Institute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan, Republic of China
共Received 26 March 2008; accepted 24 July 2008; published online 25 September 2008兲
The working principle of immunoassays is based on the specific binding reaction of an analyte-ligand protein pair in physiological environments. However, for a diffusion-limited protein, the diffusion boundary layer of the analyte on the reaction surface of a biosensor would hinder the binding reaction from association and dissociation. The formation of such association and dissociation layers thus limits the response time and the overall performance of a biosensor. In this work we have performed a two-dimensional full time scale finite element simulation on the binding reaction kinetics of two commonly used proteins, C-reactive protein共CRP兲 and immunoglobulin G 共IgG兲. By applying a nonuniform ac electric field to the flow microchannel of the biosensor, the electrothermal force can be generated to induce a pair of vortices to stir the flow field. With the aid of the vortices and a suitable choice of the location of the biosensor, the fluids flowing over the reacting surface can be accelerated fast enough to depress efficiently the growth of the diffusion boundary layer on the reaction surface, and enhance the association or dissociation of analyte-ligand complex. The interference patterns of the flow field due to the existence of the sensor at different locations of the microchannel could cause different degrees of enhancement to the association and the dissociation. By changing the location of the sensor the largest enhancement is found at the position near the negative electrode. For the configuration of the microchannel we studied, the initial slope of the curve of the analyte-ligand complex versus time can be raised up to 5.17 for CRP and 1.93 for IgG in association, and 3.74 for CRP and 1.28 for IgG in dissociation, respectively, under the applied ac field 15 Vrmspeak-to-peak and operating frequency 100 kHz. At this optimal sensor location, we also studied the effect of various settings of temperature boundary conditions on the top and bottom walls, including the two limiting cases, namely, constant temperature and thermal insulation on both walls. We show that varying the temperature boundary conditions can cause an essential effect on the enhancement of the binding reaction and can be employed to find an optimal binding enhancement. Utilizing these simulation results, an improved design incorporating a pair of electrodes and a neck region near the reaction surface is demonstrated. The sensor is fixed to locate at the middle of the bottom side. With the existence of the stirring flow field, the association rate of the 30 m neck is 2.73 times faster than that of the original channel with no neck. © 2008 American Institute of Physics.关DOI:10.1063/1.2981195兴
I. INTRODUCTION
Recent rapid advances in micro-/nanotechnologies have given impetus to the development and design of microfluidic biosensors for health-care applications, such as immunoas-says. The possibility of fabricating and integrating microsize biosensors of multiple functions has led to the idea of per-forming real-time monitoring or diagnostics on a portable laboratory on a chip. The three most commonly used devices in detecting biomolecules are the microcantilever beam based biosensor, the surface plasmon resonance 共SPR兲 sen-sor, and the quartz crystal microbalance 共QCM兲 sensor. Al-though the mechanisms of detection are different, they all involve the same kinetics of specific binding of analytes, such as C-reactive protein1 共CRP兲 and immunoglobulin G 共IgG兲, and immobilized ligands, such as CRP and anti-IgG. The concentration of the formed analyte-ligand
com-plex during the chemical binding is the major quantity cor-relating with the measured data from various sensors, and reflects the concentration of analytes in the flow, which is the physical quantity most concerned in clinic applications.
The specific recognition of analytes and immobilized ligands occurs at the reaction surface of a biosensor, which is a solid-liquid interface. The reaction kinetics can be de-scribed as a two-step process.2
共1兲 Mass-transport process: the analyte is transported by dif-fusion from the bulk solution toward the reaction surface 关A兴bulk 关A兴surface.
共2兲 Chemical reaction process: the binding of the protein pair takes place
关A兴surface+关B兴
kd ka
关AB兴,
where the concentration关A兴bulkis for the analyte in the bulk, 关A兴surfaceis for the analyte at the reaction surface, a兲Electronic mail: [email protected].
b兲Electronic mail: [email protected].
关B兴 is for the ligand, and 关AB兴 is for the analyte-ligand complex, respectively. Here ka is the association rate
constant, and kd is the dissociation rate constant.
