Study of the electro and thermophysical properties of composite ceramic materials containing nickel nanoparticles

. The regularities in the behavior of the electrophysical and some thermophysical properties of composite ceramic materials containing both micro-and nanoparticles of nickel in the region of the percolation threshold have been studied. Certain regularities in the behavior of the electrical conductivity, dielectric constant, thermal conductivity, and thermoelectric power of composite ceramic materials as a function of the volume content of nickel particles are revealed. Near the percolation threshold, the experimental results of the behavior of the electrical conductivity and static permittivity as a function of the nickel volume content in these materials differ from the dependences calculated within the framework of the percolation theory in that the curve of the dependence of the permittivity has the form similar to the electrical conductivity curve. The origin of this discrepancy is explained by the formation of a continuous spatial structure of tunnel-connected conductors.


Introduction
The world is currently on the threshold of great change developments in science and technology that will affect all aspects of the economy and human social activity.Of course, these changes will affect materials science to a large extent, and as well as all industries related to the technology of materials and their use.Therefore, the number of experimental and theoretical studies in the field of creating materials with special and practically important electrical and thermal properties based on composite materials containing metal nanoparticles has increased significantly, mainly due to the unique physical characteristics of these materials, which differ significantly from the properties of the corresponding materials.Composite materials, among which electrically conductive ceramic materials containing micro -and nanoparticles of metals are very promising, have a great applied potential.In such compositions, at a certain critical concentration of metal particles (percolation threshold), the electrical, thermophysical, magnetic, and other properties change sharply.
Despite the progress made in the study of ceramic materials containing metal microand nanoparticles, their practical application is still limited.This is apparently due to the fact that so far many physical phenomena occurring in them remain unexplained.The electrical conductivity and permittivity of inhomogeneous materials containing metal microparticles are well explained by the classical theory of percolation.However, the application of the classical theory of percolation to describe the electrical properties of materials containing metal nanoparticles randomly distributed in a dielectric matrix below the percolation threshold faces some problems; their conductivities differ from those calculated within the framework of this classical theory.Studying this discrepancy is one of the main tasks of researchers in this field.
Currently, the creation of new generation materials is very relevant.The energy model of the structure that explains the behavior of charge carrier conductivity is not clear, and the topology of an infinite cluster in such systems has not been studied.The solution of the above issues makes it possible to obtain a clearer idea of the structure and mechanism of charge transfer in composite systems containing micro-and nanoparticles of metals, as well as to determine the possible area of their practical use.
Much attention is paid to accelerating scientific and technological progress and creating new generations of materials.Among these materials, the most important place will be occupied by new varieties of ceramics.The formation of nanocrystals in the matrix of the studied materials will make it possible to manufacture various products on their basis in the energy sector, mechanical engineering, as well as to conduct fundamental research in this direction at the world level.The fundamental results obtained in the study of the electrical properties of ceramic composite materials containing nickel nanoparticles are of great importance and can become the basis for the development and application of science.
In the literature, there are a number of studies by leading scientists, where it was shown that below the percolation threshold, the behavior of the conductivity of heterogeneous materials (matrices -ceramics, polymers, glass) containing metal nanoparticles (Fe, Ni) cannot be explained within the percolation theory.This issue is discussed in detail in the review article by Balberg  This article discusses the results of studies of the electrical conductivity of heterogeneous materials containing micro-and nanoconductive particles over the past 30 years.In this work, the behavior of the conductivity of composites containing metal nanoparticles is discussed within the framework of a spatial-structural hierarchical model.
Unfortunately, there are no studies confirming Balberg's assumption for samples of the same composition, but with different sizes of metal particles.There are also no research results on the behavior of the static permittivity as a function of particle size in the region of the percolation threshold in such systems.The dependences of the behavior of electrophysical properties in composites containing metal nanoparticles on the content of nanodispersed filler have not been established.

