Magnetic nanocellulose: influence of structural features on conductivity and magnetic properties

Magnetic cellulose (MC) is prepared by hydrolysis of iron precursors in an aqueous dispersion of cellulose nanofibers. A thin, flexible film was then prepared by removing the water and drying the sample in a hot press at 110 °C followed by the removal of the water. Structural analysis of MC was performed and correlated with the measurement of electromagnetic properties. The magnetic cellulose showed high magnetic saturation of 68 emu/g with characteristic superparamagnetic behaviour, and conductivity in a range of semiconductors, with an increase of direct current (DC) conductivity with increasing temperature. Modification of cellulose with Fe3O4 has a positive effect on the DC conductivity and lower limit that needs to be exceeded to achieve a stable and sustainable conductivity in the range of ~ 5–20 × 10–9 (Ω cm)−1 @30 °C is 65 wt% of the Fe3O4 for studied MC composites. The surface roughness of the magnetic cellulose shows a dynamic change with increasing temperature, which is closely related to the enhancement of MC conductivity. A theoretical model of the conductivity is calculated based on continuous 2D percolation and shows an interesting agreement with the experimental results.

Merging of these two materials results in a new material with synergistic contributions of both components, where Fe 3 O 4 nanoparticles (NPs) provide magnetic properties (superparamagnetic), and high surface area, while CNFs provides a solid, flexible, and stretchable matrix. In one word the nanoparticles (with high surface ratio, and superparamagnetic properties) are obtained in the self-standing form.
It is well known that Fe 3 O 4 NPs can be easily controlled by an external magnetic field (Mustapić et al. 2016;Mandić et al. 2021; Krzyminiewski addition of (semi)conductive conjugated polymers, polyaniline and polypyrrole in cellulose by various methods. Cellulose was found to be an excellent alternative to improve the various material properties, either of natural or synthetic origin.
In the theoretical part, we have carried out a model of the percolation theory of samples related to the conductivity properties of the material. We consider the system of nanocellulose fibres with attached Fe 3 O 4 NPs as a system of conducting rods of the same length, randomly distributed over a rectangle of a certain size. Depending on their density, the rods form clusters of different sizes. By simulating the positions of the rods in large numbers, we were able to numerically calculate the probability distribution of the size of the largest cluster. Using this probability, we can construct the canonical partition function (or moment generating function) and calculate its Yang-Lee zeros. Analysis of zero closest to the real axis leads us to the density of the percolation threshold at n t = 5.65\pm 0.05.

Preparation of cellulose nanofibres
Cellulose nanofibres were synthesised as described in a previous study (Amiralian et al. 2020). Purified sugarcane pulp was prepared following the method reported before. (Amiralian et al. 2015a, b;Amiralian et al. 2015a, b). Briefly, washed and ground sugarcane fibres were treated with a 2% (w/v) sodium hydroxide solution using a 10:1 solvent to sugarcane ratio at 80 °C for 2 h and then rinsed with hot water.
The alkali-treated fibres were then bleached twice using an acidic solution of 1% (w/v) sodium chlorite (pH = 4, the pH decreased with glacial acetic acid) at 70 °C for one hour at a 30:1 solvent to fibre ratio. The cellulose content is 60%, hemicellulose 24%, and lignin 11%. To produce cellulose nanofibres, a 0.3% (w/v) dispersion of bleached sugarcane pulp was passed through a high-pressure homogenizer (GEA Niro-Soavi Panda NS1001 L 2 K homogenizer), once at 800 bar and three times at 1100 bar.
Additionally, cellulose nanofibers were decorated with magnetic nanoparticles in a wide content range, from 60 wt% up to a total of 83 wt% of Fe 3 O 4 . The mixture of the cellulose nanofibers (CNFs) and Fe 2+ / Fe 3+ salts (molar ratio of 1:2) were first well dispersed in DI water for 30 min under a nitrogen atmosphere, and room temperature. NPs Fe 3 O 4 were synthesised by co-precipitation method by adding of 2.2 g FeCl 3˙6 H 2 O and 0.45 g FeCl 2 in aqua solution with well dispersed cellulose nanofibers. After 30 min of vigorous stirring, 10 ml of 25% NH 4 OH was added to the reaction solution. The formation of black particles started immediately upon adding the ammonium solution. The reaction time of formation NPs in cellulose solution lasted for 15 min. The formed magnetic cellulose nanofibers were collected and washed several times with DI water and ethanol. The concentration of CNF was varied in each sample, 25 ml, 50 ml, 75 ml, and 100 ml of 0.7% (w/v) nanofiber dispersion were used to make samples 1-4, respectively, while the molar ratio of [Fe 3+ ]/[Fe 2+ ] was kept constant. (Amiralian et al. 2020). In the last step, after vacuum filtration magnetic cellulose was dried at 110 °C for 1.5 h in a hydraulic hot press under no pressure. The samples were designated in accordance with the decrease in Fe 3 O 4 amount, for instance, S1 (83 wt% Fe 3 O 4 ) up to S4 (60 wt% Fe 3 O 4 ).

