A Strategy for Dimensional Percolation in Sheared Nanorod Dispersions**

As the aspect ratio of a unit (here, a stiff high-aspect-ratio rod) increases, the tendency increases for ensembles of these units to form hierarchical structures on scales quite removed from that of the single unit (particles). This is due in large part to geometrical packing considerations (based on excluded volume analysis) arising from the non-spherical shape. Nematic ordering of rod ensembles is one illustration of this phenomenon, and percolation of rod or platelet inclusions is another. It happens that for thin rods, equilibrium percolation of isotropic distributions occurs well below the nematic transition. The introduction of a flow-induced distribution function leads one out of the highly studied domain of universal scaling laws for equilibrium, isotropic percolation thresholds. For materials processing applications, one is compelled to explore the impact of volume fraction of particles beyond the threshold of any desirable percolation-induced property transition and ascertain the role of anisotropy on macroscopic property tensors. This is required for practical applications in order to avoid statistical fluctuations across the preferred side of the threshold. There is a large quantity of literature on the onset of percolation among rods or “sticks” in two and three space dimensions, including seminal contributions of Balberg and collaborators, Bug et al., Munson-McGee, Philipse, Celzard et al., Neda et al.. A key factor in all rigorous percolation analyses of thin rod ensembles is that the orientational distribution function can be approximated by a classical equilibrium distribution (random, Gaussian, Boltzmann, or a special separable form in spherical coordinates). Percolation thresholds for these special equilibrium distributions are intimately connected with universal scaling laws and self-similarity, and the scale-invariant nature of the behavior at threshold. These fundamental scaling laws have played a dominant role in providing the basis for estimates of property percolation thresholds due to spanning networks formed by percolating rod clusters; we note the recent analysis of Hu et al. on nanowires in a poorly conducting medium. The thresholds are statistical in nature; for example, at some critical volume fraction, there is a finite probability that a spanning network cluster will form, and the cluster features will depend on the underlying lattice construct (e. g. triangular lattice, Bethe lattice, continuum, etc.). Anisotropic percolation arises when the percolation threshold depends on spatial direction, which in nanorod composites arises due to processing history. In this paper, the departure from universality and self-similarity of the percolation thresholds is explored, associated with non-classical and non-equilibrium rod distribution functions arising from an imposed shear flow. There are no theoretical scaling results to rely upon for sheared distributions. Yet there is mounting evidence that a combination of shear rate, rod volume fraction and aspect ratio determine percolation thresholds in stiff-rod composites (Ounaies et al., Xu et al., Lahiff et al.). Although never completely characterized, the orientation distributions of the rods in these systems are probably non-classical and are far from equilibrium due to processing procedures. Thus, ascertaining the interrelationships between processing (shear rate) and composite composition (volume fraction and aspect ratio of rods) is not straight forward. It should be noted that experimentally, percolation thresholds are often inferred by measuring sharp jumps in conductivity, rather than by actual rod-rod contact percolation. This identification introduces an uncontrollable error into contact percolation prediction since property percolation (e. g. conductivity) occurs prior to contact percolation via processes such as electron tunneling and carrier hopping. This effect, which is dependent on details of the particular structure, also leads to non-universality of critical exponents in conductive percolation scaling laws (Grujicic et al., Vionnet-Menot et al., Heaney). Clearly, there is a gap between universal scaling laws derived from rigorous percolation analysis and the percolation C O M M U N IC A TI O N