Experiments and Research Analysis
Development of composite ceramic materials containing micro-and nanoparticles of nickel; Study of the concentration dependence of electrical conductivity, dielectric permeability, thermal conductivity and thermo-emf of the developed composites on the content of micro-and nanoparticles of nickel and the concentration-temperature dependence of the conductivity of composite ceramic materials containing nickel nanoparticles; Study of the topology of an infinite cluster depending on metal particle size and energy diagrams; Identification of a possible area of practical application of the developed composites.
Two types of composites were prepared for research.One of them is a ceramic composite containing nickel nanoparticles.Another is a ceramic composite containing micronized nickel particles.Ceramic materials containing nickel nanoparticles were obtained by thermal decomposition of nickel formate pre-mixed with ceramics.Mixing was carried out in an agate ball mill for 7 hours.Thermal decomposition was carried out in vacuum at a temperature of 400 0 C for 5 hours [16].
As is known, the method of small-angle X-ray scattering makes it possible to study inhomogeneities of a substance, the sizes of which exceed interatomic distances and range from 5-10 to 104 Å.The size of the nickel nanoparticles was not determined because The method (small-angle X-ray diffraction), which we used for other composite materials, is inapplicable due to the fact that the densities of Si, Fe, Al and other ceramic components and the density of nickel nanoparticles are close.Therefore, polymer composites containing nickel formate were obtained in a similar way.After the decomposition of nickel formate salts in a heat-resistant polymer based on phenylone and using small-angle X-ray cameras of the KRM-1 type, the radius of nickel particles in composites was calculated, the diameter of which did not exceed ~ 20 nm.
Ceramic materials containing microdispersed nickel particles are obtained by mixing nickel powder with ceramics in an agate ball mill for 7 hours.In this case, in the samples used, the nickel particle size was in the range from 1 to 3 μm.This was established using transmission electron microscopy on a BS242E (Tesla) microscope.The volume fraction of nickel in the composition varies from 0.02 to 0.6.The electrical properties were measured on samples in the form of tablets 15 mm in diameter and 2 mm thick.The samples were prepared by pressing a ceramic powder at a pressure of 2 • 10 2 MPa and sintering the resulting pellets in a vacuum at a temperature of 1000 ° C.
In both cases, the volume fraction of metals (V1) was calculated based on the concentration of the metal in the original metal-containing compound.

