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3D modeling of transport properties on the surface of a textronic structure produced using a physical vapor deposition process

  • Mariusz Mączka EMAIL logo , Ewa Korzeniewska EMAIL logo , Marcin Lebioda and Stanisław Pawłowski
Published/Copyright: May 12, 2025
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AUTEX Research Journal
From the journal AUTEX Research Journal Volume 25 Issue 1

Abstract

This study presents a numerical model designed to simulate the transport properties of textronic structures produced by physical deposition from the gas phase. For the numerical implementation of the model, the method of fundamental solutions was used, which was implemented using the author’s iterative algorithm. Such an approach is rarely used due to the difficulty of obtaining accurate solutions with optimal computational cost. The developed algorithm ensures good convergence of solutions (δ k ≈ 5%) with acceptable computational time (t ≈ 180 s for 17 × 103 iterations). The software tool has options for calculating the electric field and density of current distributions for the given system power conditions. The ability to define a conductive path in three geometric dimensions makes it possible to study the effect of changes in the surface geometry on the resistance of the textronic structure. The accuracy of the model was verified by measuring the resistance of selected samples of simulated materials. The measurements confirmed the conclusions of the simulations about the clear dependence of the resistivity of the structure on the roughness of the conducting surface. The range of the largest changes in resistivity as a function of the surface roughness of the conducting path was also determined.

Keywords: Textronics; wearable electronics; method of fundamental solutions; iterative algorithm; thin films

1 Introduction

Wearable electronics is one of the fast-growing industries that deals with the design, manufacture, and application of electronic devices that can be worn on the body or built into clothes. Wearable electronics have many potential applications, such as health monitoring, improving sports performance, facilitating communication, entertainment, and fashion. It is also linked to other fields, such as the Internet of Things, artificial intelligence, nanotechnology, and biotechnology [1,2,3,4,5,6,7,8]. According to a report by IDTechEx, the wearable electronics market is expected to reach USD 150 billion by 2030 [9].

Examples of wearable electronics include smart watches, glasses, jewelry, fitness bands, implants, and sensors for monitoring vital functions. In addition to the miniaturization of electronic devices, the aim is to combine electronic circuits with textiles. One of the issues related to such a connection is the selection of the textile substrate and the method of manufacturing elements with given electrical parameters. There are many methods of producing thin electroconductive layers on textiles. Thin films can be created through various techniques, depending on the specific requirements and desired properties. Methods for producing such structures include the method of physical vapor deposition (PVD) [10,11], chemical vapor deposition (CVD) [12,13], electrochemical deposition [14,15], spin coating [16,17], sol–gel process [18,19], ink-jet printing [20,21], and embroidery with conductive threads [22,23]. PVD is a technique that involves the evaporation or sputtering of a material onto a substrate in a vacuum chamber. Heat can be used to vaporize the material, which then condenses on the substrate to form a film. In CVD, a chemical reaction occurs between precursor gases and a substrate, resulting in the deposition of a thin film. The reaction can be initiated by heating the precursors or by using plasma or laser energy. Electrochemical deposition is also known as electroplating and involves the use of an electric current to deposit a thin film onto a substrate. The film is formed through the reduction of metal ions present in a solution onto the electrode. In the spin coating technique, a liquid precursor or solution is deposited onto a spinning substrate. The centrifugal force causes the liquid to spread evenly and form a thin film as it evaporates or solidifies. The sol–gel process involves the hydrolysis and polycondensation of precursor molecules in a liquid solution. The resulting gel is then heated or dried to form a solid thin film. Inkjet printing is the method of creating a thin film in the printing process using ink with electroconductive particles. During the creation of thin films, defects in the structure can occur. They have a significant role in the resistance of such layers [23]. The choice of the method depends on factors such as the material being deposited, the desired thickness and uniformity, the specific application requirements, and the type of substrate on which the layer is created.

In the case of using textile substrates for electroconductive layers, the three-dimensional structure of the fabric must be taken into account (Figure 1).

Figure 1 Cross-sections of the composite fabric used for the creation of textronic structures.
Figure 1

Cross-sections of the composite fabric used for the creation of textronic structures.

Due to the uneven nature of the surface of such a substrate, in order to ensure the continuity of the electroconductive structures, a buffer layer is additionally used. This layer is placed between the fibers constituting the fabric and the electroconductive structures produced in the process of physical vacuum deposition. The buffer layer may be a film of Teflon, acrylic, or polyurethane applied to the fibers at the stage of textile composite production. One of the textile composite substrates with high mechanical strength is Cordura. This material is widely used in sports and tourist clothing due to its very high mechanical strength. It is made of polyamide with a polyurethane coating. Because it has a twill weave, the surface of the created fabric is not very plain [24,25]. The layer of polyurethane applied on the surface of the braided fibers takes the shape of the surface of their weave, causing deviations from the flat surface. The applied metal vapors in the PVD technology are distributed stochastically over the entire surface, creating a layer that reflects the shape of the surface of the composite substrate. For this reason, the shape of the produced layer may depend on the density of the weave, which may be reflected in the surface resistance of the produced electroconductive layer, as well as the current density distribution in the structures. The aim of this study is to determine the influence of the surface structure on the surface resistance of the layers created in the PVD process, which is important in the design of wearable electronics components. This is especially important in the process of creating sensory structures, where even a small change can affect the output values. To achieve the purpose of this work, a numerical model of the textronic structure was created. It allows simulations of the distribution of static fields inside the conductive path for given power supply conditions. A detailed description of this model, together with the simulation results, is presented in this study.

2 Formulation of the problem

A representative illustration of the created model of the textronic structure is shown in Figure 2. Its main advantage is the ability to reflect the geometry of the conductive layer produced with the PVD method. As can be seen in Figure 2(a), the conductive area (Ω) is defined in three geometric dimensions (x, y, z coordinate system) and limited by six boundary surfaces (S1–S6). The corrugated lower (S5) and upper (S6) surfaces express the effect of stochastic deposition of conductive particles on the fabric fibers. Figure 2b, which is a cross-section of the conductive path in the Cr plane, shows its geometry in the direction of current flow I (0 x-axis), created after applying a constant voltage U between the flat edge surfaces S1 and S2 with the distance d x between them. The size of the conductive path in the direction perpendicular to the current flow (0 y-axis) is limited by the flat surfaces S3 and S4, with the mutual distance described as the d y parameter. The thickness of the conductive path is determined by the d z parameter, which is also the mutual distance between the S5 and S6 surfaces.

