yao, santiago 2003-porous glass electroosmotic pumps theory

8/11/2019 Yao, Santiago 2003-Porous Glass Electroosmotic Pumps Theory

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Journal of Colloid and Interface Science 268 (2003) 133142www.elsevier.com/locate/jcis

Porous glass electroosmotic pumps: theory

Shuhuai Yao and Juan G. Santiago

Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA

Received 17 January 2003; accepted 17 July 2003

Abstract

This paper presents an analytical study of electroosmotic (EO) pumps with porous pumping structures. We have developed an analyticalmodel to solve for electroosmotic ow rate, pump current, and thermodynamic efciency as a function of pump pressure load for porous-structure EO pumps. The model uses a symmetric electrolyte approximation valid for the high-zeta-potential regime and numerically solvesthe PoissonBoltzmann equation for charge distribution in the idealized pore geometry. Generalized scaling of pumping performance isdiscussed in the context of a parameterization that includes porosity, tortuosity, pore size, bulk ionic density, and the nonuniform conductivitydistribution over charge layers. The model also incorporates an approximate ionic-strength-dependent zeta potential formulation. 2003 Elsevier Inc. All rights reserved.

Keywords: Electroosmotic pump; Porous glass; Thermodynamic efciency; Zeta potential

1. Introduction

EO pumps are devices that generate both ow rate andsignicant pressure capacity using electroosmosis throughpores or channels. The term electroosmosis refers to thebulk motion of an electrolyte caused by Coulombic forcesacting on diffuse ions near a ow channels solid/liquid in-terface [1]. Although electroosmosis has been studied fornearly two centuries [2], its application to miniaturized de-vices for the generation of high-pressure ow streams hasbeen of interest only in the last three decades. Electroos-motic (EO) pumps have no moving parts and offer distinctadvantages over other micropumps including high pressure(well over one atmosphere of pressure is readily achievable

at 100 V) and large ow rate (greater than 30 ml / min at100 V). These devices have the potential to impact a varietyof applications including microelectronics cooling and bio-analytical applications [35]. Interest in this research area isincreasing.

In 1974, Pretorius et al. [6] rst describedusing electroos-motic ow to drive a liquid chromatographic separation asan alternative to high-pressure pumping, but did not demon-strate the ability to generate high pressure. They demon-strated a ow velocity of 0.2 mm / s at a eld of 2000 V / cm

* Corresponding author.

E-mail address: [email protected] (J.G. Santiago).

in 5-cm-long, 1-mm-inner-diameter glass columns packedwith polydisperse silica particles with dimensions rangingfrom 1 to 20 m. We estimate that their porous struc-ture should have been able to generate pressures in ex-cess of 40 atm at a 1-kV applied potential. That sameyear, Theeuwes [7,8] patented and reduced to practice theconcept of miniaturized electroosmotic pumps for the gen-eration of relatively high-pressure ow streams and dis-cussed their application to a variety of controlled drug de-livery systems. Glass frits 0.2 cm thick with pore diameter0.1 m were used as the pumping medium, and they demon-strated a ow rate of 8 .3 102 l/ min (for applied eld160 V / cm) and a pressure capacity of 0.7 atm at 50 V.Theeuwes noted that pump pressure is linearly dependent onapplied voltage. Two and a half decades later, Gan and co-workers [9,10] described the development of a borosilicateporous glass EO pump device with pore diameter 25 mfabricated using high-temperature sintering. They demon-strated the pumping of several uid chemistries and the ef-fect of these chemistries on ow stability. Their pump gen-erated 3.5 ml / min and 1.5 atm at 500 V. Two years later,Paul and Rakestraw [11] built packed-column EO pumpsand demonstrated that large (12-kV) applied potentialscouldbe used to generate pressure capacities of 340 atm using a75-m inner diameter capillary packed with 3-m-diameterbeads. They used a fabrication method similar to that of Yan [12] for electrochromatographiccolumns. Extrapolation

of this performance suggests the ow rates of these pumps0021-9797/$ see front matter 2003 Elsevier Inc. All rights reserved.doi:10.1016/S0021-9797(03)00731-8
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134 S. Yao, J.G. Santiago / Journal of Colloid and Interface Science 268 (2003) 133142

