Electrokinetic Transport through Rough Microchannels - Analytical

Anal. Chem. 2003, 75, 5747-5758

Electrokinetic Transport through Rough Microchannels Yandong Hu,†,‡ Carsten Werner,†,‡ and Dongqing Li*,†

Department of Mechanical & Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, Ontario, Canada, M5S 3G8, and Department of Biocompatible Materials, Institute of Polymer Research, Hohe Strasse 6, 01069 Dresden, Germany

Surface roughness is present in most microfluidic devices as a result of the microfabrication techniques or particle adhesion. It is highly desirable to understand the roughness effect on microscale transport processes. In this study, we developed a 3-D, finite-volume-based numerical model to simulate electroosmotic transport in microchannels with rectangular prism rough elements on the surfaces. Various configurations of roughness were investigated, and the results show different degrees of an even-out effect on liquid transport due to the roughnessinduced local pressure field and the variation of the electroosmotic slip boundary velocities. 3D-sample transport through rough microchannels was analyzed. The results demonstrate that the sample’s transport under the electrical field is much faster in the pathway between the rough elements; the concentration field in the height and width direction is not uniform. The influence of the electrokinetic properties on liquid flow and sample transport was studied. It was found that the increase of the electroosmotic mobility or the decrease of the electrophoretic mobility can dramatically enhance the uniformity of the concentration field.

Lab-on-a-chip devices have drawn great attention recently because of their ever-increasing applications in biomedical diagnosis and analysis, such as clinical detection, DNA hybridizations,1-3 and electrophoretic separation.4-5 The characteristic dimensions of most microchannels in lab-on-a-chip devices range from 10 to 100 µm. Some devices designed for the sizing and sorting of DNA6 have a channel depth of only 3 µm. Fundamental understanding of liquid flow through the microchannels is important to the design and operation of these lab-on-a-chip devices. In addition to * Corresponding author. Fax: (416) 978-1282. E-mail: [email protected]. † University of Toronto. ‡ Institute of Polymer Research. (1) Sadana, A.; Ramakrishnan, A. J. Colloid Interface Sci. 2001, 234, 9. (2) Zeng, J.; Almadidy, A.; Watterson, J.; Krull, U. J. Sens. Actuators, B 2003, 90, 68. (3) Erickson, D.; Li, D.; Krull, U. J. Anal. Biochem. 2003, 317, 186. (4) Ren, L.; Li, D. J. Colloid Interface Sci. 2002, 254, 384. (5) Ermakov, S. V.; Jacobson, S. C.; Ramsey, J. M. Anal. Chem. 1998, 70, 4494. (6) Chou, H.-P.; Spence, C.; Schere, A.; Quake, S. Proc. Natl. Acad. Sci. U.S.A. 1999, 96, 11. 10.1021/ac0347157 CCC: $25.00 Published on Web 09/09/2003

© 2003 American Chemical Society

electrokinetic properties related to the intrinsic surface chemistry of the channel walls, surface roughness plays an important role in various microfluidic processes. Generally, the microchannel surface may exhibit certain degrees of roughness generated by the manufacturing techniques or by adhesion of biological particles from the liquids. The reported surface roughness ranges from 0.1 to 2 µm.7-9 Electroosmotic and the electrophoretic transport in microchannels have been investigated theoretically10,11 and experimentally.12-14 For lab-on-a-chip applications, many groups studied electrokinetic transport processes in complicated microchannel structures, for example, focusing and dispensing samples in crossing channels,4-5,15 mixing different streams in T-shaped channels,5,16 flows in channels with a 90° turn and a 180° turn.17-18 Because of the complexity of the problems, most of these studies consider only 1D and 2D systems with ideal channel walls that are smooth and homogeneous (i.e., with uniform ζ potential). There are several reported works regarding more realistic channel surfaces, such as the studies of electroosmotic flows in capillaries with surface defects19,20 and in microchannels with surface heterogeneity.16,21 Ajdari’s works22-24 predicted that the presence of surface heterogeneity could result in regions of bulk flow circulation. This behavior was later observed experimentally in (7) Mala, G. M.; Li, D. Int. J. Heat Fluid Flow 1999, 20, 142. (8) Qu, W.; Mala, G. M.; Li, D. Int. J. Heat Mass Transfer 2000, 43, 353. (9) Qu, W.; Mala, G. M.; Li, D. Int. J. Heat Mass Transfer 2000, 43, 3925. (10) Mosher, R. A.; Saville, D. A.; Thormann, W. The Dynamics of Electrophoresis; VCH: Weinheim, 1992. (11) Everaerts, F. M.; Beckers, J. L.; Verheggen, T. P. E. M. Isotachophoresis: Theory, Instrumentation and Application; Elsevier: Amsterdam, 1976. (12) Ross, D.; Johnson, T. J.; Locascio, L. E. Anal. Chem. 2001, 73, 2509. (13) Devasenathipathy, S.; Santiago, J. G. Anal. Chem. 2002, 74, 3704. (14) Zeng, S.; Chen, C.-H.; Mikkeksen, J. C., Jr.; Santiago, J. G. Sens. Actuators, B 2001, 79, 107. (15) Patankar, N. A.; Hu, H. H. Anal. Chem. 1998, 70, 1870. (16) Erickson, D.; Li, D. Langmuir 2002, 18, 1883. (17) Molho, J. I.; Herr, A. E.; Mosier, B. P.; Santiago, J. G.; Kenny, T. W.; Brennen, R. A.; Gordon, G. B.; Mohammadi, B. Anal. Chem. 2001, 73, 1350. (18) Mohammadi, B.; Santiago, J. G. Math. Model. Numer. Anal. 2001, 34, 513. (19) Long, D.; Stone, H. A.; Ajdari, A. J. Colloid Interface Sci. 1999, 212, 338. (20) Stroock, A. D.; Dertinger, S. K.; Whitesides, G. M.; Ajdari, A. Anal. Chem. 2002, 74, 5306. (21) Potocek, B.; Gas, B.; Kenndler, E.; Stedry, M. J. Chromatogr., A 1995, 709, 51. (22) Ajdari, A. Phys. Rev. Lett. 1995, 75, 755. (23) Ajdari, A. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1996, 53, 4996. (24) Ajdari, A. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2002, 65, 016301.

