Int. J. Appl. Ceram. Technol., 8 [4] 793–799 (2011)
DOI:10.1111/j.1744-7402.2010.02506.x

Ceramic Product Development and Commercialization

Percolation-Type Chemical Sensor: Electrical- andHumidity-Sensing Properties of Carbon–Montmorillonite
Composites
Isao Tsuyumoto* and Yuuki Iida
Department of Applied Chemistry, College of Bioscience and Chemistry, Kanazawa Institute ofTechnology, 7-1 Ohgigaoka, Nonoichi, Ishikawa 921-8501, JapanA chemical sensor based on the percolation principle is newly developed by fabricating a composite comprising of aconducting phase and an insulating phase whose volume increases by the incorporation of a desired substance. We find acomposite of montmorillonite with carbon shows a remarkable resistivity increase in a humid atmosphere, because the volumeof the montmorillonite phase increases by the incorporation of water. The resistivity increases in response to stepwise water
vapor increases like a humidity sensor. The response time is substantially improved by fabricating a composite thick film usingclassified fine montmorillonite particles.

Introduction
The electrical resistivity of a composite prepared byrandomly mixing conducting and insulating particles isclosely dependent on the volume fraction of the conducting phase. When the volume fraction of the conducting phase is relatively small, the resistivity of thecomposite shows a gradual decrease with the increasing
fraction. However, the resistivity decreases drastically by*tsuyu@neptune.kanazawa-it.ac.jp
r 2010 The American Ceramic Society

several orders of magnitude around a certain volume
fraction of the conducting phase. This drastic resistivity
decline is due to the percolation of the conducting
phase, that is, the formation of the conducting paths.
This volume fraction is termed a critical threshold (pc)and experimentally estimated from the inflection pointof the volume fraction dependence of the resistivity. The
percolation properties of composite materials have been
investigated extensively by many researchers from boththeoretical and practical aspects.1–6
The volume fraction dependence of the resistivityof cermets has been quantitatively analyzed using a

794International Journal of Applied Ceramic Technology—Tsuyumoto and Iida

general effective media equation, which combines percolation and effective medium theories.7 Influence ofparticle size ratio on the electrical resistivity of the compacted mixtures has been investigated, and it has beentheoretically and experimentally confirmed that pc is reduced when the particle size of the conducting phase ismuch smaller than that of the insulating phase.8–10
Composite conducting polymers have been developedby loading an insulating polymer matrix with a conductive filler based on the percolation principle.11 Suchcomposite materials have potential applications to sensors or transducers, because an extrinsic change that
affects the volume fraction of the conducting phase gives
rise to the resistivity change of the composite. Carbonloaded polymers have been found to show resistivity
increase when exposed to solvents and hydrocarbons
due to the volume change of the polymer matrix.12–15
Carbon-dispersed rubber has been put to practical use aspressure sensors, because the stress to the rubber brings
about the decrease in the volume fraction of the rubber
matrix, and consequently the decrease in the resistivity
due to the percolation. Furthermore, one of the authors
has first observed an anomalous resistivity increase ofhexagonal tungsten bronzes in a humid atmosphere atroom temperature.16–18 The resistivity increase was percolation-like behavior, because the conduction pathalong the c-axis was separated by water incorporation
into oxygen vacancies existing every six layers perpendicular to the c-axis.
Chemical sensor materials have been widely investigated and a number of operating principles have been
proposed. With regard to the humidity sensor, many
researchers focused on the resistivity decrease of materials including clay minerals due to water adsorption or
hydration.19–21 The principle of the resistivity decrease
due to the dissociation of H2O molecule into H1 andOH at the interfaces between electron donor and acceptor in composite materials was also proposed.22 The
composite of TiO2 and MgCr2O4 has been put to practical use in the humidity sensor of microwave ovens.23
The permittivity change of polymers due to water
adsorption has also been applied to electric hygrometers.24 Although the conventional humidity sensors arewidely used to measure relative humidity (RH), theycannot detect a slight amount of humidity in a dry atmosphere and are not sensitive to humidity change.
This is because the conductivity or permittivity of
the conventional sensor materials changes linearly inresponse to the adsorbed amount of water molecules.

