Chaotic micromixer utilizing electro-osmosis and induced charge electro-osmosis in eccentric annulusHuicheng Feng, Teck Neng Wong, Zhizhao Che, and Marcos Citation: Physics of Fluids 28, 062003 (2016); doi: 10.1063/1.4952971 View online: http://dx.doi.org/10.1063/1.4952971 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/28/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Suppression of a Brownian noise in a hole-type sensor due to induced-charge electro-osmosis Phys. Fluids 28, 032003 (2016); 10.1063/1.4943495 Complete chaotic mixing in an electro-osmotic flow by destabilization of key periodic pathlines Phys. Fluids 23, 072002 (2011); 10.1063/1.3596127 Chaotic mixing in electro-osmotic flows driven by spatiotemporal surface charge modulation Phys. Fluids 21, 052004 (2009); 10.1063/1.3139162 Symmetry breaking in induced-charge electro-osmosis over polarizable spheroids Phys. Fluids 19, 068105 (2007); 10.1063/1.2746847 Electro-osmosis at inhomogeneous charged surfaces: Hydrodynamic versus electric friction J. Chem. Phys. 124, 114709 (2006); 10.1063/1.2177659

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PHYSICS OF FLUIDS 28, 062003 (2016)

Chaotic micromixer utilizing electro-osmosis and inducedcharge electro-osmosis in eccentric annulus

Huicheng Feng,1 Teck Neng Wong,1,a) Zhizhao Che,2 and Marcos11School of Mechanical and Aerospace Engineering, Nanyang Technological University,50 Nanyang Avenue, Singapore 639798, Singapore2State Key Laboratory of Engines, Tianjin University, Tianjin 300072, China

(Received 1 September 2015; accepted 14 May 2016; published online 2 June 2016)

Efficient mixing is of significant importance in numerous chemical and biomedicalapplications but difficult to realize rapidly in microgeometries due to the lack ofturbulence. We propose to enhance mixing by introducing Lagrangian chaos throughelectro-osmosis (EO) or induced charge electro-osmosis (ICEO) in an eccentricannulus. The analysis reveals that the created Lagrangian chaos can achieve a homo-geneous mixing much more rapidly than either the pure EO or the pure ICEO. Oursystematic investigations on the key parameters, ranging from the eccentricity, thealternating time period, the number of flow patterns in one time period, to the specificflow patterns utilized for the Lagrangian chaos creation, present that the Lagrangianchaos is considerably robust. The system can obtain a good mixing effect with wideranges of eccentricity, alternating time period, and specific flow patterns utilized forthe Lagrangian chaos creation as long as the number of flow patterns in one time periodis two. As the electric field increases, the time consumption for homogenous mixingis reduced more remarkably for the Lagrangian chaos of the ICEO than that of theEO. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4952971]

I. INTRODUCTION

Efficient mixing is of critical importance for many lab-on-a-chip systems for chemical reac-tions,1,2 biological analysis,3,4 particle synthesis,5,6 medical analysis,7 colloid science,8 etc. Therefore,developing effective techniques for a rapid and homogenous mixing remains a topic of extensivescientific and technological interest, which, however, has long been a challenge in micro/nanofluidicsdue to the commonly small Reynolds number and laminar flow. So far various mixing methods, eitherpassive or active, have been proposed and studied in the effort to enhance mixing.9,10 Passive mi-cromixers have advantages as free of moving part and no additional energy consumptions, while theircommonly employed geometries are complicated and pose fabrication challenges.11,12 Compared tothe passive mixing method, active micromixers utilize external fields, such as pressure,13,14 elec-tric,15,16 magnetic,17 acoustic,18 and thermal fields,19 to create secondary flows for mixing enhance-ment. The flexibility in tunable mixing control and the resulting better mixing effect often enable theactive mixing to be preferable.20

Electrically induced mixing, as a type of active mixing method, has been confirmed to be simpleand effective in prompt mixing,21,22 among which induced charge electro-osmosis (ICEO) presentsa great potential in mixing enhancement thanks to its inborn nature of vortex generation. Generally,when a charged non-conducting (non-polarizable) surface is in contact with electrolyte solution, thecounterions in the electrolyte solution are attracted to the surface, forming an electric double layer(EDL). With an external electric field applied, the ions within the EDL are driven into motion. Cor-respondingly, a fluid flow is aroused, referred to as electro-osmosis (EO). When the surface is con-ducting (ideally polarizable), it polarizes immediately after the external electric field applied. The

a)Electronic mail: mtnwong@ntu.edu.sg

1070-6631/2016/28(6)/062003/19/$30.00 28, 062003-1 Published by AIP Publishing.

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counterions are attracted to the surface, establishing an induced EDL with a nonlinear zeta potential.The interactions between the induced EDL and the applied electric field lead to a flow in the bulk fluid,known as induced charge electro-osmosis (ICEO).23 Different from electro-osmosis, ICEO producesmicrovortices due to the nonlinear induced zeta potential. Such microvortices in ICEO around polar-izable circular cylinders have been both theoretically predicted23–25 and experimentally observed.26,27

Some studies have been carried out on the mixing behavior of ICEO.28–31 Harnett et al.,29 and Wuand Li28 developed ICEO micromixers by positioning many triangular posts in microchannels. How-ever, ICEO microvortices are intrinsically laminar, therefore, molecular diffusion is required for thetransportation across streamlines. Hence, the mixing enhancement is still limited by the moleculardiffusion. In order to conquer such limitations, we hereby propose to realize an efficient mixing byintroducing Lagrangian chaos into the EO and the ICEO micromixers.

Since Aref32 reported that Lagrangian chaos could be created by periodically alternating two ormore closed-orbit flows, plenty of studies have been carried out on Lagrangian chaos under variousflow conditions.33–37 It has been demonstrated that Lagrangian chaos can significantly enhance mixingin various flow conditions, such as secondary flow induced by herringbone structure on the bottomof channel wall,38 pulsed source-sink systems,39 and droplet flow in meandering microchannels.12,40

Techniques to quantify the mixing effect have been developed.41 Lagrangian chaos created by peri-odically alternating two ICEO flows in concentric annulus has been reported.42 Such ICEO flowsare produced by changing the applied electric field directions. Compared to the concentric annulus,the eccentric annulus, however, is a more general case (with a concentric annulus as a special casewhen the eccentricity becomes zero), which has been more frequently encountered in reality, thus,the examination of which could have more practical implications. Questions are raised in terms of theeffect brought by the eccentricity: Could Lagrangian chaos be generated by periodically alternatingEO or ICEO flows as well in the eccentric annulus? If so, does the presence of eccentricity harm orbenefit the mixing? Besides the eccentricity, many other factors, including the alternating time period,the number of flow patterns in one time period, and the specific flow patterns utilized for Lagrangianchaos creation, deserve systematic investigations and optimizations for either concentric or eccentricannuli.

