The Resistor Network Approach to Modeling Screen-Printed Silver Ink Under Uniaxial Stretch

Flexible electronic devices are used in a wide variety of applications including in structural health monitoring, automotive and aerospace industries, and wearable health monitoring [1–6]. The ability of flexible electronics to stretch, bend, twist, etc. opens the doors to a variety of applications and form factors which traditional rigid electronics cannot match. This ability to accommodate high strains is accomplished through the use of flexible substrates like polyimide, PET, TPU, fabrics, and PDMS among others. Various types of flexible, printed conductive inks [7–10] have also been developed for use as sensors, electrodes, or the primary circuit interconnections. One particular type of conductive ink is polymer thick film ink. These inks typically consist of conductive filler particles in a nonconductive polymer binding matrix. Screen printing is most commonly used to print this ink type into a circuit but other techniques such as direct-write printing can also be used. While many manufacturers provide resistivity data in data sheets, this resistivity is only useful when the ink is in the unstrained condition. However, as mentioned above, the utility of these flexible electronics is in their ability to strain extensively, and therefore the electrical performance of printed inks under strain is of interest [11]. Our previous work investigated other modes of strain including bending, twisting, and biaxial strain [12–15]. This paper investigates the piezoresistive behavior of screen-printed silver ink under uniaxial stretch through experimental and numerical methods. There are three main objectives of this paper. First, this paper aims to experimentally quantify the piezoresistive behavior of DuPont 5025 ink on polymer substrates and in doing so provide a repeatable experimental framework for quantifying other inks. The second objective is to develop a predictive numerical model that relates the composition and microstructure of a conductive ink to its electrical performance under strain. The last objective is to begin a parametric study using the numerical model to determine which ink parameters have a significant effect on the electrical performance and reliability of the ink. While others have numerically modeled the piezoresistive behavior of conductive polymers [16–19], they primarily focus on long conductive filler fibers or spherical fibers. In this work, two-dimensional (2D) platelets are used to model flake-like conductive particles.

A set of six samples were designed with straight-line conductors of various widths as shown in Fig. 1(a). Each individual sample is 25 mm wide by 175 mm long. The gauge length when gripped for stretch testing is 100 mm. These dimensions were chosen to be compliant with ASTM D882 [20] which is used to test the mechanical properties of the substrate. A pair of probe pads at each end of the trace is provided to allow for in situ four-wire resistance measurements during stretch testing. The ink trace widths vary from 0.25 mm to 10 mm. Additionally, tick marks are provided along the edge of each sample to assist in aligning the samples in the grips as well as an optical means to observe localized strain. DuPont's commercially available 5025 silver ink was used to print the test samples onto both Kapton HN polyimide substrates (Fig. 1(b)) and Melinex ST506 PET substrates (Fig. 1(c)) at DuPont's facilities. The 5025 ink consists of conductive silver flakes in an acrylic-based polymer matrix. DuPont's 5018 dielectric is used as an encapsulant layer to protect the printed silver ink. Both substrates are 125 μm thick, the ink is approximately 12 μm thick and the encapsulant layer is approximately 26 μm thick. The inks were printed by DuPont using a 325-mesh screen and then cured at 130 °C for 15 min. The maximum elongation of thin plastic films is particularly sensitive to edge defects. Hand-cutting samples with straight razor blades is not recommended as it is difficult to produce clean edges. The samples here are cut using a Dahle 550 professional rolling trimmer to ensure samples edges are straight and undamaged.

Custom fixtures shown in Fig. 2 were designed for secure gripping during stretch as well as access for in situ resistance measurements. Mechanically gripping thin plastic films during stretching can be a challenge, as the films tend to slip out of the grips during stretching. In order to ensure secure gripping, several grip insert materials and designs were tested including printer paper, rubber, SiC abrasive paper, as well as line grips. 800 grit SiC abrasive paper glued to both sides of the gripping fixture as shown in Fig. 2(a) was found to have satisfactory gripping performance. While many researchers have chosen [11] to place soldered or epoxied electrical connections to the traces within the active (strained) area, this can disturb the strain distribution throughout the test sample and subject the electrical connections to strain. Thus, in this sample and fixture design, the electrical connections were placed outside of the actively strained areas. Instead of soldering wires to each sample or making connections with conductive epoxy, reusable spring-loaded electrical connectors, commonly called pogo pins, were used to facilitate quicker experimental setup as shown in Fig. 2(b). The use and design of the four-wire resistance measurements avoids the effects of any contact resistance between the pogo pins and the sample pad as well as any lead wire resistance.