Vol. 8, No. 4, 2011

The conductivity increases due to an increase of ionic
conduction derived from water molecules, and the permittivity increases due to an increase of orientation polarization of water molecules. Enhancement mechanismof slight change is not incorporated in the conventional
humidity sensor materials. It is essential to detect andcontrol the slight amount of humidity in materials processing plants such as battery factories, semiconductor
plants, and thus the inexpensive humidity sensors sensitive to the slight amount of humidity and to the smallchange of RH are highly desirable. We considered thepercolation-type chemical sensor to be appropriate forthis purpose.
In this paper, we design and present a percolationtype chemical sensor. A desired substance can be detected by monitoring the resistivity of a composite comprising of a conducting phase and an insulating phase
whose volume increases by the incorporation of thesubstance, according to the percolation principle. In order to verify the applicability of this principle, we fabricate a composite using a montmorillonite as an
insulating phase and carbon as a conducting phase.
Here, we exploit the well-known property of montmorillonite that it swells by water incorporation. Contrary
to the conventional humidity sensors, the resistivity of
the composite increases in a humid atmosphere, because
the volume of the insulating phase increases by water
incorporation.

Experimental Procedure
The sample composites were prepared by mixing
montmorillonite particles [particle size: 5–120 mm, average: 40 mm, typical chemical formula: Na2/3Si8(Al10/3
Mg2/3)O20(OH)4, Kunipia-F, Kunimine Industries,
Tokyo, Japan] and carbon powders [average particle
size: 5 mm, Kojundo Chemical Laboratory, Saitama,
Japan] in an agate mortar. The mixtures were molded
into pellets by monoaxial pressing at 450 MPa. The
pellet was 13 mm in diameter and 0.40–0.68 mm inthickness. After gold electrodes were coated on bothsides of the pellets using gold paste [No. 8563, Tokuriki
Chemical Research, Tokyo, Japan], the pellets weredried in an oven at 1151C for 24 h. The density ofthe pellet was 2.0 g/cm3 (relative density 82%).
Moreover, the montmorillonite powder was classified by particle size in order to improve the response
time. The powder was suspended in water and centri-

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Percolation-Type Chemical Sensor

fuged at 1000 rpm for 10 min. The turbid supernatant
was separated by decantation and carbon powder was
mixed with the suspension. Thick films of carbon–montmorillonite composites were prepared by coating anddrying this suspension on a slide glass.
The resistivity was measured by the dc two-probe
method using a digital multimeter (34401A, Agilent
Technologies, Palo Alto, CA). Resistivity measurementsof the pellet as a function of the volume fraction of themontmorillonite were carried out after drying at 1151C
for 24 h and subsequently cooling down to room temperature in a dry atmosphere. The time course of theresistivity at RH 60% and 201C was also monitored
after the drying process, and thus this measurement
started from the water-free state. The resistivity change
in response to the stepwise increase of RH was monitored after leaving the pellet at RH 60% and 201C for
several days and subsequently at RH 0% and 201C for
24 h. At the starting point of this measurement, thepellet already incorporated water to some extent, because some incorporated water does not desorb reversibly at room temperature. Water incorporation into thesample was monitored by gravimetry. The desired water
vapor pressure ðP H2 O Þ was prepared by mixing a dry gas
with a wet one using mass flow meters. The morphologyof the sample composites was observed using a SEM(S-3000N, Hitachi, Tokyo, Japan) after carbon coating
and an optical microscope (VHX-900, Keyence, Osaka,
Japan). The structure change of montmorillonite bywater incorporation was also investigated by powder
X-ray diffraction (XRD) (Ultima IV, Rigaku, Tokyo,
Japan) using monochromatized CuKa radiation, andthe water content was determined by DTA/TG measurement (DTG-50, Shimadzu, Tokyo, Japan) at aheating rate 201C/min.

795

Fig. 1. Resistivitiy of a montmorillonite–carbon composite as afunction of the volume fraction of the montmorillonite.