Figure 2 Simplified concept of the 3D textronic structure model (3D TSM): boundary surfaces and voltage polarization system (a), view of the model in a cross-section with plane Cr (b), and top view of the model (c).
Figure 2

Simplified concept of the 3D textronic structure model (3D TSM): boundary surfaces and voltage polarization system (a), view of the model in a cross-section with plane Cr (b), and top view of the model (c).

The mathematical description of the conductive area Ω can be presented in the form of the following relationships:

(1) f ( Ω ) : 0 ≤ x ≤ d x 0 ≤ y ≤ d y A 2 ( 1 + sin k x x sin k y y ) ≤ z ≤ A 1 ( 1 + sin k x x sin k y y ) + d z .

As can be seen, there is a possibility of modeling surface unevenness by changing the amplitudes (A 1 and A 2) of the functions, which shape the peripheral surfaces S5 and S6. It may correspond to different thicknesses of the fabric fiber on which the conductive layer is deposited. At the same time, the distances between the fibers can be represented by the periods (T x = 2π/k x and T y = 2π/k y ) of these functions, which also has a significant impact on the geometry of the conductive path.

Assuming that the area Ω is filled with a homogeneous, isotropic and linear conductor (γ = const.), in which there are no unbalanced electric charges (ρ = 0), its surroundings are an ideal dielectric (γ ο = 0), and there is a constant voltage U between surfaces S1 and S2, the following formula can be written:

(2) E → = − grad → ϕ ,

where E → is the electric field strength and φ is the potential satisfying the Laplace equation:

(3) Δ ϕ = 0 ,

and mixed boundary conditions:

(4) ϕ = U ,

on the S1 surface

(5) ϕ = 0 ,

on the S2 surface

(6) ∂ ϕ ∂ n = 0 ,

on the S surface (i = 3, 4, 5, 6).

After solving the problem formulated above, using Ohm’s local law and expression (2), the current density distribution J → can be determined according to the following relationship:

(7) J → = γ E → ,

and then the total current I of the simulated structure:

(8) I = ∬ S J → ⋅ d s → ,

where the surface S is the cross-section of the conducting path in the direction perpendicular to the current flow. The static resistance of the tested sample is determined by Ohm’s basic law:

(9) R = U I .

A possible change in resistance caused by various factors (e.g., change in the thickness of the conductive path or the formation of unevenness on the sample surface) was the main goal of numerical research, which will be presented later in this work.

3 Solution method

To solve the problem described in the previous section, the iterative method of fundamental solutions [26,27,28] was used. It involves the iterative calculation of subsequent approximations of the function, which is being looked for, which in our case is described by the following relationship:

(10) ϕ ∼ k ( r → ) = ϕ ∼ k − 1 ( r → ) + ∑ n = 1 L f q k , n F k , n ( r → ) ,

where k is the number of the next iteration, r → = ( x , y , z ) is any point in the area Ω, q k,n is the n-dimensional vector of approximation sum coefficients for the next iteration k calculated on the basis of boundary conditions, and L f determines the number of fundamental solutions in a single iteration step. The function defining the fundamental solutions of Laplace’s equation has the following form:

(11) F k , n ( r → ) = 1 ∣ r → − r → k , n ∣ ,

where r → k , n represents the singular point of the function F k , n ( r → ) lying outside the area Ω. According to the assumptions of the method, in each iteration step, which is randomly selected from among many points generated at the beginning of the procedure, evenly surrounding the area Ω.

A linear function was used to initialize the search for solutions of type (10):

(12) ϕ ∼ 0 ( r → ) = U 1 − x d x ,

which satisfies equation (3) and boundary conditions (4), (5). The used approach requires determining the values of the coefficients q k,n in such a way as to obtain the best possible corrections to the fulfillment of the boundary conditions by function (10) in a given iteration step. The measure of this fulfillment is assessed on the basis of the value of the pseudonorm:

(13) δ k = ∑ i = 1 6 1 L i ∑ j = 1 L i ε i , k 2 ( r → j ) ,

using relative marginal errors in the following form:

(14) ε 1 , k ( r → ) = 1 U ( U − ϕ ∼ k ( r → ) ) , r → ∈ S 1 ,

(15) ε 2 , k ( r → ) = 1 U ϕ ∼ k ( r → ) , r → ∈ S 2 ,

(16) ε i , k ( r → ) = 1 E 0 ∂ ϕ ∼ k ∂ n , r → ∈ S i , i = 3 , … , 6 ,

where E 0 = U d x is the strength of the primary (initial) electric field. The process of searching for the smallest value of the above functional (13) consists of its minimization with respect to the set of parameters q k,n, which can be written as

(17) ∂ δ k ∂ q k , m = 0 where m = 1 , … L f .

It leads to a linear system of equations of the form:

(18) A k , m , n q k , n = B k , m where m = 1, … L f ,

where the coefficient matrix is described by the following relationship:

(19) A k , m , n = ∑ i = 1 2 1 L i ∑ j = 1 L i F k , m ( r → j ) F k , n ( r → j ) + d x 2 ∑ i = 3 6 1 L i ∑ j = 1 L i G k , m ( r → j ) G k , n ( r → j ) ,

while the vector of intercepts is determined by the formula:

(20) B k , n = − U ∑ i = 1 2 1 L i ∑ j = 1 L i ε i , k − 1 ( r → j ) F k , n ( r → j ) + d x ∑ i = 3 6 1 L i ∑ j = 1 L i ε i , k − 1 ( r → j ) G k , n ( r → j ) .

