Piezoresistive performance prediction method for multidirectional carbon nanotube resin-based strain sensor

Abstract

The invention discloses a method for predicting the piezoresistive performance of a multidirectional carbon nanotube resin-based strain sensor, and the method comprises the steps: based on a spherical model, considering the conductivity and volume fraction of CNTs, considering the permeation threshold value of a composite material, and considering the weakening of the conductivity caused by the tunneling effect between CNTs in the composite material; starting from several aspects of influence of strain on the volume fraction of CNTs and the conductivity of a tunnel, a strain relation in three directions is established, so that a piezoresistive performance prediction model is obtained. Compared with a test result, the error is within 15%, the minimum prediction error of the sensitivity of the sensor is-0.56%, and the method has the advantages of high efficiency and accuracy.

Description

Technical Field

[0001] The invention belongs to the technical field of carbon nanotube (CNT) strain sensors, and in particular relates to a method for predicting the piezoresistive performance of a multi-directional CNT resin-based strain sensor. Background Art

[0002] The appearance of damage in engineering structures is accompanied by disorder and irregularity, so the development of multi-directional strain sensors that can monitor the magnitude and direction of strain is crucial for structural health detection and long-term monitoring. Multi-directional CNT resin-based strain sensors use the piezoresistive effect, that is, the CNT conductive network changes with the action of external force, thereby causing the sensor resistance to change and realize strain sensing. The sensor has a wide detection range, good stability and excellent directional sensitivity, and can be used to monitor strain from all directions of the structure. Although the piezoresistive properties of the material can be directly measured through experiments, it is time-consuming, costly, and difficult to cover the full strain range. Therefore, customized theoretical analysis and prediction of the piezoresistive performance of the sensor are required.

[0003] In recent years, the prediction of piezoresistive performance of CNT strain sensors has mainly considered the influence of factors such as filler physical properties, temperature, and mechanical strain. First, the three-dimensional random orientation of nanofillers, the waviness and entanglement of CNTs, and the electrical properties of the material in the undeformed state were considered to model the conductive network of CNTs in the polymer matrix (“Analytical model for the prediction of the piezoresistive behavior of CNT modified polymers”, Panozzo et al., page 53, Composites Part B: Engineering, 2017). Further considering some key factors such as CNT size, spacing, and dispersion state, a comprehensive analytical model was developed to predict the electromechanical response of composite sensors based on conductive CNTs (“Comprehensive evaluation of the piezoresistive behavior of carbon nanotube-based composite strain sensors”, Tang et al., page 108761, Composites Science and Technology, 2021). The self-temperature compensated piezoresistive behavior of CNT / GNP polymer nanocomposites is predicted by a 3D Monte Carlo algorithm and a percolation network model (“Developing a high-efficiency predictive modelfor self-temperature-compensated piezoresistive properties of carbon nanotube / graphene nanoplatelet polymer-based nanocomposites”, Haghgoo et al., page 107380, Composites Part A: Applied Science and Manufacturing, 2023). In addition, an analytical model is proposed to describe the effect of externally applied mechanical strain on the conductivity of CNT reinforced composites (“Piezoresistive modelling of CNTs reinforced composites under mechanical loadings”, Fang et al., page 108757, Composites Science and Technology, 2021). However, existing theoretical research focuses on unidirectional strain sensors, and it is impossible to theoretically predict the piezoresistive properties of multi-directional CNT resin-based strain sensors.

[0004] This patent mainly involves a method for predicting the piezoresistive performance of a multi-directional CNT resin-based strain sensor. Based on the conductivity and micromechanics theory of composite materials, a spherical model that is more in line with the actual situation is used to study the conductivity and piezoresistive properties of CNT composite materials. Based on Hooke's law, the relationship between strains in different directions of the sensor is studied, which can theoretically predict the piezoresistive performance of CNT resin-based multi-directional sensors in different directions, providing a theoretical basis and data support for the design of multi-directional strain sensors. Summary of the invention

[0005] Purpose of the invention: In order to overcome the shortcomings of the prior art, the present invention provides a method for predicting the piezoresistive performance of a multi-directional CNT resin-based strain sensor based on the conductivity and micromechanics theory of composite materials and Hooke's law. Compared with the experimental results, the errors are all within 15%, and the minimum prediction error of the sensor sensitivity is -0.56%, which has the advantages of high efficiency and accuracy.

