Two Stage Electrical Percolation of Metal ... - ACS Publications

Subscriber access provided by UNIV OF DURHAM

C: Physical Processes in Nanomaterials and Nanostructures

Two Stage Electrical Percolation of Metal Nanoparticle-Polymer Nanocomposites Guotong Wang, Chengyuan Wang, Chun Tang, Faling Zhang, Tiger Sun, and Xiaozhu Yu J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b01079 • Publication Date (Web): 03 Apr 2018 Downloaded from http://pubs.acs.org on April 3, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Two-Stage Electrical Percolation of Metal Nanoparticle-Polymer Nanocomposites Guotong Wang1, Chengyuan Wang*1, Chun Tang*1, Faling Zhang1, Tiger Sun1 and Xiaozhu Yu2 1

Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, China

2

Faculty of Science, Jiangsu University, Zhenjiang 212013, China

Abstract: Recent experiments showed that gold nanoparticle (NP)-polymer composite exhibits excellent properties such as high strechability and electron conductivity, rendering this novel material promising for bendable and stretchable electronics and optoelectronics. Theoretical models have been proposed to investigate the conduction mechanism, however, the role of the quantum tunneling effect in electrical percolation remains unclear. Here we used a numerical approach together with Monte-Carlo sampling to investigate the percolation of the gold NP-polymer system. The effects of the electron tunneling and the inter-NP van der Waals interaction were considered in the model. A distinct two stage electrical percolation behavior is identified due to the effect of electron tunneling at the nanoscale. Such an effect is found to be dependent on the radii of gold NPs and becomes negligible when the radius is larger than 195 nm. The observed behavior is also sensitive to the potential barrier height of the hosting polymer. Our result therefore not only provides new insights into the conduction mechanism of the gold NP-polymer composites, but also offer new strategy to designing metal NP-polymer system with desired properties.

1

Corresponding author: [email protected], [email protected]

1 / 22 ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 22

1. Introduction Flexible electronics and optoelectronics have become increasingly important in the development of innovative techniques in the 21st century. Typical examples are foldable LED displays,1 conformable biosensors,2 soft energy storage devices,3,

4

organic transistors5 and smart textiles6. One major challenge for the techniques mentioned above is the capability of fabricating bendable electrodes, flexible electrical connections and foldable or stretchable integrated circuits. To remove this hurdle, a great deal of effort has been devoted to fabricating flexible conductors by distributing conductive nanofillers into insulating polymer elastomers.7 The conductivity is achieved via the percolation of the nanofillers.8 These highly conductive elastomers thus have stimulated extensive studies in the last decade. Up till now, the investigation has been mainly focused on the composites filled with carbon nanotubes (CNTs)9,10 and silver nanowires (AgNWs)11,12. Experimental studies11, 13-15, computer simulations11, 15-19 and theoretical modelling20-23 were carried out to measure their electrical and mechanical properties, and disclose the underlying physics of the high conductivity achieved. Herein, the electron tunneling and electrical contact are identified as two physical origins of the overall electrical conductivity. In addition to the slender nanofillers, e.g., CNTs and AgNWs, metal nanoparticles (NPs) were also employed to fabricate such flexible and conductive nanocomposites25-27, among them is the gold nanoparticle (AuNP)-polymer composite27 that exhibits an electrical conductivity greater than 104 S/cm and a large elongation up to 480%. This conductivity is orders of magnitude higher than the conductivity of most composites filled with the slender nanofillers.7 Moreover, piezo-resistive effect is achieved,27 which provides a new pathway to conformable sensor network for the stress/strain distributions on curvilinear or dynamics surfaces. Such a flexible stress/strain sensing film is promising for potential applications in sensing skin, smart fibres and clothes, the morphing wings of airplane, the structural health monitoring and non-destructive inspection of mechanical structures. Despite of its superior properties and a broad range of potential applications, the percolation behavior and its underlying physics so far have not been investigated in detail except for a recent Monte Carlo simulation.28 In this work,28 the percolation behavior at low volume fraction however was not discussed as the technique used is unable to give reliable results when the number of AuNPs become small. As a result, a 2 / 22 ACS Paragon Plus Environment

Page 3 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


theoretical model is required to understand the simulation results and gain the whole picture of its percolation behavior (from a low volume fraction to a high one), identify the determinants of its conductivity and piezo-resistivity, and disclose the underlying physics of its excellent properties. The present paper is intended to take an initiative to address these issues. In what follows, theoretical models were developed in Sec.2 for the equivalent spherical conductors at the nanoscale, where the electron tunneling is taken into consideration, Sec. 3 presents the simulated percolation behavior of the nanocomposite and reveals its underlying physics. Finally, the conclusions were drawn in Sec.4.

