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Eccentric electrophoretic motion of a rectangular particle in a rectangular microchannel
Journal of Colloid and Interface Science 342 (2010) 638–642
Contents lists available at ScienceDirect
Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Eccentric electrophoretic motion of a rectangular particle in a rectangular microchannel Dongquing Li *, Yasaman Daghighi Department of Mechanical and Mechatronics Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1
a r t i c l e
i n f o
Article history: Received 18 September 2009 Accepted 20 October 2009 Available online 24 October 2009 Keywords: Electrophoretic mobility Eccentric motion Particle Microchannel Electrokinetics Microfluidics
a b s t r a c t Understanding of the effects of the boundary – the channel walls – on the electrophoretic motion of particles in microchannels is very important. This paper developed an analytical solution of the electrophoretic mobility for eccentric motion of a rectangular particle in a rectangular microchannel. The simple geometry of the system does not limit the generality of the qualitative prediction of the model and the analytical solution. Several special cases are studied, and the effects of the degree of the eccentricity, the particle’s size relative to the channel’s size, and the relative zeta potentials on the particle’s mobility are discussed. For the case where the particle’s cross-section area is close to the cross-section area of the microchannel, the model’s predictions are compared with the published experimental results and good agreement was found. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Manipulation of cells or particles in microchannels is an essential step in many microfluidic lab-on-a-chip devices. Among the variety of techniques covering optical [1], electrical [2,3], acoustic [4], and magnetic [5] means, electrophoresis may be the simplest method that can accomplish this task. Distinct from the classical electrophoresis of a colloidal particle suspended in an unbounded and stationary electrolyte solution [6], the presence of channel walls causes at least three effects on the electrophoretic motion of cells or particles in microfluidic devices. Firstly, the surface charge on the channel walls gives rise to the electroosmotic flow of the suspending liquid; secondly, the electrically insulating walls alter the distribution of electric field around cells or particles, which in turn affects the local electrophoretic force; thirdly, the channel walls influence the flow field in the gap between the particle and the wall, and hence the viscous retardation of cells or particles. Therefore, it is highly desirable to understand the effects of the microchannel walls on the particle’s electrophoretic motion. Recent analytical works of the boundary effects on a particle’s motion are limited to a particle moving near a large planar surface. They fall into two categories: thick electric double layers assumption [7–10] and thin electrical double layers assumption [11–14]. In most practical applications, including applications in microfluidic systems, the thickness of the electric double layers, Debye length, is of the order of nanometers, and is usually much smaller * Corresponding author. E-mail address: [email protected] (D. Li). 0021-9797/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2009.10.047
than the characteristic length scale of the system (e.g., channel width or particle dimension). Hence, the velocity variation in such a thin electric double layers can be neglected and the liquid in the electric double layers is assumed to move at the electroosmotic flow velocity relative to the particle, given by ueof ¼ efp E=lf , where e and lf are the permittivity and viscosity of the electrolyte solution, respectively, fp is the zeta potential of the surface of the particle or the channel walls, and E is the externally applied electric field. Therefore, the Stokes equation with a ‘slip’ boundary condition is used to describe such a flow based on the thin electric double layers assumption. Keh and Anderson [12] studied the wall effects on electrophoretic motion of colloidal spheres using the method of reflection. However, this method is not valid for the cases where the particle is very close to the wall. Later, Keh and Chen [13] developed an exact analytical solution for a particle’s electrophoretic motion in the proximity of a large non-conducting plane, which is applicable for any particle–wall separation distances. They found that the particle velocity increases as the separation distance decreases. Recently, Yariv and Brenner [14] computed the asymptotic behavior of the electrophoretic motion of a sphere very close to a wall using a reciprocal theorem which can calculate the force and torque without solving the actual Stokes equation explicitly. Their results are complementary to the exact solutions of Keh and Anderson [12]. There are few analytical papers devoted to the investigation of wall effects on particle electrophoretic motion in microchannels. However, numerical simulations of particle electrokinetic motion in microchannels have been reported [15–19]. It was found, for example, for a spherical particle moving eccentrically in a
D. Li, Y. Daghighi / Journal of Colloid and Interface Science 342 (2010) 638–642
