Dynamic Drag Force based on Iterative Density Mapping:
A new numerical tool for three-dimensional analysis of particle trajectories
in a dielectrophoretic system
M. Knoerzer 1, 2, †, C. Szydzik 1,†, F. J. Tovar-Lopez 1 , X. Tang 1, A. Mitchell 1, K. Khoshmanesh 1
1
2
School of Electrical and Computer Engineering, RMIT University, VIC 3001, Australia
Institute for Optofluidics and Nanophotonics (IONAS), Karlsruhe University of Applied Sciences,
76133 Karlsruhe, Germany
† These authors have equally contributed to this work.
* Corresponding author:
Khashayar.khoshmanesh@rmit.edu.au
1
Supplementary S1. Values of parameters A, B, C, D(x) and E(z) used in our numerical model
Table S1. Values of parameters used in our numerical model
Table S2. Values of Coefficients A, B, C, D(x) and E(z) used in our numerical model
Parameters of
differential
equation (7)
Values obtained by the parameters
introduced in Figure 2(b)
Relevant force
Inertial
Drag
DEP
Variable
Drag
Sedimentation
2
Supplementary S2. Numerical calculation of DEP force
To calculate the distribution of electric field, the Laplace equation was solved within the PDMS chamber
by applying a 15 Vpp electric potential at the microelectrodes while zero electric flux at other surfaces of
the chamber, including the bottom, top and sidewalls:
 2rms  0
(S1)
Next, the electric field was calculated by differentiating the electric potential:
E  rms
(S2)
Next, the DEP force was obtained by calculating the gradient of electric field square:
FDEP  E 2
(S3)
3
Supplementary S3. Characterisation of
vs Velocity profile
In order to identify an optimal value for the
scaling factor
, its influence over both
trajectory and velocity profiles were investigated for a sweep of values.
The effect of ramping
was similar for both frequencies investigated; initial low values of
have little influence on the accuracy of simulations. Increasing
causes both velocity and
trajectory plots to approach a closer approximation of experimental results. Accuracy increases to peak
performance at a similar value for both trajectory and velocity profiles, and then deteriorates past this
point.
The optimal value of
was determined empirically using the method outlined in the following
sections S2 and S3. Characterisation of
vs Velocity profile was accomplished by comparing
experimental data obtained by μPIV with corresponding profiles generated with a sweep of
values at corresponding heights.
Direct comparison between experimental and simulated data is not practical due to the lateral
displacement experienced by simulated results. This displacement is most likely due to the inability of
this simulation approach to consider particle-particle contact effects, as well as particle-particle and
particle-sidewall friction. Lack of contact effects allows particles to move through one another, resulting
in regions of artificially high particle density, while the lack of frictional effects results in the model
excluding granular effects that would slow particles in contact with sidewalls or high-density regions of
particles in close proximity to these.
In order to effectively compare experimental and simulated results, assessment was made of the
magnitude of several peaks and troughs present in velocity profiles of both data sets, at each horizontal
plane in which a measurement was made. Figure S1 illustrates this comparison.
Figures S1 and S2 presents the experimental velocity profile, as well as the comparable velocity profiles
attained with ‘Dynamic Drag Force based on Iterative Density Mapping’ simulation method. These
values are compared at various planes within the channel, for each instance of
and were assessed
at an excitation frequency of 20 MHz and 100 kHz.
Figures S1 and S2 illustrates the experimental velocity profile measured using µPIV as a solid thick line
coloured according to the measurement plane (which is ~9 µm for 1 µm particles, and ~24 µm for 5 µm
particles). The thickness of the measurement plane introduces a velocity profile window within which
particles are expected to be travelling, this is illustrated as a the space between the two dotted lines
showing the highest and lowest velocity profile covered in the measurement plane. The simulated results
are shown with large circular data points coloured according to the measurement plane.
4
Figure S1. Characterisation of
- Comparative velocity profiles at 20 MHz
Velocity profile at 20 MHz
1e-3
2e-3
3e-3
Velocity profile at 20 MHz
5
4e-3
5e-3
1e-2
6
Velocity profile at 20 MHz
2e-2
3e-2
4e-2
7
Figure S2. Characterisation of
- Comparative velocity profiles at 100 kHz
Velocity profile at 100 kHz
1e-3
2e-3
3e-3
8
Velocity profile at 100 kHz
4e-3
5e-3
1e-2
9
Velocity profile at 100 kHz
2e-2
3e-2
4e-2
10
Supplementary S4. Characterisation of
Characterisation of
vs Trajectory
vs particle trajectory was accomplished by comparing particle density
distribution of the experimental image taken downstream of the electrodes at both 20 MHz and 100 kHz
with a simulated trajectory plot generated at the corresponding location. Influence of
was
assessed through comparison of features within the experimental particle distribution and simulated
trajectory plots.
A key feature in comparison of 20 MHz experimental and simulated data was the particle-free region
present in the centre of the channel. This was measured to be approximately 165 µm in the experimental
image, and ranged from 135 µm to 360 µm with ramping the value of
seen in Figure S3. Each table shown in Figure S3 explores an instance of
, this comparison can be
, and shows a plan
view trajectory plot against the original experimental data, as well as a three dimensional view. In
addition these tables contain vertical slices at various axial positions throughout the channel, which show
the effects of
on vertical movement of particles.