Analytical Chemistry, Vol. 75, No. 21, November 1, 2003 5747

Figure 1. (a) An example of a silicon surface with microfabricated, symmetrically arranged prism elements. (b) and (c) The symmetrical and asymmetrical roughness arrangements on the homogeneous microchannel wall and the corresponding computational domains from the top view.

slit microchannels by Stroock et. al.25 Long et al.19 also developed an analytical model for an isolated heterogeneous spot in a flat plate. However, so far, there has been no published work regarding electroosmotic flow in rough microchannels. The kinetics of a wide range of chemical/biochemical reactions in lab-on-a-chip applications depends on the reagents’ concentration in a thin layer near the reacting surface. For example, in a DNA sensor chip3, the probing DNA molecules are immobilized on a microchannel surface, and the target DNA molecules are transported with a buffer solution over the sensing surface. The kinetics of DNA hybridization is dominated by the target DNA concentration in a thin liquid layer next to the surface. Generally, the higher the concentration in the surface layer, the more opportunities for the reactions to occur.2,3 There are two common modes of transporting liquids through microchannels: pressure-driven flow and electroosmotic flow. With respect to sample transport, the pressure-driven flow causes significant sample dispersion as a result of the velocity gradient near the wall. This leads to a high concentration gradient at the wall. Electroosmotic flow results in a flat, plug-like velocity profile12 that is preferable to sample transport and, therefore, is widely (25) Stroock, A. D.; Weck, M.; Chiu, D. T.; Huck, W. T. S.; Kenis, P. J. A.; Ismagilov, R. F.; Whitesides, G. M. Phys. Rev. Lett. 2000, 84, 3314.

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Analytical Chemistry, Vol. 75, No. 21, November 1, 2003

used in biological and chemical analysis.26 However, because of the sample diffusion, electroosmotic flow still produces a relatively large concentration gradient at the wall. One way to enhance the biochemical reactions on the sensing surfaces is to increase the reaction-sensing surface area. In a microchannel, creating many three-dimensional roughness elements on the microchannel walls can increase the surface area. These 3D roughness elements can be produced by photolithography-based microfabrication techniques, and an example is shown in Figure 1a. However, these 3D roughness elements will inevitably influence liquid flow and sample transfer in the microchannel. The purpose of this study is, therefore, to examine the influence of micrometer-sized rough elements on electroosmotic flow and the associated mass transfer in microchannels by 3D numerical simulations. We consider the rough microchannel wall as a homogeneous surface with uniformly distributed rough elements. Furthermore, to simplify the modeling and the numerical simulation, we consider the rough elements to be cylinders with a square cross section (rectangular prisms). Without loss of generality, we consider two types of distributions: symmetrical and asymmetrical arrangement, as illustrated in Figure 1b and c. The electroosmotic flow field and the sample concentration field (26) Sinton, D.; Erickson, D.; Li, D. J. Micromech. Microeng. 2002, 12, 898.

Figure 2. Illustration of 3D coordinates, computational grid systems for the case of symmetrical arrangement with a/H ) 0.6, b/H ) 1 and h/H ) 0.2.

are investigated in terms of the influence of the rough element size, height, density, the element arrangement, the channel sizes, and the electrokinetic mobility. MATHEMATICAL MODEL The rough microchannel studied in this work is formed by two parallel surfaces with rectangular prismatic rough elements, as shown in Figure 2. The parameters describing the rough microchannel include (i) the roughness element’s size, a; (ii) the separation distance, b, between the roughness elements in both x and z directions; (iii) the roughness element’s height, h; (iv) the microchannel height, H; and (v) the rough elements’ arrangement: symmetrical and asymmetrical arrangement. Because our goal is to examine the roughness effect on the electrokinetic transport processes through microchannels, the flow through a virtual smooth microchannel will be used as a comparison base. This virtual smooth channel is defined as the smooth channel having the same channel volume, width, and length as the rough channel. Under these conditions, the virtual smooth channel height, Hs, and the virtual smooth channel cross-sectional area, Ac,s can be obtained. In this way, the comparison can avoid the effect of the reduction in the flow passage due to the presence of the roughness elements. The same electric potential difference and zero pressure difference are applied to the inlets and the outlets of both the nonconducting virtual smooth channel and the nonconducting rough microchannel. Further simplifications are required for developing a proper model. First, in the study, the flow is limited to a low Reynolds regime (0.001 < Re < 10); turbulence and wake region at the rough elements’ tail region can be neglected. Second, we assume that the influences of the applied electric field and the flow field on the EDL field can be neglected. This is a common treatment in essentially all published works. It means that the ion distributions in the EDL region are determined by the EDL field only. This is a necessary condition to derive the Smoluchowski equation: Veof ) µeoE ) (ζ/µ)E for electroosmotic flow. The validity of this assumption can be understood as follows. EDL is a thin layer at the solid-liquid interface, and its characteristic thickness, for example, is ∼9.6 nm for a 10-3 M KCl solution. Within this thin layer, the electrical potential drops from the ζ

