1 Introduction

Recently, electro-conductive woven and knitted fabrics have received attention due to their potential applications as electronic components such as transmission lines having the form of conductors connecting individual electronic systems [1], textile antennas being the components to be integrated in electronic systems implemented in smart garments [2], or textile sensors for monitoring human physiological parameters [3, 4]. Most textile-based sensors rely on the change in resistance. The relatively wide range of resistance values is needed for samples intended for textile-based sensor such as strain sensors [3] or temperature sensors [4]. Fixed values at a certain level are required for flat textile products intended for the manufacture of medical electrodes [5], electromagnetic interference shielding textiles [6] or textiles conveying electrical signals [1, 2]. It means that the product is stable in working conditions. Good electrical properties can be obtained applying electro-conductive thin layers by printing [7], magnetron sputtering [8] or electroless plating [9] on surface of an insulative textile substrate. Applications of textile materials depending on their surface resistance are shown in Fig. 1. The insulative textile materials have also a wide range of applications. They can provide protection against heat and fire or against cold. The materials are used for industrial filtration, geotechnical engineering or sustainable energy protection [10]. However, they are not the subject of interests in the paper.

Fig. 1
figure 1

Possible applications of flat textile materials

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Electrical conductivity of flat textile materials results from the electrical conductivity of their components i.e. fibers and threads [11,12,13,14]. It is obvious that electrical conductivity depends on textile material structure. Generally, electro-conductive woven and knitted fabrics can be compared to metal-dielectric composites [14, 15]. Such structures are composed of interlaced conductive threads and pores filled with dielectric air. There are contact points and contact surface between the interlaced threads. Thus, electrical resistivity of woven or knitted structure is dependent on electrical resistivity of linear components, contact resistance of the components and volume fraction of pores. McLachlan model can be used to predict electrical resistivity of fabrics [14, 15]. Contact resistance between two electro-conductive linear components (for example, threads or textile strips) depends on their surface roughness. Two smooth surfaces will be more adhering to each other. For this reason, a lower contact resistance will be expected. Rough surfaces will make the flow of current on the contact surface difficult. Therefore, assessment of fabric surface roughness is important from the point of view of its electro-conductive properties. Textile strips can be used to design woven pressure sensor matrix [16].

Most of textile materials show anisotropic electrical resistance [15, 17, 18]. In order to determine the resistance of this kind of materials and calculate biaxial anisotropy coefficient, most authors apply van der Pauw (vdP) method to solve different problems [15, 17,18,19,20]. Van der Pauw [21] developed the method for homogenous isotropic thin uniform and arbitrarily-shaped semiconductor material in order to assess its resistivity. The resistivity ρ of such sample is as follows:

$$\exp \left( { - \pi h\frac{{{R_1}}}{\rho }} \right)+\exp \left( { - \pi h\frac{{{R_2}}}{\rho }} \right) - 1=0$$
(1)

where h—the sample thickness; R1 and R2—the resistances in the mutually perpendicular directions of the two-dimensional sample plane.

The vdP Eq. (1) can’t be used for textile materials that exhibit anisotropic electrical resistance. Circular and rectangular samples are usually taken into account in studies [22,23,24]. Wasscher [22] modified the equation for the case of anisotropic samples.

In this paper a new approach to conductivity assessment based on the electrical resistivity determination in combination with analysis of surface roughness of textile material has been shown. Electro-conductive woven and knitted fabrics were taken into consideration due to their different structure. Electro-conductive properties of the fabrics were analyzed based on the vdP equation modified by Wasscher to the case of samples with anisotropic resistivity. The modified equations were used to determine electrical resistivity of circle-shaped woven fabrics and square-shaped knitted fabrics. Analysis of textile sample surface roughness on its electro-conductive properties was presented. It is important from the point of designing woven structure composed of strips cut out from electro-conductive fabrics where contact resistance occurs between textile strips.

2 Materials

In the research electro-conductive woven and knitted fabrics were the subject of interest. The two groups of flat textile products varied in terms of their structure have been selected to show the anisotropy of their electro-conductive properties and the surface roughness. Microscopic images of the fabrics and basic information given by producers are presented in Table 1.

Table 1 Raw material composition of fabrics
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Structural parameters of the textile materials are presented in Tables 2 and 3.

Table 2 Structural parameters of woven fabrics
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Table 3 Structural parameters of knitted fabrics
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Additionally, parameter D was calculated as a quotient of higher and lower densities and it was used for future analysis.

Next to the estimate of chosen parameter (see Tables 2, 3) the relative expanded uncertainty is given. It was calculated according to the guide [25] and assuming confidence level equal to 0.95.

