A lattice Boltzmann algorithm for electro-osmotic flows in microfluidic ...

THE JOURNAL OF CHEMICAL PHYSICS 122, 144907 共2005兲

A lattice Boltzmann algorithm for electro-osmotic flows in microfluidic devices Zhaoli Guo, T. S. Zhao,a兲 and Yong Shi Department of Mechanical Engineering, Hong Kong University of Science and Technology, Kowloon, Hong Kong

共Received 23 August 2004; accepted 26 January 2005; published online 14 April 2005兲 In this paper, a finite-difference-based lattice Boltzmann 共LB兲 algorithm is proposed to simulate electro-osmotic flows 共EOF兲 with the effect of Joule heating. This new algorithm enables a nonuniform mesh to be adapted, which is desirable for handling the extremely thin electrical double layer in EOF. The LB algorithm has been validated by simulating a problem with an available analytical solution and it is found that the numerical results predicted by the algorithm are in good agreement with the analytical solution. The LB algorithm is also applied to modeling a mixed electro-osmotic/pressure driven flow in a channel. The numerical results show that Joule heating plays an important role in EOF. © 2005 American Institute of Physics. 关DOI: 10.1063/1.1874813兴 I. INTRODUCTION

Electro-osmotic flow 共EOF兲 is created by applying an external electrical field on electrical double layers 共EDL兲 formed due to the interaction between an electrolyte solution and a dielectric surface.1 This type of flow has recently been utilized in various biomedical lab-on-a-chip devices to transport and manipulate liquids 共buffer or sample兲 for different purposes, such as sample injection, chemical reactions, and species separation. Due to these important applications, EOF in microchannels has recently received increasing attention.1–5 The lattice Boltzmann method 共LBM兲 is a recently developed efficient numerical tool for simulating fluid flows and transport phenomena based on kinetic equations and statistical physics. Because of its distinctive advantages over conventional numerical methods, the LBM has found success in a variety of fields since its emergence,6–8 and recently its applications to microfluidics have attracted much attention from researchers in a variety of fields.9 Particularly, some efforts have been made to apply the LBM to electrokinetic systems,10–14 and the simulation results for isothermal electrokinetic flows are satisfactory and encouraging. The existing lattice Boltzmann models for EOF fall into two categories: single-fluid models and multifluid/ component models. In the first category, electrolyte solution is treated as a single fluid, whose motion is governed by the incompressible Navier–Stokes equations incorporated with an electrostatic force. Warren10 made the first attempt to apply the LBM to charged fluid systems following this idea and proposed to solve the Navier–Stokes equations for the solution using the standard LBM, while the conservation equation for each ionic species and the Poisson equation for the electrical potential were solved via the “moment propagation” method. A similar single-fluid LBM was proposed by He and Li,11 except that the conservation equations for difa兲

Author to whom correspondence should be addressed; Electronic mail: [email protected]

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ferent ionic species were solved using some independent lattice Boltzmann equations 共LBE兲. In He and Li’s method, the electrical potential is described by a Poisson equation under the assumption of locally electrical neutrality, which is solved using another LBE. In the multifluid category, each species is treated as a component of the electrolyte mixture, and the motion of the mixture is determined by the combination of the motions of all species. The LBM proposed by Horbach and Frenkel was the first method of this type,12 in which the motion of each ionic species is described by a “moment propagation” LBE, and the motions of the mixture are directly determined from the motions of all species, while the Poisson equation for the electrical potential is solved with a conventional numerical method. On the other hand, Melchionna and Succi proposed a three-component LBM following the so-called “top-down” approach,13 in which each species is described by a LBE with a redefined equilibrium distribution function. Recently, Capuani et al. proposed another multi-fluid LBM, in which the fluxes between lattice nodes are taken as the basic physical quantities, and the nonideal effect of solutions is also taken into account.14 All of the existing LBMs mentioned above are designed on the Poisson–Boltzmann level and, except for the method proposed in Ref. 14, these methods treat the solution as an ideal fluid. Such treatments are sufficient if we are only interested in the hydrodynamics of the fluid. Indeed, these LBMs have been successfully used to study a variety of electrical systems, including electroviscous transport phenomena, the electrical chemical reactions, sedimentation of charged colloids, and electrorheology. In spite of their successes, however, there still exist several limitations in these existing LBM models for electrokinetic flows. First, all of these models are designed for isothermal flows. However, strictly speaking, the isothermal assumption is questionable for electrical kinetic flows, because Joule heating always exists due to the applied electric field. Joule heating may produce a significant temperature

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© 2005 American Institute of Physics

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Guo, Zhao, and Shi

gradient within the flow, especially when the applied electric voltage is high. Some recent studies have revealed that the thermal effects on EOF are rather important.15–17 Therefore, it is desirable to propose a LBM that includes the Joule heating effect. Another disadvantage of the existing lattice Boltzmann models for EOF is that they all employ a regular uniform lattice. It is well known that the coions and counterions in the liquid near a charged surface in EOF are redistributed due to electrostatic interactions and eventually an EDL forms. The EDL is usually rather thin in comparison with the size of the flow domain. The thickness of EDL is in the order of 1 – 100 nm, while the typical size of the flow domain in many microdevices is in the order of 1 – 100 ␮m. Although thin, the EDL plays an important role in EOF, since the electro-osmotic forces in EOFs are mainly concentrated within the EDL. Therefore, the transport phenomena in the EDL should be accurately tracked in any numerical simulations to produce reasonable results. Clearly, a LBM using a uniform mesh would be too expensive for EOFs with a thin EDL. The isothermal assumption and the use of a uniform mesh severely limit the applications of the LBM for practical EOFs. In this work, following the idea of the single-fluid model, we propose a LB algorithm for EOFs with the Joule heating effect using nonuniform meshes. In this new algorithm, the electrolyte is still treated at the Poisson– Boltzmann level, and the pressure relating to the density through an ideal equation of state is assumed. The rest of the paper is organized as follows. In the following section, we present the macroscopic governing equations for EOFs. In Sec. III, three discrete-velocity Boltzmann equations 共DVBEs兲 that can recover the macroscopic governing equations for EOF are proposed. In Sec. IV, a finite-difference-based lattice Boltzmann method 共FDLBM兲 is obtained by discretizing the corresponding DVBEs on a nonuniform mesh using certain finite-difference schemes. Some numerical tests are conducted in Sec. V, and finally a brief summary is given in Sec. VI. II. MACROSCOPIC HYDRODYNAMIC EQUATIONS