microchannel [18], the particle velocity increases as the separation distance between the particle and the channel wall decreases. This is attributed to the net effect of the strongest electric field in the small gap region. There are few experimental studies of the effects of microchannel’s walls on particle electrophoretic motion. Recently, Xuan et al. [20,21] visualized the near-wall electrophoretic motion of spherical polystyrene particles in glass capillaries of circular cross-section. The velocities of two groups of particles were found to increase with the decrease of capillary diameter, which is in qualitative agreement with Ye and Li’s [18] numerical simulation. It is very difficult, if possible, to obtain an analytical solution of the effects of the curved microchannel walls on the electrophoretic motion of spherical particles. This paper therefore considers the simplest model: a rectangular particle moving eccentrically in a rectangular microchannel. However, the qualitative effects derived from this simple model remain valid for other geometries of particles and channels. Several special cases of this model are discussed in this paper. 2. Modeling Consider a rectangular particle moving in an aqueous solution in a rectangular microchannel under an applied electrical field, as illustrated in Fig. 1. The particle’s dimensions are a (height), b (width), and c (length), respectively. The dimensions of the crosssection of the microchannel are W (width) and H (height), respectively. The particle moves symmetrically with respect to the two side walls of the rectangular channel. However, because of the density difference and the gravity effect, the particle moves close to the bottom of the channel wall with a separation distance d. For simplicity, we consider that the particle bears uniform surface charge or has a uniform zeta potential around the particle surface. Furthermore, we consider that the particle is made of a non-conducting material, and the particle’s size is comparable with the cross-section dimensions of the microchannel. Under the applied electric field, there are two forces acting on the particle, the electric force and the hydrodynamic (flow friction) force. Consider that the particle’s motion is in a steady state. This requires that the net force on the particle is zero. That is,
Fe ¼ Ff
ð1Þ
Use the flat plate model of electric double layer (EDL), the surface charge density is given by, for symmetrical ions,
r0 ¼
4n1 ze
j
sinh
ze1p 2kb T
r0 ffi
2n1 ðzeÞ2 f j kb T p
ð3Þ
The total surface charge of such a rectangular particle is given by
Q ¼ r0 A ¼
2n1 ðzeÞ2 fp ½2ab þ 2ac þ 2bc j kb T
ð4Þ
The electric force acting on the particle is
F 2 ¼ QE ¼
4n1 ðzeÞ2 fp ½ab þ ac þ bcE j kb T
ð5Þ
where E is the local electric field strength at the position of the particle. It should be noted that the particle is made of a non-conducting material, and only the liquid conducts electricity. If the particle’s size is comparable with the cross-section dimensions of the microchannel, the applied electric field strength around the particle is different from the rest part of the channel. Consider a section of the microchannel without the particle; the electric current through the liquid in the microchannel is given by
I¼
DV 0 DV 0 DV 0 ¼ ¼ kS0 ¼ E0 kS0 R ðL=kS0 Þ L
ð6Þ
where E0 is the electric field strength (V/m) in the section of the microchannel without the particle, k is the liquid electrical conductivity, and S0 is the microchannel cross-section area occupied by the liquid without the presence of the particle. For the section of the microchannel with the non-conducting particle, the electrical current through the liquid in the microchannel is given by
I¼
DV DV DV ¼ ¼ kS ¼ EkS R ðL=kSÞ L
ð7Þ
where E is the electric field strength (V/m) in the section of the microchannel with the particle, k is the liquid electrical conductivity, S is the cross-section area of the microchannel that is occupied by the liquid in the presence of the non-conducting particle. Realizing that the electric current is constant throughout the microchannel, we equalize the above two equations and obtained
or
E¼
S0 HW E0 E0 ¼ HW ab S
ð8Þ
Using Eq. (8) to replace E in Eq. (5), we have
F 2 ¼ QE ¼
ð2Þ
4n1 ðzeÞ2 fp HW E0 ðab þ ac þ bcÞ HW ab j kb T
ð9Þ
2.2. Flow friction force, Ff, on the particle
where j is the Debye–Huckel parameter.
Under an applied electric field, the particle undergoes electrophoretic motion while the liquid in the microchannel moves electroosmotically. The particle will experience the flow friction force. As the distances between the particle’s surfaces and the microchannel walls are small, the shear stress on the particle surface can be estimated as:
(H-d-a) H
(W-b)/2
For the cases of small zeta potential, i.e., f < 25 mV,
E S0 ¼ >1 E0 S
2.1. Electric force, Fe, on the particle
639
a
c b
d W Fig. 1. Schematic of a rectangular particle moving eccentrically in a rectangular microchannel. d is the separation distance between the particle and the bottom wall of the microchannel.
s¼l
dV DV ¼l dy Dy
As illustrated in Fig. 2, the velocity difference is given by
DV ¼ V e of ;W þ V P ¼
e0 e fw E þ VP l
ð10Þ
where Veof,W is the electroosmotic flow velocity at the microchannel walls and Vp is the particle’s velocity.
640
D. Li, Y. Daghighi / Journal of Colloid and Interface Science 342 (2010) 638–642
2
4n1 ðzeÞ2 4 j e0 ek b T
Veof, W (H-d-a) VP
HW ðab þ ac þ bcÞ ðHWabÞ 4ac Wb
bc þ Hda þ bcd
HW ðHW abÞ
3
1w 5 1p
reflect the influences of the boundary (the microchannel’s walls) on the electrophoretic motion of the particle. It shows the effects of the bc þ bc ), the particle’s size relative to degree of the eccentricity (Hda d HW the channel’s size (HWab), and the relative zeta potentials (11w ) on
H a
p
the particle’s mobility. As indicated by the negative sign of the second term in the bracket, the zeta potential of the channel wall, 1w , tends to reduce the particle’s mobility. When considering the electrophoretic retardation effect, Eq. (11), the mobility equation becomes:
d Fig. 2. Illustration of the flow field above and below the particle.
