Materials Science and Engineering A328 (2002) 67–79
Relations between microstructure, electrical percolation andcorrosion in metal—insulator composites
E. Thommerel a,b, J.C. Valmalette a, J. Musso a, S. Villain a, J.R. Gavarri a,*,D. Spada b
a Laboratory ‘‘Materiaux Multiphases & Interfaces‘‘, Faculte des Sciences et Techniques, Uni�ersite de Toulon-Var, BP 132, F-83957,La Garde Cedex, France
b M.A.D.E. Society, Espace Frioul, F-83160, La Valette, France
Received 17 October 2000; received in revised form 31 May 2001
Keywords: Metal/insulator composites; Transport properties; Electrical complex impedance spectroscopy; Effective medium approximation;Percolation theory; Grain size, Corrosion
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1. Introduction
Much of the recent progress in industry (aeronautics,electronics, corrosion protection…) is the result of theuse of both structural and functional composite materi-als. These composites can be made of insulating andconducting particle mixtures, which are subjected toboth mechanical and thermal treatment during themanufacturing process. To obtain electrical compo-nents that might be good conductors, which are simul-taneously chemically and mechanically stable, theclassical manufacturing process consists of compactingpowders of metal and polymer constituents. Below acertain volume fraction of metal defined as the critical
threshold �c, the composite is an insulator, while abovethis composition it becomes an electrical conductor. Ifelectrical currents can percolate through the bulk mate-rial, then varying degrees of corrosion may occur alongthe electrical paths existing in the composite. To limitthis corrosion and increase the lifetime of the materialone can manufacture conducting composites havingmetal compositions, �, just above the critical threshold.Previous works have described the transport propertiesin composites and discussed the various modeling ap-proaches that allow the interpretation of the sigmoidelectrical responses in these composites [1–10]. To opti-mize the properties of these composite systems, it isnecessary to model the theoretical dependencies linkingthe concentration, the particle size and the compositeconductivity. A review on various models was given byClerc [11]. The effect of particle size on the electrical
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properties was previously determined by using percola-tion theory [12,13].
In compacted composites, it is well established that,for a given composition, the electrical propertiesstrongly depend on grain size, morphology and appliedpressure [14–18]. The calculation of the equivalentconductivity necessarily involves all of the neighboringparticles, too [19].
Three major models have been developed previouslyto make predictions on the electrical behavior of idealcomposites: (i) the Effective Medium Approximation(noted as the EMA model) [11,20,21], (ii) percolationtheory [1,11], (iii) the micro-structural approach[14,22–24].
The conductivity of a composite can be described byBruggeman [25] symmetric and asymmetric mediumequations. The symmetric expression forms the basisfor the effective medium approximation, which will berepresented in Section 3. The classical EMA modelgives a global description of the electrical properties,however, it involves a critical composition �infl. which isdifferent from the effective percolation composition�perc. This theory is used usually to describe the proper-ties of binary mixtures far from the percolationthreshold when the ratio �insulator/�conductor is sufficientlyhigh [26,27]. In the case of the classical EMA approach,the major difficulty resides in the fact that the finalgranulometry and texture are never well known, andthat this model itself is based on simple and regulardistributions of ideal particles. This is not the case in‘real’ materials.
Percolation theory delivers a good description of thepercolation features near the threshold �c but does notexplain the experimental data far from the percolationfraction �perc. This model gives a good description of acomposite system when the �insulator/�conductor ratio isvery small [26]. In our case, we are trying to describethe conductor/insulator system for volume fractions �ranging from 0 to 1.
The classical EMA and a modified EMA model areused first to determine the parameters governing theconductor to insulator transition. Then a comparisonbetween the EMA and percolation theory parameters ispresented.
2. Experimental
2.1. Sample preparation
Poly-dispersed metal/polymer composites were pre-pared by mixing powdered metallic (Al, Fe, Ni, W andZn) particles with an insulator polymeric matrix ofpoly-phenylsulfur [-C6H4S-]n PPS. This polymer used inpowder form is a chemically inert insulating material.All composites were prepared under the same condi-
tions: the constituents were mixed in well-defined pro-portions and pressed in a cylindrical 13-mm die under auniaxial pressure of 10�1 kbar. Both the initial PPSand the metal constituents are granular. Hence, for allvolume fractions the composite pellets present residualcavities (�cavities). Finally, the macroscopic propertieswill be functions of the volume fractions (�, �PPS,�cavities), where:
�+�PPS+�cavities=1.