A sample clamping jig shown in Fig. 3 was used to ensure accurate and consistent gauge lengths as well as ensure that the sample was centered and aligned in the test fixture before the grips were tightened. The clamping jig consists of a rigid aluminum bar with dowel pins that align the clamping fixtures to a set gauge length of 100 mm. Another smaller aluminum bar is fixed to the jig that is the exact same width as the samples and is centered left to right between the clamps and provides a reference for aligning the sample in the grips. A precision torque wrench was used to torque all the gripping fasteners to a torque of 5.65 N m (50 in.-lbf).

The Kapton and Melinex samples were stretched in a universal testing machine under uniaxial strain until the samples ruptured. While stretching, in situ four-wire resistance measurements were collected with a 6.5-digit benchtop multimeter. Images of the testing progression of a Kapton sample are shown in Fig. 4.

Load versus displacement and stress versus strain results are shown in Figs. 5(a) and 5(b) for Kapton and Melinex samples, respectively.

The Kapton samples show an average maximum elongation of 59.5% and standard deviation of 3.6%. The ultimate tensile strength is about 220 MPa. These values are close to the datasheet values from DuPont. In contrast, the Melinex samples have an average maximum elongation of 126.2%, about double that of the Kapton samples, and standard deviation of 10.3%. In all cases except one, the samples ruptured at a location away from the end grips and toward the middle of the samples. The ultimate tensile strength of the Melinex samples is about 165 MPa. The samples were tested at two displacement rates, 10 mm/min and 50 mm/min, which is equivalent to strain rates of 1.67 × 10⁻³ s⁻¹ and 8.33 × 10⁻³ s⁻¹. In these tests, there did not seem to be significant differences in the load versus displacement data with the two different strain rates.

If one were to consider strain induced conductor geometry change alone, the resistance ratio would be less than 2.89 for a stretch ratio of L/L₀ = 1.7 or ε = 70%. However, The R/R₀ behavior in Ag-flake inks does not follow the constant volume (L/L₀)²line like solid metallic films and likely nano-Ag inks. The resistance in Ag-flake inks is typically dominated by contact resistance comprising a constriction resistance and a tunneling resistance. Physical damage, such as cracking or necking, can also occur during stretching, all of which will significantly increase the effective electrical resistivity of the ink during strain. For all 12 of the samples shown, the Kapton substrate failed before the conductor showed an open resistance measurement. Also, it is seen from Fig. 6, that the resistance change does not appear to be dependent on strain rate, based on the two strain rates tested. The 12 sets of sample data were fitted with a second-order polynomial as shown in Fig. 7. The fit appears to match the experimental data well with an R² of 0.9886.

Figure 8 shows normalized resistance data up to 60% strain for the Melinex samples tested at 50 mm/min. This initial part of the normalized resistance data is very similar in shape to what is seen for Kapton in Fig. 6. Figure 9 shows that for the Melinex substrates, the inks failed (open resistance measurement) before the substrate ruptured. The narrowest conductors failed first while the widest conductors failed last. Figure 9 also shows that the conductors often exhibit erratic electrical behavior before completely failing. This is especially obvious for the 2 mm sample that oscillated between an R/R₀ of above 30,000 to one below 2,000 in Fig. 9(b) but is also seen to a lesser extent in the other conductors. These R/R₀ values are extremely high and so would likely be considered a performance failure even though the conductor is not yet fully electrically broken. Figure 10 shows a linear increase of conductor failure strain with increasing conductor width up until a conductor width of about 2 mm.

The initial portion of the Melinex sample data, up to 60% strain, was fitted with a second-order polynomial as shown in Fig. 11. The fit appears to match the experimental data well with an R² of 0.9645. Since both the Kapton and Melinex samples have the same exact ink and UV encapsulant, a comparison of the ink's performance on both substrates might give some insight into how substrate properties affect ink performance. Figure 12 shows such a comparison for both using the polynomial fits of the experimental data up to 60% strain. In general, the curves follow the same trend regardless of substrate material. As seen in the figure, the ink behaves practically identically on both substrates up to about 30% elongation. This may be because the Young's modulus of Kapton and Melinex are similar in magnitude. Above 30% strain, R/R₀ increases faster for the Melinex substrate samples.