[0006] The present invention adopts the following technical solution: a method for predicting the piezoresistive performance of a multi-directional carbon nanotube resin-based strain sensor, comprising the following steps:

[0007] Step 1: The parallel model and the series model reflect the limit state of the conductivity of the composite material. In order to be more in line with the actual situation, the spherical model is used to consider the influence of the percolation threshold of the composite material and the tunneling effect between CNTs on the conductivity. The effective conductivity of the composite material is:

[0008]

[0009] Where V i is the volume fraction of CNT, σ io is the conductivity of CNT, σ it is the tunneling conductivity of CNT, 1≥β≥0 is a constant representing the tunneling contribution, α is a constant, V i,th is the percolation threshold of the composite or the threshold volume fraction of CNTs, λ is the aspect ratio of CNTs;

[0010] Step 2: The resistance change of the composite material due to mechanical deformation can be expressed as:

[0011]

[0012] where s and k are constants that can be determined from experimental data, ε x is the normal strain of the composite material in the x direction, ν i is the Poisson's ratio of CNT, ν c is the Poisson's ratio of the composite material, α is a constant, V i is the volume fraction of CNT, V i,this the percolation threshold of the composite or the threshold volume fraction of CNTs, ε x is the normal strain of the composite in the x-direction, R is the electrical resistance of the undeformed composite, and ΔR is the corresponding change in electrical resistance of the composite due to mechanical deformation;

[0013] Step 3: Obtain the piezoresistive performance prediction model based on the strain relationship in the three directions of 0°, 45°, and 90° with respect to the force direction:

[0014] Prediction model of piezoresistance performance in force direction:

[0015]

[0016] Prediction model of piezoresistance performance at 45° to the force direction:

[0017]

[0018] Prediction model of piezoresistance performance in the direction perpendicular to the force direction:

[0019]

[0020] where s and k are constants that can be determined from experimental data, ε x is the normal strain of the composite material in the x direction, ν i is the Poisson's ratio of CNT, ν c is the Poisson's ratio of the composite material, α is a constant, V i is the volume fraction of CNT, V i,th is the percolation threshold of the composite or the threshold volume fraction of CNTs, ε x is the normal strain of the composite material in the x direction, v is the Poisson’s ratio of the stretched material, R is the electrical resistance of the undeformed composite material, and ΔR is the corresponding change in electrical resistance of the composite material due to mechanical deformation.

[0021] Furthermore, in step 1, according to the Landauer-Buttiker formula, the tunneling conductivity of the inclusions can be approximated as follows:

[0022]

[0023] Where e is the electron charge, h is Planck's constant, M is the total number of conduction channels, and J is the transmission probability of electrons through the matrix barrier between two CNTs, which can be obtained by solving the Schrödinger equation or the WKB approximation:

[0024]

[0025] where d is the minimum distance between two adjacent CNT axes where electron tunneling is most likely to occur, and d vdW and dcutoff They represent the lower and upper bounds of the distance d, respectively. tunnel is a constant representing the characteristic length of the tunneling action, and D is the diameter of CNTs.

[0026] The present invention is based on a spherical model and starts from considering the conductivity and volume fraction of CNTs, the permeation threshold of the composite material, the weakening of the conductivity caused by the tunneling effect between CNTs in the composite material, and the influence of strain on the volume fraction of CNTs and tunnel conductivity, to establish a strain relationship in three directions, thereby obtaining a piezoresistive performance prediction model.

[0027] Beneficial effects of the present invention:

[0028] (1) Based on the conductivity and micromechanics theory of composite materials, a conductivity model of CNT resin-based composite materials was proposed according to the spherical model. The weakening of the CNT conductive properties by the tunneling effect was taken into account, and a theoretical model that is more in line with the actual situation was established.

[0029] (2) Considering the effect of externally applied mechanical load on the conductivity of CNTs-reinforced composites, a conductivity model of CNT resin-based composites under mechanical load was proposed to accurately predict the piezoresistive properties of the sensor under load and monitor the health status of the structure in real time.

