Departamento de Física Teórica de la Materia Condensada and IFIMAC Univeridad Autónoma de Madrid, Madrid 28049, Spain.
J Chem Phys. 2013 Dec 7;139(21):214113. doi: 10.1063/1.4834696.
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.
Stokes-Einstein 关系是关于在不可压缩流体中悬浮的球形颗粒的自扩散系数的渐近结果，即在 Schmidt 数很大的极限情况下，即当动量扩散速度远快于颗粒时。当 Schmidt 数适中时，这在大多数用于流体力学的颗粒方法中都会发生，预计会出现偏离 Stokes-Einstein 预测的情况。我们使用最近开发的一种方法来计算这些修正，该方法用于在二维和三维中将颗粒与不可压缩的脉动流体耦合，该方法最小程度地解析了颗粒与流体的相互作用。我们发现，对于中等 Schmidt 数，扩散系数相对于 Stokes-Einstein 预测的减小量与 Schmidt 数成反比，无论是在二维还是三维中都是如此。然而，我们发现爱因斯坦公式在所有 Schmidt 数下都成立，与线性响应理论一致。这种不匹配是由于热波动由于流体-颗粒耦合的非线性性质而影响颗粒的阻力系数。数值数据与近似自洽理论非常吻合，该理论可用于估计各种方法中的有限 Schmidt 数修正。我们的结果表明，对 Stokes-Einstein 公式的修正主要来自于这样一个事实，即颗粒本身与动量一起扩散。我们的研究将来自于非滑移流体动力学修正的影响与时间尺度的有限分离的影响分开，从而可以更好地理解分子动力学等颗粒方法中普遍存在的偏离 Stokes-Einstein 预测的情况。