Electrical and thermophysical properties of composite ceramic materials containing micro and nanoparticles of nickel
In figure 1 shows the experimental dependences of the conductivity σ on the concentration of nickel particles for both investigated ceramic materials, as well as those calculated within the framework of the percolation theory using the formulas below.
According to the percolation theory, the conductivity (σ) of systems containing metal particles distributed randomly in a dielectric matrix using the boundary conditions (V1 = 0 and V1 = 1) is described by the following formulas [17]: here σ1 is the conductivity of metal particles; σ2 -conductivity of the dielectric matrix; Vc is the critical concentration (percolation threshold) at which an infinite cluster (IC) of filler particles is first formed; t and q are parameters called critical indices.Calculated values σ of composites calculated using ( 1) and ( 2) formulas at calculated values Vc = 0.15, t = 1.6, q = 1, and experimentally determined values σd = 1.2 •10 -12 Ω - 1 •m -1 and σ1 = 2.4•10 -1 Ω -1 •cm -1 (σ value of Ni powder at P = 1.5 •10 2 MPa) do not coincide with the experimental results.Therefore, the quantities Vc, t, q and σ1 were determined as follows.For the studied ceramic materials, Vc was determined by differentiating logσ with respect to V1.The critical indices t and q were obtained from the experimental data, presenting them as a graph in the coordinates logσ-log [(V1 -Vc) / (1-Vc)] and logσ -log [(Vc-V1)/Vc], the angle of inclination of these graphs, there are t and q.The σ1 value was obtained by extrapolating this graph to V1 = 1.
Found that Vc = 0.355; t = 2.21 and 1 = 4.3  10 2 Ω -1  m -1 for a ceramic material with nanosized nickel particles and Vc= 0.443; t = 1.81; q = 1.02 and1 =3.210 2 Ω -1  m -1 for a ceramic material with microdispersed nickel particles.The value of 1 determined by extrapolating the graph in the coordinates lgσ-lg [(V1-Vc) / (1-Vc)] is an order of magnitude lower than the value of 1 determined at a pressure of 1.510 2 MPa.These results show that σ1 determined by extrapolation is not the conductivity of Ni metal particles, but is the conductivity of an infinite cluster of nickel particles in the composite in the region Vc V1 0.5.
As seen from Figure 1 for both types of the studied composite materials, the correspondence between the calculated and experimental data is observed at V1> Vc.In the case of V1<Vc, the correspondence between theoretical and experimental dependences is observed only in composite materials with microdispersed nickel particles.For a composite material with nano sized nickel particles, there is an additional contribution to the electrical conductivity in the region below Vc.These results can be explained on the basis of the model of electrical conductivity in composites proposed by Balberg [1].According to this model, all metal particles in composites, in which metal particles are randomly distributed in a dielectric matrix, are electrically coupled, and the conductivity of these composites is determined both by tunneling of charge carriers between neighboring particles and by tunneling between particles located at a distance.Percolation behavior is observed when the contribution of tunneling between particles located at a distance from each other to the macroscopic conductivity is negligible.This occurs when the particle radius (b) is much larger than the tunneling region parameter (or the tunneling decay parameter) (d).In the case when b ~ d, the tunneling of charge carriers between nonneighboring particles contributes to the macroscopic conductivity, along with tunneling between neighboring particles, and the dependence of the macroscopic conductivity on the concentration of metal-containing particles differs from that dictated by the classical percolation theory.
In Balberg's work [1], the manifestation of these two types of conductance behavior in composite materials was demonstrated by studying carbon nanotubes localized in polymer composites and Ni-SiO2 cermets.A feature of the results obtained in our work is that these two types of behavior of conductivity were observed in composites of the same composition, demonstrating the dependence of the manifestation of these types of conductivity on the size of metal-containing particles in the composites studied.
In composites, in which the contribution to the electrical conductivity from the tunneling of charge carriers between neighboring particles is observed, there are two percolation thresholds.One of them is observed at high values of V1; it is the percolation threshold Vc determined above.Another (additional percolation threshold Vcd) is observed at low values of V1, and it is the critical fractional volume of metal particles that initiates the first infinite cluster of tunnel-coupled conductors.By fitting the section (for V1<Vc) of the experimental curve 1 (Fig. 1) for a ceramic material with nickel nanoparticles to the functional dependence determined by Eq. (1) (denoting the percolation threshold as Vcd and the critical index as t in this equation), it was found that Vcd = 0.145 and t = 3.2.
The study of the concentration -frequency dependence of the dielectric constant ε of composites shows that on the dependence of ε on frequency (f), in the frequency range from 20 to 10 8 Hz of the alternating field, two regions are observed.At low frequencies (20  1000 Hz)  of composites decreases significantly, a further increase in frequency to 10 8 Hz in such systems leads to a weak dependence of their  on f.The decrease in  of composites at low frequencies is explained by the Maxwell-Wagner capacitor model [18].
In Fig. 2 shows the experimental and calculated dependences of the static dielectric constant ε on V1 for the studied composites.The experimental dependences of ε on V1 were obtained by extrapolating the frequency dependences of ε in the range of 20-200 Hz to zero frequency.Theoretical dependencies are calculated by the formula: (3) at V1<Vc; here εd is the dielectric constant of the ceramic.In these calculations, the same Vc values were used that were obtained from the experimental dependences of σ on V1.
As seen from Figure 2, for a composite with microdispersed nickel particles, the experimental dependence of ε on V1 is well described by formula (3).For a composite with nickel nanoparticles, the experimental dependence of ε on V1 does not agree with the dependence calculated using this formula and shows an additional contribution to ε at V1, which lie below the percolation threshold.
Based on a qualitative interpretation of the sharp increase in near the percolation threshold and on the physical concepts of the "hierarchy" of electrically coupled spatial structures in composites, taking into account that for the corresponding composite materials studied here, the V-dependence curves for  and  are similar, for the first time proposed interpretation of the behavior of  composites containing nickel nanoparticles.
The thermal conductivity of composite ceramic materials containing micro-and nanoparticles of nickel has been investigated.It is shown that the thermal conductivity of composite ceramic materials can be described by almost any formula derived within the framework of the effective medium theory.
The dependence of the thermos EMF (α) of the composites on the concentration of micro and nanoparticles of nickel has been studied.The concentration dependence of the thermo EMF of the investigated compositions has a clearly pronounced critical character (Figure 3).
For the case when the electrical conductivity of the metal 1 is much higher than the electrical conductivity of the dielectric 2 with close values of the thermal conductivity , i.e. m>d and md, the concentration dependence is described by a power law: At k=0.8±0.1, at -thermo-EMF, infinite cluster.To describe the character of the dependence of a on V1 of composites within the framework of the percolation theory, the experimental results were plotted in the coordinates lga versus log (V1-Vc) / (1-Vc).From the slope of the dependence, the critical index was determined, k, whose values for composites were 0.71 and 0.80, respectively.The obtained results of the index values are in good agreement with the values calculated by formula (4).
It is known that the methods of percolation theory make it possible to establish the topology of the resistance grid (topology of an infinite cluster).The density of IC, the volume fractions of the skeleton and dead ends belonging to the IC, as well as their dependence on the volumetric content of the filler were determined.It was found that near the percolation threshold; the volume fraction of the skeleton is negligible compared to the fraction of dead ends (Table 3.1) [19].
The Z / R values at t = 1.81 and t = 2.21 are shown in the table.As can be seen from the table, near the percolation threshold in composites containing nickel nanoparticles, an infinite cluster is more sinuous as compared to IC in composites containing highly dispersed particles.These results show that the higher the dispersion, the more tortuous the infinite cluster in such composites.Z / R shows how many times the skeleton is longer than R due to tortuosity.As can be seen from the table, the value of P (V1) increases with distance from the percolation threshold towards large V1.This means that the infinite cluster gradually joins the final clusters formed between the nickel particles and becomes more and more "dense".