potential (on the wall) to zero (at the edge of the EDL). Generally, the ζ potential is on the order of 100 mV. The field strength of EDL is approximately on the order of 107 V/m (i.e., 100mV/10 nm). On the other hand, the applied electrical field strength to generate electroosmotic flow in microchannels ranges approximately from 103 to 105 V/m. Therefore, the EDL field is dominant. Generally, the electroosmotic flows in microchannels are low Reynolds number laminar flow (0.001 < Re < 10). A detailed study 27 demonstrated that a weak flow field (Re < 10) will not influence appreciably the ion distributions in the EDL. In addition, we assume that the entrance effect and the edge effect are neglected, and hence, the flows through rough microchannels are considered to be fully developed periodical flows. The advantage and the numerical methods for simulating this kind of flow can be found elsewhere.28 Furthermore, the flow-induced streaming potential is considered negligible. During electroosmotic flow, the transport of the net charge in EDL region to downstream will build up a potential against the applied driving potential and may cause reverse ion conduction and back-electroosmosis. However, this induced streaming potential is usually less than 1V. In comparison with the externally applied driving potential (on the order of 1000 V), this induced potential is too small and can be safely neglected. The top view of the computational domains is shown in Figure 1b and c, that is, we take the region between two symmetrical planes with one roughness separation distance as the computational domain width. The height of the computational domain is chosen as one-half of the microchannel height because of the symmetry. For simplicity, we consider the separation distances between roughness elements in both the x and z directions are the same, denoted as b. It should be noted that two periods are chosen as the computational domain along the main flow direction. This treatment has particular benefit to simulation convergence.29 Electrical Field. We consider a thin double layer and no net charge density in the bulk liquid. According to the theory (27) Erickson, D.; Li, D. Langmuir 2002, 18, 8949. (28) Patankar, S. V.; Liu, C. H.; Sparrow, E. M. ASME J. Heat Transfer 1977, 99, 180. (29) Amano, R. S. ASME J. Heat Transfer 1985, 107, 564.


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of electrostatics, the applied electrical potential established in the rough microchannel, φ, can be described by the Poisson equation.

∂2φ ∂2φ ∂2φ + 2 + 2 )0 ∂x2 ∂y ∂z

(1)

Since the inlet and the outlet planes of the computational domain are geometrically symmetrical planes, the electrical potential is uniformly distributed on these planes. We introduce the following dimensionless parameters

φ* )

φ - φout φin - φout

x* )

x y z , y* ) , z* ) H H H

(2)

where φin and φout are the electrical potential values on the inlet and outlet of one period computational domain, respectively; H is the height of the microchannel and chosen as the characteristic length of the system studied here. The dimensionless Poisson equation and its boundary conditions are given below.

∂2φ* ∂2φ* ∂2φ* + + )0 ∂x*2 ∂y*2 ∂z*2

(3)

Because the dimensionless electrical potential field is identical in each period, which is half of the computational domain of length 2b, it is only necessary to solve the electrical potential field in one period. The corresponding boundary conditions are listed here. At the inlet of the period: x* ) 0, φ* ) 1; at the outlet of this period: x* ) b/H, φ* ) 0; at the other boundary planes, that is, the symmetrical planes y* ) 0.5, z* ) 0, z* ) b/H, and the wall y* ) 0, ∇φ* ) 0 is applied (see Figure 2). Once the electrical field in the rough microchannel is known, the local electric field strength can be calculated by

E ˜ ) - ∇φ˜

E ˜ ∂φ* ∂φ* b ∂φ* ˜ı + ˜j + k˜ )Ex,s H ∂x* ∂y* ∂z*

(

)

(5)

where Ex,s is the modulus of the tensor E˜ x,s, and ˜,ı ˜,j and k˜ are the unit tensors in the x, y, and z directions, respectively. Flow Field. The basic equations describing the electroosmotic flow through microchannels are the continuity equation and the Navier-Stokes equation with the electric body force in the righthand side,

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˜ V ˜ eo,slip ) µeoE

(8)

Introducing the following dimensionless group, V ˜ /eo ) V ˜ eo/ Veo,s, p* ) p/(µVeo,s/H), and define the Reynolds number Re ) FVeo,sH/µ, where V˜ eo,s is the characteristic velocity determined by Ex,s, that is, V ˜ eo,s ) Veo,s˜ı ) µeoEx,s˜,ı the governing equations of electroosmotic flow through rough microchannels can be changed from eqs 6 and 7 into

{

∂u* ∂v* ∂w* + + )0 ∂x* ∂y* ∂z*

(9) 2

2

2

∂u* ∂u* ∂p* ∂ u* ∂ u* ∂ u* + v* + w* + )+ + ( ∂u* ∂x* ∂y* ∂z* ) ∂x* ∂x* ∂y* ∂z* ∂v* ∂v* ∂v* ∂p* ∂ v* ∂ v* ∂ v* + v* + w* ) ) + Re(u* + + ∂x* ∂y* ∂z* ∂y* ∂x* ∂y* ∂z* ∂w* ∂p* ∂ w* ∂ w* ∂ w* ∂w* ∂w* Re(u* )+ + + v* + w* + ∂x* ∂y* ∂z* ) ∂z* ∂x* ∂y* ∂z*

Re u*

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

(10)

(4)

