#### Abstract

Force sensing resistors (FSR) based in conductive polymer composites (CPC) are a cost-effective alternative to load cells for force measurement. Nevertheless, their physical characteristics related to rheological features provoke some drawbacks such as hysteresis, drift, low repeatability, and sensitivity degradation (SD), which is a concerning issue when final application involves periodic loading. Although it is already known that SD is a voltage-related phenomenon and practical considerations to avoid it have been published, a more theoretical approach was yet missing. This study provides a set of finite element analysis (FEA) simplified models to shed light on the situations that favor degradation based in a time-dependent mechanical study considering rheological parameters and how they influence interparticle tunneling conduction. Also, a stationary electrostatic analysis with special scope in contact resistance and its influence in equipotential surfaces arousal. Simulation results show how the effect of contact resistance bends equipotential surfaces favoring conduction through time-increasing interparticle gaps, which is a direct consequence of the low Poisson ratio of considered polymers.

#### 1. Introduction

Force sensing resistors (FSR) exhibit a notorious change in its electrical resistance when subjected to mechanical stress, what makes them a suitable device for electrical transduction of force or pressure. The brand to analyze in the present study, Flexiforce A201-X, offers a FSR product composed by two electrode terminals sandwiching a conductive polymer composite (CPC) with a disk-like geometry. Its electrical model is just a resistance whose ohmic value depends on the exerted force. Thus, with a simple conditioning circuit, they provide a low-cost reliable alternative to strain gauges and force cells, for measuring force or pressure [1].

CPC are composed of an electrically insulating matrix which contains conductive nanoparticles randomly dispersed in it. The electrical conduction relies on the quantum tunneling (QT) phenomenon, where conduction between neighboring nanoparticles can be possible although a thin insulating film physically separates them [2]. This film behaves as a potential barrier that apparently hinders charge carriers to pass from a nanoparticle to its neighboring ones. Nevertheless, QT phenomenon considers that there are probabilities for some charge carriers to overcome the potential barrier, what establishes an electrical current between nanoparticles and throughout the thin film. Since the matrix is composed of polymer material, deformation is observed once the device is subjected to mechanical stress and the film between neighboring particles shrink due to deformation. Conduction probability increases when the separating thin film is reduced. Thus, the device allows a higher electrical current to flow because of the exertion of mechanical stress [3].

Simmons’ proposal of a generalized formula for electrical conduction through thin insulating films [4] has been the basis for a series of contributions and studies made to broaden the knowledge of CPC devices. The effect of contact resistance [5], conduction area [6], random behavior [7], electrode alignment [8], rheological response [3], piezocapacitance [9–12], hysteresis [13–15], and also the proper considerations when conducting elements are nanotubes [16, 17] have been widely studied for a more accurate understanding. Therefore, more accurate models have been published, which allow to understand better the performance of FSR in several operating conditions.

FSRs, and generally CPCs, due to its conditions and the phenomena involved offer advantages regarding to cost, size, and weight. They are noninvasive devices suitable for plenty of applications in robotics [18, 19], industrial/biomedical instrumentation [1, 20], human-machine interfaces (HMI) [21, 22], catalysis [23], and energy storage [24, 25]. However, disadvantages such as hysteresis, low repeatability, and drift are undesirable characteristics of FSRs when compared, for instance, to force cells. Research has been conducted to propose techniques to lessen the reading errors provoked by phenomena such as hysteresis, repeatability [14], or drift [26]. Nevertheless, when the application involves exertion of periodic force patterns, sensitivity degradation (SD) remains an important concern to be addressed. Although practical considerations have been proposed to lessen the effect of SD, to the best of authors’ knowledge a theoretical explanation is yet missing. This phenomenon hinders a precise calibration of sensor as applications demand nowadays. Hence, more precise modeling and studies are worth to be carried out to support experimental observations. Finite element analysis has been considered a pathway to build more accurate models relying on solving Dirichlet problems with the observed boundary conditions considering coupled electrical and mechanical phenomena. A simplified “four-particle-model” has been built in COMSOL Multiphysics® Modeling Software to simulate FSR operation of dynamic loading conditions that favors the observation of SD. Rheologic and electrostatic equations have been solved to serve as a basis for its physical explanation seen by periodic loading.

