Artificial fast-adapting mechanoreceptor based on carbon nanotube percolating network

Most biological sensors preferentially encode changes in a stimulus rather than the steady components. However, intrinsically phasic artificial mechanoreceptors have not yet been described. We constructed a phasic mechanoreceptor by encapsulating carbon nanotube film in a viscoelastic matrix supported by a rigid substrate. When stimulated by a spherical indenter the sensor response resembled the response of fast-adapting mammalian mechanoreceptors. We modelled these sensors from the properties of percolating conductive networks combined with nonlinear contact mechanics and discussed the implications of this finding.

www.nature.com/scientificreports/ From a biomimetic perspective, the availability of artificial sensors with intrinsic phasic properties is highly desirable according to the parsimonious principle of "morphological computation" where the physics of the embodiment of an organism perform functions that are relevant to perceptual or motor behaviour 34,35 . We describe here a mechano-sensing cell that exhibits an intrinsic phasic response without resorting to electronic circuitry. This attribute ties the delivered signal to changes in deflection rather than to deflection per se. These mechano-sensing cells, manufactured by embedding multi-wall carbon nanotubes (MWCNT) in a compliant polymer matrix, are soft, miniaturisable, printable, and compatible with large-scale production processes 24 .

Results
Compliant tactile sensing cells were manufactured by embedding MWCNT films in a Polydimethylsiloxane (PDMS) matrix. The films formed a conductive percolation network that acted as a piezoresistive sensor with a resting resistance of ≃ 30 k . The films were 12.5 × 2.5 × 0.5 mm strips embedded in a 1.3 mm-thick elastic matrix at a distance of 0.65 mm from the upper surface. The elastic layer was set on a rigid aluminum support, as in Fig. 1a. Several sensing cells were tested using a universal testing machine ("Methods") that imposed sequences of indentations by a 5 mm spherical probe with a loading phase, a hold phase, and an unloading phase, as in Fig. 1b. In a given specimen, indentations produced the load variations shown in Fig. 1c and the relative variation of resistance of the film shown in Fig. 1d. Cells tested through thousands of loading cycles did not exhibit signs of damage, as shown in Fig. 1e. We tested ten samples of the same structure and same fabrication process. The maximal changes of resistance during the loading phases had a standard deviation of 2.3%, showing good precision across samples. The accuracy of the samples remained under 9.5% over 1500 cycles.
Surprisingly, the measurements indicated that relative change of resistance was the greatest during the loading and the unloading phases and remained low during the holding phase. In other words, the sensor responded mostly to changes in normal indentation. In addition, during steady loading, the response exhibited slow relaxation. Thus, the sensor cell had a response bearing an intriguing similarity with that of rapidly adaptive sensor cells found in mammalian skin 19 . A second experiment where indentation was varied for a same speed of indentation and where the speed of indentation was varied to reach the same indentation gave the results shown in Fig. 2a,b, respectively. The change of resistance during the loading and unloading phases was clearly a function both of the final magnitude of the indentation and of the rate of indentation, prompting us to investigate the mechanisms responsible for this surprising behaviour.