When the flow rate 共hence the convective velocity for analyte兲 in a biosensor is fixed, the required experiment time of a specific biomolecular recognition usually depends on its Damköhler number 共Da number兲, which is a dimensionless
parameter to measure whether a reaction is diffusion limited or reaction rate limited. The Danumber is the ratio of
reac-tion velocity 共i.e., product of the association rate constant and the initial concentration of the ligand兲 to diffusion veloc-ity共i.e., ratio of the diffusion coefficient of the analyte in the buffer flow to the height of microchannel兲.3
When the Da
number is greater than unity, the whole reaction is restrained by diffusion, i.e., diffusion transport limited.
The development of immunoassays is to place emphasis on the high sensitivity and the real-time detecting ability for different mechanism-based biosensors. When the analyte takes a longer time to transport by convection and diffusion to the reaction surface than chemical reaction, the whole re-action is restrained by mass-transport limitation and it usu-ally causes the formation of a diffusion boundary layer.4The formation of such layers would limit the response time and the overall performance of the biosensor. In practice, it often takes hours to complete a detection cycle, which is the main technical problem to be solved. Sigurdson et al.5 found that the electrothermal microstirring effect in a microchannel is useful to enhance the binding efficiency for diffusion-limited molecules共large Daproteins兲. It is promising in our previous
study6 that several fabrication parameters can be tuned to improve the performance of a biosensor.
Microelectrode structures are commonly used in ac elec-trokinetics to generate the high strength ac electric field, which is required to move suspended particles in liquid.7 Particles with a wide range of sizes have been dielectro-phoretically manipulated in this manner, from cells 共⬃10 m兲 and bacteria down to viruses 共⬃100 nm兲 and protein molecules.8,9Ac electrokinetics can be classified into three kinds of force: dielectrophoresis, electrothermal force, and electro-osmosis.10 While a nonuniform ac electric field can move suspended particles using dielectrophoretic forces, it can also move the fluid through the electrothermal effect or ac electro-osmosis.11–13Ac electro-osmosis is only influential at the frequency below 10 kHz and is likely to induce hy-drolysis at the surface of microelectrodes at low frequency.14 Dielectrophoresis does not significantly affect the motion of the particle at the submicrometer scale. In contrast, the elec-trothermal effect is dominant in the bulk fluid at higher frequency.15Therefore, only the electrothermal force is dis-cussed for the simulation of protein immunoassay in our study.
In this paper we simulate a two-dimensional 共2D兲 full time scale process of association and dissociation in a bio-sensor immunoassay with the samples of CRP pairs and IgG pairs to predict the surface concentration of the analyte-ligand complex versus time relationship. The induced elec-trothermal force by the ac electric field can cause a vortex field which will reduce the thickness of the diffusion
bound-ary layer and significantly increase the reaction rate to accel-erate both the association and dissociation processes. Inter-ference patterns of the vortex field due to the existence of the sensor at various locations in the microchannel could cause different degrees of enhancement of association and disso-ciation and will be discussed in this work.
Due to the nature of the detecting mechanism, many biosensors 共such as SPR and QCM兲 place the reacting sur-face in the bottom position of the microchannel. As an illus-trating example of designing a biosensor with better perfor-mance, we propose a neck design of the microchannel in addition to including the electrothermal effect by introducing a pair of electrodes on the top position of the microchannel opposite to the reacting surface. The neck design of the mi-crochannel reduces the distance of diffusion of analyte in the bulk solution to the reaction surface and also raises the ve-locity of the solution flowing over the reaction surface. Com-bining this passive mechanism by changing geometries with the active mechanism provided by the electrothermal effect, we will show that the rate of association and dissociation can be effectively raised.
II. THEORY
In this section the equations governing the electrother-mal force, the electric field, the temperature field, the flow field, the concentration field, and the biochemical reaction are described. Detailed geometry, flow properties, binding constants, electrodes, and initial and boundary conditions that are required for simulation are described in Sect III.