The cavities are insulating regions for electron cur-rents; hence we will not distinguish between PPS andcavities. The metal volume fractions � after pressingwere then calculated taking into account these cavities.The volume fractions are then calculated using theinitial weights of the constituents, the individual densityof each constituent and the effective volume of thecylindrical pellets. Different metal/PPS composites wereprepared with metal compositions � ranging from 0 to1. Increments of ��=0.10 were chosen; however, nearthe inflexion threshold, samples are prepared with vari-able metal compositions where ��=0.01. For eachcomposite, two sets of samples were prepared to avoiddiscrepancies in measurement.
2.2. Grain size
The initial grain size (before compaction) was con-trolled using a Malvern sizemeter in polyphase modeanalysis. Using spherical particle approximation, parti-cle size analysis was performed in the 0.3–300 �mrange, using pure water as a solvent.
Each sample was subjected to electrical compleximpedance measurements 10 min after pressing, forfrequencies ranging from 10−1 to 107 Hz. The electricalcomplex impedance was measured using a potentiostat/galvanostat Model 273A from EGG, coupled with aHF frequency response analyser SI1255 from Schlum-berger. Each sample was placed between two goldelectrodes under a constant pressure of 50 mbar atambient temperature (20 °C). Results are usually plot-ted either as a (log �(real part), �) graph or as a Nyquistgraph (X=Z �, Y= -Z ��), where the complex impedanceis Z=1/�=Z �+ jZ ��. Nyquist representations showclearly the difference between the insulating and con-ducting state responses. In the [log�, �] plots, twocritical volume fractions (�) are defined: the inflexionpoint of the EMA model �=�infl. (see annex) and thepoint �=�c ��infl., where a rapid evolution of log�starts. This inflexion point is defined as corresponding
to the maximum ofd2log�
d�2 . These two critical volume
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Fig. 1. Determination of the critical compositions. (a) Nyquist repre-
sentation of Z ��= f(Z �). (b) Representation ofd2log�
d�2= f(�v).
phology and size distribution after the compaction pro-cess. These observations showed the evolution of grainsize as a function of metal concentration in the samples.The microscope was linked to an EDS X-ray micro-analysis system, which allowed to investigate the mi-crostructure (composition and distribution of theclusters) of the samples. In the case of corroded elec-trodes, the sample was dried before analysis.
2.5. Corrosion analysis
A simple electrochemical device associated with theimpedance spectroscopy analyzer was used (Fig. 2). Thecorrosion of Zn/PPS composite electrodes in hydro-chloric acid solution at constant pH (pH 2) and underan alternating potential was analyzed as a function ofthe working time t and of the Zn volume fraction in thecomposite. This device consisted of two electrodes Aand B connected to the analyzer. The two electrodeswere identical (dimensions, shape and composition). Allthe experiments were performed at room temperature.The working frequency � varied automatically between1 and 106 Hz. The applied a.c. voltage was V=100mV.
3. Results of the modeling approach
3.1. Initial EMA model
In real compacted composites, the critical composi-tion and the evolution of impedance can differ fromone composite system to another. To understand betterthe conductivity of our metal/polymer composites, twotypes of modeling approach were proposed: a modifica-tion of the EMA model and the application of percola-tion theory. Generally, these models are valid for idealcomposites in which all constituents are characterizedby a unique size. It is well known, however, that theexperimental percolation threshold depends on the par-ticle size distribution in the composite.
Fig. 2. Device used for ECIS measurements: two composite electrodesimmersed in acid solution, connected to impedance analyzer (alternat-ing potential).
fractions are discussed in the modified EMA model. Itcan be determined either by calculating the maximum
ofd2log�
d�2 , see Fig. 1b, or from the strong changes in
the Nyquist representation: a significant change froman insulating (pseudo-linear curve) to a conducting(circular curve) behavior occurs over a very small com-position range.
Such a critical value �c might be associated stronglywith the percolation threshold of percolation theory(see Fig. 1a). In this theory, only the percolationthreshold is clearly defined.
2.4. Scanning electron microscopy
Scanning electron microscopy (SEM) was carried outon a Philips XL30 microscope to determine grain mor-
Table 1Initial size distribution of metal and polymer powders
Note: D10 is the value of the particle size that separates the distribution in 10/90%. D50 is the value of the particle size that separates thedistribution in 50/50%. D90 is the value of the particle size that separates the distribution in 90/10%. Span: breadth of the distribution.Span= [D90−D10]/D50.
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Fig. 3. Nyquist plots for a series of Fe/PPS composites. (a) Below; (b)above the percolation threshold.
where �1 and �2 are the volume fractions of the twoconstituents with �1+�2=1 and �1,2=�1,2+ (d-1)�,where d is a dimensional parameter associated with theconnectivity of the system [11]. Generally, using thisEMA model, the inflexion point of the curve (log�,�) ischaracterized by a composition noted as �infl. with�infl.=1/d in the case of an ideal two-phase composite.This composition differs considerably from the criticalcomposition �c.
