Conjugation length dependent transport in conducting polymers from a resistor network model.

The effect of conjugation length (L) upon electronic conductivity components of conducting polymers is derived using a generalized resistor network model. Results are obtained for polymers which contain a statistical distribution of defects which limit conjugation, as well as for regular copolymers which have a fixed phase relationship between interruptions in conjugation on neighboring chains. The short-conjugation-length limits of the derived equations are identical with those previously obtained by evaluating molecular aspects of charge transfer. More specifically, when interchain transport fully limits both chain-direction conductivity (σ1) and an orthogonal conductivity (σ2), the calculated electrical anisotropy is σ1/σ2=L2/6d2F, where d is the interchain separation in a hopping direction, and the product d2F is the mean square average projection of the interchain vector on the electric field direction. The present analysis extends predictive capabilities over the entire range from short conjugation lengths to infinite conjugation lengths. For long conjugation lengths terminated by effectively infinite barrier defects, σ1/σ1(∞) and σ2/σ2(∞) are calculated from the parameters which define polymer structure and, for the former ratio, the ratio of infinite chain conductivities parallel [σ1(∞)] and orthogonal [σ2(∞)] to the chains. A general relationship, appropriate for a still wider range of conjugation lengths, is derived between σ1/σ1(∞) and [σ2/σ1(∞)]1/2(L/d)/F1/2, where the geometrical parameter F is of order unity in directions of high σ2. Using this relationship for a polymer of known structure, the chain-direction electrical conductivity in the infinite-chain limit can be derived from measurements of σ1, σ2, and average conjugation length. Good agreement is obtained between the calculated and observed dependence of... [ABSTRACT FROM AUTHOR]