Force-sensing resistor

The percolation phenomenon dominates in the conductive polymer when the particle concentration is above the percolation threshold ${\displaystyle \phi _{c}}$. A force-sensing resistor operating based on percolation exhibits a positive coefficient of pressure, and therefore, an increment in the applied pressure causes an increment in the electrical resistance ${\displaystyle R}$,[12][13] For a given applied stress ${\displaystyle \sigma }$, the electrical resistivity ${\displaystyle \rho }$ of the conductive polymer can be computed from:[14]

${\displaystyle \rho =\rho _{0}(\phi -\phi _{c})^{-x}}$

where ${\displaystyle \rho _{0}}$ matches for a prefactor depending on the transport properties of the conductive polymer, and ${\displaystyle x}$ is the critical conductivity exponent.[15] Under percolation regime, the particles are separated from each other when mechanical stress is applied; this causes a net increment in the device's resistance.

Quantum tunneling is the most common operation mode of force-sensing resistors. A conductive polymer operating on the basis of quantum tunneling exhibits a resistance decrement for incremental values of stress ${\displaystyle \sigma }$. Commercial FSRs such as the FlexiForce,[16] Interlink [17] and Peratech [18] sensors operate based on quantum tunneling. The Peratech sensors are also referred to in the literature as quantum tunnelling composite.

The quantum tunneling operation implies that the average inter-particle separation ${\displaystyle s}$ is reduced when the conductive polymer is subjected to mechanical stress; such a reduction in ${\displaystyle s}$ causes a probability increment for particle transmission according to the equations for a rectangular potential barrier.[19] Similarly, the contact resistance ${\displaystyle R_{C}}$ is reduced amid larger applied forces. To operate based on quantum tunneling, particle concentration in the conductive polymer must be held below the percolation threshold ${\displaystyle \phi _{c}}$.[6]

Several authors have developed theoretical models for the quantum tunneling conduction of FSRs,[20][21] some of the models rely upon the equations for particle transmission across a rectangular potential barrier. However, the practical usage of such equations is limited because they are stated in terms of electron energy, ${\displaystyle E}$, that follows a Fermi Dirac probability distribution, i.e., electron energy is not a priori determined or can not be set by the final user. The analytical derivation of the equations for a rectangular potential barrier including the Fermi Dirac distribution was found in the 60`s by Simmons.[22] Such equations relate the current density ${\displaystyle J}$ with the external applied voltage across the sensor ${\displaystyle U}$. However, ${\displaystyle J}$ is not straightforward measurable in practice, so the transformation ${\displaystyle I=JA}$ is usually applied in literature when dealing with FSRs.

Just as in the equations for a rectangular potential barrier, the Simmons' equations are piecewise regarding the magnitude of ${\displaystyle U}$, i.e., different expressions are stated depending on ${\displaystyle U}$ and the height of the rectangular potential barrier ${\displaystyle V_{a}}$. The simplest Simmons' equation [22] relates ${\displaystyle I}$ with ${\displaystyle U}$,${\displaystyle s}$ when ${\displaystyle U\approx 0}$ as next:

${\displaystyle I(U,s)={\frac {3A{\sqrt {2mV_{a}}}}{2s}}({\frac {e}{h}})^{2}U\exp(-{\frac {4{\pi }s}{h}}{\sqrt {2mV_{a}}})}$

where ${\displaystyle V_{a}}$ is in units of electron volt, ${\displaystyle m}$, ${\displaystyle e}$ are the electron's mass and charge respectively, and ${\displaystyle h}$ is the Planck constant. The low voltage equation of the Simmons' model [22] is fundamental for modeling the current conduction of FSRs. The most widely accepted model for tunneling conduction has been proposed by Zhang et al.[23] based on such equation. By re-arranging the equation above, it is possible to obtain an expression for the conductive polymer resistance ${\displaystyle R_{Pol}}$, where ${\displaystyle R_{Pol}}$ is given by the quotient ${\displaystyle U/I}$ according to the Ohm's law:

${\displaystyle R_{\it {Pol}}={\frac {s}{A{\sqrt {2mV_{a}}}}}({\frac {h}{e}})^{2}\exp({\frac {4{\pi }s}{h}}{\sqrt {2mV_{a}}})}$

When the conductive polymer is fully unloaded, the following relationship can be stated between the inter-particle separation at rest state ${\displaystyle s_{0}}$,the filler volume fraction ${\displaystyle \phi }$ and particle diameter ${\displaystyle D}$:

${\displaystyle s_{0}=D{\Big [}{\Big (}{\frac {\pi }{6\phi }}{\Big )}^{\frac {1}{3}}-1{\Big ]}}$

Similarly, the following relationship can be stated between the inter-particle separation ${\displaystyle s}$ and stress ${\displaystyle \sigma }$

${\displaystyle s=s_{0}(1-{\frac {\sigma }{M}})}$

where ${\displaystyle M}$ is the Young's modulus of the conductive polymer. Finally, by combining all the equations above, the Zhang's model [23] is obtained as next:

${\displaystyle R_{\it {Pol}}={\frac {D{\Big [}{\Big (}{\frac {\pi }{6\phi }}{\Big )}^{\frac {1}{3}}-1{\Big ]}(1-{\frac {\sigma }{M}})}{A{\sqrt {2mV_{a}}}}}{\big (}{\frac {h}{e}}{\big )}^{2}\exp {\Big (}{\frac {4{\pi }D}{h}}{\Big [}{\Big (}{\frac {\pi }{6\phi }}{\Big )}^{\frac {1}{3}}-1{\Big ]}(1-{\frac {\sigma }{M}}){\sqrt {2mV_{a}}}{\Big )}}$

