In this dissertation, a multiscale moving contact line (MMCL) theory is proposed, to simulate liquid droplet spreading, capillary motion and in particular, to study cell motility on the extracellular matrix. The proposed multiscale moving contact line theory combines a coarse-grained contact model (CGCM) with a generalized Gurtin-Murdoch surface elasticity theory, so that it can couple the molecular scale adhesive interaction with the macroscale motion. The intermolecular adhesive force (van der Waals force) separates and levitates the liquid droplet from the supporting solid substrate, such that the proposed MMCL theory can avoid the singularity problem caused by the no-slip condition in the conventional hydrodynamics of moving contact line theory.

The proposed MMCL theory is formulated as a variational principle and implemented within a Lagrangian finite element method. Two different implementations with different ways of calculating the contact/adhesion forces, regarding the computational efficiency, are proposed. Numerical examples are presented to illustrate the applicability of the MMCL theory. Several simulations of complete three-dimensional water droplet spreadings upon various elastic substrates are performed. The numerical results are in good agreement with those of molecular dynamics simulations and experiments reported in literature. In addition, the capillary motion around a spherical cap is captured using the MMCL theory. The contact model (CGCM) used in the MMCL theory is initially designed to simulate contact and adhesion at nano or submicro scale. By using a second level coarse graining technique, it can be employed to simulate problems at meso or even macro scales and thus make the MMCL theory available to simulations of cell motility, which is usually at a scale ranging from ten micrometer to several millimeter. At last, the MMCL theory is used to study the interactions between cells and their extracellular matrices. In our framework, a cell is modeled as Nematic liquid crystal or liquid crystal elatomer and the extracellular matrix is treated as an elastic substrate. Cell spreading upon various extracellular substrates are successfully simulated, aiming to improve the understanding of the mechanotransduction mechanism that is in charge of mechanical information exchange between a cell and its surrounding environment. Through the numerical simulations, it is demonstrated that the cell can sense the substrate elasticity in many different ways. In fact, together with a proposed scheme that resembles a linkage between the traction forces and cell substrate elasticity based on experimental observations, self-propelled movement of a cell on the gradient of substrate elasticity (called durotaxis) is successfully captured.