Mechanism
Empirical model. Measurements showed nonlinear stiffening typical of Hertzian contact mechanics and rate-dependent hysteresis loops that were attributed to the viscoelasticity of the matrix as shown in Fig. 3a. A three-parameter generalised Maxwell model described this behaviour (see Supplementary Text S1). The load www.nature.com/scientificreports/ rate as a function of deflection, Fig. 3b, suggests that a change of deformation regime took place at around 0.4 mm of indentation. Let κ(δ) relate the deflection, δ , to the relative change in electrical resistance, κ = �R/R 0 . Figure 3b,c plots this map for different indentation rates (same data as in Fig. 2b). The dependency on deflection is well captured by a smooth approximation (red line in Fig. 3c) expressed by, where a is a scaling factor and σ describes the spread of the response. The modelled response (1) did not depend on whether the sensor was loaded or unloaded. The rate of change of κ with respect to δ (red line in Fig. 3d) is, Since δ is a function of time, .
. www.nature.com/scientificreports/ Equation (3) captures the sensitivity of the sensor cell to the rate of indentation as it could be observed in Fig. 2b. The sensor response was quicker for higher indentation rates. Likewise, (3) predicts that for a same indentation rate, the response is higher for greater deflections, which can also be observed in Fig. 2a.
Physical mechanism. The response captured by (1) must have been grounded in the physics of the sensor.
The simplest model could follow from standard piezoresistivity (see Supplementary Text S2) which describes the relative variation of resistance of an elongated element. The model combines the intrinsic variation of specific resistance of the material, �ρ/ρ , owed to the strain, ε , to which the element is exposed and the effect of dimensional changes represented by axial and transverse strains, ε a and ε t , respectively. If ν is the Poisson ratio of the material, then, Thus, standard piezoresistivity model (4) cannot explain the reversal of the sign of the change of resistance empirically observed in Fig. 3c,d. A more detailed model could appeal to the properties of percolation networks. Their behaviour is modelled by evaluating the average length and number of junctions established in an heterogenous conductor-insulator mixture 36 . When the number of junctions is assumed to be constant, the relative change of resistance takes the form (see Supplementary Text S3), where γ is a constant involving the mass and charge of an electron, Planck's constant, and the potential barrier between nanotubes, and where s 0 is the effective distance between nanotubes in an undeformed film. An increase of resistance is attributed to an increase of the average distance between nanotubes that exponentially reduces the probability of electrons to tunnel through junctions. Slabs of CNT films were shown to verify this model 37,38 . Nevertheless, the nonlinear relationship expressed by (5) is still unable to explain the observed sign reversal of the change of resistance.
To elucidate the mechanism behind the observed reversal of resistance, we set up a finite element simulation (2D planar case, COMSOL Multiphysics, Stockholm, Sweden) shown in Fig. 4a since there is no known  www.nature.com/scientificreports/ closed-form solution to this contact problem 39,40 . The simulation suggested that the spherical probe deformed the sensitive film in a manner that initially resembled a Hertzian contact problem. However, the elastic layer mounted on a rigid support caused an infinite half-space approximation to no longer be applicable in the later stages of the indentation. The principal strains grew increasingly fast and a larger portion of film was recruited when the probe approached the support. This was confirmed by plotting the surface pressure profile, p(r), which at 0.7 mm of indentation departed significantly from the Hertz model, p(r) ∝ 1 − r 2 /R 2 . Figure 4b shows the axial and transverse strains along the length of the piezoresistive film during indentation. They clearly show that deformation rapidly transitions to a large strain regime over an increasingly larger region owing to the effect of the rigid support. The observed non-monotonic response can now be explained as follows. Kang et al. subjected SWCNT nano-composite films to large tensile and compressive loads and found the relative resistance to increase with axial strain and decrease with compressive loads 37 . Their analysis suggested that changes in the tunnelling resistance were combined with an alteration of the band-gap in the junctions. Thus, the contact mechanics induced in an elastic film trapped between an indenter and a rigid support may have provoked a switch in the material state from a dominantly axial deformation that increased resistance to transverse deformation that decreased resistivity.
The model (1) describes the empirical response of the sensor cell by the combination of these two effects, Fig. 5a. The viscoelasticity of the polymer matrix caused material relaxation under explaining the slow response at constant indentation and the asymmetry between the loading an unloading phases in the empirical measurement and in the simulation.

