Anomalous electrical conductivity and percolation in carbon

Anomalous electrical conductivity and percolation in carbon...

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LETTER Anomalous electrical conductivity and percolation in carbon nanotube composites Chunsheng Lu Æ Yiu-Wing Mai Received: 18 October 2007 / Accepted: 6 March 2008 / Published online: 8 August 2008 Ó Springer Science+Business Media, LLC 2008 Transport properties of disordered multiphase materials, such as electrical and thermal conductivities, have been an active research area in statistical physics for decades. In a composite consisting of conductive Fllers dispersed in an insulating matrix, there is a well-deFned insulator– conductor transition when an inFnite conductive network or path throughout the matrix is formed. This process can be well described by percolation theory [ 1 , 2 ]. Recently, carbon nanotube (CNT)-reinforced composites and sus- pensions have attracted a great deal of attention due to their excellent properties and many potential applications. CNTs have a unique set of mechanical and physical properties, including extremely high Young’s modulus, strength, electrical, and thermal conductivities. The current experi- ments showed that CNT-reinforced composites exhibit an electrical percolation with addition of 0.1 vol.% or less Fllers, at which electrical conductivity rises sharply by several orders of magnitude [ 3 7 ]. Here, the percolation threshold of CNTs is closely dependent on their geometric factors (e.g., volume fraction, size, shape, and orientation) and the interaction between them. It is a critical issue in producing conductive composites for use in Flms, coatings, and paints since the lower percolation threshold can reduce the loading of expensive CNTs, leading to lighter com- posites. In comparison with composites reinforced with isotropic particles, however, the percolation threshold of composites containing highly anisotropic conductor Fllers such as CNTs is still not well understood. The term ‘‘percolation’’ refers to the onset of a sharp transition or an inFnite network (or cluster) at which long- range connectivity suddenly appears [ 1 ]. The electrical conductivity near the percolation threshold is anomalously greater than that predicted by traditional theoretical models, such as Maxwell, Hamilton-Crosser models [ 8 ]. Intuitively, the electrical percolation process in CNT composites is similar to traditional ones with the addition of isotropic conductive particles, but with an ultra-low percolation threshold. As shown in ±ig. 1 , near the percolation thresh- old, the probability or fraction of a CNT, P , on the inFnite cluster obeys a power law and can be described as P ±½ / ÿ / c ð a Þ² b ð a Þ ; ð 1 Þ where / is the volume fraction of CNT Fllers (in vol.% or wt.%), / c is the percolation threshold, and b is the con- nectivity exponent. Similarly, the electrical conductivity, r , in the system increases monotonically for / [ / c but ð / ÿ / c Þ³ 1 and follows a universal power law r / ÿ / c ð a Þ² t ð a Þ ; ð 2 Þ where t is the conductivity exponent. Here, both the per- colation threshold and connectivity (or conductivity) exponents are a function of aspect ratios of CNTs, a = L / D , with L and D being length and diameter, respectively. It is
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Anomalous electrical conductivity and percolation in carbon...

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