The numerical approach taken in this work is to develop a representative volume element (RVE) that can capture the particle microsctructure of the conductive ink and create a resistor network that can accurately describe the ink's DC electrical performance both in the as-printed state as well as when subjected to strain. As previously mentioned, the 5025 polymer thick film ink used in this study consists of conductive silver flakes in a nonconductive, acrylic-based, polymer binding matrix as shown in Fig. 13. In these types of conductive inks, electrical conductivity is achieved when the volume fraction of the conductive particles surpasses a critical volume fraction called the percolation threshold [21]. Initially, 2D RVEs, where planar particles only interact in the planar dimensions, are created and investigated. However, 2.5D RVEs, where multiple 2D RVEs are stacked on top of each other, are ultimately used to allow particle interaction in both the planar and thickness dimensions.

Examining the SEM images in Fig. 13 of the microstructure of the ink, it is observed that the silver flake particles have roughly the same thickness but vary primarily in their planar dimensions. Therefore, for computational simplicity, each silver particle is modeled as a circular platelet whose radius can be varied depending on the desired size distribution of the particles. The thickness is assumed to be a uniform 1 μm although it could be changed, if desired. Square RVEs with equal lengths and widths of 25 μm, 50 μm, and 100 μm are used in this work. Since the thickness of the particles is nearly uniform, and all of the particles are in the same plane, a single layer of particles can be approximated with a 2D RVE.

Generating a 2D RVE is a two-step process. An initial seed RVE is generated first as shown in Fig. 14(a). This seed RVE is filled by placing particles in random locations in proportion to the desired size distribution until correct volume fraction is achieved. In this work, the RVE is made up of particles from four potential particle radius ranges as shown in Table 1, with the total percentage of particle area from each radius range being equal. This size distribution results in fewer numbers of larger particles and a much greater number of smaller particles.

In this seed RVE, particles are allowed to overlap with one another. Without initially allowing particles to overlap, higher volume fraction RVEs become practically impossible to generate when randomly placing particles. This is because higher volume fraction RVEs require some optimization of particle packing that cannot be achieved through random placement.

To generate the final RVE, as shown in Fig. 14(b), the overlapping particles generated in the seed RVE are repulsed from each other by applying a force to each particle in proportion to its overlapped area in a direction that moves the particle center directly away from the center of the particle it overlaps. These forces are shown as arrows on the particles in Fig. 14(a). The particles are allowed to move for a time-step according to the resultant forces and then the force calculation and movement process is repeated iteratively until the particles are separated. A similar approach to RVE generation has been used successfully in other work [22] and has similarities to discrete element method modeling [23].

Initial investigation of the 2D RVE revealed that the 2D representation severely underpredicted actual performance of inks. This is likely due to the fact that the 2D RVE only consists of a single layer of particles. However, as Fig. 13(b) shows, these inks actually consist of multiple layers. While full three-dimensional (3D) RVE modeling with 3D platelets would provide more accurate results, the computational expense would be very high, especially in determining particle-to-particle contact interactions. Therefore, we propose a 2.5D RVE approach where multiple 2D RVEs are stacked on top of each other to better model the true 3D nature of the conductive inks. An example of a 3-layer, 2.5D RVE is shown in Fig. 15. This compromise allows accurate results while maintaining computational efficiency. Thus, the particles will have electrical connectivity not only in a given layer but also across different layers.

The volume fraction of ink varies based on the particular ink formulation chosen by the manufacturer. The actual volume fraction of the 5025 ink was estimated by taking the area fraction of the particles across multiple cross-section SEM images similar to image shown in Fig. 16. The estimated volume fraction of 5025 silver ink was found to be 62.4% in the as-printed and unstrained condition. The volume fraction used in generating RVEs was assumed to be 50%. Particles are seen to range in size from less than 1 μm to about 10 μm.

