Online Supplementary Material
Forces acting on a particle in a concentration gradient under an externally
applied oscillating electric field
Yuan Luo, and Levent Yobasa)
Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong,
P. R. of China
Contents:
I.
Fitting parameters
S-2
II.
Governing Equations
S-2
III.
Boundary Conditions
S-4
IV.
Numerical Simulations
S-5
V.
Simulation Results
S-5
VI.
References
S-7
I.
Fitting parameters
⃑ 𝒑 | = 𝑘1 𝑉 3 + 𝑘2 𝑉 to describe the curves in Fig. 3(a)
The least-square fitting parameters, 𝑘1 and 𝑘2 , used in the relation |𝒗
are as listed in Table S-I.
⃑ 𝒑 | = 𝑘1 𝑉 3 + 𝑘2 𝑉.
TABLE S-I. Least-square fitting parameters |𝒗
𝑘1 × 1010 ,
mm/s (V-rms)3
𝑘2 × 105 ,
mm/s V-rms
10
3.29
2.54
25
4.37
1.54
50
3.59
0.94
100
1.55
0.86
𝑐,
μS/cm
II.
Governing Equations
Both the immersion medium and the electrolytes are based on the phosphate buffered saline (PBS) solution and thus
predominantly contain NaCl, a symmetric binary electrolyte (𝓏 + = −𝓏 − ≡ 𝓏0 ; 𝓏 + and 𝓏 − are the valance of the ionic species
Na+ and Cl−, respectively), for which the flux of ions is given by the Nernst-Planck equations:
𝑱± = −𝐷 ± ∇𝑐 ± ∓
𝓏0 𝑒 ± ±
𝐷 𝑐 ∇𝜙 + 𝑐 ± ⃑𝝊
𝑘𝐵 𝑇
(S-1)
where 𝐷 + = 1.4 × 10−9 m2/s and 𝐷 − = 2.0 × 10−9 m2/s are the diffusion coefficients of Na+ and Cl– ions, 𝑐 ± are their molar
concentrations, 𝑒 is the elementary charge, 𝑘𝐵 is the Boltzmann constant, 𝑇 is the absolute temperature, ⃑𝝊 is the solution
velocity vector, ∇ is the gradient operator, and 𝜙 is the electrical potential governed by the Poisson equation:
𝛻 2𝜙 = −
𝑁𝐴 𝑒𝓏0 +
(𝑐 − 𝑐 − )
𝜀𝑟 𝜀0
(S-2)
where 𝑁𝐴 is the Avogadro constant, 𝜀𝑟 = 79 is the relative permittivity of water, and 𝜀0 = 8.85 × 10−12 F/m is the vacuum
dielectric constant. The bold letters denote vectors.
In Eq. (S-1), the three terms on the right hand side refer to the contributions due to diffusion, electromigration and
convection of ions. In a typical microfluidic system, both the convective and diffusive contributions to the total current are
⃑ , where ⃑𝑬 is the
much smaller than that from electromigration.1 In view of these considerations, we can simply state 𝑱 = 𝜎𝑬
electric field vector given by −∇𝜙 and 𝜎 is the electrolyte conductivity by
𝓏0 𝑒
𝑘𝐵 𝑇
(𝐷 + 𝑐 + − 𝐷 − 𝑐 − ). Furthermore, the charge
conservation equation in the stationary state, 𝛻 ∙ 𝑱 = 0, for an oscillating electric field can be written as:
S-2
⃑)=0
𝛻 ∙ ((𝜎 + 𝑗𝜔𝜀𝑟 𝜀0 )𝑬
(S-3)
⃑ , and 𝑗 = √−1.
where 𝜔 = 2𝜋𝑓 is the angular frequency with 𝑓 being the temporal frequency of the applied electric field 𝑬
⃑ is a complex vector expressed as 𝑬
⃑ = Re[∇𝜙 sin(𝑗𝜔𝑡)] whereas 𝜙 is the electrical potential phasor.
Note that 𝑬
Under the quasi-electroneutrality assumption, the conductivity of a binary symmetric electrolyte (e.g. NaCl) can be
described by the convection–diffusion equation:2
𝐷𝑒𝑓𝑓 𝛻 2 𝜎 =
𝜕𝜎
⃑ ∙ 𝛻)𝜎
+ (𝝊
𝜕𝑡
(S-4)
where 𝐷𝑒𝑓𝑓 = 2𝐷 + 𝐷 − ⁄𝐷 + + 𝐷 − is an effective diffusivity ( 𝐷𝑒𝑓𝑓 = 1.65 × 10−9 m2/s). In the stationary state, the time
derivative in Eq. (S-4) is zero.