were 100 l/ min for 10-kV applied potentials. Paul and co-workers patented the use of these pumps as miniaturizedactuation devices and high-pressure valves [13,14]. Zeng etal. [15] presented an analytical model and characterizationof EO pumps fabricated by packing 3.5-m-diameter beadsin fused silica capillaries with inner diameters of 530 m.These pumps achieved maximum pressure and ow ratesof 23.5 atm and 4.8 m / min, respectively, at 2-kV poten-tials. In an effort to increase the ow rate of such pumps,Zeng et al. [16] used polymerized frit structures and slurrypacking of nonporous silica particles to demonstrate EOpumps that delivered maximum ow rates and pressures of 0.8 ml / min and 2.0 atm, respectively, at 1.0 kV. More re-cently, we demonstrated sintered glass frit pumps that pro-vide maximum ow rates and pressure capacities on the or-der of 7 ml/ min and 2.5 atm, respectively, at 200 V, withan active pumping volume of less than 2 cm 3 [17]. In par-allel efforts, Chen and Santiago [18] and Laser et al. [19]

demonstrated EO pumps fabricated using planar microma-chined structures in glass and silicon substrates, respectively.Chen and Santiago also presented an analytical model for theestimationof thermodynamicefciencyin these slot struc-ture pumps, including a detailed energy balance of a pumpand load closed-loop system.

Despite the conference, journal, and patent literature de-scribed above, research work on EO pumps has not yetprovided full insight into the operational behaviorand funda-mental ow principles behind EO pumps. In particular, gen-eralized scaling (e.g., geometric scaling) analyses, detailedmodels of electrical double layer (EDL) physics, advectivecurrent effects, and thermodynamic efciency models havenot been presented. In this study, we present an analyticaland numerical model that can be used to predict pump owrate, pressure, and thermodynamic efciency across varia-tions in geometry, applied potentials, working uid chem-istry, and pressure load conditions. The model treats theporous pumping structure of the pump as a network of manyow channels in parallel and the case of electrical doublelayers with a thickness on theorder of thecapillary radius (inthis paper we refer to this regime as nite electrical doublelayers). We have used this model to guide the design of ourEO pumps for high-heat-load heat transfer applications [3].

2. Theory

2.1. Electroosmotic ow model

We treat electroosmotic ow (EOF) in porous mediaas ow through a large number of idealized tortuous mi-crochannels in parallel, as depicted schematically in Fig. 1.In this model, tortuous microchannels (i.e., pores) with cir-cular cross sections are assumed to have equal pore radii,a , and an equal value of zeta potential, . The channelshave a tortuosity and the porous pumping structure has

a porosity . This model is suggested by the work of Mazur

Fig. 1. Schematic of the porous pump model with idealized cylindricalpores of uniform diameter. Flow is modeled within each pore (as shownon the right) and then ow rate and current are integrated over all pores inthe system. The structure can be characterized by its total volume, , thevoid volume,

e, its length L , and the tortuous characteristic length of the

pores L e . The ratios e/

and (L e /L) 2 are dened as the porosity and tor-tuosity, respectively.

and Overbeek [20], which leveraged a similar formulationfor electroosmotic ow in porous diaphragm structures. Themodel can be used to derive a set of ow equations that aresimilar to the corresponding equations for EOF in a singlecapillary [21]. By applying both a pressure load and poten-tial gradients along the axes of the pores of the pumpingstructure, expressions can be derived for the velocity eldin a single cylindrical pore [21]. Implicitly, in terms of theelectric potential within the pore, the general velocity pro-le within the circular cross section of a pore with a highaxial-length-to-radius ratio is

(1)u(r) =a2

P x4 1 r2

a 2 E x 1 .Here, P x = P/L e is the streamwise pressure gradient andE x = V eff /L e is the streamwise electric eld within thepores. The calculation of ow rate and pressure drop there-fore reduces to nding an adequate model for the potentialdistribution associated with the EDL in the pore. This ve-locity prole can be integrated over the cross-sectional area,A , and axial length of the porous material, L , to yield thefollowing expression for the ow rate of the entire struc-ture [15,17],

(2)Q = PAa2

8L AV eff L f ,where A is the cross-sectional area of the porous structure, =(L e /L) 2 is the tortuosity, and =e / is the poros-ity. We dene L e as the characteristic length of travel forow along the pore path and L as the physical length of theporous pumping structure. e and are the void and totalvolumes of the porous medium, respectively. The term onthe right contains the integral

(3)f =a

01

2ra 2

d r.


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S. Yao, J.G. Santiago / Journal of Colloid and Interface Science 268 (2003) 133142 135

The solution to this integralcan be determinedby solving thePoissonBoltzmann (PB) equation for the potential, where

(4) 2=

1r

d dr

rddr =

1

i

n,i zi e exp zi ekT

.

