Efeitos inerciais no Movimento Browniano de partículas imersas em meios viscoelásticos de Maxwell e Kelvin-Voigt

Abstract

Rheological behaviors are found in all materials, from brittle to ductile ones, from flexible to rigid ones, in living and inert systems, and are related to other physical and chemical behaviors of matter. The relationship between physical properties and chemical activities in cells, for instance, has gained interest in the literature for providing a new approach in the study of diseases. The way cells respond to external mechanical stimuli influences a number of biological functions, such as motility and division, and can be used to distinguish between healthy and abnormal cells. Experimental measurement and theoretical understanding of such mechanical properties, however, are not trivial in these microscopic systems due to the complexity of the components they are made of. One of the most used experimental techniques in these studies is the so-called Particle-tracking Microrheology, where small fluorescent beads are inserted into the material to perform Brownian motion. The way these beads spread out through the medium allow us to obtain those mechanical properties in the interior, in contrast with other techniques where mechanical measurements are performed on the surface of the material, such as the Atomic Force Microscopy. Since most biological materials (and other soft materials) hold viscoelasticity, our work aims to describe the diffusion of Brownian particles immersed in viscoelastic media when the system is in thermodynamic equilibrium and when the inertial contribution to the movement becomes relevant, so that the results obtained can be used in experimental studies. We then use the Generalized Langevin Equation, which describes Brownian motion in friction memory media, along with known rheological models, such as Kelvin-Voigt and Maxwell. The velocity correlation function, the time-dependent diffusion coefficient and the mean squared displacement are then obtained analytically for these types of viscoelastic medium, where we studied different damping regimes, namely, overdamping (when viscous forces dominate the motion), underdamping (when elastic forces dominate), and the critical damped regime (when the elastic and viscous forces are of the same intensity), and we discuss the limit behaviors in each case.