The dimensionless form of electrical field strength in rough channels is defined as the ratio of the local electrical field strength in the rough microchannel to the overall electrical field strength over one period: E ˜ x,s ) Ex,s˜ı ) - φout - φin/bı˜, as derived as follows:

E ˜* )

where V ˜ eo ) ueo˜ı + veo˜j + weok˜ is the electroosmotic velocity in a rough microchannel, p is the pressure, Fe is the net charge density, and F and µ are the density and the viscosity of the bulk fluid, respectively. Generally, the numerical solution of electroosmotic flow is complicated by the simultaneous presence of three separate length scales: the channel length (mm), the channel depth or width (µm) and the double layer thickness (nm). The simplest way to avoid this multiple-scale problem is to apply a slip boundary condition at the channel wall (Vwall ) -µeo∇φ where µeo ) ζ/µ is the electroosmotic mobility and ζ is the zeta potential of the channel wall) to solve for the bulk liquid motion. Since the net charge density is zero within the bulk liquid, this treatment eliminates the electrical body force term in eq 7 and, thus, the need to solve the equation for the double layer field. The slip boundary velocity on the roughened microchannel wall, V˜ eo,slip, can be written as the following:

∇V ˜ eo ) 0

(6)

F(V ˜ eo∇)V ˜ eo ) - ∇p ˜ + µ∇2˜ V ˜ eo + FeE ˜

(7)


For electroosmotic flow through rough microchannels, three types of boundary conditions should be applied to eqs 9 and 10. They are called the fully developed periodic conditions 29 applied at the inlet and outlet of the computational domain, the slip boundary condition applied at the rough wall, and the symmetrical conditions applied at the two sides and the top plane of the computational domain (see Figure 2), respectively. It is worthwhile to note that the periodic conditions here are different from the conventional fully developed periodic conditions. Because no pressure difference is imposed on the inlet and outlet of the rough microchannels, we have the following period conditions

V ˜ *eo(x*, y*, z*) ) V ˜ *eo(x* + b/H, y*, z*) p*(x*, y*, z*) ) p*(x* + b/H, y*, z*)

(11)

In the meantime, V ˜ /eo and p* at each point should satisfy the governing equations, eqs 9 and 10, especially at the inlet and the outlet planes of the computational domain. The slip boundary condition is applied at the channel wall (including the surface of roughness elements). According to the

definition of the dimensionless governing equations, the dimensionless electroosmotic slip boundary velocity can be derived from eq 8.

V ˜ *eo,slip )

V ˜ eo,slip Veo,s

)

E ˜ )E ˜* Ex,s

(12)

At the rest of the computational domain boundaries, symmetrical conditions are applied. That is, for the velocity component parallel to the symmetric plane, the change of the velocity component normal to the plane is zero; for the velocity component normal to the plane, the velocity component itself is 0. Details can be found in reference 29. Concentration Field. Study of the concentration field of sample species is of great importance in various microfluidics applications, for example, dye-based microflow velocity profile measurements,26 microscale species mixing,16 microfluidic dispensing,4-5 and surface reaction kinetic studies3. Consider that a sample with concentration C0 is released into the rough microchannel at time t0, the sample transport in time and space can be described by the law of mass conservation, which takes the form5,30

∂Ci ˜ ep,i)∇ ˜ ]Ci ) Di∇ ˜ 2Ci + [(V ˜ eo + V ∂t

(13)

where Ci is the concentration of ith species; V ˜ ep,i is the electro˜; phoretic velocity of the ith species and is given by V˜ ep,i ) µep,iE Di and µep,i are the diffusion coefficient and the electrophoretic mobility of the ith species, respectively. In eq 13, the first term on the left-hand side is the transient term; the second term includes two parts: the first part represents the convection effect resulting from the electroosmotic flow, the second part stands for the electrophoretic effect; the right-hand side term is the diffusion term. Introducing the dimensionless group,

C* ) C/C0, V ˜ /ep ) V ˜ ep/Veo,s )

µep E ˜ *, µeo t* ) (t - t0)/(H/Veo,s) (14)

and defining the Peclet number as Pe ) Veo,sH/Di, we can nondimensionalize eq 13 as

Pe

{

}

∂C/i / + [(V ˜ /eo + V ˜ ep,i )∇ ˜ ]C/i ) ∇ ˜ 2C/i ∂t*

(15)

Because releasing sample species into a microchannel is a transient process, initial condition and boundary conditions are required. The initial condition is as follows: when t* ) 0, C/i (x*, y*, z*, 0) ) 0. The boundary conditions are these: at the inlet plane, C/i (0, y*, z*, t*) ) 1; at the outlet plane, we assume the concentration at each point is influenced only by its upwind control volume and, hence, have ∇C/i ) 0; at the symmetrical boundaries and the walls, no flux condition ∇C/i ) 0 is applied. As seen from the above-defined dimensionless parameters, the electrokinetic property of the microchannel surface (zeta potential (30) J. M. MacInnes, Chem. Eng. Sci. 2002, 57, 4539.