[0030] (3) Based on Hooke's law, the relationship between strains in different directions is analyzed, and a piezoresistive model in three directions (direction of force, direction at 45° to the direction of force, and direction perpendicular to the direction of force) is proposed. This model can monitor strains in different directions and promote the development of multi-directional strain sensors. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 It is a schematic diagram of the stretching direction of the present invention;

[0032] Figure 2a , 2b 2c and 2c are schematic diagrams of the analytical model of the electrical conductivity of the two-component composite material of the present invention; wherein Figure 2a For the parallel model, Figure 2b It is a spherical model; Figure 2c It is a series model;

[0033] Figure 3a and 3b This is a schematic diagram of a spherical model of a CNT two-component composite material considering the tunneling effect of the present invention; Figure 3a RVE model for CNT reinforced polymer composites; Figure 3b A spherical model of a two-component composite material considering the tunneling effect of inclusions;

[0034] Figure 4a It is the plane stress state diagram of the present invention;

[0035] Figure 4b This is the stress state diagram of the α surface of the present invention. DETAILED DESCRIPTION

[0036] The present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0037] A method for predicting the piezoresistive performance of a multi-directional carbon nanotube resin-based strain sensor comprises the following steps:

[0038] Step 1: The main raw materials of CNT multidirectional sensor are CNT and resin, so the conductivity is derived based on the composite material made of two components. Assume that one component (resin) is a uniform medium and the other component (CNT) represents a limited number of inclusions. Then the composite material model can be divided into three types, namely parallel model, series model and spherical model, such as Figure 2a , Figure 2b , Figure 2c shown.

[0039] The parallel model gives the effective conductivity σ e The upper limit of the series model gives σ e The lower limit of the conductivity of the composite material reflects the limit state. In order to be more in line with the actual situation, the subsequent derivations are based on the spherical model.

[0040] According to the theoretical formula of the spherical model, the effective conductivity of the composite material is calculated as follows:

[0041] Spherical model:

[0042]

[0043] Among them, σ e is the effective conductivity of the composite material; σ i and V i is the conductivity and volume fraction of the inclusions; σ m and V m are the conductivity and volume fraction of the medium, respectively.

[0044] There are two formulas in the spherical model. One is obtained by assuming that the medium is the outer layer of the sphere, and the other is obtained by assuming that the inclusion is the outer layer of the sphere. The former is more suitable for V i <V m , while the latter is more suitable for V i >V m . Therefore, it is also possible to use a combination of them as shown below:

[0045]

[0046] Define f(V i ) is the weight function:

[0047]

[0048] Where α is a constant, V i,th is the percolation threshold of the composite material or the threshold volume fraction of inclusions, and λ is the aspect ratio of the inclusions. For composite materials where inclusions and dielectrics have different electrical properties, that is, σ i >>σ m , f(V i ) into the above formula and simplify it to get:

[0049]

[0050] This formula is derived based on the assumption of perfect connection between inclusions. Figure 3a and 3b , when considering tunneling between inclusions in a composite material, the conductivity of the inclusions should be reduced. Assuming that the inclusions are connected in series in the spherical model, the conductivity at the connection is represented by the representative tunneling conductivity. In this case, the effective conductivity of the inclusions in the spherical model can be calculated as follows:

[0051]

[0052] In the formula, σ i is the effective conductivity of inclusions in the spherical model considering the tunneling effect, σ io is the conductivity of the inclusion material, σ it is the tunneling conductivity of the inclusion, 1≥β≥0 is a constant representing the tunneling contribution; σ i Substitute V into the above formula i >V i,th Part, we can get:

[0053]

[0054] According to the Landauer-Buttiker formula, the tunneling conductivity of the inclusion can be approximated as follows:

[0055]

[0056] Where e is the electron charge, h is Planck's constant, M is the total number of conduction channels, and J is the transmission probability of electrons through the matrix barrier between two CNTs, which can be obtained by solving the Schrödinger equation or the WKB approximation:

[0057]

[0058] where d is the minimum distance between two adjacent CNT axes where electron tunneling is most likely to occur, and d vdW and d cutoffThey represent the lower and upper bounds of the distance d, respectively. tunnel is a constant representing the characteristic length of the tunneling action, and D is the diameter of CNTs.