It is assumed that all input quantities xi needed to determine the output quantity y are independent. Then the combined variance is as follows:

$$u_{C}^{2}(y)=\sum\limits_{{i=1}}^{N} {\left[ {{{\left( {\frac{{\partial f}}{{\partial {x_i}}}} \right)}^2}\left( {{u_A}^{2}({x_i})+{u_B}^{2}({x_i})} \right)} \right]}$$
(2)

where y—the estimate of output quantity; uA(xi)—the type A standard uncertainty; uB(xi)—the type B standard uncertainty; N—the number of input quantities xi (i = 1, 2, …, N); y—the estimate of output quantity; f—the measurement model.

The type A standard uncertainty estimated from ni independent repeated observations xi is as follows:

$${u_A}({x_i})=\sqrt {\frac{{\sum\limits_{{k=1}}^{{ni}} {{{\left( {{x_{ik}} - \overline {{{x_i}}} } \right)}^2}} }}{{{n_i}({n_i} - 1)}}}$$
(3)

where \(\overline {{{x_i}}}\)—the estimate of xi; xik—the k-th observation of xi.

The type B standard uncertainty is evaluated by scientific judgement based on all of the available information on the possible variability of input quantity. Assuming a rectangular distribution of possible values, the Type B uncertainty can be determined from the following formula:

$${u_B}({x_i})=\frac{{{d_e}}}{{\sqrt 3 }}$$
(4)

where de—the resolution of measuring instrument.

The expanded uncertainty is expressed as follows:

$$U={k_p}{u_C}(y)$$
(5)

where kp—the coverage factor; for confidence level equal to 0.95, the coverage factor equals 2.

The relative expanded uncertainty was used as a measure of the inaccuracy of the chosen parameters of woven and knitted fabrics given as:

$${U_{rel}}=\frac{U}{y}100\%$$
(6)

The uncertainty budgets are presented in Tables 4, 5, 6, 7, 8, 9 and 10.

Table 4 Uncertainty budget of fabrics thickness determination
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Table 5 Uncertainty budget of woven fabrics warp density determination
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Table 6 Uncertainty budget of woven fabrics weft density determination
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Table 7 Uncertainty budget of knitted fabrics course density determination
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Table 8 Uncertainty budget of knitted fabrics wale density determination
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Table 9 Uncertainty budget of fabrics aerial density determination
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Table 10 Uncertainty budget of fabrics apparent density determination
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3 Methods

Circle-shaped samples with diameter of 100 mm were prepared from the electro-conductive woven fabrics and square-shaped samples with side of 100 mm were prepared from the electro-conductive knitted fabrics. In order to determine the surface resistivity of the fabrics vdP method was applied. Four sufficiently small electrodes (A, B, C, D) were arranged at sample in the shape of a square with a side of 60 mm (Fig. 2).

Fig. 2
figure 2

a Electrodes arrangement for a circle-shaped sample, b electrodes arrangement for a square-shaped sample

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Line connected electrodes A and B (or D and C) was parallel to the warp or course direction (Fig. 2a). Line connected electrodes A and D (or B and C) was parallel to the weft or wale direction (Fig. 2b). The measurement conditions were in accordance with the standard [26]. Resistance R11 was obtained when direct current IDC was fed through the electrodes D and C and the voltage drop UAB between electrodes A and B was measured (variant 1). Resistance R12 was obtained when direct current IAB was fed through the electrodes A and B and the voltage drop UDC between electrodes D and C was measured (variant 3). Resistance R21 was obtained when direct current IBC was fed through the electrodes B and C and the voltage drop UAD between electrodes A and D was measured (variant 2). Resistance R22 was obtained when direct current IAD was fed through the electrodes A and D and the voltage drop UBC between electrodes B and C was measured (variant 4).

Based on analysis of sample geometry and electrodes arrangement it was found the distance between single electrode and sample edge. The distances are equal to dc = 7.6 mm and dr = 28.3 mm in case of circle-shaped and square-shaped samples respectively (see Fig. 2). To predict resistances on the sample edges additionally electrodes were arranged in the shape of a square with a side of 40 mm and next with a side of 20 mm. The mean resistances R1 and R2 were determined for all electrodes arrangements based on following dependences:

$${R_1}=0.5\left( {{R_{11}}+{R_{12}}} \right)$$
(7)
$${R_2}=0.5\left( {{R_{21}}+{R_{22}}} \right)$$
(8)

Based on results of measurements and using procedure proposed by Tokarska and Orpel [27] the resistances R1 and R2 on sample edges were predicted in order to determine the resistivity ρ and surface resistance Rs for the electro-conductive fabrics.