The EDL is formed due to the interaction of the ionized solution with the charged solid surfaces. According to electrostatic theory, the induced electric potential of ions ␾ satisfies the Poisson equation

␳e ⵱ · 共␧ ⵱ ␾兲 = − , ␧0

共1兲

where ␧0 is the permittivity of the vacuum and ␧ = ␧共T兲 is the dielectric constant of the electrolyte dependent on temperature T. ␳e represents the net charge density and is defined by ␳e = 兺i zieni, where ni is the ionic number concentration of species i, z is the valence, and e is the electron charge. For flows over a nonconducting stationary surface, the ion distribution can be well approximated by the Boltzmann distribution,1 ni = ni⬁exp共−zie␾ / kBT兲, where n⬁ is the ion density in the bulk solution and kB is the Boltzmann constant. Particularly, for a symmetric electrolyte 共zi = z and ni⬁ = n⬁兲

considered in this study, the charge density can be given by

冉 冊

␳e = − 2n⬁ze sinh

ze␾ . k BT

共2兲

Substituting Eq. 共2兲 into Eq. 共1兲 leads to the Poisson– Boltzmann equation ⵱ · 共␧ ⵱ ␾兲 =

冉 冊

2n⬁ze ze␾ sinh . ␧0 k BT

共3兲

The thickness of the EDL is characterized by the Debye length ␭D, which is defined by ␭D =

冑

␧共T0兲␧0kBT0 , n ⬁e 2z 2

共4兲

where T0 is the reference temperature. The reciprocal of ␭D, ␬ = 1 / ␭D, is also termed as the Debye–Hückel parameter. The driving force in EOF is due to the interaction between the net charge density within the EDL region and the applied external electric field. Under the assumptions that the electrolyte is incompressible and the density fluctuations due to temperature variations are negligible, the motion of the fluid is governed by the Navier–Stokes equations ⵱ · u = 0,

␳

冉

共5a兲

冊

⳵u + u · ⵱ u = − ⵱ p + ⵱ · 共␮ ⵱ u兲 + ␳eE, ⳵t

共5b兲

where u and p are the velocity and pressure of the flow, respectively. Although the density ␳ of the electrolyte solution is assumed to be a constant, the shear viscosity ␮ = ␮共T兲 is assumed to depend on the local temperature. In Eq. 共5b兲, E = − ⵱ 共⌽ + ␾兲 is the strength of the total electric field, where ⌽ is the applied external electrical potential. Since in EOF, the induced electric potential ␾ is usually much weaker than the applied external potential ⌽, we can approximate E as E ⬇ − ⵱ ⌽. The applied electric field and the motion of ions can induce Joule heating in EOF, which may cause a temperature gradient in the fluid. Assuming that the compression work and the viscous dissipation are negligible, we can write the energy equation as

␳c p

冉

冊

⳵T + u · ⵱ T = ⵱ · 共k ⵱ T兲 + q˙ , ⳵t

共6兲

where the specific heat c p is assumed to be constant, but the thermal conductivity k = k共T兲 of the fluid is assumed to be temperature dependent; q˙ represents Joule heating, which is given by q˙ = KE2 ,

共7兲

where K = K共T兲 is the electrical conductivity of the electrolyte. The Poisson–Boltzmann equation 共3兲, Navier–Stokes equation 共5兲, and energy equation 共6兲, can be rewritten in dimensionless form as

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Electro-osmotic flows in microfluidic devices

冉 冊

¯ ⵱ ␸兲 = ␤ sinh ⵱ · 共␧

␣␸ , 1+␪

⵱ · U = 0,

共8a兲

The most widely used collision operator in the LBM is the Bhatnagar–Gross–Krook 共BGK兲 or single-relaxationtime model,22

共8b兲 ⍀i共f兲 = −

⳵U 1 ¯ ⵱ U兲 ⵱ · 共␮ +U· ⵱U=− ⵱P+ Re ⳵t

冉 冊

␤ ␣␸ + sinh , Re 1+␪ 1 ⳵␪ ¯ ⵱ ␪兲 + J K ¯, ⵱ · 共k +U· ⵱␪= Pr Re Pr Re ⳵t

共8c兲

共8d兲

where ␸ = ␾ / ␨ is the dimensionless potential normalized with the ␨ potential of the EDL, P = p / ␳us2 is the dimensionless pressure, and ␪ = 共T − T0兲 / T0 is the dimensionless temperature; ␣ = ez␨ / kBT0 is the ionic energy parameter and ␤ = 共␬h兲2 / ␣ relates to the ratio of the Debye length to the characteristic length h; U = u / us is the dimensionless velocity normalized by the Helmholtz–Smoluchowski electroosmotic velocity us = −␧共T0兲␧0␨E / ␮; Re= ␳hus / ␮共T0兲 is the Reynolds number and Pr= c p␮共T0兲 / k共T0兲 is the Prandtl number; J = K共T0兲h2E2 / kT0 is the Joule number, ¯␧ = ␧共T兲 / ␧共T0兲, ¯ = K共T兲 / K共T 兲 are the normalized ¯ = ␮共T兲 / ␮共T0兲, and K ␮ 0 temperature-dependent coefficients, respectively.