This paper is organized as follows: Section 2 will briefly discuss all considerations that drove authors to propose the most accurate model for commercial FSR based in CPC. Section 3 will address evidence of SD in dynamic loading conditions with special focus on its dependence on applied voltage. Section 4 explains the simplified “four-particle-model” to be executed in COMSOL Multiphysics® Modeling Software for the finite element analysis. Sections 5 and 6 address, respectively, the outcome of the mechanical dynamic and the electrical stationary studies with the scope in SD. Section 7 summarizes the conclusions and recommendations for future work.

#### 2. Conduction Model of Force Sensing Resistors Manufactured from Conductive Polymer Composites

##### 2.1. Comprehensive Model Embracing Rheological Phenomena

Any pressure exerted with a nonzero normal component on a FSR will provoke its deformation since its matrix is composed with low stiffness polymers. Inner conductive nanoparticles come closer together because of pressure, making the thin insulating film shrink in the direction of the exerted force and expands in the perpendicular direction. The thinner the separating film is, the higher the probability is for conduction. This explains why the total device conductance increases with pressure. Figure 1(a) shows the scheme of conduction, where is the physical axis resembling the charge carriers (*e ^{-}*) would move in a vertical path form the upper electrode towards the lower through the CPC. The height of the potential barrier depicted in Figure 1(a) is a consequence of the insulating characteristics of the polymer matrix, which would avoid electrical conduction if the separating film were not that thin. Since this film is not thicker as some nanometers, quantum tunneling (QT) happens and charge carriers (

*e*) may overcome the barrier. Let be the interparticle separation thickness when the FSR is not subjected to mechanical stress, and be the thickness of interparticle separation when the device is subjected to stress, which accomplishes . Figure 1(b) shows a simplified scheme of a FSR, where the black dots represent the conductive nanoparticles. Pressure causes the devices to shrink in the direction of force, as well as interparticle separation between neighboring nanoparticles. Since deformation depends on the exerted stress, it is depicted in Figure 1(a) as a function , where is the exerted stress. Equations (1) through (3) are Simmons’ approach that plenty of CPC models rely upon [4]. Based on Simmons’ model for current density through thin insulating films, relevant contributions have been proposed as follows:

^{-}**(a)**

**(b)**

The authors’ previous work [6, 7] has proposed a conduction model for CPC-based FSRs, which predicts the value of device’s resistance (or the current flowing throughout the device) for specific values of voltage and interparticle film thickness, and thus, applied pressure. Also, it includes the contact resistance, which is the opposition to the pass of charge carriers to the nanoparticle(s) physically contacting upper or lower electrodes or also contacting among each other. (i)Contact resistance: according to the proposed electrical model in Figure 2, voltage in as a function of the sourcing voltage can be written as , where is the circuit current ( by the effective conductive area). Contact resistance has been modeled by Kalantari et al. [5] and authors’ contributions have also led to an expression for contact resistance as a function of stress in (4). For the sake of simplicity, this equation has not been plugged in Equations (1) through (3). However, the authors’ previous work [6] has managed to solve the equations as a function of and

where is the particle resistance, the contact resistance at rest state (unloaded), and is a form factor (ii) depends on and as the most independent variables according to Equations (1) through (3). At this stage, this approach has been used to analyze FSR performance for a relative wide range of sourcing voltages. The interparticle separation as a function of is shown in

where is the interparticle separation at rest state and is the polymer’s Young modulus. The behavior of as a function of time when dynamic forces are exerted on the FSR has been thoroughly studied by authors’ in [7], where rheological responses have been considered through the Burgers model.