Using the ECIS measurements represented inNyquist diagrams, the conductance value for each com-posite was determined. In the case of insulating mix-tures, the circles can be modeled by an RC parallelcircuit. For conducting mixtures, the adequate model isan RL series circuit. In all cases, it is possible toevaluate the resistance and the conductance of thesamples. Fig. 3a and 3b show a selection of resultsbelow and above the percolation threshold, for someFe/PPS composites. The experimental and theoreticalconductances (for all composites) calculated with theEMA model are represented in Fig. 4a–4e, where the dvalues represent a mean value according to d=1/�infl..As expected, we note that this simple EMA model fitsthe data well when far from the percolation (or inflex-ion) compositions; however no good fit is obtainedclose to these compositions. This can be observed forthe Al, Fe, Ni and Zn /PPS composites. Even if the�infl. critical composition can be simulated, the experi-mental evolution close to this critical composition dif-fers greatly from the calculated curves. Generally, theexperimental insulator–conductor transition is lesssteep than the calculated transition predicted by the
This size distribution of the metallic particles in ourcomposites is shown in Table 1. In addition, the finalsize distribution is the result of two factors: the originalparticle size and the modification due to the com-paction process.
To model the composites, we first use the effectivemedium approach. This approach allows to describethe behavior of the transport properties for dilute com-posites (�1 and �2 for �=0 or 1):
�−�1
�1
�1+�−�2
�2
�2=0 (1)
Fig. 4. Experimental and theoretical conduction of metal/PPS composite (EMA model) as a function of metal volume fraction �. Conductances� in logarithmic scale. (a) Al/PPS; (b) Fe/PPS; (c) Ni/PPS; (d) Zn/PPS; (e) W/PPS.
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Fig. 5. SEM micrograph showing the formation of a metal cluster dueto compaction.
coupled to polymer particles only. This means that,after compaction each metal/metal couple is changedinto one unique particle, and that the notion ofmetal particle pairs ‘in contact’ vanishes. As a result,it is possible to distinguish independent regions inthe sample in which each population of grains has itsown connectivity parameter d.
� Due to surface contributions (partial oxidation), themetal grain conductances can depend strongly ongrain size. Due to these surface effects, in compactedsamples, the smallest isolated particles have thehighest resistance. This is the reason why we haveassigned a � value to each particle. This last value isdetermined by computation.
� The mechanical effects involved in the compactionprocess condition the final grain size distributions(after compaction) which depend strongly on theinitial total volume fraction of metal. The last majorfact is that the ‘final’ grain size of the composite,involved in the compaction process, depends on thetotal volume fraction of the metal (�v). Actually, themore the volume fraction of the metal increases, themore the small grains agglomerate. The proportionof large particles increases with the volume fraction.This volume fraction depends on the extent of metalparticle clustering. In Fig. 5 the formation of ag-glomerates of metal particles due to the compactionprocess is shown.However all these hypotheses are also questionable
because they cannot take into account the statisticalnotion of percolation. Fig. 6 summarizes our varioushypotheses schemastically.
The modified equation we propose is:
�−�1
�1
�1+�−�2
�2
�2+ ...+�−�n
�n
�n=0 (2)
with �i=�i+ (di-1)�; �1+�2+…+�n=1; dn
=�
n−1
i=1
di�i
�n−1
i=1
�i
Each di value will represent the ‘effective dimension-ality’ for a metal couple (i)/PPS and will be associatedwith a size, a volume fraction �i of the metal and itsconductivity �i. In our case, we arbitrarily divided thegrain size diagrams into three parts. This implies that,from Eq. (2), we have to refine six parameters: d1, d2,d3, �1, �2, �3. The parameter �4 is the conductivity ofPPS. In our approach d4 is obtained arbitrarily fromthe inflexion point (see Appendix) by assuming that:d4=1/�infl.. Such a choice is coherent with the initialEMA model applied to an ideal two-phase system.
The initial �i values are determined from grain sizemeasurements.
Fig. 6. Model of composite divided into sections.
EMA model. However, let us remark that the W/PPScomposite electrical behavior agrees well with the EMAmodel, this case will be discussed later.
3.2. New fitting parameters for EMA model
The laser grain size analysis (Table 1) showed arelatively high dispersion of particle sizes. Moreover, ithas been previously demonstrated [12,13] that particlesize is an extremely important parameter in conduction.Therefore, it is clear that the classical EMA model fails.It was therefore necessary to control this particle sizedispersion to improve the model.
