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Modeling and optimization of colloidal micro-pumpsTo cite this article: D Liu et al 2004 J. Micromech. Microeng. 14 567

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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF MICROMECHANICS AND MICROENGINEERING

J. Micromech. Microeng. 14 (2004) 567–575 PII: S0960-1317(04)71367-4

Modeling and optimization of colloidalmicro-pumpsD Liu, M Maxey and G E Karniadakis

Division of Applied Mathematics, Brown University, Providence, RI 02912, USA

E-mail: gk@dam.brown.edu

Received 3 November 2003Published 19 January 2004Online at stacks.iop.org/JMM/14/567 (DOI: 10.1088/0960-1317/14/4/018)

AbstractManipulating micro-particles is very important in microfluidic applications,such as biomedical flows and self-assembled structures. Here, flowsgenerated by the forced motion of colloidal micro-particles in amicrochannel are investigated. The force coupling method combined withthe spectral/hp element method is used to numerically simulate thedynamics of the flow, while a penalty method is used to determine therequired forces on the particles. The pumping motion is investigated for twospecific systems: a peristaltic micro-pump and a gear micro-pump. Weverify the accuracy of the simulations and then for each system, weinvestigate the net flow rate as a function of pump frequency and channeldimension, and present optimization results. The results for the net flow rateare comparable to and within the range of the experimental data.

1. Introduction

This work was motivated by the experiments of Terray et al[1], who have designed and operated colloidal micro-devices(pumps and valves) for microfluidic control. Pump designsthat employ inertial, centrifugal action, e.g. impeller-basedpumps, are inappropriate for microfluidic applications due tothe very small Reynolds number. However, designs based onpositive-displacement methods can be quite effective. To thisend, Terray et al [1] have developed two positive-displacementmicro-pumps by using colloidal microspheres as the activeflow-control element. Small microspheres, at colloidal scales,can be manipulated remotely using electric or magnetic fieldsor optical trapping; the latter was employed by Terray et alin their experiments. Remote activation techniques areparticularly attractive in controlling micro-devices as there isno need to interface directly the microfluidic application withthe macroscopic environment—a major drawback in manyrecent designs of micro-devices.

In these devices, gravitational settling of the microspheresis negligible and the effect of Brownian motion is smallcompared to the imposed control forces. The number ofcolloidal microspheres used is also small. In the first micro-pump of Terray et al there are six particles forming a chainalong the channel. In the second micro-pump there are fourparticles, separated in two pairs that counter-rotate with respectto each other. The chain of six microspheres is activated by

optical trapping that induces a smooth traveling wave motion.In the second design the motion of the ‘meshing gears’ of themicro-pump is not as smooth. Both designs, however, wereshown to operate successfully and induce about the same netflow rate, of the order of 1 nl hr−1 for a rotation of a few Hz.

The objective of the current work is first to determinehow accurate the modeling of such devices is, and secondto use simulation to improve and possibly optimize theirperformance. Particulate microflows are difficult to modelaccurately [2] and modeling assumptions cannot be verifieddirectly. However, recent work based on atomistic simulationsin [3] has shown that the continuum hypothesis may be valideven for particles as small as 2 nm. To this end, we usecontinuum-based methods to model the two aforementionedmicro-pumps at various conditions. In particular, we employthe force coupling method that has been validated extensivelyfor microflows in [4, 5].

In the following, we first present some details of themathematical modeling and subsequently study the peristalticmicro-pump and the two-lobe gear micro-pump. In particular,we investigate their performance for a wide range of rotationalspeeds and also with respect to the width of the channel.We then conclude with a summary and a discussion on thelimitations of these micro-pump designs as well as of themodeling approach. Finally we have some direct numericalsimulation (DNS) validations in an appendix.

0960-1317/04/040567+09$30.00 © 2004 IOP Publishing Ltd Printed in the UK 567

D Liu et al

2. Mathematical modeling

The particles that we consider in this work are smaller than3 µm but larger than about 100 nm. In this size rangewe can simulate particulate microflows as a continuum withpossible corrections due to partial slip at the wall or due torandom effects. The robustness of continuum calculationsin this context has been demonstrated previously in [6, 7]for spheres approaching a plane wall. More recently, asystematic molecular dynamics (MD) study was undertakenby Drazer et al [3] for a colloidal spherical particle passingthrough a nanotube containing a partially wetting fluid. Fora generalized Lennard–Jones liquid, it was demonstrated thatthe MD simulations are in good agreement with the continuumsimulations of [8] despite the large thermal fluctuations presentin the system. This is true even for very small size particles ofthe order of 2 nm, i.e. much smaller than the ones we considerhere.