Figure 3 shows a plot of the full FSR resistance, which is the sum of and , for several values of sourcing voltage (0 V-7 V), and for a fixed stress of 250 kPa: (, where is the effective conductive area of the sensor. Contact resistance, as shown in Equation (4), does not depend on the sourcing voltage, so it remains unchanged for all voltages. Thus, FSR full resistance depends solely on the quantum tunneling (QT) phenomenon present in . Moreover, Figure 3 also shows the sharp decrement of and thus the full resistance for high values of sourcing voltage. This is a consequence of the QT phenomenon, where the higher the sourcing voltage is, the higher the probability of conduction across thin insulating films is due to the higher quantity of charge carriers provoked by higher sourcing voltages. Thus, decreases dramatically when sourcing voltage is high because of this high probability of tunneling conduction.

For very low () Figure 3 shows what can be seen in Equation (1). Both sides of equation can be divided by and multiplied by , so an expression for conductance (can be obtained. Conductance/resistance happens to be nondependent on voltage, so this first part of the graph is constant, and the sensor behaves ohmic. Once the voltage rises, resistance becomes to be more dependent on voltage (see Equations (2) and (3), where no matter what operation is done, dependence on voltage cannot be eliminated) so the sensor is no longer ohmic and also very sharp in the drop of resistance over voltage. Due to the existing exponential functions in Equations (2) and (3), it is worth to remark how little the contribution of the contact resistance is for low voltages, where it only sums less than 10^{4} *Ω* for a total resistance of approximately 10^{12} *Ω*. This is caused by the high-resistance polymer thin films offer when tunneling conduction probability is low due to low sourcing voltages. Conversely, contact resistance provides up to the half of the full resistance when sourcing voltage (and thus tunneling conduction probability) is high. This is an observation worth to bear in mind for the following explanations towards equipotential surface analysis.

#### 3. Sensitivity Degradation

Sensitivity degradation (SD) is mainly observed in periodic loading application such as gait analysis. Measurements could be seriously misled if sensors loose sensitivity during the try. Figure 4(a) shows a constant pace gait pattern acquired with insoles where Flexiforce A201-1 sensors were deployed. However, it wrongly suggests that the pressure of lasts steps has experienced a lowering trend. Sensitivity degradation consists in a progressive reduction of the voltage/force sensitivity ratio due to periodic loading. This phenomenon complicates calibration. Characterization of the reducing trends of sensitivity ratio would be needed to compensate degradation for calibration or postprocessing in offline applications. However, authors have observed and reported sensitivity degradation as a voltage-dependent phenomenon [27]. For low sourcing voltages, sensitivity degradation has not been observed for long tries of cycling loading, whereas the degraded pattern such as that in Figure 4(b) are typical for a sourcing voltage higher than 7 V (9 V for the precise case of Figure 4(b)). Thus, the practical recommendation has been to select low sourcing voltages to feed FSRs specially for periodic loading applications. Attention on sourcing voltage has been already called to enhance quality of force readings with FSRs. For instance, nonconstant voltages dropping in FSRs hinder repeatability. This fact relies on the dependence of FSR’s resistance on voltage as shown in Equations (1) through (3). Hence, voltage dividers as driving circuits have been discouraged by authors when high repeatability of readings is needed [27]. Driving circuits used for experiments in [28] guarantee constant voltage feeding for all FSRs in the insoles. However, a different signal was acquired when sourcing voltage was lower (1.27 V), *ceteris paribus*, see Figure 4.

**(a)**

**(b)**

Sensitivity degradation has also been reported in plenty of works [26, 29, 30] as a worrying and not fully understood phenomenon.

It is worth to remark that neither the deployed sensors nor the insoles presented any damage during the experiments. In fact, the non-descending trend observed when low sourcing voltage was applied is evidence of no degradation. Experimentally, dependence of sensitivity degradation on voltage has been clear as the only variable which causes degradation by periodical loading. Nevertheless, a theoretical explanation for this relation is still missing. The authors have hypothesized on the causes for sensitivity to degrade, and hence, a simplified model has been proposed to support it. Validation of hypothesis will be performed next through a finite element method approach since it relies upon the rheological behavior of the polymer matrix in an important extent.