III. LATTICE BOLTZMANN ALGORITHM

In a LBM, the motion of the fluid is modeled by a LBE for the distribution function of the fluid molecules. Historically, the LBM originates from the lattice gas automata method.18 Later it was shown that the LBM can also be derived from the continuous Boltzmann equation19 or from a discrete-velocity Boltzmann equation.20 The viewpoint that LBE is a special discretization scheme for the Boltzmann equation enables one to discretize the particle velocity space and the physical space independently,21 and a nonuniform mesh can thus be employed. In this section, we first propose three DVBEs, corresponding, respectively, to the Navier– Stokes equations, Eq. 共5兲, the energy equation, Eq. 共6兲, and the Poisson–Boltzmann equation, Eq. 共3兲. Based on these DVBEs, a finite-difference-based lattice Boltzmann method is constructed in two-dimensional curvilinear coordinates.

共10兲

where ␭u is the relaxation time and f 共eq兲 is the local equilibi rium distribution function. The fluid density ␳ and flow velocity u are defined as

␳ = 兺 f i,

␳u = 兺 ci f i .

共11兲

The discrete velocities and the EDF must be chosen properly such that the mass and momentum are conserved and the symmetry requirements are satisfied. The resulting macroscopic equations derived from Eq. 共9兲 can recover the Navier–Stokes equations, Eqs. 共5兲. For example, in the nine discrete velocity model,22 the EDF is defined by

冋

f 共eq兲 = ␻ i␳ 1 + i

ci · u cs2

+

uu:共cici − cs2I兲 2cs4

册

,

F i = ␻ i␳ e

冋

ci − u cs2

+

ci · u cs4

册

ci · E.

where f i共x , t兲 is the single-particle distribution function 共DF兲 for particles moving with velocity ci at position x and time t, Fi is the forcing term corresponding to the applied external electric field, and ⍀i is the collision operator representing the rate of change due to collisions in the DF f i.

共13兲

With the EDF given by Eq. 共12兲 and the forcing term given by Eq. 共13兲 , the macroscopic equations can be derived from the DVBE 共9兲 through the Chapman–Enskog procedure as24

The DVBE corresponding to the Navier–Stokes equations can be written as 共9兲

共12兲

where the discrete velocities are given by c0 = 0, and ci = ␭i共cos ␪i , sin ␪i兲 with ␭i = ␴, ␪i = 共i − 1兲␲ / 2 for i = 1 – 4, and ␭i = 冑2␴, ␪i = 共i − 5兲␲ / 2 + ␲ / 4 for i = 5 – 8. The weights are given by ␻0 = 4 / 9, ␻i = 1 / 9 for i = 1 – 4, ␻i = 1 / 36 for i = 5 – 8, and cs = 冑RT0 = ␴ / 冑3 is the sound speed of the model with R being the gas constant. The EDF 共12兲 can also be derived from the Boltzmann–Maxwellian distribution via a Taylor expansion in u / cs ⬇ M up to the second order,19 where M represents the Mach number. This indicates that M is required to be small in the LBM. The forcing term Fi in Eq. 共9兲 also needs to be chosen carefully, as discussed in Ref. 23. Following the approach presented in Ref. 23, we express the forcing term as

⳵␳ + ⵱ · 共␳u兲 = 0, ⳵t

A. Discrete velocity Boltzmann equations

⳵ fi + ci · ⵱ f i = ⍀i + Fi , ⳵t

1 关f i − f 共eq兲 i 兴, ␭u

␳

冉

冊

⳵u + u · ⵱ u = − ⵱ p + ⵱ · 共␮S兲 + ␳eE, ⳵t

共14a兲

共14b兲

where S = ⵱ u + 共⵱u兲†, p = cs2␳, and ␮ = cs2␳␭u. It is clear that in the incompressible limit, Eqs. 共14兲 reduce to the Navier– Stokes equations 共5兲. The DVBE corresponding to the energy equation can be constructed as

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Guo, Zhao, and Shi

1 ⳵ gi + ci · ⵱ gi = − 关gi − g共eq兲 i 兴 + Qi , ⳵t ␭T

共15兲

where gi共x , t兲 represents a “temperature” distribution funcis its equilibrium distribution function tion 共TDF兲, and g共eq兲 i and is given by

冋

= ␻ i␳ c pT 1 + g共eq兲 i

ci · u cs2

+

uu:共cici − cs2I兲 2cs4

册

,

共16兲

共17兲

i

The term Qi in Eq. 共15兲 represents the effects due to the presence of the electric field on the TDF gi, and can be expressed as

冋

Qi = ␻iq˙ 1 +

ci · u cs2

册

+ ␻ ic pT

c i · ␳ eE cs2

.

共18兲

Clearly, Qi defined in Eq. 共18兲 satisfies the following properties:

兺i Qi = q˙ , 兺i ciQi = uq˙ + cpT␳eE.

