## Abstract

Mass transport driven by oscillatory electroosmotic flows (EOF) in a two-dimensional micro-channel is studied theoretically. The results indicate that the velocity and concentration distributions across the channel-width become more and more non-uniform as the Womersley number *W*, or the oscillation frequency, increases. It is also revealed that, with a constant tidal displacement, the total mass transport rate increases with the Womersley number *W* due to both the stronger convective and the transverse dispersion effects. The total mass transport rate also increases with the tidal displacement (with a fixed oscillation frequency) because of the associated stronger convective effects. The cross-over phenomenon of the mass transport rates for different species becomes possible with sufficiently large Debye lengths and at sufficiently large values of *W*. Consequently, with proper choices of the Debye length, oscillation frequency and tidal displacement, oscillatory EOF may become a good candidate for the first-step separation of the mass species.

## 1. Introduction

Biochips are the essential elements of bioengineering designs and are often applied in drug diagnosis, tumour cell analyses, DNA sequencing systems and biological/environmental-monitoring sensors. Because of this important role in its own discipline, the technical integration and the design of ‘Lab-on-a-Chip’ devices become a major research interest both academically and industrially. Micro-fluidic components such as micro-channels, micro-mixers, micro-pumps and micro-reaction-chambers are commonly implemented in biochip designs. In order to build biochips with high efficiency and quality, the physical factors and phenomena that will affect the performance of the micro-fluidic components are needed to be studied carefully and thoroughly. For example, electrokinetic (EK) effects, originated from the presence of the electrical double layer (EDL) at the contact interface of the working liquid and the solid substrate, are long discussed in literature (Hunter 1981; Probstein 1994; Karniadakis & Beskok 2002), and are usually referred as a promising candidate to be applied in micro-fluidic devices. EK effects are recommended because of the great potential in designing and building the devices with non-moving parts so that flow control with a higher working reliability can be achieved (Dutta & Beskok 2001*a*; Karniadakis & Beskok 2002).

EDLs are formed at the solid–liquid interfaces when electrolytes or liquids of aqueous nature, such as water, are brought into contact with the solid surfaces which possess electro-static charges. The counter-ions (ions that possess opposite charges to those on the solid surface) in the liquid are, thus attracted and the co-ions are then repelled by the solid surfaces. Since the counter-ions at the very vicinity of the charged-wall are actually adhered to the wall, they are motionless when the liquid has relative motion to the solid surface. This non-mobile layer is called the Stern layer, and the characteristic electrical potential of this layer is regarded as the Zeta potential denoted by *ζ*. Beyond the Stern layer, counter-ions are relatively free to move about in a diffusive manner, however, most of the counter-ions are confined within a region near the boundary, which is determined by the balance between electrical forces and thermal motion, namely, the Brownian motion. The Debye length, *λ*_{D}, is thus defined by the above energy balance and depicts how far most of the counter-ions can diffuse into the bulk liquid away from the boundary (table 1).

When an external electrical field is applied parallel to the solid surface, the co- and counter-ions will be attracted towards the anode and cathode, respectively; liquid particles in the neighbourhood of the ions will then be dragged and accelerated by the migrating ions. Net flow motion will, thus be observed for the amount of counter-ions is excessive to that of the co-ions near the solid walls in the presence of the EDL. This phenomenon is called ‘electroosmosis,’ which was first observed by Reuss who performed a series of experiments on EK effects in 1809 (Probstein 1994). The resulting flow is called the electroosmotic flow (EOF) with a shear layer formed between the motionless Stern layer and the electric-active diffusive region. Many novel studies about EOF have been carried out recently. For instance, laminar oscillatory EOFs were described and discussed in detail by Dutta & Beskok (2001*b*) and Erickson & Li (2003); unstable oscillatory EOFs and their application to the design of high efficiency micro-mixers were presented by Oddy *et al*. (2001); in later studies performed by Lin *et al*. (2004) and Suresh & Homsy (2004), the physical mechanisms of electrokinetic instability (EKI) were investigated systematically and thoroughly.