In this paper, we addressed the aforementioned questions by carrying out a comprehensive anal-ysis on the mixing behavior of the Lagrangian chaos created by either the EO or the ICEO. Our studyshows that using the proposed micromixing method, a homogenous mixing can be achieved rapidly,e.g., within 1 s under a moderate electric field. Furthermore, due to its simple geometry, the proposedmicromixer is easy to fabricate and integrate to other lab-on-a-chip devices. We believe our studycould facilitate the understanding of the flow dynamics and the chaotic mixing mechanism for boththe EO and the ICEO in the eccentric annulus, and provide an important insight for the design andoptimization of the microchannels aiming for an effective mixing.

II. MATHEMATICAL FORMULATION

A. Problem statement

An eccentric annulus is formed with two two-dimensional cylinders and filled with electrolytesolution. We hereby define a Cartesian coordinate system in which the centers of the inner and theouter cylinders are located on the positive x-axis. We adopt a bipolar coordinate system (τ, σ) toconveniently describe the eccentric geometry, as shown in Fig. 1(b). The relationship between theCartesian and the bipolar coordinate systems is given by43

x = asinh τ

cosh τ − cosσ, y = a

sinσcosh τ − cosσ

, (1)

where −∞ < τ < ∞, 0 < σ ≤ 2π; (τ, σ) and a are the coordinates and a positive constant of thebipolar coordinate system, respectively. τ = τi and τ = τo represent the surfaces of the inner and theouter cylinders, respectively (Fig. 1(b)).

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FIG. 1. (a) Schematic diagram of the eccentric annulus micromixer. (b) Definition of the bipolar coordinates in the Cartesiancoordinates. (±a,0) are the two foci of the bipolar coordinates. The surfaces of the inner and the outer cylinders arerepresented by τ =τi and τ =τo, respectively. Ri = a/sinh τi and Ro = a/sinh τo are the radii of the inner and the outercylinders, respectively. (a coth τi, 0) and (a coth τo, 0) are the centers of the inner and the outer cylinders, respectively. ε isthe distance between the centers of the inner and the outer cylinders. eτ and eσ are the unit vectors in the bipolar coordinatesnormal and tangent to the cylinder surface, respectively. Inset: Variation of the electrical potential on the outer cylinder alongthe circumferential direction, i.e., the σ-direction, with a phase angle σ0.

Two parameters are introduced to quantitatively describe the eccentric geometry, namely theradius ratio Rr and the eccentricity ε,

Rr =Ri

Ro, ε =

ε

Ro − Ri, (2)

where Ri and Ro are the radii of the inner and the outer cylinders, respectively; ε is the distance be-tween the centers of the inner and the outer cylinders as illustrated in Fig. 1(b). The annulus becomesconcentric when the eccentricity becomes zero.

B. Electric field

Positioning the annulus into an electric field, the outer cylinder possesses a circumferentiallyvarying electrical potential (Fig. 1(b)). Thus, an electric field is established over the electrolyte solu-tion, which satisfies the Laplace equation,

∇2φ = 0, (3)

where φ is the electrical potential of the electrolyte solution.We assume that the established electric field is very small so that the Faradaic reaction on the inner

cylinder can be neglected.42 Therefore, no electrical current enters the EDL on the inner cylinder,

eτ · ∇φ = 0 at τ = τi, (4)

where eτ is the unit vector in the bipolar coordinates normal to the cylinder surface as shown inFig. 1(b). The electrical potential on the outer cylinder is

φ = E0Ro sin (σ + σ0) at τ = τo, (5)

where E0 is the electric field strength, and σ0 is the phase angle of the applied electric field, whichdefines the direction of the applied electric field.

Solving Eq. (3) together with Eqs. (4) and (5) in the bipolar coordinates, the electrical potentialis obtained

φ = E0Rocosh τi cosh τ − sinh τi sinh τ

cosh (τi − τo) sin (σ + σ0) . (6)

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When the inner cylinder is non-conducting (non-polarizable), the zeta potential is fixed, ζ f ; whilewhen the inner cylinder is conducting (ideally polarizable), the zeta potential is induced by the appliedelectric field,28

ζi = −φ + φc, (7)

where ζi is the induced zeta potential; φ is the applied electrical potential on the inner cylinder surface;φc is an integral constant defined as φc =

A φdA/A, where A is the surface of the inner cylinder.

Substituting Eq. (6) into Eq. (7), we obtain the induced zeta potential of the conducting innercylinder,

ζi = −E0Rosin (σ + σ0)cosh (τi − τo) . (8)

C. Flow field

Generally, the Reynolds number is very small in the EO and the ICEO23 and the fluid is incom-pressible. Thus, the flow field satisfies the Stokes equations, which can be written in terms of streamfunction ψ,

∇4ψ = 0, (9)

with the velocities in the bipolar coordinates defined as

uσ = h∂ψ

∂τ, uτ = −h

∂ψ

∂σ, (10)

where h = (cosh τ − cosσ) /a.The general solution of the stream function ψ in the bipolar coordinates suitable for the present

case was given by Jeffery,44

hψ = A0 cosh τ + B0τ(cosh τ − cosσ) + C0 sinh τ

+D0τ sinh τ +∞n=1

[an cosh(n + 1)τ

+ bn cosh(n − 1)τ + cn sinh(n + 1)τ+ dn sinh(n − 1)τ] cos nσ + h1τ sinσ

+

∞n=1

[en cosh(n + 1)τ + fn cosh(n − 1)τ

+ gn sinh(n + 1)τ + hn sinh(n − 1)τ] sin nσ. (11)

The established electric field exerts an electrical force on the ions within the EDL, which drivesthe ions into motion. Consequently, a flow is aroused in the bulk fluid. In the limit of the thin EDL(∼nm)23 and the weak electric field (ζ is smaller than the thermal voltage φT , ∼25 mV)45 approx-imations, the slip velocity on the inner cylinder can be described by the Helmholtz-Smoluchowskiformula,46

us = −εwζ

µEtet, (12)

where εw and µ are the dielectric permittivity and the viscosity of the electrolyte solution, respec-tively; ζ is the zeta potential of the inner cylinder; Et is the electric field component that acts on and istangential to the inner cylinder surface; et is the unit vector tangent to the inner cylinder surface. Sub-stituting the tangential electric field (obtained from Eq. (6) through E = −∇φ) and the zeta potentials(ζ f and ζi for the non-conducting and conducting inner cylinders, respectively) on the inner cylinderinto Eq. (12), we obtain the slip velocities on the non-conducting and conducting inner cylinders,respectively.

