of random switching in a regular signal.A positive Lyapunov exponent of 0.22 predicts a fairly rapid divergence of trajectories in phase space, which corresponds to an unstable regime of deterministic chaos.
Thus, the entire variety of dynamics of the relaxation signal of the accumulated electric potential after excitations by electric and magnetic fields is a response to the exit from the equilibrium state of a dissipative system which is characterized by three related nonlinear first-order differential equations.
The impact of an alternating electric field may consist in the reorientation of the dipoles of ferroelectric domains into a state of unstable equilibrium with their slow return to their original stable position.In this case, the field frequency must be low enough so that the dipoles of individual polar molecules and domains of different sizes, being in the gelified environment of coal vitrinite, have time to shift along the field.
The impact of a constant magnetic field is obviously determined by the interaction of an external magnetic field with magnetization generated by the dynamic multiferroic effect due to time-varying electric polarization during the movement of ferroelectric domain walls [9].
In crystalline multidomain ferroelectrics, displacement of domain boundaries leads to a sharp change in the polarization of the sample.In this case, the value of the dielectric constant, the greater, the weaker the domain walls are fixed on the defects and on the surface of the crystal.Nanosized crystallites in the amorphous environment of coals are weakly bound to the carbon matrix.Therefore, strong electric fields are not needed to move their domain boundaries and increase the size of domains oriented along the field.In this case, the contribution to the polarization of the sample can be very strong.

Conclusions
Treatment of tablets of most bituminous coal samples with a constant magnetic and low-frequency alternating electric field leads to strong deviations from the control exponential depolarization curve.
The observed fluctuations are caused by changes in the topology of the stable ferroelectric domain structure of the coal capacitor.In a non-equilibrium state, nonlinear dynamics of the slow depolarization process are observed in the form of bifurcations, intermittency, and chaotic behavior.

Figure 1 .
Figure 1.The impact of magnetic, alternating electric fields and their combinations on the process of relaxation of the electric charge of coals.

Figure. 2 .
Figure. 2. Reconstructed phase space of differential depolarization of coal 1 after excitation by a constant magnetic field.

Figure 3 .
Figure 3. Reconstructed phase space of the differential depolarization of coal 2 after excitation by a constant magnetic field.

Figure 4 .
Figure 4. Reconstructed phase space of the differential depolarization of coal after excitation by an alternating electric field.

Figure 5 .
Figure 5. Reconstructed phase space of the differential depolarization of coal after excitation by a combined field.

Table 1 .
Parameter estimation of a nonlinear dynamic system.