#### 4. Simplified “Four-Particle Model” for Finite Element Analysis

This section explains the so called “four-particle-model,” which is a simple representation of a FSR based in CPC with only four nanoparticles. This approach is considered enough for simulating phenomena of interest through finite element analysis (FEA), associated to contact, potential, and dynamics [31]. Moreover, the use FEA tools for MEMS devices such as FSR has been a helpful for plenty of research tasks [32–34]. Figure 5 shows four versions of this model, which is very similar to the one shown in Figure 1. Thus, polymer matrix and electrodes terminal are represented the same way. This section is intended to explain and justify each detail that composes this simplified model.

**(a)**

**(b)**

**(c)**

**(d)**

Figure 6 shows four versions of the model, where the fundamental difference among them is the different positioning of the particles in the matrix. Each version will show a different scenario of interparticle separation dynamics and how it affects the electrical response of the device. Some of them embrace the scenario of considerable contact resistance with the electrodes (6(a) and 6(d)) or among them (6(b)), which is an ohmic phenomenon that contributes to the full FSR resistance according to the electric model in Figure 2. As already shown, contact resistance needs to be considered apart since it does not rely upon QT and only depends on the mechanical stress and is voltage independent [5]. Further particles represent those QT occurs throughout. They are isolated from one another solely by polymer thin films that shrink due to the mechanical stress the device could be subjected to. It is considered in a sandwich-like fashion as the Flexiforce A201-X, which has been one of the most characterized devices in authors’ recent research [6, 7, 27, 35].

**(a)**

**(b)**

**(c)**

**(d)**

These “four-particle models” are the simplest approach to consider so the electrical response by periodic loading could be analyzed. It could be considered that these four particles are the smallest spatial sample to consider in the FSR device where the coupled mechanical and electrical phenomena that could provoke SD can be simulated.

All versions of the “four-particle model” will be subjected to electrical and mechanical boundary constraints in a FEA framework performed in COMSOL Multiphysics® Modeling Software to analyze their response under periodic loading conditions. The analysis involves an understanding of the electrical equipotential surfaces that arise due to the sourcing voltage and the dielectric conditions existent in CPCs. However, it is worth to pay attention on the fact that the dielectric role played by the polymer insulating matrix could be also conditioned by the mechanical conditions of periodic loading, contact resistance, and the rheological conditions of polymers [36, 37]. This framework allows to analyze both coupled effects and thus, the shown versions of the “four-particle model” have been implemented in the framework this software provides. FEA has been widely used for microelectromechanical systems (MEMS) due to the possibility of analyzing both coupled electrical and mechanical physics and particularly for sensing devices [38]. The technical details of the model and the executions of simulations to validate the aforesaid ideas are exposed next.

For FEA model geometry, Figure 5 shows the model geometry of the four-particle model built in COMSOL Multiphysics® with the physical parameters from studies in [6, 7]. Some other values have been measured directly from Flexiforce A201-X commercial sensors or taken from datasheets. Also, default parameters of materials library COMSOL Multiphysics® offer have been considered.

For FEA model materials, Figure 5 shows the domains that represent the polymer matrix, whose material has been configured to PDMS (polydimethylsiloxane), present in the COMSOL Multiphysics framework. Silver has been assigned to be the material the nanoparticles, and terminals are made of since it is an isotropic material widespread used for commercial force sensing devices [39, 40]. This setting provides electrical conductivity and mechanical parameters to the simulations to perform. Parameters to set have been presented in Table 1.

Two interfaces were added to the model to simulate the most important physical phenomena occurring when sensor is subjected to mechanical stress and sourcing voltage. *Electrical currents* and *solid mechanics* interfaces’ conditions are described next.

Electric current interface–current conservation consists in a low-voltage stationary study where equations to solve are

where is the current density; is the electric charge; is any electrical external current density; is the electrical conductivity, whose value comes from the previously configured material; is the electrical field; and *V* the electrical potential. Room temperature of 293.15 K has been set for all simulations.