[0059] Step 2: Considering the effect of externally applied mechanical load on the effective conductivity of CNTs reinforced composites, the volume of the composite RVE unit per unit volume will change after being subjected to mechanical load in the x direction. The new volume of RVE can be expressed as follows:

[0060] V c,new =(1+ε x )(1-v c ε x ) 2

[0061] Among them, V c,new is the new volume of the composite material after deformation, ε x is the normal strain of the composite material in the x direction, ν c is the Poisson's ratio of the composite material.

[0062] Assuming that the CNTs are uniformly distributed on the x-axis, y-axis, and z-axis, the strain ε in the composite material is x The new volume of CNTs in the induced RVE can be approximately expressed as:

[0063]

[0064] Where V i,new is the volume of CNTs in the deformed composite material, ν i is the Poisson’s ratio of CNTs. Using the small strain assumption, the new volume fraction V of CNTs in the deformed composite is i * It can be expressed as:

[0065]

[0066] The strain ε of the composite material x It not only causes a change in the volume fraction of CNTs in the composite material, but also causes a change in the transmission probability of CNTs, resulting in a change in tunnel conductivity. Assume that the new minimum distance between two adjacent CNT axes in the deformed composite material is expressed as (dD)(1+ε x )+D, the new tunneling conductivity of CNTs in the deformed composite material can be expressed as:

[0067]

[0068] where σ it * is the new tunneling conductivity of CNTs in the deformed composite, γ = (dD) / d tunnelis a dimensionless parameter. vdW <(dD)(1+ε x )+D <d cutoff If (dD)(1+ε x )+D <d vdW ,σ it * =σ i , if (dD)(1+ε x )+D>d cutoff ,σ it * = 0. Considering that RVE involves multiple tunnel distances in parallel or series, a correction factor needs to be introduced, so the formula is corrected to:

[0069]

[0070] where μ is a dimensionless constant representing the effect of the group tunneling distance in the RVE on the calculated representative tunneling conductivity of the CNT. In addition, due to σ it * and ε x are direction-related parameters, σ it * and ε x In the same direction, the value of γ is the same as σ it * and ε x The values ​​are different in different directions.

[0071] Using the above definition of V i * and σ it * Replace V i and σ it , we can get:

[0072]

[0073] where σ e * is the new effective conductivity of the deformed composite material; given by the strain ε x The relative change in the effective conductivity of the composite material caused by can therefore be expressed as:

[0074]

[0075] For most CNT-reinforced composites, the volume fraction of CNTs in the composite is very small, i.e., V i <<1; in this case, σ e It can be simplified as follows:

[0076]

[0077] By replacing σ in the formula e V i and σ it Differentiating, we get:

[0078]

[0079] ΔV i and Δσ it With strain ε x Therefore, the above formula can be rewritten as:

[0080]

[0081] For most CNT-reinforced composites σ it <<σ io , so the second term in the right bracket can be approximately taken as μγ, that is:

[0082]

[0083] According to general physics, γ should be related to V ith / V i Proportional; Assume Then the above formula becomes:

[0084]

[0085] Where s and k are constants that can be determined from experimental data; according to the inverse relationship between resistance and conductivity, the resistance change of the composite material due to mechanical deformation can be expressed as:

[0086]

[0087] Where R and R new are the resistances of the undeformed and deformed composites, respectively, and ΔR is the corresponding change in resistance of the composite due to mechanical deformation.

[0088] Step 3: According to the three directions of 0°, 45°, and 90° to the force direction (such as Figure 1 ) strain relationship to obtain the piezoresistive performance prediction model:

[0089] The test of CNT multi-directional sensor is based on unidirectional tensile test, so Hooke's law under simple stress state can be used. Since the test is carried out with thin film, only the x and y directions need to be considered:

[0090]

[0091] Among them, σ x and ε xis the stress and strain in the direction of force, σ y and ε y are the stress and strain in the direction perpendicular to the x direction, E is the elastic modulus of the tensile material, and v is the Poisson's ratio of the tensile material.

[0092] If the stress on a certain face of a unit cell is zero, the unit cell can be represented by only a flat square, such as Figure 4a , the stress acting on it can be simplified to a plane stress state. To find the stress component on plane α, intercept Figure 4b The triangular plate shown.

