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  Analytica Chimica Acta 507 (2004) 27–37 Numerical analysis of the thermal effect on electroosmotic flowand electrokinetic mass transport in microchannels G.Y. Tang, C. Yang ∗ , C.K. Chai, H.Q. Gong School of Mechanical and Production Engineering, Nanyang Technological University, Singapore 639798, Singapore Received 25 August 2003; received in revised form 24 September 2003; accepted 30 September 2003 Abstract Joule heating is present in electrokinetically driven flow and mass transport in microfluidic systems. Nowadays, there is a trend of replacingcostly glass-based microfluidic systems by the disposable, cheap polymer-based microfluidic systems. Due to poor thermal conductivity of polymer materials, the thermal management of the polymer-based microfluidic systems may become a problem. In this study, numericalanalysis is presented for transient temperature development due to Joule heating and its effect on the electroosmotic flow (EOF) and massspeciestransportinmicrochannels.TheproposedmodelincludesthecouplingPoisson–Boltzmann(P–B)equation,themodifiedNavier–Stokes(N–S) equations, the conjugate energy equation, and the mass species transport equation. The results show that the time development for boththe electroosmotic flow field and the Joule heating induced temperature field are less than 1s. The Joule heating induced temperature field isstronglydependentonchannelsize,electrolyteconcentration,andappliedelectricfieldstrength.ThesimulationsrevealthatthepresenceoftheJoule heating can result in significantly different characteristics of the electroosmotic flow and electrokinetic mass transport in microchannels.© 2003 Elsevier B.V. All rights reserved. Keywords:  Joule heating; Electroosmotic flow; Capillary electrophoresis; Electrokinetic mass transport; Microfluidics thermal management 1. Introduction During recent years, due to the rapid development of Lab-on-a-chip (or Biochip) technology, the electroosmosisis being extensively utilized as the driving forces to manipu-late liquid flows and to transport and control liquid samplesof nanovolumes in microfluidic devices used for chemicaland biological analysis and medical diagnosis [1–4]. In theliterature, a great deal of information has been generatedon electroosmotic flow (EOF) in microcapillaries of variousgeometric domains such as cylindrical capillary [5], annu- lus [6], elliptical pore [7], slit parallel plate [8], rectangular microchannel [9], and T shape and Y shape microchannel structures [10]. Moreover, electrophoretic transport has been intensively investigated both theoretically and experimen-tally. Many studies have been reported on sample transportand mixing [11–15] based on capillary electrophoresis (CE). However, all aforementioned studies of the EOF and CEhave exclusively neglected the Joule heating effect. It is wellknown that the so-called Joule heating is generated when ∗ Corresponding author. Tel.:  + 65-6790-4883; fax:  + 65-6791-1859.  E-mail address:  mcyang@ntu.edu.sg (C. Yang). an electric field is applied across conductive liquids. SuchJoule heating not only can cause temperature increase butalso may create temperature gradient. The change of liquidtemperature and the presence of temperature gradient wouldhave an impact on the EOF and impose limitations on sepa-ration performance in the CE [16,17]. Variations in solutiontemperature affect many important parameters, including thepH of the buffer, peak shapes, migration times and repro-ducibility [18]. Previous studies have amply demonstrated that the Joule heating can result in low column separationefficiency, reduction of analysis resolution, and even lossof injected samples. Furthermore, temperature rise can giverise to the decomposition of thermally labile samples andformation of gas bubbles as well.In the literature, studies of the Joule heating and itseffect on sample separation have been reported using the-oretical and experimental approaches. Jones and Grushka[19] used a simplified one-dimensional model to verifythat the parabolic temperature profile is valid for low in-put powers, but the parabolic profile is distorted for highinput powers. Bosse and Arce [20,21] carried out system-atic studies of Joule heating effect on dispersive mixing inelectrophoretic cells, focusing on both hydrodynamic and 0003-2670/$ – see front matter © 2003 Elsevier B.V. All rights reserved.doi:10.1016/j.aca.2003.09.066  28  G.Y. Tang et al./Analytica Chimica Acta 507 (2004) 27–37  Nomenclature c p  specific heat capacity of the electrolyte(J/kgK) C   ionic concentration of the electrolyte (M) C  i  mass concentration of the  i th specie (M)CE capillary electrophoresisCFD computational fluid dynamics  D i ( T  ) diffusion coefficient of the  i th specie (m 2  /s) e  fundamental charge (1 . 