Gold nanoparticle (AuNP)-polymer composite has attracted considerable attention due to its high stretchability, metal-like electrical conductivity and substantial piezoresistivity. In the nanocomposite, the effect of the van der Waals (vdW) interaction increases greatly between AuNPs, which may significantly change its overall mechanical and electrical properties. In examining this issue, the nanocomposite with randomly distributed AuNPs is constructed by Monte Carlo method, tensile tests on the material are then performed via molecular dynamics simulations and finally, its piezo-resistivity is studied based on an effective resistor model. The effects of AuNP interaction are examined for the mechanics, dynamics and piezoresistivity of the nanocomposite by comparing the results obtained in the presence and absence of the vdW interaction. It is found that the AuNP attraction tends to hold the AuNPs together, leading to enhanced Young’s modulus, yield and fracture stress even at the low volume fraction 5% to 10% of AuNPs. The piezoresistive effect of the composite is also improved as the AuNP attraction substantially affects AuNP dynamics in large deformation. It is expected that similar effects of NP vdW interaction can also be obtained for the nanocomposites based on copper or silver NPs embedded in polymer.

Flexible electronics and optoelectronics1–9 have become a trend in the development of innovative techniques in the 21st century. Flexible bio-electronics,3 artificial skin,4 stretchable display7 and strain sensors9 are some examples that have a wide range of applications in various industrial disciplines. Traditional electrical conductors, e.g., metals, are rigid and heavy, and thus not suitable for bendable electrodes, flexible electrical connections and foldable or stretchable integrated circuits required for flexible electronic devices. Efforts thus are made to synthesize flexible conductors by distributing conductive nanofillers into insulating polymer elastomers.2 In addition to nanotubes, nanowires and nanosheets, gold nanoparticles (AuNPs) are added to polymer matrix leading to the nanocomposites with metal-like electrical conductivity, high stretchability and substantial piezoresistivity.1,2 It is observed experimentally1 that AuNPs in the composite can be reorganized to form long NP-chains, which can greatly decrease the percolation threshold associated with the small aspect ratio of NPs and further improve the electrical conductivity.10 The AuNP-polymer composites thus have attracted considerable attention in recent research.1,2,10–15

For such nanocomposites, there exist two major issues associated with (1) the vdW interaction between NP and polymer matrix, i.e., the NP-matrix interface and (2) the inter-NP vdW interaction that is possibly responsible for the self-assembly of NPs observed in the experiment.1 Previously, the NP-matrix interface has been studied by several groups based on the continuum mechanics models and molecular dynamics (MD) simulations.11,12,16–18 In parallel with these, the dilute solution is assumed tacitly or explicitly in previous studies, i.e., the volume fraction of NPs is so low that the inter-NP vdW interaction is negligible due to the large distance between NPs. This, however, may not be true11 as inter-AuNP interaction is found to be responsible for their self-reorganization in the polymer matrix.1 This suggests that the AuNP interaction may have significant impact on the mechanical properties of the nanocomposites, such as Young’s modulus, yield stress and fracture strength. In addition, the AuNP interaction may also exert influence on the dynamics of AuNPs, e.g., AuNP trajectory or AuNP distribution during deformation, and thus change the piezo-resistive and viscoelastic properties of the nanocomposite.

Motivated by this idea, the present paper aims to investigate the effects of inter-AuNP vdW interaction on the mechanical properties, AuNP trajectory and piezoresistivity of the AuNP-polymer nanocomposites. To this end, MD simulations and Monte Carlo method are combined to implement the tensile tests on the composites and the effective resistor model established in Refs. 10, 13, and 14 are used to calculate the strain-dependence of electrical conductivity. The layout of the paper is as follows. The MD simulations, Monte Carlo method and effective resistor model are introduced in Sec. II. The results and discussions are presented in Sec. III and the new findings are summarized in Sec. IV.

The simulation methods used in the present study includes the model development of the nanocomposite, the simulations on the tensile tests of the AuNP-polymer composites and the effective resistor model for the study of piezo-resistivity. Here polyethylene (PE) is considered as a typical example of polymer matrix.

