111 years of Brownian motion

Bian, Xin; Kim, Changho; Karniadakis, George Em

doi:10.1039/c6sm01153e

Cited by 165 publications

(155 citation statements)

References 180 publications

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“…The self-diffusion coefficient of a colloidal particle is usually equal to k B T divided by the drag coefficient (Sutherland 1905;Einstein 1905), where k B is the Boltzmann constant and T is the temperature of the fluid. This relation (Einsten's relation) can be derived from a linear Langevin equation for the particle velocity even if it is generalized to have the memory kernel representing the back-flow effect (Zwanzig & Bixon 1970;Widom 1971;Case 1971;Kubo et al 1991;Bian et al 2016). A Brownian particle has been used experimentally as a probe for local environments in the field of microrheology (Brau et al 2007;Pesce et al 2009;Kimura 2009;Bertseva et al 2012;Grebenkov et al 2013;Domínguez-García et al 2014).…”

Section: Introductionmentioning

confidence: 99%

Drag coefficient of a rigid spherical particle in a near-critical binary fluid mixture, beyond the regime of the Gaussian model

Yabunaka¹,

Fujitani²

2020

J. Fluid Mech.

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The drag coefficient of a rigid spherical particle deviates from the Stokes law when it is put into a near-critical fluid mixture in the homogeneous phase with the critical composition. The deviation (∆γ d ) is experimentally shown to depend approximately linearly on the correlation length far from the particle (ξ ∞ ), and is suggested to be caused by the preferential attraction between one component and the particle surface. In contrast, the dependence was shown to be much steeper in the previous theoretical studies based on the Gaussian free-energy density. In the vicinity of the particle, especially when the adsorption of the preferred component makes the composition strongly off-critical, the correlation length becomes very small as compared with ξ ∞ . This spacial inhomogeneity, not considered in the previous theoretical studies, can influence the dependence of ∆γ d on ξ ∞ . To examine this possibility, we here apply the local renormalized functional theory, which was previously proposed to explain the interaction of walls immersed in a (near-)critical binary fluid mixture, describing the preferential attraction in terms of the surface field. The free-energy density in this theory, coarse-grained up to the local correlation length, has much complicated dependence on the order parameter, as compared with the Gaussian free-energy density. Still, a concise expression of the drag coefficient, which was derived in one of the previous theoretical studies, turns out to be available in the present formulation. We show that, as ξ ∞ becomes larger, the dependence of ∆γ d on ξ ∞ becomes distinctly gradual and close to the linear dependence.

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Section: Introductionmentioning

confidence: 99%

Drag coefficient of a rigid spherical particle in a near-critical binary fluid mixture, beyond the regime of the Gaussian model

Yabunaka¹,

Fujitani²

2020

J. Fluid Mech.

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“…It is worth mentioning that crossovers between diffusive regimes can also be described by generalized Langevin equations [67] and fractional (with the Riemann-Liouville operator) Kramers equations [19], among other approaches [68,69]. In particular, the usual Langevin equation [70] predicts a crossover between ballistic and usual diffusion, which has been experimentally observed only in 2011 [71]. However, the diffusion equation in terms of these new operators lead to these crossovers without explicitly considering external forces, inertial effects, and reaction terms.…”

Section: Diffusion and Fractional Operatorsmentioning

confidence: 99%

The Role of Fractional Time-Derivative Operators on Anomalous Diffusion

2017

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The generalized diffusion equations with fractional order derivatives have shown be quite efficient to describe the diffusion in complex systems, with the advantage of producing exact expressions for the underlying diffusive properties. Recently, researchers have proposed different fractional-time operators (namely: the Caputo-Fabrizio and Atangana-Baleanu) which, differently from the well-known Riemann-Liouville operator, are defined by non-singular memory kernels. Here we proposed to use these new operators to generalize the usual diffusion equation. By analyzing the corresponding fractional diffusion equations within the continuous time random walk framework, we obtained waiting time distributions characterized by exponential, stretched exponential, and power-law functions, as well as a crossover between two behaviors. For the mean square displacement, we found crossovers between usual and confined diffusion, and between usual and sub-diffusion. We obtained the exact expressions for the probability distributions, where non-Gaussian and stationary distributions emerged. This former feature is remarkable because the fractional diffusion equation is solved without external forces and subjected to the free diffusion boundary conditions. We have further shown that these new fractional diffusion equations are related to diffusive processes with stochastic resetting, and to fractional diffusion equations with derivatives of distributed order. Thus, our results suggest that these new operators may be a simple and efficient way for incorporating different structural aspects into the system, opening new possibilities for modeling and investigating anomalous diffusive processes.