The correct formulation for the model depends on the typeof buffer used and the magnitude of the zeta potential of theEDL [1]. The zeta potential of the glass pumping structuresof interest here are typically of order 100 mV which is sig-nicantly higher than the term ze/kT (equal to 25.7 mVfor z =1 at T =20 C) in the exponential functions above.As suggested by Verwey and Overbeek [22] and Hunter [1],electrokinetic systems with such potential distributions canbe treated as having symmetric electrolytes with the prop-erties of the counterion (in our case, the sodium ions of our sodium tetraborate buffer). The physical interpretationof this approach is that, at zeta potential values well above25.7 mV, the potential distribution in the EDL is determinedmostly by the attraction of counterions to the charged wall,while the repulsion of co-ions is less important [1]. For highzeta potential, the counterion density near the wall is signi-cantly higher than that of the co-iondensity. In this paper, wewill treat our buffers as symmetric, monovalent electrolyteswith the properties of the sodium ion in determining De-bye lengths and potential distributions. In determining totalpump current, we will use the charge distributions obtainedwith this model for the advective current component, andmeasured values of bulk conductivity to estimate electro-migration outside of the EDL. This treatment of the bulk conductivity leverages a single measurement to account for

all of the ions (and their respective ionic mobilities) presentin the buffer.For a symmetric electrolyte with z+=z, the PB equa-tion can be expressed as

(5)1r

d dr

rddr =

2nze

sinhzekT

.

The solution to this nonlinear equation was obtained numer-ically using a substitution of the form =4 arctanh () , assuggested by Bowen and Jenny [23]. The resulting equa-tion is also nonlinear but less stiff. We employed a fourth-order RungeKuttaalgorithm [24]with a shooting procedureto solve this boundary value problem. The numerical solu-tion for is plotted in Fig. 2. As a comparison, we alsopresent in Fig. 2 a solution to a linearized form of Eq. (5),where the term sinh (ze/kT ) is approximated as ze/kT (=) . This linearization is the so-called DebyeHckel ap-proximation valid for small zeta potentials, which results ina second-order-accurate formulation of Eq. (5) for a sym-metric electrolyte. Rice and Whitehead [21] rst used thisapproximation to solve for potential distributions in cylindri-cal tubes, which yields simply = I 0(r)/I 0(a) , whereI 0 is the zero-order modied Bessel function of the rstkind. The nondimensionalization in this solution is as fol-lows: r=r/ and a=a/ . is the Debye length which,for a symmetric electrolyte is (kT / 2e

2z

2n)

0.5. As shown

Fig. 2. Numerical solution to the PoissonBoltzmann equation (Eq. (5)) fora symmetric electrolyte, shown together with the analytical solution to thelinearized DebyeHckel approximation. The DebyeHckel approxima-tion is accurate only for ratios of a/ of order 100 or larger, and for valuesof a/ of order 0.1 and lower.

Fig. 3. Nondimensional model parameters f and g as a function of nondi-mensional pore size. The plots are generated using sample parameter val-ues of = 3.9 and =6.8 assuming a typical zeta potential value of 100 mV.

in Fig. 2, discrepancy between the numerical and linearizedsolutions is particularly apparent for values of a of orderunity.

Given the numerical solution for the potential, the integralin the ow rate expression, f , can also be calculated numer-ically (using a simple Simpsons rule integration scheme),and the result is plotted in Fig. 3. Shown together with this

numerical solution is the DebyeHckel approximation re-sulting from an integration of Eq. (3), where

(6)f (a )=1 2I 1(a)

aI 0(a).

I 1 is a rst-order modied Bessel function of the rst kind.As shown in the plot, the results show the discrepancy be-tween the two solutions for a zeta potential value of 100 mV,which is typical of our electroosmotic pumps. As expected,both the numerical and linearized solutions of f tend tounity for large a systems which have uniform uid veloc-ity throughout most of the ow cross section. The behavior

at low a, however, highlights the important difference be-


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tween the two solutions and shows how the linearized solu-tion underpredicts pump ow rate performance.

The maximum ow rate, Q max , and maximum pressure,P max , of the pump are derived from Eq. (2) as

(7)Q max

=

AV eff

Lf,

(8)P max =8 V eff

a 2 f.

These equations show that, for a given value of f , highelectric elds and large cross-sectional areas lead to highow rate, while pressure performance is linked to small porediameters and high voltage. This formulation can be nondi-mensionalized as

(9)V eff = V eff

,

(10)Eeff

=

V eff

L

,

(11)Qmax =Q max

2A,

(12)P max =P max a 2

8 2 ,

so that the equations become

(13)Qmax =

f E eff ,

(14)P max =f V eff .Equations (7) and (8) give the ratio

(15) = Qmax

P max =

Aa 28L

.

The parameter can be useful in determining the tortuosityof thepump experimentally from pressureand ow rate mea-surements and is independent of pump zeta potential. Forexample, can be obtained using dry and wet weight mea-surements of the porous structure and a can be estimatedusing mean pore radius measurements performed using amercury intrusion porosimeter [25]. Given these parametersand a measurement of temperature during experiments of Q max , and P max (to calculate the value of the bulk vis-cosity ), Eq. (15) can be used to estimate .