In constructing AuNP-PE composite structures, an AuNP sphere is built with diameter of 10 Å. The center coordinates of seven AuNPs are located in a cubic box based on the Monte Carlo method in which it is ensured that the AuNPs do not penetrate into each other. Subsequently, the obtained AuNP and the geometrically optimized PE chains are assembled in the box (Fig. 1(a)). The geometric optimization of the whole AuNP-PE system is then performed, which concludes the construction of the nanocomposite model. The initial PE mass density 0.49 g·cm-3 is chosen, and the MS (Material Studio) software is employed to achieve the bulk mass density of PE in the simulation box. To study the effect of volume fraction of the AuNPs, two different simulation box sizes are used (i.e. 8.37×8.37×8.37Å3 for VF=5% and 6.64×6.64×6.64Å3 for VF=10%) while the number of AuNPs remains unchanged.

FIG. 1.

(a), Atomic model of the AuNP-polymer used in this work, the yellow particles represent the 7 AuNPs and the grey curves are PE chains, the model was constructed by MS. (b), geometrical model used in the Monte Carlo simulation for the piezoresistive responses, the color schemes are the same as that in (a), a voltage is applied in x direction to generate electric current. (c), illustration of AuNP pair distances for the evaluation of the conduction mechanism, dij is the distance between the two AuNP surfaces, D is the AuNP diameter and Lij is the distance between the centers of the AuNPs.

FIG. 1.

(a), Atomic model of the AuNP-polymer used in this work, the yellow particles represent the 7 AuNPs and the grey curves are PE chains, the model was constructed by MS. (b), geometrical model used in the Monte Carlo simulation for the piezoresistive responses, the color schemes are the same as that in (a), a voltage is applied in x direction to generate electric current. (c), illustration of AuNP pair distances for the evaluation of the conduction mechanism, dij is the distance between the two AuNP surfaces, D is the AuNP diameter and Lij is the distance between the centers of the AuNPs.

Close modal

The PE matrix considered in this study contains 12000 CH2 units (36000 carbon and hydrogen atoms). To perform simulations on the large-scale nanostructures without losing the main physics, the united atom model19–21 is employed, where each -CH2- of PE chain is treated as an equivalent united atom. The atom mass is 14.02 g·mol-1. As shown by Huang et al.,21 the united atom model is effective in reproducing the dynamic behaviour of PE at different temperatures and strain rates with highly improved efficiency. The Dreiding potential19 is used to describe the interaction between the united atoms in PE, the embedded atom method (EAM) potential22 is chosen for Au-Au bonding interactions and the Lennard-Jones (L-J) potential (in the form of V=4εσd12σd6) is selected for the non-bond interactions, where ε and σ are the parameters associated with energy well depth and the equilibrium distance between two atoms. For the interaction between gold (Au) atoms and united atoms of PE, ε and σ were obtained based on the mixture law, i.e., σ=12(σAu+σunit) and, ε=εAuεunit23 where εAu and σAu are the parameters for Au atoms and εunit and σunit are those for the united atoms. All the values can be found in Reference 12.

The large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) is used to perform the molecular dynamics (MD) simulations on the tensile responses of the nanocomposite. To make sure the configurations are at equilibrium state, the structural relaxation is firstly performed for the AuNP-PE systems by following procedure below. At the beginning, the system temperature is set to 500K and equilibrated for 1 nst = 1fs) using the Langevin thermostat, and then additional 0.5 nst = 0.5fs) under the NPT ensemble. Subsequently, the system is cooled from 500K to 100K using the NPT ensemble within 0.5 nst = 0.5fs). The final relaxation of (Δt = 0.5fs) is performed at 100K. Here, the total annealing time 3 ns is adapted to cope with the long PE chains considered in the system. This annealing process is implemented to achieve the equilibrium state of the AuNP-PE system. The final PE mass density ρPE ≈ 0.98 g·cm-3 is obtained, which is close to the experimentally measured mass density (0.89 ∼ 0.925 g·cm-324) of low density PE. This MD simulation procedure is used in our previous study12 where the bond length, bond angle and dihedral angle obtained for PE are found to be in good agreement with existing data for bulk PE. After the structural relaxation, we apply a uniaxial tensile stain to the equilibrated AuNP-PE system at 100K. The low temperature is chosen to keep the system in a (more) steady state, i.e., the internal energy variation with time becomes smaller. Low temperature may change the values of parameters obtained in the simulations but the trend and physics of the NP interaction effect will remain primarily unchanged. In addition, following previous studies19–21 we mainly use strain rate 109s-1 and the time step 1 fs. Simulations with faster strain rates (up to 1010s-1)are also conducted to examine the strain rates effect and the main physics are found to be the same within the range of choice. The entire process is performed under the NPT ensemble.

