Influence of carbon black concentration on piezoresistivity for carbon-black-filled silicone rubber composite

CARBON

4 7 ( 2 0 0 9 ) 3 1 5 1 –3 1 5 7

available at www.sciencedirect.com

journal homepage: www.elsevier.com/locate/carbon

Influence of carbon black concentration on piezoresistivity for carbon-black-filled silicone rubber composite Wang Luheng a b

a,b,*

, Ding Tianhuai b, Wang Peng

b

College of Information Science and Engineering, Northeastern University, Shenyang 110004, PR China Department of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084, PR China

A R T I C L E I N F O

A B S T R A C T

Article history:

The piezoresistivity of carbon-black-filled silicone rubber composite with different carbon

Received 8 October 2008

black content are studied. The experimental results show that carbon black concentration

Accepted 24 June 2009

has great effects on the relation between uniaxial pressure and the electrical resistance of

Available online 30 June 2009

the composite. Based on the shell model and the theory of tunneling current, a mathematical model for piezoresistivity is developed. The experimental phenomena are explained by analyzing the changes in effective conductive paths composed of carbon black.  2009 Elsevier Ltd. All rights reserved.

1.

Introduction

Carbon black can be used as a conductive phase of conductorfilled polymer composites. The research on the electrical properties of the composites has attracted a lot of interest [1–28]. With an appropriate amount of carbon black, the composites possess flexibility and piezoresistivity. This kind of material can be used as the sensing element of a flexible pressure sensor [1–9]. The study on the relation between external pressure and the composite resistance is the key to fabricating this kind of sensor. Zhang et al. [29] and Yi [30] researched the piezoresistivity of conductor-filled polymer composites. They found that the composite resistance decreases with increasing uniaxial pressure. To explain this phenomenon, they used a concept of ‘‘conducting path’’ to describe the microstructure of the composites. Each conducting path is composed of conductive particles. They considered that the total resistance in conducting composites is decided by the resistance between adjacent conductive particles. Based on the theory on tunneling current, the total resistance of the composite can be calculated by:    L 8phs expðcsÞ; ð1Þ R¼ 2 2 N 3a ce

where L is number of particles forming a single conducting path, N is number of conducting path(s), h is Plank’s constant, s is thickness of insulating film, a2 is effective cross-sectional area, e is electron charge, and c is calculated by: 4p pffiffiffiffiffiffiffiffiffiffiffi 2mu; ð2Þ c¼ h where u is height of potential barrier between adjacent particles, m is electron mass. They considered that the thickness of insulating film (the gap between conductive particles) decreases if a stress is applied to the sample, which is caused by the compressibility difference between filler particle and matrix. The decrease of the inter-particle separations gives rise to the decrease of the composite resistance. Based on their analysis on the relation among uniaxial pressure, inter-particle separation, strain, compressive modulus of polymer matrix, particle diameter, and filler volume fraction, a mathematical model was built for the piezoresistivity as: RðrÞ  r ¼ 1 Rð0Þ M (  exp 

" # ) pffiffiffiffiffiffiffiffiffiffiffi 13 4p 2mu p r 1  D ; h 6/ M

ð3Þ

* Corresponding author: Address: College of Information Science and Engineering, Northeastern University, Shenyang 110004, PR China. Fax: +86 24 83687266. E-mail address: [email protected] (W. Luheng). 0008-6223/$ - see front matter  2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2009.06.050

3152

CARBON

4 7 ( 2 0 0 9 ) 3 1 5 1 –3 1 5 7

where r is applied stress, R(r) and R(0) are the electrical resistances of the composite under the pressure r and zero-pressure, respectively, D is filler particle diameter, / is filler volume fraction, M is compressive modulus of polymer matrix. This model can be used to explain the phenomenon that the piezoresistivity is monotonically decreasing. Maris Knite et al. [6] studied the relation between stress and electrical resistance of carbon-black-filled polyisoprene composite. Chen et al. [31] researched the piezoresistivity of conductive graphite nano-sheets filled silicone rubber matrix. They all found that the composite resistance increases with the increase of stress. To explain the phenomenon, they not only considered the changes in particle separations under applied stress, but also studied the destruction of conducting paths resulting from the deformation of matrix. Based on their analysis on the relation among the deformation of matrix, the particle separation, and the number of conducting path(s), Eq. (4) was used to fit the changes in the electrical resistance of the composite R. ln R ¼ ln R0 þ ln½1 þ ðDl=l0 Þ þ