ζ) and the properties of the solution (density and viscosity F and µ) are considered in the Reynolds number. For most electroosmotic flows in microfluidics, Re is in the range of ∼0.001 to 10. The property of the sample, the diffusion coefficient Di, is accounted for in the Peclet number. Because the species electrophoresis and the liquid electroosmosis occur simultaneously under the same electrical field, the ratio µep,i/µeo instead of the individual values determines the relative electrophoretic velocity, as eq 14 shows. Therefore, in addition to the parameters describing the surface roughness, the Reynolds number is the basic dimensionless parameter determining the characteristic of the electroosmotic flow; the Peclet number and the ratio µep,i/µeo are the basic dimensionless parameters determining the characteristics of the sample transport in the microchannels. The mobility ratio µep,i/µeo represents the ratio of the electrophoretic force to the electroosmotic force. If the two mobilities have a different sign (i.e., the electrical charge of the sample and the charge of the microchannel wall have different sign, one positive and the other negative), that is, µep,i/µeo < 0, the sample molecules will move in the same direction as the electroosmotic flow and in a speed faster than the liquid. If µep,i/µeo > 0 (i.e., the electrical charge of the sample and the charge of the microchannel wall have the same sign), the sample molecules will move in the direction opposite to the electroosmotic flow and in a speed slower than the liquid when ignoring the movement caused by diffusion. The absolute value of this ratio, |µep,i/µeo| > 1 or |µep,i/µeo| < 1, signifies the speed of the sample’s electrophoretic motion relative to that of the bulk liquid electroosmotic flow. In this study, we use the properties5 of Rhodamine 6G as a representative sample to calculate the Peclet number and the ratio µep,i/µeo, where µep,i ) 1.4 × 10-4 cm2/(Vs), and Di ) 3.0 × 10-6 cm2/(s). To solve the electrical potential field, the electroosmotic flow field, and the concentration field during the species transport process numerically, we developed a 3D-computation code based on the finite volume approach31 and SIMPLEC algorithm.31 As a part of the computational domain, the virtual space in the roughness is also solved using the extension of computational region approach.31 That is, when solving the electrical potential field, a conductivity function is set up so that it will have a value of 0 for the control volumes inside the nonconducting roughness prisms, and a value of 1 for the control volumes in the bulk liquid phase. When solving the electroosmotic flow field, we consider the control volumes in the roughness elements as a kind of fluid with extremely high viscosity. When solving the species transport process, we consider the control volumes in the roughness elements as species with a diffusion coefficient of 0. RESULTS AND DISCUSSION The Electroosmotic Flow and the Associated Sample Transport in Smooth Channels. Because the same electrical potential difference is applied to both the smooth and the rough microchannels with the same channel length, the electrical potential drop over the length of each period, ∆φ ) φout - φin, is the same for the rough and the smooth microchannels. It is known that in a smooth channel, the applied electrical potential varies linearly along the length direction, and hence, the electrical field (31) Patankar, S. V. Numerical Heat Transfer and Fluid Flow; Hemisphere: New York, 1980.


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Figure 3. The electrical potential field in the symmetrically arranged rough microchannels with a/H ) 0.6, b/H ) 1 and h/H ) 0.2. The surfaces of the constant electrical potential are distorted as a result of the presence of the rough elements. The color represents the value of the dimensionless potential, φ*.

strength and electroosmotic flow velocity are uniform in the whole channel, as given below.

E ˜ x,s ) Ex,s˜ı ) -

φout - φin ˜ı b

V ˜ eo,s ) Veo,s˜ı ) µeoEx,s˜ı

(16) (17)

Referring to the nondimensional groups used for the rough channels, the values Ex,s and Veo,s are the characteristic applied electrical field strength and the characteristic electroosmotic velocity. Therefore, the dimensionless values φ*, E*, and V/eo in rough channels represent the relative values of electrical potential, the applied electrical field strength, and the electroosmotic velocity at a certain point in the rough channel to the values of the smooth channel, respectively. Substituting the above eqs 16-17 into eq 14, the dimensionless governing equation for the associated sample transport through smooth channels can be derived from eq 15.

{ ( ) }

Pe

∂C/i µep ∂C/i + 1+ )∇ ˜ 2C/i ∂t* µeo ∂x*

(18)

The Applied Electrical Potential Field and the Electroosmotic Flow Field. Because of the existence of the rough elements, the surfaces of the constant electrical potential are not parallel to each other. For example, the electrical potential field for the case of symmetrically distributed roughness elements is shown in Figure 3, where the colors represent the values of the dimensionless potential, φ*. These isoelectric surfaces are distorted by the rough elements, and therefore, the electrical strength 5752


is not uniform in the microchannels. Thus, the calculated local slip boundary velocities are different in the values and in the directions. As shown in Figure 4a and b, the top view of the slip boundary velocity on the bottom channel surface, the color scale shows the value of the dimensionless velocity component in the x (the main flow) direction. The values >1 at certain regions indicate that the slip velocity component in the main flow direction is larger than the electroosmotic velocity in the smooth channels. Figure 4a and b clearly shows that the slip velocity >1 is mainly distributed in the lengthwise pathway formed between two neighboring rough elements (which will be called the “pathway” in the following sections), and the maximum value of u/eo is present around the corner in the pathway, whereas the slip velocity ,1 is mainly located at the gap between the front and back surfaces of two neighbor rough elements (which will be called the “gap” in the following sections). It should also be noted that at the rough element’s surface, especially at the front and the back side of the prism’s surface (refer to Figure 4c), the distorted electrical potential builds up a varying slip boundary velocity field. On the backside surface of the roughness elements, the velocity vectors are directed toward the center of the intersection line of the roughness element and the substrate surface. The magnitude of the velocity decreases gradually as it approaches this point. On the front side, the slip boundary velocities have the same distribution, except with the opposite directions. This indicates the driven force distribution around the rough element’s surface. The velocity field of the electroosmotic flow and the induced pressure field in a rough microchannel with symmetrically arranged roughness elements are shown in Figure 5. Even though no pressure differences are applied to the inlets and outlets of the whole channels, local pressure gradients are induced by the

