A DEP force is often used to generate motion orthogonal to the direction of the applied field. This force is complementary to the DEP generated by the global non-uniformity of the electric field.
Objectives
Outline
In this report, we find that the enhancement effect is comparable to the viscous effect introduced by the second wall. In this report, we characterize the phenomenon by observing the finite chain angle formed and its dependence on the particle aspect ratio.
Electric double layer
Because of this, the zeta potential is commonly used as an approximation of the electrical potential at the interface [3]. The electric potential distribution within the diffuse EDL layer is governed by the Poisson-Boltzmann (PB) equation.
Electroosmosis and electrophoresis
Under an electric field, the particle will be driven towards one of the electric fields, while the fluid in the EDL is driven towards the other end. The driving force behind this motion is the Coulombic force acting on the net charge of the particle.
Dielectrophoresis
Point-dipole method
The DEP force acting on the particle comes from the derivatives of the electric field at the point where the center of the particle would be located. Any distortion of the field from the dielectric volume of the particle is not considered.
Maxwell stress tensor integration
It is important to note that this technique is only valid in cases where the particle is sufficiently smaller than its surroundings. In the studies performed here, the particle is assumed to be a dielectric, so that εp << εm.
Electro-orientation
For a stretched spheroid where b = c, the depolarization factors L⊥ and Lk reduce to the following expressions. Analogous to the previous section, there is a more computationally demanding technique that provides a more direct approach to calculating the DEP torque.
Influence of electric fields on cells
Finally, it is important to note the importance of the dielectric properties of the cell membrane. One of the main objectives of this work is to extend and promote the application of the boundary element method within the field of electrokinetics-based microfluidics.
Derivation of the boundary-integral
Each of the electrokinetic phenomena discussed in Chapter II is included in the solver. After using divergence theorems to change. the area integral into the linear integral, we get,. in the case of x0, a singular point located within the control region.
Boundary-integral formulation for Laplace’s equation
Two-dimensional formulation
The boundary element method is a technique for solving the solution of a function on the boundary of a domain. Once the boundary solution is known, one has the option to use Eq. 3.8) to find the value of φ at any point within AC explicitly.
Three-dimensional formulation
If we tried to solve the boundary solution for φ in this example, we would find the integral in Eq. 3.10) with a finite number of boundary elements Ei. where i = 1, ..,N represents the total number of elements and x lies on Ei. By approximating the integrals over D as sums of integrals over surface elements, Ei, we can Eq.
Boundary-integral formulation for Stokes equation
Two-dimensional formulation
To solve the flow in two dimensions, we use the boundary integral formulation for the Stokes equation and express u at the point x0 which lies inside the fluid. As with the integral equations representing electric potential, the integrals in Eq. 3.19) can be approximated over a finite number of elements to develop a system of linear equations.
Three-dimensional formulation
If the point x0 lies on the boundary D, the integral equation takes the form 3.22) will be used to formulate the algebraic system to obtain the hydrodynamic drag f on the particle and cylinder surfaces (note that the fluid velocity on these surfaces will be based on the slip velocity and can be obtained, once available φ and its gradient) . Once f is obtained, the fluid velocity at any internal point of the flow can be calculated by evaluating equation (3.20) in post-processing.
Integral identities
If the surface normal was directed outside the control volume or region, the sign of the terms on the right-hand side would be reversed. Again, if the surface normal were directed outside the control volume or region, the sign of the terms on the right-hand side would be reversed.
Background
An important situation for this issue is the electrophoresis of a spherical particle moving between two parallel walls, which was studied analytically by Unni et al. In Unni et al., particle translation was found to be enhanced when both walls are close to the particle, implying that the increasing electrophoretic effect has overcome the hydrodynamic retardation for the narrow particle-wall separation.
Problem specification
In the laboratory coordinates, the fluid velocity on the particle surface is the combination of the particle's rigid body motion and the slip velocity. The particle is assumed to be neutrally buoyant in the fluid, and its inertia can also be ignored.
Formulation
Finally, (4.14) and (4.15) are solved together with the unknown translational and rotational velocities of the particle. That is, uDis zero at the four corners of the channel since the disturbance potential and.
Validation
Results
Note that although the particle's translation is reduced as the channel narrows, its rotation is not. Here, two stagnation points can be seen on the boundary of the particle in the area near the bottom wall.
Conclusion
The potential undergoes a rapid change across the narrower gap between the particle and the channel, causing a high fluid slip velocity on both the particle surface and the walls in the region.
Background
In the present study, the small particle-wall gap, the minimum of which is on the order of 1% of the particle radius, is well resolved using the highly accurate boundary element method, while in Ai et al. Finally, another important goal of the present work is to perform a comparison of the full numerical simulation with predictions based on the point-dipole method (PDM) in an attempt to provide a clear picture of the limitations associated with the latter method.