2. Methodology 2.1 Equivalent Nanoconductor Model for AuNPs As shown in Fig. 1, a composite model is considered, where AuNPs are randomly distributed in the polymer matrix. For simplification, the radii of the AuNPs in the model are set to be uniform. Under external electric field, electrons are conducted through metallic contacts in most cases when two AuNPs are close enough. While at the nanoscale, it is also known that quantum confinement effect9-12,20-24 enables electrons to tunnel through a thin layer of insulating body between two conductive objects and thus, allows the electron transportation between the two conductors that are not in physical contact. In particular, electron tunneling was identified as one of the conduction mechanisms in nanocomposites filled with CNTs9,10,20-23 and AgNWs11,12. However, the role of such a small-scale effect and its size dependency have not been examined in the percolation of AuNP-polymer composite. Thus, in the present study the electron tunneling effect is considered for the AuNP embedded in the polymer matrix. To consider the tunneling effect in the theoretical model, an equivalent nanoconductor in the AuNP-polymer composite is treated as a composite nanosphere comprising of two components (see the right panel of Fig.1), i.e. the inner sphere representing an AuNP of radius ( rnp ) and the outer spherical thin shell representing the interphase with radial thickness (tint) between the AuNP surface and the surrounding matrix (or neighboring AuNP). Such a virtual interphase reflects the electron tunneling effect between two adjacent AuNPs and its radial thickness (tint) 3 / 22 ACS Paragon Plus Environment


Page 4 of 22

equals half of the maximum distance the electrons can tunnel through. Note in this model, the spherical polar coordinator system (θ , ϕ , z ) is used for the AuNP with its origin coinciding with the center of the spherical nanoconductor. When a uniform electric field (strength E0) in the z-axis direction is applied to the equivalent spherical particle, the electrical potential φ in the equivalent conductor satisfies the Maxwell equation.

1 ∂  2 ∂φ  1 ∂  ∂φ  1 ∂ 2φ + sin + =0 r θ     ∂θ  r 2 sin 2 θ ∂ϕ 2 r 2 ∂r  ∂r  r 2 sin θ ∂θ 

(1)

Here, r denotes the radial coordinate. In Fig.1, an axisymmetric structure about the zaxis is considered, thus, the potential function φ

of the AuNP should be

axisymmetric and independent of the angular coordinate ϕ , i.e.,

∂φ = 0 . As a result, ∂ϕ

Eq.1 reduces to the following equation. ∂  2 ∂φ  1 ∂  ∂φ  r +  sin θ =0 ∂r  ∂r  sin θ ∂θ  ∂θ 

(2)

The general solution to Eq.2 is as follows.22,29  

φ ( r , θ ) = ∑  an r n + n

bn   ⋅ Pn ( cos θ ) r n +1 

(3)

where an and bn are some coefficients, Pn ( cos θ ) is the LaGrange function and n is an integer. Solving Eq.2 yields the following potential distribution in the equivalent spherical conductor

φnp = anp ,0 +

φm = am,0 +

bnp ,1   +  anp ,1r + 2  cos θ r r  

bnp ,0

b   +  am,1r + m2,1  cos θ r  r 

bm,0

φint = aint,0 +

b   +  aint,1r + int,1  cos θ r r2  

bint,0

(4)

Throughout this paper, the subscripts np, int and m represent the parameters of the AuNP, the interphase and the matrix, respectively. The boundary conditions for this


Page 5 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


problem are prescribed as:

φnp |r =0 = contant,

Em |r =∞ = E0 cosθ

φnp | r = r = φint | r = r ,

−σ np

np

φm | r = r

np + tint

np

= φint | r = rnp + tint , −σ m

∂φnp ∂r

| r = rnp = −σ int

∂φint | r = rnp ∂r

∂φm ∂φ | r = rnp +tint = −σ int int | r = rnp +tint ∂r ∂r

(5)

Here, σ np , σ int and σ m denote the electrical conductivity of the AuNP, the interphase and the matrix, respectively. Solving Eq. 4 with the corresponding boundary conditions in Eq. 5 gives the electric potential distribution shown below.

