Introduction

Conducting particle-reinforced insulating matrix composites (CPICs) are an important class of multifunctional materials with widespread applications. Notable materials that belong to this family include carbon-black-polymer systems [1–6], metal-inorganic and -organic insulator composites [7–11], and oxide-based thick film resistors (TFRs) [12–16]. Transport behavior of these materials can be described by so-called tunneling-percolation models, in which tunneling coexists with percolation behaviors. Thereby, the electrical resistivity of CPICs follows a percolation-like power law [17, 18]:

$$ \rho = \rho_{0} (x - x_{\text{c}} )^{ - t} $$
(1)

where ρ0 is a constant, x the concentration of the conducting phase, xc the critical concentration of the conducting phase below which the composite has the resistivity of the matrix phase, and t the critical exponent. According to the standard percolation theory, the critical exponent t is material independent and has a universal value t0 ≈ 2 for three-dimensional systems [19–21]. However, it has been found that for tunneling-percolation systems t becomes material dependent and can be significantly different from t0 [22]. Such nonuniversality has been attributed to specific distributions of both conducting and insulating phases [23].

One unique feature of CPICs is that they exhibit a high piezoresistive effect, i.e., the sensitivity of their resistivity upon applied stress. The significant piezoresistivity can be qualitatively understood in terms of the tunneling-percolation mechanism, in which the tunneling conductance is proportional to exp(2d/ξ) [24], where d is the interparticle spacing between the conductive particles and ξ is the tunneling factor which depends only on the potential barrier which is the property of the matrix material. Consequently, stress-induced changes in the interparticle spacing can significantly change the resistivity of the material, leading to a high piezoresistive coefficient.

The piezoresistivity of CPICs has been extensively studied since it is very important for the applications of the materials in various sensors [25]. However, fundamental understanding of the piezoresistivity of CPICs is rather difficult, because it strongly depends on the detailed microstructures and elastic properties of the materials. Carmona et al. [2] suggested that the piezoresistivity reversely depends on the concentration of conducting phase at the conductivity threshold by simple considering the stress-induced concentration changes. It has also been demonstrated that the nonuniversality of t can yield a logarithmic concentration dependence of piezoresistive response in the systems with conducting phase closer to the critical points [26]. For the systems with their insulating phases having Young’s modulus lower than conducting phases, the piezoresistive factors are enhanced due to the local deformation within the insulating phases being higher than the overall deformation [27].

Close examination on experimental data revealed that the resistivities of CPICs generally exhibited a non-linear relationship with the applied stress. This implies that the piezoresistive stress coefficient, defined as d ln ρ/dσ (where σ is the applied stress), is not a constant value, but depends on the applied stress level. This is more profound in carbon-black-polymer systems, where the rate of the change in resistivity decreases significantly with increasing the applied stress [1–4, 28–30]. This phenomenon is worth detailed investigations since it becomes important when comes to sensor design and fundamental understanding of transport mechanisms.

In this paper, we show that for the materials where the Young’s modulus of the insulating matrix phase is much less than that of conducting phase, the piezoresistive stress coefficient depends on the applied stress reversely at lower stresses and logarithmically at higher stresses at the percolation threshold. We obtain these results by an analytic solution of the piezoresistive stress coefficient based on the power law described by Eq. 1. We demonstrate that the prediction of the model agrees well with experimental results.

Model

The piezoresistivity of a material can be described by its piezoresistive factor, K (also known as gauge factor) or piezoresistive stress coefficient, Π, defined as

$$ K = \frac{{{\text{d}}\rho /\rho }}{{{\text{d}}\varepsilon }} = \frac{{{\text{d}}\ln (\rho )}}{{{\text{d}}\varepsilon }} $$
(2a)
$$ \Uppi = \frac{{{\text{d}}\rho /\rho }}{{{\text{d}}\sigma }} = \frac{{{\text{d}}\ln (\rho )}}{{{\text{d}}\sigma }} $$
(2b)

where ε is the strain and σ is the applied stress. The two parameters are related to each other by K = E · Π, where E is the Young’s modulus of the material.