For the fluid motion in stationary state, Stokes equation is used due to low Reynolds number (𝑅𝑒~10−3 , for a typical
flow velocity of ~1 mm/s and ~50 µm channel dimension):
⃑𝑒
𝜂𝛻 2⃑𝝊 − 𝛻𝑃 = −𝒇
(S-5)
𝛻 ∙ ⃑𝝊 = 0
(S-6)
⃑ 𝒆 is the electrical body force, given by:
where 𝜂 = 10−3 Pa s is the dynamic viscosity of water, 𝑃 is the pressure and 𝒇
1 2
1
𝜕(𝜀 𝜀 )
2
⃑ 𝑒 = 𝜌𝑓 𝑬
⃑ − |𝑬
⃑ | 𝛻(𝜀𝑟 𝜀0 ) + 𝛻(𝜌𝑤 ( 𝑟 0 ) 𝑇 |𝑬
⃑| )
𝒇
2
2
𝜕𝜌𝑤
(S-7)
where 𝜌𝑓 and 𝜌𝑤 are the free electrical charge density and the mass density, respectively.
We assume that the permittivity of the electrolyte does not experience a noticeable change during the experiment and
hence consider ⃑𝒇𝒆 depends solely on the Columbic contribution 𝜌𝑓 ⃑𝑬.
The free charge density under an oscillating electric field in the bulk can be written from (S-2) and (S-3) as:
𝜌𝑓 = −
𝜀𝑟 𝜀0
𝛻𝜎 ∙ ⃑𝑬
𝜎 + 𝑗𝜔𝜀𝑟 𝜀0
(S-8)
and the time-average electrical body force is
⃑𝒆=
𝒇
1
⃑ ∗]
Re[𝜌𝑓 𝑬
2
where ⃑𝑬∗ is the complex conjugate of the electric field.
S-3
(S-9)
III.
Boundary Conditions
Fig. S-1 shows a 3D illustration and a 2D schematic layout of the computational domain whereby each labeled segment
(e.g. 𝑎𝑎′ , 𝑎𝑏) in the latter denotes a particular surface for which the boundary conditions were defined according to Table SII. The potential at the boundary 𝑎𝑎′ was assigned to the applied rms voltage (71 V-rms, 500 kHz) with respect to the ground
potential set at the boundary 𝑓𝑓 ′ . The walls facing the microcapillaries and the channel top and bottom were assumed
electrically insulating. The boundaries 𝑎𝑏, 𝑐𝑑 and 𝑒𝑓 were maintained at the prescribed ionic concentration and normal
inflow velocity while those 𝑎′ 𝑏 ′ , 𝑐 ′ 𝑑 ′ and 𝑒 ′ 𝑓 ′ were given by the convective flux and atmospheric pressure. No normal flux
(insulation) and no slip were assumed for all the remaining boundaries.
FIG. S-1. The computational domain: (a) 3D illustration and (b) 2D schematic layout (not true to scale).
TABLE S-II. Boundary conditionsa set for the computational domain in Fig. S-1
Boundaries
𝑎𝑎′
Eq S-3
𝜙 = 71
𝜙=0
Eq S-4
⃑)=0
⃑ ∙ (−𝐷𝑒𝑓𝑓 ∇𝜎 + 𝜎𝝊
𝒏
⃑)=0
⃑ ∙ (−𝐷𝑒𝑓𝑓 ∇𝜎 + 𝜎𝝊
𝒏
Eqs S-5 and S-6
⃑𝝊 = 0
⃑𝝊 = 0
𝑓𝑓 ′
𝑎𝑏, 𝑒𝑓
⃑ ∙𝑱=0
𝒏
𝜎 =15 mS/m
𝑣𝑥 = 4
𝑐𝑑
⃑ ∙𝑱=0
𝒏
𝜎 = 100, 50, 25 or 10 µS/m
𝑣𝑥 = 1
𝑎′𝑏′, 𝑐′𝑑′, 𝑒′𝑓′
⃑ ∙𝑱=0
𝒏
⃑ ∙ (−𝐷𝑒𝑓𝑓 ∇𝜎) = 0
𝒏
𝑃atm
Others
⃑ ∙𝑱=0
𝒏
⃑)=0
⃑ ∙ (−𝐷𝑒𝑓𝑓 ∇𝜎 + 𝜎𝝊
𝒏
⃑𝝊 = 0
a
⃑ ) = 0; Convective
Surface normal vector: ⃑𝒏; Electric insulation: ⃑𝒏 ∙ 𝑱 = 0; Insulation: ⃑𝒏 ∙ (−𝐷𝑒𝑓𝑓 ∇𝜎 + 𝜎𝝊
flux: ⃑𝒏 ∙ (−𝐷𝑒𝑓𝑓 ∇𝜎) = 0; The velocity component in the x direction (mm/s): 𝑣𝑥 ; Applied voltage in V-rms:
𝜙; Atmospheric pressure: 𝑃atm = 1.01 × 105 Pa
S-4
IV.