For the conducting inner cylinder, we aim at investigation on flow dynamics of the ICEO and mix-ing behavior of the corresponding Lagrangian chaos. We assume the zeta potential on the outer cylin-der is small, thus, the EO aroused on the outer cylinder is negligible and not taken into

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consideration. Accordingly, in order to compare the flow field and the mixing effect of EO and ICEO,we also do not consider EO arisen on the outer cylinder for the non-conducting inner cylinder case.Hence, the no-slip boundary condition is applied on the outer cylinder,

u = 0 at τ = τo. (13)

To take EO generated on the outer cylinder into consideration, one can readily determine the flow fieldand superimpose it to the flow fields obtained later in this paper. Substituting Eq. (11) into Eqs. (12)and (13) through Eq. (10), all the unknown coefficients are determined. Hence, the flow fields of EOand ICEO are obtained. For the detailed expressions of the coefficients, one may refer to the Appendix.

To facilitate the following discussion, we hereby introduce the translational velocity scale of theinner cylinder Ui,

Ui = ζεwE2

0 Ri

µ, (14)

where

ζ =

ζ f

E0Ri, for the non-conducting cylinder,

1, for the conducting cylinder.(15)

The obtained flow fields of the electro-osmosis (EO) and the ICEO are periodic functions of theelectric field phase angle σ0 with periods 2π and π, respectively. Fig. 2 presents the flow patterns ofthe EO and the ICEO in the annulus with electric field applied in different directions, which is definedby the electric field phase angle σ0. Clearly, the EO and the ICEO form two and four counter-rotatingmicrovortices in the annulus, respectively. By changing the electric field direction, i.e., the electricphase angle σ0, different flow patterns are formed in the annulus. From the flow fields of the EOat σ0 = 0, as shown in Fig. 2(a), we can see that the microvortex on the right-hand side is muchlarger than that on the left-hand side and encircles the inner cylinder. When the electric phase angleσ0 = π/2, two microvortices symmetric to the x-axis are formed at the upper and the lower sides of

FIG. 2. Flow fields (streamlines) of (a) the EO and (b) the ICEO within the annulus. Here the radius ratio Rr = 0.1,the eccentricity ε = 0.5, and the dimensionless zeta potential of the non-conducting inner cylinder ζ f /(E0Ri)= 1. Theelectric field phase angle σ0= 0, π/2 in (a); σ0= 0, π/4 in (b). The increment between the neighbouring streamlines isψ/(UiRi)= 0.2. The flow directions are indicated by the arrows.

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the inner cylinder. The microvortices of the ICEO, as shown in Fig. 2(b), are smaller on left-hand sideof the inner cylinder than those on the right-hand side. They are symmetric to the x-axis at σ0 = 0,while asymmetric to both the x-axis and the y-axis at σ0 = π/4 as illustrated by the flow directionsindicated by the arrows in Fig. 2(b).

D. Mixing evaluation

Aref32 demonstrated that Lagrangian chaos could be created by periodically alternating two ormore closed-orbit flows. As shown in Fig. 2, both the EO and the ICEO can form different closed-orbitflows in the annulus by choosing different electric field phase anglesσ0. Hence, the Lagrangian chaoscould be created by periodically alternating two or more values of σ0.

The particle tracking method12 is adopted to quantify and visualize the mixing effect. The nonin-teracting, passive tracer particles are initially located in the flow field (xk, yk) and advected by theEO or the ICEO,

dsdt= u(t,σ0), (16)

where σ0 is varied periodically with time,

σ0 =

σ1, 0 ≤ t < T/Nfp,

. . . , . . . ,

σNfp, (Nfp − 1)T/Nfp ≤ t < T,(17)

which corresponds to periodically activate a series of electric fields, and consequently producesdifferent flow patterns. Hence, the chaotic movement of fluid elements in the annulus is created. HereNfp is the number of flow patterns in one time period; σ1 ∼ σNfp are the electric field phase angles ofthe flow patterns 1 ∼ Nfp, respectively; T is the time period for flow pattern alternating.

Integrating Eq. (16) with the fourth order Runge-Kutta method, the positions (x∗k, y∗

k) of the tracer

particles after advection at each time step are determined. Different time steps are tested until thetracer particles faithfully follow the streamlines of the given flow pattern. The dimensionless timestep adopted in the calculation is ∆t = ∆t/t0 = 10−3 with time scale t0 = Ri/Ui = µRi/(εwζ f E0) andµ/(εwE2

0) for the EO and the ICEO, respectively. The velocities (uτ,k, uσ,k) of the tracer particles atany spatial location within the annulus can be obtained from Eq. (11) through Eq. (10) with the givenσ0 and the determined coefficients in the Appendix. The dimensionless zeta potentials ζ = 1 for boththe non-conducting and conducting inner cylinders in the following discussion.

The diffusion of the tracer particles is simulated through random walk,47

xk = x∗k + αx,k

2Di∆t, yk = y

∗k + αy,k

2Di∆t, (18)

where Di is the diffusivity of the species in the electrolyte solution; αx,k and αy,k are two independentrandom numbers with the standard normal distribution for particle k at current time step; (x∗

k, y∗

k)

and (xk, yk) are the particle positions before and after diffusion. The tracer particles are ensured tobe within the annulus by imposing a mirror-reflection47 if any of them goes out of the annulus.

The annulus is positioned in a 2Ro × 2Ro square, which is equally meshed in the Cartesian coor-dinates with ∆x = (xmax − xmin)/Nx and ∆y = (ymax − ymin)/Ny, where xmax, xmin and ymax, ymin arethe upper and lower limits of the square in the x− and y− axes, respectively. The mixing index η48

is utilized to characterize the mixing effect. To calculate the mixing index η at time t, the standarddeviation of the particle concentration at time t is defined as

SD(t) =

1Nm

Nmj=1

(cj(t) − c)2, (19)

where Nm is the number of bins (Nm = Nx × Ny = 40 × 50 in the calculation); cj(t) is the particleconcentration in bin j at time t, which is obtained from particle positions determined through Eqs. (16)and (18); and c is the particle concentration at ideal mixing,

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c =Np

Nm, (20)

where Np is the total number of particles in the particle tracking (Np = 30 000 in the calculation).Initially, all the tracer particles are located in a single bin, thus, the initial standard deviation

SD0 is large. As time elapses, SD(t) decreases gradually and approaches zero. The mixing index η isdefined as the ratio of SD(t) and SD0,

η =SD(t)SD0

. (21)

Ideally, η decreases from one and eventually diminishes to zero when the tracer particles are homoge-neously mixed in the whole annulus. In the simulation with equal-sized mesh, η approaches an asymp-totic limit ηasymp = 1/

Np when the particles achieve complete spatial randomness, i.e., the homoge-

nous mixing.48 As the annulus is unequally meshed (bins at the boundaries of the annulus are smallerthan the others within the annulus), the analytical ηasymp is unavailable. However, since the tracerparticles are uniformly distributed at the complete spatial randomness, ηasymp can be numerically ob-tained by imposing a uniform distribution to the tracer particles within the annulus, which is around1.3 × 10−2. Hence, we stop the calculation of η when it reaches this value. The initial position oftracer particles influences the variation of η. Generally, η approaches ηasymp earlier when the tracerparticles are initially located nearer to the inner cylinder. In order to keep the results consistent, thetracer particles are located into the bins that are of same distance to the inner cylinder in the analysisof eccentricity, and located into the same bin in the evaluation of other parameters.