φnp ( r ,θ ) = 3 Aσ int z ，

0 ≤ r ≤ rnp

3   rnp   φint ( r ,θ ) =A ⋅ σ np + 2σ int + (σ int − σ np ) ⋅    ⋅ z ，  r   

rnp ≤ r ≤ rnp + tint

3     r +t    (σ np + 2σ int ) ⋅  np int  +   3     rnp + tint   r   φm ( r ,θ ) =  A ⋅   + E0  r  − E0  ⋅ z ， r ≥ rnp + tint 3    rnp       (σ int − σ np ) ⋅  r         

(6) where z = r cos θ and A=

3σ m E0 ( rnp + tint )

3

2 (σ int − σ np ) (σ int − σ m ) rnp3 − ( 2σ m + σ int ) ( rnp + tint ) (σ np + 2σ int ) 3

(7)

Subsequently, the electric field along the z-axis for the system shown in Fig. 1 can be determined by taking the derivative of potential with respect to r: ∂φ 1 ⋅ np = − 3 A σ int cos θ ∂ r

(8)

 2(σ int − σ np ) ⋅ rnp3  1 ∂φint ⋅ = − A(σ np + 2σ int ) −  cosθ ∂r r3  

(9)

E np , z = −

Eint, z = −



Page 6 of 22

2.2 Tunneling between AuNPs We now discuss how to evaluate the tunneling currents when two AuNPs are close enough but not in physical contact. The separation distance d between the adjacent 0 0 AuNPs of the same radius rnp can be defined as d = d − 2rnp where d is the

distance between the AuNP centers. This is actually the shortest distance between the surfaces of two NPs. It is well-known that the effect of the van der Waals (vdW) interaction becomes substantial for the nanomaterials due to their largely increased surface-to-volume ratio. As a result of the repulsive force, there will be a distance between two AuNPs, i.e., the AuNPs cannot be in real physical contact. From molecular dynamics simulation (Supporting Information), we found the equilibrium distance between two AuNPs, i.e. the vdW distance is d vdw =0.273nm , it is thus assumed that when d is equal or less than dvdw (i.e. when the composite is at zero strain or under compression), two AuNPs are in Ohmic contact, therefore the current density is determined only by σ np . When d is greater than dvdw but less than a critical value (i.e., the cut off distance), tunneling transport takes place according to quantum mechanics theory. The cut off distance for electron tunneling in AuNP is estimated to be dtun =1.5nm based on our Monte Carlo simulations (Supporting Information). This situation occurs when the volume fraction of AuNP is small therefore the AuNPs are not sufficiently packed in the matrix. Accordingly, we can define the thickness of the interphase as tint =

d . When d > dtun , no electron transport occurs between two 2

adjacent AuNPs. The tunneling-type resistance between the interphases of two adjacent AuNPs (of radius rnp) can be evaluated approximately by using Simmons’ formulation derived for the electron tunneling between two electrodes separated by a thin insulating film.21, 22,24

d − d vdw ) ⋅ h 2  4π ( d − d vdw ) ( 1/2  Rint ( d , λ ) = 2 exp  ( 2mλ )  ， 1/2 h Se ( 2mλ )  

(10)

−31 Here λ is the potential barrier height of the insulating matrix; m ( 9.10938291×10 kg )


Page 7 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


and e ( 1 .602176565 × 10 −19 °C ) are the mass and the electric charge of an electron, respectively; S is the contact area of the two equivalent nanoconductors and h ( 6.626068×10

−34

m2 kg / s ) is the Planck constant. The electrical conductivity of the

interphase surrounding an AuNP in the polymer can then be calculated by

σ int =

d − d vdw ， S Rint ( d , λ )

(11)

2.3 AuNPs in polymer matrix The electric current density in the AuNP and the interphase can be derived as:

jnp , z = σ np Enp , z and jint, z = σ int Eint, z

(12)

As shown in Eqs. 6 to 9 and 12，j and E are only the functions of r. By introducing the average current density

and the average electrical field strength Ez we have，

jz

jz = σ E z

where

jz =

1 V

Ez =

1 V

∫

V

∫

V

j z dV =

(13)

1   ∫V jnp , z dV + ∫V jint, z dV  ， np int  V

E z dV =

1   ∫V Enp , z dV + ∫V Eint, z dV  , np int  V

(14)

Finally, the effective electrical conductivity (σep) of the equivalent spherical nanoconductor (Fig.1) can be derived as follows based on Eqs. 1-14.

σ ep =

r +t 3 3σ npσ int rnp3 + (σ np + 2σ int ) ( rnp + tint ) − rnp3  σ int − 6σ int (σ int − σ np ) rnp3 ln np int   r np

3σ np rnp3 + (σ np + 2σ int ) ( rnp + tint ) − rnp3  − 6 (σ int − σ np ) rnp3 ln   3

rnp + tint rnp (15)

In addition, the volume fraction Vep of the equivalent spherical nanoconductors (Fig.1) can be obtained in terms of the volume fraction Vf of the AuNPs in the polymer.

 rnp + tint Vep =   r  np

3

  V f 


(16)


Page 8 of 22

According to Bruggeman’s effective media theory about two-phase composites consisting of spherical particles, we obtain an equation for the effective electrical conductivity of the nanoparticle-polymer system:29-31

σ eff − σ m σ − σ ep + Vep eff = 0， 2σ eff + σ ep eff + σ m

(1 − V ) 2σ ep

(17)

3. Results and discussions In what follows the model developed in Sec. 2 will be used to study the percolation behavior of the AuNP-polymer composite. The radius rnp of AuNPs varying from 5nm to 1µ m and the tunneling energy barrier of the polymer matrix 0.5eV, 1.0eV and 2.0eV,15, 23 are considered in the calculations. In addition, the electrical conductivities

4.17 ×105 S / cm and 10−9 S / cm 21,32are used for the AuNPs and the polymer matrix, respectively.