To study the piezoresistive response of the nanocomposite under tensile test, an effective resistor model is used to calculate the conductance of the system under different strain states when an electric field is applied. The schematic model is presented in Fig. 1(b), where the yellow sphere represents the AuNP with its geometry equivalent to that obtained from the MD simulations and the curved grey lines represent the PE chains. The overall conductance depends strongly on the conductive pathways formed by the AuNP pairs, therefore, we first evaluate the resistance between each pairs of AuNPs. As shown in Fig. 1(c), the distance between the ith and the jth AuNP is dij = Lij-D, where Lij is the distance between the centres of the AuNPs and D is the diameter. For each pair of AuNPs, there are three regions in dij for the electrons to transport across the interfaces. 1). When two AuNPs are in vdW contact, i.e., at the lowest energy state, dij = dvdW, contact conductance occurs, 2). when two AuNPs are too far away than a cutoff distance, dij > dcut, there is no electron transport between them, here this cutoff distance dcut is taken as 15 Å. While 3). when the distance is in-between dvdW and dcut, i.e., dvdW< dij< dcut, tunneling transport dominates the conduction mechanism. This conductance model is proposed in Ref. 25 and used in Refs. 10, 26, and 27. In particular, a good agreement has been achieved between the simplified model and experiments27 in studying the piezo-resistivity of the material made of silver nanowires.

Based on Kirchhoff ‘s current law, Ohm’s law and the above conduction model between two AuNPs the current equation of each AuNP can be established in terms of the electrical potentials of all AuNPs considered. Solving the group of the algebraic equations for the whole composite system by considering the electrical boundary conditions (i.e., the electrical voltage U applied) one can obtain the overall current I of the AuNP-PE composite and evaluate electrical conductivity σeff=IULS where L and S are the distance between two electrodes and the cross sectional area of the simulation box, respectively. For the details of this resistor model readers may refer to Ref. 10. At a given strain, the positions of AuNPs (or the electrical circuit) are obtained from MD simulations and the corresponding σeff is calculated by using the resistor model. The relationship between σeff and the applied strain quantifies the piezo-resistive responses of the composite system, as will be presented in Sec. III.

In this section, MD simulations and the effective resistor model introduced in Sec. II are used to study the mechanical and piezoresistive properties of AuNP-PE composite. To examine the effect of AuNP vdW interaction we consider two different scenarios, i.e., the nanocomposite (1) with and (2) without the AuNP-AuNP vdW interaction.

Here tensile tests are performed on the AuNP-PE composites with VF of AuNPs 0% (pure polymer), 5% and 10%, respectively. The stress-strain relation is obtained and the results associated VF =10% are shown in Fig. 2. The stress-strain curve in small strain range is enlarged in the inset of Fig. 2. For the sake of comparison, the corresponding results obtained without the AuNP interaction are also shown in Fig. 2. It is seen from Fig. 2 that, for the two cases considered the stress and strain show a linear relation at the strain smaller than 3%, where the slope represents Young’s modulus and is obtained by a linear fitting to the simulation data (the inset) achieved for the same nanocomposite in three MD simulations. Here, the linear least square technique is used and the coefficient R2 of determination obtained is 0.948±0.011 at VF = 0, 0.923±0.015 at VF = 5% and 0.931±0.013 at VF = 10%. Beyond the linear elastic region, the stress further increases with strain and then hits its local maximum where the stress and strain are defined as the yield stress and strain, respectively. After that the stress decreases with rising strain to touch a local minimum. This is followed by the strain hardening stage where the stress again rises with increasing strain. The stress-strain curve finally reaches the global maximum at which the stress and strain are considered as the fracture stress and strain, respectively. The results associated with VF = 0 (pure polymer) and 5% show nearly the same trend and thus are not shown here.

FIG. 2.

Stress-strain relation obtained for the AuNP-PE composite with the volume fraction of AuNP 10%; The inset shows the enlarged stress-strain curve at strain smaller than 30%.

FIG. 2.

Stress-strain relation obtained for the AuNP-PE composite with the volume fraction of AuNP 10%; The inset shows the enlarged stress-strain curve at strain smaller than 30%.