A. Electrothermal force
Temperature gradients occur as a result of Joule heating due to the nonuniform ac electric field. The gradient of tem-perature T in the liquid causes inhomogeneities of the per-mittivity and the conductivity of the medium, which in turn gives rise to the forces causing fluid motion. The body force FជE is given by10 FE ជ= −1 2
冋
冉
ⵜ − ⵜ 冊
· Eជ Eជ 1 +共兲2+ 1 2兩Eជ兩 2ⵜ 册
, 共2.1兲where=/is the charge relaxation time, andis angular frequency of the electric field Eជ, respectively. The local variations in temperature change the gradients of conductiv-ity and permittivconductiv-ity
ⵜ = 共/T兲 ⵜ T, 共2.2兲
ⵜ=共/T兲 ⵜ T. 共2.3兲
The force induced by the permittivity gradient is the di-electric force. The force induced by the conductivity gradient is the Coulomb force. IfⰆ/, the force is dominated by the Coulomb force. IfⰇ/, the force is dominated by the
dielectric force. For water, 1/共/T兲=
−0.4% , 1/共/T兲=2% per degree Kelvin,16 Eq. 共2.1兲 becomes
FE ៝= −1 2
冋
0.024共ⵜT · Eជ兲 Eជ 1 +共兲2 +兩Eជ兩 2 2 共− 0.004兲 ⵜ T册
. 共2.4兲B. The electric field
Because the electrothermal force is a time-averaged en-tity, it is sufficient to consider the quasistatic electric field which is the root mean square共rms兲 value of the ac field. The electrostatic field is related to the electrical potential⌽ by17
Eជ= −ⵜ⌽, 共2.5兲
and⌽ satisfies
ⵜ2⌽ = 0, 共2.6兲
where⌽ the electrical potential.
C. The temperature field
A small amount of Joule heating could give rise to a temperature increase in the fluid. In order to estimate the temperature rise for a given electrode system, the following energy balance equation must be solved:18
cp
T
t +cpVជ·ⵜT = kⵜ
2T +兩Eជ兩2, 共2.7兲
where , cp, Vជ, and k are the density of the fluid, specific
heat, velocity of the fluid, and thermal conductivity of the fluid, respectively. Here兩Eជ兩2is the Joule heating as a source term.
D. The flow field
In this work it is assumed that the fluid is incompressible so that
u x+
v
y= 0, 共2.8兲
where u andv are the x and y velocity components, respec-tively. The equations of motion are
u t +
冉
u u x+v u y冊
−ⵜ 2u +p x= FE,x, 共2.9兲 v t +冉
u v x+v v y冊
−ⵜ 2v +p y= FE,y, 共2.10兲 where is the dynamic viscosity of the fluid and p is the pressure. In this work it is assumed that the density and viscosityof the modeled incompressible fluid are constant independent of temperature and concentration.E. The concentration field
Transport of analytes to and from the reaction surface is assumed to be described by Fick’s second law with convec-tive terms 关A兴 t + u 关A兴 x +v 关A兴 y = D
冉
2关A兴 x2 + 2关A兴 y2冊
, 共2.11兲 where关A兴 is the concentration of analyte and D is the diffu-sion coefficient of analyte.F. The reaction surface
The reaction between immobilized ligand and analyte is assumed to follow the first order Langmuir adsorption model.4,19 During the reaction, the analyte-ligand complex 关AB兴 increases as a function of time according to the reaction rate
关AB兴
t = ka关A兴surface兵关B0兴 − 关AB兴其 − kd关AB兴, 共2.12兲 where关A兴surfaceis the concentration of the analyte at the re-action surface by mass-transport,关B0兴 is the surface concen-tration of the ligand, and关AB兴 is the surface concentration of the analyte-ligand complex, respectively.
III. SIMULATION DETAILS
Sketch of the 2D model of the microchannel is shown in Fig.1. In this work, the dimensions of the reaction surface of biosensor and microchannel are 40⫻3 and 500⫻150 m2, respectively. The thickness of electrode is neglected. The buffer solution mixed with the analytes flows from the left to the right. On the reaction surface are the immobilized ligands. A pair of electrodes is put on the top of the micro-channel. An ac electric field is applied through the elec-trodes.
Figure2 shows the unstructured mesh consisting of tri-angular elements generated in the calculation. It is noticed that the regions nearby the two electrodes and the reaction surface are refined with a better mesh quality.
