Supplementary materials
Journal: Applied Physics Letters
Authors: Y. J. Lo and U. Lei
Title: Experimental validation of the theory of wall effect on dielectrophoresis
Appendix I: Analytical expressions for the approximate electric fields
in the test sections
In order to apply Eqs. (1) and (3) to calculate the DEP forces, we need expressions
for the electric fields. For the present approximate uniform field in Fig. 1(a) and the
approximate radial field in Fig. 2(a), the analytic expressions are derived as follows.
(A) The approximate uniform field
Consider the rectangular channel in Fig. 1(a). When H is much less than L and Le ,
the field in the test section is approximate uniform except near the inlet and the outlet,
and can be expressed as Eu  E0e jt x . The magnitude E0 is 2V0 / L if the
potentials
 V e jt 
0
are applied at the inlet (x  0) and the outlet (x  L) planes of
the test section instead of at the electrodes. In practice, the magnitude of the potential
at the inlet (also the outlet) plane is slight less than V0 . The field near the inlet and
the outlet are two-dimensional (under the condition H << w), and is a function of x
and y. In order to account for such an inlet/outlet effect, we modified the field
magnitude as
E0  CEU 2V0 / L,
(S1)
where CEU is a modified factor, which can be obtained through a successive
approximation as discussed below. As H  L and Le in Fig. 1(a), we may replace
Le by  for problem solving if our goal is to find an approximate uniform field
inside the test section. The problem was solved by separating it into three parts: (i) a
two-dimensional potential problem in the xy-plane from   x  0 , called
 I  x, y  , (ii) a one dimensional potential problem along x from 0  x  L , called
 II  x  , and (iii) a two-dimensional potential problem in the xy-plane from
L  x  , called  III  x, y  . A successive approximation procedure is adopted for
1
obtaining the solution as follows. (a) Solve the one-dimensional Laplace equation for
 II  x  with specified potential Vin and Vout at x = 0 and L, respectively. (b) Solve
the two-dimensional Laplace equation for I  x, y  with  I  V0 at y  0,
 I / y  0 at y  H ,  I / x  d  II / dx at x 0, and finite  I as x  .
(c) Solve
 III  x, y  with
 III  V0
y  0,
at
 III / y  0
at
y  H,
 III / x  d  II / dx at x L, and finite  III as x  . (d) Calculate the average
values of  I  0, y  and  III  L, y  at the inlet and outlet planes of the test section,
and take them as the updated values for Vin and Vout , respectively. (e) Repeat
procedures (a) – (d) until convergent result is obtained. The modified factor CEU in
Eq. (S1) is Vin  Vout  /  2V0  . We started the above iteration with Vin  V0 and
Vout  V0 . The i  approximation was obtained as
th
i
i 1
CEU
 1  DCEU
,
(S2)
1
 1 and
with CEU



D  32 /  3  H / L    2n  1  1.0855  H / L  .
3
n 1
(S3)
D is a parameter of the length ratio, H/L. It generally takes three to four iterations for
a converged result.
(B) The approximate radial field
Similar idea and method as above were applied to obtain the approximate radial
field, E  r  e jt r , in the test section of Fig. 2(a). The field magnitude E  r  is
E r  E
0
r 
2V0 1
,
ln  r2 / r1  r
(S4)
if the applied potentials are applied at the inlet and the outlet planes of the test section.
Here we modify it as
E  r   CER
2V0 1
,
ln  r2 / r1  r
(S5)
by taking into account the inlet/outlet effect. The modified factor CER for the
radial field was obtained through a successive approximation similar to that for the
2
uniform field, except that the x-coordinate is replaced by the r-coordinate, and the
xy-plane is replaced by the ry-plane. The result is
i
i 1
CER
 1  BCER
,
(S6)
1
 1 and
where C ER
B