Based on Wasscher’s extension of the method of vdP to the case of circular sample the resistivity ρ is given by relationship [22, 23]:

$$\rho =\pi h\sqrt {\frac{{{R_1}{R_2}}}{{\ln \left( {\frac{2}{{1 - k}}} \right)\ln \left( {\frac{2}{{1+k}}} \right)}}}$$
(9)

wherein for R1 ≥ R2

$$\frac{{{R_1}}}{{{R_2}}}=\frac{{\ln \left( {\frac{{1 - k}}{2}} \right)}}{{\ln \left( {\frac{{1+k}}{2}} \right)}}$$
(10)

while the resistivity for the rectangular sample is given by relationship [22, 24]:

$$\rho =\frac{{\pi h}}{2}\sqrt {\frac{{{R_1}{R_2}}}{{\ln \left( {\frac{{1+k}}{{1 - k}}} \right)\ln \left( {\frac{{1+k}}{{2\sqrt k }}} \right)}}}$$
(11)

wherein for R1 ≥ R2

$$\frac{{{R_1}}}{{{R_2}}}=\frac{{\ln \left( {\frac{{1+k}}{{1 - k}}} \right)}}{{\ln \left( {\frac{{1+k}}{{2\sqrt k }}} \right)}}$$
(12)

where k—the modulus of the complete elliptic integral of the first kind and

$$k=f\left( {{R_1},{R_2}} \right)\,\,\,\forall 0<k<1$$
(13)

Surface resistance Rs is given by the following formula:

$${R_s}=\rho /h$$
(14)

where h—the sample thickness; ρ—the resistivity of anisotropic sample.

4 Results and discussion

Results of resistance measurements for different electrode distances from the sample edge corresponding to variants 1–4 are shown in Figs. 3, 4, 5, 6, 7 and 8.

Fig. 3
figure 3

Results of resistance measurements for fabric WF-1

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Fig. 4
figure 4

Results of resistance measurements for fabric WF-2

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Fig. 5
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Results of resistance measurements for fabric WF-3

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Fig. 6
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Results of resistance measurements for fabric KF-1

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Fig. 7
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Results of resistance measurements for fabric KF-2

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Fig. 8
figure 8

Results of resistance measurements for fabric KF-3

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Distances d1, d2, d3 and d4 (Figs. 3, 4, 5) between electrode and the nearest circle-shaped fabric edge (see Fig. 2a) were determined and are: d1 = 35.9 mm, d2 = 21.7 mm, d3 = 7.6 mm = dc, d4 = 0.0 mm.

Distances d1, d2, d3, d4 and d5 between electrode and the nearest square-shaped fabric corner (see Fig. 2b) were determined and are: d1 = 56.6 mm, d2 = 42.4 mm, d3 = 28.3 mm = dr, d4 = 14.1 mm, d5 = 0.0 mm.

In variants 1 and 3, the voltage and current electrodes have been changed. Resistance values are comparable, because in both cases the current flows along warp direction in the case of woven fabric and along course direction in the case of knitted fabric. In variants 2 and 4, the voltage and current electrodes have also been changed. Resistance values are comparable, because in both cases the current flows along weft direction in the case of woven fabric and along wale direction in the case of knitted fabric.

The resistances R1 and R2, the modulus k determined from Eqs. (10) and (12) and the resistivity ρ calculated from Eqs. (9) and (11) for circular sample and square-shaped sample respectively, the surface resistance Rs calculated from Eqs. (14) and the coefficient R/R are presented in Table 11. The R/R is a biaxial anisotropy coefficient and is calculated as a quotient of higher and lower of resistances R1 and R2.

Table 11 Results of textile materials research
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The values of resistivity obtained for woven and knitted fabrics indicated that the textile materials have electrical resistivity similar to metals or semiconductors. It was found that all fabrics show electrical anisotropy with anisotropy coefficient from 1.1 to 7.2. Influence of warp and weft densities for woven fabrics and course and wale densities for knitted fabrics are presented in Fig. 9.

Fig. 9
figure 9

a Structure analysis for woven fabrics, b structure analysis for knitted fabrics

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Values of resistances R1 and R2 are dependent on principal axes of textile materials. The main axes in woven fabric were defined by warp and weft directions. The main axes in knitted fabric were defined by course and wale directions. The resistance R1 was determined when current flows between electrodes arranged on the line parallel to the warp or course direction. It was found that the resistance R1 is lower than the resistance R2 where current flows parallel to the weft or wale direction. The higher warp density means occurrence of more contact resistances between warp and weft threads and therefore an increase in woven fabrics resistances (samples WF-1 and WF-2). In case of sample WF-3 it is not so obvious. The comparable values of resistivity R1 and R2 (the biaxial anisotropy coefficient R/R was equal to 1.1) may result from a fabric weave other than a plain weave observed in WF-1 and WF-2. The higher course density means occurrence of more contact points of the same thread resulting in occurrence of contact resistances and therefore an increase in all knitted fabrics resistances.