It should be noted that, if we consider the effect of the electrophoretic retardation, the electroosmotic flow, Veof,P, around the particle (in the particle’s EDL) should be considered. That is,
e0 e fp DV ¼ V e of ;W þ ðV P V e of ;p Þ where V e of ;p ¼ E l
e0 efw E Dy
þl
VP 1 ¼ ½e0 efw E þ lV P Dy Dy
Apparently, the retardation effect is reflected by the second term in the bracket. If fw < fp, this term will contribute to increase the particle’s mobility.
ð12Þ
3. Discussion 3.1. Special case – slit microchannel
The total flow friction force on the particle is then given by
Ff ¼
Z
s dA ¼ 2F f ; side þ F f ; up þ F f ; low
ð13Þ
where the subscripts side, up, and low represent the side surfaces, the upper surface, and the bottom surface of the rectangular particle, respectively.
2ac 2e0 e fw E 2l V p F f ; side ¼ þ ac ¼ e0 e fw E þ l V p W b W b W b bc F f ; up ¼ e0 efw E þ V p l Hda bc F f ; low ¼ e0 e fw E þ lV p d
ð14Þ ð15Þ ð16Þ
Finally, the total friction force acting on the particle is
Ff ¼
e0 efw E þ lV p
4ac bc bc þ þ W b Hda d
ð17Þ
HW Recall E ¼ SS0 E0 ¼ HWab E0 , we have
Ff ¼
e0 e fw E0
HW ab
ðHWÞ þ lV p
4ac bc bc þ þ W b Hda d
ð18Þ
2.3. Particle mobility From the force balance, F e ¼ F f , Eqs. (9) and (18), the mobility of the eccentric electrophoretic motion of a rectangular particle in a rectangular microchannel can be derived as
V p e0 e1 ¼ E0 l
2
4n1 ðzeÞ2 p 4 j e0 ek b T
HW ðab þ ac þ bcÞ ðHWabÞ 4ac bc bc þ Hda þ d Wb
HW ðHW abÞ
3
1w 5 1p
ð19Þ
Comparing Eq. (19) with the classical mobility equation without the boundary effects,
V p e0 e1p ¼ E0 l Clearly, the terms in the bracket
ð20Þ
ð11Þ
Thus, from Eq. (10),
s¼
2 3 4n1 ðzeÞ2 HW V p e0 e1p 4j e0 e kb T ðab þ ac þ bcÞ ðHWabÞ HW ð1w 1P Þ5 ¼ 4ac bc ðHW abÞ E0 l 1P þ Hda þ bcd Wb
If W H, W a, W b, W c, W d, and HW ab, that is, a slit channel, Eq. (20) is reduced to
2 3 4n1 ðzeÞ2 V p e0 e1p 4j e0 e kb T ðab þ ac þ bcÞ ð1w 1P Þ5 ¼ bc E0 l 1P þ bcd Hda
ð20aÞ
bc That is, in a slit channel, only the eccentricity (Hda þ bc ) and the reld
1w 1p 1p )
ative zeta potentials (
affect the particle’s mobility.
If the particle is very close to the bottom surface of the slit microchannel wall, i.e., d (H – d a), then, the above equation is reduced further to
2 3 4n1 ðzeÞ2 V p e0 e 1p 4j e0 e kb T ðab þ ac þ bcÞ d ð1w 1p Þ5 ¼ bc E0 l 1p
ð20bÞ
where (ab + ac + bc) is half of the total surface area of the particle; bc is the particle’s bottom surface area facing the bottom channel wall; d is the separation distance. In this case, only the separation distance from the bottom wall (d) and the relative zeta potentials 1 1 ( w1 p ) affect the particle’s mobility. For a given particle (i.e., fixed p a, b, and c) and given zeta potential values, fw and fp, the mobility will be lower when the particle moves closer to the wall (i.e., smaller d). If a DNA molecule is modeled as a thin cylinder, i.e., a c, b c, then ab ac and bc. For further simplifying the analysis, let a = b, the (ab + ac + bc) (ac + bc) = 2ac. The above equation (Eq. (20b)) is further reduced to:
" # ð1w 1p Þ V p e0 e1p 8n1 ðzeÞ2 ¼ d E0 l j e0 e kb T 1p
ð20cÞ
This equation describes the mobility of a thin and long cylinder aligned parallel to the axis of a slit flow channel or parallel to the applied electrical field and the flow field. It shows that the mobility is independent of the cylinder length (e.g., the length or number of base pair of a DNA molecule). This is because of the assumption of c (a, b). This implies that if c > 100a or 100b, all cylinder will have the same mobility. From Eq. (20c), it is clear that the closer the particle moves along the bottom channel wall (i.e., smaller d), the smaller the particle’s mobility.
