A Simple Model Relating Gauge Factor to Filler Loading in Nanocomposite Strain Sensors

Conductive nanocomposites are often piezoresistive, displaying significant changes in resistance upon deformation, making them ideal for use as strain and pressure sensors. Such composites typically consist of ductile polymers filled with conductive nanomaterials, such as graphene nanosheets or carbon nanotubes, and can display sensitivities, or gauge factors, which are much higher than those of traditional metal strain gauges. However, their development has been hampered by the absence of physical models that could be used to fit data or to optimize sensor performance. Here we develop a simple model which results in equations for nanocomposite gauge factors as a function of both filler volume fraction and composite conductivity. These equations can be used to fit experimental data, outputting figures of merit, or predict experimental data once certain physical parameters are known. We have found these equations to match experimental data, both measured here and extracted from the literature, extremely well. Importantly, the model shows the response of composite strain sensors to be more complex than previously thought and shows factors other than the effect of strain on the interparticle resistance to be performance limiting.


S2
between 800 and 200 nm, and a 4 mm path length-reduced volume quartz cuvette was used for the measurement.

Gauge factor derivation
The definition of the gauge factor: The resistance of a sample can be related to its conductivity by: Assuming that the volume remains constant at low strain,

Model derivation
In classical percolation theory, composite conductivity,  is related to the filler volume fraction,  , by 1 Figure S1: Literature data 3-20 for Gauge factor (G) as a function of zero-strain conductivity (0) for a range of nanocomposites. Here we observe that for almost all composites G decreases as conductivity increases. A look at the data indicates an approximate power law relationship between these parameters for well-defined data sets. Figure S2: Electro-mechanical characterisation: Representative resistance-strain measurements for G-putty (A) and Graphene-sylgard (B). As graphene volume fraction approaches the percolation threshold composite resistance becomes increasingly sensitive to strain, this leads to larger Gauge factor, G values, which can be obtained by linear fits to the data as in Figure 3C,D (main text). Gr-Sylgard -6.2%

Fit parameters
As described in the main text, standard strain sensor measurements lead to three distinct data sets (0 vs. , G vs.  and G vs. 0) that can be fit using equations 1, 5a and 5b yielding nine (main text) fit parameters as listed To expand our data set, we examine not only the fit parameters obtained from figure 3 (main text) but also parameters found by fitting literature data (Fig S3-7).
We first plot the percolation threshold found by fitting the G vs.  data using equation 5a with that obtained by fitting the 0 vs.  data using equation 1 (Fig S8 A). Here we find extremely good agreement with all data very close to the line defining y=x. We then plot the percolation exponent found by fitting the G vs. 0 data using equation 5b with that obtained by fitting the 0 vs.  data using equation 1 (Fig S8 B). Here we find the data points lying in the vicinity of the y=x line, although note some non-trivial deviations. For example, one of the graphenepolymer composites G-putty, 4 shows exponents of 10 and 7 which is a non-trivial deviation.
However, by and large, we believe the agreement shown in figure S8 B is reasonable.
Shown in figure 3C are data for 0, G  obtained by fitting G vs.  data using equation 5a plotted versus 0, G  obtained by fitting G vs. 0 data using equation 5b. While these parameters should be equal as described above, we expect this data to show the greatest scope for deviation. The reason for this is that accurate values of these parameters require good data for G at high values of  (and so 0). However, most papers do not report such data while aggregation effects can cause large errors. Yet, we do find reasonable agreement between these parameters with only two samples, both graphene-based giving significant deviation. Taken together, the results presented in figure S8 A-C give us confidence that the model above can accurately describe data. In addition, we note that equation 5 a-b show that large, positive values of 0, G  and 0, G  lead to higher gauge factors. Interestingly, while most of the data is within 5 units of the origin, S11 two data points, both representing soft, graphene-siloxane composites lie far from the origin.
One of these data points has large, positive values of 0, G  and 0, G  , which will act to boost G, the other has large negative values, which will act to decrease G.
Assuming    Table S1. Comparing these values (Table S2) to the expected values (determined obtained from linear fits at low strain to the percolation data presented in Figure 4 (C-H) main text) shows reasonable agreement in some values however others are far from the expected values from or have large errors. We mainly attribute this to a limited number of data points and suggest that producing composites with a larger number of filler volume fractions will yield more reliable derivative values.