A. The electric field configuration
In an immunoassay experiment, phosphate buffer saline is usually used to be a neutral buffer solution 共pH=7.2兲,
FIG. 1. 共Color online兲 Sketch of the 2D model. Size of reaction surface is 40⫻3 m2.
which is mixed with analytes as a carrier fluid. So the fluid in the microchannel can be assumed that its physical properties are similar to water. The relative permittivityrand the
elec-trical conductivityare 80.2 and 5.75⫻10−2 S m−1, respec-tively.
The applied voltage is 15 Vrmspeak-to-peak with an op-erating frequency of 100 kHz. Boundary conditions are ⌽ =⫾Vrms/2 at the two electrodes and electrical insulated else-where.
B. The temperature field configuration
The fluid in this domain is assumed to have properties as water. The density, specific heat, and thermal conductivity are 103 kg/m3, 4184 J/kg K, and 0.6 W/m K, respec-tively. The temperature at the inlet is maintained constant and the temperature boundary condition at the outlet is set as convective heat flux.
The parts of top wall with the horizontal location coin-ciding with the two electrodes are kept in constant tempera-ture, say T = 300 K, by using a thermoelectric cooler. Else-where on the top wall and the whole bottom wall are kept thermally insulated. It is noted that different settings for the temperature boundary conditions on the top and bottom walls will produce different temperature fields in the reaction channel20 and result in different degrees of enhancement of the binding reaction. We will discuss the effect of varying the boundary conditions on the enhancement of the binding re-action in Sec. IV.
C. The flow field configuration
The value of dynamic viscosityis set as that of water, 10−3 Pa s. Since the flow in the microchannel is in low Rey-nolds number condition, it is assumed as a laminar flow. The average velocity of the parabolic profile is set to u = 10−4 m/s at the inlet. Boundary conditions are p=0 at the outlet, and nonslip elsewhere.
D. The concentration field configuration and kinetics of the specific binding
The diffusion coefficients of human CRP 共=2.175 ⫻10−11 m2/s兲 and IgG 共=5⫻10−11 m2/s兲 are obtained from Refs.21and22. The inlet concentration of analyte is chosen as关A兴=6.4 nM. The initial surface concentration 关B0兴 is as-sumed as 1.4⫻10−8 mole/m2.23
The diffusive flux at the reaction surface should be bal-anced against the reaction rate
− D
冉
关A兴y
冊
surface= ka关A兴surface兵关B0兴 − 关AB兴其 − kd关AB兴. 共3.1兲 The association rate constant ka and dissociation ratecon-stant kd of the protein pairs can be found in Ref.24. The ka
and kd for CRP-anti-CRP binding interactions are 1.0
⫻107 M−1s−1 and 2.6⫻10−2 s−1, respectively. The k
a and
kd for IgG-anti-IgG binding interactions are 2.5
⫻105 M−1s−1and 3⫻10−4 s−1, respectively.
The initial conditions for both the concentrations of the analyte in the bulk, 关A兴, and the concentrations of the analyte-ligand complex on the reaction surface,关AB兴, are all zero, respectively.
We simulate the dissociation phase of the binding reac-tion of the two protein pairs by terminating the supply of the analyte at a time after the binding reaction is saturated. The time of discontinuing the analyte is set manually.
IV. RESULTS
The 2D simulation to evaluate the binding curves during the association and dissociation with or without an applied voltage has been performed by the finite element analysis software,COMSOL MULTIPHYSICSTM.25 The results are exam-ined to be mesh independent by means of a convergence test.
FIG. 3. 共Color online兲 The distribution of the flow velocity field when the applied voltage is 15 V with different positions of the reaction surface:共a兲 共250,1.5兲, 共b兲 共250,75兲, 共c兲 共281,133兲, and 共d兲 共277.5,145.5兲, respectively.
