Sondak David, Hawley Cory, Heng Siyu, Vinsonhaler Rebecca, Lauga Eric, Thiffeault Jean-Luc
Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA.
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom.
Phys Rev E. 2016 Dec;94(6-1):062606. doi: 10.1103/PhysRevE.94.062606. Epub 2016 Dec 16.
Artificial phoretic particles swim using self-generated gradients in chemical species (self-diffusiophoresis) or charges and currents (self-electrophoresis). These particles can be used to study the physics of collective motion in active matter and might have promising applications in bioengineering. In the case of self-diffusiophoresis, the classical physical model relies on a steady solution of the diffusion equation, from which chemical gradients, phoretic flows, and ultimately the swimming velocity may be derived. Motivated by disk-shaped particles in thin films and under confinement, we examine the extension to two dimensions. Because the two-dimensional diffusion equation lacks a steady state with the correct boundary conditions, Laplace transforms must be used to study the long-time behavior of the problem and determine the swimming velocity. For fixed chemical fluxes on the particle surface, we find that the swimming velocity ultimately always decays logarithmically in time. In the case of finite Péclet numbers, we solve the full advection-diffusion equation numerically and show that this decay can be avoided by the particle moving to regions of unconsumed reactant. Finite advection thus regularizes the two-dimensional phoretic problem.
人工携带粒子利用化学物质（自扩散泳）或电荷及电流（自电泳）中的自生成梯度游动。这些粒子可用于研究活性物质中集体运动的物理特性，并且在生物工程领域可能有广阔的应用前景。在自扩散泳的情况下，经典物理模型依赖于扩散方程的稳态解，从中可以导出化学梯度、携带流以及最终的游动速度。受薄膜中及受限条件下盘状粒子的启发，我们研究了二维情况下的扩展。由于二维扩散方程在正确的边界条件下没有稳态，必须使用拉普拉斯变换来研究该问题的长期行为并确定游动速度。对于粒子表面固定的化学通量，我们发现游动速度最终总是随时间呈对数衰减。在有限佩克莱数的情况下，我们通过数值求解完整的平流 - 扩散方程，并表明粒子移动到未消耗反应物区域可以避免这种衰减。有限平流因此使二维携带问题正则化。