3.1 Percolation of the AuNP-polymer composite In this section, the first scenario considered is that the volume fraction of the AuNPs is small and the equivalent conductors (i.e., AuNPs wrapped by a spherical shell of the interphase with thickness tint=dtun /2) have a constant radius (rnp + tint) and the electrical conductivity given by Eq. 15. Here dtun represents the maximum distance of electron tunneling and rnp is fixed at 5nm. Specifically, by using Eq.17 we assumed that the conductors are solid spheres which cannot penetrate each other. Fig. 2(a) shows that when Vf is below a critical value Vcr1 (the first percolation threshold), the conductivity of the composite is around 10-8 S/cm and steadily increases as the volume fraction rises. At Vcr1 (22%), the first jump of conductivity occurs raising it from about 10-8 S/cm to 10-4 S/cm by 4 orders of magnitudes. This observation suggests that the conductive pathways are established in the matrix by the equivalent conductors, where the adjacent two conductors are in electrical contact via the electron tunneling. Herein, it is worth mentioning that at Vcr1= 22%, Eq.16 yields the average distance of the AuNPs about 1.8nm, which is greater than the cut off distance 1.5nm, this suggests that not all AuNPs are involved in the tunneling conducting. Thus, the conductive pathways achieved only consist of a part of the AuNPs embedded in the polymer matrix. At Vcr1> 22%, further raising the volume 8 / 22 ACS Paragon Plus Environment

Page 9 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


fraction leads to the gradual growth of the conductivity with the increasing number of the conductors involved in the electrical circuits. This situation is demonstrated by the results shown in Supporting Information Fig.S1 where it is seen that from before to after Vcr1, the number of AuNP pairs that have inter-particle distance less than the threshold of 1.5 nm significantly increased. This percolation behavior is found to be analogous to similar behaviors reported for CNT-polymer composites where only one-stage percolation is observed while the value of conductivity is smaller (up to 10 S/m possibly due to different material properties of CNTs).20-23 In the second scenario, we consider the equivalent conductors with radius rnp + 0.1365nm (0.273nm is the vdW equilibrium distance between the AuNPs of radius 5nm) and the conductivity of pure gold. The corresponding percolation curves are represented by the blue dashed lines in Fig. 2., it can be seen that in the early stage (when the volume fraction is lower than 31%) the conductivity of the AuNP-polymer composite remains lower than the conductivity due to electron tunneling. Then, an abrupt increase occurs as the volume fraction reaches the second critical value Vcr2 (31%) (i.e., the second percolation threshold). In this case, the conductivity is raised from 10-4 S/cm up to 104S/cm by 8 orders of magnitudes. Such a phenomenon can be interpreted as a result of the construction of conductive pathway at Vcr2, where adjacent AuNPs are in conductive contact and thus, electron transportation is implemented via the formation of ohmic conducting channels in the composite. To summarize, the percolation of the AuNP-polymer composite is characterized by the two-stage conductivity changes represented by the red solid lines shown in Fig. 2(a), which are obtained as the superposition of the two percolation curves for the above two cases studied. The first conductivity jump (10-8 S/cm to 10-4 S/cm) at Vcr1 (22%) is primarily an effect of electron tunneling at the nanoscale while the second sudden growth (10-4 S/cm to 104 S/cm) at Vcr2 (31%) is predominantly controlled by the conductive contact of the AuNPs. In the experiment27 and our Monte Carlo simulation28 only one conductivity jump is observed, where the conductivity rises from 10-4 S/cm27 or10-2 S/cm28 to higher than 104 S/cm. The observation thus reflects the percolation behavior due to the conductive contact of the AuNPs.