Close modal

Young’s modulus obtained for the polymer (VF = 0) and the nanocomposite with VF = 5% and 10% are summarized in Table I. As noted in the table, in the absence of the AuNP vdW interaction, adding AuNPs into polymer can still raise the Young’s modulus Y1 from 2358 MPa to 2809 MPa and to 3213 MPa when VF rises from 0% to 5% and 10%. The relative increase is 19% and 34%, respectively, which is mainly due to the stiffening effect of the AuNP-polymer interface.12 In the presence of the AuNP vdW interaction Young’s modulus can be further enhanced from Y1 = 2809 MPa (without the vdW interaction) to Y2 = 3056 MPa when VF is equal to 5%. The relative increase is 8.8%. Such an effect of the AuNP vdW interaction turns out to be more substantial at VF = 10%, which raises the Young’s modulus from Y1 = 3213 MPa (without the vdW interaction) to Y2 = 3606 MPa by 12.2%.

TABLE I.

Young’s modulus of the AuNP-polymer composite with (Y2) or without (Y1) the vdW interaction between AuNPs and the relative difference (ΔY21) between them. Here, volume fraction (VF) of AuNP is 0%, 5% and 10%, respectively.

ParametersY1 (MPa)Y2 (MPa)ΔY21
VF 0% 2358 2358 0% 
5% 2809 3056 8.8% 
10% 3213 3606 12.6% 
ParametersY1 (MPa)Y2 (MPa)ΔY21
VF 0% 2358 2358 0% 
5% 2809 3056 8.8% 
10% 3213 3606 12.6% 

The yield stress and strain obtained are tabulated in Table II where the yield strain is less than 14%. It is noted that, in the absence of the AuNP vdW interaction the AuNPs embedded in polymer increase its yield stress but slightly decrease the yield strain. The enhanced yield stress can be attributed to the strengthening effect of the AuNP-polymer interface.12 Based on the data in Table II the relative change with respect to the yield stress of the polymer (187.4MPa) is 3.4% and 16.1% at VF = 5% and 10%, respectively. In Table II, the comparison between the results with and without the AuNP interaction indicates that the AuNP vdW interaction further increases the yield stress from σy1= 193.7 MPa (without the vdW interaction) to σy2= 208.3 MPa (by 7.5%) at VF = 5% and from σy1= 217.9 MPa to σy2= 246.2 MPa (by 13%) at VF = 10%. It is clearly seen from the above results at small strain (< 14%) that even with a low VF of 5%, the vdW interaction between AuNPs is strong enough to exert substantial stiffening and strengthening effects on the nanocomposites, which are comparable to the effects of the AuNP-PE interface.

TABLE II.

Yield stress and strain of the AuNP-polymer composite with (σy2,εy2) or without (σy1,εy1) the vdW interaction between AuNPs and the relative difference (Δσy21, Δεy21) between them. Here, volume fraction (VF) of AuNP is 0%, 5% and 10%, respectively.

Parametersσy1 (MPa)σy2 (MPa)Δσy21εy1 (%)εy2 (%)Δεy21
VF 0% 187.4 187.4 17.3 17.3 
5% 193.7 208.3 7.5% 14.1 15.0 6.4% 
10% 217.9 246.2 13% 12.8 13.6 6.3% 
Parametersσy1 (MPa)σy2 (MPa)Δσy21εy1 (%)εy2 (%)Δεy21
VF 0% 187.4 187.4 17.3 17.3 
5% 193.7 208.3 7.5% 14.1 15.0 6.4% 
10% 217.9 246.2 13% 12.8 13.6 6.3% 

In addition, Table III shows that at large strain (0∼600%), the inter-AuNP interaction raises the fracture stress by 11% to 21.6% when the VF grows from 5% to 10%. In the same process, the fracture strain increases by 2.5% and 1.7%, respectively. Table III also shows that in the absence of the AuNP interaction, adding AuNPs into the polymer weakens the nanocomposite. The higher VF of AuNPs leads to the lower fracture stress. This is due to the fact that the AuNP-PE interface has a low strength and thus reduces the fracture stress of the nanocomposites.12 In other words, the AuNP interaction and the AuNP-PE interface have reverse effects on the fracture stress of the nanocomposite.

TABLE III.

Fracture stress and strain of the AuNP-polymer composite with (σf2,εf2) or without (σf1,εf1) the vdW interaction between AuNPs and the relative difference (Δσf21, Δεf21) between them. Here, volume fraction (VF) of AuNP is 0%, 5% and 10%, respectively.

