Model-Based Optimal Design of Polymer-Coated Chemical Sensors

G. W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, ... and Analytics, Vienna University of Technology, 1040 Vienna, Aus...
0 downloads 0 Views 202KB Size
Download PDF
Anal. Chem. 2003, 75, 1106-1115

Model-Based Optimal Design of Polymer-Coated Chemical Sensors Cynthia Phillips,† Michael Jakusch,‡ Hannes Steiner,‡ Boris Mizaikoff,§ and Andrei G. Fedorov*,†

G. W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, Institute for Chemical Technologies and Analytics, Vienna University of Technology, 1040 Vienna, Austria, and School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332

A model-based methodology for optimal design of polymercoated chemical sensors is developed and is illustrated for the example of infrared evanescent field chemical sensors. The methodology is based on rigorous and computationally efficient modeling of combined fluid mechanics and mass transfer, including transport of multiple analytes. A simple algebraic equation for the optimal size of the sensor flow cell is developed to guide sensor design and validated by extensive CFD simulations. Based upon these calculations, optimized geometries of the sensor flow cell are proposed to further improve the response time of chemical sensors. Independent of the specific transduction mechanism, all chemical sensors generate signals upon molecular interaction of their selective chemical recognition interface with the desired target analyte. To increase the detection threshold, the analyte of interest is usually preconcentrated by various techniques, utilizing hydrophobic polymer layers as preferred implementation in many sensing applications operated in aqueous environments. In-depth understanding of analyte enrichment in the polymers due to bulk solvation effects has been a major issue of physical and analytical chemistry over several decades. In the chemical sensing community, major efforts have been invested into diffusion modeling and optimization of polymer-based enrichment layers,1,2 whereas optimization of the sensor flow cell geometry and mass transport in the flow cell was almost completely untouched. This is a surprising fact, in particular since data acquisition for chemical sensors can be a complex process; thus, appropriate simulation models of chemical sensors can serve as highly valuable tools for sensor design and data interpretation. During the past decade, only a few relatively simplistic models have been developed for different types of chemical sensors, ranging from the fiber-optic chemical sensor3-5 to the dopamine biosensor6 and thermoelectric gas sensors.7 It is widely believed that miniaturization of the cross* To whom correspondence should be addressed. E-mail: andrei.fedorov@ me.gatech.edu. Phone: 404-385-1356. Fax: 404-894-8496. † G. W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology. ‡ Vienna University of Technology. § School of Chemistry and Biochemistry, Georgia Institute of Technology. (1) Heinrich, P.; Wyzgol, R.; Schrader, B.; Hatzilazaru, A.; Luebbers D. W. Appl. Spectrosc. 1990, 44, 1641-1646. (2) Helstern, U.; Hoffman, J. J. Mol. Struct. 1995, 349, 329-332. (3) Zhou, X.; Arnold, M. A. Anal. Chem. 1996, 68, 1748-1754. (4) Blair, D. Anal. Chem. 1997, 69, 2238-2246.

1106 Analytical Chemistry, Vol. 75, No. 5, March 1, 2003

Figure 1. Schematic of the baseline flow cell and coordinate system for the model.

sectional area of the sensor flow cell results in reduction of the response time, as well as the threshold of detection. This ”intuitive” rule of thumb is questioned in this paper, aiming at establishing a sound theoretical basis for optimal design of the flow cell for polymer-coated chemical sensors. In addition, multicomponent mass-transfer effects are rigorously modeled to establish validity limits for commonly used pseudobinary diffusion approximation4 for treating mixtures containing multiple analytes of interest. In this study, we use the example of infrared evanescent wave (IR-EW) chemical sensors8,9 to illustrate the power of the developed model-based optimal design methodology. However, the obtained conclusions are of a general nature and independent of the specific signal transduction technique. A scheme of the IREW chemical sensor flow cell is shown in Figure 1. Aqueous solutions of analyte(s) of interest flow over a polymer-coated attenuated total reflection (ATR) crystal acting as a mid-infrared waveguide.10 Induced by bulk solvation effects, the analyte (5) Meriaudeau, F.; Downey, T.; Wig, A.; Passian, A.; Buncick, M.; Ferrell, T. L. Sens. Actuators, B 1999, 54, 106-117. (6) Chen, Y.; Tan, T. C. Chem. Eng. Sci. 1996, 51, 7, 1027-1042. (7) Astie´, S.; Gue´, A. M.; Scheid, E.; Lescouze`res, L.; Cassagnes, A. Sens. Actuators, A 1998, 69, 205-211., (8) Mizaikoff, B.; Go ¨bel, R.; Krska, R.; Taga, K.; Kellner, R.; Tacke, M.; Katzir, A. Sens. Actuators, B 1995, 29, 58-63. (9) Mizaikoff, B.; Lendl, B. Sensor Systems Based on Mid-Infrared Transparent Fibers. In Handbook of Vibrational Spectroscopy; Chalmers, J. M., Griffiths, P. R., Eds.; John Wiley & ,Sons Ltd.: London, 2002; Vol. 2, pp 1560-1573. 10.1021/ac020471z CCC: $25.00