共19a兲

共19b兲

With the aid of Eq. 共19兲 and using the Chapman–Enskog expansion technique, we can prove that the DVBE 共15兲 recovers the macroscopic energy equation 共6兲 with k = ␳c pcs2␭T. The second term on the right-hand side of Eq. 共18兲 implies that the evolution of the TDF is affected by the applied electric field not only through the Joule heating term, but also through the force field. In the cases in which the electrical conductivity K is negligibly small, the influence of the source term Qi still exists. Finally, the DVBE corresponding to the Poisson– Boltzmann equation 共3兲 can be written as

⳵ hi 1 + ci · ⵱ hi = − 关hi − h共eq兲 i 兴 + Hi , ⳵␶ ␭␾

⳵ ␺i 1 + ci · ⵱ ␺i = − 关␺i − ␺共eq兲 i 兴 + Si , ⳵s ␭

共21兲

¯ 共x , t , ␶兲 ⬅ 兺ihi共x , t , ␶兲 and ␧ = c2␭␾ / D. Clearly, at where ␾ s

共22兲

共eq兲 共eq兲 = f 共eq兲 where s = t or ␶ ; ␺ = f i, gi, or hi; ␺共eq兲 i i , gi , or hi ; ␭ = ␭u, ␭T, or ␭␾; and Si = Fi, Qi, or Hi, respectively. To solve Eq. 共22兲 numerically, we discretize it following the method proposed in Ref. 25, where the collision term is discretized implicitly but the convection term and forcing term are treated explicitly. With a simple transformation, the implicity of the scheme can be canceled to obtain a completely explicit scheme. Specifically, on a nonuniform mesh in general curvilinear coordinates ␰, the final finite-difference-based lattice Boltzmann equation 共FDLBE兲 for Eq. 共22兲 can be written as

˜␺ 共x,s + ⌬s兲 = 共1 − ␻兲␺ 共x,s兲 + ␻␺共eq兲共x,s兲 i i i − ⌬sci · ⵱h␺i共x,s兲 + ⌬sSi ,

共23兲

where ⌬s is the time step, ␻ = ⌬s / 2␭, and ˜␺i = ␺i + ␻共␺i − ␺共eq兲 i 兲 is a new distribution function. Due to the conservation nature of the BGK collision operator, the physical variables are now given by

␳ = 兺 ˜f i, i

␳u = 兺 ci˜f i, i

␳c pT = 兺 ˜gi, i

␾ = 兺 ˜hi . i

共24兲 In the curvilinear coordinate ␰, the discrete spatial gradient ⵜh in Eq. 共23兲 can be written as

共20兲

where ␶ is a pseudotime, hi = hi共x , t , ␶兲 is the electrical potential distribution function, h共eq兲 = ␻i␾, and Hi = D␻i␳e / ␧0, with i D ⬎ 1 being a constant parameter. Again, through the Chapman–Enskog method we can obtain the following macroscopic equation: ¯ 1 ⳵␾ ¯ 兲 + ␳e , = ⵱ · 共␧ ⵱ ␾ D ⳵␶ ␧0

B. Finite-difference-based lattice Boltzmann method

The three discrete-velocity Boltzmann equations, Eqs. 共9兲, 共15兲, and 共20兲, can take the following general form:

with the temperature being defined as

␳ c pT = 兺 g i .

¯ 共x , t , ␶兲 → ␾共x , t兲, and the Poisson– steady state as ␶ → ⬁, ␾ Boltzmann equation 共3兲 is recovered. Note that a similar strategy has been adopted to solve the Poisson equation under the framework of the LBM in previous studies.10,11

c i · ⵱ h␺ i =

⳵␺

⳵␺

兺␣ ci␣ ⳵hx␣i = 兺␤ ei␤ ⳵␰h ␤i ,

共25兲

where ei␤ = ci␣ ⳵ ␰␤ / ⳵x␣ and ⳵h␺i / ⳵␰␤ is a combination of the second-order upwind difference and the central difference,

冏 冏

⳵h ⳵h =⑀ ⳵␰␤ ⳵␰␤

+ 共1 − ⑀兲 u

冏 冏 ⳵h ⳵␰␤

,

共26兲

c

where 0 艋 ⑀ 艋 1 is a combination parameter, and

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冏 冏 ⳵ h␺ i ⳵␰␤

and

冏 冏 ⳵ ␺i ⳵␰␤

= u

= c

冦

1 关3␺i共␰␤, · 兲 − 4␺i共␰␤ − ⌬␰␤, · 兲 + ␺i共␰␤ − 2⌬␰␤, · 兲兴 2⌬␰␤

共ii兲 共iii兲 共iv兲

if ei␤ 艌 0

1 关3␺i共␰␤, · 兲 − 4␺i共␰␤ + ⌬␰␤, · 兲 + ␺i共␰␤ + 2⌬␰␤, · 兲兴 if ei␤ ⬍ 0. − 2⌬␰␤

1 关␺i共␰␤ + ⌬␰␤, · 兲 − ␺i共␰␤ − ⌬␰␤, · 兲兴, 共28兲 2⌬␰␤

where ⌬␰␤ is the mesh spacing in the ␰␤ direction. Clearly, the FDLBE 共23兲 with ⵜh defined above is of first-order accuracy in time and second-order accuracy in space. The solution procedure of the above FDLBM is as follows. 共i兲

J. Chem. Phys. 122, 144907 共2005兲

Electro-osmotic flows in microfluidic devices

Determine the relaxation times ␭u共x , t兲, ␭T共x , t兲, and ␭␾共x , t兲. Compute the density DF ˜f i共x , t + ⌬t兲, the fluid density ␳共x , t + ⌬t兲, and velocity u共x , t + ⌬t兲. Compute the temperature DF ˜gi共x , t + ⌬t兲 and the temperature T共x , t + ⌬t兲. Compute the electrical potential ␾共x , t + ⌬t兲 in the following way.