III. RESULTS AND DISCUSSION

To facilitate the following discussion, we hereby introduce the Peclet number Pe, defined as,

Pe =UiRi

Di, (22)

which indicates the relative effect of fluid transport due to advection and diffusion. The velocity scaleUi = ζ f εwE2

0 Ri/µ, thus, Pe = ζ f εwE20 R2

i /µDi. As the weak electric field approximation is adopted,E0Ri < φT (25 mV), a small Pe is required. The area of the annulus is defined by the radius ratio Rr .The mixing is faster at a larger radius ratio Rr , which is reasonable since a larger Rr represents asmaller annulus space. We take Pe = 100 and Rr = 0.1 in the following study.

A. Typical mixing process

To analyze the mixing behavior of the Lagrangian chaos, we hereby present the variation of themixing index η with the dimensionless time t/t0, the particle distributions at different times, and thePoincaré sections in Fig. 3.

Poincaré section is a good tool to visualize the chaotic nature of mixing, which is created byrecording the particle positions at the end of every time period and presenting them in one figure.49

Generally, a random distribution of dots in the Poincaré section indicates a chaotic state, whereas thewell-defined curves, i.e., the Kolmogorov-Arnold-Moser (KAM) curves,49 represent a regular state.In the random regions, the tracer particles can arrive at any position within the same region regardlessof their initial locations. In contrast, the KAM curves indicate a quasi-periodic motion of the tracerparticles. The regions encircled by the KAM curves may indicate a periodic motion or contain bothregular and chaotic regions. However, the tracer particles initially released at any point of one (regularor chaotic) region can never cross the KAM curve to enter another (regular or chaotic) region becausethe KAM curves act as walls preventing the fluid mixing among the separated regions if the diffusionis not considered.

Clearly, the random regions almost occupy the whole annulus in the Poincaré sections of Fig. 3.The random regions demonstrate that the chaotic behavior of the tracer particles can be induced byperiodically alternating either the EO or the ICEO. As the major area of the annulus is occupied by

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FIG. 3. Variation of the mixing index η with the dimensionless time t/t0, where the mixing is realized through theLagrangian chaos created by (a) the EO and (b) the ICEO within the annulus. The dashed horizontal line ηasymp= 1.3×10−2

indicates the state of homogenous mixing. Here the eccentricity ε = 0.5; the dimensionless time period T /t0= 20 and 10 forthe Lagrangian chaos created by the EO and the ICEO, respectively; the number of flow patterns Nfp= 2; and the electricfield phase angle σ0= 0, π/2 and σ0= 0, π/4 for the Lagrangian chaos created by the EO and the ICEO, respectively. Note:The parameters are of same values in other figures unless otherwise noted. (Multimedia view) [URL: http://dx.doi.org/10.1063/1.4952971.1]

the random regions, a good mixing can be expected for the Lagrangian chaos created by either theEO or the ICEO.

From Fig. 3 we can also see that the mixing index η of both the EO and the ICEO reduces grad-ually from 1 and approaches the constant value ηasymp = 1.3 × 10−2, where the homogenous mixingis obtained. Clearly, a homogeneous mixing is achievable by the Lagrangian chaos created througheither the EO or the ICEO. The particle distributions at different times in Fig. 3 demonstrate that thetracer particles are initially located in a single bin (t/t0 = 0), then dispersed by the Lagrangian chaos(t/t0 = 20, 100, 300) and eventually distributed homogeneously (t/t0 = 500). The animation of themixing process of the Lagrangian chaos created by the ICEO can be viewed in the multimedia inFig. 3 caption.

As introduced previously, the time scales of the Lagrangian chaos created by the EO and theICEO are µRi/(εwζ f E0) and µ/(εwE2

0), respectively. These two time scales are same only whenζ f = 1, i.e., ζ f = E0Ri. When ζ f is not 1, the presented results still hold by simply using the cor-responding time scale. The electric field of order 104 V/m is commonly used in the EO50 and theICEO.26 Given the inner cylinder radius Ri = 10 µm, the zeta potential of the non-conducting innercylinder ζ f = 20 mV, the fluid viscosity µ = 1 × 10−3 kg/(m s), the dielectric permittivity of fluidεw = 7 × 10−10 kg m/(V2 s2), and the electric field strength E0 = 1 × 104 V/m, the time scales of theLagrangian chaos created by the EO and the ICEO are 50 ms and 10 ms, respectively. Thus, homoge-nous mixing can be achieved by the Lagrangian chaos of the EO and the ICEO within 25 s and 5 s (att/t0 = 500 in Fig. 3), respectively. When E0 is increased to 2.5 × 104 V/m, the time scales are reducedto 20 ms and 1.6 ms, and the homogenous mixing is realized within 10 s and 0.8 s, respectively. Asthe applied electric field increases, the time needed for a homogenous mixing through the Lagrangianchaos of either the EO or the ICEO can be significantly reduced. The time consumption of the ICEOreduces more significantly than that of the EO as the electric field increases.

B. Mixing effectiveness validation

To illustrate the effectiveness of the Lagrangian chaos for mixing, we hereby present the varia-tion of the mixing index η with time t/t0 for the mixing achieved by several methods in Fig. 4. TheEO and the ICEO are simulated by considering both the advection and the diffusion with the flowpatterns shown in Fig. 2. The curves of the mixing index η clearly show that the Lagrangian chaos,created by either the EO or the ICEO, produces the best and fastest mixing (homogeneous mixingrealized around t/t0 = 450). The mixing realized by the diffusion is the slowest. The EO and ICEOachieve homogeneous mixing slower than the Lagrangian chaos but much faster than the diffusion.The mixing effect of the EO and ICEO shows a dependence on the specific flow patterns (defined by

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FIG. 4. Variation of the mixing index η with the dimensionless time t/t0, where the mixing is realized through the diffusion,the ICEO atσ0= 0, the ICEO atσ0= π/4, the EO atσ0= 0, the EO atσ0= π/2, the Lagrangian chaos created by the ICEO,and the Lagrangian chaos created by the EO. Here the EO and the ICEO are simulated by considering both the advection andthe diffusion. Inset: Variation of η with the mixing methods at t/t0= 500.

the electric field phase angle σ0). The ICEO at σ0 = 0 mixes faster than that at σ0 = π/4, while theEO at σ0 = π/2 mixes faster than that at σ0 = 0. From Fig. 2, we can see that the microvortices inthe EO at σ0 = π/2 (the ICEO at σ0 = 0) are more uniform than those in EO at σ0 = 0 (the ICEO atσ0 = π/4). In the EO and the ICEO, the tracer particles are transported by two mechanics, advectionand diffusion. The advection strictly moves the particles along the streamlines, while the diffusioncan disperse them in any direction including across streamlines. The more uniformly distributed mi-crovortices make it easier to disperse tracer particles in the annulus. With the help of the diffusion, thetracer particles can be transported into one microvortex from the other microvortex, and eventuallyhomogenous mixed. From the inset of Fig. 4, it can be concluded that the mixing effect of the EOshows a more pronounced dependence on the specific flow patterns than that of the ICEO.