© 2003 American Chemical Society Published on Web 01/24/2003

molecules partition from the aqueous phase into the hydrophobic polymer and the polymer layer becomes enriched with analyte molecules as the flow continues to pass over the membrane. IR radiation coupled in the ATR waveguide incident to the crystal/ polymer interface at angles larger than the critical angle creates an evanescent field guided along that interface. The polymer thickness is designed such that the evanescent field, which decays exponentially into the adjacent medium, attenuates completely before reaching the aqueous solution, thus effectively eliminating interference from strong IR absorption bands of water in the midinfrared (MIR) part of the spectrum (3-20 µm). Analyte molecules enriched in the polymer absorb energy of the evanescently guided field at wavelengths in resonance with molecule-specific vibrational transitions. In particular for organic molecules, the MIR spectrum serves as a chemical fingerprint allowing identification and quantification of individual components and multiple analytes.11-13 Signal detection and processing are usually achieved by Fourier transform infrared (FT-IR) analysis. In summary, the goal of this work is to perform a detailed study of fluid mechanics and mass transfer in polymer-coated chemical sensors aiming to establish a simple, yet fundamentally sound methodology for model-based sensor design. The first major part of the study discusses binary diffusion, considering a single analyte of interest present in solution. The case of multicomponent mass transport with several analytes of interest present in aqueous solution is also investigated, and its implications to optimal sensor design are briefly discussed. Results obtained during the presented theoretical study are in accordance with experimental observations reported in the literature. Furthermore, it should be mentioned that detailed validation of established models is the subject of currently ongoing complimentary experimental studies based on polymer-coated IR-EW chemical sensors with variableflow cell geometry. Physical Arrangement and Model Formulation. A schematic of the modeled baseline flow cell and coordinate system is shown in Figure 1. Transport of analyte in the flow cell of the chemical sensor is governed by the following mass and momentum conservation equations and boundary/interface conditions for analyte concentration and flow velocity:

mass conservation ∂Ca ∂Ca ∂Ca +u +v ) Da∇2Ca ∂t ∂x ∂y

aqueous phase

∂Cp/∂t ) Dp∇2Cp

polymer

(1) (2)

boundary conditions Ca ) Ci and Cp ) 0 ∂C/∂n ) 0

-Da

|

∂Ca ∂y

int

at the inlet

(3)

at all wall boundaries except for the fluid/polymer interface (4) ) -Dp

|

∂Cp ∂y

int

and KCa ) Cp

at the fluid/polymer interface (5) Here, n is an outer normal at the boundaries, Ca and Cp are the concentrations of the analyte in the aqueous phase and polymer, (10) Harrick, N. J. Internal Reflectance Spectroscopy, Wiley-Interscience: New York, 1967.

respectively, Da and Dp are the diffusivities of analyte in the aqueous phase and polymer, respectively, and K is the partition coefficient for given analyte and polymer matrix.

momentum conservation for the fluid velocity vector b v ) {u, v} aqueous phase polymer

1 ∂v b v +b v ‚∇ b v ) - ∇P + υ∇2b ∂t F b v)0

everywhere

(6) (7)

boundary conditions b v)0

at all solid walls and fluid/polymer interface (no slip) (8)