TABLE I. The initial slopes and enhancement factors of CRP binding reaction. CRP binding reaction
Curve Initial slope
共association兲⫻10−11 Initial slope 共dissociation兲⫻10−11 Enhancement factor (association) Enhancement factor (dissociation) 0 V,共250,1.5兲 1.80 −1.41 0 V,共250,75兲 2.48 −1.81 0 V,共281,133兲 1.15 −0.95 0 V,共277.5,148.5兲 1.80 −1.39 15 V,共250,1.5兲 2.13 −1.61 1.18 1.14 15 V,共250,75兲 4.02 −2.69 1.62 1.48 15 V,共281,133兲 5.94 −3.54 5.17 3.74 15 V,共277.5,148.5兲 3.95 −2.49 2.20 1.79
TABLE II. The initial slopes and enhancement factors of IgG binding reaction. IgG binding reaction
Curve Initial slope
共association兲⫻10−11 Initial slope 共dissociation兲⫻10−11 Enhancement factor (association) Enhancement factor (dissociation) 0 V,共250,1.5兲 1.26 −2.96 0 V,共250,75兲 1.43 −3.16 0 V,共281,133兲 0.91 −2.60 0 V,共277.5,148.5兲 1.25 −2.94 15 V,共250,1.5兲 1.34 −3.05 1.06 1.03 15 V,共250,75兲 1.63 −3.27 1.14 1.04 15 V,共281,133兲 1.76 −3.32 1.93 1.28 15 V,共277.5,148.5兲 1.53 −3.07 1.22 1.04
FIG. 4. 共Color online兲 The surface concentration of CRP complex as a function of time with or without applying voltage for the four locations defined in Fig.3.
FIG. 5.共Color online兲 The surface concentration of IgG complex as a func-tion of time with or without applying voltage for the four locafunc-tions defined in Fig.3.
A. Stirring vortices and position effect of reacting surface for binding enhancement
The electrothermally driven stirring flow field would construct a pair of vortices, which can enhance the binding reaction of the protein pair by a strong convection of the analyte. The existence of the reaction surface at different positions would interfere with the flow field, especially the shape of the vortices, to different degrees. This effect influ-ences the binding reaction. An optimal position of the reac-tion surface can be found to have the largest enhancement.
Figures 3共a兲–3共d兲 show the flow fields and the vortices in-duced by the nonuniform electrothermal force, when the ap-plied voltage is 15 Vrmspeak-to-peak with an operating fre-quency of 100 kHz, for four positions of the reacting surface. Notice that the reaction surfaces for Figs.3共a兲–3共c兲are faced upward and that for Fig.3共d兲is faced downward. It is inter-esting to observe that a “squeezing” effect has caused the velocity of the flow between the boundary and the reaction surface to be largely increased when the reaction surface is located near the negative electrode, i.e., the position 共281,133兲, as shown in Fig.3共c兲, which is in fact the optimal one selected from 30 studied positions. The largely acceler-ated flow over the reaction surface causes the efficient trans-port of analyte to the reaction surface and largely increases the association and dissociation speed.
FIG. 6. 共Color online兲 The development of the diffusion boundary layer of the CRP binding reaction without共the left panel兲 or with 共the right panel兲 applying voltage 15 Vrms. The reaction surface is located at共250,1.5兲.
No-tice that for the left or right panel, the first three illustrations are in associa-tion phase at the time of 500, 1000, and 1500 s, and the last three illustra-tions are in dissociation phase at the time of 3000, 4000, and 5000 s. Notice that the density scales are different to increase the plot visibility.
FIG. 7. 共Color online兲 The development of the diffusion boundary layer of the CRP binding reaction without共the left panel兲 or with 共the right panel兲 applying voltage 15 Vrms. The reaction surface is located at共281,133兲.
No-tice that for the left or right panel, the first three illustrations are in associa-tion phase at the time of 500, 1000, and 1500 s, and the last three illustra-tions are in dissociation phase at the time of 3000, 4000, and 5000 s. Notice that the density scales are different to increase the plot visibility.
In Figs.4 and5, the full time scale simulations of the binding reaction curves for CRP and IgG, corresponding to the configurations shown in Fig.3, are presented. Consistent results can be observed that the binding reaction for reacting surface located at 共281,133兲 with electrothermal force ap-plied presents the largest reaction speedup. In contrast, the binding reaction for reacting surface located at 共250,1.5兲 without electrothermal force applied has the smallest reac-tion speedup. In addireac-tion, although the dissociareac-tion equilib-rium constant共KD= kd/ka兲 of the two proteins are similar as
medium affinity 共CRP:2.6⫻10−9, IgG: 1.2⫻10−9兲, the characteristic behaviors in association phase or dissociation phase are different. The response of CRP is apparently faster
than IgG although the magnitudes of their equilibrium con-stants are similar. The major reason is that both ka and kdof
CRP are greater than IgG. This indicates that using electro-thermal effects one can separate the responses of different proteins with similar bioaffinity.
