Electrical properties of random checkerboards at finite scales

Articles you may be interested in Randomized scalable checkerboard geometries: The electrostatic problem Effect of lanthanum doping on the electrical properties of solgel derived ferroelectric lead–zirconate–titanate for ultralargescale integration dynamic random access memory applications Under investigation is the scale dependent electrical conductivity (and resistivity) of two-phase random checkerboards at arbitrary volume fractions and phase contrasts. Using variational principles, rigorous mesoscale bounds are obtained on the electrical properties at finite scales by imposing a boundary condition that is either uniform electric potential or uniform current density. We demonstrate the convergence of these bounds to the effective properties with increasing length scales. This convergence gives rise to the notion of a scalar-valued scaling function that accounts for the statistical nature of the mesoscale responses. A semi-analytical closed form solution for the scaling function is obtained as a function of phase contrast, volume fraction and the mesoscale. Finally, a material scaling diagram is constructed with which the convergence to the effective properties can be assessed for any random checkerboard with arbitrary phase contrast and volume fraction. C 2015 Author(s). All article content , except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License.