The functions in formulas (19) and (20) are defined as

(21) G k , n ( r → ) = ∂ F k , n ∂ n = n ^ ⋅ grad → F k , n ( r → ) ,

where n ^ is the normal term to the surface S i at point r → . It should be noted that the rate of convergence of the method depends on the number L f of fundamental solutions taken into account in each iteration step (formula (10)). This number can be set arbitrarily, starting from L f = 1, but it should be remembered that it is related to the size of the computer’s operating memory, which must accommodate L f (L f + 1) of the coefficients of the system of equations (18). On the other hand, the greater the number L f , the fewer iteration steps are needed to obtain the required accuracy of the procedure and, consequently, fewer fundamental solutions, i.e., fewer terms of the approximation sum in expression (10). It is also significant that the L f value greatly affects the time of a single iteration step. This time is determined by the calculation of the sums in expressions (19) and (20). Taking into account all of the above aspects, numerous numerical experiments were carried out to find the optimal solution. One of the conclusions of these studies is that the adoption of L f = 1 gives the fastest convergence of the procedure in real time because, in this case, it avoids the numerical solution of the system of equations (16), reducing it to an equation with one unknown.

4 Results of simulations

The numerical implementation of the solution described in the previous section consisted of creating a computer program operating in accordance with the algorithm illustrated in Figure 3.

Figure 3 Algorithm of the program that performs the simulations of the TS PVD.
Figure 3

Algorithm of the program that performs the simulations of the TS PVD.

The input parameters introduced to the program are described in the next section devoted to the simulation results. However, the next block of the algorithm in which the P{…} cloud is created, the so-called source points, requires a few words of explanation. In our case, these points can be interpreted as hypothetical electric field charges acting outside the conducting area Ω.

This is illustrated in Figure 4, where the mentioned source points (represented by circles with a plus sign) are defined near the edge of the conductive path. From the entire set of source points P {p 1 … p i }, in each iteration k, a subset P k {p 1…p n } is randomized, which is highlighted in a color other than white in Figure 4. For example, iteration k corresponds to all locations marked in red, iteration k + 1 corresponds to points marked in blue, etc. The geometric coordinates of the randomly selected source points are used to check the boundary conditions of equation (10) at the nodal points marked in the figure as the set R {r 1…r j }. This is preceded by determining the matrix of coefficients A k , m , n (formula (19)) and the vector of intercepts B k , n (formula (20)), and then the system of equations (18) is solved.

Figure 4 An illustrative presentation of the creation of a cloud of source points and the definition of nodal (boundary) points of a conductive area Ω.
Figure 4

An illustrative presentation of the creation of a cloud of source points and the definition of nodal (boundary) points of a conductive area Ω.

The size n of the P k vector is set depending on the assumed calculation accuracy and the expected simulation time. The optimization of the two above-mentioned parameters is the subject of research and will be described in a separate article. Preliminary numerical tests have shown that one of the possible options is to minimize the size of the mentioned vector to the value n = 1, which maximally limits the time of calculating the sums of Equation (10) at the expense of increasing the number of iterations k needed to achieve the assumed error δ max. As a result of this approach, by running simulations of a sample containing two periods (T = 5) of function (1) on a standard PC machine with an Intel Core 7 processor under Windows 10, it is possible to obtain simulation results even after several thousand iterations (k ≈ 17 × 103) at an acceptable level error (δ k ≈ 5%) and all these in a few minutes (t ≈ 180 s). However, it should be noted that the solutions of the obtained equation with the used method belong to the weak category. This means that the sequence of subsequent values of δ k does not converge to zero. As a consequence, there is a risk of setting too high a solution accuracy in the program (too low δ max value), which may result in a lack of convergence of the calculation procedure. For this reason, the program takes into account an additional condition for implementing the iteration loop related to their maximum number (k < k max). After agreeing on the solution to equation (3), the program moves to the stage of determining the distributions of given physical quantities. The first step in this direction is to determine the space in three geometric dimensions for which the process of calculating the value of the E field will take place (according to formula (3)) and the preceding process of calculating the distribution φ according to the following relationship:

(22) ϕ ( r → ) = ϕ 0 ( r → ) + ∑ k = 1 L i ∑ n = 1 L f q k , n F k , n ( r → ) ,

where L i is the number of iterations for which the convergence of the procedure for agreeing the solution of equation (3) was achieved. It is worth noting that the coordinates of the calculation points, along with the values of the indicated parameters, are stored in the program’s operating memory as multi-dimensional arrays of double-precision floating-point numbers. Therefore, it is important to skillfully determine the simulation areas in order to prevent computer memory deficits.

The average total current I av of the conductive path calculated according to equation (8) is the arithmetic mean of 100 evenly distributed cross-sections of the area Ω in the direction perpendicular to the transport of charge carriers. The static resistance of the structure is determined based on Ohm’s law (formula (9)).

5 Results of simulations

Using the created program, a number of simulations of TS-PVD structures (textronic structures created in a physical vacuum deposition process) were performed, the initial goal of which was to check the correctness and effectiveness of the calculations.

5.1 Potential distribution

The starting point in this matter was the analysis of potential distributions in the tested structure, which was carried out based on the simulation results shown in Figure 5.

Figure 5 Potential distributions in the conductive path of TS PVD. Illustrations (a), (e), and (i) are maps prepared for the upper surfaces of the Ω region for cases where the amplitudes of the conductive path roughness are A = 0.01, 1, and 0.5, respectively. The diagrams in the right column are maps for sections marked in the figures in the left column: (b–d) correspond to the sections indicated in illustration (a); (f–h) are the sections indicated in figure (e); (j–l) correspond to the sections indicated in image (i).
Figure 5

Potential distributions in the conductive path of TS PVD. Illustrations (a), (e), and (i) are maps prepared for the upper surfaces of the Ω region for cases where the amplitudes of the conductive path roughness are A = 0.01, 1, and 0.5, respectively. The diagrams in the right column are maps for sections marked in the figures in the left column: (b–d) correspond to the sections indicated in illustration (a); (f–h) are the sections indicated in figure (e); (j–l) correspond to the sections indicated in image (i).

