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Nonlinear pressure-dependent conductivity of magnetorheological elastomers

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2010 Smart Mater. Struct. 19 117001

(http://iopscience.iop.org/0964-1726/19/11/117001)

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Smart Mater. Struct. 19 (2010) 117001 (5pp) doi:10.1088/0964-1726/19/11/117001

TECHNICAL NOTE

Nonlinear pressure-dependentconductivity of magnetorheologicalelastomersXuli Zhu1,2, Yonggang Meng1,3 and Yu Tian1

1 State Key Laboratory of Tribology, Tsinghua University, Beijing 100084,People’s Republic of China2 College of Mechanical and Electronic Engineering, Shandong University of Science andTechnology, Qingdao 266510, People’s Republic of China

E-mail: mengyg@tsinghua.edu.cn

Received 26 March 2010, in final form 29 July 2010Published 23 September 2010Online at stacks.iop.org/SMS/19/117001

AbstractThe nonlinear pressure-dependent conductivity of magnetorheological elastomers (MREs) ispresented in this note. The effective cross-sectional area of current flowing through twoadjacent particles is taken as a key parameter to analyze the electric field assisted tunnel currentand the conduction current. The theory predicts the nonlinear dependence of conductivity ofMREs on electric field intensity and compressive stress. The MRE samples were prepared atlow particle fractions and low magnetic fields. The conductances of the MRE samples underdifferent electric fields and compressions have been measured, and the experimental resultshave shown a good agreement with the theoretical predictions.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Conductive polymer composites based on conductive filler inan insulating polymer matrix have attracted a great deal ofscientific and industrial interest for several decades [1–3]. Thepercolation and effective medium theories have been developedto understand the mechanism of electric conductivity ofconductive polymer composites [4, 5]. In the percolationregime the conductive particles aggregate to produce particlechains to form conductive paths inside the matrix. Electronscan tunnel through gaps between particles under a high electricfield. The external stimuli, e.g., stress and temperature,can affect the particle separation, and consequently giverise to great changes in conductance. The correspondingpiezoresistive effect and positive temperature coefficient effecthave been utilized for sensors and smart devices [6, 7].

Metal powders have been widely used as fillers ofconductive polymer composites. Some of them are

3 Author to whom any correspondence should be addressed.

ferromagnetic, e.g., Ni, Fe. If ferromagnetic particles aremixed with a polymer solution and then exposed to a magneticfield during the vulcanization of the polymer, the particlesalign themselves along the direction of the field and arelocated at fixed positions when the vulcanization is finished.Structures such as chains, networks and columns can be formedalong the direction of the magnetic field. These compositesare called magnetorheological elastomers (MREs) or field-structured composites [8, 9]. There are seldom isolatedparticles in the matrix because of the aggregation induced bythe magnetic field. The local particle volume fraction aroundthe chain is very high, therefore the percolation thresholdis very low. In this way, it is easy to produce transparentconductive polymer composites [10]. Most of the studies onMREs focus on their mechanical properties, and only a littleattention has been paid to the electrical properties.

In MREs the conductive path is usually along thechain structure and results in anisotropic conductivity. Themaximum conductance direction is along the direction of thecuring field. The conductivity in this direction is regarded

0964-1726/10/117001+05$30.00 © 2010 IOP Publishing Ltd Printed in the UK & the USA1

Smart Mater. Struct. 19 (2010) 117001 Technical Note

E

2ri

rp

h

h r

Figure 1. The longitudinal section of the RVU.

as the conductivity of the MREs generally. Leng et alfound that the electrical resistance of a polyurethane shape-memory polymer/carbon-black composite drops significantlyby forming Ni particle chains in it [11]. Kchit et al studied thepiezoresistivity [12], thermoresistance and magnetoresistanceof MREs [13]. In this study, we investigated the dependenceof the conductivity of MREs on the pressure and voltagetheoretically and experimentally. A new model for low particlevolume fraction MREs with a high conductivity has beenestablished and verified.

2. Theory

The typical particle structure of an MRE is particle chainsaligned along the magnetic field when the particle volume islow [14]. A representative volume unit (RVU) consists of twoneighboring hemispheres and the surrounding polymer matrix,which can be regarded as the minimum element of an MRE. Alongitudinal section of the unit is shown in figure 1, in which rp

is the average radius of the particles and h is the gap betweenthe neighboring particles. The cross-section of the RVU (notshown in figure 1) can be any polygon to form a plane. Theheight of the RVU hr = 2rp + h ≈ 2rp because rp � h. Giventhe particle volume fraction φ, the area of the cross-section ofthe RVU is

sr = 4πr 3p

3φhr≈ 2πr 2

p

3φ. (1)