Since it is desirable to predict the final saturated concen-tration of the analyte-ligand complex in a rapid way, the initial slope of the binding curve is a good indicator for biosensor performance. Furthermore, we quantify the en-hancement of the association or dissociation of the binding reaction with an enhancement factor. The enhancement fac-tor is defined as the ratio of the initial slope of the binding reaction curve with an applied voltage of 15 Vrms to the initial slope without applying voltage. Tables IandII show
FIG. 8. 共Color online兲 The development of the diffusion boundary layer of the CRP binding reaction without共the left panel兲 or with 共the right panel兲 applying voltage 15 Vrms. The reaction surface is located at共277.5,148.5兲.
Notice that for the left or right panel, the first three illustrations are in association phase at the time of 500, 1000, and 1500 s, and the last three illustrations are in dissociation phase at the time of 3000, 4000, and 5000 s. Notice that the density scales are different to increase the plot visibility.
FIG. 9. 共Color online兲 The development of the diffusion boundary layer of the IgG binding reaction without共the left panel兲 or with 共the right panel兲 applying voltage 15 Vrms. The reaction surface is located at共250,1.5兲.
No-tice that for the left or right panel, the first three illustrations are in associa-tion phase at the time of 1000, 2000, and 3000 s, and the last three illustra-tions are in dissociation phase at the time of 6000, 7000, and 8000 s. Notice that the density scales are different to increase the plot visibility.
the initial slopes and the enhancement factors in association and dissociation for CRP and IgG protein pairs, respectively. It is found that the enhancement is higher in the association of the binding reaction than dissociation. The largest en-hancement factor is 5.17 in the association of CRP protein pair. In contrast, the IgG protein pair is less sensitive to the effect of electrothermal stirring field since the enhancement factor is less than 2.
B. The diffusion boundary layer
Since the reaction surface of the biosensor is a nonslip boundary in the flow field, the diffusion boundary layer can
be easily formed in a laminar flow configuration. Figures
6–8 show the variations of diffusion boundary layer of the CRP protein pairs from the association phase to the dissocia-tion phase without/with applying electrothermal force for the reaction surface at the position 共250,1.5兲, 共281,133兲, and 共277.5,148.5兲, respectively. Instead, Figs. 9–11 are for IgG protein pairs. The left and right panels of these figures are for the cases of without and with electrothermal force applied, respectively. The top three subfigures are for association phase, and the bottom three subfigures are for dissociation phase, in each panel. In Figs. 6 and 9, where the reaction
FIG. 10.共Color online兲 The development of the diffusion boundary layer of the IgG binding reaction without共the left panel兲 or with 共the right panel兲 applying voltage 15 Vrms. The reaction surface is located at共281,133兲.
No-tice that for the left or right panel, the first three illustrations are in associa-tion phase at the time of 1000, 2000, and 3000 s, and the last three illustra-tions are in dissociation phase at the time of 6000, 7000, and 8000 s. Notice that the density scales are different to increase the plot visibility.
FIG. 11.共Color online兲 The development of the diffusion boundary layer of the IgG binding reaction without共the left panel兲 or with 共the right panel兲 applying voltage 15 Vrms. The reaction surface is located at共277.5,148.5兲.
Notice that for the left or right panel, the first three illustrations are in association phase at the time of 1000, 2000, and 3000 s, and the last three illustrations are in dissociation phase at the time of 6000, 7000, and 8000 s. Notice that the density scales are different to increase the plot visibility.
surface is located at the bottom surface of the microchannel and opposite to the electrode pairs, the shapes of the diffu-sion boundary layer are distorted consistently with the flow field shown in the first subfigure of Fig.3 with the electro-thermal force applied. However since the vortices are least strong at this position, the sizes of the diffusion boundary do not change a lot, as expected, and the binding reaction is least enhanced relative to the other positions, as revealed in Figs.4and5.