D. Li, Y. Daghighi / Journal of Colloid and Interface Science 342 (2010) 638–642
641
3.2. Special case – flow over a flat surface or a very small particle Let us consider a special case when H ? 1, W ? 1. This is equivalent to the case where a particle moves close to an infinitely large flat surface with a distance d. In this case,
HW ! 1; HW ab
4ac ! 0; W b
bc !0 Hda
Eq. (20) becomes Eq. (20b),
2 3 4n1 ðzeÞ2 V p e0 e1p 4j e0 e kb T ðab þ ac þ bcÞ d ð1w 1p Þ5 ¼ bc E0 l 1p
ð20bÞ
where (ab + ac + bc) is half of the total surface area of the particle; bc is the particle’s bottom surface area facing the bottom channel wall; d is the separation distance. For the case of a particle with a very small cross-section (ab) in comparison with the cross-section of the channel (WH), i.e., WH ab, and W and H a, b, c, d, ac, and bc, the particle mobility equation, Eq. (20), is also reduced to Eq. (20b). As indicated by Eq. (20b), for a particle moving parallel to a flat wall or a very small particle moving in a large microchannel, the electrokinetic mobility is affected only by the separation distance from the bottom wall (d) and the relative zeta potentials 1 1 ( w1 p ). The particle’s mobility is proportional to the separation disp tance d. 3.3. Special case – the particle’s cross-section area similar to that of the channel Consider the case where the particle’s cross-section area is close to the cross-section area of the microchannel. To discuss qualitatively the effect of the ratio of the cross-section areas, let us assume a = b = c, ab = a2, H = W, HW = A2, and a d, H d. Eq. (20) can be rewritten as
2 3 12n1 ðzeÞ2 V p e0 e1p 1 ð 1 1 Þ j e e k T P 5 0 b 4 ¼ w 5 E0 l ð1 a2 =A2 Þ Að1a=AÞ 1P þ 1d For simplicity, assume ¼ A2 ð1 a=AÞ. d ¼ Aa 2
A
(=H) = a + 2d,
2 3 12n1 ðzeÞ2 V p e0 e1p 1 ð 1 1 Þ j e e k T w P 0 b 4 5 ¼ 5 2 E0 l ð1 a2 =A2 Þ Að1a=AÞ 1p þ Að1a=AÞ
and
hence
ð20dÞ
Clearly, in addition to the relative zeta potential, the relative cross-section area ratio of the particle to the channel, ð1 a=AÞ, plays an important role in determining the effective mobility in this case. In order to examine the correctness of this model, the electrophoretic mobility of rectangular particles moving in rectangular microchannels (Fig. 1) is calculated and plotted by using Eq. (20d). Fig. 3 shows the comparison of the model predication with the published experimental data of the electrophoretic mobility of spherical polystyrene particles in different capillaries of circular cross-section [20]. In the experiments, the near-wall electrophoretic motion of spherical polystyrene particles in fused silica capillary tubes of circular cross-section is visualized and measured. It was found that the particles move faster when the particles’ cross-section area becomes closer to that of the capillary tubes. In the model calculation, zeta potentials of the particle and the wall are chosen to be 25 and 40 mV, respectively. As seen from Fig. 3, the qualitative agreement between the model predictions and the experimental results is very good. The quantitative differences can be largely attributed to the fact that the model (Eq. (20d)) is based on a rectangular particle in a rectangular microchannel; the
Fig. 3. Comparison of the prediction of Eq. (20d) with the experimental results [20] of the electrophoretic mobility of spherical polystyrene particles in different capillaries filled with 25 mM sodium carbonate (pH = 8.5) buffer. The symbol points are the experimental data points. The solid lines are the model prediction of the electrophoretic mobility of rectangular particles in rectangular microchannels.
geometry is significantly different from the spherical particles in circular cross-section channels. 4. Summary An analytical model of the electrophoretic mobility of a rectangular particle moving in a rectangular microchannel is derived in this paper. Using this model, the effects of the boundary – the microchannel walls – on the particle’s mobility are discussed for several special cases. In the case of a slit microchannel, only the separation distance from the bottom wall (d) and the relative zeta 1 1 potentials ( w1 p ) affect the particle’s mobility. For a given particle p and given zeta potential values, the mobility is lower when the particle moves closer to the wall (i.e., smaller d). If a DNA molecule is modeled as a thin and long cylinder aligned parallel to the axis of a slit flow channel, the model indicates that the mobility is independent of the cylinder length (e.g., the length or number of base pair of a DNA molecule). For a particle moving parallel to a flat wall or for a very small particle moving in a large microchannel, the electrokinetic mobility is affected only by the separation distance 1 1 from the bottom wall (d) and the relative zeta potentials ( w1 p ). p The particle’s mobility is proportional to the separation distance d. For the case where the particle’s cross-section area is close to the cross-section area of the microchannel, the model reveals that 1 1 the relative zeta potential, ( w1 p ), and the relative cross-section p area ratio of the particle to the channel, ð1 a=AÞ, are key parameters in determining the electrokinetic mobility. This is in agreement with the published experimental results. References [1] [2] [3] [4] [5] [6]
D.G. Grier, Nature 424 (2003) 810. P.C.H. Li, D.J. Harrison, Anal. Chem. 69 (1997) 1564. P.R.C. Gascoyne, J. Vykoukal, Electrophoresis 23 (2002) 1973. J.J. Hawkes, W.T. Coakley, Sens. Actuators, B 75 (2001) 213. K.E. McCloskey, J.J. Chalmers, M. Zborowski, Anal. Chem. 75 (2003) 6868. R.J. Hunter, Zeta Potential in Colloid Science: Principles and Applications, Academic Press, New York, 1981. [7] R.G. Cox, J. Fluid Mech. 338 (1997) 1. [8] J. Ennis, J.L. Anderson, J. Colloid Interface Sci. 185 (1997) 497.