Figure 4. The slip boundary velocity distribution on (a) the bottom plane in a symmetrically arranged rough microchannel, (b) the bottom plane in an asymmetrically arranged rough microchannel, and (c) the surface of one-quarter of the rough element, with rough channel parameters a/H ) 0.6, b/H ) 1, and h/H ) 0.2. The arrows are local velocity vectors. The color represents the value of the dimensionless velocity.

existence of the roughness elements in order to satisfy the flow continuity. The induced pressure in electroosmotic flow through microchannels has also been discussed by other researches.32,33 Figure 5 shows that there are high-pressure zones at the upwind regions and the low-pressure zones at the tail regions of the rough

elements; the pressure in the central flow above the roughness has little changes. The side view of the flow and the pressure (32) Liu, R. H.; Stremler, M. A.; Sharp, K. V.; Olsen, M. G.; Santiago, J. G.; Adrian, R. J.; Aref, H.; Beebe, D. J. J. Microelectromech. Syst. 2000, 9, 190. (33) Ren, L.; Li, D. J. Colloid Interface Sci. 2001, 243, 255.


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Figure 5. The top view (upper) and the side view (lower) of the electroosmotic velocity and the induced pressure distribution in symmetrically arranged rough microchannels with parameters a/H ) 0.6, b/H ) 1, and h/H ) 0.2. The arrows are local velocity vectors. The color represents the value of the local dimensionless pressure.

the liquid from the gap to the flow pathway. To demonstrate the roughness effects on the flow rate, we define the corresponding electroosmotic flow rate in the virtual smooth channel as Qs ) Veo,sAc,s. Since the pressure variations in the x directions are key to the flow, we define the average dimensionless pressure in the cross-sectional area as

p/av ) Figure 6. The 3D streamlines of the electroosmotic through symmetrically arranged rough microchannels with parameters a/H ) 0.6, b/H ) 1, and h/H ) 0.2.

fields also shows that the maximum values of the velocity component ueo at each cross section all occur on the roughened wall. Similar phenomena are found for the electroosmotic flow in microchannels with asymmetrically arranged rough elements. Figure 6 shows the 3D streamlines of the electroosmotic flow through the symmetrically arranged rough microchannels. One can clearly see that the streamlines are curved with the roughness shapes. This is true for all the simulations we conducted for cases of symmetrically and asymmetrically arranged roughness under various conditions. There are no flow recirculation zones in the gap, no matter what the roughness elements’ sizes, separation distances, heights, the channel heights, and the Reynolds numbers are. This implies that there is no sample molecule trap or accumulation of sample molecules in the gap regions between rough elements when sample transporting with the electroosmotic flows. The Influence on the Electroosmotic Flow Rate and the Induced Pressure. The roughness-induced pressure has a direct effect on the electroosmotic flow rate and the ability to remove 5754


1 Ac

∫

Ac

p* dAc

(19)

where Ac is the cross-sectional area in the rough microchannels. The Rough Element’s Size Effect on the Induced Pressure Field. Figure 7a shows the roughness size effect on the crosssectional averaged dimensionless pressure (p/av) in two-periodlength symmetrically arranged rough channels. From Figure 7a, one can see that the induced pressure is periodically undulating along the rough channel, and the local maximum locates at the upwind region of the rough element, while the local minimum locates at the element’s tail region. A higher maximum value of p/av indicates a larger pushing force acting on the fluid element near the upwind region to push the fluid from the gap to the central flow. Similarly, at the tail region of the roughness element, the low pressure attempts to suck the fluid from the central flow into the gap. Such an exchange of fluid between the gap and the central flows is referred as the even-out effect. From Figure 7a, one can clearly see that this even-out effect exhibits a maximum value at a/H ) 0.4 when the roughness element’s size changes. When a/H < 0.4, the electrical potential field distortion due to the presence of the roughness elements is rather small, and hence, the difference in the slip velocities and the induced pressure gradients at the front and the back faces of the roughness elements is small, resulting in a relatively small even-out effect.

Figure 8. The effect of roughness parameters a, b, h, and H on the electroosmotic flow rate with default parameters a ) 6 µm, b ) 10 µm, H ) 10 µm, and h ) 2 µm.

Figure 7. The roughness size effect on the induced cross-sectional averaged dimensionless pressure p/av in (a) two periods of the symmetrically arranged rough microchannel, and in (b) a half period of the asymmetrically arranged rough microchannel with parameters b/H ) 1 and h/H ) 0.2.

When a/H > 0.4, the high-pressure zone and the low-pressure zone are very close to each other, and the overlapping effect reduces the absolute value of the induced pressure in the gap, reducing the fluid exchange between the gap and the central flow. Figure 7b shows the roughness size effect on p/av for the asymmetrically arranged rough microchannels. To show the pressure fluctuation clearly, only one-half of a flow period is shown in Figure 7b. Similar to the symmetrically arranged rough channels, the maximum of p/av occurs at the upwind region, and the minimum of p/av occurs at the tail region of the roughness, as shown by the curve a/H ) 0.2 in Figure 7b. It is obvious that there is a large difference between parts a and b of Figure 7. This difference has its origin in the roughness element’s arrangement. For the asymmetrically arranged roughness, as illustrated in Figure 4b, the locations of the high-/low-pressure zones on the opposite sides of the flow pathway are asymmetrical (i.e., at different x positions) along the flow direction. For the symmetrical arrangement, as seen in Figure 5, the high-/low-pressure zones on the opposite sides of the flow pathway are at the same x position. Thus, the curve p/av ∼ x* for the asymmetrical arrangement is more complicated, and the distance between the highpressure zone and the next low-pressure zone is much smaller than that of the symmetrically arranged rough channels.