2.2. Electrical current in electroosmotic ow

In this paper, we present the derivation of an expressionthat can be used to predict total current in an EO pump us-ing a symmetric electrolyte model. As discussed by Riceand Whitehead [21], the total current in an electroosmoticpumping channel is the sum of advective and electromigra-tion currents I adv and I em , respectively, for the typical caseof negligible ion diffusion. In the electromigration compo-nent, Rice and Whitehead made the very rough approxima-tion that ion conductivity throughout the cross-section of the

pore is uniform throughout the uid up to the wall of the

capillary and made no allowance for the modied ionic con-centration in the electric double layer (EDL) near the wall.Such an assumption is justied when zeta potential is low(e.g., || < 1) and the molar conductivities of cations andations are approximately equal (e.g., as in the case of sim-ple KCl electrolytes). However, as suggested by the work of Morrison and Osterle [26], we can derive the advective andelectromigration components of current from the charge dis-tribution, velocity eld, and conductivity distribution withinthe pores as

(16)I adv =2a

0

u(r)(r)r dr,

(17)I em =2a

0

(r)E x r dr,

where (r) is the charge density at a point distance r fromthe axis and (r) is the conductivity at this point. For nitevalues of and symmetric electrolytes in which the mo-lar conductivities of cations and ations can be individuallyspecied, the charge density and conductivity terms can bewritten as

(18)(r) =r

d dr

rddr

,

(19)(r) =c +exp zekT + exp

zekT

,

where + and are the cation and anion molar conduc-

tivities, respectively. Note that these expressions for and can be approximated using Taylor series expansions of thesinh term in Eq. (5) and the exponents in Eq. (19). To second-order accuracy, the advective and electromigration currentdensities, i , in the channel are then

iadv = P x a 2

4 21

r 2

a 2I 0(r)I 0(a)

(20)+ E x I 0(r)I 0(a)

1 I 0(r)I 0(a)

,

(21)iem = E x 1 ( + )

I 0(r)I 0(a) +

2

2I 21 (r)

I 20 (a),

where

=ze kT

, = 2 2

2 .

In Eq. (21), = ++ . The parameter can be inter-preted as the ratio of EOF advective current to electromi-gration current, and describes the maximum advective cur-rent density associated with EOF. In applying these simple,symmetric electrolyte equations for current density to ourbuffered electrolyte case, we use the valence of the sodiumion (+1) and the molar conductivity of the sodium ion: +=5 103 S m2/ mol [27]. Conductivity measurementsshow that our bulk solution, sodium tetraborate buffer, has


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(a)

(b)

Fig. 4. Variation of current density across a single cylindrical pore withP x = 0 for nominal values of = 3.9 and = 6.8. (a) Numericalsolution for total current density is compared with the second-order ap-proximations given by Eqs. (20) and (21). (b) Numerical solution of the

contributions of advective current, and electromigration current, along withtotal current distribution, for a/ =1 and 50.

a nominal conductivity of =7.5 103 S m2/ mol [28],so that the effective molar conductivity of buffer ions otherthan sodium (e.g., B(OH) 4 and B3O3(OH)4 ) was estimatedas =2.5 103 S m2/ mol.Figure 4a shows the variation of the total current densityacross a single capillary from the PB solution in Fig. 2 forthe case of zero pressure gradients ( P x =0). In this calcula-tion, we assume a typical zeta-potential value of 100 mV andtypical parameter values of

= 3.9 and

=6.8. Shown

together with the numerical solution are the approximateexpressions Eqs. (20) and (21). The second-order-accurateexpressions for current density clearly underpredict the con-ductivity in the EDL for nite values of . Figure 4b showsthe individual contributions of advective and electromigra-tion current as derived from the numerical solution of thenonlinear PB equation. The advective current componentdecreases to zero at the wall because of the imposed no-slip condition on liquid motion. The electromigration cur-rent component is proportional to local ion density and soincreases as r approaches unity. The EDL can have a sig-nicant effect on current densities and area-average current

density can be a strong function of a.

The total current in a single pore is obtained by inte-gration of Eqs. (16) and (17) (which are respectively areaintegrals of Eqs. (20) and (21)) to yield

I p =a 2 P x

a

01

2ra 2

d r

+a 22E x

a

0

ddr

2 2ra 2

d r

+a 2cE xa

0

+exp zekT

(22)+ expzekT

2ra 2

d r.