[0093] By ∑F n =0, we get:

[0094] σ α A-σ x Acos 2 α+τAcosαsinα+τAsinαcosα-σ y Asin 2 α=0

[0095] By ∑F t =0, we get:

[0096] τA+σ y Asinαcosα+τAsin 2 α-σ x Acosαsinα-τAcos 2 α=0

[0097] Solve the above equations together to get:

[0098]

[0099] Where A is the oblique cross-sectional area, A x and A y is the area of ​​the other two perpendicular surfaces, A x =Acosα,A y =Asinα. Since this paper requires the stress state of the plane with an angle of 45° to the vertical, α=45°, σ y =0 Substituting into the above formula, we can get:

[0100]

[0101] According to Hooke's law, the strain in the 45° direction is:

[0102]

[0103] The strain relationship in the three directions can be obtained by combining:

[0104]

[0105] Therefore, the prediction model of piezoresistance performance in the force direction is:

[0106]

[0107] Prediction model of piezoresistance performance at 45° to the force direction:

[0108]

[0109] Prediction model of piezoresistance performance in the direction perpendicular to the force direction:

[0110]

[0111] It should be noted that, for those skilled in the art, several improvements and modifications can be made without departing from the principles of the present invention, and these improvements and modifications should also be considered as the protection scope of the present invention. All components not specified in this example can be implemented using existing technologies.

Claims

1. A method for predicting the piezoresistive performance of a multi-directional carbon nanotube resin-based strain sensor, characterized in that: The following steps are involved: Step 1: The parallel model and the series model reflect the limit state of the conductivity of the composite material. In order to be more in line with the actual situation, the spherical model is used to consider the influence of the percolation threshold of the composite material and the tunneling effect between CNTs on the conductivity. The effective conductivity of the composite material is: Where V i is the volume fraction of CNT, σ io is the conductivity of CNT, σ it is the tunneling conductivity of CNT, 1≥β≥0 is a constant representing the tunneling contribution, α is a constant, V i,th is the percolation threshold of the composite or the threshold volume fraction of CNTs, λ is the aspect ratio of CNTs; Step 2: The resistance change of the composite material due to mechanical deformation is expressed as: where s and k are constants determined from experimental data, ε x is the normal strain of the composite material in the x direction, ν i is the Poisson's ratio of CNT, ν c is the Poisson's ratio of the composite material, α is a constant, V i is the volume fraction of CNT, V i,th is the percolation threshold of the composite or the threshold volume fraction of CNTs, ε x is the normal strain of the composite in the x-direction, R is the electrical resistance of the undeformed composite, and ΔR is the corresponding change in electrical resistance of the composite due to mechanical deformation; Step 3: Obtain the piezoresistive performance prediction model based on the strain relationship in the three directions of 0°, 45°, and 90° with respect to the force direction: Prediction model of piezoresistance performance in force direction: Prediction model of piezoresistance performance at 45° to the force direction: Prediction model of piezoresistance performance in the direction perpendicular to the force direction: where s and k are constants determined from experimental data, ε x is the normal strain of the composite material in the x direction, ν i is the Poisson's ratio of CNT, ν c is the Poisson's ratio of the composite material, α is a constant, V i is the volume fraction of CNT, V i,th is the percolation threshold of the composite or the threshold volume fraction of CNTs, ε x is the normal strain of the composite material in the x direction, v is the Poisson’s ratio of the stretched material, R is the electrical resistance of the undeformed composite material, and ΔR is the corresponding change in electrical resistance of the composite material due to mechanical deformation.

2. The method for predicting the piezoresistive performance of a multi-directional carbon nanotube resin-based strain sensor according to claim 1, characterized in that: In step 1, according to the Landauer-Buttiker formula, the tunneling conductivity of the inclusions is approximately as follows: Where e is the electron charge, h is Planck's constant, M is the total number of conduction channels, and J is the transmission probability of electrons through the matrix barrier between two CNTs, which is obtained by solving the Schrödinger equation or the WKB approximation: Where d is the minimum distance between two adjacent CNT axes, d vdW and d cutoff They represent the lower and upper bounds of the distance d, respectively. tunnel is a constant representing the characteristic length of the tunneling action, and D is the diameter of CNTs.

Images