Numerical Simulations
Governing equations were numerically solved for the 3D geometry (Fig. S-1a) through the finite element method (FEM)
analyses in COMSOL Multiphysics Software 3.5 (Comsol Inc., MA). The electric field and convection–diffusion equations
were solved first to obtain the ionic concentration profile, which was subsequently used as an initial condition to the coupled
electric field, convection–diffusion, and flow equations. In the simulations, the electrolytes were set at an injection
conductivity of 15 mS/cm, while the particle immersion was assigned at 10, 25, 50 or 100 µS/cm. A tetrahedral mesh was
used with a maximum element size of 0.8 µm for the microcapillary domains and of 10 µm for the channel domains. The
chosen mesh was sufficiently fine. A finer mesh produced no significant change, indicating that the solutions are convergent
and grid independent.
V.
Simulation Results
Fig. S-2 shows the steady-state distribution of ionic concentration (conductivity) at a horizontal plane situated 10 µm
above the channel floor (electrolyte streams: 15 mS/cm; particle stream: 50 µS/cm; excitation: 71 V-rms 500 kHz). A
considerable influx of ions can be seen through the microcapillaries into the particle stream, which is in agreement with the
experimental observations (Fig. 2 of the main text).
15
Conductivity (mS/cm)
0.05
Electrolyte
y
Particle
x
Electrolyte
FIG. S-2. Steady-state concentration (conductivity) distribution across the horizontal plane situated 10 µm above the channel floor. The
dark dashed lines on the partitions represent the sidewalls of the microcapillaries located out of the horizontal plane. The electrolyte and
particle streams were set at an injection conductivity of 15 mS/cm and 50 µS/cm, respectively. Excitation: 71 V-rms, 500 kHz.
S-5
Fig. S-3a describes the electric field strength variation along a representative trajectory of 10 μm particles immersed in
distinct conductivities (legend) undergoing a trapping event (71 V-rms 500 kHz). The trajectory follows a diagonal path
originating at 𝑥 = 30, 𝑦 = 25, 𝑧 = 30 μm and terminating at 𝑥 = 10, 𝑦 = 5, 𝑧 = 10 μm in reference to the center point of a
microcapillary opening. As shown, the field intensity toward the microcapillary opening (origin) continues to monolithically
increase despite the immersion conductivity rise due to the electrolyte (15 mS/cm) influx under Maxwell-Wagner
polarization. We also evaluated the ∇ log(𝑐) averaged along the same trajectory in response to the increased applied rms
voltage and found a linear correlation (𝑅2 = 0.944) between the two (Fig. S-4).
Fig. S-3b shows the fluid velocity component at distinct immersion conductivity levels (legend) along the line that
represents the projection of the longitudinal symmetry axis of the microcapillary pair on the horizontal plane situated 10 μm
above the channel floor (dashed line in Fig. S-2). The flow velocity switches sign (direction) as the particle stream midpoint
(origin) is crossed, and it vanishes once the applied voltage is removed as shown for the simulated case corresponding to the
lowest immersion conductivity (10 μS/cm).
FIG. S-3. Line plots: (a) electric field strength along a representative trajectory of 10 μm particles undergoing a trapping event with the
applied oscillating electric field (following a diagonal line that originates at 𝑥 = 30, 𝑦 = 25, 𝑧 = 30, and terminates at 𝑥 = 10, 𝑦 = 5, 𝑧 =
10, all in μm in reference to the center point of a microcapillary opening located at the origin of the plot); (b) the fluid velocity component
along the line that represents the projection of the longitudinal symmetry axis of the microcapillary pair on the horizontal plane situated 10
μm above the channel floor (dashed line in Fig. S-2). The electrolyte streams were set at an injection conductivity of 15 mS/cm while the
particle stream was assigned at 10, 25, 50, and 100 μS/cm (legend). Excitation: 71 V-rms 500 kHz.
S-6
log (c) / (m–1)
10
×104
8
6
4
2
0
0
20
40
60
Voltage/(V-rms)
80
FIG. S-4. The concentration gradient (∇ log 𝑐) averaged along the representative trajectory (same as the one defined for Fig. S-3a) as a
function of the applied voltage rms magnitude. The solid line represents the best least-square fit (𝑅2 = 0.944). The electrolyte streams
were set at an injection conductivity of 15 mS/cm while the particle stream was assigned at 50 μS/cm. Excitation: 500 kHz.
References
1
A. Castellanos, A. Ramos, A. Gonzalez, N. G. Green, and H. Morgan, J. Phys. D: Appl. Phys. 36, 2584 (2003).
P. García-Sánchez, M. Ferney, Y. Ren and A. Ramos, Microfluid. Nanofluid. 13, 441 (2012).
2
S-7