Discussion
A specific nonlinearity confers a phasic character. The complete response to indentation could be explained by a nonlinear model formed by the product of a Gaussian function, g(δ) , with a parabola, δ 2 , as in model (1). The dynamic response was then the product of the temporal rate of indentation with the rate of change of the response to indentation as in (3). The crucial point is the change of sign of the response's rate of change, which, through mechanical nonlinear filtering, gave the sensor a phasic character. It only responded to changes in indentation. This property could also be captured by a Gaussian wavelet centred around a specific value of the indentation, δ r , where the change of resistance as a function of indentation changes sign ( δ r ≈ 0.4 mm for the sample we tested). The derivative of the response with respect to indentation would then be given by a Gaussian multiplied by the first Hermite polynomial, −σ −2 (δ − δ r ) g(δ − δ r ) . Rational expressions, splines, and others options are also available to represent the empirical results, aδ 2 /(b + δ 6 ) , for example.
Connection to biological sensors. Knibestöl 18 found a diversity of responses to indentation rate in fastadapting mechanoreceptors, with thresholds distributed over a hundred-fold span. He could developed a threeparameter sigmoidal-log model that accounted for the individual responses of twenty units. Iggo and Ogawa 19 when attempting to disentangle the response to magnitude of indentation from rate of indentation, found a significant relationship between the rate of discharge and the rate of indentation and a weak relationship with magnitude. Burgess et al. 20 also found a relationship between indentation rate and receptor response. Hayward et al. 41 investigated the dependency of the neuronal response on the temporal-rate of mechanical stimuli in second-order cells. In mammals, second-order cells receive mono-synaptic inputs from a large number of primary afferents which, statistically, are mostly fast-adapting. Certain types of stimuli elicited responses that were strongly dependent on stimulus variations due to movement and therefore depend on phasic sensing. It may be speculated that the biological mechanoreceptors gain phasic properties through static nonlinearities of the type of model (1). Models of molecular mechanisms that produce specific phasic responses in mechanoreceptors are as of yet largely unknown 42-44 . Derivative designs. The sensory response described earlier can vary according to numerous design factors including the geometry of the surface impinging on the MWCNT film, the thickness of the film, the concentration of MWCNT, the material properties of the elastomer matrix in which they are embedded, the geometry and the material properties of the viscoelastic medium in which the film is embedded. The design space of sensing units based on similar principles is large.
Practical realisations could differ from the sensing unit studied in our experiments. For example, an inverted design with protuberances on the rigid support surface to obtain the desired nonlinear characteristics, as in Fig. 5b, would have advantages for use as an artificial skin. www.nature.com/scientificreports/ Relationship with event-based vision. In computer vision, "event-based cameras" 45 that encode the change of brightness per pixel, and which are also bio-inspired sensors, share many characteristics with the sensing principle we have described. These characteristics include high temporal resolution, data transmission reduction, and the ability to serve as front-end to learning-based techniques such as spiking neural networks 46 . Application to e-Skins. The need to provide biomimetic artificial skins with sensitivity to changes rather than static stimulation has been recognised since the 1970's 47,48 . Recent designs continue to attribute great importance to phasic sensing in robot proprioception 49 , tactile imaging 32 , prosthetics 50 , neurorobotics 51 , human-like sensing 52 , artificial discriminative touch 53 , or neuroprosthetics 54 , wearable Human-Machine interfaces 55 , inter alia. Conductive polymer. Conductive polymer was PDMS based with Long-MWCNT carboxylic acid groups (Ad Nano Technologies LTD, Karnataka, India). Aqueous solutions were obtained by diluting PDMS and Long-MWCNT in CHCl 3 . The solution was shear-mixed at 300 r.p.m., adding CHCl 3 to preserve viscosity. Curing agent was then added in proportion of 10:1.

Methods
Sensor fabrication. MWCNT films were deposited by spray-coating aqueous solutions holding the substrate at 130 • C and airbrushing at 1.7 bar. Shadow masks were realised by laser cutting. Multiple coatings achieved desired resistivity. MWCNT films formed a 1.5 µm thick conductive network (Fig. 6a). Scanning Electron Microscopy revealed the conductive network (Fig. 6e). Conductive polymer stretchable interconnects were screen printed (Fig. 6b). Interconnects were cured at 110 • C to connect to flexible printed circuits and the sensors encapsulated in PDMS (Fig. 6c,d).

Test bench and mechanical sequences. Resistance was measured with Keithley 2400 series SMU
Instruments. In-house LabView program communicated with universal testing machine and Keithley SMU to command test sequences and log data. Compressive mechanical tests included deflections from 0.1 to 1.5 mm, loads from zero to 10 N and strain rates from 0.1 to 5mm s −1 at room temperature. Each trial comprised at least three cycles.

Data availability
The full set of data used to produce the results in this paper is available from the corresponding author upon request.