K X ½Ai  ðDl=l0 Þi ;

ð4Þ

i¼1

where R0 is the initial resistance of the composite, Dl is the deformation of sample, l0 is the initial thickness of sample, K is an integral number, Ai(i = 1,2, . . . , K) is coefficient. This model can be used to explain the phenomenon that the resistance increases with increasing stress. The research results aforementioned presented two opposite behaviours: a monotonically decreasing piezoresistivity and a monotonically increasing piezoresistivity. The previous research indicated that this difference has relation with external pressure range, conductive filler type, structure, volume fraction and extent of swelling, etc. However, the existing models can not be used to explain these two opposite phenomena at the same time. In this paper, we research quantitatively the piezoresistivity of carbon-black-filled silicone rubber composite with different carbon black content. Our experimental results show that if the carbon black content is low, the piezoresistivity is monotonically decreasing. If the carbon black content is moderate, the electrical resistance of the composite decreases first and then increases. If the carbon black content is high, the pizoresistivity is monotonically increasing. These piezoresistive phenomena are more complicated than previous research results [6,29–31]. To explain the complicated piezoresistive phenomena aforementioned, three kinds of changes in the conductive network of the composite are researched in this paper. Firstly, based on the research achievements in the references [29,30], the change in the electrical resistance between adjacent car-

bon black particles is considered. Secondly, based on the research achievements in the references [6,31,32], the destruction of effective conductive paths is taken into account. Thirdly, the increase of the number of effective conductive path(s) caused by the recovery of conductive network is studied. Besides, the experimental data of piezoresistivity are fitted by analyzing the changes in the electrical resistance of a single effective conductive path and the number of effective conductive path(s).

2.

Experimental

2.1.

Materials

Carbon black powder (SL36, China Rubber Group Carbon Black Research and Design Institute (CCBI)) is used as a conductive phase. CCBI uses novel technology and special active agent to fabricate this kind of carbon black (SL36) to make it possess the following characteristics: large specific surface area, high structure, excellent dispersion, and good conductivity, etc. Room temperature vulcanized liquid silicone rubber (107, Beijing Chem. Plant, China) is used as an insulating matrix. The properties of carbon black and silicone rubber are presented in Table 1. Hexane is used as solvent to mix the fillers with the rubber. Mechanical stirring along with ultrasonic vibration is also used for better particle dispersion. After 5 h of vigorous mixing, the solvent is evaporated and the viscous mixture is molded into disks (4 mm · 6 mm · 0.08 mm) at 30 C for 40 h. Seventeen samples with different carbon black contents are prepared. The mass ratio of carbon black to silicone rubber (marked by F) ranges from 0.08:1 to 0.24:1. Fig. 1 is the SEM (scanning electron microscope) micrographs of the fractured surfaces for the samples, showing a good dispersion of carbon black particles in silicone rubber matrix.

2.2.

Measurements

The experimental set-up for the measurement of piezoresistivity is shown in Fig. 2. The sample with two electrodes is placed between the sensing element of digital force gauge and the elevator platform. The area of electrode plate is a little less than that of sample, so the transverse area keeps invariant during compression. The sample is compressed up to a certain pressure through the uniaxial movement of the elevator platform, and then the displacement is kept invariant. The values of the pressure and the electrical resistance at the moment immediately after compression are recorded by the digital force gauge and the digital multimeter (HP3458A), respectively.

Table 1 – Properties of carbon black and silicone rubber. Specific surface area Carbon black Silicone rubber

2

780 (m /g) Dielectric constant 3.0

pH 6.25 Hardness 35 (Shore)

a Condition: drying temperature: 105 C, drying time: 1 h.

Volatile contenta 3.0 (%) Tearing resistance 4 (kg/cm)

Light transmittance of toluene extract 100 (%) Dielectric strength 18 (kV/mm)

CARBON

4 7 ( 20 0 9 ) 3 1 5 1–31 5 7

3153

Fig. 3 – Piezoresistivities of the samples with different carbon black contents.

black content; If F P 0.14, the electrical resistance decreases with the increase of uniaxial pressure, as shown in Fig. 3 (3) and 3 (4).

4. Fig. 1 – SEM micrographs of the fractured surfaces for the composites with different carbon black contents.

3.