The current flowing through the polymer inside the RVUmainly concentrates on the small area close to the tips of thehemispheres. The thickness of the sandwiched polymer filmis almost invariant at this position because the radius of theparticles is much bigger than the radius of the current cross-section. The conductivity of the particles is much higher thanthat of the polymer, so the electric potential drops on theparticles are negligible. The current flows through a polymerfilm with a thickness of h and a radius of the cross-sectionalarea of ri . When an electric field E is applied on an RVU, thelocal electric field intensity at the central area is approximatelygiven by,

Eloc = hr

hE ≈ 2rp

hE . (2)

The insulating polymer film between the two neighboringiron particles is very thin because of the magnetic attractionduring preparation, across which the electrical field assistedtunnel current can occur, which can be expressed by theFowler–Nordheim formula [15]. On the other hand, iron ionsmay be dispersed in the polymer matrix during the preparation

of the MREs, which contributes to the conductivity of thepolymer film through conduction. Then the total currentdensity is the sum of the tunnel density jt and the conductiondensity jc,

j = jt + jc = AE2loc exp

(− B

Eloc

)+ σf Eloc, (3)

in which A and B are constants determining the tunnel current,and σf is the conductivity of the polymer film.

The current flows through a small area around the tipsof particles, but the nominal current density of the RVU iscalculated by its own cross-section, jr = (πr 2

i /sr) j . Theapparent conductivity of the RVU is σr = jr/E . Since theRVU is the base element making up the MRE, the electric fieldintensity, the current density and the conductivity of the MRESare all equal to the counterpart parameters of the RVU. Thenthe conductivity of the MRE is

σm = 3φr 2i

[2A

h2E exp

(− h B

2rp E

)+ σf

rph

]. (4)

An important feature of conductive polymer compositesis piezoresistivity, also known as pressure-dependent con-ductivity. When an MRE is compressed, its conductivityincreases. Two factors affecting the conductivity of an MREare considered in this study when it is compressed. One is theincrement of the conductive area induced by the deformationof particles. The other is the decrement of the thickness of thepolymer film sandwiched between two neighboring particlesunder compression. The latter factor is less significant becauseit is difficult to compress the film further for the high ratio ofri/h, and the high elastic modulus of the very thin polymerfilm. So we consider that the increment of the conductive areais the major reason for the increment of conductivity undercompression. When an MRE is compressed by a constantpressure σ , the compressive force between the particles isFp = πr 2

p σ . The radius a of the contact spot can be calculatedwith the Hertz theory as:

a = 3

√3πσ(1 − ν2)

2Eprp, (5)

where Ep is the Young’s modulus and ν is the Poisson’s ratioof particles.

The influence of magnetic force on the deformation ofparticles during the preparation of MREs should also be takeninto account. The magnetic force can produce considerablestress on the contact spot because the polymer is still in itsfluidic state when the magnetic field is applied. Furthermore,the time period in which the particles undergo the magneticforce is several hours or longer during preparation. Aftercuring, a residual compressive stress σ0 would act betweenthe particles, resulting in an initial contact radius of a0,which satisfies equation (5). If the radius of the effectivecross-section for the current to flow through is ri0 whenthe compressive stress is zero, it increases along with theincrement of compressive stress,

ri = ri0 + ( 3√

σ0 + σ − 3√

σ0)3

√3π(1 − ν2)

2Eprp. (6)

2

Smart Mater. Struct. 19 (2010) 117001 Technical Note

Figure 2. Photo of a sample.

So the dependence of the conductivity of MREs on theelectric field intensity and compressive stress is

σm = 3φ

[k1 E exp

(−k2

E

)+ k3

](k4 + k5

3√

σ0 + σ)2, (7)

which is derived by defining k1 = 2Ah2 , k2 = hB

2rp, k3 = σf

rph ,

k4 = ri0 − 3

√3πσ0(1−ν2)

2E rp, and k5 = 3

√3π(1−ν2)

2E rp.

3. Experiment

Equation (7) shows that the conductivity of MREs dependsnonlinearly on the electric field intensity and compressivestress. It increases along with the electric field intensity andcompressive stress. MRE samples were prepared and tested toverify the relationship described by equation (7). In order toobtain an appropriate resistance, a low particle volume fractionof 1% and low magnetic induction of 15 mT were used toprepare the MRE samples. Silicon rubber (Sylgard® 184,Dow Corning) was used as the polymer matrix. Carbonyliron powder with a diameter of 0.5–6 μm (MPS-MRF-35,Jiangsu Tianyi Super Fine Metal Powder Co. Ltd) was usedas the ferromagnetic particles, and the average diameter was3.14 μm.