For electric current interface–terminals, the potential comes from the configuration made for both terminals. The lower is set to ground whereas the upper is set to the voltage , whose setting depends on the simulations to run. In order establish a comparison with the results in [28], two simulations must be considered: One at low voltage (1.27 V) and other at higher voltage (9 V).

For electric current interface–contact impedance, when any particle physically contacts the upper terminal or touches another particle, there is contact resistance that must be considered. This is set differently depending on the study to perform and has been set according to the explanations of Section 2. (iii)If the voltage to set is 1.27 V, since the resistance is up to eight orders of magnitude higher, what makes contact resistance negligible. It has been set to zero in the corresponding COMSOL Multiphysics® node(iv)If the voltage to set is 9 V, the contact resistance cannot be neglected, so it has been set to a non-zero value in the corresponding node. Section VI in this paper provides a more detailed explanation in this regard

For electric current interface–other boundary conditions, external boundaries of all models have been set to be electrically isolated, except the lower one, which was set to ground as previously explained. All initial values have been set to zero, which means that every component of the model starts at the same electric potential and there is not electrical charge in any part of the model.

For solid mechanics–linear elastic material, a Burgers model (Figure 7) has been set for the mechanical dynamic study at room temperature. It is worth to recall that these settings are made for the PDMS matrix, which is the violet domain shown for each version of the model in Figure 5. Burgers model parameter is taken from [6]. Since electrodes and particles are set to be made of a highly stiff material such as silver, viscoelastic features are secondary to the extent that some of them were not needed to be set.

For solid mechanics–boundary load, to simulate some gait-like loading patterns, Equation (7) was set as the dynamic periodic pressure to exert on the upper electrode. Hence, pressures from 0 to 300 kPa have been simulated with a frequency of one cycle per second.

For solid mechanics–other boundary conditions, lower electrodes are not allowed to move in any direction whereas the upper only is allowed to move along the vertical axis. All components in the model start from rest.

For electrical probes, there are measuring probes to acquire the voltage between neighboring nanoparticles and, when needed, between nanoparticles and electrodes. These readings are needed to be the voltage in Equations (1) through (3). For every version of the model, there are interparticle or electrode-particle gaps (from now on called tunneling gaps) to take into consideration. Since potential condition does not change over time, they have been acquired with two “Stationary Studies” nodes in COMSOL Multiphysics®, one for each sourcing voltage (1.27 V and 9 V). Figure 8 shows how they have been labeled in this work.

**(a)**

**(b)**

**(c)**

**(d)**

For mechanical probes, as well as voltage needs to be measured with probes for each tunneling gap, their distance at any moment of the simulation is also needed since it is the parameter *s* in Equations (1) through (3). They have been acquired with a “Time Dependent Study” node in COMSOL Multiphysics. Just one node is needed for this since mechanical dynamics does not depend on whether the sourcing voltage is 1.27 V or 9 V.

For all studies, the mesh has been set to the “extra fine” option COMSOL Multiphysics® offers, which provided convergence for all versions. Coarser options did not in some executions.

#### 5. Mechanical and Dynamics Analysis

The first aspect to consider in a mechanical and dynamic analysis must be the rheologic nature of the material the insulating matrix is composed of not only the brand in scope (FlexiForce A201-X) has a PDMS substrate but also most of further products consider different sorts of polymer to manufacture force sensing devices. Logically, this material provides the needed flexibility to deform considerably when subjected to mechanical stress. As discussed previously, this deformation boosts the probability of tunneling conduction between neighboring conductive particles since interparticle gaps shrink when an external force compresses the device. On the other hand, FSR must deal with the typical features of rheology such as creep. In a device such as those belonging to the FlexiForce A201-X series, creep is responsible for pitfalls for measurement such as drift and hysteresis. Precisely, by the unloading half-cycle when the device is subjected to periodic stress, recovering the original shape takes longer for the polymeric device as a natural feature of rheologic behavior. Hysteresis relies on this fact, but also this can affect interparticle distances for some loading patterns. If loading period is short enough, the recovering phase at unloading half-cycle may have not ended when the next loading half-cycle begins. This provokes a striking trend in inter-particle separation by periodic loading which affects also the tunneling conduction between neighboring particles.