Problem specification
Using the thin-EDL assumption, we express the fluid velocity adjacent to the particle and channel walls using a slip velocity proportional to the local tangential gradient of the electric potential [84]. In laboratory coordinates, the fluid velocity on the particle surface is a combination of the rigid body motion of the particles.
Formulation
Note that since we are modeling a spherical particle, the Maxwell stress contributes nothing to the torque on the particle. The mesh element size used in this study is flexible and based on the particle's proximity to the wall.
Validation
Note that the particle is translated outside the channel to better visualize the mesh. In the second test, the lateral dielectrophoretic force of the particle is calculated and compared with that reported by Young and Li [60].
Simulation setup
Here, we can see the distortion in the flow field caused by the presence of particles. Therefore, PDM can lead to significant errors in the DEP mobility estimation of the particle.
Results .1 E ff ect of the initial location
E ff ect of particle size
We also calculated the particle's trajectory based on PDM to evaluate its validity. Both the BEM calculation (dashed line) and the point-dipole approximation (solid line) of the DEP force are plotted.
E ff ect of the electric field
From Figure 5.7(d) we can see the combined effect of the electric field strength on the largest particle, a/b = 0.4. As shown in Figure 5.7(f), this effect is captured by the current BEM simulation, but not by the PDM.
Conclusion
If weaker fields and/or longer channels are used, the importance of particle motion due to gravity can no longer be neglected. Depending on the orientation of the bent pore, the sedimentation of the particle can change its trajectory.
Background
In this study, they were able to show how the DEP force of two spherical particles gradually aligns the particles so that the line connecting their centers is parallel to the direction of the electric field. In this study, we investigate the DEP interaction between two non-conducting, ellipsoidal particles in an electric field.
Problem specification and governing equations
The entire flow field is neutral and the distribution of the electric potential, φ, is determined by the Laplace equation, Eq. This shows that the fluid velocity at a point x on the particle surface is equal to the motion of the rigid body of the particle due to the anti-slip condition.
Boundary-integral formulation and numerical approach
As in the previous study, the total traction force is defined as. where σ is the hydrodynamic stress tensor and T is the Maxwell stress tensor given by Eq. noting that x0 falls on the boundary D. To solve for the flow, we use the boundary-integral formulation for the Stokes equation in 3D, Eq. where f = σ·n is the component of the hydrodynamic drag.
Code validation
This allows us to recover Stokes' resistance when the particle is far away from the planar surface. As the particle gets closer to the wall (d/a→ 1), the increasing effect of the correction can be seen as λ.
Results
Electro-orientation of a single prolate spheroid
A diagram of the electro-orientation process for a/b = 3.0 when the electric field is directed from left to right. Equating the two torques, we obtain the electro-orientation velocity of a single particle as a function of the aspect ratio, as shown in Figure 6.9, where the angular velocity at the orientation angleα= π/4 is defined as ω∗ = 16πµω / (εmE02).
Two particles: combined electro-orientation and global reorientation Next, we shall observe the interaction of two particles and will start by considering
Each of the particles is initially oriented with respect to the electric field such that α0 =85◦. Furthermore, when the particles are far apart, e.g. θ0 = 20◦ and d0/a= 5.0, the chain angle can also be small (less than 2◦), and the particles are almost parallel to the electric field.
Particles interacting without su ffi cient separation
Conclusion
The transient motion of the particles and their chain angle are generalized to a trajectory map for particles with a constant aspect ratio. We have further studied the relationship between the initial particle spacing and the final chain angle.
Summary of present work
There is thus a need for an alternative numerical technique in the study of electrokinetic effects. With these modifications in place, in Chapter VI we were able to study the DEP interaction that takes place between two slender particles in close proximity to each other.
Limitations of present work
Contributions of present work
By implementing these codes we have demonstrated the effectiveness of the BEM as an accurate alternative to other numerical techniques. This equation illustrates the inaccuracies obtained without considering the finite size of the particle.
Directions for future work
Characterization of hurdle-based DC-DEP for particle manipulation With the three-dimensional BEM code in place and validated, there are several di-
As an example, Figure 7.2 shows variables that could be used in a simple study to design and optimize barrier conditions. The problem specification in this study would be identical to those of the studies conducted in this dissertation.
Characterization of embedded electrode AC-DEP for particle manipula- tion
In this design, specified electrical potential will be used on the surface of each electrode. To simulate this, different values of traction will be specified at the channel inlet and outlet.
Other future studies
Electrophoresis of a colloidal sphere in a spherical cavity with arbitrary zeta potential distribution and arbitrary double layer thickness. Continuous particle size separation by ac-dielectrophoresis using a lab-on-a-chip device with 3D electrodes.