3.2 Size-dependence of the electron tunneling effect It is clearly demonstrated in Fig. 2 that the AuNP-polymer composite exhibit the



two-stage percolation behavior representing two different conducting mechanisms, i.e. the electron tunneling and conductive contacts of the AuNPs. Herein, the width of this mechanism transition can be measured by ∆Vcr = Vcr2 - Vcr1. Within this region, the variation of conductivity with respect to AuNP volume fraction are mainly due to the enhanced tunneling effect due to increasing number of AuNPs approaching the vdW contact region but not in physical contact. It is understood that the tunneling effect is a physical mechanism emerging at the nanoscale. Its effect represented by ∆Vcr might change when the radius of AuNPs varies and lead to substantial variation in the percolation behavior of the composite. It is thus of great interest to exam the effect of AuNP size on ∆Vcr or the percolation of the composite. It is interesting to note that the phenomenon observed in Sec.3.1 is qualitatively similar for AuNPs of different radii, as representative examples shown in Figs.2(b) and (c), where the two-stage transition of conductivity versus AuNP volume fraction is also seen for AuNP radii of 10nm and 50nm. However, it is also seen that the width of the transition, ∆Vcr, decreases with rising nanoparticle size. Accordingly, the ratio ∆Vcr / Vcr1 changes monotonically from 40.2% to 19.26% to 3.75% for AuNP radii of 5 nm, 10 nm and 50 nm, respectively. It is understood that the percolation threshold is primarily determined by the geometric size of AuNPs when the same distribution is assumed in the matrix. Both the present study and our previous Monte Carlo simulations28 showed that the percolation threshold generally increases with rising AuNP size. Thus, increasing the size of AuNP will significantly raise both Vcr1 and Vcr2. In this process, since the maximum tunneling distance dtun (1.5nm) and the equilibrium vdW distance dvdw (0.273nm) remain unchanged, the radii of the two equivalent conductors associated with Vcr1 and Vcr2, i.e., rnp + dtun /2 and rnp + dvdw /2, are getting closer to the radius of the AuNP rnp or the tunneling effect turns out to be smaller on the percolation relation of the composite. Accordingly, Vcr1 is moving towards Vcr2 leading to decreasing ∆Vcr = Vcr2 - Vcr1. In other words, comparing to small radius AuNP-polymer composites, the electron tunneling effect on the two-stage percolation character for large radius AuNP composites turns out to be smaller in the sense that the two-stage feature almost vanishes and appears similar to that of one-stage percolation behavior. To provide a quantitative picture of this correlation, we have calculated the ratio of ∆Vcr/Vcr1 as a function of AuNP radius in Fig.3. This ratio ∆Vcr/Vcr1 measures the effect of the tunneling transport on the electrical percolation of the AuNP-polymer 10 / 22 ACS Paragon Plus Environment

Page 10 of 22

Page 11 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


composite. An inverse relationship is clearly seen in Fig.3 and the value is 40.2% for AuNPs with a radius of 5 nm, and steadily decreases to 5% when the radius is about 37.5 nm. Whether the tunneling effect is significant depends on how we define the threshold of ∆Vcr/Vcr1. We therefore included in the inset of Fig.3 a zoomed in plot of the same curve, with a threshold value of 1% marked by the dash line. It can now be seen that the critical radius for the tunneling effect to be neglected is 195nm if the critical value of ∆Vcr/Vcr1 is set to be 1%. Indeed, the quantum effect primarily takes effect at the nanoscale and sub-nanoscale for condensed matter systems, but vanishes when the AuNP radius is sufficiently large, e.g., greater than 195nm.

3.3 Transition zone between the two conducting mechanisms It has to be noted that the above model is a simplified description of the AuNP-polymer system because the quantum effect has close relationship with the inter-particle distance and thus, the volume fraction of AuNPs, i.e. the greater the volume fraction, the closer the two particles reside (i.e., the reduced tunneling distance) and the stronger the tunneling current. In this case, the thickness tint of the interphase on the outer surface of AuNPs and thus, the radius of the equivalent conductors will decrease with the rising volume fraction. This physical effect however is not taken into account in the above model when the volume fraction rises in the transition zone between Vcr1 and Vcr2. To approximately evaluate this effect, we calculate the percolation curves (dashed lines in Fig. 4(a)) for the equivalent conductors comprising AuNPs of given radius 5nm wrapped by the interphase whose thickness varies from 0.75nm (i.e., dtun/2) to 0.70nm, 0.55nm and to 0.35nm. According to Eqs. 10 and 11, decreasing the thickness (= d/2) leads to the lower tunneling resistance, in other words, increased tunneling current. Therefore, while the volume fraction grows from Vcr1 to Vcr2, Fig.4(a) indicates that the conductivity jump associated with the interphase thickness 0.75nm is followed by a series of the sudden increases of the conductivity originating from the percolation of the equivalent conductors with decreasing interphase thickness. Herein, the envelope plot of these percolation curves (dashed lines) can better approximates the real conducting mechanism and thus, more accurately describe the dependence of the conductivity on the volume fraction in the transition zone. As can be seen in Fig.4(a), although the shape of the red solid line shows a slightly