Additionally, rheological behavior also involves related responses to its low Poisson ratio. Polymer matrix material exhibits considerable expansion in its radial direction when the exerted force goes in the normal direction. Thus, this slow recovery phase by unloading half-cycle is not observed as an expansion in the normal direction but also as a contraction in the radial one. The interparticle separations experience different trends because of this dynamic behavior depending on the interparticle tunneling gap under scope. Hence, a time-dependent study has been carried out, as described before in COMSOL Multiphysics®, to understand how interparticle distances evolve under periodic loading. Equation (7) shows how much pressure has been considered and how frequent the loading pattern is. Figures 9 and 10 show the von Mises stress for every point of all versions built of the four-particle model.

**(a)**

**(b)**

**(c)**

**(d)**

**(a)**

**(b)**

**(c)**

**(d)**

Although some measurements of the von Mises stress are shown in this work, the most important result of the mechanical simulation has been the distance between neighboring particles with special scope in the labeled tunneling gaps depicted in Figure 8 and organized in Table 2. For version in Figure 8(a), some distances of tunneling gaps are shown in the following time plots in Figure 11. Gaps (with its particles positioned on a vertical line) and (with its particles positioned on a nonvertical line) in Figure 11(a) have been selected to be shown since they represent two of the different trends observed in the simulation. Gap maximum of each cycle is slightly lower that the former one, whereas Gap shows a totally different trend: Its maxima are each time slightly higher. In Figure 11(b), similar trends have been also observed. Gap has an increasing trend whereas Gap decreases. For all time plots, increments or decrements are perceptible after long enough simulations.

**(a)**

**(b)**

Gap distance is highly correlated with gap as well as Gap with Gap , and Gap with . This relies upon the fact of the high symmetry this version of the model exhibits. Gaps , , and correspond symmetrically with , , and , respectively. Hence, only time plots for , , , and are shown in Figure 12.

**(a)**

**(b)**

For the rest of the versions these both trends have been also observed with a relevant detail. Increasing trends are observed for interparticle tunneling gaps whose particles are placed in a mostly vertical fashion, whereas decreasing trends show up when the conducting path between particles has any considerable horizontal component (or radial component, due to the disc shape of the sensor, which would be radial in cylindric coordinates). The more horizontal (radial) component the path has, the more the trend of the distance decreases with every cycle. This observation matched with the expectations due to the rheologic characteristics of the polymer matrix. The observed increasing trend relies upon low Poisson’s ratio since the exerted force goes in the normal direction, so the polymer expands in the horizontal (radial) one.

These mostly vertical gaps, called from now on in this paper “vertical gaps” present this decreasing behavior due to the creep features the polymer matrix present. When the unloading half-cycle starts, it begins to recover to the rest positions and distances. Due to creep, recovery takes longer, and the device does not reach the rest position before the next loading half-cycle begins. Thus, two neighboring vertical particles facing a periodic loading force pattern would be closer with each cycle. Conversely, this uncomplete recovery is also observed in the horizontal (radial) axis. However, these “nonvertical gaps” exhibit quite different trends among them for each cycle. The less vertical the gap is (see timeplot for gap in Figure 11(a) and timeplot for gap in Figure 11(b)), the more ascending the pattern is. Thus, two particles next to each other at similar heights, what could be called a “horizontal gap,” with each loading cycle would be each time further away from each other. Nevertheless, horizontal (radial) component of the gap must be considerable, so the resulting distance can exhibit an increasing trend, see Figures 13 and 14.

**(a)**

**(b)**

**(c)**

The most relevant fact related to these trends of distance between neighboring particles is its dependence on the tunneling resistance. The FEA simulation has also provided the voltage for each tunneling gap, and their analysis will be presented in the next section. However, once the interparticle separation and voltages for each gap are computed, the current density can be calculated with Equations (1) through (3). Thus, the contribution of all gaps in scope can be known. Since mean distances for all gaps in all versions of the model are very different, normalized will be presented next. Normalization has been performed with the maximum of the first loading cycle.