2.1. Force coupling method

We therefore adopt a continuum description of the flow anduse the force coupling method (FCM) [9, 10] to represent thedynamics of the particles in the flow. The governing equationsof fluid motion for the fluid velocity u(x, t) are given by

ρDuDt

= −∇p + µ∇2u + f(x, t) (1)

∇ · u = 0 (2)

where ρ, p and µ are the fluid density, pressure and viscosity,respectively. The source term f(x, t) represents the sumof two-way coupling forces from each spherical particle ncentered at Y(n)(t). For a single particle we have

fi(x, t) = Fi(σ1, x − Y(t)) + Gij

∂(σ2, x − Y(t))

∂xj

(3)

where both (σ1, x) and (σ2, x) are Gaussian distributionfunctions of the form

(σ1, x) = (2πσ 2

1

)−3/2exp

(−x2/(

2σ 21

)). (4)

The first term in equation (3) is a finite force monopole ofstrength F while the second term is a force dipole of strengthGij . The monopole strength is set by the sum of the externalforces, due to optical trapping, magnetic field or gravity, actingon the particle as well as the inertia of the particle. If mP is themass of the particle and mF is the mass of the displaced fluidthen

F = Fext − (mP − mF)dVdt

. (5)

Additionally, contact forces between particles or a particle anda wall may be included in determining F.

The second term in (3) is the force dipole, which hassymmetric and antisymmetric components. The symmetric,stresslet component of the force dipole Gij is chosen so that∫

1

2

(∂ui

∂xj

+∂uj

∂xi

)(σ2, x − Y) d3x = 0. (6)

This is set through a dipole iteration scheme to minimize theintegral-averaged strain rate inside the volume occupied byparticles, down to 1% of the ratio of the particle velocity

to its radius, see [4]. The antisymmetric components ofthe force dipole correspond to external torques acting on theparticle. The values of the length scales σ1 and σ2 in (3)are directly related to the particle radius, with a/σ1 = √

π ,and a/σ2 = (6

√π)1/3. These values ensure that in viscous-

dominated flows the correct Stokes drag and torque on a singleisolated particle are achieved (details are given in [9, 10]).

The velocity of each particle V(t) is evaluated from avolume integral of the local fluid velocity with the monopoleGaussian distribution function as

V(t) =∫

D

u(x, t)(σ1, x − Y(t)) d3x. (7)

The angular velocity Ω of the particle is computed by alocal volume average of the vorticity W (x, t) with the dipoleGaussian distribution function as

Ω = 1

2

∫D

W(x, t)(σ2, x − Y(t)) d3x. (8)

The new position of each particle is calculated from

V(t) = dY(t)

dt. (9)

A spectral/hp element method has been used to solve forthe primitive variables u, p in the Navier–Stokes equations andfurther details can be found in [11]. In these micro-devices theflows are dominated by viscous forces and a Reynolds numberbased on the diameter of a particle and the particle velocity isof the order of O(10−4). It is appropriate then to neglect thefluid inertia in (1) and to solve the corresponding equation forStokes flow

0 = −∇p + µ∇2u + f(x, t). (10)

Similarly, we can neglect the influence of particle inertia in (5)and the external force required for each particle is equal to themonopole force F.

2.2. Penalty method

The standard FCM scheme is based on a mobility formulation,where the force (5) acting on the particle is prescribed andthe velocity is determined from the computed flow (7). Inthe pump micro-devices considered here the motion of theparticles and their velocities are prescribed, and finding theforces and matching the motion are our goals. But the actualforces from the optical trapping acting on the particles areunknown. In order to solve this resistance problem within theFCM scheme, we incorporate a penalty method.

For example, in order to find the force needed for theparticle numbered by n to generate the desired motion, weformulate a time-dependent model for the force F(n)(t) andthe particle velocity difference between the target value V(n)

T (t)

and the computed value V(n)(t) as

dF(n)(t)

dt= λ

[V(n)

T (t) − V(n)(t)]

(11)

where λ is the penalty parameter. The magnitude of λ controlsthe rate of convergence; the larger its value the tighter thecontrol of the particle motion and the faster the convergence.However, for stability reasons, λ cannot exceed a certainlevel. Upon convergence, the difference

[V(n)

T (t) − V(n)(t)]

isbasically zero and the force F(n)(t) becomes independent oftime t, and the initially unknown value of the force is obtained.This procedure may be applied to both steady and unsteadyflows.