In Figs.7and10, where the reaction surface is located at the position共281,133兲, the diffusion boundary layer are sup-pressed very much and in fact almost eliminated as shown in the third subfigure 共at the time of 1500 s兲 of the right panel of Fig. 7, when the induced vortices “squeeze” the flow to pass rapidly through the reacting surface. In the association phase, the flow transports sufficient analyte molecules to the reacting surface. In addition, the gap between the reacting surface and the top surface of the microchannel is relatively much smaller and so it is easier for the analyte molecules to diffuse to the reacting surface. In the dissociation phase, the dissociated analyte molecules are also carried away quickly, which again limits the growth of diffusion boundary layer for dissociation. By efficiently suppressing the growth of the dif-fusion boundary layers in both phases of association and dissociation, the electrothermal force is truly an effective way to enhance the binding reaction, as revealed in Figs. 4
and5.
C. Effect of the height of microchannel
The Damköhler number Dais the ratio of reaction
veloc-ity共ka关B0兴兲 to diffusion velocity 共D/h兲. In this section we fix
the inlet velocity 共100 m/s兲, ka关B0兴, and D to investigate
how the height of channel affects the enhancement factor. It is noted that in each channel the flow rate is varied but the average velocity is kept to be constant. In addition, the reac-tion surface is set at the posireac-tion共250, 1.5兲. Figures 12and
FIG. 12.共Color online兲 The enhancement factor of CRP binding reaction as a function of Da.
FIG. 13.共Color online兲 The enhancement factor of IgG binding reaction as a function of Da.
FIG. 14. 共Color online兲 Temperature 共left panel兲 and velocity 共right panel兲 fields in the reaction channel for various temperature conditions of the top and bottom walls. For the left or right panel, the seven illustrations from top to the bottom in descending order correspond to the seven cases of different temperature boundary conditions, namely, cases 1–7, as described in Sec. IV.
13, for the CRP and IgG binding reaction curves, respec-tively, show the enhancement factors with different Da
num-bers, which is only related with the height of channel. The height of channel varies from 50 to 250 m. There is a higher enhancement factor for lower Da numbers 共lower
channel height兲 for both CRP and IgG protein pairs. The reason for this effect is that the time required for diffusing the analyte to the reaction surface is shorter with a lower height of channel.
D. Effect of temperature boundary conditions
The stirred electrothermal flows generated in the reac-tion channel are driven by the inhomogeneous temperature field. As noted in Ref. 20the temperature boundary condi-tions significantly influence the temperature field and hence could result in a stronger or weaker enhancement depending on the choice of temperature boundary conditions. In this subsection, we consider six additional temperature boundary conditions on the top and bottom walls. However, the bound-ary conditions of the flow inlet and outlet of the channel remain unchanged, namely, constant temperature at the flow inlet and convective heat flux condition at the flow outlet. It is noted that the center of reaction surface is located at 共281,133兲, which is the optimal position found above with the temperature boundary conditions described in Sec. III. In addition, only the case of CRP-anti-CRP sample is consid-ered here.
These seven cases of different settings of temperature boundary conditions 共including the one used above兲 are listed as follows:
• case 1: top and bottom walls are both thermally insu-lated;
• case 2: top and bottom walls are both maintained in constant temperature 300 K;
• case 3: top wall is kept in constant temperature 300 K, but bottom wall is thermally insulated;
• case 4: bottom wall and the parts of top wall with the same horizontal location as the two electrodes are kept in constant temperature 300 K and the rest parts of top wall are thermally insulated;
• case 5: the parts of top wall with the same horizontal location as the two electrodes are kept in constant tem-perature 300 K, and the rest of top wall and the bottom
wall are thermally insulated共this is the setting used in the above simulation兲;
• case 6: the parts of top wall from the left end of the left electrode to the right end of the right electrode and the whole bottom wall are kept in constant temperature 300 K, and the rest parts of the top wall are thermally insulated;
• case 7: the parts of top wall from the left end of the left electrode to the right end of the right electrode are kept in constant temperature 300 K, and the rest parts of top wall and the whole bottom wall are thermally insulated.
