Spectral Analysis of Brownian Motion with its Rheological Analogues

Abstract

The power spectrum of the Brownian motion of probe microparticles with mass m and radius R immersed in a viscoelastic material reveals valuable information about repetitive patterns and correlation structures that manifest in the frequency domain. In this paper, we employ a viscous viscoelastic correspondence principle for Brownian motion and we show that the power spectrum of Brownian motion in any linear, isotropic viscoelastic material is proportional to the real part of the complex dynamic fluidity of a linear rheological network that is a parallel connection of the linear viscoelastic material within which the Brownian particles are immersed and an inerter, with distributed intrance with mass mR. The synthesis of this rheological analogue simplifies appreciably the calculation of the power spectrum for Brownian motion within viscoelastic materials such as Maxwell fluids, Jeffreys fluids, subdiffusive materials, or in dense viscous fluids that give rise to hydrodynamic memory.

Figures (11)

• Figure 1: Normalized power spectra of Brownian motion within a Maxwell fluid with a single relaxation time $\eta/ G$, for different values of the dimensionless parameter $\omega_R \tau = \frac{1}{\eta}\sqrt{\frac{Gm}{6\pi R}}$ as a function of the dimensionless frequency $\omega \tau = \omega \frac{m}{6\pi R \eta}$. As the stiffness of the in-series spring increases (large $G$ or large $\omega_R \tau$), the spectra converge to the power spectrum of Brownian motion within a memoryless, viscous fluid $(\frac{1}{1+\omega^{2}\tau^{2}})$.
• Figure 2: Inertoviscous fluid which is a parallel connection of an inerter with distributed inertance $m_R$ with units $[M][L]^{-1}$ and a dashpot with viscosity $\eta$ with units $[M][L]^{-1}[T]^{-1}$. In analogy with the traditional schematic of a dashpot that is a hydraulic piston, the distributed inerter is depicted schematically with a rack--pinion--flywheel system.
• Figure 3: The inertoviscoelastic solid which is a parallel connection of a linear spring with elastic shear modulus $G$, a dashpot with shear viscosity $\eta$, and an inerter with distributed inertance $m_R$.
• Figure 4: Normalized power spectra of Brownian motion of particles suspended in a Kelvin solid (harmonic trap) for different values of the dimensionless parameter $\omega_R \tau = \frac{1}{\eta}\sqrt{\frac{G m}{6\pi R}}$ as a function of the dimensionless frequency $\omega \tau = \omega \frac{m}{6\pi R \eta}$.
• Figure 5: Rheological analogue for Brownian motion in a Maxwell fluid. It consists of the Maxwell element with shear modulus $G$ and shear viscosity $\eta$ that is connected in parallel with an inerter with distributed inertance $m_R = \frac{m}{6\pi R}$.