Continuous Percolation in Organic Conducting Blends

J. Planes1Þ, S. Bord, and J. Fraysse

Laboratoire de Physique des Metaux Synthetiques, Departement de Recherche Fondamentalesur laMatiere Condensee, UMR 5819 CEA-CNRS-Universite Joseph Fourier,CEA-Grenoble, 17 rue desMartyrs, F-38054Grenoble cedex 9, France

(Received September 6, 2001; accepted October 4, 2001)

Subject classification: 72.80.Le; 72.80.Ng; 72.80.Tm; S12

Electronic transport properties of organic conductive blends made of doped poly(aniline) and con-ventional insulating matrices exhibit non-standard dependences on temperature T and conductingphase contents p. We propose a new analysis of conductivity data that accounts for the interplay ofmicroscopic mechanisms – essentially hopping – and percolation effects. The continuous percola-tion scheme allows us to understand the T dependence of the conductivity exponent of percola-tion. This in turn is responsible for the apparent p dependence of the hopping parameters. It ispossible to derive these effective parameters from those of the pure poly(aniline). We thus end upwith a closed form formula to fit the whole set of conductivity data s p;Tð Þ. The consequences ofthis analysis are discussed.

Introduction Numerous experimental systems exhibit a non conventional electricalpercolation behavior [1], meaning that the percolation scaling law

s / p� pcð Þt ð1Þ

with s the conductivity, p the volumic fraction of the conductive phase (expressed in %throughout the paper) and pc its critical value, is obeyed with exponent t differing fromits theoretical or computed value [2]. Among them, conducting organic blends are ofparticular interest because this effect can be studied in a large temperature range. Inpoly(aniline)/poly(methyl methacrylate) (PANI/PMMA) blends, it was shown [3] thatthe critical fraction (or percolation threshold) pc remains constant over the range of 10to 300 K, whereas t diverges approximately like

t ¼ a þ bT�1 : ð2Þ

Such a divergence is consistent with the continuous percolation model [3] under theassumption that the distribution of local conductances diverges for low values, thedivergence being stronger at low T. Moreover, the geometrical percolation networkremains ‘stable’ as T varies (conversely to a ‘fuse’ network where p (and thus pc)changes with T because of differential dilation of components for instance (PTCeffect)). However no connection was made with the thermal dependence of s at givenp, which follows a generalized hopping law

ln s / � T0=Tð Þg ð3Þ

at least for low temperatures. In particular the p dependence of exponent g was notelucidated: was it the sign of a change in conduction mechanism with dilution? In the

phys. stat. sol. (b) 230, No. 1, 289–293 (2002)

# WILEY-VCH Verlag Berlin GmbH, 13086 Berlin, 2002 0370-1972/02/23003-0289 $ 17.50þ.50/0

1) Corresponding author; e-mail: jplanes@cea.fr

following, we address this question with the analysis of new data measured for a verysimilar system.

Experimental PANI is doped with 1,2-benzenedicarboxylic acid, 4-sulfo, 1,2-di-(2-ethylhexyl) ester (DEHEPSA) and blended with plasticized PMMA as described inRef. [4]. PANI content p is defined as the fraction of emeraldine base in the sample. Dcconductivity sðp;TÞ of the films (a few tens of mm thick) is measured by a standardfour probe technique between liquid helium and room temperature in a continuousflow cryostat. Electrical contacts are improved by evaporation of gold pads and theOhmic behavior is carefully checked.

Results Figure 1 displays the raw conductivity data for the whole set of samples withPANI contents ranging from p ¼ 0:13% to 100%. As already observed for camphorsul-fonic acid doped PANI blends [3, 5], a maximum is reached at Tm, which continuouslydecreases as p increases. Above Tm, ds=dT < 0 is indicative of a metallic behavior. Foreach value of p, data are correctly fitted by a series combination of quasi-1D metallicand hopping resistances. Exponent g (Eq. (3)) takes the following values: 0.91, 0.88,0.83, 0.81, 0.76, 0.74, 0.31 for p ¼ 0:13%, 0.3%, 0.45%, 0.50%, 1.0%, 1.5%, 100%.The percolative behavior is shown in Fig. 2. Log–log plots of s versus p� pc with

pc ¼ 0:06% are displayed as inset for T ¼ 310 and 10 K. The slope of the curvesgives exponent t of Eq. (1), which is reported versus T in the experimental range inFig. 2. The continuous variation of tðTÞ follows Eq. (2) with a ¼ 1:98� 0:01,b ¼ ð10:15� 0:15Þ K.