7 vol% (6 wt%) from an inflection point of the curve.
This is much lower than the reported experimental values of pc such as 47% for WAl2O3 films, 35% for
Bakelite–silver mixtures.9,25 The low pc is due to thesegregation of carbon powders around the montmorillonite particles,10 because the carbon powder is muchsmaller than the montmorillonite. Figures 2 and 3 show
the surface morphology of the pellet. In Fig. 2, the carbon phase was not randomly distributed in the pellet,
but it was segregated and partly continuous between themontmorillonite phase. In Fig. 3, the flake-like montmorilloninte particles were oriented perpendicular

Results and Discussion
The resistivity of the carbon–montmorillonite
composites is shown in Fig. 1 as a function of the volume fraction of the montmorillonite. The resistivity wasincreased gradually with the increasing volume fraction
when the fraction was below about 80%. Above about
90%, however, the resistivity was drastically increased
by eight orders of magnitude from 5.55 O cm to273 MO cm, and became almost steady in the vicinityof 100%. The percolation critical threshold (pc), thevolume fraction of the conducting phase, is estimated as

Fig. 2. Optical microscope image of the surface of the composite
pellet. The white points are the carbon powders naked on thesurface, and the black nets are the segregated continuous carbon
phases buried shallowly in the pellet.

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International Journal of Applied Ceramic Technology—Tsuyumoto and Iida

Fig. 3. Scanning electron microscopic image of the composite
pellet. The flake-like particles of the montmorillonite were oriented
and agglomerated. The carbon powder was not observed in the
agglomerate.

to the c-axis, and the carbon powder was not distributed
inside the agglomerates of the montorillonite powder.
The pc is theoretically predicted as 7.6% for the present
particle size ratio (RM/RC 5 8) according to the segregation model.9,10 This is in good agreement with
our present experimental value. It is appropriate tofabricate the composites containing around 7 vol% ofcarbon to obtain a sensitive resistivity change by water
incorporation.
As shown in Fig. 4, the composite containing
7.7 vol% of carbon showed a remarkable resistivity increase from 6 to 1200 O cm at 60% RH at 201C. This
unusual resistivity increase is due to the separation of the
conducting paths based on the percolation principle,

Fig. 4. Resistivity change of a composite pellet containing
7.7 vol% of carbon at RH 60% and 201C after drying at 1151C
for 24 h.

Vol. 8, No. 4, 2011

whereas almost all of inorganic materials including
the montmorillonite show resistivity decrease by water
incorporation. Interestingly, the time evolution of theresistivity showed two inflection points, although resistivity as a function of the volume fraction showed only
one inflection point in Fig. 1. This is probably because
the curve reflects two factors: the rate change of water
incorporation and the volume fraction dependence ofthe resistivity. The two inflection points are explainedby the faster water incorporation rate at the initial stage
and the greater volume fraction dependence at the middle stage.
The composites with the mixing ratios far from thevalue of pc showed a relatively small resistivity change ina humid atmosphere (RH 85%, 201C), for example, thecomposite with carbon 22 vol% changed from 10 to
15 O cm. In addition, the composites with carbon
2 vol% or below showed resistivity decrease in humid
atmosphere due to the proton conductivity of montmorillonite similar to a pure montmorillonite pellet,
whereas those with carbon 3 vol% or above showed resistivity increase due to the percolation effect. The decrease of electrical conductivity due to the separation of
the conducting path was balanced with the increase of
the proton conductivity through the montmorillonite
phase, when the carbon phase was 2–3 vol%.
After the resistivity almost reached a saturated value
at room temperature, a dry gas was introduced to thepellet and an equilibrium was established between thepellet and the dry gas at room temperature. The resistivity at this point was greater than that immediately
after drying at 1151C, and we investigated a humidity
sensor property from this point to reduce the response
time. As shown in Fig. 5, the resistivity of the composite(6.6 vol% carbon) was increased and reached an equilibrium in response to stepwise water vapor increases(RH 20%, 40%, and 60%). The resistivity was decreased again by the introduction of a dry gas. The 90%
response time was estimated to be 540, 512, and
476 min for RH 20%, 40%, and 60%, respectively.
The mass change of the montmorillonite powder in response to the stepwise water vapor increases showed almost the same tendency as the resistivity change (Fig.
6). The 90% response time was estimated to be 1010,
434, and 400 min for RH 20%, 40%, and 60%, respectively. The equilibrium values of water content forP H2 O 0.467, 0.935, and 1.40kPa were 1.06, 1.99, and2.79 wt%, respectively. This relationship followed theLangmuir isotherm written aswww.ceramics.org/ACT

Percolation-Type Chemical Sensor

Fig. 5. Resistivity change of a composite pellet containing
6.6 vol% of carbon in response to stepwise relative humidity change(RH 0%, 20%, 40%, and 60%) at 201C.

