### Abstract

The Stokes-Einstein relation for the self-diffusion coefficient of a spherical particle suspended in an incompressible fluid is an asymptotic result in the limit of large Schmidt number, that is, when momentum diffuses much faster than the particle. When the Schmidt number is moderate, which happens in most particle methods for hydrodynamics, deviations from the Stokes-Einstein prediction are expected. We study these corrections computationally using a recently developed minimally resolved method for coupling particles to an incompressible fluctuating fluid in both two and three dimensions. We find that for moderate Schmidt numbers the diffusion coefficient is reduced relative to the Stokes-Einstein prediction by an amount inversely proportional to the Schmidt number in both two and three dimensions. We find, however, that the Einstein formula is obeyed at all Schmidt numbers, consistent with linear response theory. The mismatch arises because thermal fluctuations affect the drag coefficient for a particle due to the nonlinear nature of the fluid-particle coupling. The numerical data are in good agreement with an approximate self-consistent theory, which can be used to estimate finite-Schmidt number corrections in a variety of methods. Our results indicate that the corrections to the Stokes-Einstein formula come primarily from the fact that the particle itself diffuses together with the momentum. Our study separates effects coming from corrections to no-slip hydrodynamics from those of finite separation of time scales, allowing for a better understanding of widely observed deviations from the Stokes-Einstein prediction in particle methods such as molecular dynamics.

Original language | English (US) |
---|---|

Article number | 214113 |

Journal | Journal of Chemical Physics |

Volume | 139 |

Issue number | 21 |

DOIs | |

State | Published - Dec 7 2013 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Physical and Theoretical Chemistry

### Cite this

*Journal of Chemical Physics*,

*139*(21), [214113]. https://doi.org/10.1063/1.4834696

**The Stokes-Einstein relation at moderate Schmidt number.** / Balboa Usabiaga, Florencio; Xie, Xiaoyi; Delgado-Buscalioni, Rafael; Donev, Aleksandar.

Research output: Contribution to journal › Article

*Journal of Chemical Physics*, vol. 139, no. 21, 214113. https://doi.org/10.1063/1.4834696

}

TY - JOUR

T1 - The Stokes-Einstein relation at moderate Schmidt number

AU - Balboa Usabiaga, Florencio

AU - Xie, Xiaoyi

AU - Delgado-Buscalioni, Rafael

AU - Donev, Aleksandar

PY - 2013/12/7

Y1 - 2013/12/7

N2 - The Stokes-Einstein relation for the self-diffusion coefficient of a spherical particle suspended in an incompressible fluid is an asymptotic result in the limit of large Schmidt number, that is, when momentum diffuses much faster than the particle. When the Schmidt number is moderate, which happens in most particle methods for hydrodynamics, deviations from the Stokes-Einstein prediction are expected. We study these corrections computationally using a recently developed minimally resolved method for coupling particles to an incompressible fluctuating fluid in both two and three dimensions. We find that for moderate Schmidt numbers the diffusion coefficient is reduced relative to the Stokes-Einstein prediction by an amount inversely proportional to the Schmidt number in both two and three dimensions. We find, however, that the Einstein formula is obeyed at all Schmidt numbers, consistent with linear response theory. The mismatch arises because thermal fluctuations affect the drag coefficient for a particle due to the nonlinear nature of the fluid-particle coupling. The numerical data are in good agreement with an approximate self-consistent theory, which can be used to estimate finite-Schmidt number corrections in a variety of methods. Our results indicate that the corrections to the Stokes-Einstein formula come primarily from the fact that the particle itself diffuses together with the momentum. Our study separates effects coming from corrections to no-slip hydrodynamics from those of finite separation of time scales, allowing for a better understanding of widely observed deviations from the Stokes-Einstein prediction in particle methods such as molecular dynamics.

AB - The Stokes-Einstein relation for the self-diffusion coefficient of a spherical particle suspended in an incompressible fluid is an asymptotic result in the limit of large Schmidt number, that is, when momentum diffuses much faster than the particle. When the Schmidt number is moderate, which happens in most particle methods for hydrodynamics, deviations from the Stokes-Einstein prediction are expected. We study these corrections computationally using a recently developed minimally resolved method for coupling particles to an incompressible fluctuating fluid in both two and three dimensions. We find that for moderate Schmidt numbers the diffusion coefficient is reduced relative to the Stokes-Einstein prediction by an amount inversely proportional to the Schmidt number in both two and three dimensions. We find, however, that the Einstein formula is obeyed at all Schmidt numbers, consistent with linear response theory. The mismatch arises because thermal fluctuations affect the drag coefficient for a particle due to the nonlinear nature of the fluid-particle coupling. The numerical data are in good agreement with an approximate self-consistent theory, which can be used to estimate finite-Schmidt number corrections in a variety of methods. Our results indicate that the corrections to the Stokes-Einstein formula come primarily from the fact that the particle itself diffuses together with the momentum. Our study separates effects coming from corrections to no-slip hydrodynamics from those of finite separation of time scales, allowing for a better understanding of widely observed deviations from the Stokes-Einstein prediction in particle methods such as molecular dynamics.

UR - http://www.scopus.com/inward/record.url?scp=84888216434&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84888216434&partnerID=8YFLogxK

U2 - 10.1063/1.4834696

DO - 10.1063/1.4834696

M3 - Article

AN - SCOPUS:84888216434

VL - 139

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 21

M1 - 214113

ER -