The illustrations in the left column of this figure present the potential distributions on the upper surface of the structure, respectively, for cases when the conductive layer is a smooth surface (a) and when there are irregularities of larger (e) and smaller (i) amplitudes. The change in amplitude is correlated with the change in the periods of unevenness, which, according to the authors’ opinion, corresponds to the change in the thickness and density of the fabric matrix on which the conductive path was deposited. The values of the most important parameters describing the geometry of the tested structure are given in Table 1. It is worth noting that each of them is defined in relation to the dimension d z = 1, determining the thickness of the conductive path.

Table 1

Basic parameters of Sym. 0, Sym. 2, and Sym. 4

Geometry parameters Convergence parameters
Sym. no. A 1 = A 2 = A T x = T y = T d x = d y d z δ k k max T s
Arbitrary units [d z] [%] [s]
Sym. 0 0.01 5 10 1 4.6 17,470 180
Sym. 2 1 4.6 22,011 226
Sym. 4 0.5 2.5 5.3 46,878 480

For example, an amplitude of 1 means that it is equal to the thickness of the conductive layer, just as the value d x = 10 determines the size of the TS model in the direction of the 0 x-axis at the level of ten times the thickness of the area Ω. This approach facilitates the analysis of changes in the conductive properties of the tested structure without the need to refer to absolute units and material parameters. Potential distribution maps were also prepared using a similar convention, where the percentage ratio of the current potential value at a given point in the structure (φ) to its maximum value (φ p ) applied to the S 1 surface is plotted.

First, the right column of the maps shown in Figure 5 presents potential distribution in selected x–z sections, i.e., along the axis of the applied voltage U. Additionally, Figure 5(b) shows an example of potential distribution in the y–z section (Figure 5(c)), so in the direction perpendicular to the direction of current flow. In general, the places of the presented cross-sections have been selected to show the diversity of the geometry of the conductive area, which changes periodically from flat surfaces (diagrams (g), (k)) to wavy surfaces with maximum amplitudes equal to 1 (diagrams (f), (h)) and 0.5 (graphs (j), (l)) for periods 5 and 2.5 of function f (Ω), respectively.

Analyzing all of the above results, it can be concluded that changes in the non-uniformity of the conductive layer have a significant impact on the potential distribution in the area of the tested textronic structure. This distribution changes from a linear form (a), characteristic of a completely flat conductive path, to an increasingly complex one, where a certain dependence of these changes on the period of the function f (Ω) describing the non-uniformity of TS PVD can be noticed. This is best seen in graphs (e) and (i), where the reduction of the period in the mentioned function by half (Table 1) results in increasing the frequency of changes in the potential distribution on the S6 surface. Cross-sectional maps in Figure 5 illustrate the process of the above changes inside the conductive path. It can be seen that the potential distribution changes from a linear form (graphs (b) and (d) for Sym. 0) to a slightly more complex one (Sym. 2), with a linear distribution for the unwaved cross-section (g) and non-linear in the cases (f) and (h) and ending with Sym. 4, where all three presented sections show non-linear potential distributions. A separate comment is required here on the map marked as (c), constituting a cross-section of the conductive path in the direction perpendicular to the axis of current flow. It was prepared to check the nature of changes in the potential distribution, which in this direction should be constant for the given supply conditions. However, the results show some small deviations from the above assumption, which may be a consequence of the numerical error (δ k ), which, as shown in Table 1, was 4.6% in this case, which is a value close to the size of the observed changes.

The accuracy and effectiveness of the simulations are described by the parameters in the last three columns of Table 1. Based on the data collected there, it can be concluded that, as expected, the fastest and most accurate calculations took place when a practically flat conductive path was simulated (line Sym. 0). Boundary conditions checked at the nodal points of the model (Figure 4) were met here with an accuracy of δ k = 4.6% after k = 17,470 iterations (the algorithm in Figure 3), carried out in time T S = 180 s using a PC running in the configuration described in the previous section. The introduction of the non-uniformity of the conductive layer with the amplitude A = 1 (line Sym. 2 in Table 1) and the period T = 5 slightly increased the number of iterations (k = 22,011) needed to achieve the same error δ k , which consequently extended the simulation time by more than half a minute (T S = 226 s). However, this time is still two times shorter than the duration of Sym. 3, for which k = 46,878 iterations were not enough to achieve the same accuracy as in Sym. 0 and Sym. 2, and finally, the calculations were finished after reaching the error δ k = 5.3%. In this case, the period of the functions describing the non-uniformity of the conductive path was reduced twice in relation to Sym. 2 (T = 2.5) as well as its amplitude (A = 0.5). All simulations described here covered an area Ω with dimensions of 10 d z in the x–y directions and 1 d z in the z direction of the global coordinate system in which the TS PVD model was defined. The size of the vector P k specifying the number of hypothetical sources of the q field (Figure 4) randomly selected in each iteration k from the set P{…} of 106 elements to take them into account to correct the value of the function (10), was in each case n = 1.

5.2 Field strength distribution

After solving equation (18), the created program proceeds to the next stage of the simulation, in which the distribution of the electric field intensity and the associated current flow are calculated. The results are presented in Figure 6. In order to facilitate the analysis, the charts in Figure 6 have been compiled in a similar convention as in Figure 5. Thus, in the left column, the distribution of the electric field strength (E) is on the upper surface of the conductive path (S6), while the right column contains a number of distributions of the same field plotted for cross-sections marked in the left column graphs. Due to the fact that the results refer to simulations that have already been analyzed in this section (Sym. 0, Sym. 2, Sym. 4), their basic input parameters can be read from Table 1. The starting point for the consideration of the studies presented here is the analysis of the intensity distribution electric field obtained under Sym. 0 (graphs (a)–(d)), which concerned the case of a straight conductive path.

Figure 6 Electric field strength distributions in the conductive path of TS PVD. Illustrations (a), (e), and (i) are maps prepared for the upper surfaces of the Ω region for cases where the amplitudes of the conductive path roughness are A = 0.01, 1, and 0.5, respectively. The diagrams in the right column are maps for sections marked in the drawings of the left column: (b–d) correspond to the sections indicated in illustration (a); (f–h) are the sections indicated in figure (e); (j–l) correspond to the sections indicated in image (i).
Figure 6

Electric field strength distributions in the conductive path of TS PVD. Illustrations (a), (e), and (i) are maps prepared for the upper surfaces of the Ω region for cases where the amplitudes of the conductive path roughness are A = 0.01, 1, and 0.5, respectively. The diagrams in the right column are maps for sections marked in the drawings of the left column: (b–d) correspond to the sections indicated in illustration (a); (f–h) are the sections indicated in figure (e); (j–l) correspond to the sections indicated in image (i).