Results

The relations between the electrical resistances of the samples and uniaxial pressure are shown in Fig. 3. If the mass ratio of carbon black to silicone rubber F is less than 0.09, the electrical resistance increases with the increase of uniaxial pressure, as shown in Fig. 3 (1); If 0.10 6 F 6 0.13, the electrical resistance decreases first and then increases with the increase of uniaxial pressure, as shown in Fig. 3 (2). The transition point is defined as a critical pressure. As shown in Table 2, the critical pressure increases with the increase of carbon

Value of Pressure

Digital Force Gauge

Sensing Element of

Discussion

4.1. Changes in effective conductive path under uniaxial pressure Fig. 4 shows the shell structure model of carbon-black-filled silicone rubber composite [33]. ‘‘Phase A’’ has active microBrownian motion. The motion of ‘‘phase B’’ has been restricted. ‘‘Phase C’’ has low level of activity. Elastic ‘‘phase A’’ and ‘‘phase B’’ are bonded to ‘‘phase C’’ which acts as a framework, forming a three dimensional network. Carbon black, the resistivity of which is far less than that of silicone rubber, acts as a conductive phase. When the gap between carbon black particles is small enough, the tunneling effect occurs, leading to the formation of local conductive path. If local conductive path penetrates insulating matrix, an effective conductive path is formed, contributing to the conductivity of composite. Fig. 5 shows the schematic views of local conductive path and effective conductive path. Carbon black is incompressible compared with silicone rubber. Therefore, the compression can induce translation and rotation of carbon black, leading to the changes in effective conductive path as follows.

Digital Force Gauge Electrode

Digital

Sample

Multimeter (HP3458-A)

Vaule of Electrical Resistance

Electrode

Elevator Platform

Fig. 2 – Experimental set-up for measurement of piezoresistivity.

(1) Change in gap size in existing effective conductive path: The compression makes the gaps between two adjacent conductive particles smaller, leading to the decrease of the electrical resistance of existing effective conductive path. (2) Formation of new effective conductive paths: The compression makes the gaps between carbon black particles smaller, leading to the formation of new effective conductive paths. This effect contributes to the increase of the number of effective conductive path(s).

3154

CARBON

4 7 ( 2 0 0 9 ) 3 1 5 1 –3 1 5 7

Table 2 – Critical pressure ranges of the samples with different carbon black contents. Mass ratio of carbon black to silicone rubber (F) Critical pressure (Pc, MPa)

F = 0.10:1

F = 0.11:1

F = 0.12:1

F = 0.13:1

0.1 < Pc < 0.3

0.3 < Pc < 0.5

0.6 < Pc < 0.8

0.8 < Pc < 1.0

Phase A

1

1

1

1

2

2

2

2

Carbon Black

Carbon Black

Particle

Phase C j

Dij

Polymer Matrix

Phase B

j+1 Effective Conductive

Fig. 4 – Shell model for carbon-black-filled silicone rubber composite. (Phase A: rubber molecule chain which is not absorbed by carbon black; Phase B: crosslinked rubber molecule chain; Phase C: macro-rubber molecule chain which is absorbed to the surface of carbon black through physisorption and chemisorption).

Path

M1

M2

Mi

MN

1

2

i

N

Fig. 6 – Schematic of the model formulation for describing the microstructure of the composite.

Effective conductive path

Ri ¼

M i 1 X

Raði;jÞ þ

Fig. 5 – Local conductive path and effective conductive path.

Rcði;j1 Þ ;

ð5Þ

j1 ¼1

j¼1

Local conductive path

Mi X

where Mi(i = 1,2, . . . , N) is the number of conductive particles in the ith effective conductive path, Raði;jÞ is the electrical resistance between the jth and (j + 1)th conductive particle in the ith effective conductive path, Rcði;j1 Þ is the electrical resistance of the j1 th conductive particle in the ith effective conductive path. As the resistivity of silicone rubber is far more than that of carbon black ðRaði;jÞ >> Rcði;j1 Þ Þ, Ri can be given by: Ri ¼

M i 1 X

Raði;jÞ :

ð6Þ

j¼1

(3) Destruction of effective conductive paths: The transverse slippage of carbon black, caused by compression, leads to the destruction of effective conductive paths. This effect contributes to the decrease of the number of effective conductive path(s). The changes in effective conductive paths aforementioned concur during compression. The change (1) and (2) contribute to the decreasing tendency of the composite resistance. The change (3) contributes to the increasing tendency of the composite resistance.