By inserting Eq. 1 into Eq. 2b, the piezoresistive stress coefficient of a tunneling-percolation system can be expressed as

$$ \Uppi = \frac{{\partial \ln \rho_{0} }}{\partial \sigma } - \ln (x - x_{\text{c}} )\frac{\partial t}{\partial \sigma } - \frac{t}{{x - x_{\text{c}} }}\frac{\partial x}{\partial \sigma } $$
(3)

The first term on the right-hand side of Eq. 3 is a constant for a given system [22], referred as to Π0 here. Its contribution to the overall piezoresistivity is typically very small for a tunneling-percolation system [26]. The second and third terms are resulted from stress dependent t and concentration, respectively.

Previous study showed that the t has the following relationship with the microstructure of the material [22],

$$ t = 0.88 + 2\lambda /\xi $$
(4)

where λ is the average interparticle spacing of neighboring conducting phase, which can be related to the stress as λ = λ0(1 + ασ/E), where λ0 is the average interparticle spacing without the applied stress and α is a factor to correct the difference between the local strain within the insulating phase and the overall strain of the composite. Consequently, the piezoresistive stress coefficient of a tunneling-percolation system can then be expressed as

$$ \Uppi = \Uppi_{0} - \frac{{2\alpha \lambda_{0} }}{\xi E}\ln (x - x_{0} ) - \frac{t}{{(x - x_{\text{c}} )}}\frac{\partial x}{\partial \sigma } $$
(5)

Equation 5 relates the piezoresistive stress factor of a tunneling-percolation system to the concentration of conducting phase.

Now, we relate the change in the concentration of conducting phase (x) to the applied stress. Assuming the composite is an isotropic material, the change in volume of the composite under uniaxial stress is

$$ \frac{{{\text{d}}V}}{V} = \frac{{{\text{d}}\sigma }}{E}(1 - 2v) $$
(6a)

where V is the volume at stress σ and v is the Poisson’s ratio of the composite. Similarly, the change in the volume under hydrostatic stress is

$$ \frac{{{\text{d}}V}}{V} = 3\frac{{{\text{d}}\sigma }}{E}(1 - 2v) $$
(6b)

For a composite with Young’s modulus of the conducting phase being different from that of the matrix, the stress-induced volume changes in the matrix phase is different from that in the conducting phase, leading to changes in the volume fraction of the conducting phase. Detailed solution of the problem needs a complicated stress partitioning analysis. Here, we consider the case where the Young’s modulus of the insulating matrix phase is much lower than that of the conducting phase so that the change in the conducting phase can be neglected. In these cases, the volume fraction of the conducting phase can be related to the applied stress as

$$ x = x_{0} \exp ( - \frac{\sigma }{E}(1 - 2v)) = x_{0} (1 - \frac{\sigma }{E}(1 - 2v)) $$
(7a)
$$ x = x_{0} \exp ( - \frac{3\sigma }{E}(1 - 2v)) = x_{0} (1 - \frac{3\sigma }{E}(1 - 2v)) $$
(7b)

where x0 is the volume fraction of the conducting phase without the applied stress. Equations 7a and 7b are for uniaxial and hydrostatic stress, respectively. It can be seen that under compressive stress (σ has a negative value), the volume fraction of the conducting phase increases with increasing the applied stress.

By inserting Eqs. 7a and 7b into Eq. 5, we obtain

$$ \Uppi = \Uppi_{0} - \frac{{2\alpha \lambda_{0} }}{\xi E}\ln ((x{}_{0} - x_{\text{c}} ) - x_{0} \frac{\sigma }{E}(1 - 2v)) + \frac{{t\frac{{x_{0} (1 - 2v)}}{E}}}{{(x_{0} - x_{\text{c}} ) - \frac{{\sigma x_{0} }}{E}(1 - 2v)}} $$
(8a)
$$ \Uppi = \Uppi_{0} - \frac{{2\alpha \lambda_{0} }}{\xi E}\ln ((x_{0} - x_{\text{c}} ) - x_{0} \frac{3\sigma }{E}(1 - 2v)) + \frac{{t\frac{{3x_{0} (1 - 2v)}}{E}}}{{(x_{0} - x_{\text{c}} ) - 3\frac{{\sigma x_{0} }}{E}(1 - 2v)}} $$
(8b)