FIG. 5. Poincaré sections of the Lagrangian chaos created by (a) the EO and (b) the ICEO at different eccentricities ε. Herethe dimensionless time period T /t0= 20 and 10 for the Lagrangian chaos created by the EO and the ICEO, respectively.

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FIG. 6. Variation of the mixing index η with the dimensionless time t/t0 at different eccentricities ε, where the mixing isrealized through the Lagrangian chaos created by (a) the EO and (b) the ICEO. Here the dimensionless time period T /t0= 20and 10 for the Lagrangian chaos created by the EO and the ICEO, respectively.

C. Effect of the eccentricity ε

How the eccentricity ε influences the behavior of the Lagrangian chaos is of great concern sincethe present eccentric annulus is more practical in reality. To evaluate the influence ε on the mixingbehavior, we therefore present the Poincaré sections and mixing index variations at different eccen-tricities in Figs. 5 and 6, respectively.

The Poincaré sections of the Lagrangian chaos created by the EO and the ICEO at different eccen-tricities ε are presented in Fig. 5. Clearly, the random regions occupy the major area of the annulus atall the presented eccentricities ε, ranging from 0.1 to 0.9. The difference among the Poincaré sectionsat these ε is insignificant. For the Lagrangian chaos of the EO, the KAM curves appear near theouter cylinder at small ε (0.1–0.5). As ε increases, the KAM curves reduce and eventually disappear(ε = 0.7, 0.9) as shown in Fig. 5(a). While for the Lagrangian chaos of the ICEO, as shown in theenlarged views of Fig. 5(b) at ε = 0.1 and 0.3, the KAM curves also show up near the inner cylin-der besides those appearing near the outer cylinder. These islands are of small size and disappear asε increases to 0.9. The KAM curves near the outer cylinder also reduce as ε increases (Fig. 5(b)).Therefore, a good mixing is expected with a wide range of ε for the Lagrangian chaos created byeither the EO or the ICEO.

The variation of the mixing index η with the dimensionless time t/t0 at different eccentricities εis presented in Fig. 6. From this figure, we can conclude that the created Lagrangian chaos can workeffectively with a wide range of ε, which is concordant with the Poincaré sections shown in Fig. 5.As the eccentricity ε becomes larger, the mixing index η reduces with a steeper slope at small t/t0.However, as t/t0 increases, the curves approach similar slope. η reaches ηasymp at the similar time,around t/t0 = 450. The difference of the mixing effects at these eccentricities ε is negligible. Clearly,the created Lagrangian chaos works robustly and effectively within a broad scope of eccentricity.

D. Effect of the time period T

The Lagrangian chaos is created by periodically alternating the EO or the ICEO, thus, the alter-nating time period T is one key factor influencing the mixing behavior. To analyze the influence ofT on the Lagrangian chaos created by the EO or the ICEO in the present model, we hereby show thePoincaré sections of the Lagrangian chaos and the corresponding mixing index variation in Figs. 7and 8, respectively.

From Fig. 7, we can see that as T/t0 increases, the random region in the Poincaré sections signifi-cantly increases, while the KAM curves and the islands reduce and eventually disappear. At T/t0 = 2,the annulus is occupied by the KAM curves in general although random regions appear among theKAM curves. Hence, poor mixing is expected at T/t0 = 2. As T/t0 increases to 5, the random regionssignificantly increase. A batch of KAM curves remain near the outer cylinder, and two islands remainaround the inner cylinder. The islands are much smaller than those at T/t0 = 2. The major area of theannulus is random for the Lagrangian chaos of either the EO or the ICEO. The difference is that the

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FIG. 7. Poincaré sections of the Lagrangian chaos created by (a) the EO and (b) the ICEO at different dimensionless timeperiods T /t0. Here the eccentricity ε = 0.5.

islands in the Poincaré section of the EO are much larger than those in the Poincaré section of theICEO. When T/t0 increases to 10, the KAM curves and the islands continue to reduce. The dots inthe annulus become more random. The islands in the Poincaré section of the ICEO almost whollydisappear. The islands in the Poincaré section of the EO also show a remarkable reduction comparedto those at T/t0 = 2 and 5. As T/t0 further increases to 20, the islands totally disappear and severalKAM curves remain near the outer cylinder. As T/t0 becomes larger than 100, the annulus is entirelyoccupied by a large chaotic region.

To quantitatively evaluate the influence of the time period T/t0 on the mixing behavior, we hereinshow the variation of the mixing index η with time t/t0 at different time periods T/t0 in Fig. 8. Clearly,as time t/t0 elapses, the mixing indices η at different T/t0 decrease significantly different for both theEO and the ICEO. As T/t0 increases, the time needed for homogenous mixing is shortened at firstbut then increased. The created Lagrangian chaos can work effectively with a wide range of T/t0.From the insets of Fig. 8, we can see that the Lagrangian chaos presents a poor mixing when T/t0 issmaller than 10 or larger than 500. This is reasonable since a large area of the annulus is taken by theregular regions in the Poincaré sections at T/t0 = 2 and 5 as shown in Fig. 7. While for the mixingat T/t0 = 1000, the Lagrangian chaos has not been created within the time range. It acts as a pureEO or ICEO flow. Therefore, although increasing the time period T/t0 can significantly increase thechaotic region in the Poincaré section (Fig. 7), it cannot necessarily shorten the time for homogenousmixing. Because a larger T/t0 needs a longer time to generate Lagrangian chaos. Thus, to achieve arapid and homogenous mixing, the time period T/t0 should be within the range 10–500.

E. Effect of the number of flow patterns Nfp

As introduced previously, the Lagrangian chaos could be created by periodically alternatingeither two or more closed-orbit flows. Thus, the influence of the number of flow patterns in one time

FIG. 8. Variation of the mixing index η with the dimensionless time t/t0 at different time periods T /t0, where the mixing isrealized through the Lagrangian chaos created by (a) the EO and (b) the ICEO. Here the eccentricity ε = 0.5. Inset: Variationof η with T /t0 at t/t0= 500.

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TABLE I. The dimensionless time period T /t0 and the electric field phase angles σ0 of the flow patterns utilized forLagrangian chaos creation at different numbers of flow patterns Nfp.