Laminar oscillatory flow fields are remarkable because of the dramatic mass dispersion enhancement when concentration gradients of miscible mass species are introduced into the flow field. The basic mechanism that contributes to the above process is the non-uniform velocity distribution between the wall boundaries, which require no-slip of the fluid, and the bulk flow region; radial or lateral concentration gradients across the flow region are therefore generated. This is the basic idea of ‘Taylor dispersion’ first proposed by Taylor (1953). Aris (1956, 1960) then extended the idea into more general flow situations by the method of moments, and broadened the adequate parametric ranges and usage of the dispersion concept. Recently, Probstein (1994) presented an inductive scaling illustration of the dispersion enhancement under various conditions when flow convection is introduced in a circular tube.

In addition to the enhancement of mass dispersion (or transfer), the separation of different species and cross-over phenomenon in which the slow diffuser eventually travels faster than the fast diffuser in an oscillatory flow under specific frequencies may exist and have been shown experimentally by Kurzweg & Jaeger (1987). Such an effect has been applied to the air revitalization process in the space station life-supporting systems as described by Thomas (2003) and Thomas & Narayanan (2002*b*). Thomas and Narayanan (2001, 2002*a*) also presented results due, respectively, to the pressure-driven and boundary-driven flow oscillations in their studies. Three dimensionless parameters, namely, the Womersley number W (the ratio of viscous diffusion time-scale to the oscillation time-scale), the Schmidt number Sc (the ratio of species diffusion time-scale to the viscous diffusion time-scale), and length ratio of the oscillation amplitude to the channel-width, were identified by them to be essential in understanding the separation process among species. This separation process was also studied analytically by Kurzweg (1988) for oscillatory liquid flows in a bundle of cylindrical capillaries. Therein, he suggested that despite the considerably low diffusivities of mass species in liquid solutions, the method using flow oscillation for species separation was still better than the traditionally used chromatography method because of the relatively low-operation pressure gradient needed in the oscillatory system. Nevertheless, little effort has been devoted to understand the fundamental physics of the enhancement of mass transport and separation of species in an oscillatory EOF.

In order to understand the fundamental physics as well as provide useful information and criteria for designing micro-fluidic devices, the present study is, thus aimed at the theoretical investigation of the transport and separation phenomena of mass species in a periodically oscillatory EOF in two-dimensional micro-channels. The flow configuration, assumptions, non-dimensionalization scheme and governing equations will be given in next section. The analytic solutions for the electrical potential, velocity distribution and species concentration field are obtained in §3, followed by the presentation and systematic discussion of the results in §4. Important aspects derived from this study will finally be summarized in §5.

## 2. Assumptions, non-dimensionalization and formulation

The isothermal system considered here is a two-dimensional rectangular channel of length *L* and width *h* filled with a liquid (the solvent) which is an electrolyte or of aqueous nature. EDLs are, thus established at the upper and lower boundaries when the carrier liquid is brought into contact with the channel walls. In addition, carried with the liquid is a neutral species of concentration *c*. An oscillatory electrical field is then imposed on the system through the anode and cathode installed at the two ends of the channel, as shown in figure 1. As a result, a periodically oscillatory flow is generated via the electroosmotic effect. By assuming no net flow in a period of oscillatory motion, the present study is aimed at investigating the effect on the mass transport due to convection of such an oscillatory EOF.

Three fields thus need to be analysed; they are the electrical, velocity and concentration fields. In this section, the appropriate assumptions, mathematical formulations and non-dimensionalization schemes for these three fields are to be presented and derived, respectively. Following in the next section are the mathematical solutions of these fields in analytical forms.

### (a) Electrical field

Due to the symmetry of the system, the EDL potentials extending from the upper and lower walls are symmetrical with respect to the centreline (depicted by *C*_{L} in figure 1) of the channel and are assumed not to overlap with each other. Such a non-overlapping condition is valid when the Debye length of EDL is much smaller compared with the width of the channel as suggested by Probstein (1994), e.g. . In the above expression, *λ*_{D} is the Debye length defined as(2.1)with *F* being the Faraday constant, *z* the valence of the co- and counter-ions in the carrier liquid (the solvent is assumed to be a 1 : 1 symmetric electrolyte), *ϵ* the permittivity of the carrier liquid, *R* the universal gas constant, *T* the absolute temperature in Kelvin and *c*′ the averaged molar concentration of the counter-ions.