The rst term in Eq. (22) is the advective current which is

proportional to the pressure gradient, P x , and the secondterm is the advective current due to EOF, and the third isthe electromigration current. The total current through all of the pores is obtained by integrating Eq. (22) over the owcross-section of the porous structure to obtain

(23)I =A

a 2 I p .We can now derive another relation which is useful in ex-perimental characterizations of pump parameters and is ex-pressed in terms of pump ow rate and current. Because of the effects of advective current, the total current of an EOpump is expected to be a linear function of backpressure.The maximum current I max is therefore obtained at the pointwhere P =0 (assuming P 0). Combining Eqs. (7)and (23),

(24)Q maxI max =

g,

where

g =f 2

a

0

ddr

2 2ra2

d r

(25)+

a

0

+ exp() +

exp()2ra2 d r .

The expression of Eq. (25) was evaluated using the nu-merical scheme described earlier with nominal parametervalues of = 3.9, =6.8. The results are plotted inFig. 3. The dimensionless ow rate per current ratio, g , re-duces to zero for small nondimensional pore diameters aselectromigration ion uxes per unit area are very high inthat regime, while ow rate per unit area is greatly reducedfrom the case of innitesimal EDLs. g also displays antici-pated trends to unity for large a. Shown together with thenumerical solution for g is the small potential, second or-

der accurate approximation of the PB equation, in which


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Eq. (25) can be derived analytically as follows:

g(a , , ) =f (a ) I 21 (a) I 0(a)I 2(a)

I 20 (a)

+1 ( +

)

I 1(a)

aI 0(a)

(26)+ 2

21

I 21 (a)

I 20 (a).

Note the discrepancy between the numerical and the second-order approximated solutions for g and the second-order ap-proximation can overpredict the parameter g by 30%. Thesetrends have important consequences in the determination of ow rate and thermodynamic efciency of the pump. In par-ticular, Eq. (22) allows for the estimation for of the pumpow rate and current measurements at the zero pressure gra-dient condition (and estimates of the uid properties , ,and ). This method is used to characterize the perfor-mance of our EO pumps in Yao et al. [28].

The maximum total current can be nondimensionalizedas

(27)I max =I max A

,

which also yields a relationship for the nondimensionalelec-trical eld,

(28)I max =

f g

Eeff .

Last, we can derive for the nonlinear problem a relationfor the ratio of the maximum values of the advected currentto the total current in a pump:

I max ,advI max =

2

a

0

ddr

2 2ra2

d r

2

a

0

ddr

2 2ra2

d r

(29)

+a

0 +

exp() +

exp() 2ra2

d r .

These current maxima occur in the case of zero pressureload. This current ratio is strongly dependent on a andweakly dependent on . As shown in Fig. 5, the numericalsolution of this ratio as a function of a is similar in shape tothe thermodynamic efciency (discussed in detail in the nextsection) but with a peak value of 0.29 at a=4 and whichasymptotes to zero for large values of a. The expected the-oretical, area-averaged advective current can be as much as

29% of the total current.

Fig. 5. Advectivetotal maximum current ratio, I max ,adv /I max , and ther-modynamic efciency, opt , as a function of nondimensional pore size for = 3.9 and =6.8.

2.3. Thermodynamic efciency of porous EO pumps

An analysis of the thermodynamic efciency of a planarEO pump has been presented recently by Chen and San-tiago [18]. Their model applies a DebyeHckel approxi-mation and uses a measured value of zeta potential on thesilica/symmetric electrolyte interface as an input variable.Here we present three extensions to their formulation includ-ing a numerical solution for the PB potential formulation (toaccount for nite values of ), the effects of advective cur-rent (which are signicant in our current pump structures),and an evaluation of the operational point of maximum ef-ciency. Thermodynamic efciency is dened as the usefulpressure work delivered by the pump over the total powerconsumption,

(30) =W P

V app I =P Q

V app I ,

where V app and I are the voltage and the current in the maincircuit. The total power consumption is expressed as the sumof pressure work, viscous dissipation and electrical dissipa-tion. As described by Chen and Santiago, the power con-sumption associated with dissociation of water molecules(electrolysis) at the electrodes is typically negligible com-pared to the overall power dissipated by the pump and is notincluded in the formulation.

Combining Eqs. (2), (23), and (30) above, the thermody-namic efciency can be derived as

(31) =(1 P ) P (/ 4) P

,

where

(32) =V appV eff

a2

2fg.

Here pressure is normalized as P = P / P max . The parameter takes into account the loss in efciency due tothe electrode-to-frit voltage drop (quantied by V app /V eff )and the effects of the EDL on ow rate, pressure, and cur-

rent generation. Although Q is a linear function of P ,


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the maximum value of is not simply the point whereP = P max / 2 since the total current, I , in the denomi-nator of Eq. (30) is also a function of pressure load. The

optimal point of operation can be determined by formulat-ing the partial derivative of Eq. (31) with respect to P while holding constant to obtain

(33)P = 4 216 4 .