Global analysis of s(p, T) In standard lattice percolation, any thermal dependence ofmacroscopic s is given by the prefactor, omitted in Eq. (1), and related to the conduct-ing phase conductance. Rewriting Eq. (1) for continuous percolation ass p;Tð Þ ¼ s0 Tð Þ p� pcð ÞtðTÞ, one sees, since t is T-dependent, that s and s0 thermaldependences are not identical. Consequently, the shape of s0 Tð Þ, extracted from experi-

mental data, depends on the choice of the percolation variable: p� pc,p� pc100� pc

,p� pcpc

,

290 J. Planes et al.: Continuous Percolation in Organic Conducting Blends

Fig. 1. Thermal dependence of dc conductivityfor PANI(DEHEPSA)/PMMA blends. In thisrepresentation, the maximum is very flat. It ishowever clearly determined and occurs at Tm,whose values range from 277 K at p ¼ 0:13%to 243 K at p ¼ 1:5% and 200 K at p ¼ 100%

or more generallyp� pc

cwith a numeric constant c. For the present data, it appears

that for c ’ 10, s0 Tð Þ is very close to the experimental thermal dependence measuredin pure PANI: sPANI Tð Þ s 100ð %;TÞ in absolute values. For other c values, s0 andsPANI do not coincide neither in absolute values nor in shape. We have thus checkedthat the complete data set could be represented by

s p;Tð Þ ¼ sPANI Tð Þ p� pcc

� �aþb=Ty : ð4Þ

phys. stat. sol. (b) 230, No. 1 (2002) 291

Fig. 2. Variation of exponent t (Eq. (1)) with T .The solid line is a fit to Eq. (2) witha ¼ 1:98� 0:01, b ¼ ð10:15� 0:15Þ K. Inset: log–log plots of s vs p� pc , pc ¼ 0:06% at T ¼ 310and 10 K, from which values of t are computed

Fig. 3. Experimental data (points) plotted as ln q (q ¼ s�1) fitted to the 2D surface defined byEq. (4) and parameters a ¼ 2:0, b ¼ 10:25 K, pc ¼ 0:059% and c ¼ 10:76

The fitting procedure has been applied to y p;Tð Þ ¼ a þ bT�1� �

lnp� pc

c

� �, where

y p;Tð Þ ¼ lns p;Tð Þ

sPANI Tð Þ. The free parameters are a, b, pc and c. The result is shown in

Fig. 3. Experimental points are displayed as ln s�1 p;Tð Þ. The surface represents theanalytical function with a ¼ 2:0, b ¼ 10:25 K, pc ¼ 0:059% and c ¼ 10:76.

Discussion According to this analysis, the respective contributions of microscopic me-chanisms and percolation to the transport properties of PANI blends can be separated.The possibility to identify the prefactor in the percolation scaling law to the conductiv-ity of the conducting component means that the dilution process does not affect themicroscopic mechanisms. The most commonly accepted description of sulfonic aciddoped PANI relies on its electronic heterogeneity [6], correlated to chemical [7] andstructural [8] heterogeneity. Lengthscales involved in the problem are in the nanometerrange (grain size around 10 nm) compatible to those of granular hopping models [9].One should thus suppose that the ‘electronic granularity’ is an intrinsic property ofPANI (for this class of dopants and their appropriate solvents), kept in the mixingprocess despite the very high dilution level. This is understood via the viscoelasticphase separation model [10]. Due to the moderate molecular weight of PMMA, theinitial co-solution is a pseudo-binary system PANI–(PMMA+solvent). The harder min-ority phase PANI supports an enhanced stress during the solvent flow. The transientgel, which, in pure PANI, is observed in the initial stages of evaporation, but collapsesin the solid state, is retained in the blend, because of PMMA. In the solid state, theresulting PANI network is evidenced by dynamical mechanical analysis, demonstratinga mechanical percolation [11]. At small scales, microstructural imaging by transmissionelectron microscopy [12, 13], atomic force microscopy with mechanical [13] or electrical[14] contrast, proves that the bonds of the percolating network are – at least partly –constituted by PANI bundles, with sharp boundaries and typical sizes of a few tens ofnanometers. On this support, the continuous percolation scheme explicited in [3] can beapplied without the former assumption that the ‘grain connectivity’ is reduced asp ! pc.We should mention that Eq. (2) is only an empirical representation of experimental

data. For the present samples, a is accidentally equal to 2.0 and we don’t have anyinterpretation for b at the moment. Parameter c acts as a renormalisation factor forPANI mass fraction p. Theoretical percolation deals with bond fraction x. Let us as-sume a proportional relationship between them: x ¼ lp. The exact percolation variablex� xc100� xc

¼ p� pc100=l � pc

, thus c ¼ 100=l � pc and xc ¼ lpc ¼100pccþ pc

’ 0:55%. This very

low percolation threshold is characteristic of a strongly anisotropic system.As a conclusion, we have shown that the transport properties of PANI blends can be

analyzed by a continuous percolation model; the anomalous behavior of conductivityexponent t is responsible for the apparent anomalous thermal dependence of conductiv-ity, which is actually governed by the same mechanisms than the pure PANI phase.

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