  The spreading of a cap-shaped spherical droplet of non-Newtonian power-law
liquids on a flat and a spherical rough and textured substrate is theoretically
studied in the capillary-controlled spreading regime. A droplet whose scale is
much larger than that of the roughness of substrate is considered. The
equilibrium contact angle on a rough substrate is modeled by the Wenzel and the
Cassie-Baxter model. Only the viscous energy dissipation within the droplet
volume is considered, and that within the texture of substrate by imbibition is
neglected. Then, the energy balance approach is adopted to derive the evolution
equation of the contact angle. When the equilibrium contact angle vanishes, the
relaxation of dynamic contact angle $\theta$ of a droplet obeys a power law
decay $\theta \sim t^{-\alpha}$ except for the Newtonian and the non-Newtonian
shear-thinning liquid of the Wenzel model on a spherical substrate. The
spreading exponent $\alpha$ of the non-Newtonian shear-thickening liquid of the
Wenzel model on a spherical substrate is larger than others. The relaxation of
the Newtonian liquid of the Wenzel model on a spherical substrate is even
faster showing the exponential relaxation. The relaxation of the non-Newtonian
shear-thinning liquid of Wenzel model on a spherical substrate is fastest and
finishes within a finite time. Thus, the topography (roughness) and the
topology (flat to spherical) of substrate accelerate the spreading of droplet.
