Ultra-sensitive Pressure sensor based on guided straight mechanical cracks

Recently, a mechanical crack-based strain sensor with high sensitivity was proposed by producing free cracks via bending metal coated film with a known curvature. To further enhance sensitivity and controllability, a guided crack formation is needed. Herein, we demonstrate such a ultra-sensitive sensor based on the guided formation of straight mechanical cracks. The sensor has patterned holes on the surface of the device, which concentrate the stress near patterned holes leading to generate uniform cracks connecting the holes throughout the surface. We found that such a guided straight crack formation resulted in an exponential dependence of the resistance against the strain, overriding known linear or power law dependences. Consequently, the sensors are highly sensitive to pressure (with a sensitivity of over 1 × 105 at pressures of 8–9.5 kPa range) as well as strain (with a gauge factor of over 2 × 106 at strains of 0–10% range). A new theoretical model for the guided crack system has been suggested to be in a good agreement with experiments. Durability and reproducibility have been also confirmed.

or a nearly identical log-logistic pdf and are parameters of the pdf.
Both of the distributions of Eqs. (S3) and (S4) belong to the class of so-called skew distributions with long tails. As we discussed in Ref. 1, the non-zero probability of large but rare contacts between crack lips lies in the essence of the mechanism of the conduction through the crack and is therefore in concordance with the tailed distributions. With Eq.
(S3), Eq. (S1) gives for the resistance = 1/ as a function of strain the following: erf ( ) is the error function. Eq. (S5) renders the normalized resistance that remarkably fits the experiment 15 for the strains up to 2%. At the same time, one can show that the log-logistic pdf of Eq. (S4) together with Eq. (S1) leads to that fits the experiment 15 with fitting parameters 0 = 0.39 and = 2.39 (see Fig S12) with the same accuracy as the log-normal pdf of Eq. (S5). Yet, the power-law function of Eq. (S6) is much simpler than the error function in Eq. (S5). We may suggest this universal power law for data fitting by experimentalists who study free parallel cracks.
Quite surprisingly, after we changed the uniform Pt film strip into a patterned one on a much more stretchable polymer ( Fig. 1) of the present work, the strain dependence of the resistance dramatically switched from the power-law of Eq.(S6) into the exponential one in a much broader strain range up to 5% and more. One can notice the straight-line behavior up to 5 % strain from the semi-log plot given in Fig 2c. Here we discuss the underlying mechanism of this phenomenon.
A crucial difference of the cracks generated in the current study from those in the previous study 15 is shown in Fig S10 b,c: the cracks between pattern patches closely follow the "crests" of the wrinkles on the metal/polymer film. That means that the crack 4 path was extremely directed and only close neighboring Pt grains were disconnected along the crack lip ( Fig S11). In this respect, the local deviations are about the size of a grain and thus, may not satisfy the scaling Eq. (S2) for free crack generation. On the other hand, the pattern patches were pressed to each other in the horizontal and perpendicular direction to the strain direction as shown in Fig S10b, because of the Poisson ratio of 0.5 2 which is an inherent characteristic of rubber-like materials. Therefore, the system remained unchanged effectively in one dimension with cut-through cracks that are now located on a train of squares lined up in the horizontal direction ( Fig S10).
In analogy with the study 15 , it is sufficient to calculate the step pdf. According to is the delta-function, and = 1,2, … The delta-function presents the microscopic pdf of a step to be constructed of positive shifts in one direction satisfying the equation 1 + ⋯ + − = 0. As we assumed, the probability of shifting a grain up (down) is ½. Therefore, defining the step as the total shift up, the probability of a given configuration with small steps of should be proportional to It is clear from Eq. (S1) that the conductance will also be the exponential function of strain at large strains, as well as the resistance One can see the difference between Eq. (S6) and Eq. (S11), the power-law and the exponential. A quite general example of ( ) = 1 that assumes arbitrary grain positions with respect to each other neighbor and then, a homogeneous distribution of the grain shifts along the crack lip in Fig S11, gives the dominating pole of 0 = 1.256 (see the Supplementary Information and Fig S12). In Fig 2e we give the normalized resistance vs strain calculated with ( ) = 1 (the red line) along with the pure exponential function of Eq. (S11) (the green line in Fig 2e) to see a close coincidence between the experimental data and the theory.

6
While fitting the experiment with, for example, the resistance vs strain calculated with the uniform pdf of grains with the asymptotic function Eq. (S11), one had to rescale the strain by = 0.7 times to match the linear slope of the experiment in Fig 2c. Physically that means that we restrict the shift of the grains by 30% and thus flatten the

Complex integration
Then, after rewriting -function as a Fourier integral, Eq. (7) reads, or after simplification of Eq. (S12) due to the independent integration over each of The geometrical series of Eq. (S2) can be directly transformed into The Cauchy integral in Eq. (S15) can be analyzed in general terms. One can notice that the decay of function ( ) at large may be nearly exponential and nearly independent of a particular form of ( ): Eq. (S17) cannot be satisfied, because it demands that f( ) = 2 > 1.
Consider a quite general example of ( ) = 1 that assumes arbitrary grain positions with respect to each other neighbor and then, a homogeneous distribution of the grain shifts along the crack lip in Fig S12. For this case Eq. (S14) gives: ( ) = (exp( ) − 1)/ (S18) 8 and then Eq.(S17) takes the form: Solutions of Eq. (S18) can be found numerically. The lowest 0 = 1.256 and the other poles are 2.789 ± 7.438 , 3.360 ± 13.866 …. (see Fig S12).

Estimating the low end detection limit
It is observed that the maximum resistance change by strain 0.04 % was 2.1468 Ω, and the noise level was below 0.7037 Ω. Signal-to-noise ratio become as ~3, so the low end detection limit is 0.04/3 = 0.013%.