different trend, a two-stage conduction picture can still be observed. For rnp = 5nm considered here, the first transition also occurs at Vcr1 = 22%, the primary difference as compared to that in Fig.2(a) is that the slope from Vcr1 = 22% to 31% is significantly increased. This behavior can now be understood as the fact that with increased volume fraction of AuNP, AuNPs become close to each other, which improves the probability for the tunneling effect to appear, the distance-dependent tunneling effect also plays a key role in the conductivity transition. Because tunneling conductivity is lower than the contact conductivity, the second transition at Vcr2 is still observed although the jumping height is now reduced (from about 101 S/cm to 104 S/cm), signifies the fact that the two-stage percolation is a key transport mechanism for the AuNP-polymer composite. Next, we shall discuss how the potential barrier height of the polymer matrix affects the percolation behavior in the transition zone. As shown in Fig.4(b), the same plot (λ =1.0 eV) as in Fig.4(a) is highlighted in red as a reference. When λ increases to 2.0 eV, the first transition disappears, whereas the two-stage percolation behavior shows up again with even higher tunneling current at λ = 0.5 eV. This effect of λ can be attributed to the fact that higher potential barrier leads to the decreased possibility of the electron tunneling and thus, the reduced tunneling current. In contrast, it is noticed in Fig. 4(b) the above -mentioned λ effect vanishes when the volume fraction approaches the second percolation threshold and the second transition occurs primarily due to the Ohmic contact between AuNPs. In addition, it is also noted in Fig.4(b) that the gradient of the percolation curves in the transition zone increases considerably with the decreasing λ, showing the stronger dependence of the conductivity on the volume fraction of AuNPs or the distance d between AuNPs. The rising d-dependence can be understood based on Eq.10 where the effect of the AuNP distance d on the tunneling-type resistance increases rapidly with rising λ. This suggests that, in the transition zone, the conductivity should be more sensitive to the variation of the AuNP distance d induced by applied strain. In other words, higher λ may lead to stronger piezo-resistive effect of the AuNP-polymer composite. Thus, the AuNP-polymer composite with high λ of polymer matrix but relatively low volume fraction of AuNPs is promising for the application in piezo-resistive strain sensors.


Page 12 of 22

Page 13 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4. Conclusions In summary, we developed a theoretical model to investigate the electrical percolation behavior of AuNPs-polymer composites by considering the tunneling effect at the nanoscale. A two-stage percolation behavior with respect to the rise of AuNP volume fraction is observed for the first time due to the two nanoscale conducting mechanisms, i.e., electron tunneling and conductive contact. The tunneling effect is sensitive to the AuNP size, which is strong at radius of AuNP smaller than 5 nm, but decreases with the rising radius. In particular, as the AuNP radius is beyond 195 nm the tunneling effect diminishes and the two-stage percolation behavior will reduce to the traditional one-stage percolation behavior observed for macroscopic composites. Thus, tunneling effect plays a vital role in the conductivity of metal nanoparticle-polymer composites and can reshape the percolation behavior of the nanocomposites. In addition, the greater potential barrier height of the polymer may lead to stronger piezo-resistive effect of the AuNP-polymer composite when the electron tunneling effect is predominant. We expect this work to stimulate further experiment and theoretical efforts to investigate the electron conduction mechanism of this new class of composites which are promising for a wide range of engineering applications in various industry disciplines. In the meantime, it should be pointed out that the two-stage transition of the conductivity achieved in the present work results from the effect of electron tunneling at the nanoscale. The general principle identified here is not limited to gold nanoparticles and the polymer matrix. Thus, we expect the unique features captured in this work to be generally applicable to conductive nanocomposites consisting of different metal nanoparticles and polymer matrix.

Supporting Information Distribution of inter AuNP distances with respect to its volume fraction, details of Monte Carlo methods used to construct the RVE model and molecular dynamics simulations in determining geometries of the AuNP-polymer composites. This information is available free of charge via the internet at http://pub.acs.org.

Acknowledgement Financial support from the General Program of Jiangsu Natural Science Foundation of



China (Grant No. SBK2015020787), Youth Fund of Jiangsu Natural Science Foundation of China (BK20140523), Start-up Research Fund for Advanced Talent of Jiangsu University (15JDG042) and Start-up Fund for Jinshan Professorship are greatly appreciated.

