Investigation and modelling of the electrical charging effect in birch wood above the fibre saturation point (FSP)

Conventional chargeability CHA(E) as defined by Schlumberger and dependent on the potential E is represented as follows:

1 CHA(E)=E2,max⁡E1,const $$CHA(E) = {{E_{2,\max } } \over {E_{1,const} }}$$

where E_2,max is the maximum potential of electrodes in volts at the beginning of the discharging process, and E_1,const is the constant potential of electrodes in volts in the charging process.

Energetic chargeability CHA(W) is defined as a ratio between stored and transmitted energy:

2 CHA(W)=W2W1 $$CHA(W) = {{W_2 } \over {W_1 }}$$

where W₂ is the total energy stored in wood in joules (J), and W₁ is the total energy (J) transmitted through wood in the charging process. The transmitted energy W₁ may briefly be called primary energy, and the energy stored in wood W₂ may briefly be called secondary energy.

Primary energy W₁ was calculated as follows (Keithley, 2004; Bard & Faulkner, 1980):

3 W1=Q1(t1)E1,const.=E1,const.∫0tI1(t)dt $$W_1 = Q_1 \left( {t_1 } \right)E_{1,const.} = E_{1,const.} \int\limits_0^t {I_1 \left( t \right)} dt$$

and secondary energy W₂ was calculated as follows:

4 W2=Q2(t2)E2(t2)max⁡=E2,max⁡∫t+0,1t2I2(t)dt $$W_2 = Q_2 \left( {t_2 } \right)E_2 \left( {t_2 } \right)_{\max } = E_{2,\max } \int\limits_{t + 0,1}^{t_2 } {I_2 \left( t \right)} dt$$

where Q₁(t₁) is the total primary charge in coulombs (C) transmitted through wood in the charging process t₁,

The number 0.1 in the bottom path of integration is the measuring interval.

The following relationship between conventional chargeability CHA(E) (as defined by Schlumberger in 1913) and energetic chargeability CHA(W) can be deduced from the above formulas:

5 CHA(W)=CHA(E)Q2Q1 $$CHA\left( W \right) = CHA\left( E \right){{Q_2 } \over {Q_1 }}$$

where Q₁ is the total charge in coulombs transmitted through the wood medium in the charging process, and Q₂ is the total electrical charge in coulombs stored in the wood medium.

The primary integral electrical capacitance corresponding to the transmitted electrical charge is represented as follows:

6 C1,int=Q1(t)/E1 $$C_{1,int} = Q_1 \left( t \right)/E_1$$

and the secondary integral electrical capacitance corresponding to the stored electrical charge is demonstrated as follows:

7 C2,int=Q2(t)/E2,max. $$C_{2,int} = Q_2 \left( t \right)/E_{2,max} .$$

To compare DC strengths in the time domain and frequency domain at the time moment of 240 seconds from the start of spectrum registration, the Ohm’s law was used:

8 IDC,240=E1|Z|240 $$I_{DC,240} = {{E_1 } \over {|Z|_{240} }}$$

where E₁ is the value of a discretely given potential for each impedance spectrum. |Z|₂₄₀ is the value of the modulus of impedance in ohms at the 240th second.

On a Nyquist and Bode phase angle plot (Figures 7 and 11a) the Warburg impedance (if the diffusion layer has an infinite thickness) appears as a diagonal line with a slope and a phase shift of 45°, respectively.

The Warburg impedance (Z_W) is obtained using

9 ZW=σ/ω1/2−jσ/ω1/2 $$Z_W = \sigma /\omega ^{1/2} - j\sigma /\omega ^{1/2}$$

where the first member of the formula describes the real part of the diffusion process and the second imaginary part of complex diffusion resistance. The corresponding faradaic impedance for mixed kinetics is expressed as a serial combination of the charge transfer resistance and Warburg impedance, also known as Randles circuit.