The nature of the conduction mechanism in composite ceramic materials containing nickel nanoparticles
The mechanism of the electric transport of charge carriers in composite ceramic materials containing nickel nanoparticles has been studied.In order to understand the nature of the temperature dependence of electrical conductivity in such systems, one should study their structure.From a physical point of view, the process of formation of ceramic composites containing metal nanoparticles can be considered as a consequence of the doping of the initial dielectric with metal nanoparticles, similar to doped compensated semiconductors.This means that electronic states similar to impurity levels appear in the band gap of the initial ceramic.
An increase in the volume content of metal nanoparticles affects not only the concentration, but also their size distribution.If this representation is correct, then the conduction mechanism in such systems is hopping, its temperature dependence should be described by the following equation E3S Web of Conferences 410, 02060 (2023) https://doi.org/10.1051/e3sconf/202341002060FORM-2023 To check the applicability of law (5), the dependence of σ on T is usually plotted in the coordinates logσТ -x .As can be seen from Figure 4, the dependence of  on T can be straightened not only in the coordinates logТ -1/2 , but also in the coordinates logТ -1/3 and logТ -1/4 .To determine x in equation ( 5) in composite materials, we used the method of analyzing the temperature dependence of the reduced activation energy of conduction.In the work of Zabrodsky [20], it was shown that to determine x, one can use the equation lgω=B-xlgT, ω=1/T•lgσ/T -1 =lgσ/T (6) at В=const,  -reduced activation energy of conduction.
Figure 5 shows the temperature dependence of ω for composites, obtained by graphical differentiation of the curves in Figure 4 in coordinates lgσ -lgТ.It can be seen from these data that in the investigated temperature range there are three regions with different temperature dependences -high -(I) and low-temperature (III), separated by a certain transition region (II).
When the density of localized states near the Fermi level is constant, i.e. g (ε) = const then n = 0, for d = 3 the value of x in ( 5) is 0.25, then equation ( 3) turns into the wellknown Mott's law (Variable range hopping, VRH).
As seen from Figure 5, in the high-temperature region ТТа (to the right of curve a) there is a region of linear dependence of logσ on logТ, corresponding to the exponential law σ(Т) with x ≈ 1, Based on the results obtained, law (7) in this case corresponds to conductivity with the activation energy of conductivity  and its value, in the temperature range from Ta (to the right of curve a) to 450 K, is constant.
As can be seen from Fig. 5, in the samples under study at relatively low temperatures ТТс (to the left of curve с), a linear dependence of logσ on logТ is also observed, corresponding to the exponential law ω (Т) сх ≈ 0.5.This corresponds to the special case of law (5) for conductivity with variable activation energy.
In Fig. 6 shows a possible energy model of composite ceramic materials containing metal nanoparticles.Occupied and free states (Fig. 6) near the Fermi level are chaotically distributed and localized due to the disorder of the system.Therefore, the tunneling transition between the states of two metal nanoparticles, which affects the value of the hopping conductivity, requires a certain activation energy.The solid line is the Fermi level at thermodynamic equilibrium.The dashes above the Fermi level are the lowest free levels, and below it (with black ones) are the highest occupied levels.0 / >0 // >0 /// are the strip widths corresponding to Т1> T2> T3.The latter is determined not only by the disordering of the system associated with the presence of metal nanoparticles of various sizes, leading to a strong spread of levels and localization of their electronic states, but also by the Coulomb interaction of an electron passing into a neighboring metal nanoparticle with a hole that has arisen in the previous place, which causes the appearance of a Coulomb gap near the Fermi level.With a decrease in temperature, in accordance with Mott's idea, one should expect a transition to the regime of hopping conduction with a variable hopping length, the temperature dependence of which is determined by the role of the Coulomb interaction in the system and its dimension.The indicated parabolic gap in such systems, as in other disordered systems, is the Coulomb gap of Efros and Shklovsky [17].The bandwidth ε0 according to law (8) is expressed by the relation Identification of (8) with VRH leads, with low anisotropy (d = 3), to the conclusion that hopping occurs in the region of the parabolic quasi gap g (ε) = g0 (ε-εF) 2  in the density of localized states in the vicinity of the Fermi level.
Some possible areas of practical use of the obtained composites are outlined, in particular, as elements with relatively high values of dielectric constant for their use in the field of radio electronics and elements with relatively high values of thermos EMF for their use as converters of thermal energy into electrical energy.
The ever-increasing need of modern technology, in particular microwave technology, in non-conductive (high-resistance) materials with increased and adjustable values of the dielectric constant necessitated the search for new solutions to this issue.At present, researchers are trying to use large amounts of fillers -ferroelectric ceramics powders in a matrix, which significantly impairs the physical and mechanical properties of the compositions.A dielectric anomaly in the region of the percolation threshold was found in a number of composite materials containing microdispersed (~ 1 ÷ 5 μm) metal powders obtained by mixing the components.This method has a number of disadvantages, one of which is that finely dispersed metal particles accumulate during mixing of the components; therefore, the electrical properties of the compositions filled with metal powder near the metal-dielectric transition are difficult to reproduce.However, this method can be used to obtain stable materials with not very high dielectric constant values.In this regard, the most acceptable are composite materials containing nanodispersed metal particles.
Fig. 2 shows the advantage of composite ceramic materials with nanodispersed metal particles over microdispersed ones.As can be seen from Fig. 2, in composite ceramic materials containing nano dispersed metal particles, far from the percolation threshold Vcd<Vc, the dielectric constant has a sufficiently high value.Studies of the behavior of the thermos EMF of the developed composites show that the concentration dependence of the thermos EMF value has a clearly pronounced critical character (Fig. 3).These results show that such composites can be used as converters of thermal energy into electrical energy.