( (y -h/2h/2) )

uinlet ) umax 1 -

2

fully developed Poiseuille

flow profile at the flow cell inlet (9) Here, F and υ are the density and viscosity of the flowing solution, respectively, P is the hydrodynamic pressure, u and v are velocity vector components in axial and transverse directions, respectively, and umax is the maximum velocity at the centerline (y ) h/2) of the flow cell. The assumptions made in the analysis are those of steady incompressible flow, isothermal conditions, and constant fluid viscosity and analyte diffusivity. Also, it is assumed that there are no mass sources present and any heating effects due to the evanescent field present in the polymer layer are negligible. Method of Solution. The governing conservation eqs 1, 2, and 6 are of parabolic type and can be effectively solved using an implicit, absolutely stable finite difference numerical integration technique.14 However, the problem in hand features one jump boundary condition for the analyte concentration at the fluid/ polymer interface (eq 5), which significantly complicates the simulation procedure. Specifically, it requires separate solution of the mass-transfer problem in the flow and polymer domains followed by iterative coupling of these solutions, which is not only very inconvenient but also computationally a very inefficient procedure.14 In contrast, if one could identify a new “modified” scalar variable equivalent to concentration, which is itself, as well as the flux associated with this variable, continuous (i.e., no jump) at the interface, then computations can be performed in an efficient noniterative manner for the combined (fluid and polymer) computational domain. To accomplish this task, we solve the complimentary heat-transfer problem by defining an equivalent “fictitious temperature”, which is continuous at the fluid/polymer interface in the equivalent thermal domain. To convert the governing mass conservation equations into equivalent energy conservation equations for the complimentary heat-transfer problem, the fictitious dimensionless fluid and polymer temperatures are introduced by defining Ta ) Ca/Ci and Tp ) Cp/KCi. Then, (11) Krska, R.; Kellner, R.; Schiessl, U.; Tacke, M.; Katzir, A. Appl. Phys. Lett. 1993 63 (14), 1868. (12) Walsh, J. E.; MacCraith, B. D.; Meany, M.; Vos, J. G.; Regan, F.; Lancia, A.; Artjushenko, S. Analyst 1996 121, 789. (13) Mizaikoff, B. Meas. Sci. Technol. 1999 10, 1185-1194. (14) Patankar, S. Numerical Heat Transfer and Fluid Flow; Hemisphere: Washington, DC, 1980.

Analytical Chemistry, Vol. 75, No. 5, March 1, 2003

1107

the energy conservation equations and boundary conditions for an equivalent thermal problem are given by

aqueous

FacPa

(

)

∂Ta ∂Ta ∂Ta +u +v ) ka∇2Ta (10) ∂t ∂x ∂y FPcPP

polymer

∂Tp ) kp∇2Tp ∂t

(11)

boundary conditions Ta ) 1 and Tp ) 0 ∂T/∂n ) 0

-ka

|

∂Ta ∂y

int

at the inlet

(12)

at all wall boundaries except for the fluid/polymer interface (13) ) -kp

|

∂Tp ∂y

int

and

Ta ) T p

at the fluid/polymer interface (14) The equivalent density, specific heat, and thermal conductivity of the fluid and the polymer in the thermal domain are determined by pairwise comparison of eqs 1 and 2 and 10 and 11:

Fa ) 1

cPa ) 1

ka ) D a

(15)

FP ) 1

cPP ) K

kp ) DPK

(16)