共1兲 Initialize hi and choose a pseudo-time-step ⌬␶. 共2兲 Compute the DF ˜hi共x , t , ␶ + ⌬␶兲, and the temporary elec¯ 共x , t , ␶ + ⌬␶兲. trical potential ␾ 共3兲 Set ␶ = ␶ + ⌬␶ and repeat step 共2兲 until certain convergence criteria is reached. ¯ 共x , t , ␶兲. 共4兲 Set ␾共x , t + ⌬t兲 = lim␶→⬁␾ C. Discussions on the algorithm

The present algorithm utilizes the equation of state of ideal gases by neglecting the excluded-volume and attractive forces between molecules of the electrolyte, whose effects become significant as the fluid undergoes a phase change. For the single-phase EOF considered in this work, the effects on the final results are insignificant. In the present algorithm, the energy equation is solved in the LBM framework. Alternatively, it can also be solved using the conventional finite-difference or finite-volume numerical scheme to form a so-called hybrid LBE method for thermal flows.26 The benefit of the present approach, however, is the easiness of programming since the computations of the velocity field and the temperature field have the same structures. The Poisson–Boltzmann equation is also solved in the LBM framework in the present algorithm. Although a relaxation method is generally thought to be inefficient for solving a Poisson equation, because of the nonlocal nature of the Laplacian operator, in this work we find that the convergence of the present LBE for the Poisson–Boltzmann equation considered here is still efficient. This is because the electrical potential distribution depends weakly on temperature; once

共27兲

the local equilibrium charge profile is achieved, the electrical potential distribution undergoes a slight change with time at a later stage. It is also found that the time to reach a local equilibrium is rather short 共usually no more than ten time steps in our simulations兲, because the response time for the electric potential field is much shorter than the convection and diffusion time scales. Similar phenomena were also reported by Capuani et al.14 In fact, the computation of the Poisson–Boltzmann equation can be turned off once the equilibrium is achieved in practical simulations. Therefore, the present LBE algorithm is still an efficient method for the Poisson–Boltzmann equation under consideration. Furthermore, as mentioned above, the LBM formulation for the Poisson–Boltzmann equation can be easily incorporated into a standard LBM code. This feature may be more useful for a multiprocessor code. However, more sophisticated methods can of course be employed. Actually, successive overrelaxation and multigrid methods have been used to solve the Poisson–Boltzmann equation in previous LB algorithms.13,14 The numerical stability of the lattice Boltzmann scheme described above depends on several parameters, including the Mach number, the relaxation time, the mixing parameter ⑀, and the Courant-Friedrichs-Levy number ⌬t兩ci兩 / ⌬x. The stability of the LBE for the density distribution function f i has been studied extensively in the absence of an external force,25 and similar arguments can be applied to the two LBEs for gi and hi. In addition to the above-mentioned parameters, the strength of the applied electrical field also affects the stability of present LB algorithm. As shown in Ref. 13, the use of the explicit Euler-forward time integration in the present algorithm requires that 兩␦ f i兩 ␳eE⌬xmin ⬃ Ⰶ1 fi ␳cs2

共29兲

for the sake of stability, where ⌬xmin represents the minimum grid size of the computational mesh. Since in EOF, the applied electrical potential takes effects mainly in the EDL, the above requirement is equivalent to ␭D Ⰷ ⌬xmin.13 Therefore, in our algorithm we can adjust the mesh such that there are enough grid points in the EDL, so that the above stability requirement is satisfied. IV. BOUNDARY CONDITIONS

We have presented a finite-difference lattice Boltzmann algorithm for simulating electro-osmotic flows in the preceding section. In practical applications, the boundary conditions for the three types of distribution function need to be specified in terms of macroscopic variables. Specifying appropriate boundary conditions for a physical problem is an

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Guo, Zhao, and Shi

FIG. 1. Schematic and boundary conditions of electro-osmotic flow in a microchannel.

important topic in the LBM, and a variety of methods have been developed based on different pictures 共e.g., see Ref. 27 and references therein兲. Most of previous boundary conditions are designed for the density distribution function f i, and usually cannot be extended directly to the temperature distribution function gi or the fictitious electrical potential distribution function hi. But the recently developed nonequilibrium extrapolation method,28 which has a good numerical accuracy and stability, may serve our purpose, and we will use this method in this work. The nonequilibrium extrapolation method is based on the fact that the nonequilibrium part of the density DF at a boundary node can be well approximated by the nonequilibrium part of the DF at the nearest neighboring node in the fluid region along the discrete velocity. For example, with xb representing a boundary node and x f representing its nearest 共xb兲 = f 共neq兲 共x f 兲. As neighboring fluid node, we can set f 共neq兲 i i such, the total DF at xb can be given by 共eq兲 * * f i共xb兲 = f 共eq兲 i 关␳ 共xb兲,u 共xb兲兴 + 关f i共x f 兲 − f i 共x f 兲兴,

共30兲

f 共eq兲 i ,

where the equilibrium part of the DF, is determined by imposing the macroscopic boundary conditions through the auxiliary density ␳* and velocity u*. For instance, for velocity boundary condition where velocity u共xb兲 is known but ␳共xb兲 is unknown, we may use ␳*共xb兲 = ␳共x f 兲 or ␳*共xb兲 = 2␳共x f 兲 − ␳共x f f 兲 共x f f is the next neighboring fluid node of xb in the same direction兲, and u*共xb兲 = u共xb兲. Similar idea can also be used to determine the DFs gi共xb兲 and hi共xb兲, 共eq兲 * * * gi共xb兲 = g共eq兲 i 关␳ 共xb兲,u 共xb兲,T 共xb兲兴 + 关gi共x f 兲 − gi 共x f 兲兴,