To further simplify the calculation of the EDL potential, other assumptions are needed. Firstly, the excessive charge distribution is assumed to vary significantly merely in the *y*-direction and hence the electrical potential of the EDL is viewed as a function of *y* only. The Boltzmann distribution of the charge density then applies, which gives(2.2)where *ρ*_{e} represents the charge density, *ψ*, the EDL potential and *c*_{0}, the ion concentration far from the charged-walls. Secondly, because of the slow velocities of EOFs (about the order of 10^{−4} m s^{−1}) and the relatively small quantities of the excessive charges, the convective effect of ions, i.e. the possible electro–magneto interactions are assumed negligible, as suggested by Karniadakis & Beskok (2002). The EDL potential is then described by the following Poisson–Boltzmann equation(2.3)Equation (2.3) can be further simplified. With the Debye–Huckel approximation, the above equation for the electrical potential distribution can then be linearized as(2.4)To non-dimensionalize the above equation, the following scheme is employed: *y*^{*}=*y*/*h*, *ψ*^{*}=*ψ*/*ζ*, where the variables with a superscript ‘*’ denote the dimensionless ones and *ζ* is the Zeta potential at the Stern layer. With such a non-dimensionalization scheme, the governing equation for the EDL potential in dimensionless form becomes(2.5)The associated boundary conditions are(2.5a)(2.5b)In equation (2.5*a*), the electrical potential at the channel wall is approximated by the Zeta potential at the Stern layer based on the argument of immobility of ions therein.

### (b) Velocity field

The flow considered is assumed to be continuum, isothermal, Newtonian, two-dimensional and incompressible. The governing equations are the continuity and momentum equations, i.e.(2.6)(2.7)In the above equations, (*u*_{x}, *u*_{y}) denotes the velocity vector * u* in the coordinate system (

*x*,

*y*);

*ρ*

_{f}and

*μ*are the density and dynamic viscosity of the carrier liquid;

*is the applied electrical field with*

**E***ρ*

_{e}being the charge density. The last term in equation (2.7), i.e.

*ρ*

_{e}

*, denotes the Lorenz force which is the main driving force to generate the EOF.*

**E**By assuming the external electrical field be applied only in the *x*-direction, * E* then reduces to the following form:(2.8)As proposed by Oddy

*et al*. (2001) and Morgan & Green (2003), the magnitude of

*E*

_{x}should be maintained below 100 V mm

^{−1}to avoid possible EK instability and significant Joule's heating for an oscillatory EOF, respectively.

By further assuming the flow be fully developed in the *x*-direction and mainly driven by the electroosmotic effect, which Dutta & Beskok (2001*b*) termed ‘pure electroosmotic flow’, the *y*-component velocity, i.e. *u*_{y}, vanishes from equation (2.6) and the pressure effect in the momentum equation, i.e. equation (2.7), becomes negligibly small when compared to the Lorenz effect. The momentum equation (2.7) then reduces to(2.9)To non-dimensionalize the above equation, the following scheme is applied: , where(2.10)is a moving referenced velocity (Thomas & Narayanan 2002*a*) suggested by the Helmholtz–Smoluchowski equation; *t*^{*}=*t*/*τ* with *τ*=2*π*/*ω*; and *y*^{*}=*y*/*h*. In terms of the above non-dimensional variables and using the linearized EDL potential, i.e.(2.11)the momentum equation then assumes the following dimensionless form:(2.12)where(2.13)is the Womersley number, a ratio of the viscous diffusion time-scale *t*_{ν} to the oscillation time-scale *t*_{ω}.

The associated boundary conditions for *u*_{x} are the no-slip condition at the channel walls, i.e.(2.12a)The above condition implies that finite Debye length effects (Dutta & Beskok 2001*a*) are taken into account in the present study so that a more general situation of the EOF can be analysed.

### (c) Concentration field

The species to be transported through the carrier liquid is assumed neutral so that the transport phenomenon will not be affected by any of the electrical potentials. In addition, the species concentration is also assumed to be infinitely dilute, so that the concentration gradients of all the species (if there are two or more species present in the system) will not interfere with each other. Meanwhile, for an infinitely dilute solution, the molar average velocity is the same as the mass average velocity which is simply the flow velocity ** u** of the carrier liquid. Assuming Fick's law apply, the mass transport equation can then be written as(2.14)where

*D*is the diffusivity coefficient and

*u*

_{y}=0 from the discussion of §2

*b*has been applied.