Conclusions
Based on the studies carried out and the results obtained, the following conclusions were made: 1. Revealed regularities in the dependences of electrical conductivity, dielectric constant, thermal conductivity and thermo-EMF of composite ceramic materials on the volume content of nickel particles.When approaching the percolation threshold, the experimentally obtained dependences of the conductivity and static dielectric constant on the fractional content of nickel in these materials differ from the dependences calculated within the framework of the percolation theory in the fact that the dependence curve for the dielectric constant is similar to the curve for electrical conductivity.The origin of this discrepancy is explained by the formation of a continuous spatial structure from tunnelconnected conductors.
2. Two types of percolation threshold have been established in composite ceramic materials of the same composition, but with different sizes of metal particles.One of them is observed at values of Vc, which is dictated by the classical theory of percolation, another percolation threshold (Vcd) is observed at Vcd<Vc, which initiates the first infinite cluster of tunnel-coupled conductors.
3. It was found that in ceramic materials containing nickel nanoparticles, in the region below the classical percolation threshold -in the high -temperature region (T≥ Ta) -in electrical conductivity, along with tunneling between the nearest neighboring particles with a constant activation energy of conduction, the carrier tunneling also contributes charge between non-nearest neighboring particles.At comparatively low temperatures (T ≤ Tc), electrical transfer in samples is carried out by tunneling of charge carriers with variable activation energy of conduction in the region of a parabolic quasigap in the density of localized states, in the vicinity of the Fermi level.
4. An energy model of the structure is proposed to explain the physical properties of ceramic materials containing metal nanoparticles.
5. The possibility of applying the percolation theory to study the structure, ie.IC topology in electrically conductive ceramic materials.The density, volume fraction, tortuosity, volume fraction of the skeleton and dead ends of an infinite cluster were determined depending on the volume content of nickel particles in such systems.Near the percolation threshold, the volume fraction of the skeleton belonging to the BC is an insignificant fraction of its total volume, and the bulk of the IC is concentrated in the dead ends.Near the percolation threshold in composites containing nickel nanoparticles, an infinite cluster is more tortuous compared to IC in composites containing microdispersed particles.These results show that the higher the dispersion, the more tortuous the infinite cluster in such composites.
6.It is shown that the high value of the dielectric constant in the region below the percolation threshold in nanocomposites allows them to be used as new materials in electrical engineering, as well as the relatively high value of thermo-EMF in them allows them to be used as converters of thermal energy into electrical energy.