EO ICEO

Nfp T /t0 σ0 T /t0 σ0

2 20 0, π/2 10 0, π/43 30 0,2π/3,4π/3 15 0, π/3,2π/34 40 0, π/2, π,3π/2 20 0, π/4, π/2,3π/45 50 0,2π/5,4π/5,6π/5,8π/5 25 0, π/5,2π/5,3π/5,4π/56 60 0, π/3,2π/3, π,4π/3,5π/3 30 0, π/6, π/3, π/2,2π/3,5π/67 70 0,2π/7,4π/7,6π/7,8π/7,10π/7,12π/7 35 0, π/7,2π/7,3π/7,4π/7,5π/7,6π/78 80 0, π/4, π/2,3π/4, π,5π/4,3π/2,7π/4 40 0, π/8, π/4,3π/8, π/2,5π/8,3π/4,7π/8

period Nfp on the Lagrangian chaos and the corresponding mixing behavior remains a great concern.The time slot for each flow pattern is kept the same in different Nfp. The dimensionless time periodT/t0 and the electric field phase angles σ0 of the EO and the ICEO utilized for the Lagrangian chaoscreation are listed in Table I.

To evaluate the effect of the number of flow patterns in one time period Nfp on the Lagrangianchaos, we hereby present the Poincaré sections of the Lagrangian chaos created by the EO orthe ICEO at different Nfp in Fig. 9. Clearly, both the random regions and the KAM curves appear inthe Poincaré sections of the Lagrangian chaos of the EO or the ICEO at all these Nfp. The areas ofthe random regions and the KAM curves vary significantly as Nfp increases. When Nfp = 2, the KAMcurves are insignificant for both the EO and the ICEO. The annulus is almost entirely occupied bya large random region. However, as Nfp increases to 3, the annulus is segmented into several smallregular or random regions. The Poincaré sections at Nfp = 3 remain chaotic near the inner cylinder butregular near the outer cylinder. Several small islands appear within the chaotic region. The sub-regionsbetween the KAM curves near the outer cylinder are also regular. Although the sub-regions confinedby the KAM curves within the chaotic region are random, the total area of chaotic regions, comparedto Nfp = 2, is significantly reduced, which implies poor mixing. As Nfp further increases, the regularregions decrease while the chaotic regions increase. When Nfp is larger than 6, the chaotic regionsregain dominance. In the Poincaré sections of EO, the islands within the chaotic regions shrink butdo not totally disappear as Nfp increases from 3 to 8; while in the Poincaré sections of ICEO, theislands totally vanish once Nfp increases to 4. The chaotic region is larger in the Poincaré sections ofEO compared to those of ICEO.

The Poincaré sections indicate the chaotic extent of the periodic flows. To quantitatively illus-trate the influence of Nfp on the mixing, we hereby present the variation of the mixing index η withthe dimensionless time t/t0 at different Nfp in Fig. 10. It is clearly illustrated that the mixing indexη gradually decreases as time t/t0 increases at all Nfp. However, the decline rates vary significantly

FIG. 9. Poincaré sections of the Lagrangian chaos created by (a) the EO and (b) the ICEO at different numbers of flowpatterns Nfp. Here the eccentricity ε = 0.5; and the dimensionless time period T /t0 is listed in Table I.

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FIG. 10. Variation of the mixing index η with the dimensionless time t/t0 at different numbers of flow patterns Nfp, wherethe mixing is realized through the Lagrangian chaos created by (a) the EO and (b) the ICEO. Here the eccentricity ε = 0.5;and the dimensionless time period T /t0 is listed in Table I. Inset: Variation of η with Nfp at t/t0= 500.

as Nfp increases. The mixing index η reaches ηasymp much earlier at Nfp = 2, which shows the bestmixing; while Nfp = 3 presents the poorest mixing. This is reasonable since the Poincaré sectionsat Nfp = 2 are random in general, while that at Nfp = 3 is regular in a large portion as illustrated inFig. 9. As Nfp further increases, the mixing effect increases. The Lagrangian chaos of the EO showsless dependence on Nfp (Fig. 10(a)) compared to that of the ICEO (Fig. 10(b)). The flow patterns usedin different Nfp differ as shown in Table I. The effect of flow patterns also contributes to the varyingmixing effect in different Nfp, which will be analyzed in Sec. III F.

F. Effect of the flow patterns

The Lagrangian chaos is created by alternating different flow patterns of the EO or the ICEO.Thus, how the flow patterns influence the Lagrangian chaos remains a hot topic of scientific interests.As stated previously, the flow fields of the EO and the ICEO are periodic functions of the electricfield phase angle σ0 with periods 2π and π, respectively. Therefore, the flow patterns of the EO orthe ICEO are determined by the values of σ0. To evaluate the influence of the flow patterns on themixing behavior, different electric field phase angles of flow pattern 2, σ2, are chosen within 0–2πand 0–π for the EO and the ICEO, respectively. Here the electric field phase angle of flow pattern 1,σ1, is set to be 0.

From the flow fields in Fig. 11(a), we can see that the streamlines in flow pattern 1 intersectwith the ones in flow pattern 2. The angles of intersection change as σ2 increases. The smallest an-gle of intersection appears at σ2 = π/100, 99π/100, and 199π/100, while the largest one appears atσ2 = π/2 and 3π/2. A good mixing can be expected atσ2 = π/2 and 3π/2 since the Poincaré sectionsare almost random in the whole annulus at these two values as illustrated in Fig. 11(a). The mixingat σ2 = π/100, 99π/100, and 199π/100 will not be good as a large portion of the Poincaré sectionsis taken by the islands or KAM curves.

Besides the angle of intersection, the flow direction is also important. The streamlines of flowpattern 2 atσ2 = π/100 and 99π/100 are identical to each other. But the corresponding flow directionsare opposite. The flow pattern 2 at σ2 = π/100 is along the direction of flow pattern 1, while the flowpattern 2 at σ2 = 99π/100 is opposite to flow pattern 1. This difference in flow directions leads toa significant variation in mixing effect. The Poincaré sections at σ2 = π/100 are random in generalalthough large islands appear within the chaotic region, while the Poincaré sections at σ2 = 99π/100are almost wholly taken by the regular region as indicated by the KAM curves.

A similar trend is presented in Fig. 11(b). The angle of intersection between the streamlines offlow pattern 1 and 2 varies asσ2 increases. The peak values appear atσ2 = π/4 and 3π/4. The Poincarésections at those two values are more random in general compared to those at other values. The stream-lines at σ2 = 49π/100 and 99π/100 are identical to each other. But the corresponding flow directionsare opposite as shown in the flow fields of Fig. 11(b). As the flow directions at σ2 = 49π/100 and

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FIG. 11. Flow fields (streamlines) utilized for the Lagrangian chaos creation and Poincaré sections of the Lagrangian chaoscreated by (a) the EO and (b) the ICEO at different electric field phase angles of flow 2, σ2. The solid black lines indicatethe flow pattern 1; the dashed blue lines indicate the flow pattern 2. The flow directions are indicated by the arrows. Here theeccentricity ε = 0.5; and the dimensionless time period T /t0= 20 and 10 for the Lagrangian chaos created by the EO and theICEO, respectively.