Transmission electron microscopy characterisation
The morphology of MC was characterised by transmission electron microscopy (TEM). 1 µL of 0.005 wt% dispersion was spotted onto a Formvarcoated 200 mesh copper/palladium grid (ProSciTech, Queensland, Australia) and allowed to dry at room temperature. The sample was then stained with a 2% (v/v) uranyl acetate aqueous solution for ten minutes in the absence of light. The grid was examined on a Hitachi HT7700 at 100 kV. The morphology of the magnetic cellulose samples and elemental mapping were studied with a JEOL JSM-6400 scanning electron microscope.
X-ray powder diffraction X-ray powder diffraction data were collected on Bruker D8 Discover diffractometer equipped with EIGER2 detector (DECTRIS AG, Baden-Daettwil, Switzerland) in a parallel beam geometry at room temperature. X-ray source was Cu tube with a wavelength of 1.54060 Å powered at 1600 W (40 kV and 40 mA)) with Ni filter, 2.5° soller slit, and fixed slit at 0.6 mm. Detector was set up in 2D mode at a distance of 166 mm from the sample. Diffraction peaks were followed from 10° to 70° with a step of 0.02° and an integrating time of 220 s/step. Rietveld structure refinement was performed in HighScore 'Xpert Plus program 3.0. Refinement was carried out by using the split-type pseudo-Voigt profile function and the polynomial background model. Isotropic vibration modes were assumed for all atoms. During the refinement, a zero shift, scale factor, half-width parameters, asymmetry and peak shape parameters were simultaneously refined. Microstructural information was also obtained during Rietveld refinement with LaB6 used as an instrumental broadening standard.

Experimental-solid-state impedance spectroscopy
The electrical properties of the cellulose modified with Fe 3 O 4 were investigated on five samples by Solid-State Impedance Spectroscopy (SS-IS). For electrical contact, gold electrodes (diameter 3.8 mm) were sputtered onto both sides of a thin sample (~ 0.1 mm) using Sputter Coater SC7620 (Quorum Technologies, Lewes, UK). Complex impedance is measured in a cross-section setup (sandwich configuration) using an impedance analyser Novocontrol Alpha-AN Dielectric Spectrometer (Novocontrol Technologies, Montabaur, Germany) in an inert atmosphere in the frequency range from 0.01 Hz to 0.1 MHz and temperatures between − 30 and 80 °C, with a 10 °C step in two heating/cooling cycles. A frequency sweep is repeated twice at each step, controlled with an accuracy of ± 0.2 °C. The obtained experimental data (impedance spectra) were analysed by modelling the equivalent electrical circuit (EEC) using the complex nonlinear least-square (CNLLSQ) fitting procedure and the corresponding parameters were determined using WinFit software version 3.2 (Novocontrol Technologies, Montabaur, Germany). This procedure is based on the fitting of experimental diagrams to a suitable equivalent circuit. The values of the resistance from the fit, R, and electrode dimensions (sample thickness d, and A as electrode area) were used to calculate the DC conductivity as follows, σ DC = d/RA. For samples where this was possible, the DC conductivity value was taken from the low frequency plateau in conductivity spectra. Good agreement was found between the DC conductivity value obtained by fitting and the plateau.

