Low-Frequency Measurement of Moistened Wood-Based Materials

Abstract

This article examines how water content is a crucial parameter for the preservation of wooden artworks and buildings, focusing on non-invasive ways of measuring water content through capacitive methods. A personalized, low-cost probe to measure the dielectric properties of oak and poplar wood at various water content levels and frequencies is described. The accuracy of the probe is confirmed by testing it with reference materials like air, PTFE, PLA, glass and Bakelite, demonstrating an accuracy error below 2%. Next, the probe is used to evaluate the relationship between water content and permittivity, indicating possible uses in conservation projects. Measurements were conducted on two types of wood, poplar and oak, at five varying levels of water content. The dielectric permittivity between 10 and 100 kHz was assessed. Using the vertical shift from the single interpolant of the dataset, a graduation curve was estimated. Finally, an R² = 0.98 value demonstrates that a sigmoidal function reflects the relationship between the percentage water content and the permittivity of materials.

1. Introduction

Water content is a critical parameter in the conservation of artwork, historical objects, and buildings [1,2], where uncontrolled levels can lead to alterations in dimensions and accelerate decay caused by chemical and biological processes. Wooden objects and structures are highly impacted by changes in water content due to their hygroscopic nature, more so than materials like stone or brick used in historic buildings.

In studying moisturized wood, it is essential to note that water adsorption has two key points that change with species: (i) the Fiber Saturation Point (FSP), which corresponds to a gravimetric water content (WC) of 28–30% where the cell wall is filled with water, and (ii) Water Saturation Point (WSP), which corresponds to a WC 140% where the water is fully embedded in the cavity [3]. Generally, below FSP, changes in WC affect the rheological, mechanical, and physical properties of wood. In WC above 5%, the risk of insect infestation increases.

Although gravimetric methods are highly accurate for evaluating water content, they are destructive and require sampling. Non-invasive and ad hoc methodologies have increased significantly over the past decade, leading to a diverse range of water level assessment methods. These methods and measurement instruments are often classified by their underlying physical principles [4,5]. The literature indicates that techniques associating water content with electrical properties are especially promising due to their sensitivity to moisture [6,7].

Methods like Near-Infrared Spectrometry (NIRS) [8], thermography [9,10], Nuclear Magnetic Resonance (NMR) [11], ultrasound (US) [12,13,14], and Ground-Penetrating Radar (GPR) [15] are frequently used for WC measurements in objects because they provide gradient distribution data instead of point-specific details.

Alternatively, methods that correlate the WC with the electric properties (i.e., resistivity, conductivity, permittivity) offer a balanced approach in terms of resolution, cost, expertise, and invasiveness.

High-frequency methods (0.3–300 GHz) use waveguides, open-ended probes, or planar resonators to emit energy and measure perturbations in the electromagnetic field caused by water molecules [16]. Errors arise from temperature, material density, and thickness [17]. Although waveguides are expensive and limited to methodological/calibration uses, recent advances in low-cost probes have enabled field applications and maintain good metrological characteristics [18].

Low-frequency electrical conductivity-based methods measure the intensity of a direct current (i_DC) flowing among two or more electrodes either in direct contact with the surface layer or embedded in the object at a given depth.

Resistance decreases as WC in wood-based items increases. Temperature gradients near the measurement points, dissolved salts, and non-homogeneity within the artifacts are all factors that can introduce errors in this method. Finally, the main restriction for this approach is the limited 1.0% accuracy within the 6% to FSP range; outside of this range, the accuracy reduces considerably. This differs from the microwave technique because the microwave allows for measuring water content accurately in the 0%—FSP range without accuracy loss [19].

Electrical capacity methods use contact probes to measure dielectric variations caused by changes in water content, with capacitance at low frequencies. The technique relies on the change in relative dielectric permittivity ε_r, which ranges from 2 to 8 for dry samples to 80 for water. Errors can arise from density variations, cavities, or internal discontinuities. It offers acceptable accuracy (3%) up to the Fiber Saturation Point (FSP) and correlates water content with dielectric permittivity at a single operative frequency [20,21].

Therefore, dielectric measurement methods can be classified as non-invasive procedures. Nonetheless, the degree of invasiveness may differ and depend on the specific technique and frequency range [22]. While resonance-based methods, such as patch resonators and coaxial probes, measure permittivity only to a limited depth and often require direct contact with the material [23], transmission antenna methods are constrained by the thickness of the material, making them unsuitable for measuring deeper layers [24]. Furthermore, waveguide techniques, though effective, are inherently invasive, as they necessitate inserting material samples into the guide [25,26]. Low-frequency dielectric methods, like capacitive probes, primarily measure surface moisture. To assess moisture content deeper within a material, the drilling of holes is often required, thus making the method invasive [27]. In contrast, the admittance cell approach proposed in this work allows for the non-invasive measurement of average water content across the material without the need for any physical alterations. This approach offers a more comprehensive and efficient means of assessing moisture levels, making it especially suitable for applications in conservation and material preservation.