4.2.

Mathematical model for the piezoresistivity

Fig. 6 shows the schematic model for the microstructure of carbon-black-filled silicone rubber composite. The total number of effective conductive path(s) is N. The total electrical resistance of the composite R is the parallel connection of the resistances of N effective conductive paths. The electrical resistance of the ith (i = 1,2, . . . , N) effective conductive path is marked by Ri. It can be calculated by:

According to the analysis [30,34–36], the tunnel current I(i,j) can be given by: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 3e2 Uði;jÞ 2muði;jÞ 4pDði;jÞ 2muði;jÞ ði;jÞ I ¼ ; ð7Þ  exp  2 h 2h Dði;jÞ where U(i,j),D(i,j) and u(i,j) are external voltage, the thickness of insulating film and the height of potential barrier, respectively. S(i,j) is the effective cross-sectional area of insulating film where tunneling effect occurs between the jth and (j + 1)th conductive particle in the ith effective conductive path. e is electron charge. m is electron mass. h is Plank’s constant. The electrical resistance of the ith effective conductive path Ri can be given by: Ri ¼

M 1 1 X j¼1

Raði;jÞ ¼

M 1 1 X j¼1

Uði;jÞ I

ði;jÞ

¼

M i 1 X j¼1

2

pffiffiffiffiffiffi

4pDði;jÞ 2h Dði;jÞ h pffiffiffiffiffiffiffiffiffiffiffi  e 3e2 Sði;jÞ 2mu

2mu

:

ð8Þ

To simplify the model, we make the assumptions as follows: Dði;j1 Þ ¼ Dði;j2 Þ ¼ Di ;

Sði;j1 Þ ¼ Sði;j2 Þ ¼ Si ;

j1 ; j2 ¼ 1; 2; . . . ; Mi ;

ð9Þ

CARBON

4 7 ( 20 0 9 ) 3 1 5 1–31 5 7

where Di is the average thickness of the insulating film between the adjacent conductive particles in the ith effective conductive path, Si is the average effective cross-sectional area of the insulating film between the adjacent conductive particles in the ith effective conductive path. Therefore, the electrical resistance of the ith effective conductive path Ri can be given by: pffiffiffiffiffiffi 2 4pDi 2mu 2h Di ð10Þ Ri ¼ ðMi  1Þ  2 pffiffiffiffiffiffiffiffiffiffiffi  e h : 3e Si 2mu To further simplify the model, we make the assumptions as follows: Mi1 ¼ Mi2 ¼ M;

Di1 ¼ Di2 ¼ D;

Ri1 ¼ Ri2 ¼ Rs ;

i1 ; i2 ¼ 1; 2; . . . N

Si1 ¼ Si2 ¼ S;

ð11Þ

where M is the average number of the conductive particles in one effective conductive path in the composite, D is the average thickness of the insulating film between the adjacent conductive particles in the composite, S is the average effective cross-sectional area of the insulating film between the adjacent conductive particles in the composite. The substitution of the Eq. (11) into the Eq. (10) yields: pffiffiffiffiffiffi 2 4pD 2mu 2h D ð12Þ Rs ¼ ðM  1Þ  2 pffiffiffiffiffiffiffiffiffiffiffi  e h ; 3e S 2mu where Rs is the average resistance of one effective conductive path in the composite. The total resistance of the composite R can be calculated by: " # pffiffiffiffiffiffi 2 4pD 2mu 1 2h D : ð13Þ R ¼  ðM  1Þ  2 pffiffiffiffiffiffiffiffiffiffiffi  e h N 3e S 2mu N and D change with uniaxial pressure, inducing that R changes with uniaxial pressure. That is to say, N, D and R are the functions of the uniaxial pressure r. Therefore, the composite resistance can also be described by: " # pffiffiffiffiffiffi 2 4pDðrÞ 2mu 1 2h DðrÞ ; ð14Þ RðrÞ ¼  ðM  1Þ  2 pffiffiffiffiffiffiffiffiffiffiffi  e h NðrÞ 3e S 2mu where R(r), N(r) and D(r) represent the composite resistance, the number of effective conductive paths and the thickness of the insulating film under the pressure r, respectively. To analyze the piezoresistivity of the composite with different conductive phase content more clearly, we define the relative resistance Rr(r) as: Rr ðrÞ ¼

