The classical approach to model collective biological cell movement is through coupled nonlinear reaction-diffusion equations for biological cells and diffusive chemicals that interact with the biological cells. This approach takes into account the diffusion of cells, proliferation, death of cells, and chemotaxis. Whereas the classical approach has many advantages, it fails to consider many factors that affect multicell movement. In this work, a multiscale approach, the Glazier-Graner-Hogeweg model, is used. This model is implemented for biological cells coupled with the finite element method for a diffusive chemical. The Glazier-Graner-Hogeweg model takes the biological cell state as discrete and allows it to include cohesive forces between biological cells, deformation of cells, following the path of a single cell, and stochastic behavior of the cells. Where the continuity of the tissue at the epidermis is violated, biological cells regenerate skin to heal the wound. We assume that the cells secrete a diffusive chemical when they feel a wounded region and that the cells are attracted by the chemical they release (chemotaxis). Under certain parameters, the front encounters a fingering morphology, and two fronts progressing against each other are attracted and correlated. Cell flow exhibits interesting patterns, and a drift effect on the chemical may influence the cells' motion. The effects of a polarized substrate are also discussed.
We introduce a framework for simulating the mesoscale self-assembly of block copolymers in arbitrary confined geometries subject to Neumann boundary conditions. We employ a hybrid finite difference/volume approach to discretize the mean-field equations on an irregular domain represented implicitly by a level-set function. The numerical treatment of the Neumann boundary conditions is sharp, i.e it avoids an artificial smearing in the irregular domain boundary. This strategy enables the study of self-assembly in confined domains and enables the computation of physically meaningful quantities at the domain interface. In addition, we employ adaptive grids encoded with Quad-/Oc-trees in parallel to automatically refine the grid where the statistical fields vary rapidly as well as at the boundary of the confined domain. This approach results in a significant reduction in the number of degrees of freedom and makes the simulations in arbitrary domains using effective boundary conditions computationally efficient in terms of both speed and memory requirement. Finally, in the case of regular periodic domains, where pseudo-spectral approaches are superior to finite differences in terms of CPU time and accuracy, we use the adaptive strategy to store chain propagators, reducing the memory footprint without loss of accuracy in computed physical observables.
This work presents a general and unified theory describing block copolymer selfassembly in the presence of free surfaces and nanoparticles in the context of Self-Consistent Filed Theory. Specifically, the derived theory applies to free and tethered polymer chains, nanoparticles of any shape, arbitrary non-uniform surface energies and grafting densities, and takes into account a possible formation of triple-junction points (e.g., polymer-air-substrate). One of the main ingredients of the proposed theory is a simple procedure for a consistent imposition of boundary conditions on surfaces with non-zero surface energies and/or non-zero grafting densities that results in singularityfree pressure-like fields, which is crucial for the calculation of forces. The generality of the theory is demonstrated using several representative examples such as the meniscus
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