Fig. 1 .
Fig. 1.Comparison of the experimental (points) and calculated (solid curves) values of conductivity as a function of the volume content (V1) of nickel particles for ceramic materials containing nanoparticles (filled points, curve 1) and micro dispersed particles (empty points, curve 2).The inset shows the curve of dLgσ/dV versus V1 (solid line for materials containing nanoparticles, dashed line for materials containing micro dispersed particles).

Fig. 2 .
Fig. 2. Comparison of the experimental (points) and calculated (solid curves) values of the dielectric constant (ε) as a function of the volume content (V1) of nickel particles for ceramic materials containing nanoparticles (filled points, curve 1) and microdispersed particles (empty points, curve2).

E3S 2023 Fig. 3 .
Fig. 3. Dependence of the Seebeck coefficient (a) of compositions on the volumetric content of nickel based on nano (1) and microdispersed (2) particles

Fig. 4 .
Fig. 4. Temperature dependence of conductivity in composite ceramic materials: Volume fraction of filler 0.220 (1); 0.275 (2) and 0.315 (3) Fig. 4 shows the experimental temperature dependences of conductivity σ in the temperature range from 100 K to 450 K for composites containing nickel nanoparticles obtained at V1<Vc.In order to understand the nature of the temperature dependence of electrical conductivity in such systems, one should study their structure.From a physical point of view, the process of formation of ceramic composites containing metal nanoparticles can be considered as a consequence of the doping of the initial dielectric with metal nanoparticles, similar to doped compensated semiconductors.This means that electronic states similar to impurity levels appear in the band gap of the initial ceramic.An increase in the volume content of metal nanoparticles affects not only the concentration, but also their size distribution.If this representation is correct, then the conduction mechanism in such systems is hopping, its temperature dependence should be described by the following equation

σ 7 )Fig. 5 .
Fig. 5. Temperature dependences of the reduced activation energy of conductivity ε of composite ceramic materials: The volume fraction of the filler in the composition is 1 -0.220; 2-0.275 and 3-0.315

Fig. 6 .
Fig. 6.Energy model of ceramic materials containing nano-dispersed metal particles

9 )
As ε0 decreases, i.e. with decreasing measurement temperature, the average hopping length of charge carriers r increases, and its temperature dependence is described by the expression 8,  -localization radius.