RðrÞ Rs ðrÞ=NðrÞ ; ¼ Rð0Þ Rs ð0Þ=Nð0Þ

ð15Þ

where Rs(r) is the electrical resistance of one single effective conductive path under external pressure r. R(0),Rs(0) and N(0) are the composite resistance, the resistance of one effective conductive path and the number of effective conductive paths under zero-pressure, respectively. The relative resistance of one single effective conductive path Rsr (r) can be given by: ffi 4ppffiffiffiffiffi  2mu DðrÞ ð16Þ  e h ½DðrÞDð0Þ ; Rsr ðrÞ ¼ Rs ðrÞ=Rs ð0Þ ¼ Dð0Þ where D(r) and D(0) are the thickness of the insulating film between the adjacent conductive particles under the pressure r and zero-pressure, respectively.

3155

As discussed aforementioned, carbon black is incompressible compared with silicone rubber matrix. Therefore, the change in the thickness of insulating film between adjacent conductive particles is only caused by the deformation of the polymer matrix. So the gap between the adjacent conductive particles D(r) can be calculated as:  r ; ð17Þ DðrÞ ¼ Dð0Þ  ð1  eÞ ¼ Dð0Þ  1  G where e is the strain, G is the compressive modulus of the polymer matrix. To simplify the arithmetic for the estimating of D(0), we made the assumptions as follows. The carbon black particle is assumed to be spherical. The sizes of the particles are the same. The carbon black network is a cubic lattice structure. Therefore, the thickness of the insulating film between the adjacent conductive particles D(0) can be given by [37]: " # 1=3 p 1 ; ð18Þ Dð0Þ ¼ d  6/ where d is the particle diameter, / is the filler volume fraction. The relation between the filler volume fraction / and F is given by: /¼F

qs ; qc

ð19Þ

where qc is the density of carbon black, qs is the density of silicone rubber. Substitute the Eqs. (17)–(19) into the Eq. (16) to yield: n4pr 1 pqc pffiffiffiffiffi ffio ½ 6Fqs  2mu  r Gh : ð20Þ Rsr ðrÞ ¼ 1  e G The relative number of effective conductive path(s) Nr(r) can be described as: 1 W  eXr þ Y  eZr W þ Y ¼ 1; W; X; Y; Z > 0: Nr ðrÞ ¼ NðrÞ=Nð0Þ ¼

ð21Þ

Substitute the Eqs. (20) and (21) into the Eq. (15) to yield: ffi pq  4prpffiffiffiffiffi  2mu c r Rr ðrÞ ¼ ½W  eXr þ Y  eZr   1   e Gh 16Fqs : G ð22Þ The Eq. (22) is used to fit the normalized experimental data. Fig. 7 shows the fitted curves. Fig. 8 shows the relation between carbon black content and the parameters (W, Y, X, and Z). If F 6 0.09, the piezoresistivity is monotonically increasing; If 0.10 6 F 6 0.13, the piezoresistivity is first monotonically decreasing, and then monotonically increasing; If 0.14 6 F 6 0.16, the piezoresistivity is monotonically decreasing. The attenuation degree of the piezoresistivity increases with the increase of carbon black content; If 0.16 6 F 6 0.24, the piezoresistivity is also monotonically decreasing. The attenuation degree of the piezoresistivity decreases with the increase of carbon black content.

4.3. Effects of carbon black content on the changes in effective conductive paths Carbon black content has great influence on the relation between uniaxial pressure and the changes in effective conductive path.

3156

CARBON

4 7 ( 2 0 0 9 ) 3 1 5 1 –3 1 5 7

Fig. 7 – Relations between the pressure and relative resistances of composites with different carbon black content.

Fig. 9 – Relation between uniaxial pressure and the relative resistance of a single effective conductive path.

Fig. 10 – Relation between stress and relative number of effective conductive path(s). Fig. 8 – Relation between carbon black content and the parameters of mathematical model of piezoresistivity.