Parametersσf1 (MPa)σf2 (MPa)Δσf21εf1 (%)εf2 (%)Δεf21
VF 0% 491.2 491.2 488 488 
5% 417.9 463.4 10.9% 602 617 2.5% 
10% 354.9 431.5 21.6% 594.1 604.3 1.7% 
Parametersσf1 (MPa)σf2 (MPa)Δσf21εf1 (%)εf2 (%)Δεf21
VF 0% 491.2 491.2 488 488 
5% 417.9 463.4 10.9% 602 617 2.5% 
10% 354.9 431.5 21.6% 594.1 604.3 1.7% 

To confirm that the difference between the two scenarios considered is purely a result of the AuNP-AuNP vdW interaction we calculate the relative change in the total energy (E2total- E1total)/E1total and the non-bond energy (E2non-bond - E1non-bond)/E1non-bond as a function of tensile strain. Here, subscript ‘non-bond’ means the energy due to the vdW interaction between all components in the composite and ‘total’ means the total energy in the composite including the energy due to the vdW interaction plus the energy from the chemical bonds of polymer chains. In addition, the superscript ‘1’ and ‘2’, respectively, show the energy calculated in the absence and presence of the vdW interaction between AuNPs. It is noted in Fig. 3 with VF = 10% that the relative change in the total energy is nearly the same as the change in the non-bond energy (i.e., the energy due to the vdW interaction) when the strain is less than 450%, i.e., before the stress reaches its maximum value (fracture stress) at 590%. This indicates that the inter-AuNP vdW interaction is primarily responsible for the total energy change and thus, the increase of Young’s modulus and yield stress. It however cannot exert significant influence on the deformation energy of the polymer chains at the strain smaller than 450%. This can be seen from the insets (a) and (b) of Fig. 3 where the strain dependence of total energy, the non-bond energy and the energy of polymer chains is shown for the two scenarios under consideration. Differently, at very large strain, i.e., greater than 450%, Fig. 3 shows the large difference in the relative changes caused by the AuNP interaction between the total energy and non-bond energy. This observation suggests that the inter-AuNP interaction not only strengthens the bonds between AuNPs but also substantially affect the deformation of the polymer chains in the matrix. As will be shown later in Fig. 4, the latter is probably attributable to the fact that, at large deformation the AuNP vdW interaction will significantly affect the trajectories of the AuNPs. Thus, at a given strain AuNP distribution in PE matrix without AuNP interaction is found to be very different from the distribution obtained with the AuNP interaction. This is expected to finally lead to the different deformation patterns of the polymer chains in the two scenarios considered here.

FIG. 3.

(a) Variation of total energy vs uniaxial strain. Definition of the variation is defined in the text, blue and purple curves are obtained from simulations with and without vdW interactions between AuNPs. The contributions from non-bond energy, bond angle energy, bond energy and dihedral energy are shown in inset (a) and (b) for cases with and without of AuNP-AuNP vdW energy considered.

FIG. 3.

(a) Variation of total energy vs uniaxial strain. Definition of the variation is defined in the text, blue and purple curves are obtained from simulations with and without vdW interactions between AuNPs. The contributions from non-bond energy, bond angle energy, bond energy and dihedral energy are shown in inset (a) and (b) for cases with and without of AuNP-AuNP vdW energy considered.

Close modal
FIG. 4.

Projection of AuNP trajectories during the tensile deformation in the X direction to the X-Y plane in the presence (a) and absence (b) of the AuNP-AuNP vdW interaction, and the projection to the X-Z plane in the presence (c) and absence (d) of the AuNP-AuNP vdW interaction. The arrows in the figures indicate the direction in which AuNPs are moving during the deformation.

FIG. 4.

Projection of AuNP trajectories during the tensile deformation in the X direction to the X-Y plane in the presence (a) and absence (b) of the AuNP-AuNP vdW interaction, and the projection to the X-Z plane in the presence (c) and absence (d) of the AuNP-AuNP vdW interaction. The arrows in the figures indicate the direction in which AuNPs are moving during the deformation.

Close modal

It is shown in the prior section that the attraction between AuNPs tends to bind them together to resist the tensile deformation and yielding of the AuNP-PE nanocomposite. The interaction could be strong enough to change the dynamics or trajectories of the AuNPs during tensile deformation and thus, affect the distribution of the AuNPs in polymer matrix. This may finally result in the change in the piezo-resistivity, i.e., the conductivity-tensile strain relation, as the electrical percolation of AuNPs depends critically on the electrical circuit formed by individual AuNPs in certain distribution.