As can be seen both on the upper surface of the conductive path (diagram (a)) and in all cross-sections of the area Ω (diagrams (b)–(d)), the vector field E has a constant value in terms of modulus and the direction is consistent with the applied voltage U. Based on the previous analysis of the potential distribution for such a case (Figure 5(a)–(d)), it can be considered as expected and, consequently, the correctness of the calculations can be confirmed. The correct electric field intensity distributions are also observed in the cross-sectional diagrams (b) and (d), where the modules and directions of the vectors E representing them are consistent with the results visible in diagram (a).

The graph marked (c) represents a cross-section of the direction of the electric field; hence, the small but still visible modulus values and the different directions of the vector field are within the numerical error range. The introduction of the unevenness of the surface of the conductive path for the same power supply conditions of the system (Sym. 2) resulted in visible changes in the field E, both in terms of modules and vector directions. These changes are periodic, which is confirmed by the results in graphs (e) and (i), where the period of E field changes clearly correlates with the period of the function describing the upper and lower surface of the conductive area (S5 and S6). The analysis of the results presented in the cross-sectional diagrams in Figure 6 leads to the conclusion that the maximum electric field strength occurs in the areas between the hills of unevenness of the conductive surface (diagrams (g) and (k)). A characteristic feature of these areas is their rectangular (non-wavy) cross-section with a width equal to the value of the conductive path thickness (d z ). At the same time, for the remaining (wavy) conductive areas, the maximum values of the vector field E are concentrated in a strip with a width also close to the width of d z (diagrams (f), (h), (j), (l)) regardless of the amplitude of the inequality (compare diagrams (h) and (l) or (f) and (j)).

The non-uniform distribution of the electric field intensity on the surface of the conductive path significantly affects the process of current flow in the structure. This was confirmed by the results of numerical research obtained, e.g., under Sym. 4, where I av decreased by 15% and in the case of Sym. 2 by as much as 23% in relation to the value of I av calculated for a straight conductive path (Sym. 0). The above changes in the current value obviously affect the changes in the resistance of the tested TS PVD sample in the same dimension. This prompted the authors to conduct further research, the purpose of which was to check to what extent the change in the thickness of the conductive path and its undulation determined by different densities of the fabric constituting the structure’s base affect the change in its resistance.

As in previous studies, the results obtained in subsequent simulations were analyzed in relation to the case of a straight conductive path, which is represented in this article by Sym. 0. The basic parameters of the calculations performed in this area are given in Table 2. The first group of simulation parameters visible here (Sym. 1–Sym. 17) illustrates the effect of the density of the fabric (Cordura), which is the substrate for the conductive path, on its resistance. The effect of different Cordura fabric densities was simulated by proportional changes of the amplitudes (A 1 and A 2) and periods (T x and T y ) of the function f ( Ω ) . These changes were carried out in accordance with the scheme in which, starting from the values A = 2 and T = 10, in each subsequent simulation, the parameters mentioned here were reduced twice. With a fixed model size d x = d y = 10, the surface of the conductive path contained from 4 to 512 “hills,” which, as shown in Figure 6, have a significant impact on the current flow through the conductive area of the structure. The direct dimension of this influence can be assessed on the basis of the analysis of the results presented in Figure 7, where the relationship between the R/R 0 ratio and the number of periods of the function modeling the area Ω of the tested sample was plotted. Each point of this graph represents the result of one simulation in which the resistance R per square of the cross-sectional area was determined, and its value was referred to the initial case when the resistance R 0 was calculated for a straight conductive path (Sym. 0).

Table 2

Basic parameters of Sym. 1–Sym. 59

Geometry parameters
Sym. no. A 1 = A 2 = A T x = T y = T d x = d y d z
Arbitrary units [d z]
Sym. 1–16 2– 0.125 10–0.625 10 1
Sym. 17–27 1 5 0.1–1
Sym. 28–38 0.5 2.5
Sym. 39–59 0–2 1
Figure 7 Simulations of changes in resistance (R/R 0) depending on the geometrical parameters of the TSPVD model: (a) changes in resistance caused by an increase in the number of periods T of the function f (Ω) describing the non-uniformity of the conductive path surface with proportional decrease in amplitudes A; (b) and (c) the effect of the thickness of the conductive layer d z on the resistance for the simulations marked in part (a) by blue and pink rectangles, respectively; and (d) changes in resistance caused by a change in the amplitude A of the function f (Ω) with a constant period T = 2.5.
Figure 7

Simulations of changes in resistance (R/R 0) depending on the geometrical parameters of the TSPVD model: (a) changes in resistance caused by an increase in the number of periods T of the function f (Ω) describing the non-uniformity of the conductive path surface with proportional decrease in amplitudes A; (b) and (c) the effect of the thickness of the conductive layer d z on the resistance for the simulations marked in part (a) by blue and pink rectangles, respectively; and (d) changes in resistance caused by a change in the amplitude A of the function f (Ω) with a constant period T = 2.5.

5.3 Study of resistivity changes

The non-uniform distribution of the electric field intensity on the surface of the conductive path significantly affects the process of the current flow in the structure. This was confirmed by the results of numerical research obtained, e.g., under Sym. 4, where I av decreased by 15% and in the case of Sym. 2 by as much as 23% in relation to the value of I av calculated for a straight conductive path (Sym. 0). The above changes in the current value obviously affect the changes in the resistance of the tested TS PVD sample in the same dimension. This prompted the authors to conduct further research, the purpose of which was to check to what extent the change in the thickness of the conductive path and its undulation determined by different densities of the fabric constituting the structure’s base affect the change in its resistance.