In the elaboration of the modified EMA model, wefirst assume that the composites are composed of cou-ples: metal/metal, polymer/polymer and metal/polymer.Then each couple could give its own contribution topercolation.
To take into account the distribution of metal parti-cle sizes, the complex medium was modeled as follows:� The composite was first divided into elemental sec-
tions or layers having the same composition;� The thickness � of these sections is linked to the size
of the largest particles after compaction;� In each section, the effective medium approximation
can be applied: each metal particle is supposed to be
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Fig. 7. SEM micrograph (over-exposed) of a Zn/PPS composite: the metal particles appear in black. (a) Below critical threshold, �v=16%. (b)Above critical threshold, �v=20%.
The evolution of particle size distribution after com-paction depends on the total volume fraction of themetal and might be described as follows:� Below the critical threshold, the particles are farther
apart. Therefore, after compaction, the grain sizewill be roughly the same as the initial grain size.
� Near the critical threshold (5% below and above),the metallic particles are more and more in contact:the grain size is modified. The number of smallergrains decreases while the number of larger onesincreases very quickly.
� Above the critical threshold the particle distributionstays unchanged irrespective of �, because a majorpart of the smaller particles has disappeared duringcompaction.A fairly good approximation of the critical threshold
can be determined from SEM analyses. Despite the factthat SEM gives a two-dimensional representation of thematerial, this approximation should be acceptable be-cause of the very high homogeneity of the samples. Fig.7 shows the distribution of the metal particles in thematrix (for Zn/PPS composites in the present case)below the critical threshold (Fig. 7a) and at the criticalthreshold (Fig. 7b). This micrograph is overexposed soas to distinguish clearly the black particles of metalfrom the white polymer.
The critical threshold as defined in this work is thecomposition from which an idealized first conductingline (the so-called infinite cluster) should appearthrough the composite (first conducting path from oneelectrode to the other). In Fig. 7a, one can observe thatno continuous black line exists: this explains the factthat no percolation of electrical current is observed inthis sample. In Fig. 7b we can observe continuousmetallic lines, which indicates that percolation occurs inthis sample. This volume fraction is associated well withthe range of definition of �c.
The SEM observation permits us to observe thecharacteristic evolution of the initial grain size: thefraction �i of each grain size family depends on thetotal metal volume fraction presently noted as �V. Thevalues �i have been adapted to the experimental data.For Al, Fe, Ni and Zn based composites, Fig. 8 showsthe functions used to refine the grain size. These func-tions are obtained from measurements.� For 0��v� (�infl.−0.05), the fractions �1 (smallest
particles), �2 and �3 (largest particles) are the sameas the fractions initially determined from laser grainsize analyses (Table 1).
� For (�infl.−0.05)��v� (�infl.+0.05), the smallestparticles agglomerate, �1 decreases and �3 increases.
� For �v��infl.+0.05, there is a complete change inthe particle size distributions because of the agglom-eration of the smallest particles: however, each frac-tion �i becomes constant again.For each composite, the initial and final grain sizes
(as a percentage of the total metal volume fraction) arepresented in Table 2.
For the W based composites, the �i are always foundto be equal to 1/3 (measurements).
The new modified equation now is:
Fig. 8. Functions used for grain size parameter definitions. �1,2,3 arevariable parameters.
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Table 2Weighted grain size �1 and �3 (as a percentage of �v) used inEq. (3)
Al Fe Ni Zn
����infl.−0.05100. �1 8 35 10 51
11 275 6100. �3
����infl.+0.0520 65 20100. �1
30100. �3 5540 35
Note: �2 is always equal to 1−�1−�3.
In Fig. 9a–9e, the experimentally obtained curves ofall metal/polymer composites can be compared to thecalculated curves obtained fromEq. (3). One can seethat close to the critical threshold, a better agreementbetween calculated and observed data is obtained thanbefore (Fig. 4a–4e).
We can also note the following features:1. For the W composites, the calculation of log� gives
d1=4.06, d2=2.96, d3=1.38 and a quite goodagreement between the calculated and measuredcurves: this may be attributed to the hardness ofeach W grain and also the initial grain size. Underthe compaction process, the grains cannot agglom-erate and the final grain size is close to the initialone (Table 1).
�−�1
�1
�1+�−�2
�2
�2+�−�3
�3
�3+�−�p
�4
�4=0 (3)
Fig. 9. Modified EMA model. Experimental and theoretical conduction of metal/PPS composite as a function of metal volume fraction �.Conductance � in logarithmic scale. (a) Al/PPS; (b) Fe/PPS; (c) Ni/PPS; (d) Zn/PPS; (e) W/PPS.