602 × 10 − 19 C)  E   strength of the applied electric field (V/m)EDL electric double layerEOF electroosmotic flow h  heat transfer coefficient outsidemicrochannel (W/m 2 K)  H   half height of the microchannel (m) k  b  Boltzmann constant (1 . 38 × 10 − 23 J/K) k  l ( T  ) thermal conductivity of the electrolytesolution (W/mK) k  s  thermal conductivity of the capillary wall(W/mK)  L  microchannel channel length (m) n + ,  n −  local number concentration of cations andanions, respectively (m − 3 ) n 0  bulk number concentration of ions in theelectrolyte solution (m − 3 )  p  hydrodynamic pressure (Pa) ˙ q  Joule heat generation (W/m 3 ) t   time (s) T   temperature (K) u  electroosmotic flow velocity in  x  direction (m/s) u mi  electrophoretic velocity of the  i th specie in  x  direction (m/s) U  ref   reference velocity (m/s) U  s  Smoluchowski velocity (m/s) v  electroosmotic flow velocity  y direction (m/s) v mi  electrophoretic velocity of the  i th specie in  y  direction (m/s)  x ,  y  coordinate in  x  direction and  y  direction, respectively  z i ,  z  charge valence of ions of the electrolytesolution Greek symbols ∆  microchannel wall thickness (m) ε 0  permittivity of vacuum(8 . 85 × 10 − 12 C/Vm) ε ( T  ) relative dielectric constant of the electrolyte ζ   zeta potential of the capillary wall (V) η + ,  η −  the number of moles of cation and anion inthe electrolyte, respectively (M) λ  electrical conductivity of the electrolytesolution (S/m) λ + ,  λ −  equivalent ionic conductivity of the cationsand anions, respectively (m 2 S/mol) µ ( T  ) viscosity of the electrolyte solution (Pas) µ ( mi ) electrophoretic mobility of the  i th speciein  x  direction (m 2  /Vs) ν  kinematic viscosity of the electrolytesolution (m 2  /s) ν mi  electrophoretic velocity of the  i th specie in  y  direction (m 2  /Vs) ρ  density of the electrolyte solution (kg/m) ρ e  local net electrical charge density in theelectrolyte solution (C/m) ψ  local electrical potential (V) Subscript  0 variables at room temperatureconvective-diffusive transport aspects. However, their anal-ysis is only limited to electrophoretic cells where the twoends of the cell are closed. Furthermore, their models donot include electroosmotic flow.Many experimental studies of Joule heating in capillaryelectrophoresis have been reported since Knox and Mccor-mack  [22]. Swinney and Bornhop [23] measured the Joule heating in a chip-scale CE by using a novel picoliter volumeinterferometer. Their results showed that the Joule heatingeffect on the CE has been underestimated and hence thereis a need to readdress theoretical modeling. The powerlimits for optimal CE have been defined by Sepaniak andCole [24] as  < 1W/m with effective electrophoretic separa-tion unlikely to occur beyond 2–3W/m. Nelson et al. [25]devised the “Ohm’s Law Plot” as a means of determiningthe “functional” buffer concentration and maximum appliedvoltage that could be utilized for CE. A linear relation-ship between the applied voltages and the system currentindicated that the generated Joule heat is being effectivelydissipated while deviation from linearity suggests that thethermostatic capacity of the system has been exceeded.Nonetheless, all these studies on the Joule heating effectare either empirical or are based on simplified theories. Inthe present study, a mathematical model on Joule heatingand its effects is developed including the coupling Poisson–Boltzmann (P–B) equation, modified Navier–Stokes (N–S)equation, conjugate energy equation and is solved numeri-cally using a finite-volume-based CFD technique. The timeand spatial developments of temperature and electroosmoticflow characteristics are analyzed, and their effect on elec-trokinetic transport of mass species is discussed. Further, inview of the fact of fast growing use of the disposable, cheappolymer-based microfluidic systems, computations for EOFvelocity profiles, temperature fields and sample concentra-tion distributions in glass and PMMA polymer-based mi-crochannels are performed under influences of the channelsize, electrolyte concentration, and electric field strength.  