642 [9] [10] [11] [12] [13] [14] [15]
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[16] [17] [18] [19] [20] [21]
C. Ye, D. Sinton, D. Erickson, D. Li, Langmuir 18 (2002) 9095–9101. C. Ye, D. Li, J. Colloid Interface Sci. 272 (2004) 480–488. C. Ye, D. Li, Microfluid Nanofluid 1 (2004) 52–61. C. Ye, D. Li, Microfluid Nanofluid 1 (2005) 234–241. X. Xuan, C. Ye, D. Li, J. Colloid Interface Sci. 289 (2005) 286–290. X. Xuan, S. Raghibizadeha, D. Li, J. Colloid Interface Sci. 296 (2) (2006) 743– 748.
Contents lists available at ScienceDirect
Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Eccentric electrophoretic motion of a rectangular particle in a rectangular microchannel Dongquing Li *, Yasaman Daghighi Department of Mechanical and Mechatronics Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1
a r t i c l e
i n f o
Article history: Received 18 September 2009 Accepted 20 October 2009 Available online 24 October 2009 Keywords: Electrophoretic mobility Eccentric motion Particle Microchannel Electrokinetics Microfluidics
a b s t r a c t Understanding of the effects of the boundary – the channel walls – on the electrophoretic motion of particles in microchannels is very important. This paper developed an analytical solution of the electrophoretic mobility for eccentric motion of a rectangular particle in a rectangular microchannel. The simple geometry of the system does not limit the generality of the qualitative prediction of the model and the analytical solution. Several special cases are studied, and the effects of the degree of the eccentricity, the particle’s size relative to the channel’s size, and the relative zeta potentials on the particle’s mobility are discussed. For the case where the particle’s cross-section area is close to the cross-section area of the microchannel, the model’s predictions are compared with the published experimental results and good agreement was found. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Manipulation of cells or particles in microchannels is an essential step in many microfluidic lab-on-a-chip devices. Among the variety of techniques covering optical [1], electrical [2,3], acoustic [4], and magnetic [5] means, electrophoresis may be the simplest method that can accomplish this task. Distinct from the classical electrophoresis of a colloidal particle suspended in an unbounded and stationary electrolyte solution [6], the presence of channel walls causes at least three effects on the electrophoretic motion of cells or particles in microfluidic devices. Firstly, the surface charge on the channel walls gives rise to the electroosmotic flow of the suspending liquid; secondly, the electrically insulating walls alter the distribution of electric field around cells or particles, which in turn affects the local electrophoretic force; thirdly, the channel walls influence the flow field in the gap between the particle and the wall, and hence the viscous retardation of cells or particles. Therefore, it is highly desirable to understand the effects of the microchannel walls on the particle’s electrophoretic motion. Recent analytical works of the boundary effects on a particle’s motion are limited to a particle moving near a large planar surface. They fall into two categories: thick electric double layers assumption [7–10] and thin electrical double layers assumption [11–14]. In most practical applications, including applications in microfluidic systems, the thickness of the electric double layers, Debye length, is of the order of nanometers, and is usually much smaller * Corresponding author. E-mail address: [email protected] (D. Li). 0021-9797/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2009.10.047
than the characteristic length scale of the system (e.g., channel width or particle dimension). Hence, the velocity variation in such a thin electric double layers can be neglected and the liquid in the electric double layers is assumed to move at the electroosmotic flow velocity relative to the particle, given by ueof ¼ efp E=lf , where e and lf are the permittivity and viscosity of the electrolyte solution, respectively, fp is the zeta potential of the surface of the particle or the channel walls, and E is the externally applied electric field. Therefore, the Stokes equation with a ‘slip’ boundary condition is used to describe such a flow based on the thin electric double layers assumption. Keh and Anderson [12] studied the wall effects on electrophoretic motion of colloidal spheres using the method of reflection. However, this method is not valid for the cases where the particle is very close to the wall. Later, Keh and Chen [13] developed an exact analytical solution for a particle’s electrophoretic motion in the proximity of a large non-conducting plane, which is applicable for any particle–wall separation distances. They found that the particle velocity increases as the separation distance decreases. Recently, Yariv and Brenner [14] computed the asymptotic behavior of the electrophoretic motion of a sphere very close to a wall using a reciprocal theorem which can calculate the force and torque without solving the actual Stokes equation explicitly. Their results are complementary to the exact solutions of Keh and Anderson [12]. There are few analytical papers devoted to the investigation of wall effects on particle electrophoretic motion in microchannels. However, numerical simulations of particle electrokinetic motion in microchannels have been reported [15–19]. It was found, for example, for a spherical particle moving eccentrically in a
D. Li, Y. Daghighi / Journal of Colloid and Interface Science 342 (2010) 638–642
