东北大学秦皇岛分校 控制工程学院, 河北 秦皇岛 066004
收稿日期:2020-03-26
基金项目:国家自然科学基金资助项目(61903069)。
作者简介:胡晟(1984-),男,云南景洪人,东北大学讲师,博士。
摘要:Velocity-Verlet或ODE算法分析微粒的运动特征,涉及的偏微分方程存在求解困难、运算量大和使用效率低的难点.采用COMSOL Multiphysics 5.3a有限元软件,通过AC/DC模块中的边界条件设定可快速求解Laplace方程,为介电泳力的求解提供先决条件.后期根据软件提供的粒子追踪模块,将介电泳力、斯托克斯拖曳力、排斥力和浮力内联进粒子追踪模块的应力参数表中,设定固定的时间步长和范围对介电泳芯片的粒子受力运动问题进行了仿真.结果表明,该方法可以有效模拟微粒受介电泳效应的运动行为,并且与Velocity-Verlet或ODE算法模拟结果相似,能有效降低计算程序的繁琐程度,提高动态模拟的人机可视化效果.
关键词:介电泳电偶极矩动力学粒子链COMSOL Multiphysics 5.3a
Dynamic Modeling and Simulation of Micro-particles Experienced Dielectrophoretic Effect
HU Sheng, WANG Ke, CAI Lu
School of Control Engineering, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China
Corresponding author: HU Sheng, E-mail: husheng@neuq.edu.cn.
Abstract: There exists tough convergence, burden computation, and low efficiency for solution to partial difference equations using Velocity-Verlet or ODE algorithms analyzing characteristics of particle motion. The finite element software COMSOL Multiphysics 5.3a, which could be used to solve Laplace equation quickly in AC/DC module when these reasonable boundaries were chosen, provides the precondition for calculating dielectrophoretic force. In terms of particle tracing module in COMSOL software, the particles experienced dielectrophoretic effect were simulated through the selection of both time step and range when dielectrophoretic, Stokes drag, repulsive, and buoyant forces were taken into the table of stress parameter in particle tracking module. The results illustrated that this approach is capable of simulating the motion of such particles exerted by dielectrophoresis effect, reducing the complexity of program and improving man-machine visualization of dynamic simulation, and are consistent with Velocity-Verlet or ODE algorithms.
Key words: dielectrophoresisdipole momentdynamicsparticle chainCOMSOL Multiphysics 5.3a
介电泳(dielectrophoresis, DEP)作为一种微/纳粒子操控、分离、富集和输运技术,已经受到国内、外****的广泛关注和学习[1-4].在非均匀电场作用下,粒子电极化的应力不平衡,因此会发生沿电场强度梯度方向的定向运动.微粒向电场强度较强区域运动,该介电泳力称为正介电泳力(positive DEP, pDEP).反之,负介电泳力(negative DEP, nDEP)则驱使粒子运动到电场强度较弱的空间位置.上述正、负介电泳力的选取主要与粒子或溶液的介电常数、电导率和激励频率有关.伴随数学物理方法和计算方法学的飞速发展,目前介电泳的理论研究也取得了长足的进步.根据经典电偶极矩法[5-6]和Maxwell应力张量法[7-8]关于介电泳力的物理描述,粒子运动轨迹和拓扑形态的仿真结果,都有效地证实了实验观测和电极设计.
然而,从当前的数学建模和仿真分析来看,粒子承受介电泳力作用的动力学研究仍然存在一些问题需要进一步研究和解决,主要由于编程的复杂以及后期代码的维护工作.特别对于粒子承受排斥力与介电泳吸附效应等问题时,粒子相互作用力的受力矩阵也导致了求解精度和迭代次数的成倍增加,而关于代码的优化和电场偏微分方程的求解仍需要一个更好的方法和工具进行优化求解.基于COMSOL Multiphysics 5.3a有限元分析软件通过各种内置的方法已经极大解决了物理域的网格离散和偏微分方程求解.通过Laplace微分方程和边界条件的设定,可以快速得到电极芯片内部的电压分布[9-10],从而掌握电场变化的方向和强度.COMSOL Multiphysics 5.3a软件提供了粒子跟踪模块(particle tracing module, PTM),该模块极大扩大了场与粒子相互作用的研究内涵.可以根据求解得到的物理场进行粒子运动特征的分析和学习.Zhao等[11]最早采用PTM和AC/DC Module结合的方法进行了粒子受介电泳效应的动力学研究.但是他们研究所提供的相关信息相对较少,并且介电泳粒子成链效应仍存在研究不足等缺点.本文重新对粒子位于介电泳芯片的粒子受力运动问题进行了仿真和研究,丰富了COMSOL Multiphysics 5.3a软件在介电泳粒子操控领域的理论研究,使该软件提供的AC/DC和PTM模块可以较好解决现有Velocity-Verlet或ODE算法编程困难、执行效率低的问题.