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“…Recent microscopic measurements, as well as large scale numerical simulations, have begun to focus on the start-up of steady shear flow and have clearly documented the evolution of particulate level microstructure under deformation [8,18,22,39,45,47,48]. The wide range of time and length scales accessible by DPD simulations, coupled with inclusion of hydrodynamic interactions between particles [25,49], now enables systematic exploration of the evolution of microstructural and rheological properties for TEVP materials as well as validation of empirical constitutive equations proposed for such systems [21,[50][51][52][53].…”

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confidence: 99%

Microstructural Rearrangements and their Rheological Implications in a Model Thixotropic Elastoviscoplastic Fluid

2017

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We identify the sequence of microstructural changes that characterize the evolution of an attractive particulate gel under flow and discuss their implications on macroscopic rheology. Dissipative particle dynamics is used to monitor shear-driven evolution of a fabric tensor constructed from the ensemble spatial configuration of individual attractive constituents within the gel. By decomposing this tensor into isotropic and nonisotropic components we show that the average coordination number correlates directly with the flow curve of the shear stress versus shear rate, consistent with theoretical predictions for attractive systems. We show that the evolution in nonisotropic local particle rearrangements are primarily responsible for stress overshoots (strain-hardening) at the inception of steady shear flow and also lead, at larger times and longer scales, to microstructural localization phenomena such as shear banding flow-induced structure formation in the vorticity direction. DOI: 10.1103/PhysRevLett.118.048003 Thixotropic elastoviscoplastic (TEVP) materials are a broad class of structured fluids that include (but are not limited to) most colloidal gels [1], nano emulsions [2], crude oils [3,4], and many biological systems such as blood clots and actin networks [5,6]. As a result of their complex underlying microstructure, TEVPs exhibit a wide range of rich and complex thermo-mechanical properties: Below a critical stress, the microstructural network formed by individual particles remains intact and resists large deformations by external forces. At this stage the macroscopic response of the material is similar to that of a viscoelastic solid. By progressively increasing the applied load, the material reaches its "yield stress" and starts to flow [7]. At this point the particle network that is responsible for solidlike response of the macroscopic sample undergoes plastic rearrangements over an increasingly wide range of length scales [8]. Upon complete yielding of this network, plastic flow results ultimately in a viscouslike response; however, as a result of constant formation and breakage events, the particle-level microstructure continues to evolve giving rise to thixotropic behavior. The many-body nature of the problem means that local forces exerted on a single particle change its energy landscape, which consequently defines its subsequent association or dissociation rate to neighboring particles [9,10]. When combined with multibody hydrodynamic effects in these fluids [11], the resulting microstructure-flow relationship becomes complex and may show long time scale transient behavior and multiple steady states [1,12]. This leads to a wide range of timedependent responses that can also be observed, including microphase separation [13], vorticity aligned structure formation [14][15][16], local rigid plug formation and shear banding [17], plus shear-induced rejuvenation of the particle network [18].Although the general form of the flow curve (relating the shear stress to shear rate) and some transient pheno...

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111 years of Brownian motion

Cited by 165 publications

References 180 publications

Drag coefficient of a rigid spherical particle in a near-critical binary fluid mixture, beyond the regime of the Gaussian model

Drag coefficient of a rigid spherical particle in a near-critical binary fluid mixture, beyond the regime of the Gaussian model

The Role of Fractional Time-Derivative Operators on Anomalous Diffusion

Microstructural Rearrangements and their Rheological Implications in a Model Thixotropic Elastoviscoplastic Fluid

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