To examine this issue, we first record the trajectories of all AuNPs for the AuNP-polymer composite (VF = 10%) subject to tension in X direction. Recording starts from initial state (strain equal to zero) and ends at the tensile strain 500%. The length step of the strain increase is 5% at the strain smaller than 25%. The strain then jumps from 25% to 100% in one step and after that the step length is fixed at 100%. The projection of AuNP trajectories to X-Y plane is obtained with and without the AuNP interaction in Fig. 4(a) and (b), respectively. The projection to X-Z plane is shown in Fig. 4(c) and (d), respectively, for the two cases studied. It is noted in Fig. 3 that AuNPs generally stretch out along the X direction but contract in the Y and Z directions mainly due to Poisson’s ratio effect. Comparison between Fig. 4(a) and (b), and Fig. 3(c) and (d) indicates that the difference caused by the AuNP interaction is relatively small at the strain smaller than 25%. It becomes larger when the strain is greater than 100%. In Fig. 4(a) and especially, Fig. 4(c) we see that, at the large strain AuNPs move more significantly in transverse directions (towards the centre of the simulation box) due to the AuNP attraction. Thus, as expected the AuNP interaction is found to be able to reshape the distribution of AuNPs and accordingly, reconstruct the electrical circuit at a given strain. Here, the stress-induced self-reorganization reported1 is not observed in the present simulations. This probably is due to the fact that the VF of AuNPs in the present study is much smaller than the VF considered in the experiment,1 another reason could be the short duration of the present simulation.

In addition to the AuNP trajectory (Fig. 4), the corresponding strain-dependence of electrical conductivity is calculated in Fig. 5 for the nanocomposite with VF = 10%. It is noted that, in both cases (with and without the vdW interaction between AuNPs) the conductivity shows the similar trend, i.e., it first increases with the strain and reaches its maximum value at a strain around 30 to 35%. The conductivity then decreases monotonically with the rising strain and approaches zero at relatively large strain. When the strain is lower than 30%, the conductivity-strain curves are quite close in the two cases studied. This is consistent with the observation in Fig. 4 that, at the small strain the effect of the AuNP interaction is relatively small on the trajectories of the AuNPs although it improves Young’ modulus and Yield stress by around 10% to 20% depending on VF (Tables I and II). Nevertheless, the AuNP attraction is found to raise the maximum conductivity from around 8 S/cm (without the vdW interaction) to 24 S/cm (with the vdW interaction) by a factor of 3. As a result, at a higher strain in [35%, 80%], the higher conductivity and greater rate of change with strain are achieved due to the AuNP interaction, i.e., the conductivity becomes more sensitive to the strain change as it decreases by an order of one magnitude when the strain rises from 35% to 80% (Fig. 5).

FIG. 5.

Strain-dependence of the electrical conductivity calculated for the AuNP-polymer nanocomposite (VF = 10%) during tensile deformation in the presence (blue line) and absence (red line) of the AuNP-AuNP vdW interaction. The insets (a) and (b) showing the corresponding strain-dependence of the fractions of type I (dij = deq), II (deq < dij < dcut) and III (dij > dcut) dij between AuNPs.

FIG. 5.

Strain-dependence of the electrical conductivity calculated for the AuNP-polymer nanocomposite (VF = 10%) during tensile deformation in the presence (blue line) and absence (red line) of the AuNP-AuNP vdW interaction. The insets (a) and (b) showing the corresponding strain-dependence of the fractions of type I (dij = deq), II (deq < dij < dcut) and III (dij > dcut) dij between AuNPs.

Close modal

In Fig. 5, the fact of conductivity increases with rising tensile strain seems to be in contradiction with experimental data1 where the conductivity decreases by an order of magnitude when the strain rises from 0 to 100%. It is noted that the experimental data are obtained for the nanocomposite with VF larger than the percolation threshold and thus high conductivity above 104S/cm. The contact conducting between AuNPs is predominant in this case. In the resistor model (Sec. II), the contact conducting is achieved at dij = deq, where deq is considered as the minimum distance between two AuNPs due to the strong repulsive vdW force between them. In tension, dij for many AuNPs increases, leading to dij > deq in the X direction immediately, i.e., losing the contact conducting between these AuNPs. Some of AuNPs may approach to each other in the Y and Z directions due to, e.g., Poisson’s ratio effect, but contact conducting cannot be established until dij reduces to deq. Thus, in this case, the tension in the matrix tends to reduce the overall conductivity of the nanocomposites.

Differently, in the present study the nanocomposite considered has a conductivity of the order of 1to 10 S/cm suggesting that the electron tunnelling is the main physical mechanism of electron transfer, which can be achieved at a wide range of dij, i.e., deq < dij < dcut. In tension, tunnelling contact maintains until dij finally increases to dcut. In the meantime, some AuNPs may get closer to each other and new tunnelling contacts can be established once dij falls in the range of deq < dij < dcut. This may lead to rising conductivity with growing strain.

