Viscous-viscoelastic correspondence principle for Brownian motion

Area of Science:

• Physics
• Materials Science
• Rheology

Background:

• Classical Brownian motion analysis relies on Newtonian fluids or harmonic potentials.
• Viscoelastic materials present complex dynamics due to time-dependent responses.
• Inertia effects become significant in Brownian motion within viscoelastic media at longer timescales.

Purpose of the Study:

• To establish a generalized viscous-viscoelastic correspondence principle for Brownian motion.
• To simplify the calculation of mean-square displacement and velocity autocorrelation functions.
• To account for non-negligible inertia effects in Brownian particles within viscoelastic materials.

Main Methods:

• Derivation of the mean-square displacement for Brownian microspheres in viscoelastic materials.
• Development of a mechanical network model involving parallel connection of viscoelastic material and an inerter.
• Introduction of distributed inertance (m_R = m/6πR) within the mechanical network.

Main Results:

• The mean-square displacement in viscoelastic materials is identical to the creep compliance of the synthesized mechanical network.
• A novel viscous-viscoelastic correspondence principle for Brownian motion is established.
• This principle simplifies calculations, particularly when inertia is significant.

Conclusions:

• The established principle provides a powerful tool for analyzing Brownian motion in complex viscoelastic fluids.
• The concept of the inerter is key to bridging viscous and viscoelastic behaviors.
• The findings are equivalent to previously established viscous-viscoelastic analogies.