microchannel [18], the particle velocity increases as the separation distance between the particle and the channel wall decreases. This is attributed to the net effect of the strongest electric field in the small gap region. There are few experimental studies of the effects of microchannel’s walls on particle electrophoretic motion. Recently, Xuan et al. [20,21] visualized the near-wall electrophoretic motion of spherical polystyrene particles in glass capillaries of circular cross-section. The velocities of two groups of particles were found to increase with the decrease of capillary diameter, which is in qualitative agreement with Ye and Li’s [18] numerical simulation. It is very difficult, if possible, to obtain an analytical solution of the effects of the curved microchannel walls on the electrophoretic motion of spherical particles. This paper therefore considers the simplest model: a rectangular particle moving eccentrically in a rectangular microchannel. However, the qualitative effects derived from this simple model remain valid for other geometries of particles and channels. Several special cases of this model are discussed in this paper. 2. Modeling Consider a rectangular particle moving in an aqueous solution in a rectangular microchannel under an applied electrical field, as illustrated in Fig. 1. The particle’s dimensions are a (height), b (width), and c (length), respectively. The dimensions of the crosssection of the microchannel are W (width) and H (height), respectively. The particle moves symmetrically with respect to the two side walls of the rectangular channel. However, because of the density difference and the gravity effect, the particle moves close to the bottom of the channel wall with a separation distance d. For simplicity, we consider that the particle bears uniform surface charge or has a uniform zeta potential around the particle surface. Furthermore, we consider that the particle is made of a non-conducting material, and the particle’s size is comparable with the cross-section dimensions of the microchannel. Under the applied electric field, there are two forces acting on the particle, the electric force and the hydrodynamic (flow friction) force. Consider that the particle’s motion is in a steady state. This requires that the net force on the particle is zero. That is,
Fe ¼ Ff
ð1Þ
Use the flat plate model of electric double layer (EDL), the surface charge density is given by, for symmetrical ions,
r0 ¼
4n1 ze
j
sinh
ze1p 2kb T
r0 ffi
2n1 ðzeÞ2 f j kb T p
ð3Þ
The total surface charge of such a rectangular particle is given by
Q ¼ r0 A ¼
2n1 ðzeÞ2 fp ½2ab þ 2ac þ 2bc j kb T
ð4Þ
The electric force acting on the particle is
F 2 ¼ QE ¼
4n1 ðzeÞ2 fp ½ab þ ac þ bcE j kb T
ð5Þ
where E is the local electric field strength at the position of the particle. It should be noted that the particle is made of a non-conducting material, and only the liquid conducts electricity. If the particle’s size is comparable with the cross-section dimensions of the microchannel, the applied electric field strength around the particle is different from the rest part of the channel. Consider a section of the microchannel without the particle; the electric current through the liquid in the microchannel is given by
I¼
DV 0 DV 0 DV 0 ¼ ¼ kS0 ¼ E0 kS0 R ðL=kS0 Þ L
ð6Þ
where E0 is the electric field strength (V/m) in the section of the microchannel without the particle, k is the liquid electrical conductivity, and S0 is the microchannel cross-section area occupied by the liquid without the presence of the particle. For the section of the microchannel with the non-conducting particle, the electrical current through the liquid in the microchannel is given by
I¼
DV DV DV ¼ ¼ kS ¼ EkS R ðL=kSÞ L
ð7Þ
where E is the electric field strength (V/m) in the section of the microchannel with the particle, k is the liquid electrical conductivity, S is the cross-section area of the microchannel that is occupied by the liquid in the presence of the non-conducting particle. Realizing that the electric current is constant throughout the microchannel, we equalize the above two equations and obtained
or
E¼
S0 HW E0 E0 ¼ HW ab S
ð8Þ
Using Eq. (8) to replace E in Eq. (5), we have
F 2 ¼ QE ¼
ð2Þ
4n1 ðzeÞ2 fp HW E0 ðab þ ac þ bcÞ HW ab j kb T
ð9Þ
2.2. Flow friction force, Ff, on the particle
where j is the Debye–Huckel parameter.
Under an applied electric field, the particle undergoes electrophoretic motion while the liquid in the microchannel moves electroosmotically. The particle will experience the flow friction force. As the distances between the particle’s surfaces and the microchannel walls are small, the shear stress on the particle surface can be estimated as:
(H-d-a) H
(W-b)/2
For the cases of small zeta potential, i.e., f < 25 mV,
E S0 ¼ >1 E0 S
2.1. Electric force, Fe, on the particle
639
a
c b
d W Fig. 1. Schematic of a rectangular particle moving eccentrically in a rectangular microchannel. d is the separation distance between the particle and the bottom wall of the microchannel.
s¼l
dV DV ¼l dy Dy
As illustrated in Fig. 2, the velocity difference is given by
DV ¼ V e of ;W þ V P ¼
e0 e fw E þ VP l
ð10Þ
where Veof,W is the electroosmotic flow velocity at the microchannel walls and Vp is the particle’s velocity.
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D. Li, Y. Daghighi / Journal of Colloid and Interface Science 342 (2010) 638–642
2
4n1 ðzeÞ2 4 j e0 ek b T
Veof, W (H-d-a) VP
HW ðab þ ac þ bcÞ ðHWabÞ 4ac Wb
bc þ Hda þ bcd
HW ðHW abÞ
3
1w 5 1p
reflect the influences of the boundary (the microchannel’s walls) on the electrophoretic motion of the particle. It shows the effects of the bc þ bc ), the particle’s size relative to degree of the eccentricity (Hda d HW the channel’s size (HWab), and the relative zeta potentials (11w ) on
H a
p
the particle’s mobility. As indicated by the negative sign of the second term in the bracket, the zeta potential of the channel wall, 1w , tends to reduce the particle’s mobility. When considering the electrophoretic retardation effect, Eq. (11), the mobility equation becomes:
d Fig. 2. Illustration of the flow field above and below the particle.