G.Y. Tang et al./Analytica Chimica Acta 507 (2004) 27–37   29 2. Problem formulation In electroosmotic flow, the motion of the fluid is mainlycaused by an applied electric field. No external pressuredifference between the inlet and outlet reservoirs is im-posed. The model developed in this study includes thePoisson–Boltzmann equation governing the electrical po-tential distribution, the modified Navier–Stokes equationsdescribing the motion of electrolyte driven by electrokineticbody forces, and the energy equation governing the temper-ature field due to Joule heating. Since the thermophysicalcoefficients including the dielectric constant in the P–Bequation, and the liquid viscosity in the N–S equations andthe energy equation are temperature dependent, the P–Bequation, the N–S equations, and the energy equation arestrongly coupled. Moreover, in view of the fact that elec-troosmotic flow is mainly used for delivering and separatingchemical or biological samples, the mass species transportequation is therefore included in the model development toexamine the Joule heating effect. 2.1. The Poisson–Boltzmann equation Consider an electroosmotic flow in a microchannel withheight 2  H  , length  L  and wall thickness  ∆  as shown in Fig. 1.When a solid surface is in contact with a polar medium, thesurface usually becomes charged [26]. Due to electrostaticinteractions, both co-ions and counter-ions are preferentiallyredistributed near the charged surface, forming an electri-cal double layer (EDL). According to the theory of electro-statics [1,26], the electrical potential distribution,  ψ(x,y)  isgoverned by the Poisson equation. ∂∂x  ε(T)∂ψ∂x  + ∂∂y  ε(T)∂ψ∂y  =− ρ e ε 0 (1)where  ε  is the dielectric constant of the electrolyte (assumedthe same as water) and is considered as a function of tem-perature ( T   in Kelvin) here, ε(T) = 305 . 7exp  − T  219   (1a) Fig. 1. Schematic diagram of the microchannel with a Cartesian coordinatesystem. Forasymmetricelectrolyte(i.e.theco-ionsandcounter-ionshave same charge valence,  | z + | = | z − | =  z ), its ionic con-centration distributions for both anions and cations are as-sumed to follow the Boltzmann distributions [27]. n +  = n 0  exp −  ze ψ/k b T  , n −  = n 0  exp  ze ψ/k b T  (2)Then the local charge density,  ρ e  is given by: ρ e (x,y) =  ze (n + − n − ) =− 2  zen 0  sinh   ze ψk b T    (3)Combining Eqs. (1) and (3), we obtain the Poisson–Boltzmann equation ∂∂x  ε(T)∂ψ∂x  + ∂∂y  ε(T)∂ψ∂y  = 2  zen 0 ε 0 sinh   ze ψk b T    (4) 2.2. The modified Navier–Stokes equations Different from the conventional pressure-driven flows,the driving force of the electroosmotic flow is due to theinteraction between net charge density in the EDL regimeand the applied electric field. Therefore, the Navier–Stokesequations including the continuity and momentum equa-tions describing a laminar, incompressible, steady flow of anelectrolyte solution with temperature-dependent viscosityare modified to [1]Continuity equation: ∂u∂x + ∂v∂y = 0 (5)Momentum equations: ρ  ∂u∂t  + u∂u∂x + v∂u∂y  =− ∂p∂x + ∂∂x  µ(T)∂u∂x  + ∂∂y  µ(T)∂u∂y  + ρ e  E − ∂ψ∂x   (6a) ρ  ∂v∂t  + u∂v∂x + v∂v∂y  =− ∂p∂y + ∂∂x  µ(T)∂v∂x  + ∂∂y  µ(T)∂v∂y  − ρ e ∂ψ∂y (6b)where  u  and  v  are the velocity of the EOF in  x  and  y  di-rections, respectively,  t   the time,  ρ  the density of solutionand is regarded as a constant in this study,  P  the hydrody-namic pressure, and  E   the strength of an externally appliedelectric field along  x  direction, and is assumed as constant.Strictly speaking, the assumption of constant field strength,  E   or uniform electric field is not valid due to Joule heatingeffect. The presence of Joule heating would produce a tem-perature variation along  x  direction, and hence it can causea variation of the electrical conductivity of electrolyte along  x  direction, resulting in non-uniform electric field.  