Fig. 9 shows the relation between uniaxial pressure and the relative resistance of a single effective conductive path Rsr(r). We can see that Rsr(r) decreases with the increase of r. The attenuation degree of Rsr(r)decreases with the increase of carbon black content. Fig. 10 shows the relation between uniaxial pressure r and the relative number of effective conductive path(s) Nr(r). We can see that there are three kinds of relations. If F 6 0.12, Nr (r) decreases with the increase of pressure. The attenuation degree of the relative number decreases with the increase of carbon black content, as shown in Fig. 10 (1); If 0.13 6 F 6 0.16, there exists a transition pressure, as shown in Fig. 10 (2). When the pressure is less than the transition pressure, Nr(r) increases with the increase of pressure. When the pressure is more than the transition pressure, Nr(r) decreases with the increase of uniaxial pressure. The transition pressure in-

creases with the increase of carbon black content; If F P 0.17, there also exists a transition pressure. The transition pressure decreases with the increase of carbon black content, as shown in Fig. 10 (3).

5.

Conclusions

Carbon black concentration has great influence on the piezoresistivity of carbon-black-filled silicone rubber composite. The electrical resistance of a single effective conductive path and the number of effective conductive path(s) are two key factors of the piezoresistivity. The electrical resistance of a single effective conductive path decreases with the increase of uniaxial pressure. The degree of attenuation decreases with the increase of carbon black content. The relation between the number of effective conductive path(s) and uniaxial pressure presents monotonically increasing, nonmonotonical, and monotonically decreasing depending on different carbon black content.

CARBON

4 7 ( 20 0 9 ) 3 1 5 1–31 5 7

R E F E R E N C E S

[19] [1] Mahmoud WE, El-Lawindy AMY, El Eraki MH, Hassan HH. Butadiene acrylonitrile rubber loaded fast extrusion furnace black as a compressive strain and pressure sensors. Sens Actuators A: Phys 2007;136(1):229–33. [2] Ding TH, Wang LH, Wang P. Changes in electrical resistance of carbon black filled silicone rubber composite during compression. J Polym Sci Part B: Polym Phys 2007;45:2700–6. [3] Job AE, Oliveira FA, Alves N, Giacometti JA, Mattoso LHC. Conductive composites of natural rubber and carbon black for pressure sensors. Synthetic Met 2003;135:99–100. [4] Wang LH, Ding TH, Wang P. Effects of conductive phase content on critical pressure of carbon black filled silicone rubber composite. Sens Actuators A: Phys 2007;135(2):587–92. [5] Clark AC, Ho SP, LaBerge M. Conductive composite of UHMWPE and CB as a dynamic contact analysis sensor. Tribology 2006;39(11):1327–35. [6] Knite M, Teteris V, Kiploka A, Kaupuzs J. Polyisoprene–carbon black nanocomposites as tensile strain and pressure sensor materials. Sens Actuators A: Phys 2004;110(1–3):142–9. [7] El Eraki MH, EL Lawindy AMY, Hassan HH, Mahmoud WE. The physical properties of pressure sensitive rubber composites. Polym Degrad Stab 2006;91(7):1417–23. [8] Wang XJ, Chung DDL. Short carbon fiber reinforced epoxy coating as a piezoresistive strain sensor for cement mortar. Sens Actuators A: Phys 1998;71(3):208–12. [9] Wang LH, Ding TH, Wang P. Effects of compression cycles and precompression pressure on the repeatability of piezoresistivity for carbon black-filled silicone rubber composite. J Polym Sci Part B: Polym Phys 2008;46(11):1050–61. [10] Souza FG, Michel RC, Soares BG. A methodology for studying the dependence of electrical resistivity with pressure in conducting composites. Polym Testing 2005;24(8):998–1004. [11] Busfield JJC, Thomas AG, Yamaguchi K. Electrical and mechanical behavior of filled rubber. III. Dynamic loading and the rate of recovery. J Polym Sci Part B: Polym Phys 2005;43(13):1649–61. [12] Busfield JJC, Thomas AG, Yamaguchi K. Electrical and mechanical behavior of filled elastomers 2: The effect of swelling and temperature. J Polym Sci Part B: Polym Phys 2004;42(11):2161–7. [13] Yamaguchi K, Busfield JJC, Thomas AG. Electrical and mechanical behavior of filled elastomers. I. The effect of strain. J Polym Sci Part B: Polym Phys 2003;41(17):2079–89. [14] Sau KP, Khastgir D, Chaki TK. Electrical conductivity of carbon black and carbon fibre filled silicone rubber composites. Die Angew Makromol Chem 1998;258:11–7. [15] Hu N, Karube Y, Yan C, Masuda Z, Fukunaga H. Tunneling effect in a polymer/carbon nanotube nanocomposite. Acta Mater 2008;56(13):2929–36. [16] Lu JR, Chen XF, Lu W, Chen GH. The piezoresistive behaviors of polyethylene/foliated graphite nanocomposites. Eur Polym J 2006;42(5):1021–51. [17] Arshak K, Morris D, Arshak A, Korostynska O. Sensitivity of polyvinyl butyral/carbon-black sensors to pressure. Thin Solid Films 2008;516(10):3298–304. [18] Wang LH, Ding TH, Wang P. Effects of instantaneous compression pressure on electrical resistance of carbon black