Figure 15 shows what was expected for gaps , , , and nonvertical gaps: The current density shows a decreasing trend of its maxima due to the increasing trend of the interparticle separation. Since the bigger the gap is, the lower the probability for tunneling conduction becomes and that is why a descending current density trends are observed for nonvertical conduction gaps. It is also worth to remark how noticeable the descending trend in is although the trends in interparticle separation are slightly increasing. Figure 16 shows for the vertical gaps and ( is not shown since it is similar to , due to symmetry), which are increasing as expected.

**(a)**

**(b)**

**(a)**

**(b)**

Although FEA simulation has been executed for two different electrical nodes with different sourcing voltages ( V and V), current density time plots acquired from each condition are not distinguishable due to normalization. However, for the following analysis, voltage is not yet relevant.

For nonvertical gaps, it is worth to pay high attention to the descending current density trend of maxima immediately recalls to sensitivity degradation. Figure 16 shows how this descending trend is more notorious for those paths with more horizontal (radial) component. However, increasing trend is dramatically increasing for vertical gaps. Thus, this analysis suggests that, if sensitivity degradation occurs because of tunneling conduction through nonvertical paths, there must be an additional condition that makes nonvertical conduction predominate. As formerly exposed, sensitivity degradation is observed for high sourcing voltages. Thus, considering sensitivity degradation as a voltage-related phenomenon, an electrical analysis must be carried out so its influence can be assessed.

Additionally, there is an extra detail to pay attention to. Figure 17 shows some sort of a secondary peak in the time plot, which has also been observed for interparticle separation in Figures 13–15. These “secondary peaks” have also been reported in [29, 30] as an extra concerning phenomenon associated to periodic loading.

**(a)**

**(b)**

#### 6. Electrical Stationary Analysis

The FEA COMSOL Multiphysics® analysis carried out for this work involves two different voltage sourcing conditions: 1.27 V, which does not show any important degradation in sensitivity, and 9 V, in which sensitivity degrades to worrying extents. It is worth to recall that contact resistance to tunneling resistance proportion relies upon sourcing voltage as explained at the end of Section 2 (see Figure 3). The following FEA electrical analysis has paid the most attention to equipotential surfaces formed in the sensing device. The behavior of these surfaces will depend on the proportion of contact resistance and the tunneling resistance . As Figure 3 shows, when sourcing voltage is low, the total device resistance is composed predominantly by , which is >10^{8} *Ω* higher than , making it negligible. Conversely, this dramatical difference between both resistances is not that large when sourcing voltage is high. Due to the higher probability of tunneling conduction at higher voltages, decreases to the extent that it could be close to values. Thus, the full device resistance will be comparable between and and no longer negligible. Two electrical conduction nodes in the COMSOL Multiphysics® Model Builder have been configured, where contact resistance is set to 0 for low sourcing voltage (1.27 V) and a higher value for high sourcing voltage (9 V). This higher resistance was calculated according with the discrete considerations presented in [41–43] about the quantum point contacts. Equations (8) and (9) are needed to understand how contact resistance has been set in the model.

where is the discrete increment of contact conductance, the length of the contact, and the Fermi level. The smallest integer that makes Equation (9) match will define how many increments will sum up the full contact conductance and its inverse hat been set as contact resistance (). For all simulations, yielded 12906 *Ω*, what is in the expected magnitude according to previous experiments where contact resistance has been calculated [5, 6].

Figure 18 shows the electric potential in the whole device when sourcing voltage is 1.27 V. In the models where any kind of contact between parts exists, same potential is observed for both contacting parts, since contact resistance is negligible. Thus, high potential differences are seen for, say, gap in version of the model in Figure 18(a), what also favors a more intense electric field. Two contacting particles are present in version of the model in Figure 18(b). Hence, both are at the same potential and thus, a more intense electrical field is present in gap (see Figure 8(b)).