共31a兲 共eq兲 * hi共xb兲 = h共eq兲 i 关␾ 共xb兲兴 + 关hi共x f 兲 − hi 共x f 兲兴,

共31b兲

where T* and ␾* at xb are specified according to the given boundary conditions for T and ␾. V. NUMERICAL RESULTS A. Validation

We first validate the proposed LB algorithm by simulating a transient EOF in a straight channel of width h and length L = 10h, where the ␨ potential on the bottom and top plates is assumed to be a constant ␨ 共see Fig. 1兲. The bottom and top plates are kept stationary and isothermal, and periodic boundary conditions are assumed at the inlet and outlet

for all physical variables. In order to obtain an analytical solution for the problem, we assume that ␧, ␮, k, and K are all constant. It is known that there exists an analytical solution for this problem if the Joule heating effect is neglected.5,29–32 As Joule heating is taken into account, however, no analytical solutions are available in the literature. Therefore, in what follows, we first present an approximate analytical solution for this problem. The full dimensionless governing equations for the problem can be written as

⳵ 2␸ ␣␸ = ␤ sinh , ⳵ ␩2 1+␪

冉 冊

共32a兲

⳵ U ⳵ 2U ⳵ 2␸ = + , ⳵ t ⳵ ␩2 ⳵ ␩2

共32b兲

Re

Pr Re

⳵ ␪ ⳵ 2␪ = + J, ⳵ t ⳵ ␩2

共32c兲

with initial condition U共␰ , ␩ , 0兲 = 0, ␪共␰ , ␩ , 0兲 = 0, and boundary condition U共␰ , 0 , t兲 = U共␰ , 1 , t兲 = 0, ␪共␰ , 0 , t兲 = ␪共␰ , 1 , t兲 = 0, and ␸共␰ , 0 , t兲 = ␸共␰ , 1 , t兲 = 1, with ␰ = x / h and ␩ = y / h. First, we solve Eq. 共32c兲 to give

冉

⬁

␪ = 兺 ␪nsin共n␲␩兲exp − n=1

冊

n 2␲ 2t , Pr Re

共33兲

with ␪n = 2J关1 − 共−1兲n兴 / 共n␲兲3. Under the conditions when the inner electrical potential is much smaller than the thermal energy of the ions, i.e., 兩ez␾ / kBT兩 Ⰶ 1 共or ␣␸ Ⰶ 1兲, Eq. 共32a兲 can be approximated by

⳵ 2␸ = a 2␸ , ⳵ ␩2

共34兲

where a = ␬h / 冑1 + ␪. Unlike the isothermal case, Eq. 共34兲 is still nonlinear as a depends on ␪ which is a function of ␩. Therefore, it is still difficult to solve Eq. 共34兲 analytically. However, if we assume that the EDL is much thinner than the channel width 共␬h Ⰶ 1兲, the solution to Eq. 共34兲 can be well approximated by

冋 冉 冊册

1 2 cosh共a/2兲

cosh a ␩ −

␸=

共35兲

.

With the same assumption, the solution to Eq. 共32b兲 can be approximated by ⬁

U共␰, ␩,t兲 = 1 − ␸ +

冋

Ansin共n␲␩兲exp 兺 n=1

−

册

n 2␲ 2t , Re

共36兲

with An = −2a2关1 − 共−1兲n兴 / 关n␲共n2␲2 + a2兲兴. We now validate the FDLBM presented in Sec. III using this problem for ␬h = 40. The computational mesh is generated by

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FIG. 3. Comparison of the LB and finite-volume results for J = 0.825 and kh = 30 共⌿ = ze␾ / kBT0兲. Lines, finite-volume results 共Ref. 16兲; symbols, LB results. Dash-dot line and 〫, velocity; dashed line and ⴰ, electrical potential; solid line and 䊐, temperature.

with this mesh. It is seen from Fig. 3 that the present LB algorithm predicts nearly the same distributions as reported in Ref. 16. B. Electro-osmotic/pressure driven flow

FIG. 2. Comparison of the LB and analytical results at different times for the transient electro-osmotic flow in a channel. 共a兲 velocity and 共b兲 temperature. Solid lines, analytical solutions; symbols, LB results.

xi = i

L , Nx

yj =

冉

tanh关c共2j/Ny − 1兲兴 h 1+ 2 tanh共c兲

冊

共37兲

for i = 0 , . . . , Nx and j = 0 , . . . , Ny, where Nx and Ny are the grid numbers in the x and y directions, respectively; c ⬎ 0 is a parameter that controls the grid distribution. Here, we choose c = 2.5 and Nx ⫻ Ny = 100⫻ 40 such that there are about six grid points in the EDL for ␬h = 40. The three distribution functions are initialized to take their equilibrium values with the initial density, velocity, temperature, and electrical potential, respectively. Periodic boundary conditions are applied to the inlet and outlet for f i, gi, and hi. At the top and bottom plates, the nonequilibrium extrapolation method described in Sec. IV is applied to specify the three DFs, in which we use ␳*共xb兲 = ␳共x f 兲, u*共xb兲 = 0, T*共xb兲 = T0, and ␾*共xb兲 = ␨ in the computations of the corresponding equilibria. A number of simulations were carried out for different dimensionless parameters with a time step set at ⌬t = 10−6. It is found that the numerical results agree very well with the analytical solutions. As an example, we present in Fig. 2 the velocity and temperature profiles at different times for the case of ␣ = 0.01, ␤ = 1.6⫻ 105, Pr= 1.2, Re= 1.0, and J = 1.0. We also note that in all simulations the time electrical potential ␸ changes very slightly with time for this case, which is consistent with the analysis presented earlier. We also simulated a mixed electro-osmotic/pressure driven channel flow as described in Ref. 16. Figure 3 shows the velocity, temperature, and electrical potential distributions across the channel center for the case of J = 0.825 and kh = 30. The mesh employed is defined by Eq. 共37兲 with c = 2.0, so that there are nine computational grid points in the EDL. We found that we can obtain grid-independent results