99π/100 are generally along and opposite to the direction of flow pattern 1, the Poincaré sectionsat σ2 = 49π/100 are more random than that at σ2 = 99π/100. Generally, the Lagrangian chaos atσ2 = π/4 and 3π/4 presents a better potential in mixing than those at other σ2.

The variation of the mixing index η with time t/t0 at different field phase angle of flow pattern 2,σ2, is shown in Fig. 12. It is evidently illustrated that η presents different decline rates as σ2 changes.For a better observation, we show the variation of η with σ2 at t/t0 = 500 as insets in Fig. 12. Clearly,as σ2 increases, η presents a W -shape variation in both the EO and the ICEO. The Lagrangian chaosof the EO can mix rapidly when σ2 is around π/2 and 3π/2, while that of the ICEO works effectivelyaround σ2 = π/4 and 3π/4. The poorest mixing is led when σ2 is around π and π/2 for the EO andICEO, respectively. Moreover, the insets of Fig. 12 show that the Lagrangian chaos of the ICEO isless sensitive to the values of σ2 compared to that of the EO.

FIG. 12. Variation of the mixing index η with the dimensionless time t/t0 at different electric field phase angles of flowpattern 2, σ2, where the Lagrangian chaos is created by (a) the EO and (b) the ICEO. Here the electric field phase angle offlow pattern 1, σ1= 0. Here the eccentricity ε = 0.5; and the dimensionless time period T /t0= 20 and 10 for the Lagrangianchaos created by the EO and the ICEO, respectively. Inset: Variation of η with σ2 at t/t0= 500.

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IV. CONCLUSION

In this paper, we have analytically studied electro-osmosis (EO) and induced charge electro-osmosis (ICEO) in the eccentric annulus, and proposed a chaotic micromixer by introducingLagrangian chaos into the EO or the ICEO micromixers. The analytical results show that EOand ICEO form two and four counter-rotating microvortices in the eccentric annulus, respectively.Different flow patterns can be generated by changing the applied electric field directions, i.e., theelectric field phase angle. The Lagrangian chaos is created by periodically alternating such EO orICEO flows. A systematic investigation is carried out on the mixing behavior of the Lagrangianchaos through the particle tracking method. Our analysis reveals that the created Lagrangian chaosperforms a much better and faster mixing than either the pure EO or the pure ICEO. Moreover, theevaluations of the key parameters show that the system can work robustly and efficiently with awide scope of eccentricity. The optimum dimensionless alternating time period of the Lagrangianchaos is fund to be around the range 10–500. Furthermore, the number of flow patterns in onetime period exerts a significant influence on the mixing behavior. Alternating two flow patterns inone time period presents the best mixing for the Lagrangian chaos created by either the EO or theICEO. The created Lagrangian chaos is more robust and less sensitive to the specific flow patternsutilized for the Lagrangian chaos creation, which are defined by the electric field phase angle, whenit is generated through the ICEO. As the electric field increases, the time needed for homogenousreduces more significantly for the ICEO than that for the EO.

The present study contributes to the understanding of flow dynamics of the EO and the ICEOin the eccentric annulus, and improves the physical insights into the corresponding chaotic mixingbehaviors. Furthermore, our investigations offer a systematic guide for the design and optimizationof chaotic micromixers. The proposed chaotic micromixer has advantages of simple geometry,low operating voltage, and easy to integrate, thus, presents a great potential in the applications oflab-on-a-chip systems for chemical, biological, and medical analysis.

ACKNOWLEDGMENTS

The authors gratefully acknowledge research support from the Singapore Ministry of EducationAcademic Research Fund Tier 2 research Grant No. MOE2011-T2-1-036. The authors thank theunknown reviewers for the helpful comments.

APPENDIX: COEFFICIENTS OF THE STREAM FUNCTIONψ

1. The non-conducting inner cylinder

The coefficients of the stream function ψ for the non-conducting inner cylinder are listed as

Anon−c0 = Knon−c

0 sinh τi��

cosh2τo sinh 2τi − τo cosh 2τi+ (2τi − τo)(cosh 2τo − 2) − sinh 2τosinh2τi

�

× sinh(τi − τo) − (τi − τo)(sinh 2τi − 2τi)× sinh τo/ sinh τi} , (A1)

Bnon−c0 = −Knon−c

0 {2(τi − τo) sinh τi cosh(τi − τo)− sinh τo sinh 2(τi − τo) + cosh τi sinh(τi − τo)× [sinh 2τo − sinh 2τi − 2(τi − τo)]} , (A2)

Cnon−c0 =

14

Knon−c0 {2τo cosh τi[sinh(τi − 3τo)

+ sinh(3τi − τo)] − cosh τo× [cosh 2(2τi − τo) − cosh 2τo+ 4τo sinh 2τi − 4τi sinh 2τo]} , (A3)

Dnon−c0 = −Knon−c

0 cosh(τi + τo) sinh τi× tanh(τi − τo) sinh 2(τi − τo), (A4)

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anon−c1 =

18

Knon−c0

�4�(τi − τo)cosh2τi cosh 3τo

− (2 + cosh 2τi + cosh 2τo)sinh2τi sinh τo�

− cosh τo [2(τi − τo)(3 − cosh 2τi) + sinh 2τi+ sinh 4τi] + sinh 2τi[3 cosh 3τo+ 2(τi − τo)(sinh 3τo − sinh τo)]} , (A5)

bnon−c1 =

14

Knon−c0 {3(2τi − τo) cosh(2τi − τo)

− (2τi − τo) cosh(2τi − 3τo)+ sinh τi[2 cosh(τi − τo) − cosh(τi − 3τo)− 2 cosh 3(τi − τo) + cosh(3τi − τo)]− 2τi[cosh τo + cosh(2τi + τo) + 4τi sinh τo]+ τo[cosh(4τi − τo) + cosh 3τo+ 8τi sinh τo]} , (A6)

cnon−c1 = −1

8Knon−c

0 {cosh 3τo + 2 sinh τo sinh 2(τi + τo)− cosh 4τi cosh τo + sinh τo[4 sinh 2(τi − τo)− sinh 4τi] + 8(τi − τo) sinh τo[cosh τo× cosh(2τi + τo) + sinh2τo] , (A7)

anon−c2 = Knon−c

2 sinh(τi + 2τo), (A8)

bnon−c2 = Knon−c

2 [sinh(τi − 2τo) − 2 sinh τi], (A9)

cnon−c2 = −Knon−c

2 cosh(τi + 2τo), (A10)

dnon−c2 = Knon−c

2 [cosh(τi − 2τo) + 2 cosh τi], (A11)

anon−cn = bnon−c

n = cnon−cn = dnon−c

n = 0, n > 2, (A12)

where

Knon−c0 = ζUi cosσ0/ {sinh τo sinh 2(τi − τo) [(τi − τo)(cosh 2τi + cosh 2τo)