It should be noted that, if we consider the effect of the electrophoretic retardation, the electroosmotic flow, Veof,P, around the particle (in the particle’s EDL) should be considered. That is,
e0 e fp DV ¼ V e of ;W þ ðV P V e of ;p Þ where V e of ;p ¼ E l
e0 efw E Dy
þl
VP 1 ¼ ½e0 efw E þ lV P Dy Dy
Apparently, the retardation effect is reflected by the second term in the bracket. If fw < fp, this term will contribute to increase the particle’s mobility.
ð12Þ
3. Discussion 3.1. Special case – slit microchannel
The total flow friction force on the particle is then given by
Ff ¼
Z
s dA ¼ 2F f ; side þ F f ; up þ F f ; low
ð13Þ
where the subscripts side, up, and low represent the side surfaces, the upper surface, and the bottom surface of the rectangular particle, respectively.
2ac 2e0 e fw E 2l V p F f ; side ¼ þ ac ¼ e0 e fw E þ l V p W b W b W b bc F f ; up ¼ e0 efw E þ V p l Hda bc F f ; low ¼ e0 e fw E þ lV p d
ð14Þ ð15Þ ð16Þ
Finally, the total friction force acting on the particle is
Ff ¼
e0 efw E þ lV p
4ac bc bc þ þ W b Hda d
ð17Þ
HW Recall E ¼ SS0 E0 ¼ HWab E0 , we have
Ff ¼
e0 e fw E0
HW ab
ðHWÞ þ lV p
4ac bc bc þ þ W b Hda d
ð18Þ
2.3. Particle mobility From the force balance, F e ¼ F f , Eqs. (9) and (18), the mobility of the eccentric electrophoretic motion of a rectangular particle in a rectangular microchannel can be derived as
V p e0 e1 ¼ E0 l
2
4n1 ðzeÞ2 p 4 j e0 ek b T
HW ðab þ ac þ bcÞ ðHWabÞ 4ac bc bc þ Hda þ d Wb
HW ðHW abÞ
3
1w 5 1p
ð19Þ
Comparing Eq. (19) with the classical mobility equation without the boundary effects,
V p e0 e1p ¼ E0 l Clearly, the terms in the bracket
ð20Þ
ð11Þ
Thus, from Eq. (10),
s¼
2 3 4n1 ðzeÞ2 HW V p e0 e1p 4j e0 e kb T ðab þ ac þ bcÞ ðHWabÞ HW ð1w 1P Þ5 ¼ 4ac bc ðHW abÞ E0 l 1P þ Hda þ bcd Wb
If W H, W a, W b, W c, W d, and HW ab, that is, a slit channel, Eq. (20) is reduced to
2 3 4n1 ðzeÞ2 V p e0 e1p 4j e0 e kb T ðab þ ac þ bcÞ ð1w 1P Þ5 ¼ bc E0 l 1P þ bcd Hda
ð20aÞ
bc That is, in a slit channel, only the eccentricity (Hda þ bc ) and the reld
1w 1p 1p )
ative zeta potentials (
affect the particle’s mobility.
If the particle is very close to the bottom surface of the slit microchannel wall, i.e., d (H – d a), then, the above equation is reduced further to
2 3 4n1 ðzeÞ2 V p e0 e 1p 4j e0 e kb T ðab þ ac þ bcÞ d ð1w 1p Þ5 ¼ bc E0 l 1p
ð20bÞ
where (ab + ac + bc) is half of the total surface area of the particle; bc is the particle’s bottom surface area facing the bottom channel wall; d is the separation distance. In this case, only the separation distance from the bottom wall (d) and the relative zeta potentials 1 1 ( w1 p ) affect the particle’s mobility. For a given particle (i.e., fixed p a, b, and c) and given zeta potential values, fw and fp, the mobility will be lower when the particle moves closer to the wall (i.e., smaller d). If a DNA molecule is modeled as a thin cylinder, i.e., a c, b c, then ab ac and bc. For further simplifying the analysis, let a = b, the (ab + ac + bc) (ac + bc) = 2ac. The above equation (Eq. (20b)) is further reduced to:
" # ð1w 1p Þ V p e0 e1p 8n1 ðzeÞ2 ¼ d E0 l j e0 e kb T 1p
ð20cÞ
This equation describes the mobility of a thin and long cylinder aligned parallel to the axis of a slit flow channel or parallel to the applied electrical field and the flow field. It shows that the mobility is independent of the cylinder length (e.g., the length or number of base pair of a DNA molecule). This is because of the assumption of c (a, b). This implies that if c > 100a or 100b, all cylinder will have the same mobility. From Eq. (20c), it is clear that the closer the particle moves along the bottom channel wall (i.e., smaller d), the smaller the particle’s mobility.