References (1) Sekitani, T.; Nakajima, H.; Maeda, H.; Fukushima, T.; Aida, T.; Hata, K.; Someya, T. Stretchable Active-Matrix Organic Light-Emitting Diode Display Using Printable Elastic Conductors. Nat. Mater. 2009, 8, 494-499. (2) Huang, X.; Yeo, W. H.; Liu, Y. H.; Rogers, J. A. Epidermal Differential Impedance Sensor for Conformal Skin Hydration Monitoring. Biointerphases

2012, 7, 1-9. (3) Wang, X. F.; Lu, X. H.; Liu, B.; Chen, D.; Tong, Y. X.; Shen, G. Z. Flexible Energy-Storage Devices: Design Consideration and Recent Progress. Adv. Mater.

2014, 26, 4763-4782. (4) Wang, X. F.; Jiang, K.; Shen, G. Z. Flexible Fiber Energy Storage and Integrated Devices: Recent Progress and Perspectives. Mater. Today 2015, 18, 265-272. (5) Sekitani, T.; Yokota, T.; Zschieschang, U.; Klauk, H.; Bauer, S. F.; Takeuchi, K.; Takamiya, M.; Sakurai, T.; Someya, T. Organic Nonvolatile Memory Transistors for Flexible Sensor Arrays. Science 2009, 326, 1516-1519. (6) Stoppa, M.; Chiolerio, A. Wearable Electronics and Smart Textiles: A Critical Review. Sensors 2014, 14, 11957-11992. (7) Yao S. S.; Zhu, Y. Nanomaterial-Enabled Stretchable Conductors: Strategies, Materials and Devices. Adv. Mater. 2015, 27, 1480-1511. (8) Hoshen J.; Kopelman, R. Percolation and Cluster Distribution. I. Cluster Multiple Labeling Technique and Critical Concentration Algorithm. Phys. Rev. B 1976, 14, 3438-3445. (9) Yu, C. J.; Masarapu, C.; Rong, J. P.; Wei, B. Q.; Jiang, H. Q. Stretchable Supercapacitors Based on Buckled Single-Walled Carbon Nanotube Macrofilms. Adv. Mater. 2009, 21, 4793–4797. (10) Shang, S. M.; Zeng, W.; Tao, X. M. High Stretchable MWNTs/polyurethane Conductive Nanocomposites. J. Mater. Chem. 2011, 21, 7274–7280.


Page 14 of 22

Page 15 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(11) Amjadi, M.; Pichitpajongkit, A.; Lee, S. J.; Ryu, S.; Park, I. Highly Stretchable and Sensitive Strain Sensor Based on Silver Nanowire Elastomer Nanocomposite. AC Nano 2014, 8, 5154–5163. (12) Liu, C. H.; Yu, X. Silver Nanowire-Based Transparent, Flexible, and Conductive Thin Film. Nanoscale. Res. Lett. 2011, 6, 1-8. (13) Ramasubramaniama, R.; Chen, J.; Liu, H. Y. Homogeneous Carbon Nanotube Polymer Composites for Electrical Applications. Appl. Phys. Lett. 2003, 83, 2928-2930. (14) McLachlan, D. S.; Chiteme, C.; Park, C.; Wise, K. E.; Lowther, S. E.; Lillehei, P. T.; Siochi, E. J.; Harrison, J. S. AC and DC Percolative Conductivity of Single Wall Carbon Nanotube Polymer Composites. J. Polym. Sci. Part B: Polym. Phys.

2005, 43, 3273-3287. (15) Hu, N.; Masuda, Z.; Yan, C.; Yamamoto, G.; Fukunaga, H.; Hashida, T. The Electrical Properties of Polymer Nanocomposites with Carbon Nanotube Fillers. Nanotechnology 2008, 19, 215701. (16) Ma, H. M.; Gao, X. L. A Three-Dimensional Monte Carlo Model for Electrically Conductive Polymer Matrix Composites Filled with Curved Fibers. Polymer

2008, 49, 4230-4238. (17) Hu, N.; Karube, Y.; Yan, C.; Masuda, Z.; Fukunaga, H. Tunneling Effect in a Polymer/Carbon Nanotube Nanocomposite Strain Sensor. Acta. Mater. 2008, 56, 2929–2936. (18) Zhang, T.; Yi, Y. B. Monte Carlo Simulations of Effective Electrical Conductivity in Short-Fiber Composites. J. Appl. Phys. 2008, 103, 014910. (19) Lu, W. B.; Chou, T. W.; Thostenson, E. T. A Three-Dimensional Model of Electrical Percolation Thresholds in Carbon Nanotube-Based Composites. Appl. Phys. Lett. 2010, 96, 223106. (20) Seidel, G. D.; Lagoudas, D. C. A Micromechanics Model for the Electrical Conductivity of Nanotube-Polymer Nanocomposites. J. Compos. Mater. 2009, 43, 917-941. (21) Feng, C.; Jiang, L. Micromechanics Modeling of the Electrical Conductivity of Carbon Nanotube (CNT)–Polymer Nanocomposites. Compos. Part A Appl. Sci. Manuf. 2013, 47, 143-149.