Equations 8a and 8b are for uniaxial and hydrostatic stress states, respectively. When x0 → xc, or stress-induced changes are larger than x0 − xc, the above equations can be simplified as

$$ \Uppi = \Uppi_{0} - \frac{{2\alpha \lambda_{0} }}{\xi E}\ln ( - x_{0} \frac{\sigma }{E}(1 - 2v)) - \frac{t}{\sigma } $$
(9a)
$$ \Uppi = \Uppi_{0} - \frac{{2\alpha \lambda_{0} }}{\xi E}\ln ( - x_{0} \frac{3\sigma }{E}(1 - 2v)) - \frac{t}{\sigma } $$
(9b)

Since t = 0.88 + 2λ/ξ and λ = λ0(1 + ασ/E), we have

$$ \Uppi = \Uppi_{0} - \frac{{2\alpha \lambda_{0} }}{\xi E}\ln [ - x_{0} \frac{\sigma }{E}(1 - 2v)] - \frac{{0.88 + 2\lambda_{0} /\xi }}{\sigma } - \frac{{2\lambda_{0} \alpha }}{E\xi } $$
(10a)
$$ \Uppi = \Uppi_{0} - \frac{{2\alpha \lambda_{0} }}{\xi E}\ln ( - x_{0} \frac{3\sigma }{E}(1 - 2v)) - \frac{{0.88 + 2\lambda_{0} /\xi }}{\sigma } - \frac{{2\lambda_{0} \alpha }}{E\xi } $$
(10b)

The last term in Eqs. 10a and 10b are very small and can be neglected,

$$ \Uppi = \Uppi_{0} - \frac{{2\alpha \lambda_{0} }}{\xi E}\ln [ - x_{0} \frac{\sigma }{E}(1 - 2v)] - \frac{{t_{0} }}{\sigma } $$
(11a)
$$ \Uppi = \Uppi_{0} - \frac{{2\alpha \lambda_{0} }}{\xi E}\ln ( - x_{0} \frac{3\sigma }{E}(1 - 2v)) - \frac{{t_{0} }}{\sigma } $$
(11b)

The equations predict that at lower stress level the piezoresistive stress coefficient of a tunneling-percolation system reversely depends on the applied stress, while at higher stress level it logarithmically depends on the applied stress.

Case studies

Carbon-reinforced insulating polymer matrix composites

Carbon-reinforced insulating polymer matrix composites are typical tunneling-percolation systems, which have been extensively studied due to their applications in a variety of sensors and others. The system is particularly suitable for testing our models since the carbon phase has much higher Young’s modulus than the polymeric matrixes. Here we select two sets of data to compare with our models: one measured under hydrostatic stresses and the other measured under uniaxial stresses.

Celzard et al. studied stress-dependent resistivities of single crystal graphite flakes (referred as to flat micronic graphite (FMG), with mean diameter of 10 μm and thickness of 0.1 μm) reinforced epoxy and polyurethane composites under hydrostatic stresses [3]. The respective critical percolation concentrations are 1.3 vol.% and 1.7 vol.% for the epoxy and polyurethane composites [31]. Here, piezoresistive stress coefficients are calculated using the stress-dependent resistivity data in Ref. [3] for two composites: one of them is epoxy-based composites containing 1.36 vol.% FMG (Fig. 7c in Ref. [3]); and the other one is polyurethane-based composite containing 2 vol.% FMG (Fig. 8d in Ref. [3]). Figure 1 plots the calculated Π as a function of the applied stress. It is seen that for the both composites the data in a low stress range follow the log Π − log σ plot, while the data in the high stress range follows Π − log σ plot.

Fig. 1
figure 1

Plots of piezoresistive stress coefficient as a function of applied hydrostatic stress in a log-log scale and b linear-log scale. Solid symbols epoxy based composites. Open symbol polyurethane based composite

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For the uniaxial stress situation, the experimental results from three silicone rubber composites are used here: two of them contain 1.35 and 1.8 vol.% carbon fibers of 9 μm in diameter and 1 mm in length [2], and the third one is reinforced with 35.5 vol.% carbon micro-balls of 8 μm in diameter [4]. The carbon-fiber-reinforced composites have the critical concentration of 1.1 vol.% [2], while the critical concentration of the carbon micro-ball-reinforced composite was not reported, but should be much less than 35.5 vol.%.Footnote 1 Figure 2 plots the calculated piezoresistive stress coefficients. For the two carbon-fiber-reinforced composites the Π exhibit the log Π − log σ plot in the whole stress range, while for the carbon micro-ball-reinforced composite, the low-stress data fit the log Π − log σ plot, but the high-stress data follow the Π − log σ plot.