Previous studies show that capacitive sensors are promising in assessing wood dielectric characteristics at low-frequency ranges in the field of restoration and conservation of old buildings; for example, Moron et al. [28] proposed capacitive sensors for larger pine samples (25 cm × 16 cm × 16 cm), roughly the size of a timber ceiling section, demonstrating that capacitive methods can be adapted to larger wooden structures.

In this paper, using a customized, low-cost, non-invasive probe, the relative permittivity (ε_r) at diverse operative frequencies of two different wood species is determined. Specifically, while air, Polytetrafluoroethylene (PTFE), and Polylactide (PLA) were used as reference materials (RM), a sample of oak and poplar were used as the material under test (MUT). The ε_r of wood at different water contents in the range of 10 kHz to 100 kHz frequency was then determined. We investigated the potential of this probe, i.e., a capacitive probe, to develop a dispersion model for wood samples and assess calibration curves for laboratory and field applications.

The manuscript is structured as follows: Section 2 outlines the proposed sensor, measurement system, moisture, and weight protocol, along with details on both reference materials and materials under test. Section 3 presents the experimental results, and Section 4 provides conclusions and future perspectives.

2. Materials and Methods

2.1. Reference and Under-Test Materials

Initially, we estimated the accuracy of the capacitive probe to distinguish the permittivity of five well-known solid materials:

• Air (ε_r = 1.00);
• Polytetrafluoroethylene or PTFE (ε_r = 2.100);
• Polylactide or PLA (ε_r = 3.50);
• Bakelite (ε_r = 4.5–5.0);
• Lead Glasses (ε_r = 7.5–10.0).

Then, we applied the capacitive probe to two different wood samples: (i) poplar and (ii) oak. In general, poplar and oak wood were found to be widely used in Italy and Germany as painting supports [29,30]. For each type of wood, a sample was cut with a length L = 15.0 cm, a width W = 10.0 cm, and a thickness T = 1.5 cm. Both samples were cut along the same grain line. The dimensions of the samples were selected to ensure that the width (W) and length (L) exceeded the sensor’s area, minimizing edge effects. The thickness was chosen to align with the typical thickness of actual painting supports [31,32], which generally spread from 5 to 35 cm.

2.2. Capacitive Probe

Starting from the literature [33,34] and past experiences [35,36], we developed a capacitive probe with guarded parallel plates able to achieve a capacitance of 2.5 pF when the plates are spaced 5.0 mm apart, and with a sensitive area of 14.2 cm². This size was chosen because it is sufficiently small relative to the sample, while also large enough to minimize the impact of minor inhomogeneities of the material in the internal humidity distribution, thus allowing for sensitivity to the ‘average’ properties of the sample.

From a mechanical point of view, the sensing plate in brass measures 27.50 mm in external radius and 2.00 mm in thickness. Also, the force plate parts are made up of brass: (i) a ground plate (guarded electrode) measuring 21.20 mm in radius and (ii) a ring with an inner radius of 23.20 mm and an outer radius of 27.50 mm. To avoid unwanted short circuits, a PTFE ring was added between the electrode and the guard ring. Both plates are surrounded by a PLA case, with an inner radius of 27.50 mm and an outer radius of 30.00 mm. Figure 1 illustrates the arrangement of the capacitive cell configuration [37].

Starting from the above-described geometry, it is possible to evaluate how the capacitance changes in function of the distance between the armors and the dielectric constant of the material (Figure 2a). Moreover, in terms of sensitivity, given (1),

$$C={\epsilon }_r{\epsilon }_0{\displaystyle \frac{A}{d}}$$

(1)

And supposing that A, d, and ε₀ do not change during the measurement, we can compute the sensitivity with respect to ε_r and d.

Sensitivity with respect to ε_r (2):

$$S\left({\mathsf{\epsilon }}_r\right)={\displaystyle \frac{\partial C}{{\partial \mathsf{\epsilon }}_r}}={\epsilon }_0{\displaystyle \frac{\mathrm{A}}{d}}$$

(2)

shows how it is higher for small distances d (closer plates), but it is constant for each value of d, while the sensitivity with respect to distance d (3)

$$S\left(d\right)=-{\epsilon }_r{\epsilon }_0{\displaystyle \frac{\mathrm{A}}{d^2}}$$

(3)

is negative (decreases as d increases) and is more pronounced for higher values of ε_r in line with the effect of materials with a high dielectric constant that amplifies the change in capacitance.