Table 3New modeling parameters for a modified effective medium approximation
2.10 2.16�d�=1/�infl. 2.701.70 5.00Quadratic error a
0.022 0.0190.0270.0431
N�n(�obs−�calc)2 0.13
0.2000.588�infl. (obs.) 0.474 0.463 0.370
a This term indicates the quality level of the refinement.
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2. For Al, Fe and Ni composites the D50 values areroughly similar and similar mean d values (from 1.7to 2.2) and similar �infl. values (from 0.588 to 0.463)(see Table 3) are observed.
3. For Zn composites, the grain sizes are much smallerthan for the other composites (D50=5 �m); �infl. issmaller, too.
4. In the case of Al and Zn composites, the ductility ofthe metal grains may be the origin of the observedprogressive evolution of the conductivity just abovethe critical threshold which can result from theagglomeration of ductile grains during the com-paction process.
The main conclusion must be that there is a strongcorrelation between the initial mean size and the criticalcomposition (the D50 and the �infl. values). In addition,the three parameters ‘d’ depend on the grain size distri-bution after the compaction process (including the vol-ume fractions associated with each size), and on thehardness of the metal grains. Moreover, conduction ischaracterized by the three di and three �i values. Fi-nally, we can conclude that the proposed model shouldimprove significantly the initial EMA model eventhough a small discrepancy persists close to the criticalthreshold.
3.3. Percolation theory
The main result of percolation theory [11,26,28]resides in the fact that a transport property (conduc-tance in our case) can be developed close to the perco-lation threshold, �perc, as follows:
�tot=A(�−�perc)=A (��) for ���c (4a)
�tot=B(�perc−�)−s=B (��)−s for ���c (4b)
where and s are the two classical critical exponentslinked to the formation of continuous clusters throughthe composite above and below the �perc volume frac-tion. A and B are constants linked to the �conductor and�insulator values. The exponents and s can be deter-mined by plotting log (�) versus log �� below andabove the percolation threshold.
3.3.1. Determination of the critical �olume fractions �c
To determine the most probable �perc value we pro-ceed as follows:� The initial �perc value is calculated as predicated in
Section 2.3. from the maximum of the derivatived2log�
d�2 . As a first approach we considered that
�perc=�c.� This �perc value is then modified to obtain an ideal
linear correlation log(�)=k log(��).� The final �perc values were found to be close to the
initial �c values.
The measured values of �c ranged between 0.190 and0.588. The evolution of this critical volume fraction canbe explained by the microstructure of the conductingpowders and the insulating polymer. Kussy [29] arguesthat in a mixture made up of small conducting andlarge insulating particles, the smallest particles will tendto coat the larger ones. This is the case with ourZn/PPS composites. The mean size of Zn grains isabout seven times smaller than that of the PPS grains.A three-dimensional conductance can occur. It is inter-esting to note that the experimental value of �c=0.190is in good agreement with the theoretical value of 0.198for a face centered cubic lattice [26,28].
For the Al, Fe, Ni and W/PPS composites, the meansizes of the conducting and insulating grains are quitesimilar. The different values of �c suggest that conduc-tion should be associated with a configuration interme-diate between a 2D and a 3D system.
3.3.2. Determination of the exponents s and �
It has been previously demonstrated in [29–32] thatthe nature of the powder, the cavities and the mi-crostructure of a composite have a strong influence onthe critical exponents. In our composites, the valuesrange between 1.45�0.29 for the W/PPS compositesand between 2.07�0.21 for the Zn/PPS composites. Itseems that the grain size dispersion and the nature ofthe material have a relatively small influence on . Likethe critical volume fraction, the observed seem to berelated to systems ranging between 2D to 3D dimen-sions. Nevertheless, a comparison between and theparticle size can be done: Table 1 shows the particle sizedistribution of the conducting particles. If we considerthe D50 characteristic, we observe that the smaller theparticle, the higher is . As previously proposed by Wuand McLachlan [33], such an evolution of could beexplained by the increase of contact resistance betweenthe grains.
Table 4 summarizes s and calculated from the bestLog–Log plots. It can be seen that a very wide range ofcalculated s values is obtained (s=0.832; s=2.282),and that no connection can be established between thes, , �c values and the particle sizes. The values s=0.832, s=0.992 and s=1.028 respectively found for theNi, W, Al /PPS composites are in quite good agreementwith the values measured for other composites (s=0.73for Ag/KCl mixture [34]). They are also congruent withthe 2D and 3D values obtained from simulations[26,35–37]. However, the very high values of s for Znand Fe/PPS composites cannot be explained.
