Influence of Humidity on the Electric Resistivity of Leather: Mathematical Modelling Matematično modeliranje vpliva vlažnosti

A mathematical model is presented to simulate the electric resistivity of leather samples as a function of humidity. It will be shown that absolute and not relative humidity is the crucial parameter. The model assumes that the leather includes channels that can absorb water from the surrounding environment. This effect primarily determines the electric conductivity of the leather samples. The theoretical results from the model are quite closely in line with experimental measurements.


Introduction
Due to 21 st century technology, we must adapt to smart products offered by the market where tablets, computers, most mobile screens and smartwatches operate with a capacitive touch screen (finger-touch method) [1]. One practical application is the use of a touch screen when wearing leather gloves. These screens use the electric conductivity of the fingertip to increase the electric capacitance in a certain area of the screen. If gloves are worn, they should also be electrically conductive. It will be shown later that pure leather is a poor electric conductor. Leather will however, become conductive by absorbing water from ambient humid air or by diffusion from the finger. This is also true, depending on weather, climate or different conditions, where the finger-touch method is not possible and a conductive glove is very important [2][3][4].
Furthermore, the rapid growth of electrical and electronic devices and accessories that emit electromagnetic energy in different frequency bands has led to an increase in exposure to EM radiation, which can be harmful for human health. It was determined from literature that electrically conductive textiles are preferred for shielding applications primarily due to their good shielding properties and numerous advantages [5]. Although there are different studies about electromagnetic shielding textiles in literature, there is an urgent need to develop electromagnetic interference (EMI) shielding materials [6]. Moreover, different groups have worked on textile antennas [7][8][9][10] and leather antennas [11] for wearables, in which the conductivity is a prerequisite. Therefore, in order to make resistive traditional leather conductive, different groups have worked with different methods [2][3][4]. In previous contributions, sheep crust leather was coated with polypyrrole using the double in-situ polymerization method and the electrical resistivity of conductive leather using the "multiple-step method" [15][16] in different air temperature and humidity conditions was measured [17]. In this paper, a theoretical analysis of the influence of humidity on the electric resistivity of leather was performed. The mathematical theory of this method will be outlined in the next section.

Mathematical model
In order to set up a mathematical model, a closer look must be taken at the experimental results shown in Figure 1. If the two figures (Figures 1a and  1b) are compared, it is clear that relative humidity RH is no longer a suitable parameter. On the other hand, Figure 1b shows a good relationship with absolute humidity. In a previously published paper, these data were fitted to H -1.042 or H -1.059 , depending on whether sheets were included or only strips were taken into account [17]. The measuring technique was especially designed for textile samples and has been described extensively in literature [15][16]. These results are not surprising at all. Relative humidity is used to describe the environmental effect of air on the human body. Moreover, relative humidity is rather easy to measure. However, this paper deals with the electric conductivity of leather. Electric conductivity is determined by the intrinsic conductivity of leather and the amount of absorbed water molecules. The latter is simply absolute humidity usually expressed in g/kg or grams of water per kilogram air. We must therefore limit our investigation to modelling electric resistivity as a function of absolute humidity H. If we extrapolate the fitting H -1.042 or H -1.059 [17] to low values of H, it is clear that the resistivity of intrinsic leather must be very high. We would expect it should be at least one order of magnitude higher than the maximum value of 500 Ωm shown in Figure 1b. Firstly, the experimental fittings H -1.042 or H -1.059 [17] suggest that, theoretically speaking, it could be just 1/H. Secondly, a closer look at Figure 1b tells us that humidity H has a tremendous influence on resistivity. The values vary between 65 Ωm and 437 Ωm. The conclusion is obvious: electric conduction is mainly determined by the amount of absorbed water, and the  [17] conductivity of completely dry leather must be very low. These results also prove that there must be channels inside the leather filled with water. These channels provide electro conducting wires between the two electrodes. The word "channel" should be interpreted in a broad sense. A surface layer of water will also conduct electricity between the two electrodes. A simple model is shown schematically in Figure  2. The two rectangular electrodes, with an area expressed as S = b × d, are at distance a. Conducting channels now connect the two electrodes. Each channel has a cross section of ΔS and a length of a. It is obvious that the conductivity of the channel is proportional to absolute humidity: σ channel = α H. Conductance G between the two electrodes is then given by the equation: Conductivity σ, which is the inverse of resistivity (σ = 1/ρ) is then found using the equation: Resistivity ρ is then the inverse of equation (2): Where ρ leather = 1/σ leather . Parameter β is given by the equation: As already stated, resistivity ρ leather is very high, so that (3) can be approximated using the equation: This result is quite close to the fittings H -1.042 or H -1.059 used in [17]. Although this mathematical model is very simple (all channels have an equal length and cross section), it will help us explain the experimental results, as will be seen below.
The experimental results shown in Figure 1b have been redrawn in Figure 3 on a double logarithmic scale. A 1/H curve, such as (5), is then represented by a straight line. The experimental data can be fitted quite well to the equation: A second fitting using (3) was only possible for ρ leather values > 10000 Ωm, as also shown in Figure  3. We can thus conclude that the fitting using (3) helps us to achieve a lower limit for the resistivity of intrinsic leather. If we assume for a moment ρ leather = 10000 Ωm, we can determine the parameter β to be given by β = 8.7108. Note that higher values of ρ leather also provide good fittings with the measurements.