The micro-channel in figure 1 is installed between two reservoirs where the concentration of species are maintained at constant values, *c*_{1} for the left reservoir and *c*_{2} for the right one, with *c*_{1}>*c*_{2}. If there is no flow motion of the carrier liquid, diffusion will be the only mechanism responsible for the transport of mass species throughout the channel, and the steady-state concentration distribution will assume a linear form in the *x*-direction and be uniform in the *y*-direction. Now, with a fully developed oscillatory EOF, i.e. *u*_{x}=*u*_{x} (*y*, *t*), as discussed in §2*b*, the concentration distribution of the species will not remain precisely uniform in any cross-section of the channel because of the non-uniform velocity distribution (no-slip required on the channel walls) across the channel-width. However, the linear variation of the species concentration along the flow direction remains unchanged except near the end regions when the system reaches steady-state. Therefore, it is reasonable to assume the following form for the species distribution when a fully developed, steady-state oscillatory EOF in a micro-channel connecting two reservoirs is considered, i.e.(2.15)where *c*_{u}(*y*, *t*) denotes the imposed flow oscillation effect on the concentration distribution. The boundary conditions at the two ends of the channel, i.e. *c*(0, *y*, *t*)=*c*_{1} and *c*(*L*, *y*, *t*)=*c*_{2}, are not satisfied exactly by the above equation due to the *c*_{u}(*y*, *t*) effect. However, it is within the approximation by neglecting the end effect of a low-aspect-ratio configuration with .

Substituting equation (2.15) into equation (2.14), it then yields(2.16)with the following non-penetration condition to be satisfied at the channel walls, i.e.(2.16a)Let the concentration of the species be non-dimensionalized by , and the *y* variable by *y*^{*}=*y*/*h*, the dimensionless form of equation (2.16) then reduces to(2.17)with *A* and W being the aspect-ratio and Womersley number as defined previously. The other two dimensionless parameters in the above equation, i.e. Sc and , are defined as follows:(2.18)(2.19)The dimensionless form of the non-penetration boundary condition at the channel walls becomes(2.17a)In summary, equations (2.5)–(2.5*b*) are to be solved to determine the electrical potential distribution. Equations (2.12) and (2.12*a*) are for the velocity distribution and the concentration field is determined by equations (2.17) and (2.17*a*). In next section, the mathematical solutions for the above equations will be pursued and presented.

## 3. Mathematical solutions

In this section, the analytic solutions for the electrical potential, velocity and concentration fields are to be determined; the time and space-averaged mass transfer rate will also be defined and calculated as an estimation of the overall mass transport. While deriving the expressions for the averaged mass-transfer rate, the Lagrangian displacement *α* and the tidal displacement *β* as suggested by Kurzweg (1985, 1988) will be employed so that a reasonable comparison of the overall mass transport among various flow situations can be achieved.

### (a) Electrical field

From equations (2.5)–(2.5*b*), the linearized EDL potential is solved as(3.1)which is a rather simple solution satisfying the symmetry condition as requested. Moreover, the ratio of channel-width to Debye length, i.e. *λ*^{*}, becomes the only parameter determining the distribution of the linearized EDL potential.

### (b) Velocity field

Due to the linearity of the *u*_{x}-momentum equation, i.e. equation (2.12), the expression for is assumed to possess the following form:(3.2)as suggested by the imposed electrical field which is the only forcing term of the equation. With the substitution of the above equation and the EDL potential (3.1) into equation (2.12), the following equation governing the space distribution of is thus obtained(3.3)The boundary conditions become as follows:(3.3a)The Green's function method is then applied to solve equations (3.3) and (3.3*a*), and the corresponding Green's function is derived as(3.4)The distribution of is thus solved and the expression of *u*_{x} is obtained, i.e.with(3.5)where the coefficients *φ*_{i}'s are given as(3.6a)(3.6b)(3.6c)and(3.6d)