For more complex cases, the generalized finite-length Warburg element for a short circuit terminus model is expressed as:

10 ZW=RDtanh⁡[ (jωT)αW ](jωT)αW $$Z_W = {{R_D \tanh \left[ {\left( {j\omega T} \right)^{\alpha _W } } \right]} \over {\left( {j\omega T} \right)^{\alpha _W } }}$$

where ω is the radial frequency, R_D is the limiting diffusion resistance, T = L2/D is the diffusion time constant, L is the effective diffusion layer thickness, D is the effective diffusion coefficient of a particle and α_w is the fractional exponent for Warburg-like diffusion impedance (Bard & Faulkner, 1980; Krause, 2003).

The Warburg coefficient σ can be determined from the slope of the Nyquist plot or by fitting it to an equivalent circuit model which includes a Warburg impedance:

11 σ=RTn2F2A2(1cox0Dox+1cred0Dred) $$\sigma = {{RT} \over {n^2 F^2 A\sqrt 2 }}\left( {{1 \over {c_{ox}^0 \sqrt {D_{ox} } }} + {1 \over {c_{red}^0 \sqrt {D_{red} } }}} \right)$$

where R is the ideal gas constant; T is absolute temperature; n is the number of electrons transferred; F is Faraday's constant; cox0$$c_{ox}^0$$ and cred0$$c_{red}^0 $$ are the concentrations of oxidant and reductant in the bulk, respectively; D_ox and D_red are diffusion coefficients of the oxidant and reductant, respectively; and A is the surface area of the electrode.

Capacitors in electrochemical impedance spectroscopy experiments often do not behave ideally; they act like a constant phase element, where the exponent α < 1 and C_dl are the double-layer capacitance, which appears at the interface between an electrode and an adjacent liquid electrolyte:

12 Z=1(jωCdl)α $$Z = {1 \over {\left( {j\omega C_{dl} } \right)^\alpha }}$$

and

13 Z=Rel+Rct1+RctCdl(jω)α $$Z = R_{el} + {{R_{ct} } \over {1 + R_{ct} C_{dl} \left( {j\omega } \right)^\alpha }}$$

where R_ct is charge transfer resistance, formed by a kinetically controlled electrochemical reaction (Bard & Faulkner, 1980; Krause, 2003).

The double-layer capacitance can be determined from the frequency at which the imaginary impedance (-Z″) is at a maximum:

14 ωmax=1RctCdl $$\omega _{max} = {1 \over {R_{ct} C_{dl} }}$$

where the top point of the complex plane semicircle corresponding to the frequency ω_max and to the characteristic time constant τ of the circuit is represented by

15 τ=RctCdl. $$\tau = R_{ct} C_{dl} .$$

(Bard & Faulkner, 1980; Krause, 2003).

The equivalent circuit adapted from Zelinka et al. (2008) and given in Figure 1a was used in modelling, which includes the Warburg element and the constant phase element (CPE). Modelling also made use of the simplified equivalent circuit presented in Figure 1b, in which the CPE was replaced with the electrical double-layer capacitance C_dl.

Figure 1.

The schematic diagram of the experiment is shown in Figure 2a.

Figure 2.

Figure 2a depicts the measuring cell (at 20 °C) which used a measuring system with two symmetrical carbon fibre pin electrodes. The dimensions of the wood specimen were as follows: 35 mm (thickness, direction R) * 150 mm (width, direction T) * 120 mm (longitudinal fibre length, direction L). The pin electrodes were partially insulated with polyurethane film (see Figure 2a). The diameter of the electrodes was 3 mm, and the distance between them was 30 mm. Green birch wood had a moisture content (relative to dry weight) of 105%. For the purpose of the experiments of this article, liquid birch sap was traditionally collected at the beginning of the spring vegetation period.

The coupling electrode (CE) and the reference electrode (RE) were coupled together (CE+RE) in the two-electrode measuring system Metrohm Autolab (2022).