Although the model from Zhang et al. has been widely accepted by many authors,[11][9] it has been unable to predict some experimental observations reported in force-sensing resistors. Probably, the most challenging phenomenon to predict is sensitivity degradation. When subjected to dynamic loading, some force-sensing resistors exhibit degradation in sensitivity.[24][25] Up to date, a physical explanation for such a phenomenon has not been provided, but experimental observations and more complex modeling from some authors have demonstrated that sensitivity degradation is a voltage-related phenomenon that can be avoided by choosing an appropriate driving voltage in the experimental set-up.[26]

The model proposed by Paredes-Madrid et al.[10] uses the entire set of Simmons' equations [22] and embraces the contact resistance within the model; this implies that the externally applied voltage to the sensor ${\displaystyle V_{FSR}}$ is split between the tunneling voltage ${\displaystyle V_{bulk}}$ and the voltage drop across the contact resistance ${\displaystyle V_{Rc}}$ as next:

${\displaystyle V_{FSR}=2V_{RC}+V_{bulk}}$

By replacing sensor current ${\displaystyle I}$ in the above expression, ${\displaystyle V_{bulk}}$ can be stated as a function of the contact resistance ${\displaystyle Rc}$ and ${\displaystyle I}$ as next:

${\displaystyle V_{bulk}=V_{FSR}-2RcI}$

and the contact resistance ${\displaystyle R_{C}}$ is given by:

${\displaystyle R_{C}=R_{\it {par}}+{\frac {R_{C}^{0}}{\sigma ^{k}}}}$

where ${\displaystyle R_{par}}$ is the resistance of the conductive nano-particles and ${\displaystyle R_{C}^{0}}$, ${\displaystyle k}$ are experimentally determined factors that depend on the interface material between the conductive polymer and the electrode. Finally the expressions relating sensor current ${\displaystyle I}$ with ${\displaystyle V_{FSR}}$ are piecewise functions just as the Simmons equations [22] are:

When ${\displaystyle V_{bulk}\approx 0}$

${\displaystyle R_{\it {bulk}}={\frac {s_{0}(1-{\frac {\sigma }{M}})}{(A_{0}+A_{1}\sigma ^{A_{2}}){\sqrt {2mV_{a}}}}}({\frac {h}{e}})^{2}\exp({\frac {4{\pi }s_{0}(1-{\frac {\sigma }{M}})}{h}}{\sqrt {2mV_{a}}})}$

When ${\displaystyle V_{bulk}<V_{a}/e}$

${\displaystyle I={\frac {(A_{0}+A_{1}\sigma ^{A_{2}})e}{2{\pi }hs_{0}^{2}(1-{\frac {\sigma }{M}})^{2}}}{\Bigg \{}(V_{a}-{\frac {V_{bulk}}{2}})\exp {\Bigg [}-{\frac {4{\pi }}{h}}s_{0}(1-{\frac {\sigma }{M}}){\sqrt {2m(V_{a}-{\frac {eV_{bulk}}{2}})}}{\Bigg ]}-(V_{a}+{\frac {V_{bulk}}{2}})\exp {\Bigg [}-{\frac {4{\pi }}{h}}s_{0}(1-{\frac {\sigma }{M}}){\sqrt {2m(V_{a}+{\frac {eV_{bulk}}{2}})}}{\Bigg ]}{\Bigg \}}}$

When ${\displaystyle V_{bulk}>V_{a}/e}$

${\displaystyle I={\frac {2.2e^{3}V_{bulk}^{2}(A_{0}+A_{1}\sigma ^{A_{2}})}{8{\pi }hV_{a}s_{0}^{2}(1-{\frac {\sigma }{M}})^{2}}}{\Bigg \{}\exp {\Bigg [}-{\frac {8{\pi }s_{0}(1-{\frac {\sigma }{M}})}{2.96heV_{bulk}^{2}}}{\sqrt {2mV_{a}^{3}}}{\Bigg ]}-(1+{\frac {2eV_{bulk}}{V_{a}}})\exp {\Bigg [}-{\frac {8{\pi }s_{0}(1-{\frac {\sigma }{M}})}{2.96heV_{bulk}}}{\sqrt {2mV_{a}^{3}(1+{\frac {2eV_{bulk}}{V_{a}}})}}{\Bigg ]}{\Bigg \}}}$

In the equations above, the effective area for tunneling conduction ${\displaystyle A}$ is stated as an increasing function dependent on the applied stress ${\displaystyle \sigma }$, and on coefficients ${\displaystyle A_{0}}$, ${\displaystyle A_{1}}$, ${\displaystyle A_{2}}$ to be experimentally determined. This formulation accounts for the increment in the number of conduction paths with stress:

${\displaystyle A=A_{0}+A_{1}\sigma ^{A_{2}}}$

Although the above model [10] is unable to describe the undesired phenomenon of sensitivity degradation, the inclusion of rheological models has predicted that drift can be reduced by choosing an appropriate sourcing voltage; experimental observations have supported this statement.[26] Another approach to reduce drift is to employ Non-aligned electrodes to minimize the effects of polymer creep.[27] There is currently a great effort placed on improving the performance of FSRs with multiple different approaches: in-depth modeling of such devices in order to choose the most adequate driving circuit,[26] changing the electrode configuration to minimize drift and/or hysteresis,[27] investigating on new materials type such as carbon nanotubes,[28] or solutions combining the aforesaid methods.