Expanding the right hand side of Eq. (33) in a Taylor series,we have

(34)P = 4

1 1 4 = 4 2 + 2 2 +O 1 3 .Taking a second-order-accurate approximation of this rela-tion, the optimum pressure condition is

(35)P opt=12 +

12

and the optimal efciency is then, to second order,

(36)opt= 2 1

3 2 2 2 .

For thin EDL systems with much greater than unity, a rst-order-accurate expression of P =0.5 can be used (so thatthe optimum pressure is P max / 2). For this simple case, optis simply

(37)opt=1 =

V eff V app

2fga2

.

We therefore see that the parameter can be interpreted asan idealized inverse thermodynamic efciency for the casewhere the advective current is small compared to the elec-tromigration current.

Note that opt is highly dependenton the size of the poresand this dependence provides insight to the design of high-efciency pumps. Figure 5 shows a plot of opt for boththe exact formulation and the small-potential, second-order-accurate limit with nominal parameter values of = 3.9and =6.8, and an ideal condition of V eff =V app . For thepurpose of this plot, we therefore assume a relative constantzeta potential of

=100mV. Although the overall trend of

both solutions is similar, a discrepancy is apparent at smallvalues of nondimensional pore sizes a. The peak opt of the numerical solution occurs at a = 2.5 as compared to avalue of a=4.2 for the approximated solution. Further, thesecond-order approximation underpredicts the increase inthe conductivity of EDLs associated with nite values of ,and therefore overpredicts the maximum efciency by a fac-tor of 1.2. For our electrolyte chemistries and the wall prop-erties predicted by the zeta potential model, the a =2.5point of the numerical solution yields opt =6.5%. For in-stance, the numerical model suggests that 48-nm-diameterpores are optimal for 1 mM ion concentrations. The ef-

ciency curves shown here are qualitatively similar to the

predictions of Chen and Santiago [18] for planar electroki-netic pumps.

The peak in the efciency curve is a result of the com-peting effects of Joule heating (which dominates at large a)and reduced hydraulic power and higher current density atlow a. We rst consider the large a limit. To demonstratethe effect of viscosity and temperature, we dene ion mobil-ity as =u d /F [29] which is a measure of the drift velocityu d acquired by an ionic species subjected to a force F . Next,if we approximate ions as hard spheres with a characteris-tic Stokes diameter, d , dened as the diameter of a spherein continuum uid ow with a drag equal to that of theion [29]. Under this approximation, the ion mobility is =1/ 3d and the molar conductivity is =z2e2N A / 3d .For a 1, f 1 and g 1, and from Eq. (37), efciencybecomes

(38)opt =6d(T) 2( pH, n , T, . . . )

2

a 2z2e2n

(a 1),

where the parameters in parenthesis have been written to re-mind the reader that permittivity and zeta potential dependon at least several independent parameters such as temper-ature and the chemistry of the working electrolyte and thesolid surface. This relation points out interesting functional-ities. First, in this regime, thermodynamic efciency is nota function of solvent viscosity and temperature dependentonly through permittivity and zeta potential. Unless temper-ature drastically affects ion density (as it might in the caseof chemical reactions in a weak electrolyte), has only aweak dependenceon temperature. Second,we see that largerions (with associated lower molar conductivities and mo-

bilities) are thermodynamically favorable as they impart thesame force density into the ow with less Joule dissipation.For the same reason, ions with valances higher than unityare unfavorable. In the regime of thin EDLs, smaller poresare therefore favored and there is a decrease of opt as aincreases. Higher conductivity yields higher current densi-ties and lowers zeta potential (see Eq. (38) and Fig. 6) andis therefore also detrimental to thermodynamic efciency.However, as discussed in Yao et al. [28], conductivity of-ten has a lower limit dictated by practical considerationsassociated with pH stability. That is, for a given electrolytebuffer, lower ion concentration implies lower buffering ca-pacity. Low buffering capacity can adversely affect pH inhigh-ow-rate, high-current applications and thereby affectzeta potential and overall pump performance.

For the large a regime, we can also use an approx-imate expression for to express thermodynamic ef-ciency in terms of surface charge density. As discussed byHunter [1], this expression follows from the PB equation as = s I 0(a)/(I 1(a)) for small potentials. Substitutionof this expression into Eq. (38) yields

(39)opt=3d 2s kT a 2z4e4n 2

(a 1, small ).