568

Modeling and optimization of colloidal micro-pumps

Direction of net flow

x2

x1

x3

01

23

45

Figure 1. Peristaltic micro-pump: sketch of the setup with sixmicrospheres initially deployed in a traveling wave chain.

3. Peristaltic micro-pump

First we simulate a peristaltic micro-pump following thedesign of Terray et al [1] based on a chain of six colloidalmicrospheres in a channel. A sketch of the system is shown infigure 1. The length of the microchannel is 24 µm, in the x1

direction, the height is 6 µm in the x2 direction and the initialwidth is 4 µm in the x3 direction. In the simulations we willvary the width of the channel to investigate its effect on thenet flow rate. Six particles with the diameter 2a = 3 µm wereinitially assembled as a sine-wave chain along the centerline ofthe microchannel. Periodic boundary conditions for the flowwere applied at the channel ends at x1 = 0 and 24 µm whileno-slip boundary conditions were specified on the four rigidsidewalls of the channel in the x2 and x3 directions.

In order to produce a pumping effect, a transversetraveling wave motion of the following form is enforced forthe nth microsphere (n = 0, 1, . . . , 5):

Y(n)2 (t) = A2 sin

[ωt + (n − 1)

π

2

](12)

Y(n)1 (t) = Y

(n)1 (0) + (−1)nA1 sin[2ωt] (13)

where A2 = 1.5 µm is the amplitude of the traveling wavemotion in the transverse x2 direction, A1 = 0.2 µm is theamplitude of the oscillatory motion along the x1 direction andω is the frequency. The amplitude of the transverse oscillationsis equal to the particle radius a. The initial offsets Y

(n)1 (0) of the

particles are 7, 9.7, 12.4, 15.1, 17.8 and 20.5 µm respectively.The motion in the x1 direction ensures that the gap betweenparticles remains small but the particles do not overlap. Usingequations (12) and (13) we can obtain the prescribed values ofthe target velocity VT(t) for each particle.

To achieve this traveling wave motion, a set of forcesF(n) for each particle, acting in both the x1 and x2 directions,are computed via the aforementioned penalty method usingequation (11), where V(n)(t) is the velocity of the microspheren computed from equation (7). The resulting forces are then theforces that, for example, the optical trapping in the experimentsof Terray et al exerts on the microspheres. The penaltycoefficient λ, which controls the speed of convergence, is inthe range of 500–1000 [12] for oscillation frequencies in therange of 1 Hz. No external torques are applied to the particles,

and the antisymmetric components of the force dipole G(n)ij are

zero. The particles are free to rotate in response to the localflow (8).

In this implementation of the FCM scheme for Stokesflow (10), an iterative procedure is used simultaneously forthe computation of the flow and the forces (11). At each phaseof the oscillation cycle, the particle positions (12) and (13) andthe target velocities VT(t) are specified. The flow and forcesare initially set from the previous phase point of the oscillationcycle. The forces are then updated from (11) and the flowis recalculated from (10). We continue this procedure untilthe difference in the velocity of each microsphere betweensuccessive iterations is less than 1%. Convergence is typicallyachieved after five iterations, see [12]. Figure 2 shows acomparison between the target and the computed velocitycomponents of the first microsphere through several oscillationcycles.

The results for the particle velocities here are given in anon-dimensional form by using scales based on the particleradius a and the angular frequency of the oscillations ω. Thisis consistent with the prescribed motion of the particles. Thetime variation is scaled by 0.5T , where T is the period of thewave motion. The motion along the length of the channel hasthe period 0.5T , as specified by (13). Details of the method,resolution and accuracy verification are given in the appendix.