The Effects of the Rough Element’s Separation Distance, and Height, and the Channel Height. When varying each parameter of b, h, and H in turn, the curves of p/av ∼ x* have similar undulating characteristics, and thus, we do not show the figures. However, it is worthwhile to note the following effects. When the separation distance between roughness elements, b, changes from 10 to 20 µm while keeping the other parameters fixed (a ) 6 µm, h ) 2 µm, H ) 10 µm, symmetrical arrangement), the simulations show an increasing even-out effect (a larger maximum value of p/av) caused by the pressure-pushing force in the pathway. When the roughness element’s height, h, increases, for example, from 2 to 5 µm, while the other parameters are fixed (H ) 10 µm, b ) 10 µm, a ) 6 µm), the even-out effect is smaller, because the velocity component on the back side of the rough element’s is smaller. When the rough channel’s height, H, increases, for instance, from 10 to 30 µm, the curves of p/av ∼ x* overlap with each other. This indicates the induced cross-sectional averaged pressure resistance, dpav/dx, is proportional to the 1/H under the same Reynolds number. Because the central flow above the roughness contributes little to the pressure undulation, the induced high (or low) pressure at the front (or back) side of the rough elements does not vary so much when changing only the rough channel’s height. The Roughness Effects on the Electroosmotic Flow Rate. The above-mentioned roughness effects on the electrical field and on the induced pressure field ultimately influence the electroosmotic flow rate. Figure 8 shows the influences of the rough microchannel geometry parameters on the electroosmotic flow rate, where Q/Qs is the ratio of the flow rate through the rough channels to that of the virtual smooth channels. Figure 8 shows that the induced pressures dramatically reduce the electroosmotic flow rates through rough microchannels, and the effect increases as the roughness size or the roughness height increases and decreases as the roughness separation distance increases. As the rough channel’s height increases, the ratio of the flow rate to that of the smooth channel decreases at first and then increases. When H decreases from 20 to 10 µm, relatively more electroosmotic driven force is produced because of the relatively larger rough element’s surface and the larger electrical field distortion. These two effects contribute to the slight increase of the flow rate compared to the electroosmotic flow rate in smooth channel. For all the asymmetrically arranged roughness, the flow rates are all larger than Analytical Chemistry, Vol. 75, No. 21, November 1, 2003

5755

Figure 9. The sample concentration field in a rough microchannel, a/H ) 0.4, b/H ) 1, and h/H ) 0.2, at the time when a prescribed amount of sample is released into the microchannels. The concentration field (a) in the computational domain, (b) on the rough surface with µeo ) 4 × 10-8 m2/Vs and µep ) 1.4 × 10-8 m2/Vs. The concentration field (c) in the computational domain, (d) on the rough surface with µeo ) 6 × 10-8 m2/Vs and µep ) 1.4 × 10-8 m2/Vs. The concentration field (e) in the computational domain and (f) on the rough surface with µeo ) 4 × 10-8 m2/Vs and µep ) -1.4 × 10-8 m2/Vs.

those of the symmetrically arranged roughness. This is because the asymmetrical arrangement produces more streamlined flows, the induced pressure resistance is divided into more steps while the fluctuation amplitude is smaller, and the pressure resisting flow mostly happens in each step. Sample Mass Transport in the Rough Microchannels. The Characteristic of Sample Mass Transport in the Rough Microchannels. Generally, sample mass transport is the result of three major mass transport mechanisms: convection, electrophoretic transport, and diffusion. In view of the characteristics of the electroosmotic flow field and the electrical potential field in rough microchannels, sample mass transport in rough microchannels is very different from most reported studies in which smooth channel walls are considered. Figure 9a shows the sample concentration field in the computational domain when a certain amount of sample is released into the computational domain at the inlet plane. In the simulations, the electrokinetic properties are chosen as µeo ) 4 × 10-8 m2/Vs and µep ) 1.4 × 10-8 m2/Vs. The color scale from dark to 5756 Analytical Chemistry, Vol. 75, No. 21, November 1, 2003

light green represents the value of dimensionless concentration from 0 to 1. For a given quantity of sample released into the computational domain, Figure 9a shows the concentration field in the computational domain during a period of t* ) 9.15. At the end of this period, 71.5% of the sample remains in the computational domain, and the rest has flown out. As seen from Figure 9a, there is no 3D enclosed isoconcentration surface in the gap region between the roughness elements. This implies that there will be no microsample molecules trapped or accumulated in the gap regions. Finite-sized particles may be trapped in the gaps; however, such particle-liquid systems are well beyond the scope of this paper. The model presented here generally is not applicable to the (micrometer-sized) particle-liquid systems. The sampleliquid systems in this study are limited to small molecule-solution systems. If there is a 3D enclosed recirculation flow zone in the gaps between roughness elements, such a recirculation flow zone is referred to as the liquid molecule trap. The even-out effect under the electroosmotic driven flow, as discussed in this work, avoids such a recirculation flow zone in the gaps. It is also noticed from

the outlet plane of the computational domain that the sample concentration at the near wall region is higher than the region above. This is due to the larger values of the electroosmotic velocity near the channel wall region, while the central velocity field above the roughness region was impaired by the induced pressure gradient. Figure 9b shows the sample concentration field on the rough microchannel surface under the same conditions as in Figure 9a. From Figure 9b, one can see that the concentration in the pathway surface is the highest. The Effect of Electrokinetic Mobility on Sample Transport in the Rough Microchannels. To examine the effects of electrokinetic mobility, we will compare the sample mass transport through the microchannel for a given quantity of sample flowing into the computation domain. The quantity of sample flowing into the computation domain is determined by the original sample concentration, C0, and the time of releasing. The amount of sample flowing into the computation domain is defined by