We now study the Joule heating effect in a mixed electro-osmotic/pressure driven flow in a channel. The geometry is the same as shown in Fig. 1. Again, the top and bottom walls are assumed to be isothermal, and the solution with temperature T0 enters the channel with a parabolic velocity profile. For a channel with a sufficient length, both the flow and temperature profiles at the channel exit can be assumed to be fully developed. Therefore, the dimensionless boundary conditions for velocity, temperature, and electrical potential, can be expressed as inlet: U = 4U0␩共1 − ␩兲, V = ␪ = 0,

outlet:

⳵U ⳵V ⳵␪ = = = 0, ⳵␰ ⳵␰ ⳵␰

⳵ 2␸ = 0, ⳵␰2

⳵␸ = 0, ⳵␰

top/bottom walls: U = V = ␪ = 0, ␸ = 1.

共38a兲

共38b兲 共38c兲

Note that the condition for ␸ at the inlet is equivalent to ⳵2␸ / ⳵␩2 = ␤ sinh共␣␸兲, which has been used in Ref. 16. Alternatively, Tang et al. directly set ␸ = 0 at the inlet.17 Our calculations indicate that the differences between the results using these three different boundary conditions for ␸ are negligible small. The boundary conditions for the three distributions, f i, gi, and hi, are specified according to the nonequilibrium extrapolation method described in Sec. IV, where the boundary conditions 共38兲 are imposed through the equilibrium parts of the DFs. For instances, at the top and bottom, we still set ␳共i , 0兲 = ␳共i , 1兲 and ␳共i , Ny兲 = ␳共i , Ny − 1兲 for i = 0 , . . . , Nx; but at the inlet, we use a second-order extrapolation scheme to determine the density, i.e., ␳*共0 , j兲 = 2␳共1 , j兲 − ␳共2 , j兲, since the density is expected to be nonuniform along the channel. For the electrical potential distribution, we set ␾*共0 , j兲 = 2␾共1 , j兲 − ␾共2 , j兲 according to the boundary condition at the inlet, and at the outlet, we set ␺*共Nx , j兲 = ␺共Nx − 1 , j兲 where ␺ = u, T, ␳, or ␾. While the other variables in the EDF are taken from the given boundary conditions 共38兲 directly.

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Guo, Zhao, and Shi

FIG. 4. Temperature distribution across the channel at ␰ = 5 for the mixed electro-osmotic/pressure flow.

The dielectric, viscosity, thermal conductivity, and electrical conductivity of the electrolyte are assumed to be functions of the temperature.33 They are given by

冉 冊 冉

␧ = 305.7 exp −

T , 219

冊

共39兲

1713 ␮ = 2.761 ⫻ 10−6exp − , T and k = 0.61 + 0.0012共T − T0兲, K = K共T0兲关1 + 0.025共T − T0兲兴,

共40兲

where T is in kelvin. The dimensionless parameters are set as follows: Reference temperature T0 = 298, Prandtl number Pr= 7.2, Reynolds number Re= 0.01. Other parameters are ␣ = 1.0 and ␤ = 900, and therefore ␬h = 30. Simulations of the problem using the proposed FDLBM were carried out on the nonuniform mesh defined by Eq. 共37兲 with c = 2.0. With these parameters, a number of simulations were conducted for U0 = 0.5 共dimensionless兲 and various values of J to study the Joule heating effect on the flow behavior. In each run, the steady state of the flow is obtained after a number of iterations. The steady normalized temperature profiles across the channel at ␰ ⬅ x / h = 5.0 are presented in Fig. 4 for different J. It is seen from Fig. 4 that as there is no Joule heating 共J = 0.0兲, the temperature distributes uniformly over the entire channel; for nonzero J, the temperature pro-

FIG. 5. Viscosity distribution across the channel at ␰ = 5 for the mixed electro-osmotic/pressure flow.

FIG. 6. Velocity distribution across the channel at ␰ = 5 for the mixed electro-osmotic/pressure flow.

file takes a parabolic shape and the peak occurs at the channel center. Furthermore, the magnitude of the peak temperature increases as J increases from 0 to 1.2. The viscosity distributions across the channel are plotted in Fig. 5 for different values of J, and the corresponding velocity distributions are shown in Fig. 6. The Joule heating effects on the flow behavior are clearly demonstrated. The viscosity gradient is caused by its temperature-dependent nature. The Joule heating effect further affects the velocity distribution through the temperature-dependent viscosity, because, for an electro-osmotic force dominated flow, the flow near the wall is predominantly driven by the electro-osmotic force, while the flow in the center region of the channel is driven by the fluid near the walls through the viscous drag force. Therefore, the velocity in the center region becomes smaller as the local viscosity becomes smaller due to the Joule heating effect, as shown in Fig. 6. On the contrary, the fluid near the wall moves faster since the flow rate at the entrance is fixed. It is interesting to note that as J is above a critical value, a counter flow is observed in the center region. We note that similar results were also reported in a previous study.16 The velocity and temperature distributions along the channel center are plotted in Fig. 7. An entrance effect is observed for the problem under consideration. As shown in Fig. 7, the velocity and temperature change rapidly in the inlet region along the channel center and both become nearly flat through most portions of the channel. It is also seen that the velocity and temperature gradients in this entrance region increases as J becomes large. We wish to point out that such “entrance effects” may be due to the specific boundary conditions described earlier.

