Coupled molecular-dynamics and finite-element-method simulations for the kinetics of particles subjected to field-mediated forces

A computational approach that couples molecular-dynamics (MD) and the-finite-element-method (FEM) technique is here proposed for the theoretical study of the dynamics of particles subjected to electromechanical forces. The system consists of spherical particles (modeled as micrometric rigid bodies with proper densities and dielectric functions) suspended in a colloidal solution, which flows in a microfluidic channel in the presence of a generic nonuniform variable electric field generated by electrodes. The particles are subjected to external forces (e.g., drag or gravity) which satisfy a particlelike formulation that is typical of the MD approach, along with an electromechanical force that, in turn, requires the three-dimensional self-consistent solutions of correct continuum field equations during the integration of the equations of motion. In the MD-FEM method used in this work, the finite element method is applied to solve the continuum field equations while the MD technique is used for the stepwise explicit integration of the equations of motion. Our work shows the potential of coupled MD-FEM simulations for the study of electromechanical particles and opens a double perspective for implementing (a) MD away from the field of atomistic simulations and (b) the continuum-particle approach to cases where the conventional force evaluation used in MD is not applicable.

Figure 1

Parabolic velocity profile for a fluid flowing through a microfluidic channel of height $h$; $u\left(0\right)=u\left(h\right)=0$; $u_{\max }=u\left(\frac{h}{2}\right)$.

Figure 2

Block diagram of the MD-FEM algorithm. $n$ and $n_{\mathrm{DEP}}$ are integers greater than or equal to zero, such that $n/n_{\mathrm{DEP}}=\mathrm{\Delta }{t}_{\mathrm{DEP}}/\mathrm{\Delta }t$.

Figure 3

(a): Spherical dielectric cell composed of the cytoplasm (inner volume) and the membrane (light brown shell indicated by the arrow). (b) Effective equivalent homogeneous sphere model ruled by the dielectric function ${\tilde{\varepsilon }}_{\mathrm{eff}}$.

Figure 4

Comparison between the values of $F^{\mathrm{MST}}$ present in Ref. [15], obtained by comsol, and the values calculated in our gmsh-fenics implementation of the electromechanical force calculation.

Figure 5

Snapshots of simulation of two particles in a parallel plate capacitor for $t=0,0.1,0.15,0.21\mathrm{s}$ [(a)–(d), respectively]. The particles attract and form a chain that aligns with the electric field and remains in this stable configuration for the rest of the simulated evolution.

Figure 6

Schematic of an interdigitated circuit. An alternating signal is applied to the electrodes in red while $V=0$ is imposed to electrodes in blue.

Figure 7

Schematic of the computational domain (limited for simplicity to only two electrodes while in the simulation $N$ electrodes are considered). The electrodes (in blue) have a width $W_e$ and are separated by a gap of width $W_g$.

Figure 8

Solution of the Laplace problem at $t=0.6\mathrm{s}$. Top panel: the schematic of the microfluidic channel in which the solution of the Laplace equation relative to the sections passing through $x=60,220,540$, and 880 $\mu \mathrm{m}$ is visible. Bottom panel: the front views of the slices themselves, where variations in potential due to the presence of particles can be observed.

Figure 9

Snapshots of simulated system ($N=20$ MDA-MD-231 cells in a flowing colloidal solution) for $t=0,0.3,0.6,1.5,2.4,3.3,4.2,5.1\mathrm{s}$ (from upper panel to lower panel). The dimensions of the microchannel are $\left(960\times 60\times 100\right)\mu {\mathrm{m}}^3$ (for the length, depth, and height, respectively) and the number of electrodes is 12.

Figure 10

Snapshots for $t=2.1,2.4,2.7,3,3.3,3.6\mathrm{s}$ (from upper to lower). The pair of particles that form the chain is circled in the upper snapshot. The dimensions of the microchannel are $\left(960\times 60\times 100\right)\mu {\mathrm{m}}^3$ (for the length, depth, and height, respectively) and the number of electrodes is 12.

Figure 11

Snapshots of simulated system (ten MDA-MD-231 cells and ten B-lymphocytes in a flowing colloidal solution) for $t=0,0.3,1.2,2.4,3.3,4.2,5.1\mathrm{s}$ (from upper to lower). The dimensions of the microchannel are $\left(800\times 60\times 100\right)\mu {\mathrm{m}}^3$ (for the length, depth, and height, respectively) and the number of electrodes is ten.

Figure 12

$z$ Coordinates of the particle with the highest altitude during the entire simulation, in a time interval where it presents a nonmonotonic change in altitude.

Figure 13

Snapshots of simulation of one MDA-MD-231cell and one B-lymphocyte in a parallel flat face capacitor for $t=0,0.1,0.15,0.21\mathrm{s}$ [(a–d), respectively]. Contrary to the case of Fig. 5, the particles repel each other.