Notice that the equivalent thermal problem features continuity in both the temperature and heat flux at the fluid/polymer interface (see eq 14), thereby allowing fast solution of the problem in the single (fluid and polymer) computational domain using a commercial CFD software package FLUENT.15 Once the solution of the equivalent thermal problem is obtained, analyte concentrations in the fluid and polymer are determined as a postprocessing procedure from the equivalent temperatures through Ca ) Ta and Cp ) TpK, respectively. Proper selection of the computational grid and of the time step is essential to have confidence in the simulation results. Several increasingly fine nonuniform grids with exponentially greater numbers of nodes near the fluid/polymer interface were created using GAMBIT software before being imported to FLUENT. A grid verification study was performed by obtaining a curve of average analyte concentration in the polymer layer versus time to steady state, which is defined as the time when the average analyte concentration in the polymer reaches 99% of the inlet value times the partition coefficient. The leanest numerical grid distribution (i.e., with smallest number of nodes) that yielded the same results was selected for CFD simulations. In addition to spatial grid verification, it was found that a time step of 1 s was sufficiently accurate to resolve the dynamic behavior within the chemical sensor. Further details on the computational algorithm and its validation can be found in ref 16. RESULTS AND DISCUSSION First, to investigate the effect of various design parameters on the sensor performance, detailed parametric simulations have (15) FLUENT CFD Software, Fluent Inc: http://www.fluent.com. (16) Phillips, C. M.S. Thesis, Georgia Institute of Technology, Atlanta, GA, 2002; p 109.

1108

Analytical Chemistry, Vol. 75, No. 5, March 1, 2003

been performed. The baseline flow cell (Figure 1) with the following dimensions and transport properties was used in all simulations: the height and polymer-coated length of the flow cell are 0.1 and 2 cm, respectively, the thickness of the polymer layer is 5 µm, and the diffusivities of the analyte in the polymer and solution (water) are considered as 10-8 and 10-5 cm2/s, respectively. A baseline partition coefficient (K) was taken as 300, which defines the relative concentration of the analyte in the polymer as compared to that in the aqueous phase at the interface. These parameters were selected in correspondence to experimentally observed values in typical polymer coated IR-EW sensors.17 The typical transient uptake curve (i.e., average analyte concentration in the polymer layer vs time) is shown in Figure 2 and exhibits initially fast response (due to advective transport of analyte) that eventually levels out (owing to diffusion-dominant behavior) to approach the steady state. Parametric simulations allowed us to identify the major parameters affecting the sensor performance (i.e., the sensor response time) and to obtain fundamental understanding of the various mass transport mechanisms governing sensor dynamics. Using this insight into sensor behavior, we then present a simple design equation that can be used to find the optimal dimensions of the flow cell, resulting in the fastest sensor response time. Finally, armed with this knowledge, we propose and demonstrate two alternative flow cell configurations which result in superior sensor performance relative to the commonly used (baseline) design of the sensor flow cell. Major Parameters. Effect of Flow Cell Height: (a) Case of Constant Volumetric Flow Rate. The influence of the flow cell height was studied by varying the channel height from 10 to 1000 µm while keeping the volumetric flow rate constant. The baseline case parameters were used, only the height was varied, and the inlet flow velocity was adjusted in order to maintain constant volumetric flow rate. Figure 3 indicates that the fastest sensor response is obtained in the case of the smallest flow cell height as long as the flow rate is maintained constant. This effect can be explained since a decrease in channel height results in increase of the flow velocity in order to maintain constant flow rates, thereby maintaining the same advective resistance to mass transfer. This fact, combined with a decreased distance for the analyte transport from the bulk to the interface and thus reduced resistance to cross-flow mass diffusion, results in significant reduction of the overall resistance of mass transport from the aqueous phase to the polymer and, thus, a faster response time. (b) Case of Constant Flow Velocity. A similar test was performed on the influence of the flow cell channel height, except that the flow velocity was held constant instead of the volumetric flow rate. Again, the baseline case parameters were used, and the height was varied from 1000 to 10 µm. Figure 4 shows that as the channel height decreases past a value of ∼50 µm, the trend is reversed and there is a sharp increase in time to reach steady state if the flow cell channel height is further decreased. The presence of a local minimum in the sensor response time indicates that there exists an optimal channel height producing the fastest sensor response, and this channel height does not necessarily correspond to the smallest channel height that can be manufactured. The (17) Jakusch, M. Ph.D. Thesis, Vienna University of Technology, Vienna, Austria, 2000; p 187

Figure 2. Typical transient uptake curve for average analyte concentration vs time.