The numerical results for the seven cases are summa-rized in Fig. 14and Table III. Figure 14 presents the tem-perature共left panel兲 and flow velocity 共right panel兲 fields for each of these cases. TableIIIcompares the enhancement fac-tors among these cases. As shown in Fig. 14, case 1 共insu-lated on both top and bottom walls兲 has a much hotter spot near the electrodes共the local temperature rises up to 36 K兲 than those of all the other six cases 共the local temperature rises only 1.7–2.7 K兲. It also exhibits a fainter vortex pattern that results in a much smaller binding enhancement factor than those of the other six cases 共see Table III兲. It is also
observed that the specific boundary temperature condition 共case 5兲 yields the best binding enhancement. Therefore it is useful to employ the settings of boundary condition to achieve a better design of biosensors.
E. The design of the microchannel
In the process of fabricating the reacting chamber, the polydimethyl-siloxane is usually used for its high manufac-turability and transparency. Since the reaction rate is highly related to the height of the channel, one can design the chan-nel with a neck region near the reaction surface to achieve a time-saving immunoassay. Figure15demonstrates the neck-ing design of the microchannel. The height of the neck re-gion, h2, varies from30, 90, up to 150 m. In addition, the
TABLE III. The initial slopes and enhancement factors of CRP binding reaction for different temperature boundary conditions.
CRP binding reaction Curve Initial slope
共association兲⫻10−11 Initial slope 共dissociation兲⫻10−11 Enhancement factor (association) Enhancement factor (dissociation) Case 1 2.05 −1.39 1.78 1.47 Case 2 4.56 −2.89 3.97 3.06 Case 3 4.62 −2.92 4.02 3.09 Case 4 5.82 −3.49 5.07 3.69 Case 5 5.90 −3.52 5.17 3.74 Case 6 4.64 −2.93 4.04 3.10 Case 7 4.55 −2.89 3.96 3.05
reaction surface is 40 m wide and fixed at the bottom of the channel. The results for CRP binding curves are shown in Fig.16. It is found that not only the neck region in terms of fast diffusion and convective velocities but also the electro-thermally driven vortices 共the squeezing effect as discussed before兲 enhance the binding reaction. With the existence of the stirring flow field, the association rate of the 30 m neck leads the original channel 共no neck兲 by a factor of 2.73. Combining both the passive mechanisms by changing geom-etries of the channel shape and the active mechanisms by the electrothermal effect, it is promising that one can achieve the best biosensor design.
V. CONCLUSION
In this work a 2D simulation on the immunoassay in a biosensor with electrothermal effect is performed by the fi-nite element analysis software, COMSOL MULTIPHYSICSTM. Two commonly used proteins, CRP and IgG, are regarded as the analytes for the analysis of the binding kinetics.
A pair of vortices is induced from the electrothermal field to stir the flow and enhance the association and disso-ciation of the protein pair. Existence of the reaction surface would interfere with the flow field and could change the shape of the vortices. This effect actively enhances the bind-ing reaction. An optimal position of the reaction surface is found to be located at 共281,133兲, yielding the largest en-hancement using the temperature boundary conditions de-scribed in Sec. III. The enhancement for association is more efficient共up to 5.17 for CRP pair, 3.74 for IgG pair兲, but less sensitive for dissociation 共up to 1.93 for CRP pair, 1.28 for IgG pair兲. By fixing the sensor locating at 共281,133兲, we also studied the effect of varying the temperature boundary con-dition on binding enhancement using six adcon-ditional settings
thermal electric forces and can be employed to enhance the binding efficiency. A specific setting of the temperature boundary conditions 共case 5兲 yields the best binding en-hancement. These results can serve as a useful reference for the design of biosensors.
The analysis for the channel with a neck region is dem-onstrated for its capability in shortening the required time to reach a steady state. The passive design 共neck region兲 and the active enhancement 共electrothermally driven stirring flow兲 need further experiments to optimize the performance of the system.
ACKNOWLEDGMENTS
This research was supported by the National Science Council in Taiwan through NSC 96-2120-M-002-014. We thank the NCHC in Taiwan for providing computing re-sources.
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COMSOLMultiphysics, Version 3.3, COMSOL Ltd., Stokhelm. FIG. 16. 共Color online兲 The surface concentration of CRP complex as a
function of time with or without applying voltage for various heights of the neck design.