Fig. 2
figure 2

Plots of piezoresistive stress coefficient as a function of applied uniaxial stress in a log-log scale and b linear-log scale. Open symbol Ref. [2]. Solid symbols Ref. [4]

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Polymer-derived ceramics

Polymer-derived ceramics (PDCs) are a new class of high-temperature multifunctional materials synthesized by thermal decomposition of polymeric precursors [32, 33]. Due to their unique manufacturing capability, PDCs have been considered for fabricating micro-electro-mechanical systems (MEMS) and micro-sensors [34–36].

PDCs exhibit a unique structure, which contains “free” carbon clusters within the amorphous matrix. Previous study demonstrated that with proper control of chemistry the precursor, the “free” carbon can form a tunneling-percolation network, resulting in anomalously high gauge factors [37]. PDCs are also suitable for testing the model of Eqs. 11a and 11b since the carbon phase has much higher Young’s modulus than the amorphous matrix phase [37].

The polymer-derived amorphous silicon carbonitride (SiCN) studied here is the same as we reported previously [37], which exhibited tunneling-percolation behavior. The stress-dependent resistivity of the SiCN is measured along longitudinal direction (parallel to the loading direction) under uniaxial compressive stress, using the procedure described in Ref. [37], and summarized in Fig. 3. It is seen that the resistivity of the SiCN decreases steeply with the applied stress, and then slowly with the higher applied tress. The piezoresistive stress coefficient of the SiCN is calculated using the data of Fig. 3 and plotted in Fig. 4. The results reveal that in the low-stress range the Π fit log Π − log σ plot, but the high-stress data follow the Π − log σ plot.

Fig. 3
figure 3

A plot of the relative resistivity of the SiCN polymer-derived amorphous ceramics versus the applied uniaxial stress. The resistivity is normalized with respect to the unstressed state

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Fig. 4
figure 4

Piezoresistive stress coefficient of the SiCN as a function of applied stress: a the log-log plot and b the linear-log plot

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Discussion and conclusion

It is seen from section “Case studies” that the experimental data from various materials fit very well with the prediction of Eqs. 11a and 11b. Note that the material systems examined in section “Case studies” covered a wide range of variations in the size, concentration, and morphology of the conducting phases, and in the elastic properties of the “insulating” matrix phases. The data used are from both hydrostatic and uniaxial loading configurations. This suggests that the effect described in Eqs. 11a and 11b should be widely valid in tunneling-percolation systems.

The transition from reverse dependence of the piezoresistive stress coefficient on the applied stress to logarithmic dependence is rather interesting. It is seen that the reverse dependent relationship always occurred at relatively low stress range, while the logarithmic dependence always occurred at high stress level. Closely examining the experimental results suggests that such transition occurred either at very high stress level when the initial concentration of the conducting phase is close to the critical value (Fig. 1) or at low stress level when the concentration of the conducting phase is much higher than the critical value (Fig. 2). In both cases, the concentrations of the conducting phases at the transition points are much higher than the critical values. Eq. 3 suggests that the reverse dependence resulted from the stress-induced change in conducting phase concentration, while the logarithmic dependence comes from the stress-induced change in the critical exponent, t. When conducting-phase concentration approaches the critical value the resistance change significantly with a small change in the concentration, thus the reverse dependence should be dominant. On the other hand, when conducting-phase concentration is much higher than the critical value the resistance change slightly with a change in the concentration, the logarithmic dependence becomes dominant.

In summary, we have shown by means of analytical results that the piezoresistive stress coefficient of a tunneling system exhibits both reversely and logarithmically dependent on the applied stress in different stress ranges. These features seem to be consistent with experimental results from various material systems, such as carbon-polymer systems and polymer-derived ceramics. The good agreements between the experimental results and theoretical prediction suggest that the stress dependence of the piezoresistive stress coefficient in tunneling-percolation systems indeed follow the effect described by Eqs. 11a and 11b.