[20]

[21]

[22]

[23] [24]

[25]

[26]

[27]

[28]

[29]

[30] [31]

[32]

[33]

[34]

[35] [36]

[37]

3157

filled silicone rubber composite during compressive stress relaxation. Compos Sci Technol 2008;68:3448–50. Qu SY, Wong SC. Piezoresistive behavior of polymer reinforced by expanded graphite. Compos Sci Technol 2007;67(2):231–7. Flandin L, Bre´chet Y, Cavaille´ JY. Electrically conductive polymer nanocomposites as deformation sensors. Compos Sci Technol 2001;61(6):895–901. Zhou JF, Song YH, Zheng Q, Wu Q, Zhang MQ. Percolation transition and hydrostatic piezoresistance for carbon black filled poly(methylvinylsilioxane) vulcanizates. Carbon 2008;46(4):679–91. Zhu SR, Chung DDL. Analytical model of piezoresistivity for strain sensing in carbon fiber polymer–matrix structural composite under flexure. Carbon 2007;45(8):1606–13. Carmona F, Ravier J. Electrical properties and mesostructure of carbon black-filled polymers. Carbon 2002;40(2):151–6. Wen SH, Chung DDL. Partial replacement of carbon fiber by carbon black in multifunctional cement–matrix composites. Carbon 2007;45(3):505–13. Yoshimura K, Nakano K, Miyake T, Hishikawa Y, Kuzuya C, Katsuno T, et al. Effect of compressive and tensile strains on the electrical resistivity of carbon microcoil/silicone–rubber composites. Carbon 2007;45:1997–2003. Novak I, Krupa I, Janigova I. Hybrid electro-conductive composites with improved toughness filled by carbon black. Carbon 2005;43(4):841–8. Das NC, Chaki TK, Khastgir D. Effect of processing parameters, applied pressure and temperature on the electrical resistivity of rubber-based conductive composites. Carbon 2002;40(6):807–16. Wang LH, Ding TH, Wang P. Research on stress and electrical resistance of skin-sensing silicone rubber/carbon black nanocomposite during decompressive stress relaxation. Smart Mater Struct 2009;18:065002. Zhang XW, Pan Y, Zheng Q, Yi XS. Time dependence of piezoresistance for the conductor-filled polymer composites. J Polym Sci Part B: Polym Phys 2000;38(21):2739–49. Yi XS. Function principle of filled conductive polymer composites. Beijing: National Defence Industry Press; 2004. Chen L, Chen G, Lu L. Piezoresistive behavior study on fingersensing silicone rubber/graphite nanosheet nanocomposites. Adv Funct Mater 2007;17(6):898–904. Song YH, Zheng Q, Pan Y, Yi XS. Effect of uniaxial pressures on the resistance of high density polyethylene/carbon black composites. Chem J Chin Univ 2000;21(3):475–9. Zhu YJ. Mechanical modification of elastomers—filler reinforcement and blending. Beijing: Science and Technology Press; 1992. Ohe K, Naito Y. New resistor having an anomalously large positive temperature coefficient. Jpn J Appl Phys 1971;10(1):99–108. Simmons JG. Low-voltage current–voltage relationship of tunnel junctions. J Appl Phys 1963;34(1):238–9. Simmons JG. Generalized formula for electric tunnel effect between similar electrodes separated by a thin insulating film. J Appl Phys 1963;34(6):1793–803. Wu SH. Polymer. Phase-structure and adhesion in polymer blends: a criterion for rubber toughening. Polymer 1985;26(12):1855–63.