Magnetic measurements
Magnetic measurements of magnetic cellulose (sample size: 1 × 1 × 0.1 mm 3 ) were performed using a Quantum Design MPMS-5 T SQUID magnetometer. Magnetic hysteresis loops M(H) were measured in an applied field within the range of 3 T at 100 K fitting and from the plateau.
Atomic force microscopy and imaging AFM imaging of cellulose nanofibres decorated with magnetic nanoparticles was performed using a Mul-tiMode Scanning Probe Microscope with a Nanoscope IIIa controller (Bruker, Billerica, USA) with a vertical engagement (JV) 125 µm scanner. Imaging was performed with a high-temperature heating controller (Digital instruments, range up to 60 °C, resolution 0.1 °C and accuracy 3%). Since the heating temperature and the actual temperature of the sample were different, calibration of each sample holder was performed as described in Šegota et al. (2015). Images were processed and analysed using NanoScopeTM software (Digital Instruments, Version V614r1, V531r1 and V1.9). All images are presented as raw data, except for the first-order twodimensional flattening. Scanning rates were normally optimized around 1 Hz. Images were acquired with a standard silicon nitride tip (RTESPA, Bruker, nominal frequency 300 kHz, nominal spring constant of 40 N m −1 ) in the air in tapping mode with scan resolution of 512 samples per line.

Theoretical calculation: definition of a model
To complement the experimental studies described in the previous sections, theoretical calculations were performed based on a simplified model of nanocellulose fibres. We imagine the whole system as consisting of N conducting width less rods of equal length, randomly distributed over a rectangle of a given size. To mimic the experimental setup, we used a 2:1 geometry for four different system sizes: (L x , L y ) = (100, 50), (200, 100), (300, 150), (400,200) with periodic boundary conditions in x-direction. The position of each rod is defined by three numbers: two of them determine the position of its centre, while the third (angle) determines its direction. Overlapping rods create clusters of different sizes. For a rectangle with dimensions (L x , L y ), the density of rods, n, is defined as.
As the density increases, the size of the largest cluster also increases, and at a certain threshold density n t , the largest cluster becomes a spanning cluster i.e. it connects two opposite sides of the rectangle. For all densities n ≥ n t , the electric current can flow through the system. A very similar model to the present one, but with square geometry and free boundary conditions, was analysed by means of Monte Carlo simulations (Li and Zhang 2009). By analysing the spanning probability as a function of stick density for different system sizes, the authors were able to locate the percolation threshold density with very high precision at the value Since conductivity is established when the largest cluster touches opposite sides of the system, it is of interest to observe the appearance of the spanning cluster.
It is shown in ref. Sen (2001) that in 2D systems defined on a lattice, the distribution of spanning clusters is numerically almost impossible to differentiate from the distribution of the largest cluster: "We first compare the distribution of the largest and the spanning cluster sizes in percolating 2D lattices and find that they are numerically indistinguishable almost always except for some cases where the size of the clusters are very small. One can ignore that difference and assume that at (1) n t = 5.63726 ± 0.00002 least for large values, the spanning cluster and the largest cluster are identical." Based on that result, we concluded that the analysis of the largest cluster is quite sufficient for determining the percolation threshold density.
To resolve the cluster structure of the system, we used a well-known find-union algorithm (Newman and Ziff 2001). For a given system size and density, we performed N steps = 10 5 simulations of random positions of N sticks and in each of them, we calculated N LC (l), i.e. the number of simulations with the largest cluster of size l. In this way, we obtained the approximate normalized probability that the largest cluster had a size equal l.
Once we have the probability distribution (2), we are able to construct the joined canonical partition function (or moment generating function).
where intensive variable Q is a complex field conjugated to extensive variable l.
The computer program for all simulations was written in FORTRAN 90, and the calculations were performed at the computer cluster placed at the Institute of Physics, Zagreb.