P H2 O P H2 O
1
¼
þ
Kx 1
x
x1

ð1Þ

where x is the amount of adsorbed water molecules andP
xN is the amount of all adsorption sites. The Hx2 O vsðP H2 O Þ plot indicated xN 5 15.2 wt% and K 5 1.61 
104 Pa1. The relationship between the chemical potential of water molecules ðmH2 O Þ and the adsorption
potential energy E is expressed by16
mH2 O ¼mH2 O  þ kT ln P H2 Oy
¼ E þ kT ln
1y

ð2Þ

where mH2 O  is the chemical potential of water molecules
at 101.3 kPa and y is the coverage of the adsorption site

Fig. 6. Mass change of montmorillonite powder in response to
stepwise relative humidity change (RH 0%, 20%, 40%, and 60%)
at 201C.

797

Fig. 7. Resistance of a montmorillonite–carbon composite thickfilm prepared from the classified fine montmorillonite particles as afunction of the volume fraction.

( 5 x/xN). The adsorption energy of water molecules
onto the montmorillonite was estimated at E  mH2 O  ¼
6:74 kJ=mol. It is important to note that the responsetime of the powder was similar to that of the pellet. This
indicates that the pellet density hardly affected the response time and that not the diffusion in the pellet but the
diffusion in the particle is crucial to the sensor response
time. We considered the response time would be diminished by the classification of the powder because it is dependent on the particle size.
Figure 7 shows the resistance of the carbon–montmorillonite thick films prepared from the fine montmorillonite particles as a function of the volume
fraction. The thickness was in the range of 0.016–
0.047 mm. The critical threshold (pc) is estimated as
35 vol%, because the resistivity was abruptly increasedin the range of 62.6–67.7 vol%. The increase of thecritical threshold compared with the pellet reflects the
reduction of the particle size of the insulating phase.9,10
The resistance change of the thick film containing
56 vol% of carbon in response to stepwise water vapor
increases is shown in Fig. 8. The 90% response time was24, 25, and 56 min for RH 20%, 40%, and 60%, respectively. The response time was substantially improved by fabricating a thinner sample comprised ofsmaller particles of the montmorillonite. The resistivity
was rapidly increased when liquid water was dropped on
the surface of the thick film, in contrast to the conventional films.
The DTA/TG measurement of the montomorillonite powder indicated that the water desorption proceeded in two stages. The interfacial water desorbed

798International Journal of Applied Ceramic Technology—Tsuyumoto and Iida

Fig. 8. Resistance change of a composite thick film containing
56 vol% of carbon in response to stepwise relative humidity change(RH 0%, 20%, 40%, and 60%) at 201C.

from room temperature to 1101C, and the structural
water desorbed at 600–7001C indicating the change
from Na2/3Si8(Al10/3Mg2/3)O20(OH)4 to Na2/3Si8(Al10/3
Mg2/3)O22. The content of the interfacial water was determined from the mass decrease below 1101C. Powder
XRD patterns of the montmorillonite are shown in Fig.
9. The lattice parameters were a 5 4.471 Å, c 5 12.48 Å
when retained at RH 60% for more than a week, andthey changed to a 5 4.503 Å, c 5 12.10 Å by drying at
1151C for 24 h. The water content of the former sample
was estimated at 14 wt% by the TG measurement. Because of the water incorporation, the lattice parameter cwas increased by 3.14%, whereas the lattice parameter a
was decreased by 0.71%. The total volume change of theunit cell corresponding to the water incorporation of
14 wt% was 1.68%. The volume change of the unit cellinvolved by the resistivity change in Fig. 5 is much lower

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than this value, because the mass change of the sample inFig. 6 was within 3 wt%. If we assume a linear relationship between the water content and the volume change ofthe montmorillonite, the resistivity change in Fig. 5 corresponds to the volume change within 0.34%. Our results demonstrate that the composite comprising of aconducting phase and an insulating phase is an effective
tool for detecting not only the incorporation of substancebut also a slight volume change.
The resistance did not return to the initial value at
RH 0% when a dry gas was reintroduced in Figs. 5 and
8, in contrast to the fact that the mass change almost
returned to the initial value in Fig. 6. This is probably
due to the change of contact resistance between theelectrode and the composite, and the change can becancelled by the improvement of the attachment of theelectrodes or the adoption of the dc four-probe method.
The resistivity changes in Figs. 5 and 8 remained within
one order of magnitude, whereas the resistivity in Fig. 4
increased three orders of magnitude. This is due to thedifference in the initial conditions. The measurement in
Fig. 4 started immediately after drying at 1151C, while
those in Figs. 5 and 8 were carried out using the samples
left in an ambient atmosphere for several days. It is appropriate to start immediately after drying at 1151C
when we prioritize the sensitivity or the detection limit
for a water vapor detector over the reversibility of theresistance. In this case, the detection limit is roughly
estimated at 0.6 ppm from the S/N ratio in Fig. 4 In
contrast, it is appropriate to use the composite almost
saturated with water vapor in application to an RH
sensor. Our results suggest that controlling the initialstate is critical for the practical use of the percolationtype sensors.