FIG. 7. Temperature and velocity distribution along the centerline of the channel. Solid line, temperature; dashed line, velocity.

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Electro-osmotic flows in microfluidic devices

FIG. 8. Electrical potential distribution across the channel at ␰ = 5 共large兲 and along the horizontal line at ␩ = 0.022 in EDL 共inset兲.

FIG. 10. Normalized pressure distribution along the centerline of the channel.

冉 冊

冉 冊

⳵U ⳵P ⳵ ␣␸ ¯ = ␮ + ␤ sinh . ⳵␰ ⳵ ␩ ⳵ ␩ 1+␪

The electrical potential distributions across and along the channel for different values of J are plotted in Fig. 8. Although the differences in electrical potentials for different J are rather small 共within 5% for the cases considered兲, the differences can still be identified from the inset, where the electrical potentials in the EDL 共␩ = 0.022兲 are plotted against channel length. The inset in Fig. 8 indicates that the electrical potential in the entrance region increases rapidly with channel length. This is due to the rapid increase in temperature in the entrance region. This is evident from the fact that in the case when J = 0 no increase in the electrical potentials is observed. In the fully developed region, ␸ is also nearly constant for each value of J. Accordingly, the net charge density ␳e for different J shows small differences in the downstream far from the entrance for different J, as shown in Fig. 9, and the entrance effect is also observed in the EDL. For the pressure field, we find that the pressure is uniform across the channel in all cases under consideration. In Fig. 10 we present the normalized pressure P* along the channel center for different J, where P* = cs2关␳ − ␳共L兲兴 / ␳0us2. It is seen that except in the entrance region, the pressure in the large downstream region increases linearly for each value J. The Joule heating effect on the pressure distribution is also clearly shown in Fig. 10. The magnitude of the pressure gradient decreases as J increases for a fixed flow rate at the entrance. This can be interpreted as follows. For this mixed electro-osmotic/pressure flow, the fluid is driven by the combination of the pressure gradient, the viscous drag force, and the applied electric force,

Note that as discussed earlier, the electrical potential ␸ depends only slightly on J 共see Fig. 8兲, and therefore the second term on the right-hand side of Eq. 共42兲 can be considered as a constant that is independent of J. The first term on the right-hand side of Eq. 共42兲, however, strongly depends on J, as shown in Fig. 5. Specifically, the gradient ⳵U / ⳵␩ at ␩ = 0 increase as J increases and, therefore, the pressure gradient ⳵ P / ⳵␰ decreases as J increases. We wish to point out that although the pressure drop across the channel is large, the density varies slightly in the whole domain. For example, as J = 0, the relative difference in density between the channel inlet and outlet, 关␳共0兲 − ␳共L兲兴 / ␳0, is only 9 ⫻ 10−6, which indicates that the fluid is nearly incompressible. Besides the viscosity, the other properties of the electrolyte, including the thermal conductivity, the dielectric permittivity, and the electric conductivity, are also functions of the temperature. Therefore, they depend on the applied electrical field, or the Joule number. As an example, in Fig. 11, we plotted the variation of the electric conductivity K at the channel center with the Joule number J. It is seen that K takes a parabolic profile across the channel, and the peak increases dramatically as J becomes larger. For instance, as J = 1.2, the Joule heating effect can lead to an increase over

FIG. 9. Net charge density distribution along the horizontal line at ␩ = 0.022 in EDL.

FIG. 11. Electric conductivity distribution across the channel at ␰ = 5 for the mixed electro-osmotic/pressure flow.

Re

共41兲

Integrating this equation over ␩ from 0 to 1 and using the symmetric property of the flow, we obtain Re

冏 冏 冕

⳵P ⳵U ¯0 = − 2␮ ⳵␰ ⳵␩

1

+

␩=0

0

冉 冊

␤ sinh

␣␸ d␩ . 1+␪

共42兲

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Guo, Zhao, and Shi

400% in the central region of the channel. This fact indicates that the Joule heating produced in the central region is much higher than that in the EDL. Finally, we wish to emphasize that the LBE algorithm for the Poisson–Boltzmann equation is efficient for the problem under consideration. For instance, for the case of J = 0.825, if we initialize ␸ = 0 in the whole domain except at the boundary, it takes about 2 ⫻ 104 iterations to obtain the converged ␸ in the first time step, where we assume that the steady state is reached as the change in the 共dimensionless兲 local electrical potential is less than 0.1% in two consecutive iterations. After that, only one iteration is needed to reach the steady state in each time step. We also find that the number of iterations for solving the Poisson–Boltzmann equation in the first step depends on the initial values of ␸. If we initialize ␸ according to Eq. 共35兲, the number of iterations in the first time step is about 2 ⫻ 103.

VI. SUMMARY

Understanding the electro-osmotic flow in a microchannel is of both fundamental and practical significance for the design and optimization of various microfluidic devices. Joule heating always exists in EOFs and may have a significant influence on the flow behavior. In this paper, we have proposed a finite-difference-based lattice Boltzmann algorithm for electro-osmotic flows where the Joule heating effect is considered. The method can be viewed as a solver for the governing equations for EOF. We have validated the method using EOF in a channel that has an approximated analytical solution. The algorithm was also applied to a mixed electro-osmotoc/pressure driven flow to study the Joule heating effect, and the numerical results indicate that Joule heating plays a significant role in EOF and should not be neglected, especially as the applied external electric field is strong. These results demonstrate that the present FDLBM can serve as a potential numerical tool to study electroosmotic flows in microfludics.

ACKNOWLEDGMENTS

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