As in previous studies, the results obtained in subsequent simulations were analyzed in relation to the case of a straight conductive path, which is represented in this article by Sym. 0. The basic parameters of the calculations performed in this area are given in Table 2.

The first group of simulation parameters visible here (Sym. 1–Sym. 17) illustrates the effect of the density of the fabric (Cordura), which is the substrate for the conductive path, on its resistance. The effect of different Cordura fabric densities was simulated by proportional changes in the amplitudes (A 1 and A 2) and periods (T x and T y ) of the function f ( Ω ) . These changes were carried out in accordance with the scheme in which, starting from the values A = 2 and T = 10, in each subsequent simulation, the parameters mentioned here were reduced twice. With a fixed model size d x = d y = 10, the surface of the conductive path contained from 4 to 512 “hills,” which, as shown in Figure 6, have a significant impact on the current flow through the conductive area of the structure. The direct dimension of this influence can be assessed on the basis of the analysis of the results presented in Figure 7, where the relationship between the R/R 0 ratio and the number of periods of the function modeling the area Ω of the tested sample is plotted.

Each point of this graph represents the result of one simulation in which the resistance R per square of the cross-sectional area was determined, and its value was referred to the initial case when the resistance R 0 was calculated for a straight conductive path (Sym. 0). The K1 curve visible in Figure 7(a) connecting the points representing the above-mentioned simulations shows that the initially large increase in the resistance of the structure (approx. 30%) compared to the case of Sym. 0, caused by the appearance of significant surface irregularities of the conductive path, gradually stabilizes at about half of this value.

Taking into account the computational error, which has been marked here for each point with red bars, it can be concluded that this is the case starting from the simulation covering four periods of the function f ( Ω ) in the TS model. The above studies lead to the conclusion that the largest increase in TS resistance can be expected in the case of very rare interlacings of the fabric constituting the substrate for the conductive layer. The structure then has unevenness of relatively large amplitude, and the path of current flow is limited by extensive areas of so-called “hills” created by alternating arrangement of textile fibers (Figure 2). As the density of the fabric increases (Sym. 2–Sym. 5), there is also a greater density of the above-mentioned hills, which paradoxically leads to the flattening of the conductive surface, and the current has more alternative paths between them (Figure 6(e) and (i)).

Thus, a significant decrease in the resistance increase in relation to a completely flat surface can be observed, which, for the case of Sym. 5, reached the level of 15%. From that moment, a slight upward trend can be observed up to Sym. 13, where the increase in resistance reaches approx. 20%, and then again for Sym. 16 falls to 15%. Of course, the numerical error cannot be omitted, which for the mentioned simulations varied from 2.5 to 3.8%, so it is comparable to the observed changes in resistance. Nevertheless, the obtained results justify the thesis that starting from point 4, the K1 curve remains in a constant trend, which confirms the belief that further increasing the density of the fabric will not fundamentally change its resistance.

In the simulations of textronic structures, the ratio of the amplitude of the unevenness of the surface of the conductive path (A) to its thickness (d z ) is also an important parameter. The influence of this parameter on the TS resistance is illustrated by the plots in Figure 7(b) obtained during the simulation Sym. 17–Sym. 27. It is worth mentioning here that the starting point for the above tests was the simulation marked in Figure 7(a) with a blue square (Sym. 2), in which the largest increase in the resistance of the structure was obtained. In addition, it should also be added that the amplitudes of surface irregularities (A 1 and A 2) for this case were the same as the thickness of the conductive track (d z ). With all this in mind, in each subsequent test, the thickness d z was reduced linearly in relation to the initial thickness (d 0 = 1), leaving all other geometrical dimensions of the structure unchanged. The K2 curve created as a result of this action, drawn in black, connecting the points representing the above-mentioned simulations, shows a tendency to converge near the value of 1.34R/R 0. This means that the increase in the resistance of the corrugated TS PVD structure caused by the reduction of the cross-section of the conductive track thickness stabilizes at a maximum value of 34% in relation to the resistance R 0. If we add the fact that this increase occurred from the level of 1.29R 0, it can be concluded that the impact of the change in the thickness of the conductive path on its resistance is relatively small and, in some cases, is comparable to the error of convergence of the calculation procedure for the initial simulation (δ k in Table 1). At the same time, it is worth adding that in order to increase accuracy, all simulations in this part of the study were adjusted to an error of δ k < 0.5% (red error bars in Figure 7b).

The impact of changes in the amplitude (A) of the unevenness of the conducting path in relation to its thickness (d z ) on the resistance of the structure results indirectly from the tests described in the previous paragraph. However, in order to facilitate the analysis in this respect, an additional curve K3 was plotted in blue in Figure 7(b) connecting points representing changes in resistance (R/R 0) obtained under the above-described Sym. 17–Sym. 27 in relation to the ratio A/d z , changing as a result of decreasing the value of d z . As can be seen, for a fixed amplitude of surface irregularities (A = 1), the largest changes in resistance occur when it is comparable to the maximum thickness of the conductive path (d z = 1). Looking at the obtained results, it can also be assumed that starting from the point A/d z = 6, further increasing this ratio by reducing the value of d z will not cause significant changes in resistance. A similar tendency of resistance changes can be observed in Figure 7(c), which illustrates the results of calculations (Sym. 28–Sym. 38) using a similar convention as Sym. 17–Sym 27. The difference was that the starting point here was the case marked with a pink square in Figure 7a (Sym. 4). The K4 curve visible in the figure connecting the R/R 0 points for the above-mentioned simulation tends to the value of 1.18 (taking into account the numerical error), as does the K5 curve, which is a combination of K4 curve simulation points converted into the A/d z ratio. The range of K5 curve values (taking into account numerical errors) is in the range of 1.14–1.18R/R 0, which is slightly smaller than the range of K3 curve values. In both cases, this is a relatively small range, which leads to the conclusion that in real structures where A/d z ≅ 50, the possible dispersion of the conductive path thickness for different TS PVD specimens should not affect their resistance.