Results and discussion
Morphology and structural analysis of Fe 3 O 4 cellulose nanofibres TEM was used to study the morphology of Fe 3 O 4 nanoparticles and their interaction with cellulose nanofibres. The cellulose nanofibres with a high aspect ratio are shown in Fig. 1.
It can be seen that fibres mat, or bundles are formed as the concentration of nanofibres increases. This is due to the formation of hydrogen bonds and the large surface area of the nanofibres, which intertwine and bond on the grid as they dry. In addition, when the cellulose concentration is high, the long fibres intertwine and form a network. The spherical Fe3O4 nanoparticles were synthesised on the surface of nanofibres, and the OH groups on the surface of the nanofibres act as nucleation sites for the formation of the nanoparticles. The average size of the magnetic nanoparticles is in the range of 9-14 nm with a narrow distribution. The size of the nanoparticles becomes slightly smaller as the content of nanocellulose increases because more OH groups are available and nanoparticles are formed there, but they cannot grow larger due to steric constraints (insufficient space). Moreover, the nanoparticles tend to attach to the nanofibres rather than interact with each other (magnetic dipole interaction), resulting in NPs adhering to the sides of the nanofibres. MC nanosheet was prepared by vacuum filtration of MC dispersion followed by drying under the press. Despite the high content of rigid Fe 3 O 4 nanoparticles, nanosheets are very flexible. Fig. 2 show a uniform distribution of elements Fe, O, and C, with a relatively smooth surface at lower magnification.

X-ray diffraction (XRD) measurements
Samples S1-S4 were recorded by X-ray powder diffraction at − 100 °C, room temperature (RT) and + 100 °C. Data were collected in the 2Θ range 10-70°, and the patterns for each sample were similar at all three temperatures. The diffractograms for all samples at RT are shown in Fig. 3a.
The XRD patterns of all samples of cellulose modified with Fe 3 O 4 NPs show diffraction maxima corresponding to Fe 3 O 4 phase. XRD pattern shows at 2θ = 18.4° (111)   This was confirmed by the Rietveld analysis using Inorganic Crystal Structure Database (ICSD) card no. 158505 as the initial model for refinement. The crystallite size for all samples was also calculated by the Rietveld method and the results are shown in Table 1.
As already mentioned, the patterns for each sample were similar for all temperatures, (albeit the small change in the position of diffraction maxima caused by the thermal expansion) indicating that heating of the samples from − 100 to + 100 °C does not change the structure of Fe 3 O 4 nanoparticles. There was also no noticeable change in the breadth of the diffraction maxima which would point to the change in the crystallite size caused by the thermal treatment. The crystallite sizes of Fe 3 O 4 NPs are all in the range between 90 and 135 Å, and the size decreases with the cellulose content. Figure 4A compares the magnetic hysteresis loops of the membranes prepared from magnetic cellulose nanofibres and Fe 3 O 4 nanoparticles with different cellulose contents (17-40 wt%). As reported in our previous research (Amiralian et al. 2015a, b), typical superparamagnetic behaviour with characteristic S-shaped hysteresis loops and negligible coercive field (H C ) was observed for all samples at 300 K. The saturation magnetisation (M S ) for sample 1 containing 83% of Fe 3 O 4 nanoparticles (17 wt% nanocellulose) was 68.5 emu g −1 , for sample 2 containing 71 wt% Fe 3 O 4 nanoparticles was 67.4 emu g −1 , and that for sample 3 (65 wt% Fe 3 O 4 nanoparticles) and sample 4 (60 wt% Fe 3 O 4 nanoparticles) with higher nanocellulose content are 53.2 emu g −1 and 38.5 emu g −1 , respectively.

Magnetic properties
In Fig. 4b a sharp transition can be seen between sample 2 and sample 3, as the cellulose content in the samples increases from 29 to 35%. This observation indicates an inflection point in the Fig. 4b which is explained by the theoretical model in the theoretical section. Indeed, in the Fig. 16a we can clearly see that the scaling analysis of the order parameter suggests n t = 5.65 as the percolation threshold density. Since we have about 10 nanoparticles per cellulose fibre, the theoretical prediction agrees well with the experimental results of about 60% nanoparticles (from Fig. 4b) required for the order parameter to occur.