Nin ) ∆t

∫

Ain

C*C0(ueo + uep) dA ) C0H3N/in

(20)

where Nin is the quantity of the sample molecule flowing into the computation domain during a period ∆t; N/in is its dimensionless form, N/in ) Nin/C0H3; Ain is the cross-sectional area at the inlet of computational domain; and ∆t is the duration of the releasing time until Nin reaches the given value. It should be noted here that the amount of sample transferred into the inlet of the computational domain by diffusion is much smaller than that by electroosmotic convection and electrophoresis and, therefore, is neglected. We focus on the effect of electroosmotic mobility µeo and the electrophoretic mobility µep on the sample transport through rough microchannels. Figure 9c shows the sample concentration field in the whole computational domain with an increased electroosmotic mobility value, µeo ) 6 × 10-8 m2/Vs, while all other parameters are the same as in Figure 9a and b. As the electroosmotic mobility increases, the electroosmotic velocity increases proportionally; therefore, the duration of releasing the same quantity of sample into the computational domain is shorter: t* ) 6.13 in Figure 9c (note t* ) 9.15 in Figure 9a). From Figure 9c, one can see that the concentration in the gap region is higher than that in Figure 9a, and there are overall more sample molecules (76% comparing with 71.5% in Figure 9a) in the whole computational domain. Figure 9d shows the corresponding concentration field on the roughened microchannel surface. Clearly the surface concentration is more uniform. When µep ) -1.4 × 10-8 m2/Vs, that is, the same absolute value of the electrophoretic mobility as in Figure 9a and c but with a different sign, while keeping the rest parameters the same, one sees a further improved concentration field in the whole computational domain, as shown in Figure 9e, and on the channel surface, as shown in Figure 9f. When the electric charge of sample molecules has a sign different from that of the channel wall, the electrophoretic motion is in the same direction as that of the electroosmotic flow, and the sample molecule velocity is larger than the electroosmotic velocity of the bulk liquid. This mechanism contributes significantly to an enhancement of the concentration uniformity in the channel’s cross-sectional area and to an increase in the number of the sample molecules in the gap regions

and the surface concentration. The duration of releasing the same quantity of sample into the computational domain is reduced further to t* ) 2.36, and more sample molecules, 81.7%, are in the computational domain. CONCLUSION The electroosmotic flow through microchannels is greatly influenced by the presence and characteristics of roughness on the channel walls. The electrical field is distorted by surface roughness elements, which makes the electroosmotic slip velocity nonuniform. Because of the presence of roughness elements in the flow passage, the electroosmotic flow in the rough microchannels induces a periodic pressure field that makes the central flow velocity smaller than that in the near wall region, and hence, the flow rate through the rough microchannel is significantly reduced. The induced high-pressure zone occurs at the upwind region of the rough elements, and the low-pressure zone is at the tail region. The induced pressure field causes the exchange of liquid between the gaps and the central flow, which is referred to as the even-out effect. The roughness element’s size, height, and separation distance have large influences on the even-out effect; the microchannel height has a small influence on the even-out effect. There is no sample concentration trap in the gap regions between roughness elements during the electroosmotic transport. The increase of the electroosmotic mobility or the decrease of the electrophoretic mobility can significantly improve the uniformity of the sample concentration field in a rough microchannel. ACKNOWLEDGMENT The authors are thankful for the financial support of the Institute for Polymer Research Dresden to Y. Hu and the support of a Research Grant of the Natural Sciences and Engineering Research Council of Canada to D. Li. NOMENCLATURE H channel’s height used as the characteristic length [m] Ci

ith species concentration [M]

C0

species drop-in concentration [M]

Di

ith species diffusion coefficient [m2/s]

E ˜, E

electrical strength vector and its modulus [V/m]

Nin

quantity of sample released into the microchannels [mol]

Pe

Peclet number, Pe ) Veo,sH/Di

Re

Reynolds number, Re ) FVeo,sH/µ

V ˜, V

velocity vector and its modulus [m/s]

a

size of the rough element’s square-shaped base [m]

b

distance between the center of the rough elements in both x and z directions [m]

h

rough element height [m]

˜,ı ˜,j k˜

unit tensor in x, y, z directions

p

pressure [Pa]; the dimensionless pressure is p* ) p/(µVeo,s/H)

t

time [s]; the dimensionless time is t* ) (t - t0)/ (H/Veo,s)

u, v, w

velocity components in x, y, and z directions, respectively [m/s]

x, y, z

coordinate variables [m]


5757

Greek Symbols

Subscript

electrical permittivity of a solution [C/Vm]

av

averaged in cross-sectional area

φ

electrical potential [V]

eo

electroosmosis

ep

electrophoresis

F

fluid density

Fe

net charge density [C/m3]

i

ith species

µ

dynamic viscosity [kg/ms]

in, out

inlet and outlet of the computational domain

µeo

electroosmotic mobility [m2/(Vs)]

s

smooth channel

µep

electrophoretic mobility [m2/(Vs)]

slip

at the slip boundary

ζ

zeta potential of a surface [V]

[kg/m3]

Superscript

Received for review June 30, 2003. Accepted August 6, 2003.

*

AC0347157

5758

dimensionless parameter


Electrokinetic Transport through Rough Microchannels - Analytical

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