The results for 100 kHz was more difficult to quantify, lacking particle-free regions allowing convenient
comparison. The key feature used to assess influence of
at 100 kHz was the funnelling effect
present with small particles in the centre of the channel. This funnelling effect becomes more pronounced
with ramping of
until at higher values, the particle stream becomes separated. Figure S4
illustrates the effects of ramping
on trajectory plots at an excitation frequency of 100 kHz.
Comparative data is shown in Figure S5. This data was collated from comparisons drawn from Figure
S3 and S4. Experimental results are shown as dotted lines, while simulated results are shown as a solid
black line for 20 MHz and a red line for 100 kHz. This comparison shows that the optimal value of
lies at the plateau between 0.002 and 0.005 m3.
11
Figure S3. Characterisation of
- Comparative particle density distribution and Trajectory
plots at 20 MHz
Velocity profile at 20 MHz
1e-3
Velocity profile at 20 MHz
12
2e-3
13
Velocity profile at 20 MHz
3e-3
14
Velocity profile at 20 MHz
4e-3
15
Velocity profile at 20 MHz
5e-3
16
Velocity profile at 20 MHz
1e-2
17
Velocity profile at 20 MHz
2e-2
18
Velocity profile at 20 MHz
3e-2
19
Velocity profile at 20 MHz
4e-2
20
Velocity profile at 20 MHz
5e-2
21
Figure S4. Characterisation of
- Comparative particle density distribution and Trajectory
plots at 100 kHz
Velocity profile at 100 kHz
1e-3
22
Velocity profile at 100 kHz
2e-3
23
Velocity profile at 100 kHz
3e-3
24
Velocity profile at 100 kHz
4e-3
25
Velocity profile at 100 kHz
5e-3
26
Velocity profile at 100 kHz
1e-2
27
Velocity profile at 100 kHz
2e-2
28
Velocity profile at 100 kHz
3e-2
29
Velocity profile at 100 kHz
4e-2
30
Velocity profile at 100 kHz
5e-2
31
Figure S5. Comparative particle density distribution – trajectory plot data for assessment of density
scaling factor performance
32
Supplementary S5. Assessing the performance of “Dynamic Drag Force based on Iterative Density
Mapping” model at a frequency of 100 kHz
Figure S6. (a-d) Comparison of particle velocity profiles obtained with the original and improved drag
models at the frequency of 100 kHz. The measured velocities obtained by µPIV are presented with solid
lines. The simulated velocities obtained with ‘improved drag model‘ are presented with diamond symbols
in different colours according to their heights
. The simulated
velocities obtained with ‘original drag model‘ are presented with circle symboles in grey
.
The velocity profile of the medium is presented with two dotted lines at the heights of z ± 5 µm to cover
the thickness of the measurement plane. (e) Presents the errors between the experimental and simulated
maximum velocities obtained with different values of
of
at a frequency of 100 kHz. The best value
is highlighted with dashed lines, which is chosen for the simulation shown in (a-d).
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Supplementary S6. Assessing the performance of “Dynamic Drag Force based on Iterative
DensityMapping” model at different flow rates and particle concentrations
In order to characterise system performance at varied flow rate and particle density, experimental data
was collected at three different flow rates with a particle density one quarter that used in primary
characterisation experiments (25 mg/ml). Figure S6 show the response of 5 μm particles at the last
microelectrode pair of the system. The conductivity of the medium was 0.0032 S/m, the magnitude and
frequency of the applied signal were 15 Vp-p and 20 MHz, respectively, while particle density was 6.25
mg/ml. The medium flow rate varied from 0.25 to 1 µL/min to examine accuracy of the simulated
trajectory plots.
Figure S6(a-c) show a comparison of experimental particle density distribution with simulated trajectory
plots calculated for corresponding flow rates, a vertical slice of the simulated channel also shows vertical
particle distribution.
Comparison of experimental and simulated results is shown in Figure S6(d), as with the 20 MHz
trajectory plot comparison conducted in Supplementary S3, the particle-free region in the centre of the
channel was used as a key feature for quantifying simulation and experimental similarity. Width of the
particle-free region in the experimental data is shown in black, while simulated results are shown in red.
Differences between simulated and experimental results are most likely due to error in the density scaling
factor
. In this work, this parameter was characterised for a particle density of 25 mg/ml at
applied excitation frequencies of 20 MHz and 100 kHz, however the effects of varied particle density was
not fully explored in this paper, as outlined in future work.
Future work is planned to characterise the influence of the density scaling factor
on trajectory
and velocity profile simulation output under a variety of different flow rates as well as particle densities
and a variety of different particle types. It is theorised form results presented in this work that this
simulation method can be adapted to a range of flow rates and densities through appropriate scaling of the
density scaling factor
.
34
Figure S7. Comparing the width of the particle-free region at different flow rates obtained from
experiments and simulations.
35
Supplementary S7. Cross-checking of “Dynamic Drag Force based on Iterative Density
Mapping” model
The integrity of the model has been verified through different checks. This includes running the model
in the absence of DEP force, leading to C=0 in equation (7), and assuming a constant velocity along the
cross section of the microchannel, leading to a constant D in equation (7). Moreover, considering the
small dimensions of particles, we can assume:
Under these assumptions, the velocity and location of suspended particles can be estimated using the
following simplified equations:
Our simulations confirm such estimations.
36