Figure 3 shows the time variation of the volume flow rate.After a very short initial transient, the flow rate reaches aperiodic state. The flow rate Q(t) fluctuates about an averagenegative value with a period T/2 equal to half the period T ofthe underlying peristaltic wave motion. The average value ofthe non-dimensional flow rate Q/a3ω is −1.426. Terray et al[1] observed in their experiments a linear relationship betweenthe flow rate and the frequency of the peristaltic wave, inagreement with the present results and scalings. We notethat there is also a quantitative agreement with the results ofTerray et al [1] in the range of 0.5 Hz to 3.5 Hz tested in theexperiments. This is illustrated in figure 4 where the averageflow rate is given in dimensional form against frequency. Wenote that in [1] the quoted maximum flow rate of 1 nl hr−1

is erroneous, given that the maximum velocity of the tracerparticle is about 3.5 µm s−1 and the cross-section of the channelis 6 µm × 3.5 µm. A revised maximum flow rate of about0.25 nl hr−1 is more appropriate.

Figures 5 and 6 present the computed forces in the x1

and x2 directions, normalized by 6πµωa2. These forces aretime-periodic and switch direction within each period. Thex2 force component is symmetric and much larger than the x1

component. The force on the leading microsphere is slightlylarger than the forces on the other five. The x1 componentsof forces are not symmetric; forces on the leading and trailingparticles are mostly positive (i.e., against the net flow direction)while forces on the intermediate particles are mostly negative,i.e. toward the direction of the traveling wave. This impliesthat the particles in the middle of the chain contribute mainlyto creating the net flow.

Next, we optimize the net flow rate with respect to thewidth of the micro-pump in the x3 direction. We vary the widthfrom 4 µm to 3.2, 10 and 20 µm. In figure 7 we plot thecorresponding normalized forces required along with the netflow rate, versus the channel width. As the width increases,

569

D Liu et al

2t/T

V1/(

a)

V 2/(a

)

0 0.5 1 1.5 2 2.5 3 3.5 4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

V1 DesiredV1 ComputedV2 DesiredV2 Computed

ωω

Figure 2. Comparison between the target and computed particle velocity components of the particle n = 0; velocities scaled by aω and timescaled by period T.

2t/T

Q/(

a3)

0 0.5 1 1.5 2 2.5 3 3.5 4-2

-1.8

-1.6

-1.4

-1.2

-1

ω

Figure 3. Time evolution of the volume flow rate Q(t) for theperistaltic micro-pump.

both force components decrease, however the net flow ratefirst increases from 3.2 to 4.0 µm and subsequently decreasesbeyond 4.0 µm. On the basis of these results, we can establishthe optimum operation point of the pump from the efficiencystandpoint. To this end, we compute the ratio of the net flowrate to the work required. The flow rate is normalized by a3ω.The work is computed from the integral average of the productof the absolute values of the required forces normalized by6πa2µω, and the particle velocity normalized by aω. Weplot the results in figure 8. We see that the best efficiency isobtained with a microchannel of width 10 µm.

Frequency (Hz)

Ave

.Flo

wra

te(

m3 /s

)

Ave

.Flo

wra

te(n

ano-

liter

/hou

r)

0 10 20 30 40 50-100

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

0

1

2

3

4

5

µ

Figure 4. Peristaltic micro-pump: linear dependence of the flowrate on frequency.

4. Two-lobe gear micro-pump

A two-lobe gear micro-pump designed by Terray et al[1] consists of two pairs of counter-rotating particles in amicrochannel with a cavity, as shown in the sketch of figure 9.The geometry of the microchannel is 19 µm in x1, 6 µm inx2 and 4 µm in x3. In the middle of the channel there is acylindrical cavity with radius 3.5 µm. Two pairs of particles(two lobes) with radius a = 1.5 µm counter-rotate with a 90

phase angle offset.As in the previous section, here too we use the inverse-

problem approach to simulate the micro-pump; that is, wespecify the motion and then compute and verify the forces to

570

Modeling and optimization of colloidal micro-pumps

2t/T

F 1/(6

a2)

0 0.5 1 1.5 2 2.5 3 3.5 4-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

Par. 0Par. 1Par. 2Par. 3Par. 4Par. 5

ωµ

π

Figure 5. Peristaltic micro-pump: normalized computed forces oneach particle in the x1 direction against 2t/T .

