Brownian Motion in the Fluids with Complex Rheology

Abstract

Theory of Brownian motion of a fine particle in a viscoelastic fluid continuum is developed. The rheology of the embedding medium is described in terms of classical structure (spring-and-damper) schemes. It is shown that a great variety of conceivable ramifications of such structures could be reduced to an effective Jeffreys model: a Maxwell chain shunted by a damper. This scheme comprises just three material parameters: two for viscosities (fast and slow) and one for elasticity. For the thermal motion of a particle in such a fluid, the set of Langevin equations is derived. In the cases of 1D translational and 2D orientational (rotation about fixed axis) motions, the time dependencies of mean-square displacements are found analytically. It is shown that in a Jeffreys fluid, the Brownian motion has three regimes: fast diffusion (short times), slow diffusion (long times) and a crossover of those resulting in virtual localization (dynamic confinement) of the particle. Experimental evidence for that taken from the literature is presented. The developed formalism is applied to the particles with "frozen-in" dipole moments, whose orientational motion is a combination of Brownian diffusion and regular excitation by an AC field. The dynamic susceptibilities are calculated for the ensembles of particles liable for either 2D or 3D rotations. Comparison shows that the out-of-phase part of the susceptibility χ ′′ (ω) in 3D case differs considerably from that of 2D case. Function χ(ω) for the forced 3D rotary diffusion in a Jeffreys fluid is found for the first time. In our view, it would be useful for realistic theoretical interpretation of microrheology data and the more so for estimating the particle-mediated AC field-induced heating (magnetic hyperthermia) in viscoelastic fluid media.

Magnetorelaxometry in the Presence of a DC Bias Field of Ferromagnetic Nanoparticles Bearing a Viscoelastic Corona

“…As a rheological model for the latter, the Jeffreys scheme (see [15], for example) is used because, unlike the plain Maxwell one, it is robust when applied to Brownian motion and is free of artifacts [16,17,18]. The viscoelastic properties of the Jeffreys model are fully rendered by three parameters (see the scheme outlined in Figure 1).…”

“…The equations of rotary motion for a Brownian particle in the inertialess limit take the form [17,18]:$\dot{\stackrel{false\to }{e}}=\left(\stackrel{false\to }{\mathrm{normal\Omega }},\times ,\stackrel{false\to }{e}\right),\overrightarrow{\mathrm{\Omega }}=0true\frac{1}{{\zeta }_N}\left(-,\stackrel{false^}{\overrightarrow{L}},U,+,\stackrel{false\to }{Q},+,{\overrightarrow{y}}_N,\left(t\right)\right),\widehat{\stackrel{false\to }{L}}=\left(\stackrel{false\to }{e},\times ,{\displaystyle \frac{\partial }{\partial \overrightarrow{e}}}\right),$ $\left(1,+,{\tau }_M,{\displaystyle \frac{\partial }{\partial t}}\right)\overrightarrow{Q}=-{\zeta }_M\overrightarrow{\mathrm{\Omega }}+{\stackrel{false\to }{y}}_M\left(t\right).$…”

“…The kinetic equation for the distribution function $W(\overrightarrow{e},\overrightarrow{Q},t)$ that corresponds to the set of stochastic Equation (1) is obtained via standard procedure (see [18,24,25]) for example:$2{\tau }_{D}0true\frac{\partial }{\partial t}W=\left((1+q^{-1}),\beta ,T,{\displaystyle \frac{\partial }{\partial \overrightarrow{Q}}},-,\stackrel{false^}{\overrightarrow{L}}\right)\left({\displaystyle \frac{1}{T}},\stackrel{false\to }{Q},+,\beta ,T,{\displaystyle \frac{\partial }{\partial \overrightarrow{Q}}}\right)W-\left(\beta ,T,{\displaystyle \frac{\partial }{\partial \overrightarrow{Q}}},-,\stackrel{false^}{\overrightarrow{L}}\right)\left({\displaystyle \frac{1}{T}},\left(\widehat{\stackrel{false\to }{L}}U\right),+,\stackrel{false^}{\overrightarrow{L}}\right)W.$…”

“…Second, in Refs. [18,26], we have developed a workable technique to obtain the pertinent dynamic susceptibility for the case of zero external field. Using those results, one may skip a considerable part of lengthy calculations and just add only the extension allowing for the bias field.…”

“…Using those results, one may skip a considerable part of lengthy calculations and just add only the extension allowing for the bias field. Besides that, in [18,26], while analyzing the results obtained from solving the kinetic equation (the analogue of Equation (4)) in an exact way, i.e., using a long multi-moment expansion, we have shown that, to get a plausibly accurate approximation, it suffices to truncate the infinite moment set to just the first two equations for the dynamic variables $\delta \overrightarrow{e\phantom{H}}=\overrightarrow{e}-{false\langle \overrightarrow{e}false\rangle }_0$ and $\overrightarrow{P}=\left(\stackrel{false\to }{Q},\times ,\stackrel{false\to }{e\phantom{0ptH}}\right)$; here, angular brackets with index 0 denote averaging over the equilibrium distribution (8). …”