Figure 3. Time to steady state vs channel height for constant flow rate.

existence of the optimal channel height suggests expecting two dominating resistances to mass transfer in the flow cell that exhibit opposite behavior as the channel height is changed. Indeed, the resistance to diffusion mass transfer in transverse (across the flow) direction becomes smaller with a decrease in the channel height, whereas the resistance to convective mass transfer in the longitudinal (along the flow) direction increases. Thus, there must exist an optimal channel height that results in the lowest overall (diffusion + convection) resistance to mass transfer in the flow cell and, in turn, leads to the fastest sensor response time. Effect of Velocity Magnitude on the Optimal Flow Cell Height. The effect of velocity magnitude on the optimal flow cell channel

height was analyzed by varying the flow velocity by 1 order of magnitude from its baseline value of 1.5 cm/s. The results are shown in Figure 5, which clearly indicate that the optimal channel height decreases with an increase in velocity. This is because an increase in the flow velocity translates into an equivalent increase of the flow rate and thus a decrease in the advective resistance to mass transfer. The decrease in the magnitude of the optimal height is also due to the decrease in thickness of the concentration boundary layer resulting from an increase in the flow velocity. It should be also noted that a decrease in the optimal flow cell channel height is accompanied by a decrease in the amount of time required for the sensor to reach steady-state conditions. Analytical Chemistry, Vol. 75, No. 5, March 1, 2003

1109

Figure 4. Time to steady state vs channel height for constant velocity.

Figure 5. Time to steady state vs channel height for various flow velocities: (diamonds) umax ) 0.15 cm/s; (squares) umax ) 1.5 cm/s; (triangles) umax ) 15 cm/s.

Effect of the Partition Coefficient on the Optimal Channel Height. Simulations were performed to identify whether the optimal channel height corresponding to minimum response time is a function of the partition coefficient. Three different partition coefficient values, 30, 300, and 3000, were tested for varying channel heights. The enrichment curves for the partition coefficients tested (not shown here) are all similar in shape and attain the minimum at the same height.16 Only the magnitude of the enrichment time is shifted by a factor of 10sthe same factor the partition coefficient was changed from the baseline case. A direct relationship between the magnitude of the partition coefficient and the time to steady state becomes apparent by realizing that the 1110

Analytical Chemistry, Vol. 75, No. 5, March 1, 2003

partition coefficient is a measure of the amount of analyte the polymer can hold. The larger the partition coefficient, the larger is the adsorptive capacity of the polymer layer, and thus, the longer it takes to saturate the polymer. This observation proves that there is a unique optimal channel height that yields the fastest time to achieve steady state and that this critical dimension is independent of the partition coefficient. Minor Parameters. Additional tests were performed in which certain parameters were determined to be noncritical to the optimization of the optimal design of the flow cell.16 The mass transport was found to be essentially independent of the inlet velocity profile, fully developed or developing, as long as the flow

Figure 6. Isolines (contours) of analyte concentration in the flow channel and polymer layer at steady state for the flow cell channel heights of (A) 10, (B) 50 (optimal), and (C) 100 µm (the concentration values given in terms of fractions of the inlet analyte concentration).

rate was maintained constant. The transport processes in the flow channel were found to be essentially independent of the analyte diffusivity in the polymer because of the extremely small thickness of the polymer layer as compared to the height of the flow cell. The relationship between the size of the polymer layer and the response time is linear, whereas the optimal channel height is independent of the polymer layer thickness. Multicomponent Mass Transport. The presence of more than one analyte in the solution may in principle affect mass transfer and, in turn, selectivity and response time of the sensor. Multicomponent diffusion may result in behavior that cannot be explained by Fick’s law for binary (noninteracting) diffusion: for example, diffusion in the absence of a driving force due to concentration gradients or absence of diffusion in the presence of a driving force or even diffusion counter to the driving force induced by the concentration gradient. To this end, transport of multiple analytes in a chemical sensor system was also investigated16 aiming at establishing the threshold for importance of multicomponent diffusion effects. Detailed analysis16 based on the Maxwell-Stefan formalism18 demonstrates the following: (a) In case of approximately equal analyte concentrations, the multicomponent effects due to secondary analyte diffusion become significant when the ratio of Maxwell-Stefan diffusivities of the secondary analyte to the primary analyte is equal to or greater than 10. (b) In case of approximately equal analyte MaxwellStefan diffusivities, the multicomponent effects due to secondary analyte diffusion become significant when concentrations of both analytes are at least 100 ppm or greater and the concentration of one analyte is 2 orders of magnitude greater than that of the other. (c) Pressure diffusion exhibits effects so small that it can be neglected unless the flow cells with extremely small flow cell channel heights are considered. In general, however, the multicomponent effects on the mass transfer were found to be small for the operating conditions and design configurations found in the chemical sensors. Thus, the transport of multiple analytes can be adequately treated as binary diffusion of multiple noninteracting species. (18) Krishna, R.; Wesslingh, J. A. Chem. Eng. Sci. 1997, 52, 6. 861-911