From s, and �c one can calculate the conductanceusing the normalized percolation equations [38]:
�=�conductor [(�−�c)/(1−�c)] ���c (5a)
�=�insulating [(�c−�)/�c]−s ���c (5b)
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Table 4Critical volume fraction and critical exponents of percolation theory. The r value represent the linearity coefficient of the linear regression usedfor the Log–Log plot calculations
Fig. 10. Plot of the conductance of metal/PPS composites using percolation theory. Conductance � in logarithmic scale. (a) Al/PPS; (b) Fe/PPS;(c) Ni/PPS; (d) Zn/PPS; (e) W/PPS.
In Fig. 10a–10e, the experimental data and the con-ductances calculated from Eq. (5a) and Eq. (5b) arerepresented as a function of the metal volume fractions�. It can be observed clearly, that such a percolationmodel fits the data well. For the W/PPS composites(and partly for the Al/PPS composites) and s are bothclose to 1. These results are in agreement with Brugge-man’s symmetric theory.
4. Percolation and corrosion
Often metal– insulator composites are intended tofunction in aggressive environments over a period ofmany years. To connect the percolation compositionsfound for electron currents with the percolation pathsavailable for corrosion currents, we carried out a corro-sion study for Zn composites in acid media. The mea-
E. Thommerel et al. / Materials Science and Engineering A328 (2002) 67–7976
surements were carried out using the electrochemicaldevice described in the experimental section. Fromthese ECIS measurements (see Fig. 11), the complexconductance modulus �� � was determined at each fre-quency �. Its variation as a function of time andcomposition is conditioned by the evolution of theinternal electrical circuits subjected to corrosion.
In Fig. 12a–c, the complex conductances �� �=1/�Z �,obtained for a fixed frequency of 10 Hz, are reported asa function of time. In Fig. 12d, the conductances arereported as a function of the Zn concentration at afixed frequency of 10 Hz for 30, 200 and 400 min. �� � islinked directly to the corrosion current. In Fig. 12a–c,different types of behavior are observed:� For t�400 min, a decrease of the conductance
modulus is observed in all cases, due to the surfacecorrosion of the Zn metal by the H+ ions.
� For 400� t�500 min, a plateau is observed in eachexperiment. This is probably due to the formation ofhydrated species (including ZnO oxide) on the sur-face and in the cavities close to the surface of thecomposite. The infiltration of the H+ ions is thenstopped because the precipitated species act as insu-lating barriers thus lowering corrosion.
� For t�500 min, different types of behavior areobserved:
� At Zn concentrations below the percolationthreshold (Fig. 12a), the conductance increaseswhich might be due to the fact that the Zn basedspecies have gone and that the acid solution canagain reach the interior of the composite. But dueto the low Zn concentration, corrosion phenom-ena are quite limited.
� At Zn compositions above the percolationthreshold (Fig. 12c), the conductance decreasesagain. H+ ions infiltrate and the Zn particles arecorroded due to their high concentration (Fig.13). After a long period of time the acid solutioninfiltrated all the bulk. Since no Zn cluster arefound, conduction due to electrons ceases.
� For compositions near the percolation threshold(Fig. 12b), conductance becomes stable since theinfiltration of the acid solution is compensated forby the formation of insulating Zn based species.
The hypothesis concerning the infiltration of acidsolution into the bulk is confirmed in Fig. 13a–c. For aZn/PPS (30/70) working electrode, these micrographsshow the interface between the parts of the electrodewhich is in the acid solution and that which is not. Thepart immersed in the acid solution has been attacked bycorrosion. After corrosion the Zn particles (white inFig. 13) are replaced by cavities into which the acidsolution can easily penetrate.
Fig. 11. Nyquist plots for a series of Zn/PPS composites as a function of corrosion time. (a) 10/90; (b)15/85; (c) 20/80.
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Fig. 12d shows that close to the inflexion thresholdand for a fixed time t, the increase of ��� is very sharp.As the electron currents circulating through the com-posite electrodes are in series with the ionic currentsmigrating through the acid solution, such a strongvariation can be ascribed to the appearance of a corro-sion current I (defined as I=U.���).