µ ( T  ) is  30  G.Y. Tang et al./Analytica Chimica Acta 507 (2004) 27–37  the temperature-dependent viscosity of electrolyte solution(here being assumed the same as water), µ(T) = 2 . 761 × 10 − 6 exp  1713 T    (6c) 2.3. Energy equation The energy equation taking into  account   of temperature-dependent thermal conductivity and heat source can be ex-pressed as ρc p  ∂T ∂t  + u∂T ∂x + v∂T ∂y  = ∂∂x  k l (T)∂T ∂x  + ∂∂y  k l (T)∂T ∂y  +˙ q  (7)where  T   is the temperature in Kelvin,  c p  and  k  l ( T  ) the spe-cific heat capacity and the temperature-dependant thermalconductivity of electrolyte solution, respectively. The Jouleheating ( ˙ q ) includes two parts. One is due to the appliedelectric field imposing on the conductive solution ( Eλ ), theother is due to the net charged density moving with the fluidflow ( uρ e ). Therefore according to the Ohm’s law, the heatgeneration due to Joule heating can be expressed as ˙ q = (uρ e + Eλ) 2 λ (8a)where  λ  is the electrical conductivity of the electrolyte andis given as: λ = λ + η + + λ − η −  (8b) λ = λ + (T)η + + λ − (T)η −  (8c) λ + (T) = λ + 0 + 0 . 025 λ + 0 (T   − 298 )  (8d) λ − (T) = λ − 0 + 0 . 025 λ − 0 (T   − 298 )  (8e)Here,  λ + (T)  and  λ − (T)  are ionic conductivity of the cationsand anions of the electrolyte at temperature  T  ,  λ + 0  and  λ − 0 are ionic conductivity of the cations and anions of the elec-trolyte at the room temperature (i.e. 298K), respectively,and  η +  and  η −  respectively denote the mole concentrationof the cations and anions of the electrolyte.In general, the temperature at the inner capillary wall isunknown. Since the heat generated by Joule heating in theelectrolyte solution is mainly dissipated through the capil-lary wall to the surrounding environment, a conjugate heattransfer problem has to be solved to simultaneously accountfor heat transfer in both the solution and the capillary wall[27]. The governing equation for heat conduction in the mi-crochannel wall is expressed as ρ s C ps ∂T  s ∂t  = ∂∂x  k s ∂T ∂x  + ∂∂y  k s ∂T ∂y   (9)where  ρ s ,  C ps , and  k s  are the density, specific heat capacity,and thermal conductivity of the channel wall, respectively,and  T  s  the channel wall temperature. 2.4. Mass species transport equation Electroosmotic flows are often used for injection and sep-aration of biological or chemical samples, such as the sep-aration of DNA [2]. Another objective of this research is to study the role of Joule heating in the mass species transport.Consider a sample species of interest needs to be trans-ported from one reservoir to the other reservoir in the mi-crochannel that is filled with an electrolyte solution. Foranalysis, assumptions are made for no adsorption of sam-ple species onto the microchannel wall and no interactionbetween sample species and the electrolyte solution com-ponents. As the species transported by electrokinetic meansin general is accomplished by three mechanisms includingconvection, diffusion, and electrophoresis, the mass trans-port equation can be formulated as [13] ∂C i ∂t  + (u + u mi )∂C i ∂x + (v + v mi )∂C i ∂y = ∂∂x  D i (T)∂C i ∂x  + ∂∂y  D i (T)∂C i ∂y   (10)where  C i  is the  i th sample species concentration,  D si (T)  isthe temperature-dependent mass diffusivity of the  i th samplespecies, and it can be expressed as D si (T) = D si 0 + 0 . 0309 D si 0 (T   − 298 )  (10a)where  D si 0  is diffusion coefficient of corresponding ions atthe room temperature (i.e. 298K) and  u mi  and  v mi  are thecomponents of the electrophoretic velocity in  x  direction and  y  directions, respectively, they can be expressed as u mi  = µ mi ×  E − ∂ψ∂x   (10b) v mi  = ν mi ×  − ∂ψ∂y   (10c)where  µ mi  and  ν mi  are the components of the electrophoreticmobility in  x  and  y  directions, respectively. According to[28], they can be expressed as µ mi  = µ mi 0 + 0 . 025 µ mi 0 (T   − 298 )  (11a) ν mi  = ν mi 0 + 0 . 025 ν mi 0 (T   − 298 )  (11b)where  µ mi 0  and  ν mi 0  are the components of the elec-trophoretic mobility in  x  and  y  direction at room temperature(i.e. 298K), respectively. 2.5. Initial and boundary conditions The above equations are subject to the following initialand boundary conditions:The initial conditions T   = T  0 , ψ = 0 , u = 0 , v = 0 , C i  = 0 . 0 (12)
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