Table 1 illustrates the capability of various measuring procedures to experimentally determine the energetic chargeability and integral electrical capacitance of wood (and sap) according to Formulas (1) to (6). To calculate the energetic chargeability, it is necessary to determine the primary energy W₁ and secondary energy W₂ in the experiment. Table 1 shows that primary and secondary energy are fully measurable with the PDM and CV measuring procedures, because these procedures allow the time dependencies of both current strength and potential in the charging and discharging phases to be determined. The disadvantage of CV is that it does not allow the open circuit potential in the discharging process to be measured, because the linear potential is provided by the measuring program Metrohm Autolab (2022).

PDM is described in detail in a previous study (Tamme et al., 2013). In practice, it proved more convenient to combine the PDM and CCD procedures. Primary charge Q₁ and secondary charge Q₂ were determined with the CCD measuring procedure using the tool found in the Autolab control software Nova 1.8 for numerical integration of charging and discharging current curves (Metrohm Autolab, 2022). PDM was used to determine E_2,max. Figure 2b shows the time-dependent current curves typical of the chrono charging-discharging (CCD) measurement procedure as Metrohm Autolab (2022) screenshots at a 0.1 second measurement interval.

The 0.1 second measurement interval was used. The upper left curve in Figure 2b is the charging current curve for the potential E = 2.6 V, and the lower right curve is the discharging curve. The transition from the first charging phase to the second discharging phase occurs at the moment of potential interruption (Figure 2b).

For PDM, the corresponding typical current and potential curves are presented in the study by Tamme et al. (2013).

The EIS (i.e. FRA impedance potentiostatic) measuring procedure was used to collect the experimental data needed for modelling in the frequency domain. Fifty frequencies between 100 kHz and 0.01 Hz were used in logarithmic sequence with the sine signal amplitude of 50 mV. Electrode potentials were provided discreetly at the registration of each spectrum with the following values: 0 V, 0.4 V, 0.8 V, 1.2 V, 1.8 V and 2.6 V. ZView ver. 2.3 (Scribner Inc., 2022) software was used for the impedance modelling.

The results obtained in the time domain are found in Tables 2 and 3.

First, the dependence of the conventional and energetic chargeability and integral capacitance on the potential applied to the electrodes was studied (see Table 2).

Figure 3a shows the values of certain comparable energies in millijoules (mJ) on the same axis as dependent on the potential applied to the electrodes. In some cases, the calculated energy has square dependence on the potential.

Figure 3.

It can be ascertained from Figure 3b that conventional chargeability is, compared to actual energetic chargeability, systematically overestimated at all electrode potentials at a rate of about 2 to 2.5 times. Both chargeabilities depend on the potential applied to the electrodes. A comparison of the chargeabilities is performed for the same measuring interval of 0.1 second.

Figure 4a shows the primary integral electrical capacitances as dependent on the potential applied to the electrodes. From the 1.2 V potential, the primary integral electrical capacitance of birch sap begins to rise sharply compared to that of wood.

Figure 4.

Figure 4b represents the secondary integral electrical capacitances C_2,int for wood and sap, which are easily comparable both in terms of their course and their numerical values.

Secondly, in addition to the dependence of CHA(W) on the electrode potential, the dependencies of energetic chargeability and stored energy on the primary energy W₁ transmitted through wood and sap were also studied (see Table 3).

Referring to Figure 5a, it may be established that the amount of energy stored in wood is systematically about 2 times higher than the energy stored in sap when comparing the transmitted primary energies W₁. The almost linear increase in the energy stored in the system electrodes + sap observed in the same figure suggests that as the transmitted primary energy grows, the secondary energy stored in the volume charge of the system (symmetrical electrodes + sap) also grows. Figure 5b depicts the course of energetic chargeability of wood and sap as dependent on the transmitted primary energy. Wood chargeability is initially (in the case of W₁ = 8 mJ) about 5 times better than that of sap chargeability, but as the transmitted primary energy increases up to more than 100 mJ, the energetic chargeabilities of wood and sap equalize.