### (c) The Lagrangian displacement and tidal displacement

After the velocity distribution is solved, the Lagrangian displacement and tidal displacement can then be calculated. Since the Lagrangian displacement represents the moving distance of each fluid particle during time interval Δ*t*=*t*−0, it can be expressed as(3.7)which is the same for every cross-section in the present study except near the two end regions. Therefore, the *x*-position of each fluid particle at time *t* is denoted by *x*=*x*_{0}+*α*. To calculate the averaged moving distance of each cross-section, the tidal displacement is thus defined and expressed as(3.8)In equation (3.8), one half-cycle of oscillation has been applied for calculation to have a maximum averaged moving distance, which will thence be used equivalently as the averaged amplitude Δ*x*′ of the periodically oscillatory EOF (Kurzweg 1985, 1988; Thomas & Narayanan 2001) in the present study. In order to perform a reasonable comparison of the averaged mass transport, the oscillation amplitude Δ*x*′ or *β* has to be kept the same for each flow situation considered. As a result, the applied electrical field , the slip velocity , and the corresponding Peclet number will no longer remain constant for all the flow situations. Instead, they have to be adjusted in accordance with the frequency so that the condition *β*=Δ*x*′=const. will be preserved, i.e.(3.9)(3.10)and(3.11)The factor *η*(*ω*) in equations (3.9) and (3.10) is given by(3.12)

### (d) Concentration field

From the previous discussion in §2*c*, the convective effect on the total concentration distribution depends only on the oscillatory flow motion. Thus, in order to be timewise consistent with the flow field, the concentration distribution due to the convective effect is assumed to possess the following form(3.13)By substituting equations (3.2) and (3.13) into equation (2.17), the time and space coordinates are decoupled, and equation (2.17) reduces to the following form(3.14)where is the complex conjugate of and is the frequency-dependent Peclet number. With the non-penetration boundary conditions(3.14a)the Green's function of equations (3.14) and (3.14*a*) is derived as(3.15)After integrating(3.16)and taking the complex conjugate of , the non-dimensional spacewise distribution of the convective part of the concentration field, i.e. , is solved as(3.17)with being the complex conjugate of , and the coefficients *γ*_{i}'s being functions of *λ*^{*}, W and Sc, i.e.(3.18a)(3.18b)(3.18c)(3.18d)(3.18e)and(3.18f)where 's are the complex conjugates of *φ*_{i}'s. From equations (2.15), (3.13) and (3.17), the total concentration distribution can then be expressed as(3.19)

### (e) The time and space-averaged mass-transfer rate

As discussed previously, although there is no net flow in each period of oscillation, there exists a net mass transport from upstream to downstream, because of either the pure diffusion effect or the convective effect. In order to calculate this net mass transport, the following time-and-space averaged mass-transfer rate, called the effective mass flux, is then defined for each cross-section of the channel in one period of oscillation (Thomas & Narayanan 2001, 2002*a*,*b*; Thomas 2003), i.e.(3.20)In the above expression, *J*_{x} denotes the instant mass flux in the *x*-direction caused by both the diffusion and convective effects, i.e.(3.21)or(3.21a)After substituting equation (3.21*a*) into equation (3.20) and then integrating it with respect to time, *Q*_{x} reduces to(3.22)where and are the complex conjugates of and , respectively. The above relation can also be expressed in the dimensionless form as(3.23)with the parameters defined as , and . The integral in equation (3.23) is expressed in terms of *φ _{i}*'s and

*γ*

_{i}'s, and their complex conjugates as well, i.e.(3.24)

## 4. Results and discussion

Results in this section will generally be presented in the dimensionless forms so that effects on the velocity distribution, mass transport, etc. can be discussed systematically through the variations of the dimensionless parameters. Dimensional results will also be given when exact values of the averaged or effective mass flux are preferred. Specific dimensions and conditions of the system as well as the transport properties of certain species used for calculations in this study are given in tables 2 and 3.

In order to maintain a similar convective contribution to the mass transport, a fixed tidal displacement, or oscillation amplitude, i.e. Δ*x*′, is required for both high and low-frequency flow oscillations in a given micro-channel filled with electrolyte buffer. The applied electric field , the slip velocity , and the Peclet number are, therefore, functions of the oscillation frequency or Womersley number. However, the detrimental effects of Joule's heating and EKI limit the maximum employable applied electric field, and thus, through equations (3.7)–(3.12), limit the maximum employable oscillation frequency for mass transport while the tidal displacement is fixed to a constant. In accordance with the above argument, the Womersley number is generally maintained below 0.6, which corresponds to approximately 570 Hz for *h*=10 μm, *ν*∼10^{−6} m^{2} s^{−1} and Δ*x*′/*h*=1.0, in the following results and discussion. Although frequencies beyond the above limit may intentionally be applied in the following parametric study for demonstrating the trends of certain physical phenomena, the employed frequencies are always smaller than 1 MHz as reported in Green *et al*. (2000) for settings of *h*=10 μm and *ν*∼10^{−6} m^{2} s^{−1}.