Master Equation for Optimal Sensor Design. Detailed numerical simulations of the fluid mechanics and mass transfer in the sensor flow cell offer unique fundamental insight into the physics of the sensor transient behavior; however, the results of such simulations would be of little practical value to the sensor developer unless they are used as a basis to obtain a simple, but sufficiently general methodology for optimal sensor design. Ultimately, the sensor practitioner would like to have a simple, preferably algebraic equation that permits calculation of the optimal flow cell dimensions (e.g., optimal height resulting in fastest response time) as a function of the operating and design parameters. To develop such an equation, it is important to understand which parameters essentially control this unusual transient behavior. To facilitate such understanding, the steadystate isolines (contours of constant analyte concentration) over the entire flow field are shown in Figure 6 for the flow cell channel heights of 10, 50, and 100 µm. The aspect ratio of the graphs is altered from that of the actual flow field to allow easier inspection. Since the analyte migrates in the direction normal to the isolines (contours), the minimum overall resistance to mass transfer corresponds graphically to the shortest path connecting the upper left corner of the flow cell (top of the inlet) and the lower right corner of the flow cell (i.e., bottom of the exit). The contour plots for the three cases considered display very different behavior, and inspection of the contours reveals an explicit relationship between the concentration boundary layer and the rate-limiting mode of analyte transport. Indeed, to yield the fastest sensor response time, the overall resistance to mass transfer needs to be minimized. From the tests of the effect of diffusivity in both the aqueous and polymer phases,16,17 the dominant resistance to mass transfer was shown to exist in the fluid rather than in the polymer. Thus, the overall resistance is defined by the combination of advective resistance in the direction of flow and diffusion resistance in crossflow direction. For the first case (Figure 6A), when the height is smaller than the optimal channel height, the contours are nearly perpendicular to the flow direction, suggesting that the dominant resistance to mass transfer is advection resistance for the analyte to travel from Analytical Chemistry, Vol. 75, No. 5, March 1, 2003

1111

occurring at the channel exit). These resistances are schematically shown in Figure 7 and can be expressed as

Radv ) 1/uavh and Rdiff ) δc/DaL

(17)

where the average velocity is related to the maximum flow velocity for Poiseuille flow:

uav ) (2/3)umax

Figure 7. Mass-transfer resistance network for chemical sensor.

the inlet to the outlet of the channel. In contrast, when the channel height is greater than the optimal height (Figure 6C), the contours are more horizontally aligned, displaying dominant diffusion resistance to mass transfer in the cross-flow direction. The contours for the optimal channel height (50 µm, Figure 6B) reveal that the analyte travels the shortest distance in a straight line from the top of the inlet to the bottom of the exit along a path that is perpendicular to the contour lines. This path provides the least possible total resistance to mass transfer in the flow cell and thus corresponds to the optimal flow cell channel height resulting in the fastest sensor response. To translate the qualitative observations discussed above into the desired master equation for optimal sensor design, the analytical framework of the resistance network is applied. The resistance to advective transport (Radv, parallel to the flow) and to cross-channel diffusion (Rdiff, perpendicular to the flow) depends on the polymer-coated length of the flow cell (L), the height of the flow cell (h), the average flow velocity (uav), the diffusion coefficient of the analyte in the aqueous phase (Da), and the maximum thickness of the concentration boundary layer (δc,

(18)