The process of creating a resistor network from an RVE of particles is shown schematically in Fig. 17. A simple initial percolation network is created by placing a node at the center of each particle. A grid-based neighbor search algorithm (i.e., cell method or linked-list method) [24] was implemented to find neighbors that are within 1 nm of the current node. Once found, connection or edge is made between neighboring nodes. This simple, undirected, and unweighted network can be used to determine whether or not a percolating network has formed that transverses the entire RVE. This type of network is useful when studying percolation thresholds or finding the strain at which the RVE is nonpercolating and thus nonconducting (infinite resistance). However, it is not able to predict absolute resistance values or changes in resistance with strain.

where d is the tunneling distance, h is Planck's constant, and m_e and q_e are the mass and charge of an electron , respectively. The barrier height of the insulating polymer matrix, λ, is assumed to be 1 eV but likely varies depending on the particular material. Since the tunneling surfaces between two particles are not flat, the tunneling resistance, R_tun, is calculated by numerically integrating $ρtun$ with respect to the tunneling area and the changing tunneling distance due to the curved surfaces. The total resistance between two nodes is the sum of the intrinsic resistance, R_intrs, of each particle as the current flows through each particle and the interface resistance, either R_tun or R_cont. The silver flakes are assumed to have the same resistivity as bulk silver, 1.59 × 10⁻⁸ Ω·m [29]. Once the resistor network has been developed (Fig. 18), a voltage difference is applied across two opposite faces along the length of the RVE, and the total current through the RVE can be determined by using all particles in serial and parallel contact both ohmic and tunneling. Ohm's laws can then be used to determine the effective resistance of the RVE.

where $x0$ and $y0$ are the coordinates of the center of the particle before strain, x and y are the coordinates of the center of the particle after strain, and ν is the Poisson's ratio of the material system. A Poisson's ratio of 0.3 was assumed for all of the models presented in this paper. An example of the RVE undergoing a uniaxial strain of 10% in the x direction is shown in Fig. 19. The left side of the RVE is fixed in the x-direction, while the bottom side of the RVE is fixed in the y-direction.

The electrical results from 33 model runs of different, randomly generated 5-layer, 100 μm × 100 μm RVEs, with a volume fraction of 50% and a size distribution shown in Table 1, that were subjected to uniaxial strain until electrical failure are shown in Fig. 20. From the figure, we see that R/R₀ results are consistent across almost all model runs through about 30% - 40% axial strain. After about 40% strain, the predicted resistance results start diverging which indicates the electrical performance at these higher strains is more dependent on the initial starting RVE.

The results of the numerical RVE models are shown in comparison to the experimental Kapton and Melinex results in Figs. 21 and 22, respectively. The comparison of model results to Kapton experimental results in Fig. 21 shows good agreement between the two throughout the entire experimental test. The Kapton experimental data ends around 60% strain due to the substrate rupturing, but several of the RVE model runs predict that the ink should still be conductive through 80%–90% strain. Comparing the modeling results to the Melinex experimental results in Fig. 22(a), one can see that the modeling results agree well with experimental results below 60% strain. However, even above 60% strain, the experimental results stay within the bounds of the RVE results (Fig. 22(b)). The RVE results underpredict the strain at electrical break, as the RVE shows electrical failure around 90% strain while the Melinex experimental results show conductors strained above 120% (Fig. 10) before electrical failure. This is likely due to the RVE models shown here only having up to 5 layers of particles while the actual inks are composed of 10 or more layers of particles. The RVE model is also substrate-independent and assumes that the substrate can impose a homogeneous strain onto the conductive ink.

where $σc$ is the electrical conductivity of the composite, $σf$ is the bulk electrical conductivity of the filler particle, $ϕ$ is the volume fraction of the conductive filler, and $ϕcrit$ is the critical volume fraction or percolation threshold, below which the composite is not conductive. The exponent t is conductivity exponent.

As noted above, this model has been used to model conductive inks, however, one of its main drawbacks is that several of the model parameters must be determined empirically. The t exponent is one of the empirical parameters. The second is $ϕcrit$⁠, which is highly dependent on the filler particle size distribution and morphology. Thus, modifying either of those parameters requires additional empirical work to determine the new $ϕcrit.$ Despite these limitations, a percolation model can be tuned to yield satisfactory results when concerned with a well-studied material set.

where $ϕ0$ is the initial volume fraction of the composite. This relation assumes uniform strain and no cracking of the ink. Starting with an initial volume fraction of 50%, the volume fraction decreases to 30.8% at an applied strain of 100% as shown in Fig. 23.

As described in Sec. 4.2, one of the features of the resistor network approach is that it can numerically determine $ϕcrit$ through Monte Carlo simulations, given the filler particle size distribution and morphology. For the RVEs used in this study, $ϕcrit$ was determined to be 26.5%. Since conductive particles in the ink are composed of silver, $σf=σsilver=6.289·107S/m$⁠.