D. Li, Y. Daghighi / Journal of Colloid and Interface Science 342 (2010) 638–642
641
3.2. Special case – flow over a flat surface or a very small particle Let us consider a special case when H ? 1, W ? 1. This is equivalent to the case where a particle moves close to an infinitely large flat surface with a distance d. In this case,
HW ! 1; HW ab
4ac ! 0; W b
bc !0 Hda
Eq. (20) becomes Eq. (20b),
2 3 4n1 ðzeÞ2 V p e0 e1p 4j e0 e kb T ðab þ ac þ bcÞ d ð1w 1p Þ5 ¼ bc E0 l 1p
ð20bÞ
where (ab + ac + bc) is half of the total surface area of the particle; bc is the particle’s bottom surface area facing the bottom channel wall; d is the separation distance. For the case of a particle with a very small cross-section (ab) in comparison with the cross-section of the channel (WH), i.e., WH ab, and W and H a, b, c, d, ac, and bc, the particle mobility equation, Eq. (20), is also reduced to Eq. (20b). As indicated by Eq. (20b), for a particle moving parallel to a flat wall or a very small particle moving in a large microchannel, the electrokinetic mobility is affected only by the separation distance from the bottom wall (d) and the relative zeta potentials 1 1 ( w1 p ). The particle’s mobility is proportional to the separation disp tance d. 3.3. Special case – the particle’s cross-section area similar to that of the channel Consider the case where the particle’s cross-section area is close to the cross-section area of the microchannel. To discuss qualitatively the effect of the ratio of the cross-section areas, let us assume a = b = c, ab = a2, H = W, HW = A2, and a d, H d. Eq. (20) can be rewritten as
2 3 12n1 ðzeÞ2 V p e0 e1p 1 ð 1 1 Þ j e e k T P 5 0 b 4 ¼ w 5 E0 l ð1 a2 =A2 Þ Að1a=AÞ 1P þ 1d For simplicity, assume ¼ A2 ð1 a=AÞ. d ¼ Aa 2
A
(=H) = a + 2d,
2 3 12n1 ðzeÞ2 V p e0 e1p 1 ð 1 1 Þ j e e k T w P 0 b 4 5 ¼ 5 2 E0 l ð1 a2 =A2 Þ Að1a=AÞ 1p þ Að1a=AÞ
and
hence
ð20dÞ
Clearly, in addition to the relative zeta potential, the relative cross-section area ratio of the particle to the channel, ð1 a=AÞ, plays an important role in determining the effective mobility in this case. In order to examine the correctness of this model, the electrophoretic mobility of rectangular particles moving in rectangular microchannels (Fig. 1) is calculated and plotted by using Eq. (20d). Fig. 3 shows the comparison of the model predication with the published experimental data of the electrophoretic mobility of spherical polystyrene particles in different capillaries of circular cross-section [20]. In the experiments, the near-wall electrophoretic motion of spherical polystyrene particles in fused silica capillary tubes of circular cross-section is visualized and measured. It was found that the particles move faster when the particles’ cross-section area becomes closer to that of the capillary tubes. In the model calculation, zeta potentials of the particle and the wall are chosen to be 25 and 40 mV, respectively. As seen from Fig. 3, the qualitative agreement between the model predictions and the experimental results is very good. The quantitative differences can be largely attributed to the fact that the model (Eq. (20d)) is based on a rectangular particle in a rectangular microchannel; the
Fig. 3. Comparison of the prediction of Eq. (20d) with the experimental results [20] of the electrophoretic mobility of spherical polystyrene particles in different capillaries filled with 25 mM sodium carbonate (pH = 8.5) buffer. The symbol points are the experimental data points. The solid lines are the model prediction of the electrophoretic mobility of rectangular particles in rectangular microchannels.
geometry is significantly different from the spherical particles in circular cross-section channels. 4. Summary An analytical model of the electrophoretic mobility of a rectangular particle moving in a rectangular microchannel is derived in this paper. Using this model, the effects of the boundary – the microchannel walls – on the particle’s mobility are discussed for several special cases. In the case of a slit microchannel, only the separation distance from the bottom wall (d) and the relative zeta 1 1 potentials ( w1 p ) affect the particle’s mobility. For a given particle p and given zeta potential values, the mobility is lower when the particle moves closer to the wall (i.e., smaller d). If a DNA molecule is modeled as a thin and long cylinder aligned parallel to the axis of a slit flow channel, the model indicates that the mobility is independent of the cylinder length (e.g., the length or number of base pair of a DNA molecule). For a particle moving parallel to a flat wall or for a very small particle moving in a large microchannel, the electrokinetic mobility is affected only by the separation distance 1 1 from the bottom wall (d) and the relative zeta potentials ( w1 p ). p The particle’s mobility is proportional to the separation distance d. For the case where the particle’s cross-section area is close to the cross-section area of the microchannel, the model reveals that 1 1 the relative zeta potential, ( w1 p ), and the relative cross-section p area ratio of the particle to the channel, ð1 a=AÞ, are key parameters in determining the electrokinetic mobility. This is in agreement with the published experimental results. References [1] [2] [3] [4] [5] [6]
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