**(a)**

**(b)**

**(c)**

**(d)**

On the other hand, Figure 19 shows the same study with a sourcing voltage of 9 V, *ceteris paribus*. In the version of the model where there is any kind of contact, the voltage drop could be up to the half of the total voltage drop through the device since both and resistance could be similar. Thus, electrical potential changes significantly, except for version of the model in Figure 19(c), in which no contact situation has been considered. Comparing Figures 18(b) and 19(b), for instance, offers an idea of how potential changes affect the arousal of electrical fields in some other parts of the devices. Instead of a more intense electrical field in gap in Figure 18(b), in Figure 19(b), this field is more notorious for gap , which is nonvertical. In Figure 18(a), the high potential difference in gap has decreased in comparison with Figure 19(a). Also, for Figures 18(d) and 19(d), there are quite notorious differences between the potential differences in gap . Thus, the solely presence of a more considerable contact resistance, which arises due to higher sourcing voltages, provokes changes in the potential spatial distribution that in the simulated cases, favor conduction through nonvertical gaps rather than through vertical ones, because the higher the voltage in a certain gap is, the higher the probability of tunneling conduction becomes. Favoring conduction through nonvertical gaps means favoring it also through time-increasing tunneling resistances. This explains why high voltages, which are responsible of bending over equipotential surfaces and forcing electric field directions to be also nonvertical, causing the tunneling current to flow through particles which experience an increasing trend in their separation and therefore an increasing resistance, which leads to observe a time-decreasing current (sensitivity degradation), as seen in Figure 4 by the insoles experiment. As it is for the sensors in the insoles, this is an only voltage related phenomenon since it needs the electric field to be nonvertical. For low voltages, simulations show a mostly vertical direction for electric field, not favoring conduction through increasing gaps. This explains why SD is not observed for low voltages and only arises when sourcing voltages are high, as seen in Figure 4.

**(a)**

**(b)**

**(c)**

**(d)**

#### 7. Conclusions and Ideas for Future Work

Calculating the whole device current with all tunneling contributions from each gap is not a straightforward task. Random dispersion of nanoparticles in the polymer matrix hinders the process of finding out a general and accurate expression for current. Moreover, the high amounts of particles in a single specimen make any full model execution computationally too expensive. However, this 2D FEA model approach is a simple beginning to shed some light to more complete endeavors. Considering four particles have been the simplest representation found to portray the two desired key phenomena that provoke SD (increasing trend of interparticle separation maxima and equipotential surfaces modification due to contact resistance). It is logical to think that taking this model to the next level would be adding a third dimension. Any 3D model built under the consideration made in this work would increase the possibilities of the so-called nonvertical tunneling conduction gaps, since the third dimension to add would be orthogonal to the conducting downwards direction. Results in this work encourages to go forward in 3D modeling, but also it is worth to bear in mind that full accuracy could anyway be not reachable, and version of the models could be never-ending.

Nevertheless, this study also opens a window to lessen undesirable consequences of SD, which relies on voltage-related phenomena according with the potential consequences for high sourcing voltage. It has been already recommended to consider low sourcing voltage for applications where SD may mislead measurements. However, manufacturing considerations to reduce contact resistance could also avoid the bending over of equipotential surfaces and permit higher sourcing voltage, which are normally desired for boosting signal-to-noise ratios in acquired readings. This works also motivates to experiment in modifying manufacture protocols aiming to lessen contact resistance.

#### Data Availability

Data can be found in the Supplementary Information files: Palacio, Carlos (2021), “Finite Element Analysis Approach for Sensitivity Degradation in Force Sensing Resistors,” Mendeley Data, V1, doi:10.17632/5z2gd75v7z.1

#### Conflicts of Interest

The authors declare that there is no conflict of interests.

#### Acknowledgments

The author Carlos Palacio acknowledges the Antonio Nariño University under project number 2021011-PI/UAN-2021-704GIFAM. The author Arnaldo Matute acknowledges the Juan De Castellanos University Foundation under project number CI00120-3.