The largest changes in resistance were obtained when, in subsequent simulations (Sym. 39–Sym. 59), the amplitude A of the function f (Ω) was changed while maintaining a constant period of this function (T = 2.5). This was to reflect situations where we are dealing with a wide range of surface irregularities, ranging from small (A < 1) to very large (A > 1). The K6 curve connecting all the simulation points in this area of research shows that, as expected, the resistance increases as the A/d z ratio increases, but the R/R 0 ratio practically does not change from the case when A/d z = 1.5. A further increase of amplitude A in relation to the value of d z confirms this fact because all other simulations are characterized by the ratio R/R 0 ≅ 2.35. At the same time, it is noteworthy that this is an increase in resistance in relation to its value in a straight conductive path by as much as 235%, which suggests the conclusion that the appearance of surface irregularities with relatively large amplitudes can significantly change the resistance of the PVD structure.

5.4 Periodicity aspect

Considering the method of determining the area of the conductive path using a periodic function (formula 1), it is worth mentioning the effects. Analyzing the obtained simulation results, certain periodicity features of the potential distributions (Figure 5) and the vector field (Figure 6) can be noticed. It is also confirmed by the simulation results shown in Figure 8, where the potential distribution on the upper conducting surface (a) and the potential distributions in cross-sections equally spaced by the period of the given f (Ω) function are plotted.

Figure 8 Potential distributions in the conductive path of TS PVD. Illustration (a) is a map prepared for the upper surfaces of the Ω region for cases where the amplitudes of the conductive path roughness are A = 1 and T = 2. The diagrams in the right column (b–d) are maps for sections indicated in image (a).
Figure 8

Potential distributions in the conductive path of TS PVD. Illustration (a) is a map prepared for the upper surfaces of the Ω region for cases where the amplitudes of the conductive path roughness are A = 1 and T = 2. The diagrams in the right column (b–d) are maps for sections indicated in image (a).

Of course, there are small differences in the shown cross-sections, which could be eliminated by applying periodic boundary conditions. Such an approach would have a significant impact on reducing the computational time. Currently, the largest model simulated for this work contained 32 periods of the f (Ω) function and its computation took 8,350 s for 2 × 105 iterations with the achieved convergence error δ k = 5.54%. On the other hand, as shown in the microscopic photograph (Figure 1), the real fabric (its warp and weft) is not perfectly periodic, and our future work will be focused on taking this feature into account.

6 Measurement verification

An important stage of work under the 3D TS simulator project was the reference of the results of numerical tests obtained in the previous section to changes in parameters occurring in real TS PVD structures. For this purpose, the results of resistance measurements of selected structures with different textures of the textile material and different substrates of the conductive path were used.

6.1 Sample preparation and measurement method

The experimental part of the work was focused on measuring the electrical properties of a silver layer deposited on a textile substrate. Four different textile composites were used as substrates (Table 3). Samples A and E were made of the same material, but the surface of sample E was coated on one side with polyurethane. The microscopic views of tested substrates are described in Table 3, and their cross-sections are presented in Figure 9.

Table 3

Parameters of the samples

Sample Name Surface mass (g/m2) Materials
Sample A Cordura® 130 Nylon threads
Sample B Cordura® 195 Nylon threads/polyurethane
Sample C Cordura® 210 Nylon threads/polyurethane
Sample D RAINCRUCIAL® 155 Nylon threads/polyurethane/Teflon
Sample E Cordura® 130 Nylon threads/polyurethane
Figure 9 Views of sample surfaces: Sample A (a), Sample B (b), Sample C (c), Sample D (d), Sample E (e), and equipment used (f). Upper part – microscopic view, lower part – cross-of the samples.
Figure 9

Views of sample surfaces: Sample A (a), Sample B (b), Sample C (c), Sample D (d), Sample E (e), and equipment used (f). Upper part – microscopic view, lower part – cross-of the samples.

The samples of a textile substrate in the form of long strips with dimensions of 130 mm × 5 mm were used in the test (Figure 10). The measurement of the sample resistance was carried out using the four-probe method using the HP34420A digital multimeter (Keysight Technologies, Santa Rosa, CA, USA). The measurement of the electrical parameters of the samples was performed at room temperature without sample deformation and mechanical stress. For each of the materials, 45 resistance measurements were made in different areas of the substrate over a length of 130 mm.

Figure 10 Schematic view of the sample geometry and the measurement method.
Figure 10

Schematic view of the sample geometry and the measurement method.

The measurement method used allows for the observation of the homogeneity level of the conductive layer and the detection of local discontinuities. In addition, the measurement of the resistance of many sections of the sample is of application importance, as it is an indicator of macroscopic continuity and layer quality.

6.2 Sample preparation and measurement method

The average value of the sample resistance R ¯ , the value of the standard deviation SD, and sheet resistance R□ are presented in Table 4. Sample D has the lowest resistance and smallest standard deviation of resistance. This is related to the surface profile of the composite. The surface of the material is smooth and devoid of significant artifacts, which facilitates the formation of a continuous metallic layer (Figure 9(d)). Moreover, the polymer layer covering the composite is not thick, and the fabric threads have a tight weave and a flattened elliptical shape (Figure 9d). The monofilaments of the nylon threads have a small diameter and are continuous, free of twists and frays. The thickness of the A, D, i, and E samples are similar, but substrate A/E has a different weave and more artifacts on the surface (Figures 9a, 8e). These materials have the lowest standard deviation of metallic layer resistance despite significant variations in the measured resistance (Table 4). The value of the layer resistance depends mainly on the surface profile and the polymeric buffer layer covering the composite. It should be noted that the surfaces of samples A, D, and E have the greatest regularity and repeatability and is related to the thickness of the thread.

Table 4

Average values of the sample’s resistance R ¯ and the value of the standard deviation SD

Sample A B C D E
R ¯ 16.0780 1.1082 6.0773 0.2120 0.5816
SD 2.2813 0.4453 4.1302 0.0108 0.1090
R □ 2.2968 0.1583 0.8682 0.0303 0.0831

Samples B and C are more than twice as thick as substrates A, D, and E and are covered with a thicker layer of polyurethane (sample B). A thick layer of polymer (Figure 9(b)) smoothens the surface of the weave but promotes the formation of local cracks and damage to the metallic layer (Figure 9b). The conductive layer is characterized by low resistance but a significant dispersion of values (Table 4). Sample C was covered with a thinner layer of polymer (Figure 9(c)), but the regularity of the weave and the smoothness of the thread is low (Figure 9(c)). As a result, the sample is characterized by a higher resistance and high standard deviation of resistance.