To confirm this theory, in the insets of Fig. 4 we compute the strain-dependence of the fraction of type I dij, i.e., dij = deq (contact conducting), (2) type II dij satisfying deq < dij < dcut (tunnelling conducting) and (3) type III dij, i.e., dijdcut (no electron transfer), which is defined as the ratio between the number of dij and the total number of AuNP pairs in the simulation box. As noted in the insets, the fraction of type II dij corresponding to the tunnelling conducting is raised from around 60% up to 80%, while the fraction of type I dij leading to contact conducting is lower than 20% at very beginning and becomes very close to zero afterward. The observation supports the presumption that the electrical conductivity of the nanocomposite is achieved primarily via the electron tunnelling between AuNPs. Specifically, it is noted that, in the two scenarios considered the trend of type II dij to change with strain (the insets of Fig. 5) is approximately the same as the strain-dependence of the electrical conductivity observed in Fig. 5. This indicates that, with rising tensile strain the increase of the electrical conductivity is a result of the increasing number of type II dij (associated with the tunnelling electron transfer).

In this study, MD simulations are combined with Monte Carlo technique and the effective resistor model to examine the effects of the vdW interaction between AuNPs on the mechanical properties, the dynamics of AuNPs and the piezo-resistivity of the AuNP-polymer nanocomposite subject to tensile deformation. Based on the obtained results the conclusions are drawn as follows.

The vdW interaction between AuNPs tends to hold the AuNPs together, which improves the ability to resist tensile deformation and yielding of the nanocomposite. At large strain, e.g., larger than 450%, it also leads to substantial changes in the deformation of polymer chains. Even at VF = 5%, the interaction between AuNPs can enhance the Young’s modulus, yield stress and fracture stress of the nanocomposite by an order of 10%. The enhancement further increases when VF rises from 5% to 10% or even larger.

The vdW interaction between AuNPs is strong enough to affect the dynamics or the trajectories of individual AuNPs during the tensile deformation and thus, reconstruct the electrical circuit formed by AuNPs. Such an effect is small at relatively low strain, e.g., < 30%. It however turns out to be substantial at higher strain and largely enhances the effect of piezo-resistivity, i.e., the sensitivity of electrical conductivity to tensile strain. In addition, it is also found that the piezoresistive responses of the composites depend substantially on the VF of the conductive NPs or the predominant conducting mechanism of the composite.

The effect of AuNP interaction on the mechanics of the nanocomposite is comparable to the effect of the AuNP-polymer interface. Its influence on the dynamics of AuNPs and piezoelectricity of the nanocomposite would be more pronounced than the interface effect. Thus, the AuNP interaction has to be taken into consideration in the study of the AuNP-PE nanocomposites as well as the nanocomposite fabricated based on copper or silver nanoparticles. We note here that other inter NP interactions may also exist but are not considered in this work such as those arise from columbic interaction or spin orbit coupling, this is because the AuNPs considered here are charge neutral while spin-orbit interactions primarily lead to band energy change at the level of only a few meV. Here, it is also expected that the AuNP attraction could affect the viscous elastic properties or even the constitutive relation of the nanocomposites when the VF and strain are sufficiently large. These issues deserve to be examined in detail in near future.

Work supported by Jiangsu Province Natural Science Foundation (SBK2015020787), Y.L. is supported by NSFC (11520101001).