These parameters can be used in the percolation model to predict the change in conductivity of the composite as shown in Eq. (9). Accounting for the change in sample length and width, and assuming no change in thickness, the conductivity can be used to predict the sample resistance and relative resistance with strain. A plot comparing the predicted R/R₀ from the percolation model, the RVE model, and experimental data is shown in Fig. 24. As shown previously, the RVE model matches the experimental data well. However, the percolation-based model severely underpredicts R/R₀.

This numerical model allows for the exploration of the effects various ink/RVE parameters have on the electrical performance of the ink during strain. An initial parametric study on the effect of the initial volume fraction on the average strain until electrical open, ε_max, and the DC resistance performance are presented below. A more comprehensive parametric study on the effects of the RVE size, the number of conductor layers, the Poisson's ratio, and the size distribution of the particles will be part of a separate paper.

The volume fraction of a conductive ink is one of the key properties of an ink formulation. For example, DuPont has multiple Ag-flake, acrylic matrix inks with varying silver contents. They range from 5043 at about 52% silver to 5029 with about 80% silver content. The amount of silver has a significant effect on electrical performance as shown in the next few sections. It also has a significant effect on the cost of the ink as the silver is one of the more expensive material components of the ink. Five different initial nominal volume fractions are considered here: 40%, 50%, 62%, 70%, and 80%. Examples of 50 μm × 50 μm RVE layers at each of these volume fractions are shown in Fig. 25.

A plot of R/R₀ at various strains versus volume fraction is shown in Fig. 26. RVEs with higher volume fractions show lower R/R₀ values, especially at higher strains. This result is intuitive as volume fractions of filler increase, there are more conductive pathways that form. Percolation theory also supports this trend of better electrical performance with greater initial volume fraction as shown in Fig. 27.

Figure 28 shows a plot of ε_max versus ϕ which indicates that ε_max increases linearly with increasing volume fraction. Here, ε_max is taken as the largest strain at which the RVE is still conductive. This trend also agrees with classical percolation theory. As shown previously in Fig. 23, the volume fraction of an RVE continually decreases from its initial starting volume fraction. At some point, the volume fraction decreases below the percolation threshold and the ink is no longer conductive. Since the rate at which the volume fraction decreases is constant, a higher initial volume fraction allows for larger applied strains before decreasing below the critical volume fraction.

This work has shown practical methods for evaluating the piezoresistive behavior of printed conductive inks under uniaxial strain. DuPont 5025 screen-printed silver ink on Kapton and Melinex substrates were stretched until substrate failure. Kapton samples were found to rupture at around 60% strain and have an R/R₀ of about 30–40 at substrate rupture. In Melinex substrates, the ink was found to electrically fail before the substrate ruptured but could be stretched to strains exceeding 130% or higher before failing. The R/R₀ values for these high strains in the Melinex samples were erratic and could exceed 1,000 and in one case more than 30,000. The ink strain to failure exhibited a dependence on conductor width with narrower conductors failing before wider ones. Finally, a 2.5D RVE model that accounts for ink volume fraction, particle size distribution, contact resistance, and electron tunneling was developed that accurately predicts the piezoresistive behavior of 5025 ink up to 60% axial strain. This RVE model was compared to classical percolation theory and was found to be an improvement. Increasing the ink volume fraction was found to result in lower relative resistances during strain and a higher tolerance to strain before electrically failing.

This material is based, in part, on research sponsored by Air Force Research Laboratory under Agreement No. FA8650-15-2-5401, as conducted through the flexible hybrid electronics manufacturing innovation institute, NextFlex. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of Air Force Research Laboratory or the U.S. Government. The authors would also like to thank various colleagues at Georgia Tech as well as collaborators at Binghamton University for their valuable input and discussion during the ongoing project. The authors would like to thank Jeffrey Meth at DuPont for providing the samples and for many fruitful discussions on the initial part of this work. This research was also supported in part through research cyberinfrastructure resources and services provided by the Partnership for an Advanced Computing Environment (PACE) at the Georgia Institute of Technology, Atlanta, Georgia.

• Air Force Research Laboratory (Agreement No. FA8650-15-2-5401; Funder ID: 10.13039/100006602).

• Georgia Institute of Technology (Funder ID: 10.13039/100006778).

The authors declare that they have no conflict of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.