To describe better the correlation between the electri-cal percolation composition (�perc=0.19 for the Zncomposites) and the percolation composition linked to
corrosion currents, we have determined the time depen-dent critical exponents s and from the conductancecurves obtained after corrosion. These critical expo-nents were calculated for each working time t, and afterhaving determined the corresponding percolationthresholds, �corr. The (s, ) couples and the correspond-ing �corr values were found to be:
�corr=�perc=0.19; (1.45�0.24; 2.07�0.12) for t=0 min,�corr=0.17; (1.31�0.17; 1.40�0.09) for t=30 min,�corr=0.15; (0.68�0.18; 0.26�0.13) for t=100 min�corr=0.14; (0.65�0.16; 0.18�0.21) for t=400min.It is interesting to note that these s and values
decrease as the corrosion time increases. The interpreta-tion of this evolution is not clear. In such granularcomposites, several phenomena could be considered.Due to corrosion (see Fig. 13), the metallic particles arefirst dissociated into ions that can migrate through theaqueous acidic solution; then, this dissociation involvesthe formation of additional cavities into which theaqueous solution can infiltrate increasingly. Finally, thecomposite medium transforms into a more complexfour-phase system (ie metal, PPS, cavities and solution):at this stage, the classical two-phase percolation theoryprobably fails. From these results, we can conclude thatthere is a satisfactory correlation between the percola-tion of ‘electron currents’ and the percolation of thecurrents due to corrosion. However, it should be re-marked that �corr is somewhat smaller than the initial�perc value. Even though the correlation is not perfect,the percolation paths for acid solutions are well corre-lated with the percolation of metal circuits in the com-posites. In other terms, the residual cavities that canfacilitate corrosive solution penetration are mainly as-sociated with metal particles.
5. Discussion and conclusion
It has been demonstrated already [39] that the effec-tive medium approximation will generally fail in granu-lar materials. In this paper, we have proposed amodified model that should improve strongly the agree-ment between experimental and calculated data. InTable 3, the main results of this paper are gathered. Itwas shown clearly that the introduction of several di, �i
values and �i linked to the metal grain size improvesthe agreement between calculated and experimental log-arithmic conductances. In other terms, a distribution ofconnectivity parameters associated with different sizesshould be at the origin of the variation of � close to thepercolation threshold. The proposed model allowed usto determine a set of different values for di and �i.
The evaluation of the percolation exponents and spermits us to better define the values of �c. A marked
Fig. 12. Electrical conductance of Zn/PPS composites. Conductance� in logarithmic scale. (a) 10/90; (b) 20/80; (c) 30/70 as a function oftime; (d) as a function of Zn composition for various working times.
E. Thommerel et al. / Materials Science and Engineering A328 (2002) 67–7978
Fig. 13. SEM (back scattered electron) micrographs of a Zn/PPS (30/70) composite electrode. (a) magnification 150; (b) magnification 350; (c)magnification of delimited section. The section of the electrode out of the acid solution is not attacked (the Zn particles are white). The sectionimmersed in the acid solution is corroded (absence of white particles, presence of black cavities). The arrows show some cavities.
correlation between particle size, percolation composi-tion �perc, percolation exponents s and and materialhardness can be noted. However, a small discrepancypersists close to the percolation threshold: it is mainlydue to the fact that the EMA approach cannot accountfor the probability of creating percolation pathsthrough the sample. The calculated curves obtainedeither from our modified EMA approach or from per-colation theory, both fit the data in a satisfactory way.Nevertheless, the correlation defined as:
R=
��n
1
(Experimental-Calculated)2/(Calculated)2
n−1(6)
shows that percolation theory always fits better the
systems than the EM model (see Figs. 9 and 10). Themain interest of the present EM model resides in thefact that microstructural parameters can be taken intoaccount directly, thus allowing a better description ofthe real heterogeneous material. This approach mightbe improved by increasing the number of di values thatare associated with the grain size distribution.
Acknowledgements
The authors gratefully acknowledge Professor J.P.Clerc (IUSTI, Marseille, France) for helpful discussionsduring this study. They also thank the ‘Conseil Re-gional Provence Alpes Cote d’Azur’ for financialsupport.
E. Thommerel et al. / Materials Science and Engineering A328 (2002) 67–79 79
Appendix A. Percolation threshold and inflexion pointdetermination
Eq. (1) gives
�=(d�1−1)�1+ (d�2−1)�2��1/2
2(d−1)where
�= (d�1−1)2�12+ (d�2−1)2�2
2
+2(d−1+�1�2d2)�1�2 with �1+�2=1.
The calculation ofd2(log�)
d�12 =0 presents only one
solution:
�infl.= −−�1−�2+�2d
(�1−�2)dor �infl.
=−�2
�1−�2
+�1+�2
(�1−�2)dIn the case of an insulator/metal composite, �2 isnegligible as compared to �1. The solution is then:
�infl.=1d
, where �infl. is the inflexion threshold of the
(log�,�) representation. The �c value can be deter-
mined from the maximum ofd2(log�)
d�12 , which is very
close to the �perc value.