Conductivity of magnetic cellulose
The conductivity spectra of the modified cellulose samples with 83 wt% (S1) and 60 wt% (S4) Fe 3 O 4 at different temperatures during heating and cooling (2 cycles) are shown in Fig. 5. The conductivity isotherms have a similar shape, and several spectral features can be observed. First, there is a frequencyindependent conductivity plateau present at low frequencies. This feature is related to the long-range transport of charge carriers and corresponds to the DC conductivity. In addition, with increasing frequency, conductivity dispersion occurs in the form of a power-law which is due to local movements of charge carriers over short distances. The dispersive behaviour is more evident at lower temperatures and shifts to the higher part of the experimental frequency window used in this study as the temperature increases. It can be seen that increasing Fe 3 O 4 concentration in the samples (S4 to S1) results in an order of magnitude increase in electrical conductivity (i.e. from 60 to 83 wt% Fe 3 O 4 ).
At the first heating run between 60 and 80 °C, a conductivity "jump" is observed which implies the structural particularity of the material and its direct influence on electrical transport at higher temperatures. The same feature was not observed during the first cooling run or repeated heating/cooling runs, suggesting that it is related to the composite properties above 60 °C remaining "locked" so that the conductivity remains higher after the first cycle.
The observed electrical processes show an Arrhenius temperature dependence and characteristic activation energy with semiconducting behaviour in our cellulose modified samples, see Fig. 6. Therefore, the activation energy for each sample is determined from the slope of log σ DC versus 1000/T using the following equation: σ DC = σ 0 exp (W DC /k B T), where σ 0 is the pre-exponential factor, W DC is the activation energy for the DC conductivity k B is the Boltzmann constant and T is the temperature (K). The linear dependence of conductivity can be divided into three regions: (i) heating 1 (− 30 to 60 °C), followed by a steeper region (ii) heating 1 (between 60 and 80 °C), and (iii) cooling 1/heating 2/cooling 2. One can see the difference in the DC values after heating up to 80 °C, i.e., higher values in cooling are observed, while heating up to 60 °C does not influence the corresponding values in the cooling runs. The values of the activation energies in the heating run up to 60 °C are in Fig. 4 a Magnetic hysteresis loops of the magnetic cellulose samples at the temperature of 300 K; b graph of dependence of nanoparticles percentage in magnetic cellulose and magnetisation Fig. 5 The frequency-dependent real part of electrical conductivity at different temperatures for samples S1 and S4 in heating (H) and cooling (C) runs. The error bars are at most of the order of the symbol size Fig. 6 Arrhenius plots of DC electrical conductivity for samples S4 and S1 (log σ DC vs. 1000/T) along with calculated values of activation energies, W DC for different marked regions. The error bars are at most of the order of the symbol size line with the activation energies during both cooling cycles as well as during the second heating run. The first heating cycle is characterised by a steep increase of activation energy in the region above 60 °C, where the value increases from ~ 25 to ~ 70 kJ/ mol, which indicates that the changes occur in the sample at that temperature range only during the first heating cycle. It seems that structural reorganization is occurring in cellulose-based composites above 60 °C slightly enhances the conductivity which is then preserved during cooling. Moreover, this effect is not dependent on Fe 3 O 4 amount, so it could be attributed to the effect of the cellulose matrix. We also investigated the possible relaxation after the measurements were finished and noticed that with time the conductivity returns to the starting value before heating/cooling runs. Although interesting, the straightforward correlation cannot be safely concluded, so the observed effect should be investigated further.
In the remainder of the analysis, we turn our attention to the comparison of the conductivity and complex impedance spectra for different cellulose modified samples, with the purpose to see how the level of Fe 3 O 4 influences the electrical transport mechanism. Conductivity isotherms at 30 °C, shown in Fig. 7a, reveal several interesting features of the investigated samples. Firstly, while commenting on the shape of the conductivity spectra, the major change is visible between treated and non-treated cellulose samples. In the pure cellulose sample, an advanced decrease is visible in conductivity at higher temperatures and in the low-frequency region, which correlates with the electrode polarization effect due to the accumulation of mobile charge carriers at the blocking metallic electrodes used in the experimental setup. In the samples treated with Fe 3 O 4 , a change in the shape of the conductivity spectra is observed between the different samples with respect to the amount of Fe 3 O 4 . In particular, in treated samples, only already mentioned two characteristic features in spectra are present (i.e. DC plateau correlated to the long-range carriers, and high-frequency dispersion correlated to the short-range carriers). With the increase of the Fe 3 O 4 amount up to 83 wt%, the DC conductivity values increase by almost 3 orders of magnitude, and saturation is observed except for the sample with the highest Fe 3 O 4 content.
Furthermore, we have presented the experimental data in a different presentation, in a complex impedance plane, the so-called Nyquist diagram. Such a formalism and presentation allow us to study our material in more detail because different processes and characteristics are highlighted. The impedance spectra of samples S1 and pure cellulose, at different temperatures, are shown in Figure 7b, c, respectively.