### (a) The velocity profile

The velocity distributions for various values of Womersley number W and the dimensionless Debye length *λ*^{*} at *tω*=(2*n*+0.5)*π* are shown in figure 2; figure 2*a* is for W=0.5, *λ*^{*}=20, 70 and 700, figure 2*b* is for W=5, *λ*^{*}=20, 70 and 700 and figure 2*c* is for W=15, *λ*^{*}=20, 70 and 700, i.e. for *h*=10 μm channels, frequencies from 400 Hz to 360 kHz, and Debye lengths from 15 to 500 nm, a Debye length range that corresponds to electrolyte buffer concentrations from approximately 10^{−4} to 10^{−6} M (Hunter 1981; Dutta & Beskok 2001*b*). From figure 2, it can be learnt that when the Womersley number is small, i.e. figure 2*a* with W=0.5, the flow remains in a plug profile generally without phase variation across the channel-width. However, the velocity distribution tends to vary more gradually near the solid wall with a smaller value of *λ*^{*}, which indicates a larger Debye length. As the Womersley number increases, the velocity profile tends to be non-uniformly distributed across the channel-width as shown in figure 2*b* and *c*, indicating phase lags between the velocity distribution and the imposed electrical field. Since the counter-ions spread farther into the bulk liquid with a larger Debye length, i.e. a smaller value of *λ*^{*}, more fluid particles are dragged by the counter-ions driven by the electrical force. Consequently, the velocity boundary layer becomes thicker with the peak value of the non-dimensional velocity distribution being smaller.

### (b) The convective concentration distribution

To calculate for the various mass species, the corresponding Schmidt numbers of these species are to be specified in advance. From table 3, the Schmidt numbers for ethanol, glycine, glucose and sucrose are all found to be around the order of 10^{3}. Therefore, the Schmidt number used in the following computations will be specified as Sc=1000.

Shown in figure 3 are the calculated concentration distributions for W=0.1, 0.3 and 0.6 at *tω*=(2*n*+0.5)*π* and with Sc=1000, *λ*^{*}=70 (*λ*_{D}∼140 nm), and Δ*x*′/*h*=1.0. As already seen in figure 2*a*, the velocity distributions around this frequency range are almost uniform across the channel-width except near the wall boundaries where a rapid decrease from a finite value to zero prevails because of the no-slip condition. Consequently, the convective concentration distributions, as shown in figure 3, also possess similar profiles with significant variations occurring only near the boundaries, which are approximately the regions of *y*^{*}=0–0.2 and *y*^{*}=0.8–1.0. However, it is these variations near the boundaries that would incur transverse dispersion through the strong concentration gradient, and thus increase the mass transport rate in the flow direction as similar to the ‘Taylor dispersion’ effect in the pressure-driven flows. Such a mass transport phenomenon will be presented and discussed in the following section. The velocity and concentration distributions for Δ*x*′/*h*=1.5 and 2.0 are qualitatively similar to those shown in figure 3 deferring only slightly in the numerical values and hence, not included.

### (c) The time/space-averaged mass transport

The dimensionless averaged mass transport rates, , due to an oscillatory EOF are determined by equation (3.23) and are shown in figure 4. The conditions used for calculations are *λ*^{*}=70, Sc=1000 and Δ*x*′/*h*=1.0, 1.5 and 2.0. One of the results described by figure 4 is that at a fixed value of W, or frequency, the mass transport rate increases as the tidal displacement enlarges. This is because a larger tidal displacement magnifies the local, instantaneous convective and dispersion effects in the concentration field and eventually enhances the mass transport rate in comparison to the cases with small tidal displacements under a fixed operation frequency. Another result from figure 4 is that for a fixed tidal displacement, the mass transport rate increases with the dimensionless frequency W. In fact, increasing W requests necessarily an equivalent amplification on the characteristic slip velocity so that a constant tidal displacement can be preserved. Meanwhile, as W enlarges, the concentration distribution near the wall boundaries becomes more non-uniform (figure 3) which thus strengthens the local dispersion effects due to the steeper concentration gradients near the wall boundaries.