Conductivity plot
Equations (2) and (3) suggest that it might be more obvious to plot conductivity σ as a function of absolute humidity H. We should then expect a straight line in a linear plot as shown in Figure 4. A linear fitting has been performed using the least squares approximation. The result is: As in the previous section, an extrapolation towards H → 0 should inform us about the intrinsic conductivity of the leather. However, the fitting (7) gives us a negative number, which physically makes no sense. However, a closer view of Figure 4 reveals that the negative number in (7) is caused by inevitable errors due to the fitting procedure. We can also interpret these results as the zero conductivity of completely dry leather. Nevertheless, the conductivity plot is not suitable for obtaining a limiting value of leather conductivity.
For higher values of H, we can omit the negative number so that: from which we get easily: which is very close to the approximation 1148/H found in (6). The conclusion is thus obvious: the conductivity plot is suitable for finding a simple relation such as (7), but we cannot find an upper limit for the conductivity of pure intrinsic leather. Using the resistivity plot presented in the previous section, it was possible to find a lower limit for leather resistivity of 10000 Ωm, which corresponds to a conductivity of 0.0001 S/m.

Discussion
In order to set up a mathematical model, a closer look must be taken at the experimental results shown in Figure 1. If we compare the model outlined so far, it is a highly simplified version of the real situation. In practice, not all conducting channels will connect two electrodes by a straight line. Some channels will be wider, and thus able to absorb more water from the surrounding air. We assume, for example, that half of the channels have a Equation (3) still remains valid. The difference is that coefficient β is now given by the equation: This proves that the theory outlined above remains valid, and only the numerical value of β must be adjusted.
Obviously, the actual situation is much more complicated than channels with lengths of a and 2a. In practice, channels are randomly distributed, may have different lengths and sections, and are oriented in all possible directions. Only channels connecting two electrodes can contribute to detectable resistivity. Even when probability distributions are introduced for channels lengths as the cross-sections in equation (2), we still arrive at the same result (3), except that the expression for coefficient β will be much more complicated than (4). The only assumption that may not be changed is: σ channel = α H. In other words: electric conductivity σ channel is proportional to absolute humidity H, and remains the basic assumption of our model.

Conclusion
A theoretical investigation of the electric conductivity of leather samples as a function of the absolute humidity is presented in this paper. In order to set up a mathematical model, a closer look must be taken at the experimental results. A mathematical model was presented to explain the electric conductivity of leather samples as a function of absolute humidity. It was also made clear that there is Figure 4: Plot of conductivity σ as a function of absolute humidity H no direct correlation between resistivity and relative humidity. Moreover, it was possible to find a lower limit for the resistivity of intrinsic leather samples. It is clear that the same approach can be used to model the resistivity of other solid materials with channels that can absorb water from ambient humid air. Alternatively, such a method can also be used to indicate the porosity of a material.