It can be seen that the impedance spectra for the sample with 83 wt% of Fe 3 O 4 (S1) consist of a well-formed semicircle at all measured temperatures, which represents the bulk electrical process in a composite material. Similar behaviour is observed for other modified cellulose samples (composites Fig. 7 a The frequency-dependent real part of electrical conductivity spectra at 30 °C for all samples investigated, and the complex impedance spectra for b S1 (83 wt% Fe 3 O 4 ), and c pure cellulose sample at different temperatures. The symbols (coloured open shapes) denote experimental values, whereas the solid blue lines correspond to the best fit. The matching equivalent circuit model is composed of parallel/series combinations of the resistor (R) and the constant-phase element (CPE), used for fitting the data of individual spectra. The error bars are at most of the order of the symbol size S2-S4). In contrast, for the pure cellulose sample, it can be seen in the Nyquist diagram that the spur is formed at the low frequencies in addition to the dominant bulk process. Such an effect at low frequencies could be related to the electrode polarization effect and/or the surface-electrode effect. This feature indicates the change in the dominant electrical mechanism when we treat cellulose with Fe 3 O 4 . In particular, the polaron conductivity mechanism, i.e., thermally activated hopping of small polarons from Fe 2+ to Fe 3+ ions, begins to dominate in composite samples where an ionic transport mechanism is also present. Consequently, the conductivity of composite material depends on the shape, size and morphology of the filler particles. (Abdel-Karim et al. 2018;Jayakrishnan 2018).
The appropriate electrical equivalent circuit (EEC) model used to fit the experimental data for both samples is shown in Figure 7b, c. Depending on the EEC used, various process(es) can be identified and separated based on the fitting parameters obtained. In general, a single impedance semicircle can be represented by an EEC consisting of a resistor and a capacitor connected in parallel. Ideally, such a semicircle passes through the origin of a complex plot and yields a low-frequency intercept on the real axis corresponding to the resistance, R, of the observed process. However, in cases where the experimentally observed semicircle is depressed, the constant-phase element (CPE) is used instead of the ordinary capacitor in equivalent circuits. The CPE is an empirical impedance function of the type: Z* CPE = 1/A(iω) α where A and α are the constants. For the samples in this study, the complex impedance spectrum is described by an equivalent R-CPE circuit for modified cellulose samples (Fig. 7b), and for pure cellulose one description is enhanced with one more CPE element for the spur at low frequencies, see Fig. 7c. The parameters for each circuit element (R, A, and α) were obtained directly from measured impedance data using the CNLLSQ method. The DC conductivity values at 30 °C calculated based on modelling are shown in Table 2.
The dependence of the DC conductivity of studied cellulose-based samples depending on the amount of Fe 3 O 4 nanoparticles is shown in Fig. 8. It can be clearly seen that modification of cellulose with Fe 3 O 4 has a positive effect on the DC conductivity which is observed up to 71 wt% of Fe 3 O 4 (S4 to S2 sample). Furthermore, saturation occurs above 71 wt % of Fe 3 O 4 , followed by a slight decrease in DC conductivity (S1 sample with 83 wt% Fe 3 O 4 ). Thus, the obtained trend could lead to the conclusion that ~ 70 wt% is the upper limit for cellulose to make an efficient composite when talking about electrical properties of the same. Moreover, we can also conclude that the lower limit of Fe 3 O 4 content, which must be exceeded to achieve a stable and sustainable conductivity in the range of ~ 5-20 × 10 -9 (Ω cm) −1 @30 °C is 65 wt% of the Fe 3 O 4 for the studied cellulose-Fe 3 O 4 composites. Regarding the activation energy, see Fig. 8., one can see that as the DC conductivity decreases, the activation energy increases as expected, i.e., with the increase of the cellulose content in the sample.
AFM characterisation AFM analysis was performed using 2D-and 3D-topographic height images shown in Fig. 9. The surface area of the sample with the highest content of cellulose nanofibres (S4 sample-60% Fe 3 O 4 ) shows a directional superordinate granular structure consisting of clearly visible directional individual nanoparticles with a height between 7 and 12 nm. The granular superordinate structure is also observed at a lower weight fraction of cellulose nanofibres (S3 sample-65% Fe 3 O 4 ), but shows a lateral orientation with Moreover, the lateral alignment of individual Fe 3 O 4 nanoparticles on the membrane surface is lower when the cellulose content is decreased. i.e., the Fe 3 O 4 nanoparticle content is increased, and for the highest Fe 3 O 4 NPs content, the lateral alignment is completely lost, resulting in inhomogeneously distributed Fe 3 O 4 NPs on the membrane surfaces. Since the interaction between the AFM probe and Fe 3 O 4 NPs is significantly different from the interaction between the AFM probe and cellulose nanofibres, it was possible to detect the orientation of the crosslinked fibres in the presence of Fe 3 O 4 NPs.
The loss of lateral orientation of cellulose fibres was confirmed by phase images in the sample in which the cellulose content had decreased from 29 to 17 wt% (S1 sample), see Fig. 10. The spatial arrangement of Fe 3 O 4 NPs bound to the cellulose fibres was reflected in the change in roughness values. Namely, loading the cellulose nanofibres with NPs increased the surface roughness from R a = 19.1 ± 0.2 nm to R a = 64.4 ± 0.5 nm for laterally directed and inhomogeneously randomly distributed NPs (see Table 3) making the surface of the membrane rougher.
In addition, the effect of increasing the temperature (from 30 to 60 °C) on the surface morphology of the membrane was investigated using cellulose nanofibres containing 71% Fe 3 O 4 (S2 sample). Table 3 shows that the roughness values decrease by 30.37%; 28.77% and 60.42% for R a , R ms and Z range, respectively when the temperature increases from 30 to 60 °C. The effect of temperature increases from 40 to 60 °C is larger for the sample of cellulose nanofibres with S1 sample with higher Fe 3 O 4 content (83%), where a decrease in roughness values of 42.82%, 43.76 and 34.99% was observed for R a , R ms and Z range, respectively. The cross-section analysis in Figures 11 and 12 shows the effects of temperature on the surface morphology and roughness parameters.