Figure 5.

Figure 6a shows the ratio W2(B)/ W2(BS) between secondary energies stored in wood and sap as dependent on the transmitted primary energy W₁. It may be observed that the initial (8 mJ) six-fold difference of W2(B) in favour of wood shrinks to one at high (more than 100 mJ) primary energies.

Figure 6.

Figure 6b represents the DC strength (I (calc)) calculated with the Equation (8) given in the same axis system, and the DC strength (I (meas)) measured with the CCD procedure at the 240th second from the start of the charging process. Both current strengths are mutually satisfactory.

In conclusion, the experimental results show (see Figures 3b and 5b) that wood energetic chargeability is systematically many times greater than that of sap, both in the comparison of the potentials applied to the electrodes and in the comparison of primary energies transmitted through the wood and sap mediums. The reason may lie in the capillary-porous structure of wood, which seems to favour the electrical charging effect in wood.

The Nyquist plots (Figure 7) were measured for birch wood- and birch sap-based systems in the AC frequency range of 300 kHz to 20 mHz in the region of potentials from 0 to 2.6 V. The complex plane plot consists mainly of at least two or three parts: from the very noticeable depressed semicircle at higher AC frequencies with the characteristic frequency f_max and to the so-called double-layer capacitance region at very low frequencies. The results also demonstrate that the shape of -Z’’, Z’-plots noticeably depends on the system studied as well as on the electrode potential.

Figure 7.

According to Formula (9), charge transfer resistance and the Warburg coefficient can be calculated from the slope of Z′ versus ω-1/2 or – Z″ versus ω^-1/2. According to Figure 8, these values are different for birch wood- and birch sap-based systems and depend also on the diffusivity and concentration of species.

Figure 8.

Calculated mass-transfer coefficients for both systems studied show (Figure 9) that there is a noticeable dependence on electrode potential as well as the system studied.

Figure 9.

The series capacitance, C_s, with values at f = 25.5 mHz, depends on the potential applied, and the highest C_s values have been established for a birch sap-based system at 2.6 V (Figure 10).

Figure 10.

The dependence (Figure 11a) of the phase angle θ on log f at fixed electrode potential shows that at ≈ 0.2 Hz for birch sap- and birch wood-based systems approaches 60° and 34°, respectively, which is characteristic of kinetically mixed processes.

Figure 11.

The linear dependence (Figure 11b) of log –Z“ on log f at a fixed electrode potential (0.8 V) clearly shows that at f < 2000 Hz for birch sap- and f < 100 Hz for birch wood-based systems, the systems are described mainly by the slow diffusion process. At higher AC frequencies slow heterogeneous adsorption takes place.

In comparing the general course of the integral capacitance C_1,int of wood and sap and the serial capacitance C_s found by impedance modelling, which are given in Figures 4a and 10, a similar trend stands out – higher potentials have higher C_1,int and C_s in sap than in wood. However, a comparison of numerical quantities of electrical capacitances at the same potentials in sap shows a significantly higher growth for C_1,int when moving to higher potentials. The reason may be that the growth in the integral capacitance of sap starting from the 1.2 V potential is facilitated by the macroscopic symmetrical volumetric charge distribution emerging around the symmetrical electrodes in addition to double-layer electrical charges on the nanoscale (i.e. 1–10 nm, according to Zuleta (2005)). The modelling of Cs according to impedance spectra only considers double-layer electrical capacitance. In the case of wood, no volume charge effect occurs, probably due to its high electrical resistance, and the numerical results of modelling are comparable both in the time domain and frequency domain (i.e. if E = 1.8V, then C_s = 0.5 mF and C_1,int = 0.7 mF).

DC strengths measured directly with the CCD measuring procedure in the time domain as well as DC strengths calculated according to Formula (8) in the frequency domain (impedance spectrum) depending on the applied potential are shown in Figure 6b. Both current strengths are mutually satisfactory.