References
[1] R. Chaim, J. Mater. Res. 12 (1997) 1828–1835.[2] L. Tortet, J.R. Gavarri, J. Musso, G. Nihoul, J.P. Clerc, A.N.
Lagarkov, A.K. Sarychev, Phys. Rev. B 58 (1998) 5390–5407.[3] A. Kopia-Zastawa, J.R. Gavarri, I. Sulliga, M.A. Fremy, M.H.
Pishedda, S. Jasienska, J. Sol. State. Chem. 145 (1999) 317–326.[4] J.R. Gavarri, L. Tortet, J. Musso, Eng. Mater. Trans. Tech.
Pub. 155-156 (1998) 447–467.[5] L. Tortet, J.R. Gavarri, G. Nihoul, Sol. State. Ionics 89 (1996)
99–107.[6] A. Benlhachemi, J. Musso, J.R. Gavarri, C. Alfred-Duplan, J.
Marfaing, Phys. C 230 (1994) 246–254.[7] A. Benlhachemi, J.R. Gavarri, Y. Massiani, J. Marfaing, Ann.
Chim. Sc. Mater. 20 (1995) 335–344.[8] A. Benlhachemi, M.A. Fremy, C. Breandon, H. Tatarenko, J.R.
Gavarri, H. Benyaich, J. Phys. IV, 8, Pr 1, May 1998[9] O. Levy, D. Stroud, J. Phys. Condens. Matter 9 (1997) 599–605.
[14] C. Alfred-Duplan, J. Musso, J.R. Gavarri, C. Cesari, J. Sol.State. Chem. 110 (1993) 6.
[15] A. Malliaris, D.T. Turner, J. Appl. Phys. 42 (1971) 614–618.[16] S. De Bondt, L. Froyen, A. Deruyttere, J. Mater. Sci. 27 (1983)
1983–1988.[17] K. Miyasaka, K. Watanabe, E. Jojima, H. Aida, M. Sumita, K.
Ishikawa, J. Mater. Sci. 17 (1982) 1610–1616.[18] D. Bouvard, F.F. Lange, Acta. Metall. Mater. 39 (1991) 3083–
3090.[19] J.C. Dyre, Phys. Rev. B 48 (1993) 15511–15526.[20] D.S. McLachlan, A. Priou, I. Chenerie, E. Issac, F. Henry, J.
Elec. Waves. Applic. 6 (1992) 1099–1131.[21] S. Kirkpatrick, Rev. Mod. Phys. 45 (1973) 574.[22] F. Zhongyun, Phil. Mag. A 73 (1996) 1663–1684.[23] A. Rahman, C.W. Bates, A. Marshall, S. Qadri, T. Bari, Matt.
Lett. 39 (1999) 343–349.[24] C. Alfred-Duplan, J. Marfaing, G. Vaquier, A. Benlhachemi, J.
Musso and J.R. Gavarri, Mater. Sci. Eng. B, 29, 1996[25] D.A.G Bruggeman, Ann. Phys. 24 (1935) 636.[26] D. Stauffer, Introduction to Percolation Theory, Taylor & Fran-
cis, London, 1985.[27] D.J. Bergmanar, D. Stroud, Sol. State. Phys, in: H. Ehrenreich,
D. Turnbull (Eds.), Advances in Research and Applications,Academic, San Diego, CA, 1992.
[28] J.F. Gouyet, Physique et Structures Fractales, Masson, Paris,1992.
[29] R.P. Kussy, J. Appl. Phys. 70 (1977) 5301.[30] F. Carmona, P. Prudhon, F. Barreau, Solid State Commun. 51
(1984) 255.[31] G.E. Pike, In Electrical Transport and Optical Properties of
Inhomogeneous Media, Ref. 2, p. 306.[32] N. Deprez, D.S. McLachlan, J. Phys. D 21 (1988) 101.[33] J. Wu, D.S. McLachlan, Phys. Rev. B 56 (1997) 1236–1248.[34] D.M. Grannan, J.C. Garland, D.B. Tanner, Phys. Rev. B 33
(1986) 904.[35] H.J. Hermann, B. Derrida, J. Vannimenus, Phys. Rev. Lett. 53
(1984) 1121.[36] X. Derrida, B. Stauffer, D. Herrmann, J. Vannimenus, J. Phys.
Lett. Paris 44 (1983) L701.[37] R.M. Ziff, G. Stell, Phys. Rev. Lett, LaSC (Ann Arbor, Michi-
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