The powder and the silicon rubber were mixed thoroughlyand degassed in a vacuum box for 30 min. Then the mixturewas poured into a cylindrical mold with two plates made ofsteel on each side. The mold was exposed to a magnetic fieldand was heated at 150 ◦C for 90 min to solidify the polymermatrix. The permeability of the steel plates is very high andtheir magnetic reluctance can be omitted. The thickness ofthe plates was 3 mm and the gap between them was 1 mm,which was also the thickness of the samples. The magneticfield between the plates was enhanced to about 105 mT whenthe external magnetic field was 15 mT. After the solidificationprocess, the mold was removed but the steel plates wereretained and used as electrodes in the later tests, as shown infigure 2. The MRE samples were cylinders with a radius of10 mm.

The samples were compressed using the compressivetester fabricated by us, as shown in figure 3. An MRE samplewas placed between a force sensor and the lifting platform ofthe tester. The sample was insulated from the sensor and theplatform by two dielectric plates. The lifting platform was

Figure 3. Photo of the experimental setup.

Figure 4. The voltage dependence of the current of an MRE.

driven up by a step-motor to compress the sample at a constantspeed. The electrodes were connected to an adjustable DCpower supply and an ammeter was connected to the circuitin series. The compressive force during compression wasmeasured by the force sensor and the current was measuredby the ammeter. The DC power supply was in the range of0–32 V, and the minimum resolution of the ammeter used was0.1 nA. The current flowing through the sample was measuredunder different compressive forces and voltages.

4. Results and discussion

The experimental results show that the current–voltagedependence is significantly nonlinear, as shown in figure 4.The results are analyzed according to the above theoreticalmodel. The constants of equation (7) are fitted and listed intable 1. Figure 4 shows that the theoretical analysis can predictthe current dependence on the electric field and compressionof the MRE very well. The current increases with incrementsof voltage when the pressure is constant. A higher current canbe obtained when an MRE is compressed by a higher pressure.The tunnel current equals the conduction current at 7.75 V, and

3

Smart Mater. Struct. 19 (2010) 117001 Technical Note

Table 1. Constants for the calculation of the current.

Constants Values Units

k1 8.61 × 105 S V−1 m−2

k2 5.39 × 103 V m−1

k3 3.33 × 109 S m−3

k4 1.37 × 10−8 mk5 4.52 × 10−10 Pa−1/3 mσ0 970 Pa

then exceeds the conduction current when the voltage exceeds7.75 V. Therefore, the main component of the total current isthe conduction current when the voltage is lower than 7.75 V,while the tunnel current is the main contribution when thevoltage is higher than 7.75 V. The calculated ri0 is 18.2 nm,a rather big radius inducing considerable initial conductanceof the MRE.

The conductivity increases monotonically with the electricfield intensity and the compressive stress. However the effectsof the electric field and compressive stress are non-coupledbecause the thickness of the polymer film is thought to bea constant in this study. So the conductivity of the MREscan be studied with the electric field or compressive stressrespectively. σre is defined as the relative conductivity, whichdepends on the field intensity only,

σre = 1

σm(E0)

[k1 E exp

(−k2

E

)+ k3

], (8)

in which E0 is the reference electric field intensity. Alsoσrs is defined as the relative conductivity depending on thecompressive stress only,

σrs = 1

σm(0)(k4 + k5

3√

σ0 + σ)2. (9)

The relative conductivities σre and σrs are shown infigures 5(a) and (b) respectively. The values in figure 5(a)are obtained by setting E0 = 10 kV m−1. All results agreewith the predicted curve well. It can be seen in figure 5(a)that the dependence of σre on E is linear when E is inthe higher region. The reason is that the exponential termexp(−k2/E) approaches the unity with a high E , resulting ina proportional relationship between σre and E according toequation (8). Figure 5(b) shows that the dependence of σrs

on the compressive stress accords with the theoretical analysisbased on the Hertz theory, and the hypothesis of the invariantthickness of the polymer film is verified.

5. Conclusions

The current of MREs consisting of the electric field assistedtunnel component and the conduction component is analyzedbased on a representative volume unit. The current increaseswith the compressive stress by increasing the effective cross-sectional area. The thickness of the polymer film throughwhich the current flows is thought to be invariant, so theeffects of the electric field and compressive stress on theconductivity are non-coupled. A model for the dependence

Figure 5. The relative conductivity of an MRE. (a) Under differentelectric fields; (b) under different compressive stresses.

of the conductivity on the electric field and compressive stressis presented. Experimental results agree with the theoreticalpredictions well. This model is helpful to design MREs withpreferable piezoresistivity.

Acknowledgments

This work is sponsored by the National Natural ScienceFoundation of China with grant nos 50525515, 50875152 and50721004.

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