2t/T

F 2/(

6a2

)

0 0.5 1 1.5 2 2.5 3 3.5 4-5

-4

-3

-2

-1

0

1

2

3

4

5

Par. 0Par. 1Par. 2Par. 3Par. 4Par. 5

ωµ

π

Figure 6. Peristaltic micro-pump: normalized computed forces oneach particle in the x2 direction against 2t/T .

generate the motion. To reproduce the counter-rotating motionfor the particle pairs we studied the motion of the lobes in thevideos provided in [1]. In particular, we first specify the motionfor each pair of particles as a pure rotation around the centerof mass of this particle pair, with a 90 phase angle offset forthe two pairs of particles. The motion of the first particle ofthe top lobe is

Y 11 = Y 12

1 + a cos[π − 2πt/T ] (14)

Y 12 = Y 12

2 + a sin[π − 2πt/T ] (15)

while the motion of the second particle is

Y 21 = Y 12

1 + a cos[−2πt/T ] (16)

Y 22 = Y 12

2 + a sin[−2πt/T ] (17)

where Y 12 is the coordinate of the center of mass of this lobe,

Channel Width (10-6m)

F/(

6a2

)

Q/(

a3)

5 10 15 200.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

1.22

1.24

1.26

1.28

1.3

1.32

1.34

1.36

1.38

1.4

1.42

1.44

1.46

1.48

1.5

1.52

F2max

F1max

Q/(a3 )ω

πµ

ω

ω

Figure 7. Peristaltic micro-pump: optimization of the average flowrate with respect to the channel width in the x3 direction.

Channel Width ( m)

6Q

/|FV

|

0 5 10 15 200.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

µ

πµ

ω

Figure 8. Peristaltic micro-pump: overall performanceoptimization.

a is the particle radius and T is the period of the pump. Themotion of the particles in the lower lobe is specified similarly.From these analytic expressions we can compute the exactvalues of the target particle velocity VT(t). To achieve thiscounter-rotational motion, a set of periodic forces Fext, in boththe x1 and x2 directions for each particle, were computed via apenalty method from equation (11), as in the previous section,and included in the force monopole term of f(x, t) in equation(10). A similar iterative procedure as before was employed.

Figures 10 and 11 present the normalized computed forceson each particle in the x2 and x1 directions. Figure 12 showsthe time evolution of the net flow rate Q(t) in the x1 directionas a function of 2t/T . After a short initial transient, the flowrate is established quickly and follows a time-periodic pattern.The direction of the net flow matches the observations from theexperiments of Terray et al, and the non-dimensional averagedvalue Q/a3ω is 1.483 which is in quantitative agreement withthe results reported in [1].

571

D Liu et al

1

2

3

4

1

23

4

x1

x2

x3

Figure 9. Gear micro-pump: configuration sketch—two pairs ofcounter-rotating particles in a lobed microchannel.

2t/T

F2/(

6πa2 µω

)

0 0.5 1 1.5 2 2.5 3 3.5 4-6

-4

-2

0

2

4

6

Par. 1Par. 2Par. 3Par. 4

Figure 10. Gear micro-pump: normalized forces on each particle inthe x2 direction.

Similar to the peristaltic micro-pump we study the effectof the width of the channel on the net flow rate. In figure 13 weplot the normalized forces and the corresponding flowrate as afunction of the microchannel width. The results are similar tothose of the peristaltic pump; however, the optimum operatingpoint from the efficiency standpoint is at a width of 4 µm. Wealso pursued a similar optimization study of a two-lobe gearpump inside a straight channel, i.e. without the cylindrical

2t/T

F1/

(6πa

2 µω)

0 0.5 1 1.5 2 2.5 3 3.5 4-4

-2

0

2

4

6

8 Par. 1Par. 2Par. 3Par. 4

Figure 11. Gear micro-pump: normalized forces on each particle inthe x1 direction.

2t/T

Q1/

(a3 ω

)

0 0.5 1 1.5 2 2.5 3 3.5 4-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 12. Gear micro-pump: time evolution of the flow rate;T = 0.5 s.

cavity, see [12]. The results are similar to those of the two-lobe gear micro-pump with the same best operating point, i.e.at 4 µm.

5. Summary and discussion

We have presented here the first three-dimensional simulationsof the colloidal micro-pumps designed by Terray et al [1].Both designs simulated here induce approximately the samenet flow rate with the gear micro-pump giving slightly highervalues. However, its drawback is the aggressive motion of itsmeshing gears that may affect adversely certain applications,e.g. transport of cells [1]. We have demonstrated a goodagreement with the experimental results and have investigatedways of enhancing the performance of these micro-devices.We have also studied the stability of the flexible chain of

572

Modeling and optimization of colloidal micro-pumps

Pump Width (10-6m)

F/(

6πa2 µω

)

Q1/

(a3 ω

)

5 10 15 202.75

3

3.25

3.5

3.75

4

4.25

4.5

4.75

5

5.25

5.5

5.75

6

6.25

6.5

6.75

7

-0.25

0

0.25

0.5

0.75

1

1.25

1.5

1.75

F2max

F1max

Q1

Figure 13. Gear micro-pump: optimization of the average flow ratewith respect to the pump width in the x3 direction.

colloidal microspheres that provide the positive displacementthat leads to net flow rate.