To obtain an expression for the optimal channel height using the resistance framework, the thickness of the concentration boundary layer at the exit of the channel needs to be related to the flow cell height. The results of the simulations indicate that the thickness of the concentration boundary layer at the exit is nearly equal to the channel height in case of optimal channel height resulting in the fastest sensor response. When the channel height is decreased to less than the optimal (critical) value, the boundary layer reaches the height of the channel before reaching the exit of the channel. From these observations, the thickness of the concentration boundary layer at the flow cell exit can be approximated by the optimal channel height (i.e., δc ) hopt) in case of optimal (fastest response) conditions. With this information, the expression for the diffusive resistance can be modified to become

Rdiff ) hopt/DaL

(19)

The diffusion resistance is dominant when the channel height is greater than optimal height, and the advection resistance is dominant at channel heights smaller than optimal height. These resistances become equal at one distinct channel height, this being the optimal channel height. This is also the point of minimal total

Figure 8. Average flow velocity vs respective optimal channel height predicted by the master equation (20) (squares) and by detailed CFD simulations (solid line). 1112

Analytical Chemistry, Vol. 75, No. 5, March 1, 2003

Figure 9. Schematic of the flow cell with top centerline flow entry.

resistance to mass transfer, which would lead to the conclusion that the corresponding channel height should yield the fastest response time. Thus, the resulting equation for optimal channel height that produces the fastest sensor response is obtained by equating the advection and diffusion resistances in eqs 18 and 19:

hopt ) xDaL/uav

(20)

Calculation of the optimal channel height for the baseline case using eq 20 yields 50 µm, which corresponds to the optimal height predicted using detailed CFD simulations (Figure 4). Detailed numerical simulations were performed for a wide range of operating conditions, and results are compared with analytical predictions obtained using eq 20. The results are shown in Figure 8, and their remarkable correspondence supports the resistance network analysis and the master equation (20) as an accurate approach for prediction of the optimal height of the sensor flow cell that produces the fastest sensor response.

Final Remarks on Optimal Design of the Sensor Flow Cell. For global optimal design of the sensor flow cell it is also important to consider the pressure drop across the flow channel in addition to the response time. The pressure drop provides a measure of the amount of power that would be needed to pump the fluid through the sensor. In contrast to the response time of the sensor, the pressure drop increases with an increase in the flow velocity and a decrease in the flow cell channel height, thereby establishing a tradeoff situation leading to the globally optimal solution. The relative importance of the sensor response time and pressure drop depends on the specific application. To determine the overall best design for the sensor flow channel, the following optimization formula can be used in which the response time and the pressure drop are the determining factors:

[global optimum criteria] )

[

]

A B + f max ∆tss/∆t /ss ∆p/∆p* (21)

The starred values represent the reference values obtained from the baseline case or could also represent target values. The optimal operating parameters depend on the application of the sensor, more specifically, on the relative importance of a small pressure drop versus a fast response time. The weighting factors are noted in the formula as the variables A and B; these weighting factors should be defined such that the sum of A and B is equal to 100%, and each is the respective measure of importance to the sensor design. Proposed New Configurations for the Chemical Sensor Flow Cell. Armed with basic understanding of the sensor transient behavior quantified by the master equation (20), two new configurations for sensor flow cells are proposed that capitalize on the effects leading to reduction of the total resistance to mass transfer. An ideal design of the flow cell should aim at decreasing

Figure 10. Time to steady state vs flow cell channel height for varying inlet velocities. Comparison of the top entry design (solid line) and baseline design (dashed line): (triangles) u ) 0.1 cm/s, (diamonds); u ) 1 cm/s; (circles) u ) 10 cm/s.

Analytical Chemistry, Vol. 75, No. 5, March 1, 2003

1113

Figure 11. Schematic of the flow cell with tapered height.