Sample A is characterized by the highest resistance, which may be due to the relatively large non-uniformity of the conductive surface. This corresponds to the cases of Sym. 49–Sym. 59, where relatively large changes in resistance result from a large value of A/d z . At the same time, the much lower resistance of sample B may be because the structure was covered with a polymer coating, which, as can be seen, significantly smoothed the conductive surface, just as the relatively smooth surface of sample D is the cause of the lowest resistance measured during the research work described here.

The lower value of the resistance of sample C in relation to sample A can be explained by the fact that the lower density of sample C fibers and the related effect of the presence of areas with difficult flows (Figure 6(e)–(h)) is compensated by the greater amplitude of the roughness of sample A, whose characteristics are roughly described in Figure 7(d).

However, all the above conclusions should be considered in qualitative and not quantitative categories because the simulations did not take into account the physical phenomena that may occur during the fabrication of structures and have a significant impact on their resistance. Nevertheless, the main goal of numerical research consisting of determining the trends of resistance changes due to changes in the geometry of the conductive path has been achieved.

7 Conclusion

A numerical model was developed to simulate the transport parameters of TS PVD structures. The ability to define a conductive path in three geometrical dimensions allowed for the calculation of potential distributions and electric field strength in the entire volume of the conductive layer. The study on the influence of the geometry of the conductive path on the resistance of TS PVD structures presented in this work is the result of these possibilities. They showed the significance of surface unevenness of the conductive layer in the current flow and allowed us to qualitatively determine the limits of changes in resistance caused by changes in the unevenness. This has been positively verified by analyzing the resistance measurements of real PVD structures and is the basis for further work toward the development of a 3D simulator of PVD structures.

The developed algorithm gives new possibilities to the method, which, despite its inconveniences of belonging to the category of so-called weak solutions of differential equations, has proven to be efficient and effective. Its high-efficiency results from the fact that it is a boundary method in which matrix equations containing nodal points of the very edge of the examined area are solved. For example, the simulation that required the largest resources of the calculating machine (Sym. 28) operated on P{…} and R{…} resources of 3.7 × 106 and 1.6 × 105 elements, respectively, reserving about 0.5 GB of PC operating memory. At the same time, the time of agreeing on the solution of equation (18) after reaching the error of δ k = 5.8% for k = 4 × 105 iterations was 16,277 s (about 4.5 h). In addition, the calculations of 3D distributions of fields in a selected section of the area Ω carried out later in the program, proceeded depending on the size of this section and the planned size of the discretization mesh. For example, the calculation of the potential distribution and electric field strength for the upper surface of the conductive path discretized by 300 × 300 calculation points and the cross-section of the conductive path in the direction of transport covered with the calculation grid in the dimension of 300 × 100 points took 14 h 32 min on the computer in the configuration described in Section 4. Taking into account the fact that the application of other popular numerical methods used in commercial software (e.g., Ansys) to the above case would require discretization of the entire researched area from the stage of solving differential equations, and thus also much greater time and, above all, financial resources, it seems reasonable to create an alternative tool based on their own experiences, which the authors plan to implement and publish in subsequent works.

  1. Funding information: Authors state no funding involved.

  2. Author contributions: Mariusz Mączka – M.M, Ewa Korzeniewska E.K. Stanisław Pawłowski S.P. Marcin Lebioda M.L. Conceptualisation: M.M., E.K., S.P., M.L.; methodology: M.M., E.K., S.P. M.L.; software: M.M. and S.P.; validation: M.L. and E.K; formal analysis, S.P., M.M., E.K., M.L. investigation: M.M., E.K., S. P., M.L.; re-sources, E.K. and M.M.; writing–original draft preparation: M.M., M.L.; writing–review and editing, M.M. E.K. All authors have read and agreed to the published version of the manuscript.

  3. Conflict of interest: Authors state no conflict of interest.

  4. Ethical approval: The conducted research is not related to either human or animal use.

  5. Data availability statement: The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Received: 2024-10-24
Revised: 2025-03-20
Accepted: 2025-03-25
Published Online: 2025-05-12

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/aut-2025-0035
Keywords for this article
Textronics; wearable electronics; method of fundamental solutions; iterative algorithm; thin films
Creative Commons
BY 4.0

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  15. Optimization of particle swarm for force uniformity of personalized 3D printed insoles
  16. Development of auxetic shoulder straps for sport backpacks with improved thermal comfort
  17. Image recognition method of cashmere and wool based on SVM-RFE selection with three types of features
  18. Construction and analysis of yarn tension model in the process of electromagnetic weft insertion
  19. Influence of spacer fabric on functionality of laminates
  20. Design and development of a fibrous structure for the potential treatment of spinal cord injury using parametric modelling in Rhinoceros 3D®
  21. The effect of the process conditions and lubricant application on the quality of yarns produced by mechanical recycling of denim-like fabrics
  22. Textile fabrics abrasion resistance – The instrumental method for end point assessment
  23. CFD modeling of heat transfer through composites for protective gloves containing aerogel and Parylene C coatings supported by micro-CT and thermography
  24. Comparative study on the compressive performance of honeycomb structures fabricated by stereo lithography apparatus
  25. Effect of cyclic fastening–unfastening and interruption of current flowing through a snap fastener electrical connector on its resistance
  26. NIRS identification of cashmere and wool fibers based on spare representation and improved AdaBoost algorithm
  27. Biο-based surfactants derived frοm Mesembryanthemum crystallinum and Salsοla vermiculata: Pοtential applicatiοns in textile prοductiοn
  28. Predicted sewing thread consumption using neural network method based on the physical and structural parameters of knitted fabrics
  29. Research on user behavior of traditional Chinese medicine therapeutic smart clothing
  30. Effect of construction parameters on faux fur knitted fabric properties
  31. The use of innovative sewing machines to produce two prototypes of women’s skirts
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