This expression shows that the dependence of opt on is

perhaps less important than suggested by Eq. (38) and em-


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Fig. 6. Nonlinear tting curve for the GCSG site-binding model for thezeta potential of KCl/silica interfaces. The curve t shown is of the form = (0.058log 10 (pH) + 0.026)(log10 (c))

1.02 which gives valueswithin 10% of the GCSG site-binding model prediction within the follow-ing range: 2 .8 pH 10, 104 c 10

2 M.

phasizes the importance of using monovalent ions to achievehigh-efciency pumps. Also, this expression shows the ex-plicit dependence of opt on temperature and surface chargedensity, which is a strong function of pH and concentra-tion [30].

At values of a of order unity and lower, the effects of -nite EDLs dominate. In the regime of a 1, using second-order-accurate series expansions for f and g , Eq. (37) yields

(40)opt=3d 2a 2z2e2n

16k2T 2 (a 1, small ).

If we assume a constant zeta potential, this equation shows

opt decreases as a decreases in the small a regime (seeFig. 5). This can be explained in terms of pressure work. Be-cause of the no-slip condition at the wall, nite EDLs implya signicant decit in mass ux as compared to the thin EDLplug ow case. This contributes to lower ow rate per unitarea, lowering hydraulic power produced. Since pressure isa linear function of ow rate, pressure forces are accordinglydecreased in this fully viscous liquid ow.

In addition to lower hydraulic power, channels with smalla also typically have higher average current density since,at high zeta potentials, the conductivity of EDLs can bemuch higher than the bulk conductivity. For example, localion conductivity near the wall for the prediction given byEq. (19) is 34 times higher than the conductivity of the bulk electrolyte, as shown in Fig. 4b. At low a, both this andthe hydraulic power reduction discussed above contribute tolower thermodynamicefciency for decreasing a. Note thatthe model described here neglects ionic current in the stag-nant layer of counterions immediately adjacent to the wallof the microchannel, as discussed by Lyklema et al. [31].However, the effects of such ion conduction would furtherincrease average current density in electrokinetic channelsand therefore should not change the trends predicted by ourmodel in the low a regime.

Lastly, we should note that the theoretical development

given in this paper for ow rate, pressure, and efciency of

electroosmotic pumping should be interpreted only as an ap-proximation for values of a of less than about unity. Themodel uses a simple form of the Boltzmann equation whichmay result in inaccurate descriptions of the potential distrib-ution (and therefore ion density distributions) for the case of overlappingchargedouble layers. Although such layers haverecently been treated theoretically by Qu and Li [32] andConlisk et al. [33], we believe that the theory for overlap-ping EDLs andassociated models for the dependence of ona has not yet been adequately developed. We are currentlyworking on a model for electrokinetic ows with overlap-ping EDLs and will discuss this topic in a future paper.

2.4. Model for pH and ion density dependence of zeta potential

Zeta potential is a key parameter in the characterizationof any EO pump system. Classical theory describes the EDLas divided into the Stern and GouyChapman diffuse lay-ers [2]. The Stern layer counterions are absorbed onto thewall, while the ions of the GouyChapman layer are diffuseand therefore available to impart work on the uid. The planeseparating these two layers is called the shear plane, and thepotential at this plane is the zeta potential, . Combiningthe Boltzmann distribution of the EDL ions with the Pois-son equation, GouyChapman (GC) theory relates the zetapotential to the effective surface charge of the shear plane,which is a strong function of pH and a relatively weak func-tion of ionic concentration.

Because of the immense difculties in predicting thevalue of from rst principles, this parameter is typicallyan empirically determined value obtained using electroos-motic ow or streaming sodium tetraborate buffer general,difcult to determine, even for a xed working uid and sur-face material. However, as we discuss in Results, the valueof varies signicantly with changes in the ionic concen-tration, c, of the electrolyte. As pointed out by Yates etal. [30], applying a simple GouyChapman diffuse EDLmodel leads to the erroneous conclusion that scales withconcentration as c0.5 . Yates et al. [30] have presented anEDL model, called the site-binding model, on the basis of GouyChapmanSternGrahame(GCSG) theory, which can

be used to predict across a signicant range of pH andionic concentration values for simple buffers on oxide sur-faces. Their model separates the surface and buffer associa-tion/dissociation reactions of the EDL into those occurringon a layer immediately adjacent to the surface (below the in-ner Helmholtz plane, IHP) and a layer corresponding to themolecules further from the wall but unable to diffuse (abovethe outer Helmholtz plane, OHP). GCSG theory takes intoaccount interfacial chemical reactions at the IHP (involvingsilanol groups at the surface and water ions) and the reac-tions which occur at the OHP (which involve water ions andthe association/dissociationreactions of the electrolyte ions).

This theory predicts a less pronounced (and more accurate)


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dependence of zeta potential on bulk ionic concentrationthan the GC theory.