Conclusions

Fig. 9. Powder X-ray diffraction patterns of montmorillonite: (a)with interfacial water of 14 wt%, and (b) water free.

We demonstrated that a percolation-type chemicalsensor can be fabricated using a composite of a conducting phase and an insulating phase whose volume
increases by the incorporation of a desired substance,
because the resistivity of the composite increases
abruptly with the increasing volume fraction of the insulating phase at a certain threshold of the volume fraction. Water vapor (4.6  102–1.4  103 Pa, 201C) was
successfully detected by monitoring the resistivity of thecomposite comprising of montmorillonite and carbon.
Contrary to the conventional humidity sensors, the re-www.ceramics.org/ACT

Percolation-Type Chemical Sensor

sistivity showed an increase by water incorporation. The
response time was substantially improved from 400–
1010 to 60–128 min by using a thick film preparedfrom classified finer montmorillonite particles. Simultaneously, the critical threshold (pc) of the volume fraction of the conduction phase increased from 7 to
35 vol% by reducing the particle size. This was in agreement with the earlier reports on the influence of particle
size ratios. This percolation-type humidity sensor has
advantages in cost, sensitivity, and durability, and thus itwill be a powerful tool to detect a water vapor at hightemperature up to about 1501C in a corrosive atmosphere. Furthermore, this new working mechanism ofchemical sensor can be easily applied to the other sensor
systems. Selective and sensitive detection of a desired
substance will be feasible by modifying the interlayer ofthe montmorillonite with functional groups. Biosensors
will also be possible by fabricating a composite of aconductor and a receptor containing an enzyme or anion channel whose volume or morphology change by
the incorporation of a desired substance. We believethat the new working mechanism will occupy an important position in the principles of chemical sensors in
the near future.
References
1.
2.
3.

4.

5.
6.

S. Kirkpatrick, ‘‘Percolation and Conduction,’’ Rev. Mod. Phys., 45 [4] 574–
588 (1973).
S. Kirkpatrick, ‘‘Classical Transport in Disordered Media: Scaling and Effective-Medium Theories,’’ Phys. Rev. Lett., 27 [25] 1722–1725 (1971).
C. G. Granqvist and O. Hunderi, ‘‘Conductivity of Inhomegeneous Materials: Effective-Medium Theory with Dipole–Dipole Interaction,’’ Phys. Rev.
B, 18 [4] 1554–1561 (1978).
M. Gajdardziska-Josifovska, R. C. McPhedran, D. R. McKenzie, and R. E.
Collins, ‘‘Silver–Magnesium Fluoride Cermet Films. 2: Optical and Electrical
Properties,’’ Appl. Opt., 28 [14] 2744–2753 (1989).
C. W. Nan, ‘‘Physics of Inhomogeneous Inorganic Materials,’’ Prog. Mater.
Sci., 37 1–116 (1993).
Y. M. Strelniker, A. Frydman, and S. Havlin, ‘‘Percolation Model for theSuperconductor–Insulator Transition in Granular Films,’’ Phys. Rev. B, 76[22] 224528 (2007).

7.