Shown in figure 5*a* are the dimensionless averaged mass transport rates, , for *λ*^{*}=20, 70 and 700 with Δ*x*′/*h*=1.0; all the other conditions are the same as those used in figure 4. Results from this figure suggest that for a fixed value of W, the mass transport rate is larger in a flow system with a larger Debye length, or a smaller value of *λ*^{*}, because of the stronger dispersion effect in the transverse direction due to the more non-uniform convective concentration distributions as shown in figure 5*b* with *W*=0.3, or ∼140 Hz for *h*=10 μm and *ν*∼10^{−6} m^{2} s^{−1}.

### (d) Cross-over phenomenon and separation of species

Shown in figure 6 are the variations of versus W with Sc=100, 500, 1000, 1500 and 2000 (mass diffusivities from 10^{−8} to 5×10^{−10} m^{2} s^{−1} in water with *ν*∼10^{−6} m^{2} s^{−1}); figure 6*a* is for *λ*^{*}=20, figure 6*b* is for *λ*^{*}=70, and figure 6*c* is for *λ*^{*}=700, all with Δ*x*′/*h*=1.0. The results indicate that an oscillatory EOF is always more conducive to the enhancement of mass transport rate of the species with a larger Sc number, or a smaller mass diffusivity and this phenomenon becomes more and more obvious with a smaller value of *λ*^{*}. Thus, with a sufficiently low value of *λ*^{*} and at a sufficiently high frequency, the cross-over phenomenon as appearing in the variations of the ‘dimensional’ mass transport rate *Q*_{x} versus the Womersley number W becomes possible.

Examples of the cross-over phenomenon are shown in figure 7*a* for glucose and sucrose and in figure 7*b* for ethanol and sucrose both with *λ*^{*}=20 and Δ*x*′/*h*=10.0, wherein the first and second cross-over points are identified. Note that when the oscillation frequency is high enough, a second cross-over point exists. When the second cross-over exists, the oscillatory EOF at sufficiently high W becomes effectively conducive to the enhancement of mass transport rate of the species with a higher mass diffusivity. With the first and second cross-over points identified, the application of oscillatory EOF in the species separation becomes very interesting. For the flow situation falling in the region beyond the second cross-over point, the species with a high diffusivity will be transported downstream even faster than that with a lower diffusivity. Under such a flow condition, the oscillatory EOF can be utilized as the first-step separation of species. However, under certain circumstances, the species with a lower diffusivity needs to be separated from the one with a higher diffusivity and transported to a second container with more appropriate environment; the flow situation falling in between the first and second cross-over points, thus becomes preferable. In conclusion, with proper choices of the Debye length, oscillation frequency, and tidal displacement, oscillatory EOFs may become a good candidate for the first-step separation of mass species.

## 5. Concluding remarks

The mass transport phenomena driven by an oscillatory EOF in a two dimensional micro-channel are studied theoretically herein. With the assumptions of non-overlapping EDLs, linearized electrical potentials, a fully developed incompressible flow in the streamwise direction, and an infinitely dilute solution with neutral species, the governing equations for the electrical potential, as well as the velocity and the concentration distributions are uncoupled so that analytical solutions become attainable. Results are then presented and discussed systematically with the following conclusions drawn from the present study.

The velocity and concentration distributions across the channel-width become more and more non-uniform as the Womersley number W increases, which might then induce a stronger dispersion effect in the transverse direction.

With a constant tidal displacement, the total mass transport rate increases with W due to both the stronger convective and transverse dispersion effects.

With a fixed value of W, the total mass transport rate also increases rapidly with the tidal displacement because of the associated stronger convective effects.

With a sufficiently low value of

*λ*^{*}and at a sufficiently large value of W, the cross-over phenomenon of the mass transport rates for different species becomes possible.With proper choices of the Debye length, oscillation frequency and tidal displacement, a first-step separation of species by means of an oscillatory EOF becomes achievable.

Future work may include the removal of the linear electrical potential assumption; the transient effect on the velocity and concentration distributions may also need to be considered for the flow situations with W≥1. Numerical results will then become inevitable.

## Acknowledgments

The authors gratefully acknowledge the financial support from the National Science Council of Taiwan, ROC, through grant no. NSC93-2212-E-002-036.

## Footnotes

- Received December 27, 2005.
- Accepted January 6, 2006.

- © 2006 The Royal Society