Therefore, for materials with thickness of 15 mm and ε_r that spreads from 5 to 20, the sensitivity with respect to ε_r, S(ε_r) = 0.84 pF, while the sensitivity with respect to d S(d)-spreads from −112 pF/mm to −27.9 pF/mm. However, with thickness t held constant, we can consider only S(ε_r).

Rearranging (1) to (4), the ε_r of a sample is proportional to the thickness t of MUT and the capacitance measured with the MUT in place C_MUT, and inversely proportional to the area A of the ground plate and to the vacuum permittivity ε₀ ≅ 8.85 × 10⁻¹² F/m.

$${\epsilon }_r={\displaystyle \frac{t\cdot C_{MUT}}{A\cdot {\epsilon }_0}}$$

(4)

In practice, ε_r can also be evaluated as the ratio of C_MUT to the capacitance of the empty capacitor C_AIR, as shown in the following Equation (2):

$${\epsilon }_r={\displaystyle \frac{C_{MUT}}{C_{Air}}}$$

(5)

which ensures that the separation between the plates has the same thickness as the MUT, even when the capacitor is unfilled.

Even if the use of the guard ring helps to reduce the impact of fringing fields and correct errors caused by a small misalignment between the sensing and force electrodes by utilizing a smaller ground electrode than the force electrode, on the other hand, small misalignments can be compensated for if there is an overlap between the sensor and ground. However, if there is no overlap, edge effects occur that cannot be compensated for. This is illustrated in Figure 3, where CST studio electromagnetic simulations of the proposed probe demonstrate how the guard ring reduces fringing fields and compensates for small misalignments, compared to a probe without a guard ring.

2.3. Measurement Chain

The measurement chain illustrated in Figure 4 is used for assessing the capacitance values. The capacitance measurements were conducted with a precision 6100 LCR meter from GW-Instek, operating at frequencies between 10 kHz and 100 kHz in “parallel mode”. For the measurements, a setup of four wires was utilized, and an additional wire connected the instrument ground to the guard electrode. A brief OPEN/SHORT calibration was conducted before each measurement. This involved connecting the probe to the instrument without a material under test (MUT) between the electrodes for the OPEN condition, and using a short-circuit for the SHORT condition. For our application, we performed SHORT calibration using a customized component consisting of a brass disk with a knob, designed to ensure proper contact between the sensor and ground probes.

The tests were conducted with an operational power supply of 0.5 Vrms applied to the electrodes. Indeed, the specifications of the instrument guarantee the most accurate measurements for a 0.1 to 1 V span. Moreover, the temperature of the MUT was observed using a k-type thermocouple since the measurements were not carried out in a controlled environment.

2.4. Moistening and Weighing Procedure

The specimens were dehydrated in an oven at a temperature of (100 ± 5) °C until the weight stayed the same for three consecutive readings. Following this procedure, the wood samples were rinsed in deionized water for a minimum of 5 repetitions lasting 90 s each, until reaching various moisture contents within the range of 0% to 5%—Scheme 1 details the measurement procedure.

The WC percentage (WC_%) was calculated following (6), as stated in EN 16682-2017.

$$W{C}_{\%}={\displaystyle \frac{{WC}_i-{WC}_d}{V\cdot {\rho }_{H_{2}O}}}\times 100$$

(6)

In the equation, WC_d denotes the dehydrated weight, while WC_i represents the WC at the i-th level. The variable V is the volume of the sample and ρ is the density of deionized water, fixed at 1 g/cm³. The weight was measured using an electronic balance with a resolution of 10 mg, both before and after measurements, to evaluate the effects of evaporation. The average value obtained from these two measurements was utilized to determine the water content.

2.5. Measurement Procedure

As stated previously, this article shows how the dielectric properties of wood change with varying frequencies (10–100 kHz) and with different water content.

At first, we assessed the frequency response of the vacant cell (air) to determine its accuracy α (4). Next, the dielectric permittivity of both reference materials, PLA and PTFE, was calculated using Equation (7), and the precision was evaluated by comparing the results with the theoretical values from references [38]:

$$\alpha ={\displaystyle \frac{C_m-C_t}{C_t}}\times 100$$

(7)

with $C_m$ as the measured capacitance and $C_t$ as the theoretical one.

Moreover, the combined uncertainty (u) was determined by combining the repeatability uncertainty u_r, namely the ratio between the standard deviation (σ) and the square root of the measurements, i.e., n = 5, and with the instrumentation uncertainty (u_inst).

