Power-law rheology behavior of soft matter with surface tension under spherical relaxation and creep indentations

Indentation serves as a prevalent experimental approach for characterizing the viscoelasticity of soft biological cells. However, at the micro- and nano-scale, surface tension can significantly alter the contact response, leading to a misestimation of mechanical properties when conventional contact models are adopted. In this study, we investigate the spherical step-hold relaxation and creep responses of a power-law rheology viscoelastic half-space with surface tension. The results demonstrate that surface tension would elevate the apparent relaxation modulus and reduce both the apparent creep compliance and the apparent power-law exponent relative to their true material properties. Furthermore, surface tension alters the time-dependent evolution of the contact radius, deviating from the classical Hertzian relation. Explicit relations linking the extracted apparent viscoelastic parameters to the indentation depth (or applied load) are established. These relations reveal that neglecting surface effects results in a markable overestimation of the referenced modulus and an underestimation of both the referenced compliance and power-law exponent, particularly at shallow indentation depths or low loads. Based on these findings, we propose a robust two-step identification strategy to simultaneously and accurately determine the true viscoelastic parameters (referenced modulus and power-law exponent) and the surface tension, applicable to both relaxation and creep indentations. This work establishes a refined framework for processing indentation data on viscoelastic solids, leading to improved accuracy in the mechanical characterization of biological cells and analogous soft matter systems.