1
8/ 2

ln  r2 / r1  n 1  2n  1 2
 1 I 0  kn r1 
1 K 0  kn r2  


,
 kn r1 I1  kn r1  kn r2 K1  k n r2  
(S7)
with I 0  kn r1  and I1  kn r1  the modified Bessel functions of the first kind,
K0  kn r2  and K1  kn r2  the modified Bessel functions of the second kind, and
kn   2n  1  /  2H  . The parameter B depends on the length ratios r2 / r1 , r2 / H
and r1 / H . It generally takes three to four iterations for a converged result.
The validity of both the analytical modified uniform and radial fields in Eqs. (S1)
and (S5) above were checked against numerical simulations. Figure S1 compares
the analytical and numerical results for the radial field. The numerical calculation
was performed with the aid of a commerical software, COSMOL. The paremeters
for the simulation are r1  300m , r2  700m , H  30m and V0  10 V . The
numerical result was calculated by solving the Laplace equation in the ry-plane in
Fig. 1(a) from r = 0 to r4 , with r2  r4  r3 and r4  r2  H , subject to specified
potentials on the electrodes and vanishing normal potential gradients at all other
boundaries. Figure S1 shows contours of the normalized gradient of the square of
the field magnitude, which is directly proportional to the DEP force parallel to the
wall at y = 0 in Fig. 2(a) as shown in Eq. (3a). The numerical result in Fig. S1(b)
shows that the distribution is approximately radial (with vertical contour lines in
the figure) between 0.2   r  r1  /  r2  r1   0.8 , which supports that we indeed can
generate an approximate radial field in the test section of Fig. (2a) where it is not
too close to both ends. However, there is a substantial shift of the contour patterns
between the zeroth order theory based on Eq. (S4) in Fig. S1(a) and the numerical
result in Fig. S1(b), which indicates that there exists certain error if the zeroth
order theory of the electric field is employed for the evaluation of the DEP force.
On the other hand, Fig. S1(c) shows that the result using the present modified
theory in Eq. (S5) agrees much better with the numerical solution. Thus Eq. (S5) is
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adequate for describing the approximate radial field generated in the test section of
Fig. 2(a) between 0.2   r  r1  /  r2  r1   0.8 . Our experimental data in Fig. 2(c)
agrees nicely with the theory for 400 m < r < 600 m, which corresponds to
0.25   r  r1  /  r2  r1   0.75 in Fig. S1.
FIG. S1. The contour lines for the normalized result of the gradient of the square


of the field magnitude, V02  r2  r1   E 2 / 2 / r , using (a) the zeroth
3
order theory in Eq. (S4), (b) the numerical calculation, and (c) the
modified theory in Eq. (S5).
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Appendix II: More results for the DEP force parallel to the wall
Figure S2 shows further experimental results for the validation of the DEP force
parallel to the wall using Eqs. (4a) and (5). There are four tests with different
combinations for two channel heights (H) and two electric frequencies (f). The result
for H  21 m and f 10 kHz was shown in Fig. 2(c) before, and is also presented
here for comparison. The results for different cases are similar, and the discussion of
Fig. 2(c) is also applied here. Table SI compares the experiment and the theory via the
wall effect coefficient, Cd , of the Stokes’ drag. Cdt is the theoretical value
calculated using Eq. (5), Cde is calculated using Eq. (4a) with the experimental
values of U, Cde is the mean value of Cde , and  is the standard deviation of Cde .


Note that Cde  Cdt / Cdt  is the discrepancy between the experiment and the DEP
theory without wall effect, as only
F(t )

is employed in Eq. (4a). The
discrepancies are within 8.1%, which implies that DEP theory in an infinite medium
(without wall effect) can be employed for predicting the DEP force parallel to the wall
in the vicinity of the wall. The wall effect is Fr in Eq. (3b), and its contribution is
within the discrepancy for all the cases in Fig. S2.
Table SI: Comparison between the theory and the experiment for the DEP force
parallel to the wall.
C

H 
f
C 
Cde
 
21 μm
1 kHz
2.16
2.21
0.17
2.3%
21 μm
10 kHz
2.16
2.23
0.21
3.2%
28 μm
1 kHz
1.86
1.95
0.12
4.8%
28 μm
10 kHz
1.86
1.71
0.11
8.1%
t
d
5
e
d
 Cdt / Cdt
FIG. S2. Experimental results for the validation of the DEP force parallel to the wall.
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