Scales et al. [34] have applied the model of Yates etal. [30] to the case of KCl electrolytes on silica surface.The site-binding model ts their experimental data very wellgiven modeling parameters suggested by Scales. However,because of the nature of the equilibrium reaction formula-tions, it is difcult to derive an explicit formula for basedon their model. For ourpump efforts, therefore,we have gen-erated the following explicit relation for as a function of pH and ion density by curve tting to the model data pre-sented by Scales:

(41) = 0.058log10(pH) +0.026 log10(c )1.02 .

This curve t predicts the value given by the side-bindingmodel to within 10% for ranges of ionic concentration of 104 to 102 M and pH values of 2.8 to 10. A compar-ison of this curve t and the site-binding model results is

shown in Fig. 6. In this paper, we apply the zeta potentialmodel for silica/KCl interfaces as an approximate predic-tion of trends in our borosilicate/borate buffer pumps. One justication for this is that, as discussed earlier, we expectthe buffers in our high zeta potential porous structures to actas symmetric electrolytes with the properties of the univa-lent sodium ion. A second justication of this is that theobserved trends of zeta potential as a function of pH andconcentration for silica and borosilicate are similar [35]. InResults, we quantitatively compare the trends we observedfrom our borosilicate-based EO pumps to the trends reportedby Scales et al. [34] for silica surfaces with KCl.

3. Conclusions

We have developed analytical expressions for the elec-troosmotic ow rate, current, and thermodynamic efciencyfor a porous structure EO pump operating under a pressureload. The model includes a numerical solution of the nonlin-ear PoissonBoltzmann equation for electric potential andthe coupling of this solution to the equations of uid motionand ionic current. The model is valid for large zeta poten-tial and accounts for the nonuniformity of ion conductivityacross the ow areas of a porous pump. We treat our buffersas symmetric, monovalent electrolytes and leverage a sin-gle measurement of bulk liquid conductivity to account forthe current contribution of all ions present in the buffer. Themodel uses determined pore sizes to predict absolute pres-sure performance. The model also leverages a curve t of the GCSG site-binding model for KCl/silica interface to es-timate the dependence of zeta potential on ionic density andpH in our borosilicate-based system. The model expressespressure, ow rate, current, and thermodynamic efciencyof EO pumps as a function of quantiable electrolyte prop-erties and the propertiesof the electrical double layer. For thenondimensional surface potential thermodynamic efciency

curve predicts a maximum is a result of the effects of Joule

heating, which dominates the trend in efciency at large a,and the effects of reduced hydraulic power and higher cur-rent density in the low a regime.

Lastly, we note that the Boltzmann equation is not di-rectly applicable for overlapping EDL elds where the cen-ter potential associated with the wall charges is nonzero, andthe ionic concentrations in the center of the pore are not nec-essarily equal to those of the original bulk concentration.Therefore, the solutions of the PB equation presented heremay result in inaccuracy in our model for the thick EDL case(e.g., a less than about unity). We are currently developingan analytical model to account for the effects of overlappingEDLs.

Acknowledgments

This work was supported by DARPA under Contract

F33615-99-C-1442 and by Intel Corporation.

Appendix A. Nomenclature

A Cross-sectional area (m 2)E x Electric eld (V m 1)F Force (N)I Current (A)L Length (m)L e Average length along the pore path (m)N A Avogadros number (mol 1)P

x Pressure gradient (Pa m

1)

P Pressure capacity (Pa)Q Flow rate (ml min 1)T Temperature (K)V Potential (V)W P Pressure work (W)

Total volume of the porous medium (m 3)

e Void volume of the porous medium (m 3)

a Pore radius (m)d Sphere diameter (m)c Electrolyte concentration (M)e Elementary charge (C)i Current density (A m 2)k Boltzmann constant (J K 1)n Electrolyte number concentration (m

3)r Radial coordinate (m)u Velocity (m s 1)z Charge number () Molar conductivity (S m 2 mol1) Nondimensional advective current parameter () Permittivity of liquid (C V 1 m1) Thermodynamic efciency () Nondimensional efciency parameter () Debye length (m) Viscosity (Pa s)

Mobility (m s 1

N1)


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(r) Charge density at a point distance r from the axis(C m3)

(r) Conductivity at a point distance r from the axis(S m1)

Electrolyte conductivity (S m 1)

s Surface charge density (C m 2) Electrical potential (V) Zeta potential (V) Tortuosity () Porosity ()

Subscripts

adv Advectionapp Applied valueeff Effective valueem Electromigrationmax Maximum value

opt Optimal valuep Single pore

+ Cation Anion

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yao, santiago 2003-porous glass electroosmotic pumps theory

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