Yang-Lee theory
In the middle of the last century, Yang and Lee (YL) proposed a novel method for phase transition detection (Yang and Lee 1954a, b;Yang and Lee 1954a, b). The essence of YL theory is based on the study of the distribution of the partition function zeros over the complex plane of the field Q, conjugated to the observable extensive variable (like magnetisation for YL zeros or the size of the largest cluster, l, in the present model). In most cases, in thermodynamic limit, zeros lie on a line which forms a gap around the Re(Q) axis. By changing the control parameter (such as temperature for YL zeros or density in the present case), the width of the gap is also changed. As a control parameter approaches the critical value from above, the gap shrinks, until it closes and remains closed for all values of the control parameter equal to or greater than the critical value. We will show that exactly the same mechanism occurs in the present model. The partition function (3) can be represented by its zeros, Q n In complete analogy to the YL method, we calculated some of the lowest Q n in a complex Q plane and traced the position of the lowest zero Q 0 as a function of the density and the size of the system. Since all P LC (l) are real and positive, all zeros Q n are either real or appear in complex-conjugate pairs. For a complex field Q = ℜ(Q) + iℑ(Q) , the partition function (3) is decomposed into its real and imaginary parts (4) Numerically, we have calculated the positions of zeros of the real and imaginary parts of the partition function separately. An example of such a calculation for a fixed value of the density, n = 5.65, is shown on the left side of Fig. 13.
The intersections of the blue and red lines give the zeros of the entire partition function. In Fig. 14a we see these zeros calculated for different values of the densities. For all densities, the zeros lie in a finite part of the Q plane, with a well-defined zero, Q 0 (N, n), closest to the real Q-axis. As the size of the system increases, the imaginary part of Q 0 (N, n) vanishes ( Fig. 14b, and red points in Fig. 15) i.e. the gap closes which is a sign of a phase transition. Figure 14 shows the convergence in N of the real (a) and imaginary (b) parts of Q 0 , for different values of density as a control parameter.
When the data from Fig. 14 are linearly extrapolated to the thermodynamic, N → ∞ limit, we obtain the results shown in Fig. 15. With increasing density, a decrease gradually and finally a disappearance of the imaginary part of Q 0 , i. e. a closing of the gap, can be seen at a percolation threshold density estimated to be. The Imaginary part of Q 0 remains approximately zero at densities greater than n t . The percolation threshold density (7) is quite close (and less accurate) to the density n t = 5.63726 ± 0.00002, obtained by a different method in Fig. 15 for a quadratic system with free boundary conditions. At the same critical value of density (4), the real part of Q 0 passes through zero, which means that the (7) n t = 5.65 ± 0.05.
transition is spontaneous, i.e. it occurs without the help of an external field.
Order parameter In addition to the calculations of the zeros partition functions, we performed the calculation of a spontaneous normalised order parameter, ⟨l⟩∕N . It was calculated as the first moment of the largest cluster distribution function, with a zero field, Q = 0, The normalized order parameter as a function of stick density and with the size of the system as a parameter, is shown in Fig. 16a. A close examination of the intersections of neighboring lines, yields data in Table 4 (also shown in Fig. 16b). Linear extrapolation of the above data in the N → ∞ limit gives which suggests to us a value of n t ≡ n t (∞) ≈ 5.65 as the percolation threshold density in the thermodynamic limit. This result agrees well with the estimate (4) obtained from the calculations of the zeros partition function.