The u_inst, hypothesizing a uniform distribution of possible values, was obtained by dividing the instrumentation accuracy provided by the instrument datasheet by the square root of three.

$$u=\sqrt{u_r^2+u_{inst}^2}$$

(8)

Finally, we evaluated the expanded uncertainty U,

$$U=t_{\nu ,p}\ast \sqrt{u^2}$$

(9)

where t is the Student’s factor computed for the associated degrees of freedom (ν) and probability (p). In this case, t_ν,p = 2.87 with ν = n − 1 = 4 and p = 95.45.

Additionally, the mean relative permittivity over the frequency range $\overline{{\epsilon }_r\left(f\right)}$ and its fluctuations δ(f) were determined for every material.

A similar procedure was repeated for the two woods at different WC contents, assessing the average $\overline{{\epsilon }_r\left(f\right)}$ over the range of frequencies and then calculating the graduation curve of $\overline{{\epsilon }_r\left(f\right)}$ as a function of the WC percentage.

3. Results and Discussions

3.1. Calibration Results

Table 1. Measurement results reporting mean, expanded uncertainty (u_r), instrumentation (u_inst), and expanded uncertainty (U).

f [kHz]	AIR				PTFE				PLA				BAKELITE				LEAD GLASS
	εr	ur	uinst	U	εr	ur	uinst	U	εr	ur	uinst	U	εr	ur	uinst	U	εr	ur	uinst	U
10	1.009	0.053	0.033	0.18	2.132	0.053	0.033	0.18	3.449	0.074	0.033	0.23	5.005	0.036	0.033	0.17	3.449	0.074	0.033	0.23
20	1.010	0.056	0.033	0.19	2.134	0.048	0.033	0.17	3.451	0.065	0.033	0.21	5.001	0.001	0.033	0.17	3.451	0.065	0.033	0.21
30	1.008	0.056	0.033	0.19	2.134	0.059	0.033	0.20	3.450	0.083	0.033	0.26	4.918	0.003	0.033	0.17	3.450	0.083	0.033	0.26
40	1.008	0.059	0.033	0.20	2.135	0.065	0.033	0.21	3.449	0.074	0.033	0.23	4.845	0.001	0.033	0.17	3.449	0.074	0.033	0.23
50	1.009	0.053	0.033	0.18	2.135	0.053	0.033	0.18	3.449	0.048	0.033	0.17	4.833	0.001	0.033	0.17	3.449	0.048	0.033	0.17
60	1.009	0.059	0.033	0.20	2.131	0.053	0.033	0.18	3.450	0.059	0.033	0.20	4.773	0.030	0.033	0.17	3.450	0.059	0.033	0.20
70	1.008	0.056	0.033	0.19	2.131	0.059	0.033	0.20	3.451	0.065	0.033	0.21	4.737	0.048	0.033	0.18	3.451	0.065	0.033	0.21
80	1.008	0.062	0.033	0.20	2.132	0.048	0.033	0.17	3.449	0.083	0.033	0.26	4.762	0.014	0.033	0.17	3.449	0.083	0.033	0.26
90	1.009	0.059	0.033	0.20	2.129	0.050	0.033	0.17	3.450	0.062	0.033	0.20	4.737	0.100	0.033	0.21	3.450	0.062	0.033	0.20
100	1.009	0.062	0.033	0.20	2.130	0.050	0.033	0.17	3.450	0.065	0.033	0.21	4.710	0.001	0.033	0.17	3.450	0.065	0.033	0.21

reports the mean, repeatability uncertainty (u_r), instrumentation uncertainty (u_inst), and expanded uncertainty (U) of the parameters involved for n = 5 for each chosen frequency.

Table 2. Measurement results for reference materials.

Materials	Temperature (°C)	${\mathit{\epsilon }}_{\mathit{r}}(\mathit{f}$) Ref	$\overline{{\mathit{\epsilon }}_{\mathit{r}}\left(\mathit{f}\right)}$ Measured
air	23 ± 1	1.00	1.01 ± 0.006
PTEF [38]	24 ± 1	2.10	2.13 ± 0.004
PLA [39]	23 ± 1	3.50	3.45 ± 0.007
Bakelite [40]	22 ± 1	4.5–5.0	4.83 ± 0.12
LEAD GLASS [41]	22± 1	7.5–10.0	9.45 ± 0.03

shows the value of ε_r, the average experimental value $\overline{{\epsilon }_r\left(f\right)}$, and its deviation. Furthermore, the recorded temperature T in °C is reported. The observed εr values closely align with typical standards. Specifically, the accuracy error for AIR is 0.9%, while PTFE and PLA exhibit accuracy errors of 1.5%. Bakelite shows an accuracy error of 1.7% computed on ${\epsilon }_r$ ref = 4.75, while lead glasses show 5% computed on ${\epsilon }_r$ ref = 10.