We have also investigated the robustness of the peristalticpump in the presence of a parasitic force along the spanwisedirection induced by the actuation, i.e. the optical trappingsystem. To this end, we introduce another traveling wavemotion in x3 with the same frequency as in x2 but withan amplitude of 0.15 µm, that is 10% of the amplitude inx2. Figure 14 compares the flow rate between disturbedand undisturbed flows in the x1 and x3 directions. We seethat the 10% disturbance induces a slight deviation of about0.3% in magnitude in the net flow rate. It also induces asmall flow along the spanwise direction, which is non-zeroinstantaneously but its time-averaged value is zero.

From the simulation results, it is possible that a flowrate of 1 nl hr−1 can be achieved at an excitation frequency

2t/T

Q1(

t)/(

a3 ω)

Q3(

t)/(

a3 ω)

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2-1.6

-1.5

-1.4

-1.3

-1.2

-1.1

-1

-0.024

-0.02

-0.016

-0.012

-0.008

-0.004

0

0.004

0.008Q1(t)

Q3(t)

Figure 14. Peristaltic micro-pump: variation of flow rate due to a parasitic force along the x3 direction.

of about 10 Hz. While optical trapping was used as theflow-activation element in [1], the use of paramagnetic beadsexcited remotely by a time-dependent magnetic field is anotherpossible choice [13]. However, achieving the high flow rates inthe higher frequency range that the simulation predicts may bechallenging in the physical laboratory. Preliminary attemptsto increase the frequency beyond 4 Hz in the experiments ofTerray et al showed that the particles tended to fall out ofthe trap, thus disrupting the pumping motion [14]. Therefore,one may need to increase the laser power or use smaller sizeparticles.

The force coupling method we have used has beenvalidated systematically for prototype particulate microflowsin [4, 5, 12]. Even at a frequency of 100 Hz the assumptions oflinear Stokes flow are still applicable. The Reynolds numberis about 10−3 and the scale (2µ/ρω)0.5 for the Stokes layerassociated with the oscillatory motion is still much larger thanthe channel width. We have also investigated the qualityof simulation results based on the monopole term only, i.e.omitting intentionally the dipole term in equation (3). Thepredicted flow rate without the dipole is 7.5% less than thatwith the dipole. This suggests that the force dipole is importantin simulating particulate flows with FCM even if the flow is inthe Stokes regime. All the simulation results reported in thispaper employed both the force monopole and dipole terms.

Acknowledgments

We would like to thank professor David W M Marr of ColoradoSchool of Mines for helpful discussions. This work wassupported by NSF and DARPA.

Appendix

In order to illustrate the flow structure in the channeland to verify the accuracy of the flow simulations withthe FCM scheme, we compare the results with a full DNS

573

D Liu et al

1

23

4

5

0

Figure 15. Flow velocity vectors in the x1, x2 plane at t = 0 from FCM computation: vectors indicate particle velocities, angular velocitiesare respectively 3/ω = 0.223, −0.009, −0.265, −0.002, 0.271, −0.047.

Figure 16. Comparison of u1 contours in the x1, x2 plane at t = 0between DNS (top) and FCM (bottom) computations. Circles,radius a, show particle positions.

Figure 17. Comparison of u1 contours in the x1, x2 plane at t = 0between DNS (top) and FCM (bottom) computations. Circles,radius 1.2a, show particle positions.

from the spectral element NEKTAR code [11]. For thecomputations with the FCM scheme, only a single spectralelement decomposition of the flow domain is required. Thisconsists typically of 128 elements, each with a polynomialexpansion of between fourth and ninth order. The expansionorder may be varied as needed to ensure accuracy. With afull simulation, a new spectral element decomposition of theflow domain is required in each time step to account for theinstantaneous particle positions. The continuous generationof new meshes is very costly and not practical for routinesimulations. Here, we compare flow results at a single instant,t = 0. The NEKTAR simulations were tested with 4088, 6950and 10 162 elements with the expansion order from three tofive.

u1/(a )

x 2(

m)

-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0

-3

-2

-1

0

1

2

3

µ

ω

x 2(

m)

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75

-3

-2

-1

0

1

2

3

µ

u1/(a )ω

Figure 18. Comparison of u1 velocity profiles at t = 0 betweenDNS (squares) and FCM (solid line): at x1 = 2 (top) and at x1 = 7(bottom).