In order to more accurately understand the impact of sample surface roughness on resistances R1 and R2, and surface resistance Rs a line profile was identified to obtain the basic structural characteristics of fabrics. Two lines of profile were received: the horizontal and the vertical ones (Tables 12, 13) based on TIF file format of images converted into grayscale. Mean file size was 1957 KB (the coefficient of variation was equal to 3%). Mean resolution of all digital images expressed by pixel count was 995684 pixels (the coefficient of variation was equal to 2%). Line profile was routed through the center of the warp/weft and parallel to the warp/weft direction for woven fabric and through the center of the course/wale and parallel to the course/wale direction for knitted fabric. Measurements were repeated three times and mean values were calculated.

Table 12 Surface roughness characteristics for woven fabrics
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Table 13 Surface roughness characteristics for knitted fabrics
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Some important parameters allowing evaluation of sample surface roughness were determined based on received characteristics. Surface area S under the line profile f(l) was calculated using following formula:

$$S=\int\limits_{{lo}}^{{lm}} {f(l)dl} \quad {l_o} \le l \le l{}_{m}$$
(15)

where l—the distance; lo—the initial distance lo = 0 px; lm—the final distance.

In formula (15) lm = 387 px was assumed. Surface areas Sh and Sv were calculated corresponding to horizontal and vertical lines respectively using (15). Next spectral analysis was carried out using Statistica® to determine the occurrence of periodicity in the characteristics. The largest peak was identified in periodogram and the corresponding surface roughness frequency fh and period Th for the horizontal line and fv and period Tv for the vertical line. All calculated parameters are given in Tables 12 and 13.

Pearson’s correlation coefficient was used in statistics to measure how strong a relationship is between two parameters characterizing features of textile materials. Pairs of chosen parameters and the corresponding significant values of Pearson’s correlation coefficient (0.05 significance level was assumed) are given in Table 14.

Table 14 Impact of surface roughness parameters on sample electro-conductive properties
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It was found strong downhill linear relationship between the surface area Sh or Sv and the surface resistance Rs. The surface resistance decreases when the surface area increases. The bigger surface area under the function f(l) being the line profile of the sample means smaller oscillation of the function f(l) and consequently smaller surface resistance. Thus, sample with a smooth surface will conduct electrical current better than sample with a rough one. It was noticed that the greater the disproportion between the densities (expressed by D) the greater the coefficient R/R. Thus, the textile materials are characterized by higher anisotropy of electro-conductive properties.

Strong uphill linear relationship was observed between the coefficient R/R and the surface roughness frequency fh. Therefore, if frequency fh corresponding to the largest peak of periodogram obtained for the horizontal line increases the coefficient R/R also increases. Strong uphill linear relationship was also observed between the above coefficient and the surface roughness frequency fv corresponding to the largest peak of periodogram obtained for the vertical line. Results of the conducted analysis confirmed the effect of sample surface roughness on its electro-conductive properties.

5 Conclusions

The main research results of this work are as follows:

  1. (a)

    The study shows that the biaxial anisotropy coefficient and the electrical resistivity are very useful in characterizing electrical anisotropy of textile materials. The biaxial anisotropy coefficient indicates whether material property depends on its testing direction with respect to the principal axes.

  2. (b)

    Resistance was higher when current flows between electrodes arranged on the line parallel to the warp and course direction in the case of woven or knitted fabrics respectively.

  3. (c)

    The value of electrical resistivity makes it possible to qualify textile material in terms of electrical conductivity. The electrical resistivity of the fabrics ranged from 1.75 × 10−4 Ω cm to 1.42 × 10−1 Ω cm. It indicates that they have properties similar to metals.

  4. (d)

    Results of analysis of fabrics surface roughness enable the first selection of fabrics from the point of view of its electro-conductive properties. It is also possible to initially conclude on the contact resistance between two linear components in a form of woven or knitted strips intended to design a matrix of pressure sensor.

  5. (e)

    The woven fabrics can be used as paths for conveying electrical signals such as textile transmission lines. Moreover they can be used as medical electrodes or e-textile antennas where low resistivity is desired.