the thickness of the concentration boundary layer to minimum at the exit of the flow cell channel. An ideal design would also cause the thickness of the concentration boundary layer to reach the height of the channel exactly and not earlier than at the exit of the flow cell. The first proposed design is one in which the inlet (flow entry) is centered in the top plate of the flow cell (Figure 9). This choice of inlet placement introduces the stream at the midpoint of the polymer length, creating two symmetric concentration boundary layers developing from the midpoint of the polymer layer toward two respective outflows. This leads to shorter flow development length on both sides of the inlet and thus smaller thickness of the concentration boundary layers at both exits from the flow cell. In turn, the overall resistance to mass transfer is reduced as well, thus speeding up the response time of the sensor. The response times for the new geometry were simulated for the height range from 10 to 1000 µm. Figure 10 shows the results of the simulations as compared to the baseline case of single inletsingle exit flow cell of the same dimensions. Unlike in the baseline case, there is no evidence of a critical height for the centerline

inlet geometry; the smallest channel height (10 µm) tested yielded the fastest response time. This is explained as the even smallest simulated channel size is still larger than the critical (optimal) channel height, predicted analytically to be less than 10 µm for given baseline volumetric flow rate (flow velocity). The analytical prediction of this, much smaller optimal channel height for the centerline inlet geometry has also been verified through detailed CFD simulations of the sensor response. The centerline vertical entry geometry clearly outperformed the baseline geometry for all channel heights tested, especially in the limit of low flow velocities. The second geometry of the flow cell (Figure 11) is one in which the channel height is linearly tapered over the length of the channel, holding the inlet height constant and equal to the baseline channel ()1000 µm), but decreasing the exit channel height over the range of 1000-10 µm. The idea behind this approach is to force thinning of the concentration boundary layer at the channel exit by confining and thus accelerating the flow along the flow cell. Different tapering angles (expressed in terms of varying exit heights) were tested for three average inlet velocities, 0.1, 1, and 10 cm/s, and the results are shown in Figure 12. The new tapered design produces very similar but slightly inferior response times as compared to those for the baseline (parallel wall) flow cell geometry when the channel height is larger than the optimal channel height. However, the tapered design shows a vast improvement in response time when the channel height is less than the optimal height, especially in the case of low flow velocities. It appears that manipulation with the tapering angle of the flow cell delays, if not completely avoids, the saturation limit when sensor response time cannot be further decreased by a decrease in the flow cell channel height. Comparing advantages offered by the two above-discussed designs of the sensor flow cell, the best design should feature an optimal combination of the top entry (inlet) and tapered channels for streaming the flow from the inlet toward two peripheral exits.

Figure 12. Time to steady state vs flow cell channel height at the exit for varying inlet velocities. Comparison of the tapered height design (solid line) and baseline design (dashed line): (triangles) u ) 0.1 cm/s; (diamonds) u ) 1 cm/s; (circles) u ) 10 cm/s. 1114 Analytical Chemistry, Vol. 75, No. 5, March 1, 2003

CONCLUSIONS Fluid mechanics and mass transfer in polymer-coated chemical sensors have been investigated with the aim to establish a sound theoretical methodology for optimal design of sensor flow cells. A resistance network was used to determine the dominant, ratecontrolling mechanism of mass transfer and to develop criteria for optimal design. Diffusion and advection resistances in the aqueous phase have been shown to be the rate-controlling parameters for the response time of the sensor. A simple master design equation, hopt ) (DRL/uav)1/2, has been developed to accurately predict optimal channel height that yields the fastest response time for a given average flow velocity. This analytical model was carefully validated by extensive CFD calculations. The following specific conclusions can be drawn from our analysis: (a) The sensor response time is highly dependent on the analyte diffusivity in the aqueous phase; on average, the time to steady state decreases 1 order of magnitude with a 2 orders of magnitude increase in diffusivity. (b) The least total resistance to mass transfer is achieved when the channel (optimal) height is equal to the concentration

boundary layer thickness at the exit of the flow cell channel. The flow velocity can be used to control the optimal channel height indirectly by altering the concentration boundary layer approaching the height of the channel at the exit. (c) A global optimal criterion has been proposed to determine the optimal design of the sensor flow cell based on the required pressure drop and response time. (d) Alternative configurations of the sensor flow cell with midpoint top flow entry and with tapered channel have been proposed and analyzed to further improve the sensor response time. ACKNOWLEDGMENT B.M., M.J., and H.S. acknowledge support by the European Union within Project IMSIS, EVK1-CT1999-00042, and the Austrian Federal Ministry of Education, Science and Culture (BMBWK). AC020471Z

Analytical Chemistry, Vol. 75, No. 5, March 1, 2003

1115