Figure 5 illustrates the measurement results for AIR, PTFE, and PLA. The calculated ε_r remains stable within the selected frequency interval for each sample. Throughout each measurement, the dispersion factor (D) was maintained below 0.003, indicating that ohmic parasitic components did not influence the results.

3.2. Wood Sample Results

The results of the measurements for five levels of WC in poplar species are presented in Figure 6a. The ε_r decreases as the frequency increases, while it increases with the WC. Furthermore, the expanded uncertainty for each sample’s five repetitions is less than 1.2.

Table 3. Measurement results for poplar.

Poplar (WC%)	$\overline{{\mathit{\epsilon }}_{\mathit{r}}\left(\mathit{f}\right)}$	$\mathit{\delta }\left(\mathit{f}\right)$	Vertical Shift
poplar 0.0%	4.97	0.062	5.27
poplar 1.3	8.49	1.38	15.6
poplar 2.0%	9.87	2.26	21.6
poplar 2.5	12.8	2.16	24.0
poplar 3.4%	13.6	2.51	26.5

shows the average $\overline{{\epsilon }_r\left(f\right)}$ and its change δ(f) as a function of the WC for the poplar. The temperature during all measurements was maintained at 23 ± 1 °C.

The results of the measurements for oak species are presented in Figure 6b. Also, in this situation, we can see how the εr decreases as the frequency increases for each WC and increases based on the WC. The combined uncertainty for each sample’s five repetitions is consistently below 1.5 for all measurements.

Table 4. Measurement results for oak.

Oak (WC%)	$\overline{{\mathit{\epsilon }}_{\mathit{r}}\left(\mathit{f}\right)}$	$\mathit{\delta }\left(\mathit{f}\right)$	Vertical Shift
Oak 0.0%	8.51	0.139	9.2
Oak 0.4%	9.73	0.584	12.7
Oak 1.1%	11.1	1.57	19.1
Oak 1.5%	12.3	1.96	22.3
Oak 2.0%	12.8	1.95	22.8

displays the average value of $\overline{{\epsilon }_r\left(f\right)}$ and its change δ(f) based on the WC. Also, in this case, the temperature during all measurements was 22 ± 1 °C. Figure 7a,b display the graduation curve for both types of wood obtained, imposing a logarithmic interpolation of raw data and using the average value of vertical shift among the interpolants. In both scenarios, one can see that the law between WC% and ε_r follows a sigmoidal function represented by (10):

$$y=b+{\displaystyle \frac{\left(t-b\right)}{1+{\left(\frac{C_{50\%}}{x}\right)}^s}}$$

(10)

with t representing the top bound of the data and b the bottom bound, C_50% as the mid-concentration between b and t, and s as the slope of the curve.

Table 5. Graduation curve coefficients.

Coefficients	Poplar	Oak
b	5.27	9.23
t	29.2	26.1
C50%	1.46	0.859
s	2.39	1.81
R2	0.98	0.99
Interpolation error	0.3	0.3

shows the coefficient of curve fitting and the R² coefficient of determination, which are both 0.98.

This result aligns with findings reported in the literature, such as in [3], where a sigmoidal relationship between water content and permittivity is evident for a pine sample.

4. Conclusions

In this study, we focused on developing a low-cost capacitive sensor to measure the dielectric properties of wood samples. Typically, capacitive sensors are costly (e.g., Keysight 16451B [34]) and require controlled laboratory environments for analyzing small-scale samples (solid or granular). However, with integration into a multi-axis setup as a scanner, this sensor could potentially be adapted for non-invasive moisture monitoring in larger wooden panels without requiring perforations.

This study examines how the WC and relative permittivity of two types of wood commonly used in artwork are related. The tests performed on the standard samples revealed an accuracy error from the literature values lower than 2% and with a dispersion $\delta \left(f\right)$ over the frequency range for the five trials consistently below 0.01.

Finally, the study quantified the correlation between WC and ε_r in poplar and oak wood. For both types of wood, the results indicated a clear trend: relative permittivity decreased with increasing frequency, while rising with higher water content. Moreover, a sigmoidal function, i.e., a dose–response curve, represented the graduation curve with an R² value of 0.98 and 0.99, respectively, for poplar and oak and the same maximum interpolation error of 0.3.

Future work will investigate the adaptation of this sensor for more complex, in situ conservation applications, including testing on larger, multi-dimensional surfaces common in heritage preservation.