Conclusion
In this research, we continued the successful production of magnetic cellulose nanofibres containing Fe 3 O 4 nanoparticles with a narrow distribution of grain size in the range of 9-14 nm. In contrast to our previous research on water purification, the focus of this research was on conductivity and electromagnetic properties, which are closely related to the structure and morphology of the samples. Impedance spectroscopy measurements show the semiconductor behaviour of the samples with negative coefficients of resistance when the temperature is increased (typical for semiconductors). Modification of cellulose with Fe 3 O 4 has a positive effect on DC conductivity observed up to 71 wt% Fe 3 O 4 . Beyond that, saturation occurs followed by a slight decrease in DC conductivity for a sample containing 83 wt% Fe 3 O 4 . At the same time, the activation energy for the DC conductivity is in the range of ~ 25-30 kJ/mol, and ~ 80 kJ/ mol for pure cellulose. The change in the dominant electrical mechanism during the treatment of cellulose with Fe 3 O 4 can be observed. In particular, the polaron conductivity mechanism begins to dominate in composite samples where an ionic transport mechanism is also present. The lower limit of Fe 3 O 4 content that must be exceeded to achieve stable and sustainable conductivity in the range of ~ 5-20 × 10 -9 (Ω cm) −1 @30 °C is 65 wt% of Fe 3 O 4 for the cellulose-Fe 3 O 4 composites studied. Detailed structural analysis by AFM, XRD, and TEM show that the shape and size of the grains (Fe 3 O 4 nanoparticles) do not change with increasing temperature (− 100 °C to + 100 °C). On the other hand, AFM analysis shows that the roughness values of all composite samples decrease with an increase in temperature from 30 to 60 °C. This indicates that the changes in morphology and consequently the conductivity properties of all composite samples (magnetic cellulose) are strictly due to the change in morphology of cellulose fibres.
In the theoretical model we can say that the YL method has been successfully extended to the solution of problem of continuous percolation in 2D: the typical YL mechanism is clearly visible, and the transition point is uniquely determined. The accuracy of the results could be further increased by increasing the number of simulation steps and the number of sticks, both of which require more time and more computer resources.
Finally, this research shows that composite materials such as magnetic cellulose (Fe 3 O 4 nanoparticles + cellulose fibres), and other combinations of different nanoparticles and cellulose fibres can be a very promising material for the electronics and semiconductor industries due to the easy synthesis, cheap starting materials, and good ability to fine tune structure and properties. Availability of data and materials The datasets generated during the current study are included in this article. Additional information, and data are available from the corresponding author on reasonable request.

Conflict of interest
The authors declare that they have no competing interests relevant to the content of this article.
Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors.

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