1.
Y.
Kim
,
J.
Zhu
,
B.
Yeom
,
M. D.
Prima
,
X. L.
Su
,
J. G.
Kim
,
S. J.
Yoo
,
C.
Uher
, and
N. A.
Kotov
,
Nature
500
,
59
63
(
2013
).
2.
S.
Yao
and
Y.
Zhu
,
Adv. Mater.
27
,
1480
1511
(
2015
).
3.
K. D.
Harris
,
A. L.
Elias
, and
H. J.
Chung
,
J. Mater. Sci.
51
,
2771
2805
(
2016
).
4.
J.
Ge
,
L.
Sun
,
F. R.
Zhang
,
Y.
Zhang
,
L. A.
Shi
,
H. Y.
Zhao
,
H. W.
Zhu
,
H. L.
Jiang
, and
S. H.
Yu
,
Adv. Mater.
28
,
722
728
(
2016
).
5.
J. A.
Rogers
,
T.
Someya
, and
Y.
Huang
,
Science
327
,
1603
1607
(
2010
).
6.
K. D.
Harris
,
A. L.
Elias
, and
H. J.
Chung
,
J. Mater. Sci.
51
,
2771
2805
(
2016
).
7.
T.
Sekitani
,
H.
Nakajima
,
H.
Maeda
,
T.
Fukushima
,
T.
Aida
,
K.
Hata
, and
T.
Someya
,
Nat. Mater.
8
,
494
499
(
2009
).
8.
M.
Amjadi
,
A.
Pichitpajongkit
,
S.
Lee
,
S.
Ryu
, and
I.
Park
,
ACS Nano
8
,
5154
5163
(
2014
).
9.
X.
Huang
,
W. H.
Yeo
,
Y. H.
Liu
, and
J. A.
Rogers
,
Biointerphases
7
,
52
(
2012
).
10.
G. T.
Wang
,
C. Y.
Wang
,
F. L.
Zhang
, and
X. Z.
Yu
,
Comput. Mater. Sci.
150
,
102
106
(
2018
).
11.
Y.
Wang
,
C. Y.
Wang
,
G. T.
Wang
, and
R. J.
Wang
,
Comput. Mater. Sci.
134
,
58
66
(
2017
).
12.
Y.
Wang
,
R. J.
Wang
,
C. Y.
Wang
 et al,
J. Appl. Phys.
123
,
214305
(
2018
).
13.
G. T.
Wang
,
C. Y.
Wang
,
C.
Tang
,
F. L.
Zhang
,
T.
Sun
, and
X. Z.
Yu
,
J. Phys. Chem. C
122
,
8614
8620
(
2018
).
14.
F. L.
Zhang
,
G. T.
Wang
,
C. Y.
Wang
,
C.
Tang
,
T.
Sun
, and
X. Z.
Yu
,
Appl. Phys. Letts.
113
,
253105
(
2018
).
15.
P. V.
Gaikwad
,
S. K.
Sharma
,
K.
Sudarshan
,
V.
Kumar
,
A.
Kshirsagar
, and
P. K.
Pujari
,
Polymer Compos
39
,
1137
1143
(
2018
).
16.
L. Y.
Jiang
,
Y.
Huang
,
H.
Jiang
,
G.
Ravichandran
,
H.
Gao
, and
K. C.
Hwang
,
J. Mech. Phys. Solids.
53
,
2436
2452
(
2006
).
17.
S. D.
Ben
,
J. H.
Zhao
, and
T.
Rabczuk
,
Compos. Struct.
117
,
288
29
(
2014
).
18.
J. H.
Zhao
,
L. X.
Lu
,
Z. L.
Zhang
,
W. L.
Guo
, and
T.
Rabczuk
,
Comp. Mater. Sci.
96
,
432
438
(
2015
).
19.
D.
Hossain
,
M. A.
Tschopp
,
D. K.
Ward
,
J. L.
Bouvard
,
P.
Wang
, and
M. F.
Horstemeyer
,
Polymer
51
,
6071
6083
(
2010
).
20.
F. M.
Capaldi
,
M. C.
Boyce
, and
G. C.
Rutledge
,
Polymer
45
,
1391
1399
(
2004
).
21.
L.
Huang
,
X. P.
Yang
,
X. L.
Jia
, and
D. P.
Cao
,
Phys. Chem. Chem. Phys.
16
,
24892
24898
(
2014
).
22.
S. M.
Foiles
,
M. I.
Baskes
, and
M. S.
Daw
,
Phys. Rev. B
33
,
7983
7991
(
1986
).
23.
J. H.
Fang
,
John Wiley & Sons Ltd.
,
West Sussex, UK
,
2010
.
24.
C.
Vasile
and
M.
Pascu
,
Practical Guide to Polyethylene
(
RAPRA Technology
,
2005
).
25.
N.
Hu
,
Y.
Karube
,
C.
Yan
,
Z.
Masuda
, and
H.
Fukunaga
,
Acta Mater
56
,
2929
2936
(
2008
).
26.
W. B.
Lu
,
T. W.
Chou
, and
E. T.
Thostenson
,
Appl. Phys. Lett.
96
,
223106
(
2010
).
27.
M.
Amjadi
,
A.
Pichitpajongkit
,
S. J.
Lee
,
S.
Ryu
, and
I.
Park
,
ACS Nano
8
,
5154
5163
(
2014
).