1 原理与物理场模型微粒的介电泳力产生机理归因于电场分布的非均匀特性,从空间角度观察粒子运动可分为全局和局部效应.全局介电泳效应主要是电极产生的电场在较大尺度空间诱导粒子输运和富集.但是局部介电泳效应,则是微粒迫使均匀电场在局部或粒子附近发生扭曲,进一步诱导周围粒子移动成链的过程.在无损耗介质溶液中,电极产生的介电泳力表达式[1]为
(1) |
(2) |
然而,粒子承受的局部介电泳力与它们之间的电偶极矩动量密切相关,表达式[5, 13]为
(3) |
另外粒子相互碰撞和迁移还应包含相互排斥力,防止粒子发生重合现象,表达式[13-14]为
(4) |
粒子在运动过程中还受Stokes拖曳力,并且粒子所受的重力分别满足式(5)和式(6).
(5) |
(6) |
(7) |
图 1 微粒受力示意图Fig.1 Schematic diagram of forces on particles |
式中:Fw0是墙壁给予粒子的最大排斥力;ri是i粒子的空间位置(xi, yi, zi).式(4)和式(7)的初始值Frep0,Fw0和1/κ分别取值2.3×10-12 N,1×10-10 N和0.01.同时,求解域中剩余5个墙壁面,设定Freeze边界条件,有利于粒子的观测和学习.
2 结果和讨论2.1 均匀电场首先进行平行极板产生均匀电场的求解,图 1中,左、右边界分别设定+3.5 V和接地边界条件,其余边界为绝缘边界条件.通过求解Laplace方程,即式(8)和式(9),可得到施加电压后芯片内部的电场分布[15].
(8) |
(9) |
表 1(Table 1)
表 1 仿真模型关键参数Table 1 Parameters of simulated model
| 表 1 仿真模型关键参数 Table 1 Parameters of simulated model |
图 2(Fig. 2)
图 2 均匀电场下,35个粒子且半径为5 μm的仿真结果Fig.2 Simulated results of 35 particles with radius of 5 μm under homogeneous electric field (a)—t=0 s;(b)—t=2 s;(c)—t=4 s;(d)—t=6 s. |
图 3(Fig. 3)
图 3 均匀电场下,15个半径为5 μm和20个半径为2.5 μm的粒子仿真结果Fig.3 Simulated results of both 15 particles with radius of 5 μm and 20 particles with radius of 2.5 μm under homogeneous electric field (a)—t=0 s;(b)—t=2 s;(c)—t=4 s;(d)—t=6 s;(e)—t=0 s时,xy平面图;(f)—t=6 s时,xy平面图. |
2.2 非均匀电场设置金属电极产生非均匀电场,诱导粒子向电场较强或较弱方向运动.激励电极如图 4a所示,右侧负电极接地.电极电压仍为+3.5 V.聚苯乙烯粒子εp为2.25, 受负介电泳力运动到电场强度较弱位置,并且局部形成短链结构,如图 4c所示.然后对粒子表面进行化学修饰, 假设微粒εp为120,高于水溶液80,电极针尖附近的粒子将运动到其附近,如图 4d所示.但是远离针尖的粒子所受正介电泳力相对较弱,则逐渐沉淀至芯片底部.
图 4(Fig. 4)
图 4 非均匀电场下,半径为5 μm的35个粒子承受负、正介电泳力仿真结果Fig.4 The 35 particles with radius of 5 μm experienced either negative or positive DEP forces under non-uniform electric field (a)—电场强度仿真结果;(b)—t=0 s初始粒子分布图;(c)—t=2 s粒子受负介电泳力的分布结果;(d)—t=2 s粒子受正介电泳力的分布结果. |
另外采用电极阵列型结构计算区域内35个半径5 μm的微粒运动结果,如图 5所示.负介电泳效应时,靠近边缘的粒子运动到电极与电极之间电场较弱区域.反之正介电泳力驱使粒子到电极边缘位置,该模拟结果也与文献[16-18]实验基本类似.
图 5(Fig. 5)
图 5 阵列型电极结构中,粒子承受负、正介电泳力的仿真结果Fig.5 Simulated results of negative or positive DEP force acting on particles in arrayed electrode structure (a)—电场强度仿真结果;(b)—t=0 s初始粒子分布图;(c)—t=2 s粒子受负介电泳力的分布结果;(d)—t=2 s粒子受正介电泳力的分布结果. |
3 结论1) 本文采用了COMSOL Multiphysics 5.3a软件,通过求解Laplace方程获得求解域内的电场分布;后期耦合了粒子追踪模块,模拟芯片内微粒受介电泳效应的运动行为.
2) 仿真结果表明,本文方法能有效降低计算程序的繁琐程度,提高动态模拟的人机可视化效果.较好解决现有Velocity-Verlet或ODE算法编程困难、执行效率低的问题.
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