− sinh 2τi − 2(τi − τo) + sinh 2(τi − τo) + sinh τo]} , (A13)

Knon−c2 =

18

ζUi cosσ0

sinh τo cosh(τi − τo)sinh2(τi − τo), (A14)

and

enon−c1 = −Λnon−c

1 [2(τi − τo) cosh 2τo − sinh 2τi + sinh 2τo], (A15)

f non−c1 = −Λnon−c

1 [−2τi + sinh 2(τi − τo) + 2τo cosh 2(τi − τo)], (A16)

gnon−c1 = −Λnon−c

1 [cosh 2τi − cosh 2τo − 2(τi − τo) cosh 2τo], (A17)

hnon−c1 = 4Λnon−c

1 sinh2(τi − τo), (A18)

enon−c2 = −Λnon−c

2 sinh(τi + 2τo), (A19)

f non−c2 = −Λnon−c

2 [sinh(τi − 2τo) − 2 sinh τi], (A20)

gnon−c2 = Λnon−c

2 cosh(τi + 2τo), (A21)

hnon−c2 = −Λnon−c

2 [cosh(τi − 2τo) + 2 cosh τi], (A22)

enon−cn = f non−c

n = gnon−cn = hnon−c

n = 0, n > 2, (A23)

where

Λnon−c1 =

14ζUi cosh τi sinσ0/ {sinh τo sinh 2(τi − τo) [(τi − τo) cosh(τi − τo) − sinh(τi − τo)]} ,

(A24)

Λnon−c2 =

18

ζUi sinσ0

sinh τo cosh(τi − τo)sinh2(τi − τo). (A25)

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2. The conducting inner cylinder

The coefficients of the stream function ψ for the conducting inner cylinder are listed as

Ac0 = Kc

0 [cosh 2τi + cosh 2(τi − τo)+ 2τi sinh 2τo − cosh 2τo − 2τo sinh 2τi − 4τi(τi − τo) − 1] , (A26)

Bc0 = 4Kc

0 [2(τi − τo) + sinh 2τi − sinh 2τo], (A27)

Cc0 = −Kc

0 [2(τi − τo) − 2τo cosh 2τi + 2τi cosh 2τo+ sinh 2τi + sinh 2(τi − τo) − sinh 2τo] , (A28)

Dc0 = −4Kc

0 sinh(τi + τo) sinh(τi − τo), (A29)

ac1 =

12

Kc0

sinh(τi − τo)�8(τi − τo)sinh2τo cosh(τi − τo)

+ 2 cosh τi sinh τo(2 − cosh 2τi − cosh 2τo)+ 3 cosh 3τo sinh τi − cosh τo sinh 3τi] , (A30)

bc1 = Kc0 {cosh 2τi + cosh 2τo − 2 cosh 2τi cosh 2τo+ 4(τi − τo)[τi − sinh τi sinh τo/ sinh(τi − τo)]+ 2[τo(sinh 2τi − sinh 2τo)+ sinh 2τi sinh 2τo]} , (A31)

cc1 =12

Kc0

sinh(τi − τo) {2 cosh(τi − 3τo) − 2 cosh(τi + τo)+ cosh(3τi + τo) − cosh(τi + 3τo)+ 2(τi − τo)[sinh(τi − τo) + 2 sinh(τi + τo)]− 2(τi − τo) sinh(τi + 3τo)} , (A32)

ac2 = Kc

2 sinh(τi + 2τo), (A33)

bc2 = −Kc2 [2 sinh τi − sinh(τi − 2τo)], (A34)

cc2 = −Kc2 cosh(τi + 2τo), (A35)

dc2 = Kc

2 [2 cosh τi + cosh(τi − 2τo)], (A36)

ac3 = −Kc

3 [sinh 4τo + 2 sinh 2(τi + τo)], (A37)

bc3 = Kc3 [3 sinh 2τi + 2 sinh 2τo − sinh 2(τi − 2τo)], (A38)

cc3 = Kc3 [cosh 4τo + 2 cosh 2(τi + τo)], (A39)

dc3 = −Kc

3 [3 cosh 2τi + 2 cosh 2τo + cosh 2(τi − 2τo)], (A40)

acn = bcn = ccn = dc

n = 0, n > 3, (A41)

where

Kc0 =

116

Ui sinh τi sin 2σ0/�sinh2τocosh2(τi − τo)

× [(τi − τo)(cosh 2τi + cosh 2τo − 2)− sinh 2τi + sinh 2(τi − τo) + sinh 2τo]} , (A42)

Kc2 =

14

Ui sinh 2τi sin 2σ0

sinh2τosinh22(τi − τo), (A43)

Kc3 =

18

Ui sinh τi sin 2σ0/�sinh2τosinh22(τi − τo)[2 + cosh 2(τi − τo)] , (A44)

and

ec1 = Λc1[2(τi − τo) cosh 2τo − sinh 2τi + sinh 2τo], (A45)

f c1 = Λc1[2τo cosh 2(τi − τo) + sinh 2(τi − τo) − 2τi], (A46)

gc1 = Λc1[cosh 2τi − cosh 2τo − 2(τi − τo) sinh 2τo], (A47)

hc1 = −4Λc

1sinh2(τi − τo), (A48)

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ec2 = Λc2 sinh(τi + 2τo), (A49)

f c2 = Λc2[sinh(τi − 2τo) − 2 sinh τi], (A50)

gc2 = −Λc2 cosh(τi + 2τo), (A51)

hc2 = Λ

c2[cosh(τi − 2τo) + 2 cosh τi], (A52)

ec3 = −Λc3[sinh 4τo + 2 sinh 2(τi + τo)], (A53)

f c3 = Λc3[3 sinh 2τi + 2 sinh 2τo − sinh 2(τi − 2τo)], (A54)

gc3 = Λc3[cosh 4τo + 2 cosh 2(τi + τo)], (A55)

hc3 = −Λ

c3[3 cosh 2τi + 2 cosh 2τo + cosh 2(τi − 2τo)], (A56)

ecn = f cn = gcn = hc

n = 0, n > 3, (A57)

where

Λc1 =

18

Ui sinh τi cos 2σ0/�sinh2τosinh22(τi − τo)

× [(τi − τo) coth(τi − τo) − 1]} , (A58)

Λc2 =

14

Ui sinh 2τi cos 2σ0

sinh2τosinh22(τi − τo), (A59)

Λc3 =

18

Ui sinh τi cos 2σ0/�sinh2τosinh22(τi − τo)

× [cosh 2(τi − τo) + 2]} . (A60)

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