Figure 15 shows the flow velocity vectors at t = 0, ascomputed from the FCM scheme. The height of the channelhas been expanded to 6.4 µm to accommodate the mesh ofthe full DNS. The flow is shown for the x1, x2 plane passingthrough the centers of the particles. The velocity of eachparticle is indicated. The particles are also rotating and thecomponents of the angular velocity 3 are non-zero. The flowvectors show how the motion of the particles forces a resultantnet flow, from right to left, in the negative x1 direction.

The prescribed particle velocities, together with theangular velocities Ω(n) computed from (8), are used to set

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Modeling and optimization of colloidal micro-pumps

the boundary conditions for the full NEKTAR simulations.Figure 16 shows the contours of the velocity component u1

computed at t = 0 from both FCM and NEKTAR. Thereis generally a good agreement between the two, except forregions close to the particle surface. A further comparison isshown in figure 17, where the circles denoting the particleshave been expanded by 20% to eliminate the near-surfaceregion. A more detailed comparison is shown in figure 18 withthe velocity profiles at two different locations in the mid-planeof the channel, x1 = 2 that lies to the left of the particles,and x1 = 7 that coincides with the initial position of thefirst particle. Again there is similar agreement between thetwo simulations, with better agreement away from the particlesurface.

The results for the flow data and the forces, in general,may be specified in a non-dimensional form by using the scalesbased on the particle radius a and the angular frequency ofthe oscillations ω. As the underlying equations for Stokesflow are linear, the flow velocities scale with aω and the netvolume flow rate Q(t) through any section of the microchannelscales with a3ω. For the flows computed above, at t = 0,the values of Q/a3ω are between −1.57 and −1.63 for theDNS results, depending on the resolution and the number ofelements used. For the FCM computations the correspondingflow rate is −1.57.

References

[1] Terray A, Oakey J and Marr D W 2002 Microfluidic controlusing colloidal devices Science 296 1841–4

[2] Karniadakis G E and Beskok A 2001 Microflows,Fundamentals and Simulation (New York: Springer)

[3] Drazer G, Koplik J, Acrivos A and Khusid B 2002 Adsorptionphenomena in the transport of a colloidal particle through ananochannel containing a partially wetting fluid Phys. Rev.Lett. 89 244501

[4] Liu D, Maxey M R and Karniadakis G E 2002 A fast methodfor particulate microflows J. Microelectromech. Syst. 11691–702

[5] Lomholt S, Stenum B and Maxey M R 2002 Experimentalverification of the force coupling method for particulateflows Int. J. Multiph. Flow 28 225–46

[6] Israelachvili J 1991 Intermolecular and Surface Forces (NewYork: Academic)

[7] Vergeles M, Keblinski P, Koplik J and Banavar J R 1996Stokes drag and lubrication forces: a molecular dynamicsstudy Phys. Rev. E 53 4852

[8] Bungay P M and Brenner H 1973 The motion of aclosely-fitting sphere in a fluid-fitted tube Int. J. Multiph.Flow 1 25

[9] Maxey M R and Patel B K 2001 Localized forcerepresentations for particles sedimenting in Stokes flow Int.J. Multiph. Flow 9 1603–26

[10] Lomholt S and Maxey M R 2003 Force coupling method forparticulate two-phase flow: Stokes flow J. Comput. Phys.184 381–405

[11] Karniadakis G E and Sherwin S J 1999 Spectral/hp ElementMethods for CFD (Oxford: Oxford University Press)

[12] Liu D 2004 Spectral element/force coupling method:application to colloidal micro-devices and self-assemblystructures in 3D complex-geometry domainsPhD Dissertation Brown University

[13] Hayes M A, Polson N A and Garcia A A 2001 Active controlof dynamic supraparticle structures in microchannelsLangmuir 17 2866–71

[14] Marr D 2003 Personal communication

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