799

D. S. McLachlan, M. Blaszkiewicz, and R. E. Newnham, ‘‘Electrical
Resistivity of Composites,’’ J. Am. Ceram. Soc., 73 [8] 2187–2203(1990).
8. R. P. Kusy and D. T. Turner, ‘‘Electrical Resistivity of a Polymeric Insulator
Containing Segregated Metallic Particles,’’ Nature Phys. Sci., 229 [1] 58–59(1971).
9. A. Malliaris and D. T. Turner, ‘‘Influence of Particle Size on the Electrical
Resistivity of Compacted Mixtures of Polymeric and Metallic Powders,’’ J.
Appl. Phys., 42 [2] 614–618 (1971).
10. R. P. Kusy, ‘‘Influence of Particle Size Ratio on the Continuity of Aggregates,’’ J. Appl. Phys., 48 [12] 5301–5305 (1977).
11. R. D. Sherman, L. M. Middleman, and S. M. Jacobs, ‘‘Electron Transport
Processes in Conductor-Filled Polymers,’’ Polymer Eng. Sci., 23 [1] 36–46(1983).
12. B. Lundberg and B. Sundqvist, ‘‘Resistivity of a Composite Conducting
Polymer as a Function of Temperature, Pressure, and Environment: Applications as a Pressure and Gas Concentration Transducers,’’ J. Appl. Phys., 60[3] 1074–1079 (1986).
13. G. R. Ruschau, R. E. Newnham, J. Runt, and B. E. Smith, ‘‘0–3 Ceramic/
Polymer Composite Chemical Sensors,’’ Sens. Actuators B, 20 269–275(1989).
14. M. C. Lonergan, E. J. Severin, B. J. Doleman, S. A. Beaber, R. H. Grubbs,
and N. S. Lewis, ‘‘Array-Based Vapor Sensing Using Chemically Sensitive
Carbon Black-Polymer Resistors,’’ Chem. Mater., 8 [9] 2298–2312(1995).
15. H. Lei, W. G. Pitt, L. K. McGrath, and C. K. Ho, ‘‘Resistivity Measurementsof Carbon–Polymer Composites in Chemical Sensors: Impact of Carbon
Concentration and Geometry,’’ Sens. Actuators B, 101 122–132 (2004).
16. I. Tsuyumoto, A. Kishimoto, and T. Kudo, ‘‘Hexagonal Tungsten Bronze
Synthesized from Potassium Peroxo-Polytungstate and Its Electrical Properties,’’ Solid State Ionics, 59 [3/4] 211–216 (1993).
17. I. Tsuyumoto and T. Kudo, ‘‘Humidity Sensor Using Potassium Hexagonal
Tungsten Bronze Synthesized from Peroxo-Polytungstic Acid,’’ Sens. Actuators B, 30 [2] 95–99 (1996).
18. I. Tsuyumoto and T. Kudo, ‘‘Humidity Sensor Using Potassium Hexagonal
Tungsten Bronze Synthesized from Peroxo-Polytungstic Acid and Its Resistivity Change Mechanism,’’ Mat. Res. Bull., 31 [1] 17–28 (1996).
19. J. E. Moneyron, A. de Roy, C. Forano, and J. P. Besse, ‘‘Realization of Humidity Sensors Based on a Screen-Printed Anionic Clay,’’ Appl. Clay Sci., 10[1–2] 163–175 (1995).
20. K. Miyazaki, M. Hieda, and T. Kato, ‘‘Development of a Novel Manganese
OxideClay Humidity Sensor,’’ Ind. Eng. Chem. Res., 36 [1] 88–91(1997).
21. P. G. Su and C. Y. Chen, ‘‘Humidity Sensing and Electrical Properties of Naand K–Montmorillonite,’’ Sens. Actuators B, 129 380–385 (2008).
22. Y. Nakamura, H. Yoshioka, M. Miyayama, H. Yanagida, T. Tsurutani, and
Y. Nakamura, ‘‘Selective CO Gas Sensing Mechanism with CuO/ZnO Heterocontact,’’ J. Electrochem. Soc., 137 [3] 940–943 (1990).
23. T. Nitta, Z. Terada, and S. Hayakawa, ‘‘Humidity-Sensitive Electrical Conduction of MgCr2O4–TiO2 Porous Ceramics,’’ J. Am. Ceram. Soc., 63 [5–6]
295–300 (1980).
24. Y. Sakai, Y. Sadaoka, and M. Matsuguchi, ‘‘Humidity Sensors Based on
Polymer Thin Films,’’ Sens. Actuators B, 35 [1–3] 85–90 (1996).
25. B. Abeles, H. L. Pinch, and J. I. Gittleman, ‘‘Percolation Conductivity in W–
Al2O3 Granular Metal Films,’’ Phys. Rev. Lett., 35 [4] 247–250 (1975).