Page 16 of 22

(22) Pal, G.; Kumar, S. Multiscale Modeling of Effective Electrical Conductivity of Short Carbon Fiber-Carbon Nanotube-Polymer Matrix Hybrid Composites. Mater. Design. 2016, 89, 129-136. (23) Takeda, T.; Shindo, Y.; Yu, K.; Narita, F. Modelling and Characterization of the Electrical Conductivity of Carbon Nanotube-Based Polymer Composites. Polymer 2011, 52, 3852-3856. (24) Simmons, J. G. Generalized Formula for the Electric Tunnel Effect Between Similar Electrodes Separated by a Thin Insulating Film. J. Appl. Phys. 1963, 34, 1793–1803. (25) Ahn, B. Y.; Duoss, E. B.; Motala, M. J.; Guo, X. Y.; Park, S. I.; Xiong, Y. J.; Yoon, J. S.; Nuzzo, R. G.; Rogers, J. A.; Lewis, J. A. Omnidirectional Printing of Flexible, Stretchable, and Spanning Silver Microelectrodes. Science 2009, 323, 1590-1593. (26) Hyun, D. C.; Park, M.; Park, C. J.; Kim, B.; Xia, Y.; Hur, J. H.; Kim, J. M.; Park, J. J.; Jeong, U. Ordered Zigzag Stripes of Polymer Gel/Metal Nanoparticle Composites for Highly Stretchable Conductive Electrodes. Adv. Mater. 2011, 23, 2946–2950. (27) Kim, Y.; Zhu, J.; Yeom, B.; Prima, M. D.; Su, X.; Kim, J. G.; Yoo, S. J.; Uher, C.; Kotov, N. A. Stretchable Nanoparticle Conductors with Self-Organized Conductive Pathways. Nature 2013, 500, 59-64. (28) Wang, G. T.; Wang, C. Y.; Zhang, F. L.; Yu, X. Z. Electrical Percolation of Nanoparticle-Polymer Composites. Comp. Mater. Sci. in press. (29) Torquato, S.; Haslach, H. Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Appl. Mech. Rev. 2002, 55, B62. (30) Mclachlan, D. S.; Blaszkiewicz, M.; Newnham, R. E. Electrical Resistivity of Composites. J. Am. Ceram. Soc. 2010, 73, 2187-2203. (31) Bruggeman, D. A. G. Dielectric Constant and Conductivity of Mixtures of Isotropic Materials. Ann. Phys. 1935, 24, 636-664. (32) Yan, K. Y.; Xue, Q. Z.; Zheng, Q. B.; Hao, L. Z. The Interface Effect of the Effective

Electrical

Conductivity

of

Carbon

Nanotechnology 2007, 18, 255705.


Nanotube

Composites.

Page 17 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60




Figures

Figure 1. Left, a nanocomposite model comprised of the polymer matrix filled with randomly distributed gold nanoparticles. Right, zoomed in model of an equivalent AuNP model with an effective tunneling layer of thickness tint, the yellow sphere represents the physical part of the AuNP while the outer grey layer represents the virtual tunneling interphase. A spherical coordinate system is used to describe the model system used in the context, the external electric field of strength E0 is applied along the z direction. Due to the axisymmetry of the AuNP system, φ is not shown here on the right panel.


Page 18 of 22

Page 19 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 2. Conductivity as a function of AuNP volume fraction for the AuNP-polymer composite for AuNP nanoparticles of three different radii: 5nm (a), 10nm (b) and 50nm(c). Red lines show the simulated conductivity-volume fraction relationship, blue dashed lines denote the relationship when tunneling effect is not taken into account, purple dashed lines are visual extension of tunneling curve before Vcr2 plotted for comparison. The two vertical black dashed lines in (a) mark the transition between different conduction stages. Other parameters used here are:

λ =1.0eV , σ np = 4.17 ×105 S / cm, σ m = 1.0 ×10−9 S / cm



Figure 3. Ratio of ∆Vcr/Vcr1 as a function of AuNP radius, inset shows the zoomed in region for AuNP radii from 102 nm to 103 nm.


Page 20 of 22

Page 21 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 4. (a) Envelope curve of conductance change trend with respect to AuNP volume fraction when considering varied tunneling thickness, dashed lines denote simulation results with constant tint, red line is obtained by fitting the envelope curve of the above-mentioned curves. (b) The same plot obtained for the polymer matrix with different potential barrier.



TOC Graphic


Page 22 of 22

Two Stage Electrical Percolation of Metal ... - ACS Publications

Recommend Documents